Content-Type: multipart/mixed; boundary="-------------0110121019673" This is a multi-part message in MIME format. ---------------0110121019673 Content-Type: text/plain; name="01-369.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-369.keywords" Schrodinger equation, dispersive estimates, Strichartz estimates ---------------0110121019673 Content-Type: application/postscript; name="dispersion.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dispersion.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: dispersion.dvi %%Pages: 70 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips dispersion -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.10.12:1051 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 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(h)f(the)i(theorem)f(applies)e(are)j Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))26 b(=)f(cos)q(\()p Fy(t)p Fz(\))p Fy(V)2847 288 y Fq(0)2887 274 y Fz(\()p Fy(x)p Fz(\))h(where)e Fu(k)p Fy(V)3390 288 y Fq(0)3430 274 y Fu(k)3475 325 y Fw(L)3533 275 y Fr(3)p 3533 283 31 3 v 3533 318 a(2)3578 325 y Fq(\()p Fp(R)3653 306 y Fr(3)3688 325 y Fq(\))3745 274 y Fy(<)h(c)3880 288 y Fq(0)3920 274 y Fz(,)-47 418 y(and)35 b(for)h(whic)m(h)e(\(1.5\))k(holds.)56 b(More)37 b(generally)-8 b(,)37 b(one)f(can)g(tak)m(e)i(p)s(oten)m(tials)d(that)h (are)h(quasi-p)s(erio)s(dic)32 b(in)j(time,)-47 531 y(suc)m(h)30 b(as)g Fy(V)21 b Fz(\()p Fy(t;)15 b(x)p Fz(\))26 b(=)f Fy(\036)p Fz(\()p Fy(t)p Fz(\))p Fy(V)870 545 y Fq(0)910 531 y Fz(\()p Fy(x)p Fz(\))31 b(with)1534 749 y Fy(\036)p Fz(\()p Fy(t)p Fz(\))26 b(=)1834 662 y Fn(X)1813 867 y Fw(\027)t Fx(2)p Fw(Z)1952 848 y Fs(d)2002 749 y Fy(c)2041 763 y Fw(\027)2085 749 y Fy(e)2127 711 y Fq(2)p Fw(\031)r(it!)r Fx(\001)p Fw(\027)-47 1048 y Fz(and)130 980 y Fn(P)225 1075 y Fw(\027)t Fx(2)p Fp(Z)362 1056 y Fs(d)411 1048 y Fu(j)p Fy(c)475 1062 y Fw(\027)519 1048 y Fu(j)g Fy(<)f Fu(1)p Fz(,)30 b Fy(!)e Fu(2)d Fz([0)p Fy(;)15 b Fz(1\))32 b(arbitrary)-8 b(.)-47 1161 y(Note)24 b(that)g(Theorem)f(1.2)h(also)g (applies)d(to)j(time)f(indep)s(enden)m(t)e(p)s(oten)m(tials)h Fy(V)2620 1175 y Fq(0)2660 1161 y Fz(\()p Fy(x)p Fz(\))i(via)f Fy(V)d Fz(\()p Fy(t;)15 b(x)p Fz(\))26 b(:=)f Fy(V)3416 1175 y Fq(0)3456 1161 y Fz(\()p Fy(x)p Fz(\).)39 b(Clearly)-8 b(,)-47 1274 y(in)29 b(that)i(case)g(the)g(conditions)e(b)s(ecome)1203 1503 y Fu(k)p Fy(V)1301 1517 y Fq(0)1341 1503 y Fu(k)1386 1554 y Fw(L)1444 1504 y Fr(3)p 1444 1512 V 1444 1547 a(2)1489 1554 y Fq(\()p Fp(R)1564 1535 y Fr(3)1599 1554 y Fq(\))1650 1503 y Fz(+)36 b(sup)1741 1587 y Fw(x)p Fx(2)p Fp(R)1876 1568 y Fr(3)1926 1380 y Fn(Z)1976 1586 y Fp(R)2024 1567 y Fr(3)2098 1442 y Fu(j)p Fy(V)20 b Fz(\()p Fy(y)s Fz(\))p Fu(j)p 2088 1483 262 4 v 2088 1566 a(j)p Fy(x)g Fu(\000)g Fy(y)s Fu(j)2375 1503 y Fy(dy)28 b(<)d(c)2630 1517 y Fq(0)2670 1503 y Fy(:)-47 1763 y Fz(Since)k(b)m(y)h(fractional)g(in)m(tegration)1188 1887 y Fn(Z)1238 2093 y Fp(R)1286 2074 y Fr(6)1350 1949 y Fu(j)p Fy(V)21 b Fz(\()p Fy(x)p Fz(\))p Fu(j)26 b(j)p Fy(V)20 b Fz(\()p Fy(y)s Fz(\))p Fu(j)p 1350 1989 514 4 v 1457 2073 a(j)p Fy(x)g Fu(\000)g Fy(y)s Fu(j)1718 2046 y Fq(2)1889 2010 y Fy(dxdy)29 b Fu(\024)c Fy(C)7 b Fu(k)p Fy(V)20 b Fu(k)2440 1973 y Fq(2)2440 2069 y Fw(L)2498 2019 y Fr(3)p 2498 2027 31 3 v 2498 2062 a(2)2543 2069 y Fq(\()p Fp(R)2618 2050 y Fr(3)2653 2069 y Fq(\))2685 2010 y Fy(;)-47 2258 y Fz(it)30 b(follo)m(ws)f(that)i(these)g (conditions)e(are)i(strictly)e(stronger)h(than)h(those)f(in)g(Theorem)g (1.1.)-47 2371 y(Whereas)i(our)e(main)g(emphasis)g(is)g(of)i(course)f (on)g(the)h(deca)m(y)g(estimate)g(\(1.8\))r(,)g(it)e(app)s(ears)h(that) h(ev)m(en)f(the)h(easier)-47 2484 y(question)e(of)h(solv)-5 b(abilit)m(y)29 b(of)j(equation)f(\(1.7\))i(for)e(rough)f(p)s(oten)m 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Fw(p)1957 2936 y Fz(.)68 b(Notice)40 b(that)g(in)e(our)h(case)h Fy(q)j Fz(=)d Fu(1)p Fz(,)h Fy(p)f Fz(=)3596 2900 y Fw(n)p 3596 2915 43 4 v 3600 2967 a Fq(2)3648 2936 y Fz(,)i(whic)m(h)-47 3048 y(corresp)s(onds)35 b(to)i(the)g(endp)s(oin)m(t)d(of)j(this)e(condition)g(not)i(co)m(v)m (ered)h(in)d([Y1)q(].)59 b(W)-8 b(e)37 b(use)g(the)f(endp)s(oin)m(t)f (Stric)m(hartz)-47 3161 y(estimate)22 b([KT)o(])f(for)g(the)g Fl(fr)-5 b(e)g(e)22 b Fz(problem)d(for)i(that)h(purp)s(ose,)f(whic)m(h) f(automatically)h(yields)e(the)j(endp)s(oin)m(t)d(Stric)m(hartz)-47 3274 y(estimate)31 b(in)e(the)i(con)m(text)h(of)e(Theorem)g(1.2.)-47 3387 y(F)-8 b(or)43 b(time-dep)s(enden)m(t)f(p)s(oten)m(tials)g(the)h (analogue)h(of)f(Kato's)g(scattering)h(result)d([Ka)q(])i(w)m(as)g(pro) m(v)m(ed)g(b)m(y)g(Ho)m(w-)-47 3500 y(land)23 b([H1)q(].)39 b(More)25 b(precisely)-8 b(,)25 b(under)e(the)i(condition)e(that)j(for) e(a)h(su\016cien)m(tly)e(large)i(time)g Fy(t)3083 3514 y Fq(0)3147 3500 y Fy(>)g Fz(0,)h Fy(V)21 b Fz(\()p Fy(t;)15 b(x)p Fz(\))26 b Fu(\024)f Fy(V)3783 3514 y Fq(0)3822 3500 y Fz(\()p Fy(x)p Fz(\))-47 3613 y(for)37 b(some)h Fl(time)i(indep)-5 b(endent)39 b Fz(p)s(oten)m(tial)e Fy(V)1501 3627 y Fq(0)1540 3613 y Fz(\()p Fy(x)p Fz(\))i(ob)s(eying)d (the)i(small)e(Rollnik)f(condition)i(\(1.4\))r(,)i(there)f(exist)f(a) -47 3726 y(unitary)h(w)m(a)m(v)m(e)k(op)s(erator)e Fy(W)52 b Fz(in)m(tert)m(wining)38 b Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))40 b(and)f Fy(e)2075 3693 y Fw(it)p Fq(\()p Fx(\000)p Fq(\001\))2298 3726 y Fz(.)69 b(In)39 b(case)i Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))40 b(do)s(es)g(deca)m(y)h(in)d (time)i(\(in)-47 3839 y(the)31 b(sense)f(of)h(a)h(small)d(amoun)m(t)i (of)g(in)m(tegrabilit)m(y\),)f(w)m(a)m(v)m(e)j(op)s(erators)d(w)m(ere)i (constructed)f(b)m(y)f(Ho)m(wland)h([H2])g(and)-47 3952 y(Da)m(vies)k([D].)53 b(In)33 b(con)m(trast)j(to)f(Theorem)f(1.2)h (they)f(do)h(not)f(require)f(smallness)f(\(the)j(latter)f(b)s(eing)f (replaced)h(b)m(y)-47 4065 y(time)c(deca)m(y)h(of)g(the)f(p)s(oten)m (tial\))h(and)e(they)i(also)f(obtain)g(strong)h(con)m(tin)m(uit)m(y)f (of)g(the)h(ev)m(olution.)-47 4178 y(One)f(of)h(the)g(di\016culties)d (in)i(this)f(case)j(is)e(the)h(absence)g(of)g(the)g(connection)g(b)s (et)m(w)m(een)g(the)g(semigroup)f(generated)-47 4291 y(b)m(y)c(the)h(Sc)m(hr\177)-45 b(odinger)25 b(equation)h(and)g(the)g (sp)s(ectral)g(prop)s(erties)f(of)h(the)h(op)s(erator)g Fu(\000)p Fz(\001)12 b(+)g Fy(V)20 b Fz(.)39 b(Recall)26 b(that)h(for)f(time)-47 4403 y(indep)s(enden)m(t)i(p)s(oten)m(tials)i Fy(V)20 b Fz(,)1511 4553 y Fy(e)1553 4516 y Fw(itH)1670 4553 y Fy(f)35 b Fz(=)1846 4429 y Fn(Z)1952 4553 y Fy(e)1994 4516 y Fw(it\025)2089 4553 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))p Fy(f)-47 4762 y Fz(where)33 b Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))36 b(is)d(the)h(sp)s(ectral)f(measure)h(of)g(the)g(op)s (erator)h Fu(\000)p Fz(\001)22 b(+)g Fy(V)e Fz(.)52 b(This)32 b(is)h(no)h(longer)g(a)m(v)-5 b(ailable)33 b(for)h(time-)-47 4875 y(dep)s(enden)m(t)29 b(p)s(oten)m(tials.)-47 4988 y(The)f(pro)s(of)g(of)36 b(\(1.8\))31 b(is)d(similar)e(to)k(that)f(of)g (Theorem)g(1.1)h(but)e(m)m(uc)m(h)h(more)g(in)m(v)m(olv)m(ed.)40 b(Since)28 b(w)m(e)h(can)g(no)g(longer)-47 5101 y(rely)j(on)h(the)g(sp) s(ectral)f(theorem,)j(resolv)m(en)m(ts,)f(and)f(Born)f(series)h(to)h (construct)f(the)g(ev)m(olution)g(of)40 b(\(1.7\))r(,)34 b(w)m(e)f(use)-47 5214 y(the)24 b(Duhamel)g(form)m(ula)f(instead)h(\(w) m(e)h(note)g(in)e(passing)g(that)i(the)g(F)-8 b(ourier)24 b(transform)f(in)g(the)i(sp)s(ectral)e(parameter)1926 5472 y(5)p eop %%Page: 6 6 6 5 bop -47 274 a Fz(establishes)22 b(an)i(equiv)-5 b(alence)24 b(b)s(et)m(w)m(een)g(the)g(represen)m(tation)g(of)h(the)f(ev)m(olution) f(in)g(terms)h(of)g(a)g(Born)g(series)g(and)f(an)-47 387 y(in\014nite)h(expansion)h(of)i(the)f(solution)f(b)m(y)i(means)f (of)h(Duhamel's)e(principle\).)37 b(One)26 b(of)g(the)h(no)m(v)m (elties)g(in)e(our)h(pap)s(er)-47 500 y(is)38 b(the)h(form)m(ula)f (represen)m(ting)g(the)h(time)g(ev)m(olution)f(of)h(the)h(Sc)m(hr\177) -45 b(odinger)37 b(equation)i(with)e(a)j(time-dep)s(enden)m(t)-47 613 y(p)s(oten)m(tial)i(as)g(an)g(in\014nite)e(series)i(of)h (oscillatory)e(in)m(tegrals)h(in)m(v)m(olving)f(the)h(resolv)m(en)m(ts) h(of)g(the)f Fl(fr)-5 b(e)g(e)43 b Fz(problem.)-47 726 y(Most)c(of)f(the)h(w)m(ork)f(in)f(the)i(pro)s(of)e(of)i(Theorem)f(1.2) h(is)f(dev)m(oted)h(to)g(estimating)f(these)g(oscillatory)g(in)m (tegrals,)-47 839 y(whose)27 b(phases)g(t)m(ypically)g(ha)m(v)m(e)i(a)f (critical)e(p)s(oin)m(t)h(with)f(degeneracies)i(of)g(the)g(third)e (order.)39 b(See)28 b(Sections)f(5{8)i(for)-47 951 y(details.)-47 1064 y(Tw)m(o)37 b(sections)g(of)g(this)f(pap)s(er)g(are)h(dev)m(oted)h (to)f(time)g(indep)s(enden)m(t)e(p)s(oten)m(tials)h(without)g(an)m(y)h (restrictions)f(on)-47 1177 y(their)27 b(sizes.)39 b(In)28 b(Section)f(3)i(w)m(e)f(pro)m(v)m(e)h(the)f(follo)m(wing)e(result.)39 b(As)28 b(b)s(efore,)g Fy(H)33 b Fz(=)24 b Fu(\0004)15 b Fz(+)g Fy(V)48 b Fz(and)28 b Fy(P)3327 1191 y Fw(a:c:)3466 1177 y Fz(refers)g(to)h(the)-47 1290 y(pro)5 b(jection)30 b(on)m(to)h(the)g(absolutely)e(con)m(tin)m(uous)h(subspace)g(of)h Fy(L)2179 1257 y Fq(2)2248 1290 y Fz(relativ)m(e)g(to)g Fy(H)7 b Fz(.)-47 1478 y Fm(Prop)s(osition)36 b(1.3.)46 b Fl(L)-5 b(et)1153 1725 y Fu(j)-15 b(j)g(j)p Fy(V)21 b Fu(j)-15 b(j)g(j)26 b Fz(:=)f Fu(k)p Fy(V)c Fu(k)1628 1739 y Fq(2)1688 1725 y Fz(+)36 b(sup)1779 1809 y Fw(x)p Fx(2)p Fp(R)1914 1790 y Fr(3)1963 1601 y Fn(Z)2014 1807 y Fp(R)2062 1788 y Fr(3)2135 1663 y Fu(j)p Fy(V)20 b Fz(\()p Fy(y)s Fz(\))p Fu(j)p 2125 1704 262 4 v 2125 1787 a(j)p Fy(x)h Fu(\000)f Fy(y)s Fu(j)2412 1725 y Fy(dy)28 b(<)d Fu(1)p Fy(:)-47 1990 y Fl(Then)33 b(for)g(every)f Fy(")26 b(>)f Fz(0)33 b Fl(ther)-5 b(e)33 b(exists)g(some)h(p)-5 b(ositive)33 b Fy(\025)1905 2004 y Fq(0)1970 1990 y Fz(=)25 b Fy(\025)2119 2004 y Fq(0)2158 1990 y Fz(\()p Fu(j)-15 b(j)g(j)p Fy(V)21 b Fu(j)-15 b(j)g(j)p Fy(;)15 b(")p Fz(\))35 b Fl(so)e(that)1221 2209 y Fu(k)p Fy(e)1308 2171 y Fw(itH)1425 2209 y Fy(\037)p Fz(\()p Fy(H)r(=\025)1693 2223 y Fq(0)1733 2209 y Fz(\))p Fy(P)1826 2223 y Fw(a:c:)1939 2209 y Fu(k)1984 2229 y Fw(L)2032 2210 y Fr(1)2067 2229 y Fx(!)p Fw(L)2186 2210 y Ft(1)2280 2209 y Fu(\024)25 b Fy(C)7 b(t)2481 2171 y Fx(\000)2546 2144 y Fr(3)p 2545 2156 31 3 v 2545 2197 a(2)2585 2171 y Fq(+)p Fw(")-47 2209 y Fz(\(1.9\))-47 2413 y Fl(for)33 b(al)5 b(l)33 b Fy(t)25 b(>)g Fz(0)p Fl(.)-47 2601 y Fz(The)41 b(pro)s(of)g(is)g (again)h(p)s(erturbativ)m(e.)73 b(F)-8 b(or)43 b(the)f(case)h(of)f (large)f(energies,)k Fl(and)f(for)g(those)g(only)p Fz(,)i(the)c (required)-47 2714 y(smallness)g(is)h(pro)m(vided)g(b)m(y)h(the)h (follo)m(wing)e(estimate)i(on)f(the)g(resolv)m(en)m(ts,)49 b(whic)m(h)43 b(can)h(b)s(e)g(view)m(ed)g(as)g(some)-47 2827 y(instance)30 b(of)g(the)h(limited)d(absorption)h(principle:)1045 3041 y Fu(k)p Fz(\()p Fu(\0004)21 b(\000)e Fy(\025)i Fz(+)f Fy(i)p Fz(0\))1664 3003 y Fx(\000)p Fq(1)1759 3041 y Fy(f)10 b Fu(k)1859 3060 y Fw(L)1907 3041 y Fr(4)1941 3060 y Fq(\()p Fp(R)2016 3041 y Fr(3)2051 3060 y Fq(\))2108 3041 y Fu(\024)25 b Fy(C)7 b(\025)2329 3003 y Fx(\000)2394 2976 y Fr(1)p 2394 2988 V 2394 3029 a(4)2438 3041 y Fu(k)p Fy(f)j Fu(k)2583 3091 y Fw(L)2641 3041 y Fr(4)p 2641 3049 V 2641 3084 a(3)2686 3091 y Fq(\()p Fp(R)2761 3072 y Fr(3)2796 3091 y Fq(\))2827 3041 y Fy(:)-2899 b Fz(\(1.10\))-47 3276 y(The)26 b(pro)s(of)f(of)34 b(\(1.10\))28 b(is)e(an)g(immediate)f (consequence)j(of)e(the)h(Stein-T)-8 b(omas)26 b(theorem)h([St].)39 b(The)26 b(app)s(earance)h(of)-47 3389 y(the)h(Stein-T)-8 b(omas)27 b(theorem)i(in)d(this)h(con)m(text)i(is)e(most)h(natural,)g (as)g(the)g(resolv)m(en)m(t)g(\()p Fu(\0004)15 b(\000)g Fy(\025)g Fz(+)g Fy(i)p Fz(0\))3423 3356 y Fx(\000)p Fq(1)3546 3389 y Fz(of)28 b(the)g(free)-47 3502 y(problem)d(is)g (closely)h(related)h(to)g(the)g(restriction)e(of)i(the)g(F)-8 b(ourier)26 b(transform)g(to)h(the)g(sphere)f Fu(j)p Fy(x)p Fu(j)f Fz(=)3409 3424 y Fu(p)p 3485 3424 54 4 v 78 x Fy(\025)i Fz(for)f Fy(\025)f(>)g Fz(0.)-47 3615 y(In)30 b(con)m(trast)h(to)h(\(1.10\))r(,)f(whic)m(h)e(hea)m(vily)h (relies)f(on)h(the)h(non)m(v)-5 b(anishing)28 b(Gaussian)i(curv)-5 b(ature)30 b(of)h(the)g(sphere,)f(the)-47 3727 y(classical)37 b(limiting)f(absorption)h(principle)e(of)k(Agmon)g(and)f(Kuro)s(da)f ([Ag)q(],)k([Ku1],)g(and)c([Ku2])i(only)f(uses)g(the)-47 3840 y(most)32 b(elemen)m(tary)g(restriction)f(prop)s(ert)m(y)g(of)h (the)g(F)-8 b(ourier)31 b(transform)h(to)g(arbitrary)f(surfaces)g(whic) m(h)g(leads)g(to)h(a)-47 3953 y(loss)e(of)239 3917 y Fq(1)p 239 3932 36 4 v 239 3985 a(2)304 3953 y Fz(+)20 b Fy(")31 b Fz(deriv)-5 b(ativ)m(es)30 b(in)f Fy(L)1089 3920 y Fq(2)1159 3953 y Fz(\(on)h(the)h(ph)m(ysical)e(side)g(this)h (translates)g(in)m(to)g(the)h(w)m(eigh)m(ts)g Fu(j)p Fy(x)p Fu(j)3378 3890 y Fr(1)p 3378 3902 31 3 v 3378 3943 a(2)3419 3917 y Fq(+)p Fw(")3540 3953 y Fz(in)f Fy(L)3709 3920 y Fq(2)3778 3953 y Fz(that)-47 4066 y(app)s(ear)j(in)g ([Ag],)j([JK)o(])e(etc.\).)53 b(F)-8 b(or)35 b(further)d(details)h(of)h (the)g(pro)s(of)f(of)h(Prop)s(osition)e(1.3)j(w)m(e)g(refer)e(the)h (reader)g(to)-47 4179 y(Section)c(3.)-47 4292 y(It)h(is)f(common)i(kno) m(wledge)f(that)g(the)h(case)g(of)f(large)g(energies)g(should)f(b)s(e)g (the)h(most)h(accessible)f(one.)43 b(F)-8 b(rom)32 b(the)-47 4405 y(p)s(ersp)s(ectiv)m(e)24 b(of)i(scattering)g(the)f(in)m(tuition)e (is)i(that)g(particles)g(with)f(high)g(energies)h(will)d(escap)s(e)k (the)g(scatterer)g(and)-47 4518 y(th)m(us)f(lead)h(to)g(extended)g (states)h(\(absolutely)e(con)m(tin)m(uous)h(sp)s(ectrum\))f(whereas)g (particles)g(with)g(smaller)f(energies)-47 4631 y(can)38 b(b)s(e)g(trapp)s(ed)f(and)g(create)j(b)s(ound)c(states)k(\(pure)d(p)s (oin)m(t)h(sp)s(ectrum\).)63 b(It)38 b(is)g(of)g(course)g(a)h(most)f (in)m(teresting)-47 4744 y(problem)30 b(to)j(extend)g(Prop)s(osition)d (1.3)j(to)g(small)e(energies)h(under)f(similar)e(conditions.)45 b(Recall)32 b(that)h([JSS)o(])f(and)-47 4857 y(particularly)26 b([Y2)q(])j(ha)m(v)m(e)h(accomplished)d(exactly)j(that,)g(but)e(under)f (conditions)g(on)i Fy(V)49 b Fz(that)29 b(are)g(b)m(y)g(far)g(stronger) -47 4969 y(than)h(those)h(in)e(Prop)s(osition)f(1.3.)-47 5082 y(W)-8 b(e)26 b(also)f(address)g(the)g(question)f(of)i(Stric)m (hartz)f(estimates)g(for)g Fy(e)2194 5049 y Fw(it)p Fq(\()p Fx(\0004)p Fq(+)p Fw(V)16 b Fq(\))2558 5082 y Fz(in)23 b(dimensions)g(greater)j(or)f(equal)g(than)-47 5195 y(three.)41 b(T)-8 b(raditionally)27 b(the)j(mixed)f(norm)g(Stric)m(hartz)h (estimates)g(\(1.2\))i(are)e(sho)m(wn)g(to)g(b)s(e)g(a)g(consequence)h (of)f(the)1926 5472 y(6)p eop %%Page: 7 7 7 6 bop -47 274 a Fz(disp)s(ersiv)m(e)32 b(estimates.)56 b(In)35 b(fact,)i(in)d([JSS)o(],)j(Journ)m(\023)-43 b(e,)37 b(So\013er,)f(and)f(Sogge)h(establish)e(the)h Fy(L)3189 241 y Fq(1)3261 274 y Fu(!)f Fy(L)3448 241 y Fx(1)3557 274 y Fz(disp)s(ersiv)m(e)-47 387 y(b)s(ound)c(and)i(therefore)h(also)f 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b(+)g Fy(V)k Fl(.)41 b(Her)-5 b(e)-47 1602 y Fy(P)11 1616 y Fw(c)78 1602 y Fl(denotes)34 b(the)f(sp)-5 b(e)g(ctr)g(al)35 b(pr)-5 b(oje)g(ction)35 b(onto)e(the)g(c)-5 b(ontinuous)34 b(states.)-47 1789 y Fz(The)29 b(deca)m(y)h(condition)e Fu(j)p Fy(V)21 b Fz(\()p Fy(x)p Fz(\))p Fu(j)26 b Fo(.)f Fz(\(1)19 b(+)f Fu(j)p Fy(x)p Fu(j)p Fz(\))1481 1756 y Fx(\000)p Fq(2)p Fx(\000)p Fw(")1692 1789 y Fz(is)29 b(v)m(ery)h(natural)e(from)h(the)g(p)s(ersp)s(ectiv)m(e)g(of)g(Kato's)i (smo)s(othing)-47 1902 y(theory)j([Ka].)50 b(In)33 b(con)m(trast)i(to)g ([JSS)n(])f(w)m(e)g(pro)m(v)m(e)h(the)e(Stric)m(hartz)h(estimates)g (directly)-8 b(,)34 b(i.e.,)g(without)f(relying)f(on)-47 2015 y(disp)s(ersiv)m(e)g(estimates.)55 b(In)34 b(fact,)j(w)m(e)e(do)g (not)g(kno)m(w)g(if)e(the)i Fy(L)2139 1982 y Fq(1)2211 2015 y Fu(!)e Fy(L)2397 1982 y Fx(1)2506 2015 y Fz(estimates)j(hold)d (under)g(the)i(conditions)-47 2128 y(of)d(Theorem)f(1.4.)46 b(It)32 b(is)f(kno)m(wn)h(that)g(\(lo)s(cal)f(in)g(time\))h(Stric)m 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b(r)-5 b(e)g(quir)g(e)34 b(that)f(b)-5 b(oth)901 521 y Fu(k)p Fy(V)21 b Fu(k)1065 483 y Fq(2)1065 543 y Fw(R)1148 521 y Fz(:=)1269 397 y Fn(Z)1320 603 y Fp(R)1368 584 y Fr(3)1402 603 y Fx(\002)p Fp(R)1505 584 y Fr(3)1569 459 y Fu(j)p Fy(V)g Fz(\()p Fy(x)p Fz(\))p Fu(j)15 b(j)p Fy(V)21 b Fz(\()p Fy(y)s Fz(\))p Fu(j)p 1569 500 504 4 v 1671 583 a(j)p Fy(x)f Fu(\000)g Fy(y)s Fu(j)1932 557 y Fq(2)2098 521 y Fy(dx)15 b(dy)87 b(<)82 b Fz(\(4)p Fy(\031)s Fz(\))2714 483 y Fq(2)2853 521 y Fl(and)-3043 b Fz(\(2.1\))1200 773 y Fu(k)p Fy(V)21 b Fu(k)1364 787 y Fx(K)1448 773 y Fz(:=)41 b(sup)1569 857 y Fw(x)p Fx(2)p Fp(R)1704 838 y Fr(3)1754 649 y Fn(Z)1804 856 y Fp(R)1852 837 y Fr(3)1925 712 y Fu(j)p Fy(V)21 b Fz(\()p Fy(y)s Fz(\))p Fu(j)p 1916 752 262 4 v 1916 836 a(j)p Fy(x)f Fu(\000)g Fy(y)s Fu(j)2212 773 y Fy(dy)87 b(<)82 b Fz(4)p Fy(\031)s(:)-2716 b Fz(\(2.2\))-47 1038 y(The)32 b(norm)f Fu(k)22 b(\001)g(k)540 1052 y Fw(R)631 1038 y Fz(on)32 b(the)g(left-hand)g(side)f(of)40 b(\(2.1\))34 b(is)d(usually)g (referred)g(to)i(as)g(the)g Fl(R)-5 b(ol)5 b(lnik)35 b(norm)p Fz(.)48 b(Kato)33 b([Ka)q(])-47 1151 y(sho)m(w)m(ed)22 b(that)h(under)e(the)h(condition)f(\(2.1\))j(the)e(op)s(erator)h Fy(H)29 b Fz(admits)21 b(a)i(self-adjoin)m(t)f(extension)f(whic)m(h)g (is)h(unitarily)-47 1264 y(equiv)-5 b(alen)m(t)37 b(to)h Fy(H)586 1278 y Fq(0)662 1264 y Fz(=)f Fu(\0004)p Fz(.)62 b(In)36 b(particular,)i(the)g(sp)s(ectrum)e(of)i Fy(H)45 b Fz(is)36 b(purely)g(absolutely)g(con)m(tin)m(uous.)62 b(Man)m(y)-47 1377 y(prop)s(erties)20 b(of)i(the)g(Rollnik)d(norm,)k (whic)m(h)e(can)h(b)s(e)f(seen)h(to)h(b)s(e)e(ma)5 b(jorized)21 b(b)m(y)h(the)g(norm)f(of)h Fy(L)3186 1314 y Fr(3)p 3186 1326 31 3 v 3186 1367 a(2)3230 1377 y Fz(\()p Fv(R)3325 1344 y Fq(3)3371 1377 y Fz(\))g(via)f(fractional)-47 1490 y(in)m(tegration,)33 b(can)g(b)s(e)f(found)f(in)g(Simon's)g (monograph)i([Si1)o(].)47 b(The)32 b(norm)g Fu(k)22 b(\001)g(k)2753 1504 y Fx(K)2844 1490 y Fz(in)32 b(\(2.2\))i(is)e(closely)g(related)g (to)-47 1603 y(the)d(w)m(ell-kno)m(wn)f Fl(Kato)33 b(norm)p Fz(,)e(see)f(Aizenman)f(and)g(Simon)e([AS],)j([Si2])f(and)g(w)m(e)h (refer)f(to)h(it)f(as)g(the)h Fl(glob)-5 b(al)33 b(Kato)-47 1716 y(norm)p Fz(.)-47 1829 y(The)38 b(main)g(result)g(in)f(this)h (section)h(is)f(Theorem)g(2.6.)68 b(The)38 b(pro)s(of)g(splits)f(in)m (to)i(sev)m(eral)g(lemmas,)i(the)e(\014rst)f(of)-47 1941 y(whic)m(h)f(presen)m(ts)i(some)g(w)m(ell-kno)m(wn)e(prop)s(erties)h (of)g(the)h(resolv)m(en)m(ts)h Fy(R)2507 1955 y Fw(V)2567 1941 y Fz(\()p Fy(z)t Fz(\))h(=)d(\()p Fu(\0004)26 b Fz(+)f Fy(V)46 b Fu(\000)26 b Fy(z)t Fz(\))3419 1908 y Fx(\000)p Fq(1)3552 1941 y Fz(under)38 b(the)-47 2054 y(condition)27 b(\(2.1\))r(.)40 b(W)-8 b(e)30 b(b)s(egin)d(b)m(y)i (recalling)e(that)i(a)g(p)s(oten)m(tial)g(with)e(\014nite)h(\(but)g (not)h(necessarily)f(small\))f(Rollnik)-47 2167 y(norm)i(is)h Fl(Kato)j(smo)-5 b(othing)p Fz(,)33 b(i.e.,)404 2381 y(sup)411 2457 y Fw(">)p Fq(0)556 2381 y Fu(k)15 b(j)p Fy(V)21 b Fu(j)750 2317 y Fr(1)p 750 2329 V 750 2370 a(2)795 2381 y Fy(R)864 2395 y Fq(0)903 2381 y Fz(\()p Fy(\025)g Fu(\006)f Fy(i")p Fz(\))p Fy(f)10 b Fu(k)1311 2405 y Fw(L)1359 2382 y Fr(2)1359 2429 y Fs(\025)1400 2405 y Fw(L)1448 2386 y Fr(2)1448 2422 y Fs(x)1517 2381 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)1830 2401 y Fw(L)1878 2382 y Fr(2)1916 2381 y Fy(;)197 b Fz(sup)2145 2457 y Fw(">)p Fq(0)2290 2381 y Fu(k)31 b Fy(R)2435 2395 y Fq(0)2474 2381 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))15 b Fu(j)p Fy(V)22 b Fu(j)2932 2317 y Fr(1)p 2932 2329 V 2932 2370 a(2)2976 2381 y Fy(f)10 b Fu(k)3076 2405 y Fw(L)3124 2382 y Fr(2)3124 2429 y Fs(\025)3165 2405 y Fw(L)3213 2386 y Fr(2)3213 2422 y Fs(x)3281 2381 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)3594 2401 y Fw(L)3642 2382 y Fr(2)-47 2381 y Fz(\(2.3\))-47 2665 y(for)32 b(an)m(y)h Fy(f)39 b Fu(2)29 b Fy(L)504 2632 y Fq(2)543 2665 y Fz(\()p Fv(R)638 2632 y Fq(3)683 2665 y Fz(\))k(and)f(with)g Fy(R)1209 2679 y Fq(0)1248 2665 y Fz(\()p Fy(z)t Fz(\))f(=)d(\()p Fu(\0004)22 b(\000)g Fy(z)t Fz(\))1877 2632 y Fx(\000)p Fq(1)1971 2665 y Fz(.)48 b(This)31 b(implies,)g(in)g(particular,)h(that)i Fu(D)s Fz(\()p Fu(j)p Fy(V)20 b Fu(j)3588 2602 y Fr(1)p 3588 2614 V 3588 2655 a(2)3633 2665 y Fz(\))29 b Fu(\033)g Fy(H)3880 2632 y Fq(2)3920 2665 y Fz(.)-47 2778 y(The)34 b(Rollnik)f(norm)h(arises)g(in)g(this)g(con)m(text)j(as)e(a)h(ma)5 b(joran)m(t)35 b(for)g(the)g(Hilb)s(ert-Sc)m(hmidt)e(norm)h Fu(k)24 b(\001)f(k)3561 2792 y Fw(H)5 b(S)3710 2778 y Fz(of)36 b(the)-47 2891 y(op)s(erators)1280 3105 y Fy(K)7 b Fz(\()p Fy(\025)20 b Fu(\006)g Fy(i")p Fz(\))27 b(:=)e Fu(j)p Fy(V)20 b Fu(j)1952 3040 y Fr(1)p 1952 3052 V 1952 3093 a(2)1997 3105 y Fy(R)2066 3119 y Fq(0)2106 3105 y Fz(\()p Fy(\025)g Fu(\006)g Fy(i")p Fz(\))p Fu(j)p Fy(V)i Fu(j)2548 3040 y Fr(1)p 2548 3052 V 2548 3093 a(2)2592 3105 y Fy(:)-2664 b Fz(\(2.4\))-47 3309 y(Indeed,)29 b(it)h(is)g(w)m(ell-kno)m(wn)f(that)i(the)g(resolv)m(en)m(t)g Fy(R)1733 3323 y Fq(0)1772 3309 y Fz(\()p 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4600 a(2)2468 4611 y Fy(R)2537 4625 y Fq(0)2577 4611 y Fz(\()5 b(\026)-50 b Fy(z)t Fz(\))p Fu(j)p Fy(V)21 b Fu(j)2827 4547 y Fr(1)p 2827 4559 V 2827 4600 a(2)2872 4538 y Fn(\003)2910 4611 y Fy(:)-47 4849 y Fz(In)29 b(view)h(of)38 b(\(2.6\))32 b(therefore)1051 5092 y Fu(k)p Fy(S)1152 5106 y Fw(z)1192 5092 y Fu(k)1237 5054 y Fq(2)1302 5092 y Fz(=)25 b Fu(k)p Fy(S)1499 5106 y Fw(z)1539 5092 y Fy(S)1600 5054 y Fx(\003)1595 5114 y Fw(z)1640 5092 y Fu(k)h Fo(.)1875 5030 y Fz(1)p 1816 5071 163 4 v 1816 5154 a Fu(j=)p Fy(z)t Fu(j)1989 5092 y(k)p Fy(K)7 b Fz(\()p Fy(z)t Fz(\))p Fu(k)27 b Fo(.)2470 5030 y Fz(1)p 2412 5071 V 2412 5154 a Fu(j=)p Fy(z)t Fu(j)2584 5092 y(k)p Fy(V)21 b Fu(k)2748 5106 y Fw(R)2806 5092 y Fy(;)-2878 b Fz(\(2.7\))1926 5472 y(8)p eop %%Page: 9 9 9 8 bop -47 274 a Fz(as)20 b(desired.)36 b(One)20 b(of)h(the)f(main)f (observ)-5 b(ations)20 b(of)h(Kato)g([Ka)q(])f(w)m(as)h(the)f(relation) g(b)s(et)m(w)m(een)h(this)e(p)s(oin)m(t)m(wise)g(condition)-47 387 y(in)27 b Fy(z)j Fz(=)25 b Fy(\025)17 b Fu(\006)f Fy(i")30 b Fz(and)e(the)h Fy(L)877 354 y Fq(2)877 415 y Fw(\025)951 387 y Fz(b)s(oundedness)d(that)j(app)s(ears)f(in)g (\(2.3\))q(.)41 b(W)-8 b(e)29 b(presen)m(t)g(a)g(short)g(pro)s(of)f(of) h(this)e(fact)j(for)-47 500 y(the)d(sak)m(e)h(of)g(completeness.)39 b(Although)27 b(it)f(is)g(standard,)i(the)f(follo)m(wing)e(argumen)m(t) j(is)e(somewhat)i(di\013eren)m(t)e(from)-47 613 y(the)33 b(usual)f(one)i(whic)m(h)e(can)i(b)s(e)f(found)f(in)g(basic)h (references)h(lik)m(e)f(Kato)h([Ka])g(and)f(Reed,)i(Simon)c([RS].)50 b(Denote)-47 739 y Fy(T)6 753 y Fw(")75 739 y Fz(:=)32 b Fu(j)p Fy(V)20 b Fu(j)336 676 y Fr(1)p 336 688 31 3 v 336 729 a(2)381 739 y Fy(R)450 753 y Fq(0)489 739 y Fz(\()p Fy(\025)j Fz(+)g Fy(i")p Fz(\))35 b(for)g Fy(")d(>)g Fz(0.)53 b(T)-8 b(runcating)33 b(the)i(large)f(v)-5 b(alues)34 b(of)h Fy(V)54 b Fz(and)34 b(then)g(passing)f(to)j(the)e(limit)e(w)m(e) -47 852 y(ma)m(y)f(assume)f(that)h Fy(V)50 b Fz(is)30 b(b)s(ounded.)38 b(Then)30 b Fy(T)1553 866 y Fw(")1615 852 y Fz(:)25 b Fy(L)1727 819 y Fq(2)1792 852 y Fu(!)g Fy(L)1970 819 y Fq(2)1970 879 y Fw(\025)2015 852 y Fy(L)2077 819 y Fq(2)2077 874 y Fw(x)2151 852 y Fz(for)30 b(ev)m(ery)i Fy(")25 b(>)g Fz(0)31 b(and)f(one)g(c)m(hec)m(ks)i(that)1294 1101 y Fy(T)1360 1064 y Fx(\003)1399 1101 y Fy(F)39 b Fz(=)1592 978 y Fn(Z)1698 1101 y Fy(R)1767 1115 y Fq(0)1806 1101 y Fz(\()p Fy(\025)21 b Fu(\000)f Fy(i")p Fz(\))p Fu(j)p Fy(V)h Fu(j)2248 1037 y Fr(1)p 2249 1049 V 2249 1090 a(2)2293 1101 y Fy(F)13 b Fz(\()p Fy(\025)p Fz(\))i Fy(d\025)-47 1354 y Fz(for)30 b(ev)m(ery)h Fy(F)38 b Fu(2)25 b Fy(L)576 1321 y Fq(2)576 1381 y Fw(\025)621 1354 y Fz(\()p Fy(L)718 1321 y Fq(2)718 1376 y Fw(x)763 1354 y Fz(\).)41 b(Th)m(us)753 1603 y Fy(T)806 1617 y Fw(")843 1603 y Fy(T)909 1566 y Fx(\003)896 1626 y Fw(")948 1603 y Fy(F)96 b Fz(=)1256 1480 y Fn(Z)1362 1603 y Fu(j)p Fy(V)21 b Fu(j)1496 1539 y Fr(1)p 1496 1551 V 1496 1592 a(2)1541 1603 y Fy(R)1610 1617 y Fq(0)1649 1603 y Fz(\()p Fy(\025)g Fz(+)e Fy(i")p Fz(\))p Fy(R)2025 1617 y Fq(0)2066 1603 y Fz(\()p Fy(\026)h Fu(\000)g Fy(i")p Fz(\))p Fu(j)p Fy(V)i Fu(j)2510 1539 y Fr(1)p 2510 1551 V 2510 1592 a(2)2555 1603 y Fy(F)13 b Fz(\()p Fy(\026)p Fz(\))i Fy(d\026)1102 1845 y Fz(=)83 b Fu(\000)1342 1721 y Fn(Z)1448 1845 y Fu(j)p Fy(V)21 b Fu(j)1582 1780 y Fr(1)p 1582 1792 V 1582 1833 a(2)1636 1783 y Fy(R)1705 1797 y Fq(0)1745 1783 y Fz(\()p Fy(\025)f Fz(+)g Fy(i")p Fz(\))i Fu(\000)d Fy(R)2233 1797 y Fq(0)2273 1783 y Fz(\()p Fy(\026)h Fu(\000)g Fy(i")p Fz(\))p 1636 1824 947 4 v 1885 1907 a Fy(\025)g Fu(\000)g Fy(\026)g Fz(+)g(2)p Fy(i")2593 1845 y Fu(j)p Fy(V)h Fu(j)2727 1780 y Fr(1)p 2727 1792 31 3 v 2727 1833 a(2)2771 1845 y Fy(F)13 b Fz(\()p Fy(\026)p Fz(\))i Fy(d\026)-3131 b Fz(\(2.8\))1102 2088 y(=)83 b Fu(\000)1342 1964 y Fn(Z)1458 2026 y Fy(K)7 b Fz(\()p Fy(\025)20 b Fz(+)g Fy(i")p Fz(\))p Fy(F)13 b Fz(\()p Fy(\026)p Fz(\))p 1458 2067 589 4 v 1528 2150 a Fy(\025)20 b Fu(\000)g Fy(\026)g Fz(+)g(2)p Fy(i")2072 2088 y(d\026)g Fz(+)2285 1964 y Fn(Z)2401 2026 y Fy(K)7 b Fz(\()p Fy(\026)20 b Fu(\000)g Fy(i")p Fz(\))p Fy(F)13 b Fz(\()p Fy(\026)p Fz(\))p 2401 2067 591 4 v 2472 2150 a Fy(\025)20 b Fu(\000)g Fy(\026)g Fz(+)g(2)p Fy(i")3017 2088 y(d\026;)-3191 b Fz(\(2.9\))-47 2342 y(where)31 b(w)m(e)g(used)g(the)g(resolv)m(en)m(t)h (iden)m(tit)m(y)f(to)h(pass)f(to)h(\(2.8\))r(.)43 b(By)32 b(the)f Fy(L)2478 2309 y Fq(2)2549 2342 y Fz(b)s(oundedness)d(of)k(the) f(\(v)m(ector)i(v)-5 b(alued\))-47 2455 y(Hilb)s(ert)28 b(transform,)1145 2700 y(sup)1152 2776 y Fw(">)p Fq(0)1282 2596 y Fn(\015)1282 2650 y(\015)1282 2705 y(\015)1332 2576 y(Z)1552 2639 y Fy(F)13 b Fz(\()p Fy(\026)p Fz(\))p 1448 2679 404 4 v 1448 2763 a Fy(\025)20 b Fz(+)g Fy(i")h Fu(\000)f Fy(\026)1877 2700 y(d\026)1979 2596 y Fn(\015)1979 2650 y(\015)1979 2705 y(\015)2030 2763 y Fw(L)2078 2741 y Fr(2)2078 2787 y Fs(\025)2119 2763 y Fw(L)2167 2745 y Fr(2)2167 2780 y Fs(x)2235 2700 y Fo(.)25 b Fu(k)p Fy(F)13 b Fu(k)2492 2724 y Fw(L)2540 2701 y Fr(2)2540 2748 y Fs(\025)2582 2724 y Fq(\()p Fw(L)2657 2705 y Fr(2)2657 2740 y Fs(x)2696 2724 y Fq(\))2728 2700 y Fy(:)-47 2956 y Fz(Using)29 b(this)h(b)s(ound)e(and)i(\(2.6\))i(in)d(\(2.9\))j (yields)1306 3158 y(sup)1313 3234 y Fw(">)p Fq(0)1458 3158 y Fu(k)p Fy(T)1556 3172 y Fw(")1594 3158 y Fy(T)1660 3121 y Fx(\003)1647 3181 y Fw(")1699 3158 y Fy(F)13 b Fu(k)1815 3178 y Fw(L)1863 3159 y Fr(2)1863 3195 y Fs(x)1932 3158 y Fo(.)25 b Fu(k)p Fy(F)13 b Fu(k)2189 3182 y Fw(L)2237 3159 y Fr(2)2237 3206 y Fs(\025)2278 3182 y Fw(L)2326 3163 y Fr(2)2326 3199 y Fs(x)2369 3158 y Fu(k)p Fy(V)21 b Fu(k)2533 3172 y Fw(R)-47 3462 y Fz(whic)m(h)29 b(implies)e(\(2.3\)) 33 b(with)c(a)h(constan)m(t)i(of)f(the)f(form)g Fy(C)7 b Fu(k)p Fy(V)20 b Fu(k)2108 3378 y Fr(1)p 2109 3390 31 3 v 2109 3431 a(2)2098 3491 y Fw(R)2156 3462 y Fz(.)-47 3649 y Fm(Lemma)32 b(2.2.)47 b Fl(L)-5 b(et)32 b Fu(k)p Fy(V)21 b Fu(k)852 3663 y Fw(R)936 3649 y Fy(<)k Fz(4)p Fy(\031)36 b Fl(as)d(in)f(De\014nition)h(2.1.)42 b(Then)33 b(for)h(al)5 b(l)33 b Fy(f)5 b(;)15 b(g)28 b Fu(2)d Fy(L)2833 3616 y Fq(2)2872 3649 y Fz(\()p Fv(R)2967 3616 y Fq(3)3013 3649 y Fz(\))514 3921 y Fu(h)p Fy(R)618 3935 y Fw(V)680 3921 y Fz(\()p Fy(\025)20 b Fu(\006)g Fy(i")p Fz(\))p Fy(f)5 b(;)31 b(g)s Fu(i)21 b(\000)f(h)p Fy(R)1390 3935 y Fq(0)1430 3921 y Fz(\()p Fy(\025)h Fu(\006)e Fy(i")p Fz(\))p Fy(f)5 b(;)31 b(g)s Fu(i)c Fz(=)2077 3807 y Fx(1)2046 3835 y Fn(X)2053 4032 y Fw(`)p Fq(=1)2178 3921 y Fz(\()p Fu(\000)p Fz(1\))2364 3883 y Fw(`)2398 3921 y Fu(h)p Fy(R)2502 3935 y Fq(0)2541 3921 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\)\()p Fy(V)i(R)3028 3935 y Fq(0)3067 3921 y Fz(\()p Fy(\025)f Fu(\006)f Fy(i")p Fz(\)\))3410 3883 y Fw(`)3444 3921 y Fy(f)5 b(;)15 b(g)s Fu(i)-3662 b Fz(\(2.10\))-47 4215 y Fl(wher)-5 b(e)28 b(the)g(right-hand)h(side)f (of)47 b Fz(\(2.10\))30 b Fl(is)d(an)h(absolutely)h(c)-5 b(onver)g(gent)29 b(series)e(in)h(the)g(norm)g(of)g Fy(L)3325 4182 y Fq(1)3364 4215 y Fz(\()p Fy(d\025)p Fz(\))h Fl(uniformly)-47 4328 y(in)j Fy(")26 b(>)f Fz(0)p Fl(.)42 b(F)-7 b(urthermor)i(e,)35 b(if)d Fu(k)p Fy(V)41 b Fu(\000)20 b Fy(V)1267 4342 y Fw(m)1334 4328 y Fu(k)1379 4342 y Fw(R)1462 4328 y Fu(!)25 b Fz(0)33 b Fl(as)g Fy(m)26 b Fu(!)f(1)p Fl(,)32 b(then)908 4573 y Fz(sup)915 4649 y Fw(">)p Fq(0)1060 4449 y Fn(Z)1151 4469 y(\014)1151 4523 y(\014)1151 4578 y(\014)1181 4573 y Fu(h)p Fy(R)1285 4587 y Fw(V)1326 4595 y Fs(m)1390 4573 y Fz(\()p Fy(\025)20 b Fu(\006)g Fy(i")p Fz(\))p Fy(f)5 b(;)31 b(g)s Fu(i)21 b(\000)f(h)p Fy(R)2100 4587 y Fw(V)2161 4573 y Fz(\()p Fy(\025)h Fu(\006)f Fy(i")p Fz(\))p Fy(f)5 b(;)31 b(g)s Fu(i)2656 4469 y Fn(\014)2656 4523 y(\014)2656 4578 y(\014)2702 4573 y Fy(d\025)26 b Fu(!)f Fz(0)-3036 b(\(2.11\))-47 4799 y Fl(as)33 b Fy(m)25 b Fu(!)g(1)p Fl(.)-47 4986 y(Pr)-5 b(o)g(of.)47 b Fz(W)-8 b(e)31 b(start)g(from)f(the)h(resolv)m(en)m(t)g(iden)m(tit)m (y)920 5189 y Fy(R)989 5203 y Fw(V)1049 5189 y Fz(\()p Fy(z)t Fz(\))22 b Fu(\000)e Fy(R)1347 5203 y Fq(0)1386 5189 y Fz(\()p Fy(z)t Fz(\))26 b(=)f Fu(\000)p Fy(R)1764 5203 y Fq(0)1803 5189 y Fz(\()p Fy(z)t Fz(\))p Fy(V)d(R)2063 5203 y Fw(V)2124 5189 y Fz(\()p Fy(z)t Fz(\))k(=)f Fu(\000)p Fy(R)2502 5203 y Fw(V)2562 5189 y Fz(\()p Fy(z)t Fz(\))p Fy(V)d(R)2822 5203 y Fq(0)2861 5189 y Fz(\()p Fy(z)t Fz(\))-3024 b(\(2.12\))1926 5472 y(9)p eop %%Page: 10 10 10 9 bop -47 274 a Fz(whic)m(h)29 b(holds)g(in)g(the)h(sense)g(of)h(b)s (ounded)d(op)s(erators)j(on)f Fy(L)2012 241 y Fq(2)2081 274 y Fz(for)g(an)m(y)h Fu(=)p Fy(z)e Fu(6)p Fz(=)c(0,)31 b(see)g(\(2.7\))r(.)41 b(It)30 b(is)f(a)i(standard)f(fact,)-47 387 y(see)g([Ka],)g(that)h(the)e(Kato)i(smo)s(othing)d(prop)s(ert)m(y)i (\(2.3\))i(remains)c(v)-5 b(alid)28 b(with)g Fy(R)2761 401 y Fw(V)2852 387 y Fz(instead)g(of)i Fy(R)3336 401 y Fq(0)3405 387 y Fz(pro)m(vided)e(that)-47 513 y Fu(k)p Fy(V)20 b Fu(k)116 527 y Fw(R)200 513 y Fy(<)25 b Fz(4)p Fy(\031)s Fz(.)41 b(Indeed,)30 b(m)m(ultiplying)e(\(2.12\))k(b)m(y)f Fu(j)p Fy(V)20 b Fu(j)1791 450 y Fr(1)p 1791 462 31 3 v 1791 503 a(2)1866 513 y Fz(leads)30 b(to)1376 688 y(\(1)21 b(+)f Fy(Q)p Fz(\()p Fy(z)t Fz(\)\))p Fy(AR)1928 702 y Fw(V)1990 688 y Fz(\()p Fy(z)t Fz(\))26 b(=)f Fy(AR)2365 702 y Fq(0)2405 688 y Fz(\()p Fy(z)t Fz(\))-2568 b(\(2.13\))-47 888 y(where)30 b Fy(Q)p Fz(\()p Fy(z)t Fz(\))c(:=)f Fy(AR)688 902 y Fq(0)728 888 y Fz(\()p Fy(z)t Fz(\))p Fy(B)5 b Fz(,)31 b Fy(A)25 b Fz(=)g Fu(j)p Fy(V)c Fu(j)1297 825 y Fr(1)p 1297 837 V 1297 878 a(2)1341 888 y Fz(,)31 b(and)f Fy(B)f Fz(=)c Fu(j)p Fy(V)c Fu(j)1902 825 y Fr(1)p 1902 837 V 1902 878 a(2)1947 888 y Fz(sign)o Fy(V)f Fz(.)41 b(In)29 b(view)h(of)37 b(\(2.6\))c(one)d(has)451 1063 y(sup)432 1142 y Fx(=)p Fw(z)s Fx(6)p Fq(=0)623 1063 y Fu(k)p Fy(Q)p Fz(\()p Fy(z)t Fz(\))p Fu(k)901 1083 y Fw(L)949 1064 y Fr(2)986 1083 y Fx(!)p Fw(L)1105 1064 y Fr(2)1168 1063 y Fz(=:)c Fy(\032)f(<)g Fz(1)61 b(so)30 b(that)97 b(sup)1918 1142 y Fx(=)p Fw(z)s Fx(6)p Fq(=0)2110 1063 y Fu(k)p Fz(\(1)21 b(+)f Fy(Q)p Fz(\()p Fy(z)t Fz(\)\))2570 1025 y Fx(\000)p Fq(1)2666 1063 y Fu(k)2711 1083 y Fw(L)2759 1064 y Fr(2)2794 1083 y Fx(!)p Fw(L)2913 1064 y Fr(2)2976 1063 y Fu(\024)25 b Fz(\(1)c Fu(\000)f Fy(\032)p Fz(\))3346 1025 y Fx(\000)p Fq(1)3441 1063 y Fy(:)-3513 b Fz(\(2.14\))-47 1303 y(In)29 b(conjunction)h(with)f(\(2.13\))k(and)c(\(2.3\))j(this)e (implies)d(that)512 1478 y(sup)520 1553 y Fw(">)p Fq(0)665 1478 y Fu(k)15 b Fy(AR)862 1492 y Fw(V)923 1478 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))p Fy(f)10 b Fu(k)1331 1501 y Fw(L)1379 1479 y Fr(2)1379 1525 y Fs(\025)1420 1501 y Fw(L)1468 1483 y Fr(2)1468 1518 y Fs(x)1537 1478 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)1850 1497 y Fw(L)1898 1479 y Fr(2)1936 1478 y Fy(;)197 b Fz(sup)2165 1553 y Fw(">)p Fq(0)2310 1478 y Fu(k)31 b Fy(R)2455 1492 y Fw(V)2516 1478 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))15 b Fy(B)5 b(f)10 b Fu(k)3013 1501 y Fw(L)3061 1479 y Fr(2)3061 1525 y Fs(\025)3102 1501 y Fw(L)3150 1483 y Fr(2)3150 1518 y Fs(x)3218 1478 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)3531 1497 y Fw(L)3579 1479 y Fr(2)-47 1478 y Fz(\(2.15\))-47 1713 y(for)30 b(an)m(y)h Fy(f)j Fu(2)25 b Fy(L)491 1680 y Fq(2)530 1713 y Fz(.)41 b(Fix)30 b Fy(f)5 b(;)15 b(g)28 b Fu(2)d Fy(L)1068 1680 y Fq(2)1108 1713 y Fz(.)40 b(Iterating)32 b(\(2.12\))h(leads)c(to)-47 2096 y Fu(h)p Fy(R)57 2110 y Fw(V)118 2096 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))p Fy(f)5 b(;)30 b(g)s Fu(i)d Fz(=)769 1982 y Fw(N)735 2009 y Fn(X)741 2207 y Fw(`)p Fq(=0)866 2096 y Fz(\()p Fu(\000)p Fz(1\))1052 2058 y Fw(`)1086 2096 y Fu(h)p Fy(R)1190 2110 y Fq(0)1230 2096 y Fz(\()p Fy(\025)20 b Fu(\006)g Fy(i")p Fz(\)\()p Fy(V)i(R)1716 2110 y Fq(0)1755 2096 y Fz(\()p Fy(\025)f Fu(\006)f Fy(i")p Fz(\)\))2098 2058 y Fw(`)2132 2096 y Fy(f)5 b(;)15 b(g)s Fu(i)21 b Fz(+)f(\()p Fu(\000)p Fz(1\))2601 2058 y Fw(N)7 b Fq(+1)2759 2096 y Fu(h)p Fy(R)2863 2110 y Fw(V)2924 2096 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\)\()p Fy(V)i(R)3411 2110 y Fq(0)3450 2096 y Fz(\()p Fy(\025)f Fu(\006)f Fy(i")p Fz(\)\))3793 2058 y Fw(N)7 b Fq(+1)3951 2096 y Fy(f)e(;)15 b(g)s Fu(i)-47 1885 y Fz(\(2.16\))-47 2355 y(for)30 b(an)m(y)h(p)s(ositiv)m(e)e(in)m (teger)i Fy(N)10 b Fz(.)40 b(By)32 b(\(2.15\))h(the)d(error)g(term)h (is)112 2530 y Fu(h)p Fy(R)216 2544 y Fw(V)277 2530 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))p Fy(B)5 b Fz(\()p Fy(AR)831 2544 y Fq(0)871 2530 y Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))p Fy(B)5 b Fz(\))1288 2492 y Fw(N)1356 2530 y Fy(AR)1493 2544 y Fq(0)1533 2530 y Fz(\()p Fy(\025)20 b Fu(\006)g Fy(i")p Fz(\))p Fy(f)5 b(;)15 b(g)s Fu(i)27 b Fz(=)e Fu(h)p Fy(R)2238 2544 y Fw(V)2299 2530 y Fz(\()p Fy(\025)c Fu(\006)f Fy(i")p Fz(\))p Fy(B)5 b(Q)p Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i")p Fz(\))3061 2492 y Fw(N)3129 2530 y Fy(AR)3266 2544 y Fq(0)3306 2530 y Fz(\()p Fy(\025)g Fu(\006)g Fy(i")p Fz(\))p Fy(f)5 b(;)15 b(g)s Fu(i)-47 2705 y Fz(and)25 b(th)m(us)g(has)h Fy(L)540 2672 y Fq(1)579 2705 y Fz(\()p Fy(d\025)p Fz(\))g(norm)f(b)s(ounded)f(b)m(y)h Fy(C)d(\032)1630 2672 y Fw(N)1697 2705 y Fz(,)27 b(see)f(\(2.15\))i(and)d(\(2.14\))s(.) 39 b(Similarly)-8 b(,)23 b(eac)m(h)k(of)f(the)g(terms)f(in)g(the)-47 2818 y(sum)f(for)g(1)i Fu(\024)f Fy(`)g Fu(\024)g Fy(N)35 b Fz(has)25 b Fy(L)926 2785 y Fq(1)965 2818 y Fz(\()p Fy(d\025)p Fz(\))h(norm)e(at)i(most)f Fy(C)c(\032)1849 2785 y Fw(`)p Fx(\000)p Fq(1)1973 2818 y Fz(.)38 b(Th)m(us)24 b(\(2.10\))j(holds)d(for)g(an)m(y)i Fy(V)45 b Fz(whic)m(h)23 b(satis\014es)i(\(2.1\))r(.)-47 2931 y(If)i Fy(m)g Fz(is)f(su\016cien)m (tly)g(large,)i(then)f(the)g(series)g(expansion)f(\(2.10\))k(holds)25 b(for)i(b)s(oth)g Fy(V)47 b Fz(and)27 b Fy(V)3119 2945 y Fw(m)3186 2931 y Fz(.)39 b(Subtracting)26 b(these)-47 3044 y(series)k(term)m(wise)h(and)f(in)m(v)m(oking)g(the)h(previous)f (b)s(ounds)e(yields)h(that)j(the)f(left-hand)f(side)g(of)38 b(\(2.11\))33 b(is)d(b)s(ounded)-47 3157 y(b)m(y)1005 3283 y Fx(1)975 3310 y Fn(X)981 3508 y Fw(`)p Fq(=1)1121 3396 y Fy(C)7 b(`\032)1278 3359 y Fw(`)p Fx(\000)p Fq(1)1416 3396 y Fu(k)p Fy(V)41 b Fu(\000)20 b Fy(V)1699 3410 y Fw(m)1765 3396 y Fu(k)1810 3410 y Fw(R)1894 3396 y Fu(\024)25 b Fy(C)7 b Fz(\(1)20 b Fu(\000)g Fy(\032)p Fz(\))2335 3359 y Fx(\000)p Fq(2)2445 3396 y Fu(k)p Fy(V)41 b Fu(\000)20 b Fy(V)2728 3410 y Fw(m)2794 3396 y Fu(k)2839 3410 y Fw(R)2898 3396 y Fy(;)-47 3650 y Fz(and)30 b(the)g(lemma)g(follo)m(ws.) p 3879 3650 4 62 v 3883 3592 55 4 v 3883 3650 V 3937 3650 4 62 v -47 3833 a(The)g(follo)m(wing)f(tec)m(hnical)h(corollary)g (deals)g(with)f(the)i(case)h Fy(")26 b Fz(=)f(0)31 b(in)e(Lemma)i(2.2.) 42 b(W)-8 b(e)32 b(state)g(it)e(in)f(the)i(form)f(in)-47 3946 y(whic)m(h)k(it)h(is)f(used)h(later)g(on.)56 b(In)34 b(particular,)h(w)m(e)h(did)e(not)h(striv)m(e)h(for)f(the)g(greatest)i (generalit)m(y)-8 b(.)57 b(Belo)m(w)36 b Fy(C)3770 3913 y Fq(0)3763 3974 y Fw(b)3809 3946 y Fz(\()p Fv(R)s Fz(\))-47 4059 y(refers)30 b(to)h(the)f(b)s(ounded)f(con)m(tin)m(uous)h (functions)f(on)h Fv(R)39 b Fz(with)29 b(the)h(suprem)m(um)f(norm.)-47 4222 y Fm(Corollary)35 b(2.3.)46 b Fl(L)-5 b(et)33 b Fy(V)45 b Fu(2)25 b Fy(C)1039 4189 y Fx(1)1032 4247 y Fq(0)1113 4222 y Fz(\()p Fv(R)1208 4189 y Fq(3)1254 4222 y Fz(\))33 b Fl(satisfy)g Fu(k)p Fy(V)21 b Fu(k)1769 4236 y Fw(R)1852 4222 y Fy(<)k Fz(4)p Fy(\031)s Fl(.)42 b(Then)33 b(for)h(al)5 b(l)33 b Fy(f)5 b(;)15 b(g)28 b Fu(2)d Fy(C)2950 4189 y Fx(1)2943 4247 y Fq(0)3024 4222 y Fz(\()p Fv(R)3119 4189 y Fq(3)3165 4222 y Fz(\))33 b Fl(the)g(limit)1172 4398 y Fu(h)p Fy(R)1276 4412 y Fw(V)1337 4398 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(i)p Fz(0\))p Fy(f)5 b(;)15 b(g)s Fu(i)27 b Fz(=)k(lim)1942 4456 y Fw(")p Fx(!)p Fq(0)2080 4398 y Fu(h)p Fy(R)2184 4412 y Fw(V)2245 4398 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(i")p Fz(\))p Fy(f)5 b(;)15 b(g)s Fu(i)-47 4598 y Fl(exists)35 b(for)h(every)f Fy(\025)c Fu(2)e Fv(R)44 b Fl(and)37 b(is)e(a)h(c)-5 b(ontinuous)36 b(function)g(in)f Fy(\025)p Fl(.)50 b(Mor)-5 b(e)g(over,)37 b(for)f(e)-5 b(ach)36 b Fy(\025)f Fl(one)h(c)-5 b(an)36 b(p)-5 b(ass)36 b(to)g(the)-47 4711 y(limit)d Fy(")25 b Fu(!)h Fz(0)33 b Fl(in)f(al)5 b(l)33 b(other)h(terms)f(in)40 b Fz(\(2.10\))35 b Fl(and)508 4956 y Fu(h)p Fy(R)612 4970 y Fw(V)674 4956 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\))p Fy(f)5 b(;)31 b(g)s Fu(i)21 b(\000)f(h)p Fy(R)1387 4970 y Fq(0)1427 4956 y Fz(\()p Fy(\025)g Fz(+)g Fy(i)p Fz(0\))p Fy(f)5 b(;)31 b(g)s Fu(i)c Fz(=)2077 4842 y Fx(1)2046 4870 y Fn(X)2053 5067 y Fw(`)p Fq(=1)2178 4956 y Fz(\()p Fu(\000)p Fz(1\))2364 4918 y Fw(`)2398 4956 y Fu(h)p Fy(R)2502 4970 y Fq(0)2541 4956 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(i)p Fz(0\)\()p Fy(V)i(R)3031 4970 y Fq(0)3070 4956 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))3416 4918 y Fw(`)3450 4956 y Fy(f)5 b(;)15 b(g)s Fu(i)-3668 b Fz(\(2.17\))-47 5223 y Fl(holds)34 b(for)f(every)g Fy(\025)f Fl(and)i(the)f(series)g(c)-5 b(onver)g(ges)33 b(absolutely)h(in)e(the)h(norm)h(of)f Fy(C)2746 5190 y Fq(0)2739 5250 y Fw(b)2785 5223 y Fz(\()p Fv(R)s Fz(\))26 b Fu(\\)20 b Fy(L)3084 5190 y Fq(1)3124 5223 y Fz(\()p Fy(d\025)p Fz(\))p Fl(.)1903 5472 y Fz(10)p eop %%Page: 11 11 11 10 bop -47 274 a Fl(Pr)-5 b(o)g(of.)47 b Fz(Fix)23 b Fy(f)5 b(;)15 b(g)29 b Fu(2)24 b Fy(C)713 241 y Fx(1)706 298 y Fq(0)788 274 y Fz(\()p Fv(R)883 241 y Fq(3)928 274 y Fz(\).)39 b(By)24 b(our)g(assumptions)e(on)i Fy(V)44 b Fz(and)23 b(the)h(explicit)f(represen)m(tation)h(\(2.5\))r(,)h Fy(V)20 b(R)3648 288 y Fq(0)3688 274 y Fz(\()p Fy(z)t Fz(\))p Fy(f)35 b Fu(2)-47 387 y Fy(C)25 354 y Fx(1)18 411 y Fq(0)99 387 y Fz(,)22 b(and)e(th)m(us)g(also)g Fy(R)744 401 y Fq(0)784 387 y Fz(\()p Fy(z)t Fz(\)\()p Fy(V)h(R)1078 401 y Fq(0)1118 387 y Fz(\()p Fy(z)t Fz(\)\))1269 354 y Fw(`)1303 387 y Fy(f)30 b Fz(for)20 b(ev)m(ery)h Fy(z)29 b Fu(2)c Fv(C)44 b Fz(with)19 b Fu(=)p Fy(z)29 b Fu(\025)c Fz(0.)38 b(Moreo)m(v)m(er,)24 b Fy(z)30 b Fu(7!)25 b(h)p Fy(R)3226 401 y Fq(0)3266 387 y Fz(\()p Fy(z)t Fz(\)\()p Fy(V)c(R)3560 401 y Fq(0)3600 387 y Fz(\()p Fy(z)t Fz(\)\))3751 354 y Fw(`)3785 387 y Fy(f)5 b(;)15 b(g)s Fu(i)-47 500 y Fz(is)34 b(a)h(con)m(tin)m(uous)f(function) g(in)f Fu(=)p Fy(z)k Fu(\025)32 b Fz(0)j(for)g(ev)m(ery)h Fy(`)c Fu(\025)h Fz(0.)54 b(As)35 b(in)e(the)i(previous)f(pro)s(of)g (one)h(obtains)f(the)h(Kato)-47 613 y(smo)s(othing)29 b(b)s(ound)521 847 y(sup)528 924 y Fw(")p Fx(\025)p Fq(0)673 723 y Fn(Z)764 742 y(\014)764 797 y(\014)764 851 y(\014)794 847 y Fu(h)p Fy(R)898 861 y Fq(0)938 847 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i")p Fz(\)\()p Fy(V)i(R)1424 861 y Fq(0)1463 847 y Fz(\()p Fy(\025)f Fz(+)f Fy(i")p Fz(\)\))1806 809 y Fw(`)1840 847 y Fy(f)5 b(;)15 b(g)s Fu(i)2011 742 y Fn(\014)2011 797 y(\014)2011 851 y(\014)2057 847 y Fy(d\025)26 b Fu(\024)f Fy(C)7 b Fz(\()p Fu(k)p Fy(V)20 b Fu(k)2549 861 y Fw(R)2608 847 y Fy(=)p Fz(4)p Fy(\031)s Fz(\))2788 809 y Fw(`)p Fx(\000)p Fq(1)2912 847 y Fu(k)p Fy(f)10 b Fu(k)3057 866 y Fw(L)3105 847 y Fr(2)3105 883 y Fs(x)3148 847 y Fu(k)p Fy(g)s Fu(k)3284 866 y Fw(L)3332 847 y Fr(2)3332 883 y Fs(x)-47 847 y Fz(\(2.18\))-47 1105 y(for)30 b(eac)m(h)h Fy(`)26 b Fu(\025)f Fz(1)30 b(\(note)i(that)f(the)f(case)i Fy(")25 b Fz(=)g(0)31 b(is)e(included)f(here\).)41 b(Moreo)m(v)m(er,)33 b(see)e(\(2.4\))h(and)e(\(2.6\))q(,)396 1325 y(sup)376 1403 y Fx(=)p Fw(z)s Fx(\025)p Fq(0)553 1220 y Fn(\014)553 1275 y(\014)553 1329 y(\014)583 1325 y Fu(h)p Fy(R)687 1339 y Fq(0)727 1325 y Fz(\()p Fy(z)t Fz(\)\()p Fy(V)22 b(R)1022 1339 y Fq(0)1061 1325 y Fz(\()p Fy(z)t Fz(\)\))1212 1287 y Fw(`)1246 1325 y Fy(f)5 b(;)15 b(g)s Fu(i)1417 1220 y Fn(\014)1417 1275 y(\014)1417 1329 y(\014)1531 1325 y Fu(\024)103 b Fz(sup)1685 1403 y Fx(=)p Fw(z)s Fx(\025)p Fq(0)1877 1325 y Fu(k)15 b(j)p Fy(V)21 b Fu(j)2071 1260 y Fr(1)p 2071 1272 31 3 v 2071 1313 a(2)2116 1325 y Fy(R)2185 1339 y Fq(0)2224 1325 y Fz(\()p Fy(z)t Fz(\))p Fy(g)s Fu(k)2431 1339 y Fq(2)2488 1325 y Fu(k)p Fy(K)7 b Fz(\()p Fy(z)t Fz(\))p Fu(k)2778 1287 y Fw(`)p Fx(\000)p Fq(1)2918 1325 y Fu(k)15 b(j)p Fy(V)21 b Fu(j)3112 1260 y Fr(1)p 3112 1272 V 3112 1313 a(2)3157 1325 y Fy(R)3226 1339 y Fq(0)3265 1325 y Fz(\()p Fy(z)t Fz(\))p Fy(f)10 b Fu(k)3481 1339 y Fq(2)1531 1541 y Fu(\024)83 b Fy(C)7 b Fz(\()p Fy(f)e(;)15 b(g)s(;)g(V)21 b Fz(\))15 b(\()p Fu(k)p Fy(V)21 b Fu(k)2291 1555 y Fw(R)2350 1541 y Fy(=)p Fz(4)p Fy(\031)s Fz(\))2530 1503 y Fw(`)p Fx(\000)p Fq(1)2654 1541 y Fy(:)-47 1738 y Fz(This)28 b(implies)g(that)1039 1999 y Fy(S)1095 2014 y Fw(f)s(;g)1192 1999 y Fz(\()p Fy(\025)p Fz(\))e(:=)1493 1886 y Fx(1)1462 1913 y Fn(X)1469 2111 y Fw(`)p Fq(=0)1594 1999 y Fu(h)p Fy(R)1698 2013 y Fq(0)1738 1999 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\)\()p Fy(V)i(R)2227 2013 y Fq(0)2266 1999 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))2612 1962 y Fw(`)2646 1999 y Fy(f)5 b(;)15 b(g)s Fu(i)p Fy(;)-2889 b Fz(\(2.19\))-47 2275 y(con)m(v)m(erges)41 b(uniformly)35 b(and)k(th)m(us)f(de\014nes)g (a)h(con)m(tin)m(uous)g(function.)65 b(F)-8 b(urthermore,)41 b(one)e(concludes)f(that)i(the)-47 2388 y(series)30 b(in)g(\(2.10\))j (con)m(v)m(erges)g(uniformly)28 b(in)i(the)h(closed)g(upp)s(er)e (half-plane)g(\(i.e.,)j(for)f(all)f Fy(\025)c Fu(2)g Fv(R)40 b Fz(and)30 b Fy(")d Fu(\025)f Fz(0\))31 b(and)-47 2501 y(therefore)25 b(de\014nes)f(the)h(limit)e Fu(h)p Fy(R)1089 2515 y Fw(V)1150 2501 y Fz(\()p Fy(\025)9 b Fz(+)g Fy(i)p Fz(0\))p Fy(f)c(;)15 b(g)s Fu(i)27 b Fz(p)s(oin)m(t)m (wise)d(in)g Fy(\025)h Fu(2)g Fv(R)r Fz(.)45 b(Also)25 b(note)g(that,)i(b)m(y)f(\(2.18\))s(,)g(the)f(series)f(for)-47 2614 y Fy(S)9 2629 y Fw(f)s(;g)106 2614 y Fz(\()p Fy(\025)p Fz(\))18 b Fu(\000)g(h)p Fy(R)440 2628 y Fq(0)480 2614 y Fz(\()p Fy(\025)h Fz(+)f Fy(i)p Fz(0\))p Fy(f)5 b(;)15 b(g)s Fu(i)31 b Fz(con)m(v)m(erges)g(absolutely)d(in)g Fy(L)1993 2581 y Fq(1)2033 2614 y Fz(\()p Fy(d\025)p Fz(\),)j(and)d(similarly)e(for)j(ev)m(ery)i Fy(z)e Fu(2)c Fv(C)53 b Fz(with)28 b Fu(=)p Fy(z)h Fu(\025)c Fz(0.)-47 2727 y(In)k(view)h(of)38 b(\(2.10\))r(,)30 b(\(2.18\))s(,)g(and)g(with) f(an)h(arbitrary)g Fy(N)35 b Fu(\025)25 b Fz(1,)437 2842 y Fn(Z)543 2966 y Fz(lim)15 b(inf)600 3024 y Fw(")p Fx(!)p Fq(0)810 2966 y Fu(jh)p Fy(R)939 2980 y Fw(V)1001 2966 y Fz(\()p Fy(\025)21 b Fz(+)e Fy(i")p 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Fz(\()p Fy(\025)f Fz(+)g Fy(i)p Fz(0\)\))3345 3443 y Fw(`)3380 3481 y Fy(f)5 b(;)15 b(g)s Fu(i)3551 3376 y Fn(\014)3551 3431 y(\014)3551 3485 y(\014)3597 3481 y Fy(d\025)983 3786 y Fz(+)p Fy(C)1223 3672 y Fx(1)1193 3699 y Fn(X)1140 3897 y Fw(`)p Fq(=)p Fw(N)7 b Fq(+1)1377 3786 y Fz(\()p Fu(k)p Fy(V)21 b Fu(k)1576 3800 y Fw(R)1634 3786 y Fy(=)p Fz(4)p Fy(\031)s Fz(\))1814 3748 y Fw(`)p Fx(\000)p Fq(1)1939 3786 y Fu(k)p Fy(f)10 b Fu(k)2084 3800 y Fq(2)2123 3786 y Fu(k)p Fy(g)s Fu(k)2259 3800 y Fq(2)283 4029 y Fu(\024)83 b Fy(C)7 b Fz(\(1)21 b Fu(\000)f(k)p Fy(V)g Fu(k)864 4043 y Fw(R)922 4029 y Fy(=)p Fz(4)p Fy(\031)s Fz(\))1102 3991 y Fx(\000)p Fq(1)1213 4029 y Fz(\()p Fu(k)p Fy(V)h Fu(k)1412 4043 y Fw(R)1470 4029 y Fy(=)p Fz(4)p Fy(\031)s Fz(\))1650 3991 y Fw(N)1719 4029 y Fu(k)p Fy(f)10 b Fu(k)1864 4043 y Fq(2)1904 4029 y Fu(k)p Fy(g)s Fu(k)2040 4043 y Fq(2)2081 4029 y Fy(;)-47 4226 y Fz(and)30 b(w)m(e)g(are)h(done.)p 3879 4226 4 62 v 3883 4168 55 4 v 3883 4226 V 3937 4226 4 62 v -47 4412 a(Next)g(w)m(e)g(turn)e(to)i(a)g(simple)d(lemma)i(that)h(is)f (basically)e(an)j(instance)f(of)g(stationary)h(phase.)-47 4594 y Fm(Lemma)h(2.4.)47 b Fl(L)-5 b(et)33 b Fy( )j Fl(b)-5 b(e)33 b(a)h(smo)-5 b(oth,)35 b(even)e(bump)g(function)h(with)g Fy( )s Fz(\()p Fy(\025)p Fz(\))27 b(=)f(1)33 b Fl(for)h Fu(\000)p Fz(1)26 b Fu(\024)g Fy(\025)g Fu(\024)g Fz(1)33 b Fl(and)h Fz(supp)o(\()p Fy( )s Fz(\))27 b Fu(\032)-47 4707 y Fz([)p Fu(\000)p Fz(2)p Fy(;)15 b Fz(2])p Fl(.)43 b(Then)33 b(for)g(al)5 b(l)33 b Fy(t)25 b Fu(\025)g Fz(1)33 b Fl(and)h(any)f(r)-5 b(e)g(al)34 b Fy(a)p Fl(,)1066 4969 y Fz(sup)1065 5047 y Fw(L)p Fx(\025)p Fq(1)1203 4864 y Fn(\014)1203 4919 y(\014)1203 4973 y(\014)1234 4845 y(Z)1325 4871 y Fx(1)1284 5051 y Fq(0)1414 4969 y Fy(e)1456 4931 y Fw(it\025)1567 4969 y Fz(sin)n(\()p Fy(a)1761 4887 y Fu(p)p 1837 4887 54 4 v 82 x Fy(\025)q Fz(\))15 b Fy( )s Fz(\()2048 4830 y Fu(p)p 2125 4830 V 2125 4907 a Fy(\025)p 2048 4948 129 4 v 2082 5031 a(L)2188 4969 y Fz(\))g Fy(d\025)2338 4864 y Fn(\014)2338 4919 y(\014)2338 4973 y(\014)2394 4969 y Fu(\024)25 b Fy(C)d(t)2610 4931 y Fx(\000)2675 4904 y Fr(3)p 2674 4916 31 3 v 2674 4957 a(2)2734 4969 y Fu(j)p Fy(a)p Fu(j)-2879 b Fz(\(2.20\))-47 5223 y Fl(wher)-5 b(e)33 b Fy(C)39 b Fl(only)34 b(dep)-5 b(ends)34 b(on)66 b Fy( )s Fl(.)1903 5472 y Fz(11)p eop %%Page: 12 12 12 11 bop -47 274 a Fl(Pr)-5 b(o)g(of.)47 b Fz(Denote)33 b(the)g(in)m(tegral)f(in)f(\(2.20\))j(b)m(y)f Fy(I)1584 288 y Fw(L)1636 274 y Fz(\()p Fy(a;)15 b(t)p Fz(\).)47 b(Clearly)-8 b(,)32 b Fy(I)2275 288 y Fw(L)2327 274 y Fz(\()p Fy(a;)15 b(t)p Fz(\))33 b(is)f(a)g(smo)s(oth)g(function)f(of)i Fy(a;)15 b(t)32 b Fz(for)g(an)m(y)-47 387 y Fy(L)25 b(>)g Fz(0)31 b(and)e Fy(I)428 401 y Fw(L)480 387 y Fz(\(0)p Fy(;)15 b(t)p Fz(\))27 b(=)e(0.)41 b(The)30 b(c)m(hange)i(of)e(v)-5 b(ariables)29 b Fy(\025)c Fu(!)h Fy(\025)2118 354 y Fq(2)2187 387 y Fz(leads)k(to)h(the)g(expression)1144 630 y Fy(I)1184 644 y Fw(L)1236 630 y Fz(\()p Fy(a;)15 b(t)p Fz(\))26 b(=)f(2)1609 507 y Fn(Z)1701 533 y Fx(1)1661 713 y Fq(0)1791 630 y Fy(\025)15 b(e)1901 593 y Fw(it\025)1991 569 y Fr(2)2046 630 y Fz(sin)o(\()p Fy(a\025)p Fz(\))g Fy( )s Fz(\()p Fy(\025=L)p Fz(\))g Fy(d\025)-47 871 y Fz(In)m(tegrating)31 b(b)m(y)f(parts)g(w)m(e)h(obtain)647 1115 y Fy(I)687 1129 y Fw(L)739 1115 y Fz(\()p Fy(a;)15 b(t)p Fz(\))26 b(=)1063 1053 y Fy(i)p 1062 1094 33 4 v 1062 1177 a(t)1120 991 y Fn(Z)1211 1017 y Fx(1)1170 1197 y Fq(0)1301 1115 y Fy(e)1343 1077 y Fw(it\025)1433 1054 y Fr(2)1472 986 y Fn(\022)1539 1115 y Fy(a)31 b Fz(cos\()p Fy(a\025)p Fz(\))15 b Fy( )s Fz(\()p Fy(\025=L)p Fz(\))23 b(+)2349 1053 y(1)p 2341 1094 62 4 v 2341 1177 a Fy(L)2428 1115 y Fz(sin)o(\()p Fy(a\025)p Fz(\))15 b Fy( )2788 1077 y Fx(0)2812 1115 y Fz(\()p Fy(\025=L)p Fz(\))3042 986 y Fn(\023)3125 1115 y Fy(d\025:)-47 1366 y Fz(Since)29 b Fy( )34 b Fz(is)29 b(assumed)h(to)h(b)s(e)f(ev)m(en,)h Fy( )1268 1333 y Fx(0)1322 1366 y Fz(is)f(o)s(dd.)39 b(Hence,)10 1610 y Fy(I)50 1624 y Fw(L)102 1610 y Fz(\()p Fy(a;)15 b(t)p Fz(\))26 b(=)449 1548 y Fy(i)p 425 1589 79 4 v 425 1672 a Fz(2)p Fy(t)529 1486 y Fn(Z)620 1512 y Fx(1)579 1692 y(\0001)724 1610 y Fy(e)766 1572 y Fw(it\025)856 1549 y Fr(2)896 1482 y Fn(\022)963 1610 y Fy(a)k Fz(cos)q(\()p Fy(a\025)p Fz(\))15 b Fy( )s Fz(\()p Fy(\025=L)p Fz(\))22 b(+)1772 1548 y(1)p 1764 1589 62 4 v 1764 1672 a Fy(L)1851 1610 y Fz(sin)o(\()p Fy(a\025)p Fz(\))15 b Fy( )2211 1572 y Fx(0)2236 1610 y Fz(\()p Fy(\025=L)p Fz(\))2466 1482 y Fn(\023)2549 1610 y Fy(d\025)319 1858 y Fz(=)440 1797 y Fy(a)p 425 1837 79 4 v 425 1921 a Fz(4)p Fy(t)529 1858 y(i)575 1735 y Fn(Z)666 1761 y Fx(1)626 1941 y(\0001)770 1858 y Fy(e)812 1821 y Fw(it\025)902 1797 y Fr(2)942 1785 y Fn(\000)999 1858 y Fy(e)1041 1821 y Fw(ia\025)1168 1858 y Fz(+)20 b Fy(e)1301 1821 y Fx(\000)p Fw(ia\025)1463 1785 y Fn(\001)1520 1858 y Fy( )s Fz(\()p Fy(\025=L)p Fz(\))15 b Fy(d\025)22 b Fz(+)2040 1735 y Fn(Z)2131 1761 y Fw(a)2091 1941 y Fq(0)2221 1797 y Fy(i)p 2198 1837 V 2198 1921 a Fz(4)p Fy(t)2316 1735 y Fn(Z)2407 1761 y Fx(1)2367 1941 y(\0001)2512 1858 y Fy(e)2554 1821 y Fw(it\025)2644 1797 y Fr(2)2683 1785 y Fn(\000)2740 1858 y Fy(e)2782 1821 y Fw(ib\025)2902 1858 y Fz(+)e Fy(e)3035 1821 y Fx(\000)p Fw(ib\025)3190 1785 y Fn(\001)3261 1797 y Fy(\025)p 3257 1837 62 4 v 3257 1921 a(L)3344 1858 y( )3406 1821 y Fx(0)3430 1858 y Fz(\()p Fy(\025=L)p Fz(\))15 b Fy(d\025)g(db:)-47 2105 y Fz(Th)m(us)29 b(it)h(su\016ces)g (to)h(sho)m(w)f(that)1255 2326 y Fy(J)1305 2340 y Fw(L)1358 2326 y Fz(\()p Fy(a;)15 b(t)p Fz(\))26 b(=)1671 2202 y Fn(Z)1762 2229 y Fx(1)1722 2409 y(\0001)1866 2326 y Fy(e)1908 2289 y Fw(i)p Fq(\()p Fw(t\025)2025 2265 y Fr(2)2061 2289 y Fq(+)p Fw(a\025)p Fq(\))2242 2326 y Fy(\036)p Fz(\()p Fy(\025=L)p Fz(\))15 b Fy(d\025)-47 2605 y Fz(ob)s(eys)31 b(the)g(estimate)h Fu(j)p Fy(J)804 2619 y Fw(L)857 2605 y Fz(\()p Fy(a)15 b(t)p Fz(\))p Fu(j)28 b(\024)f Fy(C)7 b(t)1279 2569 y Fx(\000)1344 2542 y Fr(1)p 1343 2554 31 3 v 1343 2595 a(2)1419 2605 y Fz(for)31 b(an)m(y)h(smo)s(oth)f(bump)e(function)i Fy(\036)g Fz(satisfying)f(the)i(same)f(prop)s(erties)-47 2731 y(as)f Fy( )s Fz(.)41 b(The)29 b(c)m(hange)i(of)e(v)-5 b(ariables)29 b Fy(\025)c Fu(!)g Fy(\025=L)30 b Fz(further)e(reduces)i (the)g(problem)e(to)i(the)g(estimate)g Fu(j)p Fy(J)9 b Fz(\()p Fy(a;)15 b(t)p Fz(\))p Fu(j)27 b(\024)e Fy(C)7 b(t)3836 2696 y Fx(\000)3901 2668 y Fr(1)p 3900 2680 V 3900 2722 a(2)-47 2844 y Fz(with)1331 2979 y Fy(J)i Fz(\()p Fy(a;)15 b(t)p Fz(\))26 b(=)1703 2855 y Fn(Z)1794 2881 y Fx(1)1754 3061 y(\0001)1898 2979 y Fy(e)1940 2941 y Fw(i)p Fq(\()p Fw(t\025)2057 2918 y Fr(2)2093 2941 y Fq(+)p Fw(a\025)p Fq(\))2274 2979 y Fy(\036)p Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)-47 3196 y Fz(for)26 b(all)g Fy(t)f Fu(6)p Fz(=)g(0)i(and)f(all)f(real)i Fy(a)p Fz(.)39 b(Observ)m(e)27 b(that)g Fy(J)9 b Fz(\()p Fy(a;)15 b(t)p Fz(\))28 b(is)d(a)i(smo)s(oth)g(solution)e(of)i(the)g(1-dimensional)d (Sc)m(hr\177)-45 b(odinger)-47 3309 y(equation)1405 3472 y Fy(i)1478 3411 y(@)p 1461 3451 87 4 v 1461 3535 a(@)5 b(t)1557 3472 y(J)k Fz(\()p Fy(a;)15 b(t)p Fz(\))22 b Fu(\000)1953 3411 y Fy(@)2006 3378 y Fq(2)p 1929 3451 141 4 v 1929 3535 a Fy(@)5 b(a)2030 3508 y Fq(2)2080 3472 y Fy(J)k Fz(\()p Fy(a;)15 b(t)p Fz(\))27 b(=)e(0)p Fy(;)1405 3694 y(J)9 b Fz(\()p Fy(a;)15 b Fz(0\))27 b(=)1790 3570 y Fn(Z)1881 3596 y Fx(1)1840 3776 y(\0001)1985 3694 y Fy(e)2027 3656 y Fx(\000)p Fw(ia\025)2189 3694 y Fy(\036)p Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025:)-47 3906 y Fz(By)30 b(the)h(explicit)e(represen)m(tation)h(of)h(the)f(k)m(ernel)g(of)h(the) f(fundamen)m(tal)g(solution)1108 4144 y Fy(J)9 b Fz(\()p Fy(a;)15 b(t)p Fz(\))27 b(=)e(\()p Fu(\000)p Fz(4)p Fy(\031)s(it)p Fz(\))1786 4107 y Fx(\000)1851 4080 y Fr(1)p 1852 4092 31 3 v 1852 4133 a(2)1911 4021 y Fn(Z)2002 4047 y Fx(1)1962 4227 y(\0001)2107 4144 y Fy(e)2149 4107 y Fx(\000)p Fw(i)2238 4073 y Ft(j)p Fs(a)p Ft(\000)p Fs(b)p Ft(j)2385 4052 y Fr(2)p 2238 4092 182 3 v 2301 4133 a(4)p Fs(t)2433 4144 y Fy(J)9 b Fz(\()p Fy(b;)15 b Fz(0\))g Fy(db)-47 4397 y Fz(whic)m(h)29 b(implies)e(that)k Fy(J)9 b Fz(\()p Fy(a;)15 b(t)p Fz(\))32 b(ob)s(eys)e(the)g(standard)g(one-dimensional)e (deca)m(y)k(estimate)1409 4611 y Fu(j)p Fy(J)9 b Fz(\()p Fy(a;)15 b(t)p Fz(\))p Fu(j)27 b(\024)d Fy(C)7 b(t)1936 4573 y Fx(\000)2001 4546 y Fr(1)p 2001 4558 31 3 v 2001 4599 a(2)2045 4611 y Fu(k)p Fy(J)i Fz(\()p Fu(\001)p Fy(;)15 b Fz(0\))p Fu(k)2374 4630 y Fw(L)2422 4612 y Fr(1)2464 4611 y Fy(:)-47 4810 y Fz(Since)20 b(the)i(function)f Fy(J)9 b Fz(\()p Fy(a;)15 b Fz(0\))23 b(is)e(the)h(F)-8 b(ourier)21 b(transform)g(of)h(the)g(smo)s(oth)g(bump)e(function)g Fy(\036)p Fz(,)k(the)e(desired)e(estimate)-47 4923 y(on)30 b Fy(J)9 b Fz(\()p Fy(a;)15 b(t)p Fz(\))31 b(follo)m(ws.)p 3879 4923 4 62 v 3883 4865 55 4 v 3883 4923 V 3937 4923 4 62 v -47 5110 a(The)38 b(follo)m(wing)e(lemma)i(explains)f(to)i(some) g(exten)m(t)h(wh)m(y)d(condition)g(\(2.2\))k(is)c(needed.)64 b(Iterated)40 b(in)m(tegrals)e(as)-47 5223 y(in)29 b(\(2.21\))j(will)c (app)s(ear)i(in)f(a)i(series)e(expansion)h(of)g(the)h(sp)s(ectral)e (resolution)g(of)i Fy(H)h Fz(=)25 b Fu(\0004)20 b Fz(+)g Fy(V)g Fz(.)1903 5472 y(12)p eop %%Page: 13 13 13 12 bop -47 274 a Fm(Lemma)32 b(2.5.)47 b Fl(F)-7 b(or)33 b(any)g(p)-5 b(ositive)34 b(inte)-5 b(ger)33 b Fy(k)j Fl(and)d Fy(V)53 b Fl(as)33 b(in)g(De\014nition)f(2.1)598 558 y Fz(sup)477 642 y Fw(x)517 651 y Fr(0)552 642 y Fw(;x)612 654 y Fs(k)q Fr(+1)726 642 y Fx(2)p Fp(R)821 623 y Fr(3)871 434 y Fn(Z)921 640 y Fp(R)969 622 y Fr(3)p Fs(k)1140 415 y Fn(Q)1226 441 y Fw(k)1226 510 y(j)t Fq(=1)1368 483 y Fu(j)p Fy(V)20 b Fz(\()p Fy(x)1553 497 y Fw(j)1590 483 y Fz(\))p Fu(j)p 1067 537 657 4 v 1067 568 a Fn(Q)1153 594 y Fw(k)1153 663 y(j)t Fq(=0)1295 636 y Fu(j)p Fy(x)1372 650 y Fw(j)1429 636 y Fu(\000)g Fy(x)1572 650 y Fw(j)t Fq(+1)1698 636 y Fu(j)1795 444 y Fw(k)1749 472 y Fn(X)1755 669 y Fw(`)p Fq(=0)1895 558 y Fu(j)p Fy(x)1972 573 y Fw(`)2025 558 y Fu(\000)g Fy(x)2168 573 y Fw(`)p Fq(+1)2291 558 y Fu(j)26 b Fy(dx)2441 572 y Fq(1)2496 558 y Fy(:)15 b(:)g(:)31 b(dx)2731 573 y Fw(k)2799 558 y Fu(\024)25 b Fz(\()p Fy(k)f Fz(+)c(1\))p Fu(k)p Fy(V)h Fu(k)3336 520 y Fw(k)3336 580 y Fx(K)3395 558 y Fy(:)-3467 b Fz(\(2.21\))-47 839 y Fl(Pr)-5 b(o)g(of.)47 b Fz(De\014ne)30 b(the)h(op)s(erator)f Fu(A)g Fz(b)m(y)h(the)f(form)m(ula)1385 1084 y Fu(A)p Fy(f)10 b Fz(\()p Fy(x)p Fz(\))25 b(=)1756 960 y Fn(Z)1806 1166 y Fp(R)1854 1147 y Fr(3)1927 1022 y Fu(j)p Fy(V)c Fz(\()p Fy(y)s Fz(\))p Fu(j)p 1918 1063 262 4 v 1918 1146 a(j)p Fy(x)f Fu(\000)g Fy(y)s Fu(j)2204 1084 y Fy(f)10 b Fz(\()p Fy(y)s Fz(\))15 b Fy(dy)s(:)-47 1335 y Fz(Observ)m(e)28 b(that)g(the)h(assumption)d(\(2.2\))k(on)e(the)g(p)s(oten)m(tial)g Fy(V)48 b Fz(implies)25 b(that)k Fu(A)c Fz(:)41 b Fy(L)2830 1302 y Fx(1)2929 1335 y Fu(!)26 b Fy(L)3108 1302 y Fx(1)3210 1335 y Fz(and)i Fu(kAk)3548 1349 y Fw(L)3596 1330 y Ft(1)3661 1349 y Fx(!)p Fw(L)3780 1330 y Ft(1)3874 1335 y Fu(\024)-47 1448 y Fy(c)-8 1462 y Fq(0)65 1448 y Fz(where)k(w)m(e)i(ha)m(v)m(e)g (set)f Fy(c)862 1462 y Fq(0)932 1448 y Fz(:=)d Fu(k)p Fy(V)20 b Fu(k)1221 1462 y Fx(K)1313 1448 y Fz(for)33 b(con)m(v)m(enience.)50 b(Denote)34 b(b)m(y)f Fy(<;)15 b(>)33 b Fz(the)g(standard)g Fy(L)3258 1415 y Fq(2)3330 1448 y Fz(pairing.)47 b(In)32 b(this)-47 1560 y(notation)e(the)h (estimate)g(\(2.21\))i(is)c(equiv)-5 b(alen)m(t)30 b(to)h(pro)m(ving)f (that)h(the)f(op)s(erators)h Fu(B)2879 1575 y Fw(k)2951 1560 y Fz(de\014ned)e(as)1336 1850 y Fu(B)1396 1865 y Fw(k)1439 1850 y Fy(f)34 b Fz(=)1671 1736 y Fw(k)1625 1763 y Fn(X)1614 1958 y Fw(m)p Fq(=0)1792 1850 y Fy(<)25 b(f)5 b(;)15 b Fu(A)2051 1812 y Fw(k)r Fx(\000)p Fw(m)2210 1850 y Fz(1)26 b Fy(>)f Fu(A)2450 1812 y Fw(m)2516 1850 y Fz(1)-47 2140 y(are)31 b(b)s(ounded)d(as)i(op)s(erators)h(from)f Fy(L)1267 2107 y Fq(1)1331 2140 y Fu(!)25 b Fy(L)1509 2107 y Fx(1)1614 2140 y Fz(with)k(the)i(b)s(ound)1464 2342 y Fu(kB)1569 2357 y Fw(k)1612 2342 y Fu(k)1657 2362 y Fw(L)1705 2343 y Fr(1)1740 2362 y Fx(!)p Fw(L)1859 2343 y Ft(1)1953 2342 y Fu(\024)25 b Fz(\()p Fy(k)f Fz(+)c(1\))p Fy(c)2365 2305 y Fw(k)2365 2365 y Fq(0)2408 2342 y Fy(:)-47 2544 y Fz(F)-8 b(or)31 b(arbitrary)e Fy(f)35 b Fu(2)24 b Fy(L)730 2511 y Fq(1)800 2544 y Fz(one)31 b(has)235 2828 y Fu(kB)340 2843 y Fw(k)382 2828 y Fy(f)10 b Fu(k)482 2842 y Fw(L)530 2823 y Ft(1)625 2828 y Fu(\024)778 2715 y Fw(k)731 2742 y Fn(X)721 2936 y Fw(m)p Fq(=0)888 2828 y Fu(j)26 b Fy(<)f(f)5 b(;)15 b Fu(A)1198 2791 y Fw(k)r Fx(\000)p Fw(m)1357 2828 y Fz(1)26 b Fy(>)f Fu(j)30 b(kA)1697 2791 y Fw(m)1764 2828 y Fz(1)p Fu(k)1854 2842 y Fw(L)1902 2823 y Ft(1)2027 2828 y Fu(\024)2180 2715 y Fw(k)2134 2742 y Fn(X)2123 2936 y Fw(m)p Fq(=0)2291 2828 y Fu(kA)2409 2791 y Fw(k)r Fx(\000)p Fw(m)2569 2828 y Fu(k)2614 2842 y Fw(L)2662 2823 y Ft(1)2728 2842 y Fx(!)p Fw(L)2847 2823 y Ft(1)2915 2828 y Fu(kA)3033 2791 y Fw(m)3100 2828 y Fu(k)3145 2842 y Fw(L)3193 2823 y Ft(1)3259 2842 y Fx(!)p Fw(L)3378 2823 y Ft(1)3446 2828 y Fu(k)p Fy(f)10 b Fu(k)3591 2848 y Fw(L)3639 2829 y Fr(1)2027 3147 y Fu(\024)2180 3034 y Fw(k)2134 3061 y Fn(X)2123 3255 y Fw(m)p Fq(=0)2291 3147 y Fy(c)2330 3110 y Fw(k)2330 3170 y Fq(0)2373 3147 y Fu(k)p Fy(f)g Fu(k)2518 3167 y Fw(L)2566 3148 y Fr(1)2630 3147 y Fu(\024)25 b Fz(\()p Fy(k)f Fz(+)c(1\))p Fy(c)3042 3110 y Fw(k)3042 3170 y Fq(0)3086 3147 y Fu(k)p Fy(f)10 b Fu(k)3231 3167 y Fw(L)3279 3148 y Fr(1)3318 3147 y Fy(;)-47 3425 y Fz(as)30 b(claimed.)p 3879 3425 4 62 v 3883 3367 55 4 v 3883 3425 V 3937 3425 4 62 v -47 3612 a(W)-8 b(e)31 b(are)g(no)m(w)f(in)g(a)g(p)s(osition)f(to)i (pro)m(v)m(e)g(the)g(main)e(result)g(of)i(this)e(section.)-47 3798 y Fm(Theorem)34 b(2.6.)46 b Fl(With)40 b Fy(H)45 b Fz(=)38 b Fu(\0004)24 b Fz(+)i Fy(V)59 b Fl(and)41 b Fy(V)60 b Fl(satisfying)40 b(the)g(c)-5 b(onditions)42 b(in)d(De\014nition)h(2.1)h(one)f(has)g(the)-47 3911 y(b)-5 b(ound)1509 4009 y Fn(\015)1509 4063 y(\015)1509 4118 y(\015)1559 4113 y Fy(e)1601 4076 y Fw(itH)1718 4009 y Fn(\015)1718 4063 y(\015)1718 4118 y(\015)1769 4176 y Fw(L)1817 4157 y Fr(1)1851 4176 y Fx(!)p Fw(L)1970 4157 y Ft(1)2064 4113 y Fu(\024)25 b Fy(C)d(t)2280 4076 y Fx(\000)2345 4048 y Fr(3)p 2345 4060 31 3 v 2345 4102 a(2)-47 4337 y Fl(in)32 b(thr)-5 b(e)g(e)34 b(dimensions.)-47 4523 y(Pr)-5 b(o)g(of.)47 b Fz(Let)32 b Fy( )k Fz(b)s(e)31 b(a)i(smo)s(oth)f(cut-o\013)h(function)e(as)h(in)f(Lemma)i(2.4.)47 b(W)-8 b(e)33 b(will)d(sho)m(w)i(that)g(there)h(is)e(an)h(absolute)-47 4636 y(constan)m(t)f Fy(C)37 b Fz(suc)m(h)30 b(that)1112 4838 y(sup)1111 4916 y Fw(L)p Fx(\025)p Fq(1)1249 4733 y Fn(\014)1249 4788 y(\014)1249 4842 y(\014)1279 4764 y(\012)1322 4838 y Fy(e)1364 4800 y Fw(itH)1481 4838 y Fy( )s Fz(\()1578 4757 y Fu(p)p 1655 4757 83 4 v 1655 4838 a Fy(H)7 b(=L)p Fz(\))p Fy(f)e(;)15 b(g)2016 4764 y Fn(\013)2060 4733 y(\014)2060 4788 y(\014)2060 4842 y(\014)2115 4838 y Fu(\024)25 b Fy(C)7 b(t)2316 4800 y Fx(\000)2381 4773 y Fr(3)p 2381 4785 31 3 v 2381 4826 a(2)2425 4838 y Fu(k)p Fy(f)j Fu(k)2570 4852 y Fq(1)2610 4838 y Fu(k)p Fy(g)s Fu(k)2746 4852 y Fq(1)-47 4838 y Fz(\(2.22\))-47 5110 y(for)23 b(an)m(y)g Fy(f)5 b(;)15 b(g)29 b Fu(2)24 b Fy(C)568 5077 y Fx(1)561 5134 y Fq(0)643 5110 y Fz(\()p Fv(R)738 5077 y Fq(3)783 5110 y Fz(\),)h(whic)m(h)d(pro) m(v)m(es)i(the)f(theorem.)39 b(It)23 b(will)e(b)s(e)h(con)m(v)m(enien)m (t)i(to)g(assume)f(that)h(the)f(p)s(oten)m(tial)g Fy(V)-47 5223 y Fz(b)s(elongs)33 b(to)i Fy(C)470 5190 y Fx(1)463 5247 y Fq(0)544 5223 y Fz(\()p Fv(R)639 5190 y Fq(3)684 5223 y Fz(\),)h(in)d(addition)f(to)j(satisfying)f(\(2.1\))i(and)d (\(2.2\))r(.)52 b(In)33 b(case)i(of)f(a)h(general)f(p)s(oten)m(tial)g Fy(V)54 b Fz(as)34 b(in)1903 5472 y(13)p eop %%Page: 14 14 14 13 bop -47 274 a Fz(De\014nition)27 b(2.1,)j(one)f(appro)m(ximates)g Fy(V)49 b Fz(b)m(y)28 b Fy(V)1548 288 y Fw(j)1610 274 y Fu(2)d Fy(C)1768 241 y Fx(1)1761 298 y Fq(0)1871 274 y Fz(via)j(the)h(usual)e(cut-o\013)i(and)f(mollifying)e(pro)s(cess.)40 b(Clearly)-8 b(,)-47 387 y Fu(k)p Fy(V)42 b Fu(\000)21 b Fy(V)238 401 y Fw(j)275 387 y Fu(k)320 401 y Fw(R)406 387 y Fu(!)28 b Fz(0)33 b(as)g Fy(j)h Fu(!)28 b(1)k Fz(and)g Fu(k)p Fy(V)1307 401 y Fw(j)1344 387 y Fu(k)1389 401 y Fx(K)1476 387 y Fu(\024)c(k)p Fy(V)20 b Fu(k)1738 401 y Fx(K)1826 387 y Fy(<)28 b Fz(4)p Fy(\031)s Fz(.)46 b(Since)32 b(the)g(sp)s(ectral)g(resolution)f Fy(E)3327 401 y Fw(V)3420 387 y Fz(of)h Fy(H)39 b Fz(satis\014es)-47 500 y(\(recall)30 b(that)h(the)f(sp)s(ectrum)g(of)g Fy(H)37 b Fz(is)30 b(purely)e(absolutely)i(con)m(tin)m(uous\))1229 747 y Fy(E)1301 709 y Fx(0)1296 769 y Fw(V)1357 747 y Fz(\()p Fy(\025)p Fz(\))c(:=)1664 685 y Fy(d)p 1637 726 101 4 v 1637 809 a(d\025)1748 747 y(E)1815 761 y Fw(V)1876 747 y Fz(\()p Fy(\025)p Fz(\))g(=)f Fu(=)p Fy(R)2256 761 y Fw(V)2316 747 y Fz(\()p Fy(\025)c Fz(+)f Fy(i)p Fz(0\))p Fy(;)-2699 b Fz(\(2.23\))-47 972 y(one)30 b(concludes)g(from)g (Lemma)g(2.2)i(that)1199 1072 y Fn(Z)1290 1091 y(\014)1290 1145 y(\014)1290 1200 y(\014)1320 1195 y Fu(h)p Fy(E)1427 1158 y Fx(0)1422 1218 y Fw(V)1484 1195 y Fz(\()p 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b(therefore)g(su\016ces)g(to)g(pro)m(v)m(e)h (\(2.22\))h(under)c(the)j(additional)c(assumption)-47 1958 y(that)38 b Fy(V)57 b Fu(2)37 b Fy(C)437 1925 y Fx(1)430 1982 y Fq(0)511 1958 y Fz(\()p Fv(R)606 1925 y Fq(3)651 1958 y Fz(\).)62 b(Fix)37 b(suc)m(h)g(a)h(p)s(oten)m(tial)f Fy(V)20 b Fz(,)40 b(as)d(w)m(ell)g(as)g(an)m(y)h Fy(L)f Fu(\025)g Fz(1,)i(and)e(real)g Fy(f)5 b(;)15 b(g)40 b Fu(2)d Fy(C)3401 1925 y Fx(1)3394 1982 y Fq(0)3475 1958 y Fz(\()p Fv(R)3570 1925 y Fq(3)3616 1958 y Fz(\).)62 b(Then)-47 2071 y(applying)29 b(\(2.23\))s(,)h(Corollary)f(2.3,)j (\(2.5\))r(,)e(Lemma)h(2.4,)g(and)f(Lemma)h(2.5)g(in)e(this)h(order,) 190 2297 y(sup)190 2376 y Fw(L)p Fx(\025)p Fq(1)328 2193 y Fn(\014)328 2247 y(\014)328 2302 y(\014)358 2224 y(\012)401 2297 y Fy(e)443 2260 y Fw(itH)560 2297 y Fy( )s Fz(\()657 2216 y Fu(p)p 734 2216 83 4 v 734 2297 a Fy(H)7 b(=L)p Fz(\))p Fy(f)e(;)15 b(g)1095 2224 y Fn(\013)1138 2193 y(\014)1138 2247 y(\014)1138 2302 y(\014)36 2555 y Fu(\024)83 b Fz(sup)190 2633 y 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280 y FA(with)h(an)f Fj(\017)g FA(loss)-47 482 y Fz(The)26 b(purp)s(ose)g(of)h(this)g(section)g(is)f (to)i(pro)m(v)m(e)g(a)g(disp)s(ersiv)m(e)d(inequalit)m(y)g(for)i Fy(e)2572 449 y Fw(itH)2689 482 y Fy(\037)p Fz(\()p Fy(H)7 b Fz(\))p Fy(P)2957 496 y Fw(a:c)3077 482 y Fz(where)27 b Fy(\037)g Fz(is)f(a)i(cut-o\013)g(to)-47 595 y(large)33 b(energies)g(and)g Fy(P)763 609 y Fw(a:c)888 595 y Fz(is)g(the)g(pro)5 b(jection)33 b(on)m(to)h(the)g(absolutely)e(con)m(tin)m(uous)h(part)g (of)h Fy(L)3206 562 y Fq(2)3245 595 y Fz(\()p Fv(R)3340 562 y Fq(3)3385 595 y Fz(\))g(with)e(resp)s(ect)-47 708 y(to)f Fy(H)h Fz(=)25 b Fu(\0004)20 b Fz(+)g Fy(V)g Fz(.)41 b(W)-8 b(e)31 b(will)d(assume)i(that)h Fy(V)50 b Fz(satis\014es)30 b(the)h(follo)m(wing)e(prop)s(erties:)1153 955 y Fu(j)-15 b(j)g(j)p Fy(V)21 b Fu(j)-15 b(j)g(j)26 b Fz(:=)f Fu(k)p Fy(V)c Fu(k)1628 969 y Fq(2)1688 955 y Fz(+)36 b(sup)1779 1039 y Fw(x)p Fx(2)p Fp(R)1914 1020 y Fr(3)1963 832 y Fn(Z)2014 1038 y Fp(R)2062 1019 y Fr(3)2135 894 y Fu(j)p Fy(V)20 b Fz(\()p Fy(y)s Fz(\))p Fu(j)p 2125 934 262 4 v 2125 1018 a(j)p Fy(x)h Fu(\000)f Fy(y)s Fu(j)2412 955 y Fy(dy)28 b(<)d Fu(1)p Fy(:)-2791 b Fz(\(3.1\))-47 1220 y(Under)30 b(these)i(conditions)e(w)m(e)i(will)c(pro)m(v)m(e)33 b(the)e(follo)m(wing)f(result.)42 b(As)32 b(usual,)e(w)m(e)i(let)f Fy(\037)c Fu(2)g Fy(C)3197 1187 y Fx(1)3302 1220 y Fz(with)j Fy(\037)p Fz(\()p Fy(\025)p Fz(\))e(=)e(0)32 b(if)-47 1333 y Fy(\025)25 b Fu(\024)g Fz(1)31 b(and)f Fy(\037)p Fz(\()p Fy(\025)p Fz(\))25 b(=)g(1)31 b(for)f Fy(\025)c Fu(\025)e Fz(2.)-47 1521 y Fm(Prop)s(osition)36 b(3.1.)46 b Fl(L)-5 b(et)47 b Fu(j)-15 b(j)g(j)p Fy(V)21 b Fu(j)-15 b(j)g(j)50 b Fy(<)g Fu(1)d Fl(as)g(in)53 b Fz(\(3.1\))r Fl(.)83 b(Then)47 b(for)g(every)g Fy(")j(>)h Fz(0)46 b Fl(ther)-5 b(e)48 b(exists)f(some)g(p)-5 b(ositive)-47 1634 y Fy(\025)6 1648 y Fq(0)70 1634 y Fz(=)25 b Fy(\025)219 1648 y Fq(0)259 1634 y Fz(\()p Fu(j)-15 b(j)g(j)p Fy(V)21 b Fu(j)-15 b(j)g(j)p Fy(;)15 b(")p Fz(\))34 b Fl(so)f(that)1219 1853 y Fu(k)p Fy(e)1306 1815 y Fw(itH)1423 1853 y Fy(\037)p Fz(\()p Fy(H)r(=\025)1691 1867 y Fq(0)1731 1853 y Fz(\))p Fy(P)1824 1867 y Fw(a:c:)1936 1853 y Fu(k)1981 1873 y Fw(L)2029 1854 y Fr(1)2029 1889 y Fs(x)2069 1873 y Fx(!)p Fw(L)2188 1854 y Ft(1)2188 1889 y Fs(x)2282 1853 y Fu(\024)25 b Fy(C)7 b(t)2483 1815 y Fx(\000)2548 1788 y Fr(3)p 2547 1800 31 3 v 2547 1841 a(2)2587 1815 y Fq(+)p Fw(")-47 1853 y Fz(\(3.2\))-47 2057 y Fl(for)33 b(al)5 b(l)33 b Fy(t)25 b(>)g Fz(0)p Fl(.)-47 2245 y Fz(Previously)-8 b(,)32 b(con)m(v)m(ergence)j(of)d(the)h(Born)g(series)f(w)m(as)h (guaran)m(teed)g(b)m(y)g(a)g(smallness)e(assumption)g(on)h(the)h(p)s (oten-)-47 2358 y(tial)d Fy(V)21 b Fz(.)43 b(The)31 b(follo)m(wing)e (lemma)i(will)e(allo)m(w)i(us)f(to)i(sum)f(the)g(Born)g(series)g(for)g (large)g(p)s(oten)m(tials)g(in)f Fy(L)3537 2325 y Fq(2)3576 2358 y Fz(\()p Fv(R)3671 2325 y Fq(3)3716 2358 y Fz(\),)j(but)-47 2470 y(only)d(for)g(large)h(energies.)42 b(This)29 b(lemma)h(is)g(an)g (immediate)g(consequence)i(of)e(the)h(Stein-T)-8 b(omas)31 b(theorem)g(in)f(the)-47 2583 y(form)m(ulation)f(due)h(to)h(Stein)e ([St].)-47 2771 y Fm(Lemma)j(3.2.)47 b Fl(L)-5 b(et)39 b Fy(R)764 2785 y Fq(0)803 2771 y Fz(\()p Fy(z)t Fz(\))g(=)d(\()p Fu(\0004)25 b(\000)g Fy(z)t Fz(\))1454 2738 y Fx(\000)p Fq(1)1588 2771 y Fl(for)40 b Fu(=)p Fz(\()p Fy(z)t Fz(\))d Fy(>)g Fz(0)j Fl(b)-5 b(e)39 b(the)g(r)-5 b(esolvent)41 b(of)e(the)h(fr)-5 b(e)g(e)39 b(L)-5 b(aplac)g(e)g(an.)64 b(Then)-47 2884 y(ther)-5 b(e)33 b(is)g(an)g(absolute)g(c)-5 b(onstant)35 b Fy(C)k Fl(so)33 b(that)h(for)f(any)g Fy(\025)26 b(>)f Fz(0)1182 3098 y Fu(k)p Fy(R)1296 3112 y Fq(0)1336 3098 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\))p Fy(f)10 b Fu(k)1746 3118 y Fw(L)1794 3099 y Fr(4)1830 3118 y Fq(\()p Fp(R)1905 3099 y Fr(3)1940 3118 y Fq(\))1997 3098 y Fu(\024)25 b Fy(C)7 b(\025)2218 3060 y Fx(\000)2283 3033 y Fr(1)p 2282 3045 V 2282 3087 a(4)2326 3098 y Fu(k)p Fy(f)j Fu(k)2471 3149 y Fw(L)2529 3099 y Fr(4)p 2530 3107 V 2530 3142 a(3)2574 3149 y Fq(\()p Fp(R)2649 3130 y Fr(3)2684 3149 y Fq(\))-47 3098 y Fz(\(3.3\))-47 3328 y Fl(for)33 b(al)5 b(l)33 b Fy(f)i Fu(2)24 b(S)40 b Fl(.)-47 3516 y(Pr)-5 b(o)g(of.)47 b Fz(It)30 b(is)g(w)m(ell-kno)m(wn)f(that)i (the)f(resolv)m(en)m(t)h Fy(R)1704 3530 y Fq(0)1744 3516 y Fz(\()p Fy(z)t Fz(\))26 b(=)f(\()p Fu(\0004)20 b(\000)g Fy(z)t Fz(\))2361 3483 y Fx(\000)p Fq(1)2486 3516 y Fz(for)30 b Fu(=)p Fz(\()p Fy(z)t Fz(\))c Fy(>)f Fz(0)31 b(has)f(the)h(k)m(ernel) 1344 3772 y Fy(K)1421 3786 y Fq(0)1460 3772 y Fz(\()p Fy(z)t Fz(\)\()p Fy(x;)15 b(y)s Fz(\))28 b(=)1920 3711 y(exp)o(\()p Fy(i)2124 3645 y Fu(p)p 2201 3645 47 4 v 2201 3711 a Fy(z)t Fu(j)p Fy(x)21 b Fu(\000)e Fy(y)s Fu(j)p Fz(\))p 1920 3751 625 4 v 2051 3835 a(4)p Fy(\031)s Fu(j)p Fy(x)i Fu(\000)f Fy(y)s Fu(j)-47 3772 y Fz(\(3.4\))-47 4030 y(where)30 b Fu(=)p Fz(\()317 3964 y Fu(p)p 392 3964 47 4 v 392 4030 a Fy(z)5 b Fz(\))25 b Fy(>)g Fz(0.)41 b(By)31 b(the)g(Stein-T)-8 b(omas)30 b(theorem)h(in)e(Stein's)g(v)m (ersion)h([St])h(one)f(has)989 4177 y Fn(\015)989 4232 y(\015)989 4286 y(\015)1054 4158 y(Z)1105 4364 y Fp(R)1153 4345 y Fr(3)1217 4220 y Fz(exp)o(\()p Fy(i)p Fu(j)p Fy(x)22 b Fu(\000)d Fy(y)s Fu(j)p Fz(\))p 1217 4261 503 4 v 1287 4344 a(4)p Fy(\031)s Fu(j)p Fy(x)i Fu(\000)f Fy(y)s Fu(j)1744 4282 y Fy(f)10 b Fz(\()p Fy(y)s Fz(\))15 b Fy(dy)2027 4177 y Fn(\015)2027 4232 y(\015)2027 4286 y(\015)2078 4345 y Fw(L)2126 4326 y Fr(4)2160 4345 y Fq(\()p Fp(R)2235 4326 y Fr(3)2270 4345 y Fq(\))2327 4282 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)2640 4333 y Fw(L)2698 4283 y Fr(4)p 2698 4291 31 3 v 2698 4325 a(3)2742 4333 y Fq(\()p Fp(R)2817 4314 y Fr(3)2852 4333 y Fq(\))2884 4282 y Fy(:)-2956 b Fz(\(3.5\))-47 4551 y(P)m(assing)30 b(to)h(\(3.3\))h(only)e(requires) e(c)m(hanging)j(v)-5 b(ariables)29 b Fy(x)c Fu(7!)2102 4474 y(p)p 2178 4474 54 4 v 77 x Fy(\025x)30 b Fz(and)g Fy(y)e Fu(7!)2679 4474 y(p)p 2755 4474 V 77 x Fy(\025y)s Fz(,)i(whic)m(h)g(w)m(e)g(skip.)p 3879 4551 4 62 v 3883 4493 55 4 v 3883 4551 V 3937 4551 4 62 v -47 4739 a(It)d(is)f(w)m (ell-kno)m(wn,)h(see)h(Simon)e([Si2)o(])i(Theorem)e(A.2.9,)k(that)e (for)f(an)m(y)g Fy(V)46 b Fu(2)25 b Fy(L)2637 4706 y Fq(2)2637 4767 y Fw(l)q(oc)2727 4739 y Fz(\()p Fv(R)2822 4706 y Fq(3)2868 4739 y Fz(\))i(that)h(satis\014es)f(the)g(so)h(called) -47 4852 y Fl(Kato)33 b(c)-5 b(ondition)1322 5081 y Fz(lim)1315 5139 y Fw(r)r Fx(!)p Fq(0)1486 5081 y Fz(sup)1470 5165 y Fw(x)p Fx(2)p Fp(R)1605 5146 y Fr(3)1654 4957 y Fn(Z)1705 5164 y Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p Fw()f(M)37 b Fz(with)26 b Fy(M)38 b Fz(as)28 b(b)s(efore,)g(see)g (\(B11\))i(in)c([Si2].)40 b(These)27 b(b)s(ounds)e(imply)h(the)-47 1089 y(Sob)s(olev)j(inequalities)407 1303 y Fu(k)p Fz(\()p Fy(H)f Fz(+)19 b(2)p Fy(M)10 b Fz(\))859 1266 y Fx(\000)p Fw(\014)962 1303 y Fu(k)1007 1326 y Fw(L)1055 1293 y Fs(p)1055 1336 y(x)1095 1326 y Fx(!)p Fw(L)1214 1293 y Fs(q)1214 1336 y(x)1282 1303 y Fy(<)25 b Fu(1)60 b Fz(for)30 b(an)m(y)h(1)26 b Fu(\024)f Fy(p;)15 b(q)28 b Fu(\024)d(1)30 b Fz(and)g(with)f Fy(\014)h(>)2950 1242 y Fz(3)p 2950 1282 46 4 v 2950 1366 a(2)3006 1303 y(\()p Fy(p)3087 1266 y Fx(\000)p Fq(1)3201 1303 y Fu(\000)20 b Fy(q)3336 1266 y Fx(\000)p Fq(1)3430 1303 y Fz(\),)-3537 b(\(3.7\))-47 1522 y(as)33 b(can)h(b)s(e)e(seen)i(from)e(writing)g(\()p Fy(H)d Fz(+)22 b(2)p Fy(M)10 b Fz(\))1509 1489 y Fx(\000)p Fw(\014)1645 1522 y Fz(as)33 b(the)h(Laplace)f(transform)g(of)g(the)g (heat-semigroup,)h(see)g(The-)-47 1634 y(orem)g(B.2.1.)i(in)d([Si2)o (].)52 b(Since)33 b(w)m(e)h(are)h(assuming)d(that)j Fy(V)51 b Fu(2)31 b Fy(L)2198 1601 y Fq(2)2238 1634 y Fz(,)k(Cauc)m(h)m(y-Sc)m (h)m(w)m(arz)g(implies)d(that)i(\(3.6\))i(holds,)-47 1747 y(and)31 b(th)m(us)h(so)g(do)g(all)f(aforemen)m(tioned)h(prop)s (erties.)44 b(In)31 b(addition,)g(w)m(e)i(will)c(use)j(the)g(follo)m (wing)e(result)h(of)h(Jensen)-47 1860 y(and)k(Nak)-5 b(am)m(ura,)40 b(see)e([JN])g(Theorem)e(2.1:)56 b(Supp)s(ose)35 b(that)j Fy(V)57 b Fu(2)36 b Fy(L)2386 1827 y Fq(2)2386 1888 y Fw(l)q(oc)2514 1860 y Fz(satis\014es)h(\(3.6\))q(.)62 b(Let)37 b Fy(g)j Fu(2)d Fy(C)3551 1827 y Fx(1)3544 1885 y Fq(0)3625 1860 y Fz(\()p Fv(R)s Fz(\))43 b(and)-47 1973 y(1)25 b Fu(\024)g Fy(p)g Fu(\024)g(1)p Fz(.)41 b(Then)29 b(there)i(exists)f(a)h(constan)m(t)g Fy(C)37 b Fz(suc)m(h)30 b(that)1068 2155 y Fu(k)p Fy(g)s Fz(\()p Fy(\022)s(H)7 b Fz(\))p Fu(k)1403 2177 y Fw(L)1451 2145 y Fs(p)1451 2188 y(x)1491 2177 y Fx(!)p Fw(L)1610 2145 y Fs(p)1610 2188 y(x)1678 2155 y Fu(\024)25 b Fy(C)67 b Fz(uniformly)27 b(in)i(0)d Fy(<)f(\022)i Fu(\024)e Fz(1.)-2876 b(\(3.8\))-47 2336 y(Moreo)m(v)m(er,)39 b(the)d(constan)m (t)h Fy(C)42 b Fz(is)34 b(uniform)g(for)h Fy(g)k Fz(ranging)c(o)m(v)m (er)i(b)s(ounded)c(sets)j(of)g Fy(C)2968 2303 y Fx(1)2961 2361 y Fq(0)3042 2336 y Fz(\()p Fv(R)s Fz(\).)63 b(As)35 b(an)h(immediate)-47 2449 y(corollary)22 b(of)29 b(\(3.7\))c(and)d (\(3.8\))i(one)f(obtains)f(that)h(for)f(an)m(y)h Fy(g)29 b Fu(2)c Fy(C)2196 2416 y Fx(1)2189 2473 y Fq(0)2270 2449 y Fz(\()p Fv(R)s Fz(\),)31 b(an)m(y)22 b(1)k Fu(\024)f Fy(p)g Fu(\024)g Fy(q)j Fu(\024)d(1)p Fz(,)f Fy(\014)31 b(>)3446 2413 y Fq(3)p 3446 2428 36 4 v 3446 2480 a(2)3491 2449 y Fz(\()p Fy(p)3572 2416 y Fx(\000)p Fq(1)3671 2449 y Fu(\000)5 b Fy(q)3791 2416 y Fx(\000)p Fq(1)3884 2449 y Fz(\),)-47 2562 y(there)30 b(is)g(a)h(constan)m(t)g Fy(C)37 b Fz(dep)s(ending)28 b(on)i Fy(g)s(;)15 b(V)21 b Fz(,)31 b(and)f Fy(\014)5 b Fz(,)30 b(suc)m(h)g(that)1033 2743 y Fu(k)p Fy(g)s Fz(\()p Fy(H)r(=\025)1335 2757 y Fq(0)1377 2743 y Fz(\))p Fu(k)1457 2766 y Fw(L)1505 2734 y Fs(p)1505 2777 y(x)1544 2766 y Fx(!)p Fw(L)1663 2734 y Fs(q)1663 2777 y(x)1731 2743 y Fu(\024)25 b Fy(C)7 b(\025)1952 2699 y Fw(\014)1952 2769 y Fq(0)2059 2743 y Fz(uniformly)28 b(in)h Fy(\025)2633 2757 y Fq(0)2698 2743 y Fu(\025)c Fz(1.)-2911 b(\(3.9\))-47 2925 y(This)34 b(b)s(ound)g(is)h(needed)h(for)g(the)g(follo)m(wing)f(lemma.)57 b(Recall)36 b(that)h Fy(R)2452 2939 y Fq(0)2491 2925 y Fz(\()p Fy(z)t Fz(\))g(denotes)g(the)f(resolv)m(en)m(t)h(of)f(the)h (free)-47 3038 y(Laplacean.)-47 3207 y Fm(Lemma)32 b(3.3.)47 b Fl(L)-5 b(et)37 b Fy(\021)h Fu(2)c Fy(C)943 3174 y Fx(1)936 3231 y Fq(0)1017 3207 y Fz(\()p Fv(R)s Fz(\))44 b Fl(b)-5 b(e)37 b(\014xe)-5 b(d.)57 b(Then)38 b(for)g(any)g Fy(\025;)15 b(\025)2285 3221 y Fq(0)2359 3207 y Fu(\025)34 b Fz(1)k Fl(and)h(any)f(nonne)-5 b(gative)38 b(inte)-5 b(ger)38 b Fy(k)i Fl(one)-47 3319 y(has)33 b(the)g(estimate)412 3513 y Fu(k)p Fy(\021)s Fz(\()p Fy(H)r(=\025)716 3527 y Fq(0)757 3513 y Fz(\))p Fy(R)861 3527 y Fq(0)901 3513 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\)\()p Fy(V)i(R)1390 3527 y Fq(0)1429 3513 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))1775 3475 y Fw(k)1819 3513 y Fy(\021)s Fz(\()p Fy(H)r(=\025)2078 3527 y Fq(0)2118 3513 y Fz(\))p Fu(k)2198 3533 y Fw(L)2246 3514 y Fr(1)2246 3549 y Fs(x)2286 3533 y Fx(!)p Fw(L)2405 3514 y Ft(1)2405 3549 y Fs(x)2499 3513 y Fu(\024)25 b Fy(C)7 b(\025)2730 3428 y Fr(3)p 2730 3440 31 3 v 2730 3482 a(4)2770 3456 y Fq(+)2720 3539 y(0)2844 3513 y Fy(\025)2897 3475 y Fx(\000)2962 3448 y Fr(1)p 2962 3460 V 2962 3501 a(4)3007 3513 y Fz(\()p Fu(k)p Fy(V)21 b Fu(k)3206 3527 y Fq(2)3246 3513 y Fy(\025)3299 3475 y Fx(\000)3364 3448 y Fr(1)p 3363 3460 V 3363 3501 a(4)3408 3513 y Fz(\))3443 3475 y Fw(k)-47 3694 y Fl(wher)-5 b(e)33 b(the)g(c)-5 b(onstant)35 b Fy(C)k Fl(only)33 b(dep)-5 b(ends)35 b(on)e Fy(g)j Fl(and)d Fy(V)21 b Fl(.)-47 3863 y(Pr)-5 b(o)g(of.)47 b Fz(By)30 b(Lemma)h(3.2)g(and)f(H\177)-45 b(older's)30 b(inequalit)m(y)-8 b(,)606 4054 y Fu(k)p Fy(V)21 b(R)794 4068 y Fq(0)833 4054 y Fz(\()p Fy(\025)g Fz(+)f Fy(i)p Fz(0\))p Fy(f)10 b Fu(k)1244 4121 y Fw(L)1302 4055 y Fr(4)p 1303 4063 V 1303 4098 a(3)1292 4131 y Fs(x)1376 4054 y Fu(\024)25 b(k)p Fy(R)1586 4068 y Fq(0)1626 4054 y Fz(\()p Fy(\025)c Fz(+)f Fy(i)p Fz(0\))p Fy(f)10 b Fu(k)2037 4074 y Fw(L)2085 4055 y Fr(4)2085 4091 y Fs(x)2129 4054 y Fu(k)p Fy(V)20 b Fu(k)2292 4074 y Fw(L)2340 4055 y Fr(2)2340 4091 y Fs(x)2409 4054 y Fu(\024)25 b Fy(C)d Fu(k)p Fy(V)e Fu(k)2755 4074 y Fw(L)2803 4055 y Fr(2)2803 4091 y Fs(x)2862 4054 y Fy(\025)2915 4017 y Fx(\000)2980 3990 y Fr(1)p 2980 4002 V 2980 4043 a(4)3040 4054 y Fu(k)p Fy(f)10 b Fu(k)3185 4121 y Fw(L)3243 4055 y Fr(4)p 3243 4063 V 3243 4098 a(3)3233 4131 y Fs(x)-47 4054 y Fz(\(3.10\))-47 4271 y(for)30 b(an)m(y)h Fy(f)j Fu(2)25 b(S)7 b Fz(.)41 b(Hence)454 4453 y Fu(k)p Fy(\021)s Fz(\()p Fy(H)r(=\025)758 4467 y Fq(0)799 4453 y Fz(\))p Fy(R)903 4467 y Fq(0)943 4453 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\)\()p Fy(V)i(R)1432 4467 y Fq(0)1471 4453 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))1817 4415 y Fw(k)1876 4453 y Fy(\021)s Fz(\()p Fy(H)r(=\025)2135 4467 y Fq(0)2176 4453 y Fz(\))p Fu(k)2256 4473 y Fw(L)2304 4454 y Fr(1)2304 4489 y Fs(x)2344 4473 y Fx(!)p Fw(L)2463 4454 y Ft(1)2463 4489 y Fs(x)454 4606 y Fu(\024)25 b(k)p Fy(\021)s Fz(\()p Fy(H)r(=\025)854 4620 y Fq(0)895 4606 y Fz(\))p Fy(R)999 4620 y Fq(0)1039 4606 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\))p Fu(k)1394 4673 y Fw(L)1452 4607 y Fr(4)p 1454 4615 V 1454 4650 a(3)1442 4683 y Fs(x)1498 4673 y Fx(!)p Fw(L)1617 4654 y Ft(1)1617 4689 y Fs(x)1701 4606 y Fu(k)p Fz(\()p Fy(V)h(R)1924 4620 y Fq(0)1964 4606 y Fz(\()p Fy(\025)f Fz(+)g Fy(i)p Fz(0\)\))2309 4569 y Fw(k)2353 4606 y Fu(k)2398 4673 y Fw(L)2456 4607 y Fr(4)p 2457 4615 V 2457 4650 a(3)2446 4683 y Fs(x)2501 4673 y Fx(!)p Fw(L)2630 4607 y Fr(4)p 2630 4615 V 2630 4650 a(3)2620 4683 y Fs(x)2694 4606 y Fu(k)p Fy(\021)s Fz(\()p Fy(H)r(=\025)2998 4620 y Fq(0)3039 4606 y Fz(\))p Fu(k)3119 4673 y Fw(L)3167 4654 y Fr(1)3167 4689 y Fs(x)3206 4673 y Fx(!)p Fw(L)3335 4607 y Fr(4)p 3335 4615 V 3335 4650 a(3)3325 4683 y Fs(x)454 4806 y Fu(\024)25 b(k)p Fy(\021)s Fz(\()p Fy(H)r(=\025)854 4820 y Fq(0)895 4806 y Fz(\))p Fu(k)975 4826 y Fw(L)1023 4807 y Fr(4)1023 4842 y Fs(x)1063 4826 y Fx(!)p Fw(L)1182 4807 y Ft(1)1182 4842 y Fs(x)1250 4806 y Fu(k)p Fy(R)1364 4820 y Fq(0)1404 4806 y Fz(\()p Fy(\025)c Fz(+)f Fy(i)p Fz(0\))p Fu(k)1760 4872 y Fw(L)1818 4807 y Fr(4)p 1819 4815 V 1819 4850 a(3)1808 4883 y Fs(x)1864 4872 y Fx(!)p Fw(L)1983 4854 y Fr(4)1983 4889 y Fs(x)2041 4806 y Fu(k)p Fy(V)g(R)2228 4820 y Fq(0)2268 4806 y Fz(\()p Fy(\025)g Fz(+)g Fy(i)p Fz(0\))p Fu(k)2623 4768 y Fw(k)2623 4881 y(L)2681 4815 y Fr(4)p 2683 4823 V 2683 4858 a(3)2671 4891 y Fs(x)2727 4881 y Fx(!)p Fw(L)2856 4815 y Fr(4)p 2856 4823 V 2856 4858 a(3)2846 4891 y Fs(x)2920 4806 y Fu(k)p Fy(\021)s Fz(\()p Fy(H)r(=\025)3224 4820 y Fq(0)3265 4806 y Fz(\))p Fu(k)3345 4872 y Fw(L)3393 4854 y Fr(1)3393 4889 y Fs(x)3433 4872 y Fx(!)p Fw(L)3562 4807 y Fr(4)p 3561 4815 V 3561 4850 a(3)3552 4883 y Fs(x)454 5041 y Fu(\024)25 b Fy(C)7 b(\025)685 4957 y Fr(3)p 684 4969 V 684 5010 a(8)725 4984 y Fq(+)675 5067 y(0)784 5041 y Fy(\025)837 5004 y Fx(\000)902 4976 y Fr(1)p 902 4988 V 902 5030 a(4)946 5041 y Fz(\()p Fu(k)p Fy(V)21 b Fu(k)1145 5055 y Fq(2)1185 5041 y Fy(\025)1238 5004 y Fx(\000)1303 4976 y Fr(1)p 1303 4988 V 1303 5030 a(4)1348 5041 y Fz(\))1383 5004 y Fw(k)1441 5041 y Fy(\025)1504 4957 y Fr(3)p 1504 4969 V 1504 5010 a(8)1544 4984 y Fq(+)1494 5067 y(0)1603 5041 y Fy(;)-47 5223 y Fz(as)30 b(claimed.)p 3879 5223 4 62 v 3883 5165 55 4 v 3883 5223 V 3937 5223 4 62 v 1903 5472 a(16)p eop %%Page: 17 17 17 16 bop -47 274 a Fl(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)35 b(3.1.)46 b Fz(W)-8 b(e)30 b(start)f(with)e(a)h (justi\014cation)f(of)i(the)f(Born)h(series)e(expansion)h(for)g(high)f (energies.)-47 417 y(Let)33 b Fy(\025)171 431 y Fq(0)239 417 y Fy(>)c Fz(0)k(b)s(e)f(c)m(hosen)h(so)f(that)i Fu(k)p Fy(V)20 b Fu(k)1312 436 y Fw(L)1360 417 y Fr(2)1360 453 y Fs(x)1419 417 y Fy(\025)1472 359 y Fx(\000)1537 332 y Fr(1)p 1537 344 31 3 v 1537 385 a(4)1472 443 y Fq(0)1610 417 y Fy(<)29 b Fz(1.)47 b(By)35 b(\(3.10\))r(,)e(the)g(op)s(erator)g (1)22 b(+)f Fy(V)f(R)3093 431 y Fq(0)3133 417 y Fz(\()p Fy(\025)i Fz(+)f Fy(i)p Fz(0\))34 b(is)d(in)m(v)m(ertible)-47 556 y(in)e Fy(L)131 493 y Fr(4)p 131 505 V 131 546 a(3)175 556 y Fz(\()p Fv(R)270 523 y Fq(3)316 556 y Fz(\))h(pro)m(vided)f Fy(\025)d(>)f(\025)983 570 y Fq(0)1052 556 y Fz(and)30 b(the)h(Neumann)e(series)997 830 y(\(1)21 b(+)f Fy(V)g(R)1331 844 y Fq(0)1371 830 y Fz(\()p Fy(\025)g Fz(+)g Fy(i)p Fz(0\)\))1716 792 y Fx(\000)p Fq(1)1837 830 y Fz(=)1963 716 y Fx(1)1933 744 y Fn(X)1934 941 y Fw(k)r Fq(=0)2064 830 y Fz(\()p Fu(\000)p Fz(1\))2250 792 y Fw(k)2294 830 y Fz(\()p Fy(V)h(R)2472 844 y Fq(0)2511 830 y Fz(\()p Fy(\025)g Fz(+)f Fy(i)p Fz(0\)\))2857 792 y Fw(k)-47 830 y Fz(\(3.11\))-47 1145 y(con)m(v)m(erges)32 b(in)d Fy(L)538 1082 y Fr(4)p 538 1094 V 538 1135 a(3)583 1145 y Fz(\()p Fv(R)678 1112 y Fq(3)723 1145 y Fz(\).)41 b(Therefore,)30 b(the)h(resolv)m(en)m(t)g Fy(R)1869 1159 y Fw(V)1930 1145 y Fz(\()p Fy(z)t Fz(\))26 b(:=)f(\()p Fu(\0004)20 b Fz(+)g Fy(V)41 b Fu(\000)20 b Fy(z)t Fz(\))2757 1112 y Fx(\000)p Fq(1)2882 1145 y Fz(satis\014es)296 1419 y Fy(R)365 1433 y Fw(V)425 1419 y Fz(\()p Fy(\025)h Fz(+)f Fy(i)p Fz(0\))26 b(=)f Fy(R)927 1433 y Fq(0)967 1419 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\)\(1)i(+)e Fy(V)h(R)1613 1433 y Fq(0)1652 1419 y Fz(\()p Fy(\025)g Fz(+)e Fy(i)p Fz(0\)\))1997 1381 y Fx(\000)p Fq(1)2119 1419 y Fz(=)2245 1305 y Fx(1)2215 1332 y Fn(X)2216 1530 y Fw(k)r Fq(=0)2346 1419 y Fz(\()p Fu(\000)p Fz(1\))2532 1381 y Fw(k)2575 1419 y Fy(R)2644 1433 y Fq(0)2684 1419 y Fz(\()p Fy(\025)h Fz(+)g Fy(i)p Fz(0\)\()p Fy(V)i(R)3173 1433 y Fq(0)3212 1419 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))3558 1381 y Fw(k)-47 1734 y Fz(for)j(all)g Fy(\025)i(>)g(\025) 432 1748 y Fq(0)495 1734 y Fz(and)f(is)e(th)m(us)i(a)g(b)s(ounded)e(op) s(erator)i(from)f Fy(L)2018 1671 y Fr(4)p 2018 1683 V 2018 1724 a(3)2062 1734 y Fz(\()p Fv(R)2157 1701 y Fq(3)2203 1734 y Fz(\))i Fu(!)h Fy(L)2442 1701 y Fq(4)2481 1734 y Fz(\()p Fv(R)2576 1701 y Fq(3)2621 1734 y Fz(\).)39 b(F)-8 b(urthermore,)25 b(since)f(the)g(sp)s(ectral)-47 1847 y(resolution)29 b Fy(E)5 b Fz(\()p Fu(\001)p Fz(\))31 b(of)g Fy(H)h Fz(=)25 b Fu(\0004)20 b Fz(+)g Fy(V)50 b Fz(satis\014es)30 b Fy(P)1638 1861 y Fw(a:c:)1765 1847 y Fy(E)5 b Fz(\()p Fy(d\025)p Fz(\))27 b(=)e Fu(=)p Fy(R)2265 1861 y Fw(V)2325 1847 y Fz(\()p Fy(\025)c Fz(+)f Fy(i)p Fz(0\))15 b Fy(d\025)p Fz(,)32 b(one)f(has)665 2120 y Fu(h)p Fy(P)758 2134 y Fw(a:c:)870 2120 y Fy(E)5 b Fz(\()p Fy(d\025)p Fz(\))p Fy(f)g(;)15 b(g)s Fu(i)27 b Fz(=)1437 2007 y Fx(1)1406 2034 y Fn(X)1408 2232 y Fw(k)r Fq(=0)1538 2120 y Fz(\()p Fu(\000)p Fz(1\))1724 2083 y Fw(k)1767 2120 y Fu(h=)p Fz([)p Fy(R)1962 2134 y Fq(0)2002 2120 y Fz(\()p Fy(\025)21 b Fz(+)e Fy(i)p Fz(0\)\()p Fy(V)j(R)2491 2134 y Fq(0)2531 2120 y Fz(\()p Fy(\025)e Fz(+)g Fy(i)p Fz(0\)\))2876 2083 y Fw(k)2920 2120 y Fz(])p Fy(f)5 b(;)15 b(g)s Fu(i)g Fy(d\025)-47 2435 y Fz(for)30 b(an)m(y)h Fy(f)5 b(;)15 b(g)28 b Fu(2)d Fy(L)583 2372 y Fr(4)p 583 2384 V 583 2426 a(3)627 2435 y Fz(\()p Fv(R)722 2402 y Fq(3)768 2435 y Fz(\).)41 b(No)m(w)31 b(de\014ne)f Fy(\021)s Fz(\()p Fy(\025)p Fz(\))c(:=)f Fy(\037)p Fz(\()p Fy(\025)p Fz(\))c Fu(\000)f Fy(\037)p Fz(\()p Fy(\025=)p Fz(2\).)42 b(Clearly)-8 b(,)30 b Fy(\021)e Fu(2)d Fy(C)2851 2402 y Fx(1)2844 2460 y Fq(0)2925 2435 y Fz(\()p Fv(R)s Fz(\))q(,)36 b(and)30 b(also)1398 2597 y Fx(1)1368 2624 y Fn(X)1372 2820 y Fw(j)t Fq(=0)1514 2711 y Fy(\021)s Fz(\()p Fy(\025)p Fz(2)1695 2673 y Fx(\000)p Fw(j)1788 2711 y Fz(\))c(=)f Fy(\037)p Fz(\()p Fy(\025)p Fz(\))61 b(for)30 b(all)f Fy(\025:)-47 3010 y Fz(Observ)m(e)f(that)h(at)g(most)f (three)h(terms)f(in)f(this)g(sum)g(can)i(b)s(e)e(nonzero)i(for)f(an)m (y)g(giv)m(en)h Fy(\025)p Fz(.)40 b(No)m(w)29 b(let)34 b(~)-51 b Fy(\021)28 b Fu(2)d Fy(C)3623 2977 y Fx(1)3616 3034 y Fq(0)3697 3010 y Fz(\(0)p Fy(;)15 b Fu(1)p Fz(\))-47 3123 y(ha)m(v)m(e)31 b(the)g(prop)s(ert)m(y)f(that)h Fy(\021)10 b Fz(~)-52 b Fy(\021)29 b Fz(=)c(1.)41 b(Then)29 b(for)h(an)m(y)h Fy(f)5 b(;)15 b(g)28 b Fu(2)d(S)7 b Fz(,)31 b(one)f(has)h(the)f(expansion)581 3327 y Fu(jh)p Fy(e)683 3289 y Fw(itH)801 3327 y Fy(\037)p Fz(\()p Fy(H)r(=\025)1069 3341 y Fq(0)1109 3327 y Fz(\))p Fy(f)5 b(;)15 b(\037)p Fz(\()p Fy(H)r(=\025)1502 3341 y Fq(0)1542 3327 y Fz(\))p Fy(g)s Fu(ij)427 3515 y Fz(=)581 3383 y Fn(\014)581 3437 y(\014)581 3492 y(\014)581 3546 y(\014)611 3391 y(Z)702 3417 y Fx(1)662 3597 y Fq(0)792 3515 y Fy(e)834 3477 y Fw(it\025)929 3515 y Fu(h)p Fy(E)5 b Fz(\()p Fy(d\025)p Fz(\))p Fy(\021)s Fz(\()p Fy(H)r(=)p Fz(\(2)1492 3477 y Fw(j)1533 3515 y Fy(\025)1586 3529 y Fq(0)1625 3515 y Fz(\)\))p Fy(f)g(;)15 b(\021)s Fz(\()p Fy(H)r(=)p Fz(\(2)2071 3477 y Fw(`)2106 3515 y Fy(\025)2159 3529 y Fq(0)2199 3515 y Fz(\)\))p Fy(g)s Fu(i)g Fy(d\025)2465 3383 y Fn(\014)2465 3437 y(\014)2465 3492 y(\014)2465 3546 y(\014)427 3785 y Fu(\024)581 3654 y Fn(\014)581 3708 y(\014)581 3763 y(\014)581 3817 y(\014)699 3672 y Fx(1)668 3699 y Fn(X)651 3897 y Fw(j;`)p Fq(=0)611 3968 y Fx(j)p Fw(j)t Fx(\000)p Fw(`)p Fx(j\024)p Fq(1)872 3662 y Fn(Z)963 3688 y Fx(1)923 3868 y Fq(0)1053 3785 y Fy(e)1095 3748 y Fw(it\025)1190 3785 y Fu(h)p Fy(E)5 b Fz(\()p Fy(d\025)p Fz(\))p Fy(\021)s Fz(\()p Fy(H)r(=)p Fz(\(2)1753 3748 y Fw(j)1793 3785 y Fy(\025)1846 3799 y Fq(0)1886 3785 y Fz(\)\))p Fy(f)g(;)15 b(\021)s Fz(\()p Fy(H)r(=)p Fz(\(2)2332 3748 y Fw(`)2367 3785 y Fy(\025)2420 3799 y Fq(0)2459 3785 y Fz(\)\))p Fy(g)s Fu(i)24 b Fz(~)-54 b Fy(\021)6 b Fz(\()p Fy(\025=)p Fz(\(2)2889 3748 y Fw(j)2927 3785 y Fy(\025)2980 3799 y Fq(0)3019 3785 y Fz(\)\))15 b Fy(d\025)3204 3654 y Fn(\014)3204 3708 y(\014)3204 3763 y(\014)3204 3817 y(\014)427 4179 y Fu(\024)611 4065 y Fx(1)581 4092 y Fn(X)582 4290 y Fw(k)r Fq(=0)712 4047 y Fn(\014)712 4101 y(\014)712 4156 y(\014)712 4210 y(\014)830 4065 y Fx(1)800 4092 y Fn(X)782 4290 y Fw(j;`)p Fq(=0)743 4362 y Fx(j)p Fw(j)t Fx(\000)p Fw(`)p Fx(j\024)p Fq(1)1003 4055 y Fn(Z)1094 4081 y Fx(1)1054 4261 y Fq(0)1184 4179 y Fy(e)1226 4141 y Fw(it\025)1321 4179 y Fu(h)p Fy(R)1425 4193 y Fq(0)1465 4179 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(i)p Fz(0\))15 b(\()p Fy(V)21 b(R)1969 4193 y Fq(0)2009 4179 y Fz(\()p Fy(\025)f Fz(+)g Fy(i)p Fz(0\)\))2354 4141 y Fw(k)2399 4179 y Fy(\021)s Fz(\()p Fy(H)r(=)p Fz(\(2)2685 4141 y Fw(j)2723 4179 y Fy(\025)2776 4193 y Fq(0)2815 4179 y Fz(\)\))p Fy(f)5 b(;)15 b(\021)s Fz(\()p Fy(H)r(=)p Fz(\(2)3261 4141 y Fw(`)3297 4179 y Fy(\025)3350 4193 y Fq(0)3389 4179 y Fz(\)\))p Fy(g)s Fu(i)2951 4552 y Fz(~)-51 b Fy(\021)s Fz(\()p Fy(\025=)p Fz(\(2)3206 4515 y Fw(j)3244 4552 y Fy(\025)3297 4566 y Fq(0)3337 4552 y Fz(\)\))15 b Fy(d\025)3522 4420 y Fn(\014)3522 4475 y(\014)3522 4530 y(\014)3522 4584 y(\014)-47 4552 y Fz(\(3.12\))427 4823 y(=)611 4710 y Fx(1)581 4737 y Fn(X)582 4934 y Fw(k)r Fq(=0)728 4691 y Fn(\014)728 4746 y(\014)728 4800 y(\014)728 4855 y(\014)758 4699 y(Z)849 4726 y Fx(1)808 4906 y Fq(0)939 4823 y Fy(e)981 4786 y Fw(it\025)1076 4823 y Fu(h)p Fy(R)1180 4837 y Fq(0)1220 4823 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(i)p Fz(0\))15 b(\()p Fy(V)22 b(R)1724 4837 y Fq(0)1763 4823 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))2109 4786 y Fw(k)2153 4823 y Fy(\037)p Fz(\()p Fy(H)r(=\025)2421 4837 y Fq(0)2461 4823 y Fz(\))p Fy(f)5 b(;)15 b(\037)p Fz(\()p Fy(H)r(=\025)2854 4837 y Fq(0)2894 4823 y Fz(\))p Fy(g)s Fu(i)g Fy(d\025)3125 4691 y Fn(\014)3125 4746 y(\014)3125 4800 y(\014)3125 4855 y(\014)3172 4823 y Fy(:)-3244 b Fz(\(3.13\))1903 5472 y(17)p eop %%Page: 18 18 18 17 bop -47 274 a Fz(F)-8 b(rom)31 b(the)f(previous)f(section)i(one)f (has)g(the)h(disp)s(ersiv)m(e)d(b)s(ounds)316 391 y Fn(\014)316 446 y(\014)316 500 y(\014)316 555 y(\014)346 399 y(Z)437 426 y Fx(1)397 606 y Fq(0)527 523 y Fy(e)569 486 y Fw(it\025)664 523 y Fu(h)p Fy(R)768 537 y Fq(0)808 523 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(i)p Fz(0\))15 b(\()p Fy(V)22 b(R)1313 537 y Fq(0)1352 523 y Fz(\()p Fy(\025)f Fz(+)f Fy(i)p Fz(0\)\))1698 486 y Fw(k)1742 523 y Fy(\037)p Fz(\()p Fy(H)r(=\025)2010 537 y Fq(0)2050 523 y Fz(\))p Fy(f)5 b(;)15 b(\037)p Fz(\()p Fy(H)r(=\025)2443 537 y Fq(0)2483 523 y Fz(\))p Fy(g)s Fu(i)g Fy(d\025)2714 391 y Fn(\014)2714 446 y(\014)2714 500 y(\014)2714 555 y(\014)2771 523 y Fu(\024)25 b Fy(C)d(t)2987 486 y Fx(\000)3052 458 y 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y(18)p eop %%Page: 19 19 19 18 bop -47 278 a Fz(the)30 b(notion)f(of)i(Kato's)g(smo)s(othing)e (for)h(the)g(m)m(ultiplication)d(op)s(erator)j Fu(j)p Fy(V)21 b Fu(j)2603 215 y Fr(1)p 2603 227 31 3 v 2603 268 a(2)2677 278 y Fz(relativ)m(e)31 b(to)f Fy(H)3187 292 y Fq(0)3256 278 y Fz(and)g Fy(H)7 b Fz(.)40 b(Applying)-47 391 y(the)28 b(abstract)h(result)d(to)j Fy(H)894 405 y Fq(0)958 391 y Fz(=)c Fu(\000)p Fz(\001,)j Fy(H)33 b Fz(=)25 b Fu(\000)p Fz(\001)14 b(+)h Fy(V)48 b Fz(with)27 b Fy(V)48 b Fz(ob)s(eying)27 b(the)h(estimate)h Fu(j)p Fy(V)20 b Fz(\()p Fy(x)p Fz(\))p Fu(j)26 b(\024)f Fy(C)7 b Fz(\(1)16 b(+)f Fu(j)p Fy(x)p Fu(j)3688 358 y Fq(2)3728 391 y Fz(\))3763 358 y Fx(\000)p Fq(1)p Fx(\000)p Fw(")-47 504 y Fz(requires)27 b(app)s(ealing)h(to)i(the)f(Agmon-Kato-Kuro)s(da)h (theory)f(on)h(the)f(absence)h(of)f(p)s(ositiv)m(e)f(singular)f(con)m (tin)m(uous)-47 616 y(sp)s(ectrum)i(for)h Fy(H)37 b Fz(and)30 b(a)h(separate)g(argumen)m(t)g(that)g(deals)f(with)f(the)h(p)s(oin)m(t) 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Fs(x)2146 4117 y Fu(\024)25 b Fy(C)2307 4131 y Fq(\000)2355 4117 y Fz(\()2414 4094 y(\026)2390 4117 y Fy(H)7 b Fz(\))p Fu(k)p Fy(f)j Fu(k)2653 4136 y Fw(L)2701 4118 y Fr(2)2701 4153 y Fs(x)2745 4117 y Fy(:)-2817 b Fz(\(4.5\))-47 4355 y(W)-8 b(e)35 b(shall)e(call)h Fy(C)566 4369 y Fq(\000)614 4355 y Fz(\()673 4332 y(\026)649 4355 y Fy(H)7 b Fz(\))35 b(the)f(smo)s(othing)g(b)s(ound)e(of)j(\000)f (relativ)m(e)g(to)2358 4332 y(\026)2334 4355 y Fy(H)7 b Fz(.)53 b(Let)35 b(\012)d Fu(\032)f Fv(R)43 b Fz(and)34 b(let)g Fy(P)3336 4369 y Fq(\012)3426 4355 y Fz(b)s(e)g(a)h(sp)s (ectral)-47 4468 y(pro)5 b(jection)31 b(of)515 4445 y(\026)491 4468 y Fy(H)39 b Fz(asso)s(ciated)33 b(with)d(a)j(set)f(\012.)45 b(W)-8 b(e)33 b(sa)m(y)g(that)f(\000)g(is)2327 4445 y(\026)2303 4468 y Fy(H)7 b Fz(-smo)s(oth)32 b(on)g(\012)f(if)g(\000)p Fy(P)3164 4482 y Fq(\012)3251 4468 y Fz(is)3368 4445 y(\026)3344 4468 y Fy(H)7 b Fz(-smo)s(oth.)46 b(W)-8 b(e)-47 4581 y(denote)35 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(the)e(main)f(result)h(of)g(this)g(section.)p -47 5137 1597 4 v 56 5191 a Fr(1)91 5223 y Fg(The)c(assumption)f(of)h(b)r (oundedness)g(is)g(a)g(con)n(v)n(enience)f(that)g(is)i(su\016cien)n(t)e (for)i(our)e(main)g(application)i(b)r(elo)n(w.)1903 5472 y Fz(19)p eop %%Page: 20 20 20 19 bop -47 274 a Fm(Theorem)34 b(4.1.)46 b Fl(L)-5 b(et)34 b Fy(H)839 288 y Fq(0)911 274 y Fl(and)h Fy(H)f Fz(=)27 b Fy(H)1373 288 y Fq(0)1433 274 y Fz(+)21 b Fy(B)1599 241 y Fx(\003)1638 274 y Fy(A)34 b Fl(b)-5 b(e)33 b(as)h(ab)-5 b(ove.)46 b(We)33 b(assume)i(that)g(that)f Fy(B)39 b Fl(is)33 b Fy(H)3397 288 y Fq(0)3470 274 y Fl(smo)-5 b(oth)36 b(with)-47 387 y(a)41 b(smo)-5 b(othing)44 b(b)-5 b(ound)42 b Fy(C)815 401 y Fw(B)875 387 y Fz(\()p Fy(H)986 401 y Fq(0)1026 387 y Fz(\))f Fl(and)h(that)h(for)e(some)h Fz(\012)f Fu(\032)f Fv(R)50 b Fl(the)41 b(op)-5 b(er)g(ator)45 b Fy(A)c Fl(is)g Fy(H)7 b Fl(-smo)-5 b(oth)43 b(on)f Fz(\012)e Fl(with)i(the)-47 500 y(smo)-5 b(othing)35 b(b)-5 b(ound)33 b Fy(C)710 514 y Fw(A)767 500 y Fz(\()p Fy(H)r(;)15 b Fz(\012\))p Fl(.)42 b(Assume)33 b(also)h(that)g(the)f (unitary)g(semigr)-5 b(oup)34 b Fy(e)2746 467 y Fw(itH)2853 476 y Fr(0)2925 500 y Fl(satis\014es)g(the)f(estimate)1391 689 y Fu(k)p Fy(e)1478 651 y Fw(itH)1585 660 y Fr(0)1625 689 y Fy( )1684 703 y Fq(0)1724 689 y Fu(k)1769 711 y Fw(L)1817 679 y Fs(q)1817 731 y(t)1852 711 y Fw(L)1900 693 y Fs(r)1900 728 y(x)1968 689 y Fu(\024)25 b Fy(C)2129 703 y Fw(H)2187 712 y Fr(0)2226 689 y Fu(k)p Fy( )2330 703 y Fq(0)2370 689 y Fu(k)2415 708 y Fw(L)2463 690 y Fr(2)2463 725 y Fs(x)-47 689 y Fz(\(4.7\))-47 894 y Fl(for)h(some)g Fy(q)i Fu(2)d Fz(\(2)p Fy(;)15 b Fu(1)p Fz(])27 b Fl(and)g Fy(r)h Fu(2)c Fz([1)p Fy(;)15 b Fu(1)p Fz(])p Fl(.)41 b(Then)26 b(the)g(semigr)-5 b(oup)27 b Fy(e)2193 861 y Fw(itH)2336 894 y Fl(asso)-5 b(ciate)g(d)28 b(with)e Fy(H)32 b Fz(=)25 b Fy(H)3224 908 y Fq(0)3268 894 y Fz(+)5 b Fy(B)3418 861 y Fx(\003)3457 894 y Fy(A)p Fl(,)27 b(r)-5 b(estricte)g(d)-47 1007 y(to)33 b(the)g(sp)-5 b(e)g(ctr)g(al)35 b(set)e Fz(\012)p Fl(,)f(also)i(veri\014es)e(the)h(estimate)40 b Fz(\(4.7\))r Fl(,)33 b(i.e.,)1005 1196 y Fu(k)p Fy(e)1092 1158 y Fw(itH)1210 1196 y Fy(P)1268 1210 y Fq(\012)1323 1196 y Fy( )1382 1210 y Fq(0)1422 1196 y Fu(k)1467 1218 y Fw(L)1515 1186 y Fs(q)1515 1238 y(t)1550 1218 y Fw(L)1598 1200 y Fs(r)1598 1235 y(x)1666 1196 y Fu(\024)25 b Fy(C)1827 1210 y Fw(H)1885 1219 y Fr(0)1924 1196 y Fy(C)1989 1210 y Fw(B)2049 1196 y Fz(\()p Fy(H)2160 1210 y Fq(0)2200 1196 y Fz(\))p Fy(C)2300 1210 y Fw(A)2357 1196 y Fz(\()p Fy(H)r(;)15 b Fz(\012\))p Fu(k)p Fy( )2715 1210 y Fq(0)2756 1196 y Fu(k)2801 1215 y Fw(L)2849 1197 y Fr(2)2849 1232 y Fs(x)-47 1196 y Fz(\(4.8\))-47 1560 y Fl(Pr)-5 b(o)g(of.)47 b Fz(W)-8 b(e)31 b(start)g(with)e(the)i(Duhamel)e(form)m(ula)h(\(4.3\)) 1030 1859 y Fy(e)1072 1821 y Fw(itH)1189 1859 y Fy( )1248 1873 y Fq(0)1313 1859 y Fz(=)25 b Fy(e)1451 1821 y Fw(itH)1558 1830 y Fr(0)1598 1859 y Fy( )1657 1873 y Fq(0)1717 1859 y Fu(\000)20 b Fy(i)1907 1717 y Fw(t)1854 1735 y Fn(Z)1862 1995 y Fq(0)1960 1859 y Fy(e)2002 1821 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(H)2251 1830 y Fr(0)2291 1859 y Fy(B)2365 1821 y Fx(\003)2404 1859 y Fy(Ae)2514 1821 y Fw(isH)2638 1859 y Fy( )2697 1873 y Fq(0)2752 1859 y Fy(ds:)-47 2156 y Fz(W)-8 b(e)31 b(ha)m(v)m(e)h(the)e(follo)m (wing)f(estimate)i(with)f(the)g(exp)s(onen)m(ts)g Fy(q)s(;)15 b(r)33 b Fz(describ)s(ed)c(in)g(\(4.7\))r(:)492 2455 y Fu(k)p Fy(e)579 2418 y Fw(itH)696 2455 y Fy(P)754 2469 y Fq(\012)810 2455 y Fy( )869 2469 y Fq(0)909 2455 y Fu(k)954 2478 y Fw(L)1002 2446 y Fs(q)1002 2498 y(t)1036 2478 y Fw(L)1084 2459 y Fs(r)1084 2495 y(x)1153 2455 y Fu(\024)c(k)p Fy(e)1336 2418 y Fw(itH)1443 2427 y Fr(0)1483 2455 y Fy(P)1541 2469 y Fq(\012)1596 2455 y Fy( )1655 2469 y Fq(0)1695 2455 y Fu(k)1740 2478 y Fw(L)1788 2446 y Fs(q)1788 2498 y(t)1823 2478 y Fw(L)1871 2459 y Fs(r)1871 2495 y(x)1934 2455 y Fz(+)20 b Fu(k)2138 2313 y Fw(t)2085 2332 y Fn(Z)2093 2592 y Fq(0)2191 2455 y Fy(e)2233 2418 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Fx(\003)3190 4508 y Fz(acting)k(b)s(et)m(w)m(een)h(the)-47 4621 y(spaces)j Fy(L)300 4588 y Fq(2)339 4621 y Fz(\([0)p Fy(;)15 b Fu(1)p Fz(\);)g Fy(L)712 4588 y Fq(2)712 4643 y Fw(x)758 4621 y Fz(\))38 b(and)e Fy(L)1076 4588 y Fw(q)1114 4621 y Fz(\([0)p Fy(;)15 b Fu(1)p Fz(\);)g Fy(L)1487 4588 y Fw(r)1487 4643 y(x)1533 4621 y Fz(\).)60 b(Observ)m(e)37 b(that)h(b)m(y)e(the)i(assumptions)d(of)i(Theorem)g(4.1,)i Fy(q)h(>)35 b Fz(2)-47 4734 y(and)30 b(th)m(us)g(the)g(condition)f Fy(q)f(>)d(p)30 b Fz(in)f(Lemma)i([CK])f(is)f(v)m(eri\014ed.)-47 4847 y(W)-8 b(e)31 b(can)g(rewrite)f(the)g(Duhamel)g(term)1299 5123 y Fy(D)e Fz(=)1550 4981 y Fw(t)1498 4999 y Fn(Z)1505 5259 y Fq(0)1604 5123 y Fy(e)1646 5085 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(H)1895 5094 y Fr(0)1935 5123 y Fy(B)2009 5085 y Fx(\003)2047 5123 y Fy(Ae)2157 5085 y Fw(isH)2282 5123 y Fy(P)2340 5137 y Fq(\012)2395 5123 y Fy( )2454 5137 y Fq(0)2509 5123 y Fy(ds)1903 5472 y Fz(20)p eop %%Page: 21 21 21 20 bop -47 314 a Fz(in)29 b(the)h(form)g Fy(D)e Fz(=)653 291 y(~)629 314 y Fy(K)713 186 y Fn(\022)780 314 y Fy(Ae)890 281 y Fw(i)p Fx(\001)p Fw(H)1001 314 y Fy(P)1059 328 y Fq(\012)1115 314 y Fy( )1174 328 y Fq(0)1213 186 y Fn(\023)1280 314 y Fz(.)41 b(Therefore,)862 566 y Fu(k)p Fy(D)s Fu(k)1030 589 y Fw(L)1078 556 y Fs(q)1078 609 y(t)1113 589 y Fw(L)1161 570 y Fs(r)1161 605 y(x)1230 566 y Fo(.)24 b Fu(k)p Fy(K)7 b Fu(k)1499 586 y Fw(L)1547 567 y Fr(2)1583 586 y Fq(\([0)p Fw(;)p Fx(1)p Fq(\);)p Fw(L)1851 567 y Fr(2)1851 602 y Fs(x)1889 586 y Fq(\))p Fx(!)p Fw(L)2035 567 y Fs(q)2070 586 y Fq(\([0)p Fw(;)p Fx(1)p Fq(\);)p Fw(L)2338 567 y Fs(r)2338 602 y(x)2376 586 y Fq(\))2438 566 y Fu(k)p Fy(Ae)2593 529 y Fw(isH)2718 566 y Fy( )2777 580 y Fq(0)2816 566 y Fu(k)2861 590 y Fw(L)2909 567 y Fr(2)2909 610 y Fs(t)2944 590 y Fw(L)2992 571 y Fr(2)2992 606 y Fs(x)-47 566 y Fz(\(4.10\))-47 753 y(W)-8 b(e)31 b(no)m(w)g(need)f(to)h(estimate)g(the)g(norm)e(of)i (the)f(op)s(erator)h Fy(K)7 b Fz(.)453 1033 y Fu(k)p Fy(K)g(F)13 b Fu(k)698 1055 y Fw(L)746 1023 y Fs(q)746 1075 y(t)781 1055 y Fw(L)829 1037 y Fs(r)829 1072 y(x)897 1033 y Fz(=)25 b Fu(k)1084 891 y Fx(1)1054 909 y Fn(Z)1061 1169 y Fq(0)1160 1033 y Fy(e)1202 995 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(H)1451 1004 y Fr(0)1491 1033 y Fy(B)1565 995 y Fx(\003)1604 1033 y Fy(F)13 b Fz(\()p Fy(s)p Fz(\))i Fy(ds)p Fu(k)1938 1055 y Fw(L)1986 1023 y Fs(q)1986 1075 y(t)2021 1055 y Fw(L)2069 1037 y Fs(r)2069 1072 y(x)2137 1033 y Fz(=)25 b Fu(k)p Fy(e)2320 995 y Fw(itH)2427 1004 y Fr(0)2513 891 y Fx(1)2482 909 y Fn(Z)2490 1169 y Fq(0)2588 1033 y Fy(e)2630 995 y Fx(\000)p Fw(isH)2800 1004 y Fr(0)2839 1033 y Fy(B)2913 995 y Fx(\003)2952 1033 y Fy(F)13 b Fz(\()p Fy(s)p Fz(\))i Fy(ds)p Fu(k)3286 1055 y Fw(L)3334 1023 y Fs(q)3334 1075 y(t)3369 1055 y Fw(L)3417 1037 y Fs(r)3417 1072 y(x)897 1537 y Fu(\024)25 b Fy(C)1058 1551 y Fw(H)1116 1560 y Fr(0)1155 1537 y Fu(k)1246 1395 y Fx(1)1215 1413 y Fn(Z)1223 1673 y Fq(0)1321 1537 y Fy(e)1363 1499 y Fx(\000)p Fw(isH)1533 1508 y Fr(0)1572 1537 y Fy(B)1646 1499 y Fx(\003)1685 1537 y Fy(F)13 b Fz(\()p Fy(s)p Fz(\))i Fy(ds)p Fu(k)2019 1557 y Fw(L)2067 1538 y Fr(2)2067 1573 y Fs(x)2111 1537 y Fy(:)-47 1843 y Fz(The)30 b(last)g(inequalit)m(y)f(is)g(the)i(estimate) g(\(4.7\))h(for)e Fy(e)1744 1810 y Fw(itH)1851 1819 y Fr(0)1891 1843 y Fz(.)40 b(By)31 b(dualit)m(y)282 2128 y Fu(k)373 1986 y Fx(1)343 2004 y Fn(Z)350 2264 y Fq(0)449 2128 y Fy(e)491 2090 y Fx(\000)p Fw(isH)661 2099 y Fr(0)699 2128 y Fy(B)773 2090 y Fx(\003)812 2128 y Fy(F)13 b Fz(\()p Fy(s)p Fz(\))i Fy(ds)p Fu(k)1146 2147 y Fw(L)1194 2129 y Fr(2)1194 2164 y Fs(x)1263 2128 y Fz(=)101 b(sup)1359 2210 y Fx(k)p Fw(\036)p Fx(k)1471 2237 y Fs(L)1513 2220 y Fr(2)1513 2251 y Fs(x)1557 2210 y Fq(=1)1673 2128 y Fy(<)1799 1986 y Fx(1)1769 2004 y Fn(Z)1776 2264 y Fq(0)1875 2128 y Fy(e)1917 2090 y Fx(\000)p Fw(isH)2087 2099 y Fr(0)2125 2128 y Fy(B)2199 2090 y Fx(\003)2238 2128 y Fy(F)13 b Fz(\()p Fy(s)p Fz(\))i Fy(ds;)g(\036)26 b(>)1263 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b(a)g(sp)s(ectral)g(pro)5 b(jection)33 b(of)g Fy(H)40 b Fz(corresp)s(onding)31 b(to)j(the)g(in)m(terv)-5 b(al)32 b(\012)e(=)g([0)p Fy(;)15 b Fu(1)p Fz(\).)50 b(Our)-47 1903 y(goal)28 b(is)e(to)i(pro)m(v)m(e)g(Stric)m(hartz)g (inequalities)c(for)j Fy(e)1652 1870 y Fw(itH)1797 1903 y Fz(restricted)g(to)h(the)g(absolutely)e(con)m(tin)m(uous)h(sp)s (ectrum)f(of)i Fy(H)7 b Fz(.)-47 2016 y(W)-8 b(e)31 b(no)m(w)g(state)g (the)g(result.)-47 2195 y Fm(Theorem)j(4.3.)46 b Fl(L)-5 b(et)35 b Fy(V)55 b Fl(b)-5 b(e)35 b(a)g(p)-5 b(otential)38 b(verifying)k Fz(\(4.14\))r Fl(.)49 b(In)35 b(addition,)j(we)d(imp)-5 b(ose)36 b(the)g(c)-5 b(ondition)36 b(that)g(the)-47 2308 y(p)-5 b(oint)37 b Fy(\025)30 b Fz(=)h(0)36 b Fl(in)f(the)h(sp)-5 b(e)g(ctrum)37 b(of)f(the)g(op)-5 b(er)g(ator)39 b Fy(H)f Fz(=)30 b Fu(\000)p Fz(\001)22 b(+)g Fy(V)56 b Fl(is)35 b(neither)h(an)h(eigenvalue)e(nor)h(a)g(r)-5 b(esonanc)g(e)-47 2421 y(\(se)g(e)41 b(the)h(discussion)f(b)-5 b(elow,)45 b(in)c(p)-5 b(articular)43 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b(Their)31 b(app)s(earance)i(cannot)h (b)s(e)e(ruled)f(out)j(b)m(y)e(merely)h(strengthening)1903 5472 y(22)p eop %%Page: 23 23 23 22 bop -47 274 a Fz(the)23 b(regularit)m(y)f(and)g(deca)m(y)i (assumptions)e(on)h(the)g(p)s(oten)m(tial)f Fy(V)e Fz(.)38 b(W)-8 b(e)25 b(therefore)e(imp)s(ose)f(an)g(additional)f(condition)-47 387 y(that)34 b(0)g(is)e(a)i(regular)e(p)s(oin)m(t.)49 b(There)33 b(are)h(sev)m(eral)g(situations)e(where)h(this)f(condition,) h(or)g(at)h(least)g(part)f(of)h(it,)g(is)-47 500 y(automatically)26 b(satis\014ed.)39 b(In)26 b(particular,)g(for)g(an)m(y)h(non-negativ)m (e)h(p)s(oten)m(tial)e(0)h(is)f(a)h(regular)f(p)s(oin)m(t.)38 b(In)26 b(addition,)-47 613 y(it)35 b(is)g(w)m(ell-kno)m(w)f(\(see)j (e.g.)57 b([JK]\))36 b(that)g(0)g(is)f(not)h(a)g(resonance)g(in)e (dimensions)f Fy(n)h Fu(\025)f Fz(5.)57 b(The)35 b(b)s(eha)m(vior)g(of) h(the)-47 726 y(resolv)m(en)m(t)29 b(near)g(0)g(in)f(the)h(sp)s(ectrum) f(and)g(ev)m(en)i(its)e(asymptotic)h(expansions)f(w)m(as)h(extensiv)m (ely)g(studied)e(in)h([JK)o(],)-47 839 y([J1],)j([J2],)g(but)f(their)f (assumptions)g(are)i(to)s(o)g(strong)f(for)h(our)e(purp)s(oses.)-47 1026 y Fm(Prop)s(osition)36 b(4.5.)46 b Fl(Supp)-5 b(ose)42 b Fy(V)60 b Fl(satis\014es)40 b(the)h(assumption)49 b Fz(\(4.14\))42 b Fl(and)f(assume,)h(in)e(addition,)j(that)e Fz(0)g Fl(is)e(a)-47 1152 y(r)-5 b(e)g(gular)36 b(p)-5 b(oint)36 b(of)f(the)h(sp)-5 b(e)g(ctrum)36 b(of)f Fy(H)i Fz(=)29 b Fu(\000)p Fz(\001)21 b(+)h Fy(V)e Fl(.)48 b(Then)36 b(the)f(op)-5 b(er)g(ator)38 b Fy(A)29 b Fz(=)g Fu(j)p Fy(V)21 b Fu(j)2952 1089 y Fr(1)p 2952 1101 31 3 v 2952 1142 a(2)2997 1152 y Fl(sgn)15 b Fz(\()p Fy(V)21 b Fz(\))35 b Fl(is)g Fy(H)7 b Fl(-smo)-5 b(oth)36 b(on)-47 1265 y Fz([0)p Fy(;)15 b(\016)s Fz(\))35 b Fl(for)e(some)g(su\016ciently)g (smal)5 b(l)34 b Fy(\016)29 b(>)c Fz(0)p Fl(.)94 1453 y Fz(The)35 b(\014rst)g(observ)-5 b(ation,)36 b(whic)m(h)f(follo)m(ws)f (from)h(\(4.6\))r(,)i(is)d(that)j(since)d(the)i(p)s(oten)m(tial)f Fy(V)56 b Fz(deca)m(ys)36 b(at)g(the)g(rate)-47 1566 y Fu(j)p Fy(x)p Fu(j)55 1533 y Fx(\000)p Fq(2)p Fx(\000)p Fq(2)p Fw(")294 1566 y Fz(it)21 b(su\016ces)h(to)h(pro)m(v)m(e)f(the)g (follo)m(wing)f(prop)s(ert)m(y)g(of)h(the)g(resolv)m(en)m(t)h Fy(R)q Fz(\()p Fy(\025)s Fu(\006)s Fy(i\014)5 b Fz(\))22 b(form)m(ulated)g(in)e(the)i(language)-47 1679 y(of)30 b(the)h(w)m(eigh)m(ted)g(spaces)f Fy(L)932 1646 y Fq(2)p Fw(;\013)1037 1679 y Fz(:)1192 1883 y(sup)1030 1966 y Fq(0)p Fw(<\014)s(<\016)o(;\025)p Fx(2)p Fq([0)p Fw(;\016)r Fq(\))1506 1883 y Fu(k)p Fy(R)q Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i\014)5 b Fz(\))p Fy(f)10 b Fu(k)2043 1903 y Fw(L)2091 1884 y Fr(2)p Fs(;)p Ft(\000)p Fr(1)p Ft(\000)p Fs(\015)2337 1883 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)2650 1903 y Fw(L)2698 1884 y Fr(2)p Fs(;)p Fr(1+)p Fs(\015)-47 1883 y Fz(\(4.16\))-47 2167 y(for)31 b(an)m(y)g Fy(f)36 b Fu(2)26 b Fy(L)495 2134 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)716 2167 y Fz(and)k(some)i(su\016cien)m(tly)e(small)f Fy(\016)35 b Fz(and)c Fy(\015)36 b 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a(2)2740 2596 y Fq(+)p Fw(\015)2868 2631 y Fz(and)e Fy(L)3105 2596 y Fq(2)p Fw(;)p Fx(\000)3225 2568 y Fr(1)p 3225 2580 V 3225 2622 a(2)3265 2596 y Fx(\000)p Fw(\015)3393 2631 y Fz(for)h(an)m(y)g Fy(\015)h(>)25 b Fz(0.)-47 2744 y(As)37 b(in)f(that)j(case)f(w)m(e)g(reduce)f(the)h (pro)s(of)f(to)h(the)g(same)g(estimates)g(for)f(the)h(free)g(resolv)m (en)m(t)g Fy(R)3330 2758 y Fq(0)3369 2744 y Fz(\()p Fy(\025)26 b Fu(\006)e Fy(i\014)5 b Fz(\).)63 b(The)-47 2857 y(connection)30 b(is)g(established)e(via)i(the)h(resolv)m(en)m(t)g(iden)m(tit)m(y)960 3062 y Fy(R)q Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i\014)5 b Fz(\))26 b(=)f Fy(R)1543 3076 y Fq(0)1582 3062 y Fz(\()p Fy(\025)c Fu(\006)f Fy(i\014)5 b Fz(\))21 b Fu(\000)f Fy(R)2085 3076 y Fq(0)2124 3062 y Fz(\()p Fy(\025)h Fu(\006)f Fy(i\014)5 b Fz(\))p Fy(V)21 b(R)q Fz(\()p Fy(\025)f Fu(\006)g Fy(i\014)5 b Fz(\))p Fy(:)-47 3266 y Fz(Th)m(us)29 b(formally)g(w)m(e)i(can)f(solv)m(e)h(for)f Fy(R)q Fz(\()p Fy(\025)21 b Fu(\006)e Fy(i\014)5 b Fz(\),)1050 3470 y Fy(R)q Fz(\()p Fy(\025)21 b Fu(\006)f Fy(i\014)5 b Fz(\))26 b(=)f(\()p Fy(I)i Fz(+)20 b Fy(R)1826 3484 y Fq(0)1866 3470 y Fz(\()p Fy(\025)g Fu(\006)g Fy(i\014)5 b Fz(\))p Fy(V)21 b Fz(\))2296 3432 y Fx(\000)p Fq(1)2391 3470 y Fy(R)2460 3484 y Fq(0)2500 3470 y Fz(\()p Fy(\025)f Fu(\006)g Fy(i\014)5 b Fz(\))p Fy(:)-2893 b Fz(\(4.17\))-47 3674 y(W)-8 b(e)32 b(iden)m(tify)d(the)i(b)s(oundary)e(v)-5 b(alue)30 b(of)i(the)f(free)g(resolv)m(en)m(t)g(at)h Fy(\025)26 b Fz(=)g(0)p Fy(;)31 b(\014)g Fz(=)26 b(0)31 b(as)g(the)g(op)s(erator)h(with)d(the)i(k)m(ernel)-47 3787 y(giv)m(en)f(b)m(y)g(the)h(Green's)g(function)e(\(up)g(to)i (constan)m(ts\),)1025 4030 y Fy(G)p Fz(\()p Fy(x;)15 b(y)s Fz(\))27 b(:=)e Fy(R)1523 4044 y Fq(0)1562 4030 y Fz(\(0\)\()p Fy(x;)15 b(y)s Fz(\))28 b(=)2197 3968 y(1)p 2021 4009 399 4 v 2021 4092 a Fu(j)p Fy(x)20 b Fu(\000)g Fy(y)s Fu(j)2282 4066 y Fw(n)p Fx(\000)p Fq(2)2429 4030 y Fy(;)197 b(n)25 b Fu(\025)g Fz(3)-47 4277 y(and)30 b(break)g(the)g(pro)s(of)g(in)m(to)g(a)h(series)f(of)g(lemmas.)-47 4465 y Fm(Lemma)i(4.6.)47 b Fy(G)f Fl(is)h(a)f(b)-5 b(ounde)g(d)48 b(map)g(fr)-5 b(om)47 b(the)g(weighte)-5 b(d)48 b(sp)-5 b(ac)g(e)47 b Fy(L)2511 4432 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)2747 4465 y Fl(into)g Fy(L)3011 4432 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)3301 4465 y Fl(for)g(any)g Fy(\015)56 b(>)50 b Fz(0)p Fl(.)-47 4578 y(Mor)-5 b(e)g(over,)33 b(for)h(any)f(p)-5 b(ositive)33 b Fy(\033)c(<)1214 4542 y Fq(1)p 1214 4557 36 4 v 1214 4609 a(2)1292 4578 y Fl(we)j(have)h Fy(G)26 b Fz(:)41 b Fy(L)1860 4545 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)t Fq(+)p Fw(\033)2172 4578 y Fu(!)25 b Fy(L)2350 4545 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+)p Fw(\033)2692 4578 y Fl(.)-47 4766 y Fm(Lemma)32 b(4.7.)47 b Fl(The)33 b(r)-5 b(esolvent)35 b Fy(R)1168 4780 y Fq(0)1208 4766 y Fz(\()p Fy(\025)21 b Fu(\006)g Fy(i\014)5 b Fz(\))34 b Fl(is)g(c)-5 b(ontinuous)35 b(at)f Fy(\025)27 b Fz(=)g(0)p Fy(;)15 b(\014)33 b Fz(=)27 b(0)34 b Fl(in)g(the)g(top)-5 b(olo)g(gy)36 b(of)e(the)g(b)-5 b(ounde)g(d)-47 4878 y(op)g(er)g(ators)36 b(b)-5 b(etwe)g(en)33 b Fy(L)742 4846 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)964 4878 y Fl(and)g Fy(L)1202 4846 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)1479 4878 y Fl(for)g(any)h Fy(\015)c(>)25 b Fz(0)p Fl(.)-47 5066 y Fm(Lemma)32 b(4.8.)47 b Fl(Under)34 b(the)g (assumptions)j(of)d(Pr)-5 b(op)g(osition)37 b(4.5)e(the)f(op)-5 b(er)g(ator)37 b Fy(GV)55 b Fl(is)34 b(c)-5 b(omp)g(act)36 b(as)f(an)g(op)-5 b(er)g(ator)-47 5179 y(on)33 b Fy(L)145 5146 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)422 5179 y Fl(for)g(any)g Fy(\015)d(>)25 b Fz(0)p Fl(,)33 b(and)h Fz(\()p Fy(I)27 b Fz(+)20 b Fy(GV)h Fz(\))33 b Fl(is)g(invertible)f(on)h Fy(L)2291 5146 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)2535 5179 y Fl(.)1903 5472 y Fz(23)p eop %%Page: 24 24 24 23 bop -47 274 a Fl(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)35 b(4.5.)46 b Fz(According)27 b(to)i(Lemma)e(4.8)i(the) f(op)s(erator)g(\()p Fy(I)22 b Fz(+)15 b Fy(GV)20 b Fz(\))27 b(is)g(in)m(v)m(ertible)f(on)h Fy(L)3564 241 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)3836 274 y Fz(for)-47 387 y(an)m(y)35 b Fy(\015)j(>)32 b Fz(0.)55 b(Therefore,)36 b(b)m(y)f(con)m(tin)m(uit)m(y)f(of)i Fy(R)1627 401 y Fq(0)1666 387 y Fz(\()p Fy(\025)24 b Fu(\006)e Fy(i\014)5 b Fz(\))36 b(at)g Fy(\025)d Fz(=)f(0)p Fy(;)15 b(\014)39 b Fz(=)32 b(0)k(asserted)f(in)e(Lemma)i(4.7)h(and)f (the)-47 500 y(fact)h(that)g Fy(V)55 b Fz(maps)34 b Fy(L)751 467 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)1031 500 y Fz(to)i Fy(L)1209 467 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)1433 500 y Fz(pro)m(vided)e(that)h Fy(\015)k(<)33 b(\014)5 b Fz(,)36 b(there)g(exists)e(a)i(small)e(neigh)m(b)s(orho)s(o)s(d)e Fy(\016)39 b Fz(of)c(0)-47 613 y(suc)m(h)30 b(that)g(\()p Fy(I)d Fz(+)20 b Fy(R)616 627 y Fq(0)655 613 y Fz(\()p Fy(\025)g Fu(\006)f Fy(i\014)5 b Fz(\))p Fy(V)22 b Fz(\))30 b(is)f(uniformly)e(in)m(v)m(ertible)i(for)h(all)f Fu(j)p Fy(\025)p Fu(j)p Fy(;)15 b(\014)31 b(<)25 b(\016)34 b Fz(on)c(the)g(space)h Fy(L)3269 580 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)3513 613 y Fz(.)40 b(It)31 b(follo)m(ws)-47 726 y(that)e(for)g(suc)m(h)g Fy(\025;)15 b(\014)35 b Fz(the)29 b(resolv)m(en)m(t)h Fy(R)q Fz(\()p Fy(\025)18 b Fu(\006)f Fy(i\014)5 b Fz(\))30 b(is)e(w)m(ell-de\014ned)f(via)i(the) g(iden)m(tit)m(y)h(\(4.17\))h(and)e(acts)h(b)s(et)m(w)m(een)g(the)-47 839 y(spaces)h Fy(L)294 806 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)513 839 y Fz(and)f Fy(L)752 806 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)1027 839 y Fz(as)g(desired.)p 3879 839 4 62 v 3883 780 55 4 v 3883 839 V 3937 839 4 62 v -47 1026 a Fl(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(4.6.)46 b Fz(The)38 b(resolv)m(en)m(t)h Fy(G)h Fz(=)f Fy(R)1695 1040 y Fq(0)1734 1026 y Fz(\(0\))h(is)e(a)h(m)m(ultiplier)c (with)j(sym)m(b)s(ol)f Fu(j)p Fy(\030)t Fu(j)3127 993 y Fx(\000)p Fq(2)3222 1026 y Fz(.)65 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Fz(and)30 b(pro)m(v)m(e)h(that)918 1612 y Fu(j)p Fy(\030)t Fu(j)1012 1575 y Fx(\000)p Fq(2)1147 1612 y Fz(:)56 b Fy(W)1327 1574 y Fq(1+)1314 1639 y(2)1446 1612 y Fu(!)25 b Fy(W)1661 1574 y Fx(\000)p Fq(1)p Fx(\000)1648 1639 y Fq(2)1810 1612 y Fy(;)197 b Fu(j)p Fy(\030)t Fu(j)2126 1575 y Fx(\000)p Fq(2)2261 1612 y Fz(:)56 b Fy(W)2451 1528 y Fr(3)p 2451 1540 31 3 v 2451 1581 a(2)2491 1555 y Fq(+)2428 1639 y(2)2575 1612 y Fu(!)26 b Fy(W)2791 1555 y Fx(\000)2856 1528 y Fr(1)p 2855 1540 V 2855 1581 a(2)2895 1555 y Fx(\000)2778 1639 y Fq(2)2954 1612 y Fy(;)-47 1817 y Fz(where)39 b Fu(\006)h Fz(represen)m(t)g(the)g(fact)h (that)f(w)m(e)h(do)f(not)g(pro)m(v)m(e)h(the)f(end-p)s(oin)m(t)f (results)f(themselv)m(es.)70 b(Since)39 b Fu(j)p Fy(\030)t Fu(j)3749 1784 y Fx(\000)p Fq(2)3884 1817 y Fz(is)-47 1930 y(smo)s(oth)28 b(a)m(w)m(a)m(y)j(from)d Fy(\030)h Fz(=)c(0)k(w)m(e)g(can)g(consider)e(instead)h(the)h(op)s(erator)f(of)h (m)m(ultiplication)d(b)m(y)i Fy(\037)p Fz(\()p Fy(\030)t Fz(\))p Fu(j)p Fy(\030)t Fu(j)3503 1897 y Fx(\000)p Fq(2)3627 1930 y Fz(where)g Fy(\037)-47 2043 y Fz(is)c(a)j(smo)s(oth)e(cut-o\013) i(function)d(with)g(supp)s(ort)g(in)h(a)h(unit)e(ball)g Fy(B)5 b Fz(.)39 b(W)-8 b(e)26 b(ha)m(v)m(e)h(the)f(standard)f(Sob)s (olev)g(em)m(b)s(eddings)1413 2262 y Fy(W)1512 2224 y Fq(1+)1499 2288 y(2)1631 2262 y Fy(,)-15 b Fu(!)25 b Fy(L)1853 2192 y Fr(2)p Fs(n)p 1829 2204 117 3 v 1829 2245 a(n)p Ft(\000)p Fr(2)1956 2219 y Fq(+)2015 2262 y Fy(;)-2072 b Fz(\(4.18\))1413 2446 y Fy(W)1522 2361 y Fr(3)p 1522 2373 31 3 v 1522 2414 a(2)1562 2388 y Fq(+)1499 2472 y(2)1646 2446 y Fy(,)-15 b Fu(!)26 b Fy(L)1868 2376 y Fr(2)p Fs(n)p 1844 2388 117 3 v 1844 2429 a(n)p Ft(\000)p Fr(3)1975 2446 y Fz(\()p Fy(L)2072 2408 y Fx(1)2147 2446 y Fy(;)31 b(n)25 b Fz(=)f(3\))p Fy(;)-2515 b Fz(\(4.19\))-47 2696 y(the)27 b(dual)g(v)m(ersion)g(of)34 b(\(4.18\))s(,)28 b Fy(L)1100 2626 y Fr(2)p Fs(n)p 1077 2638 116 3 v 1077 2679 a(n)p Fr(+2)1202 2653 y Fx(\000)1286 2696 y Fy(,)-15 b Fu(!)26 b Fy(W)1512 2657 y Fx(\000)p Fq(1)p Fx(\000)1499 2722 y Fq(2)1660 2696 y Fz(,)j(and)e Fy(L)1983 2626 y Fr(2)p Fs(n)p 1959 2638 V 1959 2679 a(n)p Fr(+1)2085 2653 y Fx(\000)2169 2696 y Fy(,)-15 b Fu(!)25 b Fy(W)2394 2638 y Fx(\000)2459 2611 y Fr(1)p 2459 2623 31 3 v 2459 2664 a(2)2499 2638 y Fx(\000)2381 2722 y Fq(2)2558 2696 y Fz(.)40 b(Therefore,)28 b(w)m(e)g(shall,)e(in)h(fact,)i(pro)m(v)m(e) -47 2809 y(a)i(stronger)f(result)f(that)673 3028 y Fy(\037)p Fz(\()p Fy(\030)t Fz(\))15 b Fu(j)p Fy(\030)t Fu(j)953 2990 y Fx(\000)p Fq(2)1089 3028 y Fz(:)56 b Fy(L)1266 2958 y Fr(2)p Fs(n)p 1242 2970 117 3 v 1242 3011 a(n)p Ft(\000)p Fr(2)1368 2985 y Fq(+)1453 3028 y Fu(!)25 b Fy(L)1664 2958 y Fr(2)p Fs(n)p 1641 2970 116 3 v 1641 3011 a(n)p Fr(+2)1766 2985 y Fx(\000)1825 3028 y Fy(;)197 b(\037)p Fz(\()p Fy(\030)t Fz(\))15 b Fu(j)p Fy(\030)t Fu(j)2327 2990 y Fx(\000)p Fq(2)2463 3028 y Fz(:)56 b Fy(L)2640 2958 y Fr(2)p Fs(n)p 2616 2970 117 3 v 2616 3011 a(n)p Ft(\000)p Fr(3)2743 2985 y Fq(+)2827 3028 y Fu(!)25 b Fy(L)3038 2958 y Fr(2)p Fs(n)p 3015 2970 116 3 v 3015 3011 a(n)p Fr(+1)3140 2985 y Fx(\000)3199 3028 y Fy(;)-47 3232 y Fz(Since)200 3196 y Fw(n)p Fx(\000)p Fq(2)p 200 3211 133 4 v 228 3263 a(2)p Fw(n)363 3232 y Fz(+)468 3196 y Fq(2)p 464 3211 43 4 v 464 3263 a Fw(n)542 3232 y Fz(=)648 3196 y Fw(n)p Fq(+2)p 648 3211 133 4 v 676 3263 a(2)p Fw(n)821 3232 y Fz(and)1008 3196 y Fw(n)p Fx(\000)p Fq(3)p 1008 3211 V 1036 3263 a(2)p Fw(n)1171 3232 y Fz(+)1276 3196 y Fq(2)p 1272 3211 43 4 v 1272 3263 a Fw(n)1350 3232 y Fz(=)1456 3196 y Fw(n)p Fq(+1)p 1456 3211 133 4 v 1483 3263 a(2)p Fw(n)1599 3232 y Fz(,)31 b(w)m(e)f(ha)m(v)m(e)779 3436 y Fu(k)15 b Fy(\037)p Fz(\()p Fy(\030)t Fz(\))p Fu(j)p Fy(\030)t Fu(j)1104 3399 y Fx(\000)p Fq(2)1200 3436 y Fy(f)10 b Fu(k)1300 3494 y Fw(L)1381 3437 y Fr(2)p Fs(n)p 1358 3445 116 3 v 1358 3480 a(n)p Fr(+2)1483 3457 y Ft(\000)1564 3436 y Fu(\024)25 b(k)31 b(j)p Fy(\030)t Fu(j)1830 3399 y Fx(\000)p Fq(2)1925 3436 y Fu(k)1970 3476 y Fw(L)2028 3426 y Fs(n)p 2028 3434 39 3 v 2032 3469 a Fr(2)2077 3446 y Ft(\000)2129 3476 y Fq(\()p Fw(B)s Fq(\))2244 3436 y Fu(k)p Fy(f)10 b Fu(k)2423 3437 y Fr(2)p Fs(n)p 2399 3449 117 3 v 2399 3490 a(n)p Ft(\000)p Fr(2)2526 3464 y Fq(+)2610 3436 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)2957 3437 y Fr(2)p Fs(n)p 2933 3449 V 2933 3490 a(n)p Ft(\000)p Fr(2)3060 3464 y Fq(+)-47 3660 y Fz(and)727 3773 y Fu(k)15 b Fy(\037)p Fz(\()p Fy(\030)t Fz(\))p Fu(j)p Fy(\030)t Fu(j)1052 3735 y Fx(\000)p Fq(2)1148 3773 y Fy(f)10 b Fu(k)1248 3831 y Fw(L)1329 3774 y Fr(2)p Fs(n)p 1305 3782 116 3 v 1305 3817 a(n)p Fr(+1)1431 3793 y Ft(\000)1512 3773 y Fu(\024)25 b(k)31 b(j)p Fy(\030)t Fu(j)1778 3735 y Fx(\000)p Fq(2)1873 3773 y Fu(k)1918 3813 y Fw(L)1976 3763 y Fs(n)p 1976 3771 39 3 v 1980 3806 a Fr(2)2024 3783 y Ft(\000)2077 3813 y Fq(\()p Fw(B)s Fq(\))2192 3773 y Fu(k)p Fy(f)10 b Fu(k)2337 3831 y Fw(L)2419 3774 y Fr(2)p Fs(n)p 2395 3782 117 3 v 2395 3817 a(n)p Ft(\000)p Fr(3)2526 3831 y Fq(+)2610 3773 y Fu(\024)25 b Fy(C)7 b Fu(k)p Fy(f)j Fu(k)2923 3831 y Fw(L)3005 3774 y Fr(2)p Fs(n)p 2981 3782 V 2981 3817 a(n)p Ft(\000)p Fr(3)3112 3831 y Fq(+)-47 3963 y Fz(as)30 b(desired.)p 3879 3963 4 62 v 3883 3905 55 4 v 3883 3963 V 3937 3963 4 62 v -47 4151 a Fl(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(4.7.)46 b Fz(The)23 b(result)f(of)i(Lemma)g(4.7)g(is)f(con)m(tained)h (in)e(Ginibre-Moulin)e([GM)q(])j(and)g(can)h(b)s(e)f(traced)-47 4263 y(to)31 b(the)f(earlier)g(w)m(ork)g(of)h(Kato)g([Ka].)41 b(Here)31 b(w)m(e)g(essen)m(tially)e(repro)s(duce)h(the)g(pro)s(of)g (in)f([GM)q(].)-47 4376 y(Consider)d(the)i(resolv)m(en)m(t)h Fy(R)q Fz(\()p Fy(\025)16 b Fz(+)g Fy(i\014)5 b Fz(\).)40 b(W)-8 b(e)30 b(shall)c(pro)m(v)m(e)j(that)g(it)f(is)f(con)m(tin)m (uous)h(\(in)f(fact,)i(H\177)-45 b(older)28 b(con)m(tin)m(uous\))g(in) -47 4489 y(the)21 b(upp)s(er)d(half-plane)p 763 4416 66 4 v 20 w Fv(C)828 4503 y Fq(+)887 4489 y Fz(.)38 b(The)20 b(same)h(statemen)m(t)i(also)e(holds)e(for)h Fy(R)q Fz(\()p Fy(\025)q Fu(\000)q Fy(i\014)5 b Fz(\).)39 b(W)-8 b(e)21 b(app)s(eal)f(to)i(the)f(represen)m(tation)-47 4602 y(\(4.1\))r(,)1308 4813 y Fy(R)1377 4827 y Fq(0)1417 4813 y Fz(\()p Fy(\025)f Fz(+)g Fy(i\014)5 b Fz(\))26 b(=)1891 4671 y Fx(1)1860 4689 y Fn(Z)1868 4949 y Fq(0)1966 4813 y Fy(e)2008 4775 y Fw(i)p Fq(\()p Fw(\025)p Fq(+)p Fw(i\014)s Fq(\))p Fw(t)2280 4813 y Fy(e)2322 4775 y Fw(itH)2429 4784 y Fr(0)2484 4813 y Fy(dt:)1903 5472 y Fz(24)p eop %%Page: 25 25 25 24 bop -47 274 a Fz(Therefore,)43 b(using)c(the)h(inequalit)m(y)f Fu(j)p Fy(e)1316 241 y Fw(i)p Fq(\()p Fw(\025)1408 250 y Fr(2)1444 241 y Fq(+)p Fw(i\014)1563 250 y Fr(2)1597 241 y Fq(\))p Fw(t)1681 274 y Fu(\000)27 b Fy(e)1821 241 y Fw(i)p Fq(\()p Fw(\025)1913 250 y Fr(1)1948 241 y Fq(+)p Fw(i\014)2067 250 y Fr(1)2101 241 y Fq(\))p Fw(t)2158 274 y Fu(j)43 b(\024)e Fz(min)o(\(2)p Fy(;)15 b Fz(\()p Fu(j)p Fy(\025)2723 288 y Fq(2)2791 274 y Fu(\000)27 b Fy(\025)2942 288 y Fq(1)2981 274 y Fu(j)g Fz(+)g Fu(j)p Fy(\014)3207 288 y Fq(2)3274 274 y Fu(\000)g Fy(\014)3423 288 y Fq(1)3463 274 y Fu(j)p Fz(\))p Fy(t)p Fz(\))41 b(and)f(the)-47 407 y(em)m(b)s(edding)28 b Fy(L)508 337 y Fr(2)p Fs(n)p 484 349 117 3 v 484 390 a(n)p Ft(\000)p Fr(2)611 364 y Fq(+)670 407 y Fz(\()p Fv(R)765 374 y Fw(n)818 407 y Fz(\))e Fy(,)-15 b Fu(!)25 b Fy(L)1067 374 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)1311 407 y Fz(\()p Fv(R)1406 374 y Fw(n)1459 407 y Fz(\),)31 b(w)m(e)g(obtain)f(for)g(arbitrary)f Fy(\025)2546 421 y Fq(1)2585 407 y Fy(;)15 b(\025)2678 421 y Fq(2)2743 407 y Fu(2)25 b Fv(R)39 b Fz(and)30 b Fy(\014)3153 421 y Fq(1)3193 407 y Fy(;)15 b(\014)3284 421 y Fq(2)3349 407 y Fu(2)25 b Fv(R)3495 421 y Fq(+)3560 407 y Fz(,)12 696 y Fu(k)p Fz(\()p Fy(R)161 710 y Fq(0)201 696 y Fz(\()p Fy(\025)289 710 y Fq(2)349 696 y Fz(+)20 b Fy(i\014)522 710 y Fq(2)562 696 y Fz(\))h Fu(\000)f Fy(R)q Fz(\()p Fy(\025)867 710 y Fq(1)927 696 y Fz(+)g Fy(i\014)1100 710 y Fq(1)1140 696 y Fz(\)\))p Fy(f)10 b Fu(k)1310 716 y Fw(L)1358 697 y Fr(2)p Fs(;)p Ft(\000)p Fr(1)p Ft(\000)p Fs(\015)1744 696 y Fu(\024)1870 554 y Fx(1)1840 572 y Fn(Z)1847 832 y Fq(0)1946 696 y Fu(k)p Fy(e)2033 658 y Fw(itH)2140 667 y Fr(0)2180 696 y Fy(f)g Fu(k)2280 716 y Fw(L)2328 697 y Fr(2)p Fs(;)p Ft(\000)p Fr(1)p Ft(\000)p Fs(\015)2578 696 y Fz(min)n(\(2)p Fy(;)15 b Fz(\()p Fu(j)p Fy(\025)2962 710 y Fq(2)3024 696 y Fu(\000)20 b Fy(\025)3168 710 y Fq(1)3207 696 y Fu(j)h Fz(+)e Fu(j)p Fy(\014)3419 710 y Fq(2)3480 696 y Fu(\000)h Fy(\014)3622 710 y Fq(1)3662 696 y Fu(j)p Fz(\))p Fy(t)p Fz(\))15 b Fy(dt)1744 1054 y Fu(\024)1870 912 y Fx(1)1840 930 y Fn(Z)1847 1190 y Fq(0)1946 1054 y Fu(k)p Fy(e)2033 1017 y Fw(itH)2140 1026 y Fr(0)2180 1054 y Fy(f)10 b Fu(k)2280 1112 y Fw(L)2362 1055 y Fr(2)p Fs(n)p 2338 1063 V 2338 1098 a(n)p Ft(\000)p Fr(2)2464 1074 y(+)2550 1054 y Fz(min)n(\(2)p Fy(;)15 b Fz(\()p Fu(j)p Fy(\025)2934 1068 y Fq(2)2996 1054 y Fu(\000)k Fy(\025)3139 1068 y Fq(1)3179 1054 y Fu(j)h Fz(+)g Fu(j)p Fy(\014)3391 1068 y Fq(2)3452 1054 y Fu(\000)g Fy(\014)3594 1068 y Fq(1)3633 1054 y Fu(j)p Fz(\))p Fy(t)p Fz(\))15 b Fy(dt:)-3928 b Fz(\(4.20\))-47 1364 y(W)-8 b(e)29 b(no)m(w)g(recall)e(that)i(in)e (addition)f(to)j(the)g(Stric)m(hartz)f(estimates)h(\(4.13\))h(the)f (semigroup)e Fy(e)3162 1331 y Fw(itH)3269 1340 y Fr(0)3337 1364 y Fz(also)h(v)m(eri\014es)g(the)-47 1477 y(disp)s(ersiv)m(e)g (estimates)1052 1627 y Fu(k)p Fy(e)1139 1590 y Fw(itH)1246 1599 y Fr(0)1286 1627 y Fy(f)10 b Fu(k)1386 1641 y Fw(L)1434 1623 y Fs(p)1500 1627 y Fu(\024)1699 1565 y Fy(C)1764 1579 y Fw(p)p 1606 1607 292 4 v 1606 1722 a Fy(t)1639 1677 y Fw(n)p Fq(\()1719 1650 y Fr(1)p 1719 1662 31 3 v 1719 1703 a(2)1759 1677 y Fx(\000)1825 1650 y Fr(1)p 1824 1662 33 3 v 1824 1703 a Fs(p)1866 1677 y Fq(\))1907 1627 y Fu(k)p Fy(f)g Fu(k)2052 1657 y Fw(L)2100 1638 y Fs(p)2132 1624 y Ft(0)2163 1627 y Fy(;)197 b Fu(8)p Fy(p)25 b Fu(2)g Fz([2)p Fy(;)15 b Fu(1)p Fz(])p Fy(:)-47 1880 y Fz(Inserting)29 b(this)g(b)s(ound)f(in)m(to)j(\(4.20\))h(and)e (in)m(v)m(oking)g(the)g(em)m(b)s(edding)f Fy(L)2458 1847 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)2673 1880 y Fy(,)-15 b Fu(!)25 b Fy(L)2894 1810 y Fr(2)p Fs(n)p 2871 1822 116 3 v 2871 1863 a(n)p Fr(+2)2996 1837 y Fx(\000)3055 1880 y Fz(,)31 b(w)m(e)f(infer)f(that)-32 2186 y Fu(k)p Fz(\()p Fy(R)117 2200 y Fq(0)157 2186 y Fz(\()p Fy(\025)245 2200 y Fq(2)305 2186 y Fz(+)20 b Fy(i\014)478 2200 y Fq(2)518 2186 y Fz(\))h Fu(\000)f Fy(R)q Fz(\()p Fy(\025)823 2200 y Fq(1)882 2186 y Fz(+)g Fy(i\014)1055 2200 y Fq(1)1096 2186 y Fz(\)\))p Fy(f)10 b Fu(k)1266 2205 y Fw(L)1314 2187 y Fr(2)p Fs(;)p Ft(\000)p Fr(1)p Ft(\000)p Fs(\015)1559 2186 y Fo(.)1683 2044 y Fw(M)1655 2062 y Fn(Z)1663 2322 y Fq(0)1771 2124 y Fz(\()p Fu(j)p Fy(\025)1884 2138 y Fq(2)1944 2124 y Fu(\000)20 b Fy(\025)2088 2138 y Fq(1)2128 2124 y Fu(j)g Fz(+)g Fu(j)p Fy(\014)2340 2138 y Fq(2)2400 2124 y Fu(\000)g Fy(\014)2542 2138 y Fq(1)2582 2124 y Fu(j)p Fz(\))p Fy(t)p Fz(\))p 1771 2165 940 4 v 2177 2248 a Fy(t)2210 2222 y Fq(1+)2736 2186 y Fy(dt)15 b Fu(k)p Fy(f)10 b Fu(k)2976 2243 y Fw(L)3058 2187 y Fr(2)p Fs(n)p 3034 2195 116 3 v 3034 2229 a(n)p Fr(+2)3160 2206 y Ft(\000)3236 2186 y Fz(+)3358 2044 y Fx(1)3327 2062 y Fn(Z)3315 2325 y Fw(M)3484 2124 y Fz(2)p 3443 2165 128 4 v 3443 2248 a Fy(t)3476 2222 y Fq(1+)3595 2186 y Fu(k)p Fy(f)g Fu(k)3740 2243 y Fw(L)3822 2187 y Fr(2)p Fs(n)p 3799 2195 116 3 v 3799 2229 a(n)p Fr(+2)3924 2206 y Ft(\000)1559 2499 y Fo(.)1655 2371 y Fn(\022)1722 2499 y Fz(\()p Fu(j)p Fy(\025)1835 2513 y Fq(2)1895 2499 y Fu(\000)20 b Fy(\025)2039 2513 y Fq(1)2079 2499 y Fu(j)g Fz(+)g Fu(j)p Fy(\014)2291 2513 y Fq(2)2351 2499 y Fu(\000)g Fy(\014)2493 2513 y Fq(1)2533 2499 y Fu(j)p Fz(\))p Fy(M)2691 2461 y Fq(1)p Fx(\000)2806 2499 y Fz(+)g(2)p Fy(M)3040 2461 y Fq(0)p Fx(\000)3135 2371 y Fn(\023)3202 2499 y Fu(k)p Fy(f)10 b Fu(k)3347 2518 y Fw(L)3395 2499 y Fr(2)p Fs(;)p Fr(1+)p Fs(\015)-47 2747 y Fz(for)30 b(some)h(constan)m(t)g Fy(M)36 b(>)25 b Fz(0.)41 b(Cho)s(osing)29 b Fy(M)35 b Fz(=)25 b(\()p Fu(j)p Fy(\025)1746 2761 y Fq(2)1806 2747 y Fu(\000)20 b Fy(\025)1950 2761 y Fq(1)1990 2747 y Fu(j)g Fz(+)g Fu(j)p Fy(\014)2202 2761 y Fq(2)2262 2747 y Fu(\000)g Fy(\014)2404 2761 y Fq(1)2444 2747 y Fu(j)p Fz(\))2504 2714 y Fx(\000)p Fq(1)2629 2747 y Fz(w)m(e)31 b(\014nally)d(conclude)i(that)428 2939 y Fu(k)p Fz(\()p Fy(R)577 2953 y Fq(0)617 2939 y Fz(\()p Fy(\025)705 2953 y Fq(2)765 2939 y Fz(+)20 b Fy(i\014)938 2953 y Fq(2)978 2939 y Fz(\))h Fu(\000)f Fy(R)q Fz(\()p Fy(\025)1283 2953 y Fq(1)1342 2939 y Fz(+)g Fy(i\014)1515 2953 y Fq(1)1556 2939 y Fz(\)\))p Fy(f)10 b Fu(k)1726 2958 y Fw(L)1774 2939 y Fr(2)p Fs(;)p Ft(\000)p Fr(1)p Ft(\000)p Fs(\015)38 b Fo(.)25 b Fz(\()p Fu(j)p Fy(\025)2228 2953 y Fq(2)2288 2939 y Fu(\000)20 b Fy(\025)2432 2953 y Fq(1)2472 2939 y Fu(j)g Fz(+)g Fu(j)p Fy(\014)2684 2953 y Fq(2)2744 2939 y Fu(\000)g Fy(\014)2886 2953 y Fq(1)2926 2939 y Fu(j)p Fz(\))2986 2901 y Fq(0+)3081 2939 y Fu(k)p Fy(f)10 b Fu(k)3226 2958 y Fw(L)3274 2939 y Fr(2)p Fs(;)p Fr(1+)p Fs(\015)j Fy(;)-47 3130 y Fz(as)30 b(claimed.)p 3879 3130 4 62 v 3883 3072 55 4 v 3883 3130 V 3937 3130 4 62 v -47 3315 a Fl(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(4.8.)46 b Fz(Lemma)34 b(4.6)h(implies)c(the)k(b)s(oundedness)c(of)j Fy(G)e Fz(:)g Fy(L)2637 3282 y Fq(2)p Fw(;)p Fq(1+)p Fw(\015)2858 3315 y Fu(!)f Fy(L)3042 3282 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)3321 3315 y Fz(for)i(an)m(y)i(p)s (ositiv)m(e)-47 3428 y Fy(\015)5 b Fz(.)40 b(Therefore,)28 b(since)g Fu(j)p Fy(V)20 b Fz(\()p Fy(x)p Fz(\))p Fu(j)27 b(\024)e Fy(C)7 b Fz(\(1)16 b(+)f Fu(j)p Fy(x)p Fu(j)1450 3395 y Fq(2)1490 3428 y Fz(\))1525 3395 y Fx(\000)p Fq(1)p Fx(\000)p Fw(")1707 3428 y Fz(,)29 b(the)f(p)s(oten)m(tial)g Fy(V)48 b Fz(maps)28 b(the)g(space)h Fy(L)3091 3395 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)3363 3428 y Fz(in)m(to)g Fy(L)3608 3395 y Fq(2)p Fw(;)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+2)p Fw(")3920 3428 y Fz(.)-47 3541 y(Th)m(us,)g(using)g(the)i(second)f(conclusion)f(of)i(Lemma)f (4.6,)i(w)m(e)f(obtain)f(that)1388 3732 y Fy(GV)46 b Fz(:)25 b Fy(L)1670 3695 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)1940 3732 y Fu(!)g Fy(L)2118 3695 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+2)p Fw(")2485 3732 y Fy(:)-2557 b Fz(\(4.21\))-47 3924 y(pro)m(vided)29 b(that)i(0)25 b Fy(<)g Fu(\000)p Fy(\015)g Fz(+)20 b(2)p Fy(")26 b(<)1143 3888 y Fq(1)p 1143 3903 36 4 v 1143 3955 a(2)1189 3924 y Fz(.)94 4037 y(Since)31 b Fu(\000)p Fz(\001\()p Fy(GV)20 b Fz(\))28 b(=)f Fy(V)52 b Fz(and)31 b Fy(V)21 b Fz(,)32 b(of)g(course,)g(maps)g Fy(L)1946 4004 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)2217 4037 y Fu(!)c Fy(L)2398 4004 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)2642 4037 y Fz(,)k(w)m(e)h(conclude)e(that)h Fy(GV)52 b Fz(tak)m(es)34 b(the)-47 4150 y(space)e Fy(L)259 4117 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)536 4150 y Fz(to)h Fy(H)732 4117 y Fq(2)725 4177 y(l)o Fw(oc)813 4150 y Fz(.)46 b(The)31 b(compactness)i(of)g Fy(GV)52 b Fz(on)32 b Fy(L)2072 4117 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)2349 4150 y Fz(then)g(follo)m(ws)f(from)g(the) i(observ)-5 b(ation)32 b(ab)s(o)m(v)m(e)-47 4262 y(and)25 b(the)i(extra)g(2)p Fy(")g Fz(deca)m(y)g(at)g(in\014nit)m(y)d (established)h(in)g(\(4.21\))r(.)39 b(W)-8 b(e)28 b(infer)d(that)h Fy(I)20 b Fz(+)12 b Fy(R)2910 4276 y Fq(0)2949 4262 y Fy(V)46 b Fz(is)25 b(a)i(F)-8 b(redholm)26 b(op)s(erator)-47 4375 y(on)k Fy(L)141 4342 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)416 4375 y Fz(and)f(it)h(is)g(therefore)g(in)m(v)m (ertible)f(i\013)h(its)g(n)m(ull)e(space)j(is)e(empt)m(y)-8 b(.)-47 4488 y(Let)31 b Fy(\036)f Fz(b)s(e)g(an)g Fy(L)512 4455 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)787 4488 y Fz(solution)f(of)h(the)h(equation)1683 4679 y Fy(\036)21 b Fz(+)f Fy(GV)g(\036)26 b Fz(=)f(0)-2261 b(\(4.22\))-47 4871 y(First)30 b(w)m(e)g(observ)m(e)i(that)f(b)m(y)g (\(4.21\))i(function)c Fy(\036)p Fz(,)i(in)e(fact,)j(b)s(elongs)d(to)i (the)g(space)g Fy(L)2896 4838 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+2)p Fw(")3293 4871 y Fz(for)f(some)h Fy(\015)g Fz(:)40 b(0)26 b Fy(<)-47 4984 y(\015)46 b(<)41 b Fz(2)p Fy(")p Fz(.)70 b(It)39 b(then)h(follo)m(ws)f(that)h Fy(V)21 b(\036)41 b Fu(2)f Fy(L)1510 4951 y Fq(2)p Fw(;)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+4)p Fw(")1823 4984 y Fz(.)69 b(Lemma)40 b(4.6)h(then)e(implies)e(that,)43 b(as)d(long)g(as)g(4)p Fy(")i(<)3874 4948 y Fq(1)p 3874 4963 V 3874 5015 a(2)3920 4984 y Fz(,)-47 5097 y(also)33 b Fy(GV)21 b(\036)30 b Fu(2)f Fy(L)519 5064 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+4)p Fw(")919 5097 y Fz(and,)k(using)g(\(4.22\)) j(again)d(w)m(e)g(ha)m(v)m(e)i(that)f Fy(\036)29 b Fu(2)h Fy(L)2662 5064 y Fq(2)p Fw(;)p Fx(\000)p Fq(1)p Fx(\000)p Fw(\015)t Fq(+4)p Fw(")3029 5097 y Fz(.)49 b(W)-8 b(e)34 b(can)g(con)m(tin)m(ue)f(this)-47 5223 y(argumen)m(t)e(and)e(obtain)h (that)h Fy(\036)26 b Fu(2)e Fy(L)1243 5187 y Fq(2)p Fw(;)p Fx(\000)1363 5160 y Fr(1)p 1363 5172 31 3 v 1363 5213 a(2)1403 5187 y Fx(\000)p Fw(\013)1538 5223 y Fz(for)30 b(an)m(y)h Fy(\013)26 b(>)f Fz(0.)1903 5472 y(25)p eop %%Page: 26 26 26 25 bop -47 278 a Fz(Applying)30 b Fu(\000)p Fz(\001)j(to)g(b)s(oth)g (sides)e(of)i(the)h(equation)e(\(4.22\))k(w)m(e)d(conclude)f(that)i Fy(\036)f Fz(is)f(an)h Fy(L)3069 242 y Fq(2)p Fw(;)p Fx(\000)3189 215 y Fr(1)p 3188 227 31 3 v 3188 268 a(2)3229 242 y Fx(\000)p Fw(\013)3366 278 y Fz(solution)f(of)h(the)-47 391 y(equation)1633 504 y Fu(\000)p Fz(\001)p Fy(\036)20 b Fz(+)g Fy(V)g(\036)26 b Fz(=)f(0)p Fy(;)-47 670 y Fz(and)g(th)m(us)h (either)f(an)h(eigenfunction)f(or)h(a)g(resonance)h(corresp)s(onding)d (to)j Fy(\025)e Fz(=)g(0.)40 b(Since)25 b(w)m(e)h(assumed)g(that)g Fy(\025)f Fz(=)g(0)-47 783 y(is)k(a)i(regular)f(p)s(oin)m(t,)f Fy(\036)i Fz(m)m(ust)f(b)s(e)g(iden)m(tically)e(zero)j(and)f(the)h(n)m (ull)d(space)j(of)f Fy(I)e Fz(+)19 b Fy(R)2842 797 y Fq(0)2882 783 y Fy(V)50 b Fz(is)30 b(empt)m(y)-8 b(.)p 3879 783 4 62 v 3883 725 55 4 v 3883 783 V 3937 783 4 62 v -47 1070 a FA(5)134 b(Time)46 b(dep)t(enden)l(t)f(p)t(oten)l (tials:)61 b(Reduction)46 b(to)f(oscillatory)i(in)l(tegrals)-47 1273 y Fm(De\014nition)35 b(5.1.)46 b Fl(L)-5 b(et)39 b Fy(Y)59 b Fl(b)-5 b(e)38 b(the)h(norme)-5 b(d)41 b(sp)-5 b(ac)g(e)40 b(of)f(me)-5 b(asur)g(able)40 b(functions)f Fy(V)21 b Fz(\()p Fy(t;)15 b(x)p Fz(\))39 b Fl(on)g Fv(R)3261 1240 y Fq(3)3345 1273 y Fl(that)h(satisfy)f(the)-47 1386 y(fol)5 b(lowing)39 b(pr)-5 b(op)g(erties:)55 b Fy(t)34 b Fu(7!)h(k)p Fy(V)21 b Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)1335 1436 y Fw(L)1393 1386 y Fr(3)p 1394 1394 31 3 v 1394 1429 a(2)1439 1436 y Fq(\()p Fp(R)1514 1417 y Fr(3)1549 1436 y Fq(\))1615 1386 y Fu(2)34 b Fy(L)1772 1353 y Fx(1)1847 1386 y Fz(\()p Fv(R)s Fz(\))44 b Fl(and)39 b(for)f(a.e.)g Fy(x)d Fu(2)f Fv(R)2777 1353 y Fq(3)2860 1386 y Fl(the)39 b(function)f Fy(t)c Fu(7!)h Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))39 b Fl(is)-47 1531 y(in)e Fu(S)131 1498 y Fx(0)154 1531 y Fz(\()p Fv(R)s Fz(\))q Fl(,)44 b(the)38 b(sp)-5 b(ac)g(e)39 b(of)f(temp)-5 b(er)g(e)g(d)40 b(distributions.)58 b(Mor)-5 b(e)g(over,)39 b(the)f(F)-7 b(ourier)39 b(tr)-5 b(ansform)40 b(of)e(this)g(distribution,)-47 1644 y(which)33 b(we)g(denote)g(by)g Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)11 b(;)k(x)p Fz(\))p Fl(,)33 b(is)f(a)h(\(c)-5 b(omplex\))35 b(me)-5 b(asur)g(e)34 b(whose)g(norm)f(satis\014es)1378 1896 y Fz(sup)1363 1980 y Fw(y)r Fx(2)p Fp(R)1495 1961 y Fr(3)1545 1772 y Fn(Z)1596 1979 y Fp(R)1644 1960 y Fr(3)1707 1835 y Fu(k)p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)10 b(;)15 b(x)p Fz(\))p Fu(k)2083 1849 y Fx(M)p 1707 1875 466 4 v 1809 1958 a Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)s Fu(j)2198 1896 y Fy(dx)25 b(<)g Fu(1)p Fy(:)-2581 b Fz(\(5.1\))-47 2218 y Fl(The)32 b(norm)h(in)f Fy(Y)51 b Fl(is)32 b(the)g(sum)g(of)g (the)h(expr)-5 b(ession)33 b(on)f(the)h(left-hand)g(side)f(of)51 b Fz(\(5.1\))34 b Fl(and)e(the)h(norm)g(in)e Fy(L)3655 2185 y Fx(1)3655 2241 y Fw(t)3730 2218 y Fz(\()p Fy(L)3837 2134 y Fr(3)p 3837 2146 31 3 v 3837 2187 a(2)3827 2230 y Fw(x)3882 2218 y Fz(\))p Fl(.)-47 2406 y Fz(In)e(what)i(follo)m(ws)e (w)m(e)i(study)e(the)i(Sc)m(hr\177)-45 b(odinger)29 b(equation)1496 2610 y Fy(i@)1575 2624 y Fw(t)1606 2610 y Fy( )23 b Fu(\000)d Fz(\001)p Fy( )k Fz(+)c Fy(V)g Fz(\()p Fy(t;)15 b(x)p Fz(\))p Fy( )29 b Fz(=)c(0)p Fy(;)-2598 b Fz(\(5.2\))1496 2748 y Fy( )s Fu(j)1583 2762 y Fw(t)p Fq(=)p Fw(s)1701 2748 y Fz(\()p Fy(x)p Fz(\))26 b(=)f Fy( )2004 2762 y Fw(s)2041 2748 y Fz(\()p Fy(x)p Fz(\))-47 2952 y(for)31 b(p)s(oten)m(tials)h Fy(V)47 b Fu(2)28 b Fy(Y)51 b Fz(and)32 b(with)e(initial)f(data)k Fy( )1727 2966 y Fw(s)1792 2952 y Fu(2)27 b Fy(L)1942 2919 y Fq(2)1981 2952 y Fz(\()p Fv(R)2076 2919 y Fq(3)2122 2952 y Fz(\).)45 b(An)31 b(in)m(teresting)h (case)g(is)f Fy(V)21 b Fz(\()p Fy(t;)15 b(x)p Fz(\))28 b(=)g(cos\()p Fy(t)p Fz(\))15 b Fy(V)21 b Fz(\()p Fy(x)p Fz(\))-47 3078 y(where)33 b Fy(V)50 b Fu(2)30 b Fy(L)485 3015 y Fr(3)p 485 3027 V 485 3068 a(2)563 3078 y Fz(satis\014es)j(sup) 1038 3100 y Fw(y)1095 3005 y Fn(R)1138 3110 y Fp(R)1186 3091 y Fr(3)1249 3033 y Fx(j)p Fw(V)16 b Fq(\()p Fw(x)p Fq(\))p Fx(j)p 1249 3057 191 4 v 1259 3111 a(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)1465 3078 y Fy(dx)31 b(<)f Fu(1)p Fz(.)50 b(Because)35 b(of)e(the)h(limited)d(regularit)m(y)i(of)h(p)s (oten)m(tials)e(in)h Fy(Y)20 b Fz(,)-47 3204 y(w)m(e)31 b(de\014ne)e(\(w)m(eak\))j(solutions)d Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )1344 3218 y Fw(s)1412 3204 y Fz(of)37 b(\(5.2\))c(via)d(Duhamel's)f(form)m(ula:)735 3465 y Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )1052 3479 y Fw(s)1115 3465 y Fz(=)25 b Fy(e)1253 3427 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(H)1502 3436 y Fr(0)1542 3465 y Fy( )1601 3479 y Fw(s)1658 3465 y Fz(+)20 b Fy(i)1795 3341 y Fn(Z)1887 3367 y Fw(t)1846 3547 y(s)1931 3465 y Fy(e)1973 3427 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2137 3436 y Fr(1)2173 3427 y Fq(\))p Fw(H)2258 3436 y Fr(0)2297 3465 y Fy(V)g Fz(\()p Fy(s)2448 3479 y Fq(1)2488 3465 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(U)10 b Fz(\()p Fy(s)2738 3479 y Fq(1)2778 3465 y Fy(;)15 b(s)p Fz(\))p Fy( )2955 3479 y Fw(s)3008 3465 y Fy(ds)3098 3479 y Fq(1)3137 3465 y Fy(:)-3209 b Fz(\(5.3\))-47 3715 y(In)26 b(the)h(follo)m(wing)e(lemma)i(w)m(e)g(sho)m(w)g(b)m(y)f(means) h(of)g(Keel's)g(and)f(T)-8 b(ao's)28 b(endp)s(oin)m(t)d(Stric)m(hartz)i (estimate)h([KT)o(])f(that)-47 3828 y(suc)m(h)f(w)m(eak)i(solutions)e (exist)h(and)f(are)i(unique)d(pro)m(vided)h(the)h(p)s(oten)m(tial)f(is) g(small)g(in)g(an)h(appropriate)f(sense.)39 b(The)-47 3941 y(pro)s(of)29 b(is)h(presen)m(ted)g(only)g(in)f Fv(R)1054 3908 y Fq(3)1100 3941 y Fz(,)h(but)g(it)g(carries)g(o)m(v)m (er)h(to)h(an)m(y)e(dimension)e Fy(n)d Fu(\025)g Fz(3.)41 b(W)-8 b(e)32 b(set)1325 4145 y Fy(X)h Fz(=)25 b Fy(L)1591 4107 y Fx(1)1591 4167 y Fw(t)1666 4145 y Fz(\()p Fy(L)1763 4107 y Fq(2)1763 4167 y Fw(x)1807 4145 y Fz(\()p Fv(R)1902 4107 y Fq(3)1947 4145 y Fz(\)\))c Fu(\\)f Fy(L)2181 4107 y Fq(2)2181 4167 y Fw(t)2220 4145 y Fz(\()p Fy(L)2317 4107 y Fq(6)2317 4167 y Fw(x)2361 4145 y Fz(\()p Fv(R)2456 4107 y Fq(3)2502 4145 y Fz(\)\))-2619 b(\(5.4\))-47 4349 y(and)30 b(de\014ne)f Fy(H)468 4363 y Fq(0)532 4349 y Fz(=)c Fu(\000)p Fz(\001)30 b(to)h(b)s(e)f(the)h(unp)s(erturb)s(ed)26 b(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erator)i(with)e(ev)m(olution)h Fy(e)3230 4316 y Fw(itH)3337 4325 y Fr(0)3377 4349 y Fz(.)-47 4537 y Fm(Lemma)i(5.2.)47 b Fl(Assume)32 b(that)i(the)f(p)-5 b(otential)35 b Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))33 b Fl(satis\014es)h(the)f(smal)5 b(lness)34 b(assumption)1155 4824 y Fu(k)p Fy(V)21 b Fu(k)1319 4890 y Fw(L)1367 4867 y Ft(1)1367 4910 y Fs(t)1432 4890 y Fw(L)1490 4825 y Fr(3)p 1490 4833 31 3 v 1490 4867 a(2)1480 4901 y Fs(x)1564 4824 y Fz(=)k(sup)1669 4902 y Fw(t)p Fx(2)p Fp(R)1812 4696 y Fn(\022)1894 4700 y(Z)1945 4906 y Fp(R)1993 4887 y Fr(3)2047 4824 y Fu(j)p Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))p Fu(j)2375 4759 y Fr(3)p 2376 4771 V 2376 4812 a(2)2421 4696 y Fn(\023)2498 4691 y Fr(2)p 2498 4703 V 2498 4744 a(3)2568 4824 y Fy(<)25 b(c)2703 4838 y Fq(0)-47 4824 y Fz(\(5.5\))1903 5472 y(26)p eop %%Page: 27 27 27 26 bop -47 274 a Fl(for)28 b(some)g(su\016ciently)f(smal)5 b(l)28 b(c)-5 b(onstant)29 b Fy(c)1405 288 y Fq(0)1470 274 y Fy(>)c Fz(0)p Fl(.)41 b(Then)27 b(for)h(any)g Fy(s)d Fu(2)g Fv(R)35 b Fl(and)29 b(any)e Fy( )2865 288 y Fw(s)2928 274 y Fu(2)e Fy(L)3076 241 y Fq(2)3076 296 y Fw(x)3146 274 y Fl(ther)-5 b(e)29 b(exists)e(a)h(unique)-47 387 y(we)-5 b(ak)31 b(solution)h Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )828 401 y Fw(s)896 387 y Fl(of)50 b Fz(\(5.3\))32 b Fl(with)g(the)f(pr)-5 b(op)g(erty)33 b(that)f Fy(U)10 b Fz(\()p Fu(\001)p Fy(;)15 b(s)p Fz(\))p Fy( )2428 401 y Fw(s)2491 387 y Fu(2)25 b Fy(X)38 b Fl(and)32 b(so)f(that)h Fy(t)25 b Fu(7!)g(h)p Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )3689 401 y Fw(s)3727 387 y Fy(;)g(g)s Fu(i)32 b Fl(is)-47 500 y(c)-5 b(ontinuous)33 b(for)h(any)f Fy(g)c Fu(2)c Fy(L)950 467 y Fq(2)989 500 y Fz(\()p Fv(R)1084 467 y Fq(3)1129 500 y Fz(\))p Fl(.)42 b(Mor)-5 b(e)g(over,)34 b(for)f(any)h(such)e Fy(g)k Fl(and)e(any)f Fy(t)25 b(>)g(s)p Fl(,)346 747 y Fu(h)p Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )698 761 y Fw(s)736 747 y Fy(;)g(g)s Fu(i)27 b Fz(=)1021 634 y Fx(1)990 661 y Fn(X)980 855 y Fw(m)p Fq(=0)1240 747 y Fy(i)1271 710 y Fw(m)1409 624 y Fn(Z)1515 747 y Fu(\001)15 b(\001)g(\001)1636 624 y Fn(Z)1308 885 y Fw(s)p Fx(\024)p Fw(s)1429 893 y Fs(m)1487 885 y Fx(\024)p Fw(::)p Fx(\024)p Fw(s)1670 894 y Fr(1)1703 885 y Fx(\024)p Fw(t)1783 747 y Fu(h)p Fy(e)1860 710 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2024 719 y Fr(1)2060 710 y Fq(\))p Fw(H)2145 719 y Fr(0)2184 747 y Fy(V)20 b Fz(\()p Fy(s)2335 761 y Fq(1)2375 747 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)2517 710 y Fw(i)p Fq(\()p Fw(s)2601 719 y Fr(1)2637 710 y Fx(\000)p Fw(s)2725 719 y Fr(2)2759 710 y Fq(\))p Fw(H)2844 719 y Fr(0)2883 747 y Fy(V)20 b Fz(\()p Fy(s)3034 761 y Fq(2)3074 747 y Fy(;)15 b Fu(\001)p Fz(\))g Fy(:)g(:)g(:)2155 1026 y(V)20 b Fz(\()p Fy(s)2306 1040 y Fw(m)2373 1026 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)2515 989 y Fw(i)p Fq(\()p Fw(s)2599 997 y Fs(m)2659 989 y Fx(\000)p Fw(s)p Fq(\))p Fw(H)2832 998 y Fr(0)2886 1026 y Fy( )2945 1040 y Fw(s)2982 1026 y Fy(;)g(g)s Fu(i)26 b Fy(ds)3219 1040 y Fq(1)3274 1026 y Fy(:)15 b(:)g(:)h(ds)3485 1040 y Fw(m)-47 1026 y Fz(\(5.6\))-47 1204 y Fl(wher)-5 b(e)41 b(the)f(series)g(c)-5 b(onver)g(ges)40 b(absolutely.)65 b(In)40 b(the)g(str)-5 b(ong)41 b(sense,)g(i.e.,)g(without)g(the)f(p)-5 b(airing)41 b(against)g Fy(g)s Fl(,)h(this)-47 1317 y(r)-5 b(epr)g(esentation)34 b(holds)g(in)e(the)g(sense)g(of)g(norm)h(c)-5 b(onver)g(genc)g(e)33 b(in)f(the)g(sp)-5 b(ac)g(e)33 b Fy(X)40 b Fl(\(and)33 b(thus)f(c)-5 b(an)33 b(only)f(b)-5 b(e)32 b(assume)-5 b(d)-47 1430 y(for)33 b(a.e.)g Fy(t)p Fl(\).)-47 1596 y(Pr)-5 b(o)g(of.)47 b Fz(F)-8 b(or)25 b(the)g(purp)s(oses)e(of)i(this)e(pro)s(of,)i(w)m(e)h(let)e Fy(F)39 b Fz(=)25 b Fy(F)13 b Fz(\()p Fy(t;)i(x)p Fz(\))25 b(b)s(e)f(a)h(function)f(of)g(\()p Fy(t;)15 b(x)p Fz(\))27 b Fu(2)d Fv(R)3186 1558 y Fq(1+3)3186 1620 y Fw(t)q(;x)3322 1596 y Fz(.)38 b(F)-8 b(or)26 b(simplicit)m(y)-8 b(,)-47 1709 y(w)m(e)30 b(often)h(write)e Fy(F)13 b Fz(\()p Fy(t)p Fz(\))31 b(for)f(the)h(function)d Fy(x)e Fu(7!)f Fy(F)13 b Fz(\()p Fy(t;)i(x)p Fz(\).)41 b(Recall)30 b(the)h(follo)m(wing)d (end-p)s(oin)m(t)h(Stric)m(hartz)h(estimates)-47 1822 y(for)44 b(the)h(op)s(erator)g Fy(H)734 1836 y Fq(0)818 1822 y Fz(pro)m(v)m(ed)g(b)m(y)f(Keel-T)-8 b(ao)46 b(in)e(an)m(y)h (dimension)d Fy(n)49 b Fu(\025)f Fz(3,)h(see)d([KT)o(]:)70 b(There)44 b(exists)g(some)-47 1976 y(dimensional)27 b(constan)m(t)32 b Fy(C)885 1990 y Fq(1)950 1976 y Fz(=)25 b Fy(C)1111 1990 y Fq(1)1150 1976 y Fz(\()p Fy(n)p Fz(\))31 b(so)f(that)h(for)f(all)g Fy(f)k Fu(2)25 b Fy(L)2107 1943 y Fq(2)2107 1999 y Fw(x)2181 1976 y Fz(and)30 b Fy(F)38 b Fu(2)25 b Fy(L)2602 1943 y Fq(2)2602 1999 y Fw(t)2641 1976 y Fy(L)2736 1885 y Fr(2)p Fs(n)p 2713 1897 116 3 v 2713 1938 a(n)p Fr(+2)2703 1988 y Fw(x)980 2155 y Fu(k)p Fy(e)1067 2117 y Fw(itH)1174 2126 y Fr(0)1214 2155 y Fy(f)10 b Fu(k)1314 2228 y Fw(L)1362 2205 y Fr(2)1362 2248 y Fs(t)1396 2228 y Fw(L)1478 2155 y Fr(2)p Fs(n)p 1454 2163 117 3 v 1454 2198 a(n)p Ft(\000)p Fr(2)1444 2239 y Fs(x)1614 2155 y Fu(\024)25 b Fy(C)1775 2169 y Fq(1)1815 2155 y Fu(k)p Fy(f)10 b Fu(k)1960 2174 y Fw(L)2008 2155 y Fr(2)2008 2191 y Fs(x)2051 2155 y Fy(;)-2123 b Fz(\(5.7\))980 2419 y Fu(k)1040 2295 y Fn(Z)1131 2322 y Fw(t)1091 2502 y(s)1176 2419 y Fy(e)1218 2382 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)1382 2391 y Fr(1)1417 2382 y Fq(\))p Fw(H)1502 2391 y Fr(0)1542 2419 y Fy(F)13 b Fz(\()p Fy(s)1691 2433 y Fq(1)1730 2419 y Fz(\))i Fy(ds)1870 2433 y Fq(1)1910 2419 y Fu(k)1955 2493 y Fw(L)2003 2470 y Fr(2)2003 2512 y Fs(t)2038 2493 y Fw(L)2120 2420 y Fr(2)p Fs(n)p 2096 2428 V 2096 2463 a(n)p Ft(\000)p Fr(2)2086 2503 y Fs(x)2256 2419 y Fu(\024)25 b Fy(C)2417 2433 y Fq(1)2457 2419 y Fu(k)p Fy(F)13 b Fu(k)2618 2493 y Fw(L)2666 2470 y Fr(2)2666 2512 y Fs(t)2701 2493 y Fw(L)2782 2420 y Fr(2)p Fs(n)p 2759 2428 116 3 v 2759 2463 a(n)p Fr(+2)2749 2503 y Fs(x)2893 2419 y Fy(:)-2965 b Fz(\(5.8\))-47 2649 y(Consider)28 b(the)j(op)s(erator)g Fu(K)925 2663 y Fw(s)992 2649 y Fz(de\014ned)e(b)m(y)1030 2879 y(\()p Fu(K)1134 2893 y Fw(s)1172 2879 y Fy(F)13 b Fz(\)\()p Fy(t;)i Fu(\001)p Fz(\))27 b(=)e Fy(i)1615 2755 y Fn(Z)1706 2781 y Fw(t)1666 2961 y(s)1751 2879 y Fy(e)1793 2841 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)1957 2850 y Fr(1)1992 2841 y Fq(\))p Fw(H)2077 2850 y Fr(0)2116 2879 y Fy(V)c Fz(\()p Fy(s)2268 2893 y Fq(1)2307 2879 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(F)e Fz(\()p Fy(s)2556 2893 y Fq(1)2596 2879 y Fy(;)i Fu(\001)p Fz(\))g Fy(ds)2801 2893 y Fq(1)2842 2879 y Fy(:)-47 3103 y Fz(Then)29 b(de\014nition)f(\(5.3\))k(tak)m(es)g(the)f(form)1232 3207 y Fn(\002)1270 3281 y Fz(\(1)21 b Fu(\000)f(K)1531 3295 y Fw(s)1568 3281 y Fz(\))p Fy(U)10 b Fz(\()p Fu(\001)p Fy(;)15 b(s)p Fz(\))p Fy( )1912 3295 y Fw(s)1950 3207 y Fn(\003)1988 3281 y Fz(\()p Fy(t)p Fz(\))26 b(=)f Fy(e)2255 3243 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(H)2504 3252 y Fr(0)2544 3281 y Fy( )2603 3295 y Fw(s)2640 3281 y Fy(:)-2712 b Fz(\(5.9\))-47 3459 y(Inequalit)m(y)25 b(\(5.8\))h(and)e(the)h(smallness)e(assumption)g(\(5.5\))j(imply)c (that)k(the)e(norm)g(of)h(the)g(op)s(erator)g Fu(K)3494 3473 y Fw(s)3556 3459 y Fz(:)41 b Fy(L)3684 3426 y Fq(2)3684 3482 y Fw(t)3723 3459 y Fy(L)3785 3426 y Fq(6)3785 3482 y Fw(x)3854 3459 y Fu(!)-47 3572 y Fy(L)15 3539 y Fq(2)15 3595 y Fw(t)54 3572 y Fy(L)116 3539 y Fq(6)116 3595 y Fw(x)190 3572 y Fz(satis\014es)567 3750 y Fu(kK)681 3764 y Fw(s)719 3750 y Fy(F)13 b Fu(k)835 3774 y Fw(L)883 3751 y Fr(2)883 3794 y Fs(t)918 3774 y Fw(L)966 3755 y Fr(6)966 3791 y Fs(x)1034 3750 y Fu(\024)25 b Fy(C)1195 3764 y Fq(1)1250 3750 y Fu(k)p Fy(V)36 b(F)13 b Fu(k)1500 3817 y Fw(L)1548 3794 y Fr(2)1548 3836 y Fs(t)1583 3817 y Fw(L)1641 3751 y Fr(6)p 1641 3759 31 3 v 1641 3794 a(5)1631 3827 y Fs(x)1715 3750 y Fu(\024)25 b Fy(C)1876 3764 y Fq(1)1930 3750 y Fu(k)p Fy(V)c Fu(k)2094 3817 y Fw(L)2142 3794 y Ft(1)2142 3836 y Fs(t)2207 3817 y Fw(L)2265 3751 y Fr(3)p 2265 3759 V 2265 3794 a(2)2255 3827 y Fs(x)2314 3750 y Fu(k)p Fy(F)13 b Fu(k)2475 3774 y Fw(L)2523 3751 y Fr(2)2523 3794 y Fs(t)2559 3774 y Fw(L)2607 3755 y Fr(6)2607 3791 y Fs(x)2675 3750 y Fu(\024)25 b Fy(C)2836 3764 y Fq(1)2891 3750 y Fy(c)2930 3764 y Fq(0)2969 3750 y Fu(k)p Fy(F)13 b Fu(k)3130 3774 y Fw(L)3178 3751 y Fr(2)3178 3794 y Fs(t)3214 3774 y Fw(L)3262 3755 y Fr(6)3262 3791 y Fs(x)3305 3750 y Fy(:)-3377 b Fz(\(5.10\))-47 3986 y(Moreo)m(v)m(er,)33 b(for)d(an)m(y)g Fy(g)f Fu(2)c Fy(L)908 3953 y Fq(2)947 3986 y Fz(\()p Fv(R)1042 3953 y Fq(3)1088 3986 y Fz(\),)433 4220 y Fu(jh)p Fz(\()p Fu(K)597 4234 y Fw(s)635 4220 y Fy(F)13 b Fz(\)\()p Fy(t)p Fz(\))p Fy(;)i(g)s Fu(ij)85 b Fz(=)1229 4089 y Fn(\014)1229 4143 y(\014)1229 4198 y(\014)1229 4252 y(\014)1259 4097 y(Z)1350 4123 y Fw(t)1310 4303 y(s)1380 4220 y Fu(h)p Fy(V)21 b Fz(\()p Fy(s)1567 4234 y Fq(1)1606 4220 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(F)e Fz(\()p Fy(s)1855 4234 y Fq(1)1895 4220 y Fy(;)i Fu(\001)p Fz(\))p Fy(;)g(e)2077 4183 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2296 4192 y Fr(1)2333 4183 y Fq(\))p Fw(H)2418 4192 y Fr(0)2457 4220 y Fy(g)s Fu(i)g Fy(ds)2643 4234 y Fq(1)2684 4089 y Fn(\014)2684 4143 y(\014)2684 4198 y(\014)2684 4252 y(\014)1075 4478 y Fu(\024)1229 4355 y Fn(Z)1320 4381 y Fw(t)1280 4561 y(s)1365 4478 y Fu(k)p Fy(V)20 b Fz(\()p Fy(s)1561 4492 y Fq(1)1601 4478 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(F)e Fz(\()p Fy(s)1850 4492 y Fq(1)1890 4478 y Fy(;)i Fu(\001)p Fz(\))p Fu(k)2045 4479 y Fr(6)p 2047 4491 V 2047 4532 a(5)2117 4478 y Fu(k)p Fy(e)2204 4441 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2423 4450 y Fr(1)2458 4441 y Fq(\))p Fw(H)2543 4450 y Fr(0)2582 4478 y Fy(g)s Fu(k)2673 4492 y Fq(6)2729 4478 y Fy(ds)2819 4492 y Fq(1)1075 4732 y Fu(\024)1229 4631 y Fn(\020)1283 4609 y(Z)1374 4635 y Fw(t)1334 4815 y(s)1419 4732 y Fu(k)p Fy(V)21 b Fz(\()p Fy(s)1616 4746 y Fq(1)1655 4732 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(F)e Fz(\()p Fy(s)1904 4746 y Fq(1)1945 4732 y Fy(;)i Fu(\001)p Fz(\))p Fu(k)2090 4695 y Fq(2)2100 4741 y Fr(6)p 2101 4753 V 2101 4795 a(5)2161 4732 y Fy(ds)2251 4746 y Fq(1)2290 4631 y Fn(\021)2354 4627 y Fr(1)p 2354 4639 V 2354 4680 a(2)2424 4631 y Fn(\020)2479 4609 y(Z)2569 4635 y Fw(t)2529 4815 y(s)2614 4732 y Fu(k)p Fy(e)2701 4695 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2920 4704 y Fr(1)2956 4695 y Fq(\))p Fw(H)3041 4704 y Fr(0)3080 4732 y Fy(g)s Fu(k)3171 4695 y Fq(2)3171 4755 y(6)3227 4732 y Fy(ds)3317 4746 y Fq(1)3356 4631 y Fn(\021)3420 4627 y Fr(1)p 3420 4639 V 3420 4680 a(2)1075 4986 y Fu(\024)83 b Fy(C)1294 5000 y Fq(1)1349 4885 y Fn(\020)1403 4862 y(Z)1494 4889 y Fw(t)1453 5069 y(s)1539 4986 y Fu(k)p Fy(V)20 b Fu(k)1702 4949 y Fq(2)1702 5061 y Fw(L)1750 5038 y Ft(1)1750 5081 y Fs(t)1816 5061 y Fq(\()p Fw(L)1901 4995 y Fr(3)p 1901 5003 V 1901 5038 a(2)1891 5072 y Fs(x)1946 5061 y Fq(\))1977 4986 y Fu(k)p Fy(F)13 b Fz(\()p Fy(s)2171 5000 y Fq(1)2211 4986 y Fy(;)i Fu(\001)p Fz(\))p Fu(k)2356 4949 y Fq(2)2356 5009 y(6)2412 4986 y Fy(ds)2502 5000 y Fq(1)2542 4885 y Fn(\021)2606 4881 y Fr(1)p 2606 4893 V 2606 4934 a(2)2676 4986 y Fu(k)p Fy(g)s Fu(k)2812 5000 y Fq(2)-47 4986 y Fz(\(5.11\))1075 5182 y(=)83 b Fy(C)1294 5196 y Fq(1)1349 5182 y Fu(k)p Fy(V)20 b Fu(k)1512 5249 y Fw(L)1560 5226 y Ft(1)1560 5268 y Fs(t)1626 5249 y Fq(\()p Fw(L)1711 5183 y Fr(3)p 1711 5191 V 1711 5226 a(2)1701 5259 y Fs(x)1756 5249 y Fq(\))1787 5182 y Fu(k)p Fy(F)13 b Fu(k)1948 5206 y Fw(L)1996 5183 y Fr(2)1996 5225 y Fs(t)2032 5206 y Fq(\()p Fw(L)2107 5187 y Fr(6)2107 5222 y Fs(x)2146 5206 y Fq(\))2178 5182 y Fu(k)p Fy(g)s Fu(k)2314 5196 y Fq(2)1903 5472 y Fz(27)p eop %%Page: 28 28 28 27 bop -47 274 a Fz(where)30 b(w)m(e)g(used)g(\(5.7\))i(to)f(pass)f (to)h(\(5.11\))s(.)40 b(This)29 b(sho)m(ws)h(that)1198 460 y(ess)h(sup)1478 482 y Fw(t)1507 460 y Fu(k)p Fz(\()p Fu(K)1656 474 y Fw(s)1694 460 y Fy(F)13 b Fz(\)\()p Fy(t)p Fz(\))p Fu(k)1948 474 y Fq(2)2014 460 y Fu(\024)25 b Fy(C)2175 474 y Fq(1)2230 460 y Fy(c)2269 474 y Fq(0)2309 460 y Fu(k)p Fy(F)13 b Fu(k)2470 483 y Fw(L)2518 461 y Fr(2)2518 503 y Fs(t)2553 483 y Fq(\()p Fw(L)2628 464 y Fr(6)2628 500 y Fs(x)2668 483 y Fq(\))-47 645 y Fz(whic)m(h)29 b(in)g(conjunction)g(with)g(\(5.10\))k(yields)c(that)1476 869 y Fu(kK)1590 883 y Fw(s)1628 869 y Fu(k)1673 883 y Fw(X)5 b Fx(!)p Fw(X)1900 869 y Fu(\024)25 b Fy(C)2061 883 y Fq(1)2115 869 y Fy(c)2154 883 y Fq(0)2219 869 y Fy(<)2325 808 y Fz(1)p 2325 849 46 4 v 2325 932 a(2)2381 869 y Fy(;)-2453 b Fz(\(5.12\))-47 1081 y(pro)m(vided)26 b Fy(c)363 1095 y Fq(0)431 1081 y Fz(is)h(small)f(\(see)j(\(5.4\))g (for)f(the)g(de\014nition)e(of)i Fy(X)7 b Fz(\).)40 b(Therefore,)29 b(the)f(op)s(erator)g Fy(I)22 b Fu(\000)15 b(K)3296 1095 y Fw(s)3361 1081 y Fz(is)27 b(in)m(v)m(ertible)f(on)-47 1194 y(the)k(space)h Fy(X)38 b Fz(and)30 b Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))31 b(can)f(b)s(e)g(expressed)g(via)g(the)g (Neumann)g(series)834 1449 y Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)1214 1376 y Fn(\002)1252 1449 y Fz(\()p Fy(I)i Fu(\000)20 b(K)1515 1463 y Fw(s)1552 1449 y Fz(\))1587 1412 y Fx(\000)p Fq(1)1681 1449 y Fy(e)1723 1412 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1967 1421 y Fr(0)2006 1376 y Fn(\003)2044 1449 y Fz(\()p Fy(t)p Fz(\))26 b(=)2310 1336 y Fx(1)2280 1363 y Fn(X)2269 1557 y Fw(m)p Fq(=0)2422 1376 y Fn(\002)2460 1449 y Fu(K)2530 1412 y Fw(m)2529 1472 y(s)2597 1449 y Fy(e)2639 1412 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2883 1421 y Fr(0)2922 1376 y Fn(\003)2960 1449 y Fz(\()p Fy(t)p Fz(\))-47 1716 y(whic)m(h)h(con)m(v)m(erges)k(in)d(the)h(norm)f(of)h Fy(X)7 b Fz(.)41 b(W)-8 b(riting)28 b(out)h Fu(h)p Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )2207 1730 y Fw(s)2245 1716 y Fy(;)g(g)s Fu(i)30 b Fz(explicitly)d(leads)h(to)i(\(5.6\))r(.)40 b(Next)30 b(w)m(e)f(c)m(hec)m(k)-47 1829 y(that)j(for)f(an)m(y)g Fy(F)40 b Fu(2)26 b Fy(L)710 1796 y Fq(2)710 1851 y Fw(t)750 1829 y Fz(\()p Fy(L)847 1796 y Fq(6)847 1851 y Fw(x)891 1829 y Fz(\))31 b(the)h(function)e Fy(t)c Fu(7!)h(hK)1753 1843 y Fw(s)1790 1829 y Fy(F)s(;)15 b(g)s Fu(i)33 b Fz(is)d(con)m(tin)m (uous)h(for)g(an)m(y)h(c)m(hoice)g(of)f Fy(g)g Fu(2)26 b Fy(L)3464 1796 y Fq(2)3503 1829 y Fz(.)44 b(In)30 b(fact,)j(if)-47 1941 y Fy(t)-14 1955 y Fq(1)50 1941 y Fy(<)25 b(t)179 1955 y Fq(2)218 1941 y Fz(,)31 b(then)169 2179 y Fu(jh)p Fz(\()p Fu(K)333 2193 y Fw(s)371 2179 y Fy(F)13 b Fz(\)\()p Fy(t)545 2193 y Fq(2)585 2179 y Fz(\))p Fy(;)i(g)s Fu(i)22 b(\000)e(h)p Fz(\()p Fu(K)993 2193 y Fw(s)1030 2179 y Fy(F)13 b Fz(\)\()p Fy(t)1204 2193 y Fq(1)1244 2179 y Fz(\))p Fy(;)i(g)s Fu(ij)85 b(\024)1664 2055 y Fn(Z)1755 2081 y Fw(t)1780 2090 y Fr(2)1715 2261 y Fw(s)1834 2179 y Fu(jh)p Fy(V)21 b Fz(\()p Fy(s)2046 2193 y Fq(1)2086 2179 y Fz(\))p Fy(F)13 b Fz(\()p Fy(s)2270 2193 y Fq(1)2310 2179 y Fz(\))p Fy(;)i Fz(\()p Fy(e)2462 2141 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)2593 2150 y Fr(1)2629 2141 y Fx(\000)p Fw(s)2717 2150 y Fr(1)2752 2141 y Fq(\))p Fw(H)2837 2150 y Fr(0)2896 2179 y Fu(\000)20 b Fy(e)3029 2141 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)3160 2150 y Fr(2)3196 2141 y Fx(\000)p Fw(s)3284 2150 y Fr(1)3318 2141 y Fq(\))p Fw(H)3403 2150 y Fr(0)3442 2179 y Fz(\))p Fy(g)s Fu(ij)15 b Fy(ds)3688 2193 y Fq(1)2028 2433 y Fz(+)2114 2309 y Fn(Z)2205 2335 y Fw(t)2230 2344 y Fr(2)2164 2515 y Fw(t)2189 2524 y Fr(1)2284 2433 y Fu(jh)p Fy(V)21 b Fz(\()p Fy(s)2496 2447 y Fq(1)2535 2433 y Fz(\))p Fy(F)13 b Fz(\()p Fy(s)2719 2447 y Fq(1)2759 2433 y Fz(\))p Fy(;)i(e)2876 2395 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)3007 2404 y Fr(1)3044 2395 y Fx(\000)p Fw(s)3132 2404 y Fr(1)3166 2395 y Fq(\))p Fw(H)3251 2404 y Fr(0)3290 2433 y Fy(g)s Fu(ij)g Fy(ds)3501 2447 y Fq(1)1510 2645 y Fu(\024)83 b(k)p Fy(V)21 b Fu(k)1828 2711 y Fw(L)1876 2688 y Ft(1)1876 2731 y Fs(t)1941 2711 y Fq(\()p Fw(L)2026 2646 y Fr(3)p 2027 2654 31 3 v 2027 2688 a(2)2016 2722 y Fs(x)2071 2711 y Fq(\))2103 2645 y Fu(k)p Fy(F)13 b Fu(k)2264 2668 y Fw(L)2312 2646 y Fr(2)2312 2688 y Fs(t)2347 2668 y Fq(\()p Fw(L)2422 2650 y Fr(6)2422 2685 y Fs(x)2462 2668 y Fq(\))2493 2645 y Fu(k)p Fy(g)25 b Fu(\000)20 b Fy(e)2739 2607 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)2870 2616 y Fr(2)2905 2607 y Fx(\000)p Fw(t)2985 2616 y Fr(1)3020 2607 y Fq(\))p Fw(H)3105 2616 y Fr(0)3144 2645 y Fy(g)s Fu(k)3235 2659 y Fq(2)2028 2902 y Fz(+)p Fu(k)p Fy(V)g Fu(k)2262 2969 y Fw(L)2310 2946 y Ft(1)2310 2989 y Fs(t)2376 2969 y Fq(\()p Fw(L)2461 2903 y Fr(3)p 2461 2911 V 2461 2946 a(2)2451 2979 y Fs(x)2506 2969 y Fq(\))2537 2802 y Fn(\020)2592 2779 y(Z)2682 2805 y Fw(t)2707 2814 y Fr(2)2642 2985 y Fw(t)2667 2994 y Fr(1)2762 2902 y Fu(k)p Fy(F)13 b Fz(\()p Fy(s)2956 2916 y Fq(1)2996 2902 y Fz(\))p Fu(k)3076 2865 y Fq(2)3076 2930 y Fw(L)3124 2912 y Fr(6)3124 2947 y Fs(x)3183 2902 y Fy(ds)3273 2916 y Fq(1)3312 2802 y Fn(\021)3376 2797 y Fr(1)p 3376 2809 V 3376 2850 a(2)3446 2902 y Fu(k)p Fy(g)s Fu(k)3582 2916 y Fq(2)3623 2902 y Fy(:)-47 3143 y Fz(Since)29 b(the)h(last)h(expression)e(tends)g(to)i(zero)g(as)g Fy(t)1647 3157 y Fq(2)1712 3143 y Fu(!)25 b Fy(t)1861 3157 y Fq(1)1900 3143 y Fz(,)30 b(con)m(tin)m(uit)m(y)h(follo)m(ws.)39 b(Hence)31 b(\()p Fu(K)3093 3110 y Fw(m)3092 3166 y(s)3161 3143 y Fy(F)13 b Fz(\)\()p Fy(t)p Fz(\))31 b(is)e(also)i(w)m(eakly)-47 3256 y(con)m(tin)m(uous)h(in)f Fy(t)p Fz(,)i(and)f(th)m(us)h Fu(K)1061 3223 y Fw(m)1060 3279 y(s)1128 3256 y Fy(e)1170 3223 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1414 3232 y Fr(0)1453 3256 y Fy( )1512 3270 y Fw(s)1581 3256 y Fz(is,)g(to)s(o.)48 b(Since)31 b Fu(h)p Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )2493 3270 y Fw(s)2531 3256 y Fy(;)g(g)s Fu(i)34 b Fz(is)e(a)h(uniformly)c(con)m(v)m(ergen)m(t)35 b(series)-47 3369 y(of)30 b(these)h(con)m(tin)m(uous)f(functions,)f(it) h(follo)m(ws)g(that)h(it)f(is)f(con)m(tin)m(uous.)p 3879 3369 4 62 v 3883 3311 55 4 v 3883 3369 V 3937 3369 4 62 v -47 3554 a Fl(R)-5 b(emark)34 b(5.3.)46 b Fz(The)30 b(pro)s(of)g(of)h(Lemma)g(5.2)h(sho)m(ws)f(that)g(the)g(op)s(erator)h Fu(K)2538 3568 y Fw(s)2601 3554 y Fz(:)26 b Fy(L)2714 3521 y Fq(2)2714 3576 y Fw(t)2753 3554 y Fy(L)2815 3521 y Fq(6)2815 3576 y Fw(x)2885 3554 y Fu(!)g Fy(w)d Fu(\000)d Fy(C)3253 3521 y Fq(0)3246 3576 y Fw(t)3292 3554 y Fz(\()p Fy(L)3389 3521 y Fq(2)3389 3576 y Fw(x)3433 3554 y Fz(\))31 b(maps)g Fy(L)3800 3521 y Fq(2)3800 3576 y Fw(t)3839 3554 y Fy(L)3901 3521 y Fq(6)3901 3576 y Fw(x)-47 3667 y Fz(in)m(to)f(the)h(space)g(of)f(w)m(eakly)h(con)m(tin)m(uous)f (functions)f(with)g(v)-5 b(alues)29 b(in)g Fy(L)2433 3634 y Fq(2)2473 3667 y Fz(\()p Fv(R)2568 3634 y Fq(3)2613 3667 y Fz(\).)-47 3809 y(F)-8 b(or)35 b(tec)m(hnical)g(reasons)g (connected)h(with)d(the)i(functional)e(calculus)h(in)f(the)i(follo)m (wing)f(section)h(it)f(will)e(b)s(e)i(con-)-47 3922 y(v)m(enien)m(t)k (to)h(w)m(ork)f(with)f(smo)s(oth)h(p)s(oten)m(tials)f(in)g Fy(Y)58 b Fz(rather)38 b(than)g(general)g(ones.)64 b(T)-8 b(o)38 b(appro)m(ximate)g(a)h(general)-47 4035 y(p)s(oten)m(tial)30 b Fy(V)51 b Fz(b)m(y)30 b(means)h(of)g(smo)s(oth)f(ones,)i(c)m(ho)s (ose)f(nonnegativ)m(e)h(cut-o\013)f(functions)f Fy(\037)25 b Fu(2)h(S)7 b Fz(\()p Fv(R)3298 4002 y Fq(3)3343 4035 y Fz(\))31 b(and)f Fy(\021)f Fu(2)d(S)7 b Fz(\()p Fv(R)s Fz(\))-47 4148 y(so)36 b(that)g Fy(\037)f Fz(and)42 b(^)-52 b Fy(\021)39 b Fz(ha)m(v)m(e)e(compact)g(supp)s(ort)d(and)h(satisfy) 2027 4075 y Fn(R)2070 4180 y Fp(R)2118 4161 y Fr(3)2172 4148 y Fy(\037)p Fz(\()p Fy(x)p Fz(\))15 b Fy(dx)35 b Fz(=)e(1,)2712 4075 y Fn(R)2755 4180 y Fp(R)2822 4148 y Fy(\021)s Fz(\()p Fy(t)p Fz(\))15 b Fy(dt)35 b Fz(=)f(1.)57 b(In)34 b(addition,)i(let)-47 4261 y Fy(\037)25 b Fz(=)g(1)31 b(on)f(a)h(neigh)m(b)s(orho)s(o)s(d)d(of)i(0.)41 b(F)-8 b(or)31 b(an)m(y)g Fy(V)46 b Fu(2)24 b Fy(Y)51 b Fz(and)29 b Fy(R)d(>)f Fz(1)31 b(de\014ne)1269 4458 y Fy(V)1342 4410 y Fq(\(1\))1322 4486 y Fw(R)1436 4458 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))27 b(:=)e Fy(V)20 b Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fy(\037)p Fz(\()2119 4396 y Fy(:)p 2095 4437 70 4 v 2095 4520 a(R)2176 4458 y Fz(\))21 b Fu(\003)f Fy(R)2367 4420 y Fq(3)2406 4458 y Fy(\037)p Fz(\()p Fy(R)q Fu(\001)p Fz(\))-47 4699 y(where)26 b(the)h(con)m(v)m (olution)f(is)g(in)f(the)i Fy(x)p Fz(-v)-5 b(ariable)26 b(only)-8 b(.)39 b(Note)28 b(that)f Fy(V)2319 4651 y Fq(\(1\))2299 4727 y Fw(R)2440 4699 y Fz(is)f(w)m(ell-de\014ned,)g(smo) s(oth)g(and)g(compactly)-47 4833 y(supp)s(orted)e(in)h Fy(x)p Fz(,)j(and)e(satis\014es)g Fu(k)p Fy(V)1205 4785 y Fq(\(1\))1185 4862 y Fw(R)1300 4833 y Fu(k)1345 4847 y Fw(L)1393 4825 y Ft(1)1393 4867 y Fs(t;x)1504 4833 y Fy(<)f Fu(1)h Fz(since)g Fu(k)p Fy(V)21 b Fu(k)2100 4900 y Fw(L)2148 4877 y Ft(1)2148 4920 y Fs(t)2213 4900 y Fq(\()p Fw(L)2298 4834 y Fr(3)p 2299 4842 31 3 v 2299 4877 a(2)2288 4910 y Fs(x)2343 4900 y Fq(\))2400 4833 y Fy(<)k Fu(1)p Fz(.)39 b(Moreo)m(v)m(er,)30 b(it)c(is)f(standard)h(to) h(c)m(hec)m(k)-47 4991 y(that)1409 5177 y(sup)1405 5255 y Fw(R>)p Fq(0)1564 5177 y Fu(k)p Fy(V)1683 5129 y Fq(\(1\))1662 5205 y Fw(R)1777 5177 y Fu(k)1822 5191 y Fw(Y)1909 5177 y Fu(\024)e(k)p Fy(\037)p Fu(k)2152 5191 y Fx(1)2242 5177 y Fu(k)p Fy(V)c Fu(k)2406 5191 y Fw(Y)2467 5177 y Fy(:)1903 5472 y Fz(28)p eop %%Page: 29 29 29 28 bop -47 274 a Fz(Indeed,)636 461 y Fu(k)p Fy(V)755 413 y Fq(\(1\))734 490 y Fw(R)849 461 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)1062 512 y Fw(L)1120 462 y Fr(3)p 1122 470 31 3 v 1122 505 a(2)1196 461 y Fu(\024)25 b(k)p Fy(\037)p Fu(k)1439 475 y Fx(1)1514 461 y Fu(k)15 b(j)p Fy(V)21 b Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(j)22 b(\003)f Fy(R)2024 424 y Fq(3)2063 461 y Fy(\037)p Fz(\()p Fy(R)q Fu(\001)p Fz(\))p Fu(k)2330 512 y Fw(L)2388 462 y Fr(3)p 2389 470 V 2389 505 a(2)2463 461 y Fu(\024)k(k)p Fy(\037)p Fu(k)2706 475 y Fx(1)2796 461 y Fu(k)p Fy(V)c Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)3128 512 y Fw(L)3186 462 y Fr(3)p 3188 470 V 3188 505 a(2)3236 461 y Fy(;)-47 660 y Fz(whereas)30 b(with)f(\000\()p Fy(x)p Fz(\))d(:=)f Fu(j)p Fy(x)p Fu(j)932 627 y Fx(\000)p Fq(1)1057 660 y Fz(and)30 b Fu(M)g Fz(denoting)g(measures)g(in)f(the)i Fy(\034)10 b Fz(-v)-5 b(ariable,)765 758 y Fn(\020)820 859 y Fu(k)p Fy(V)938 811 y Fq(\(1\))918 888 y Fw(R)1033 859 y Fz(\()5 b(^)-50 b Fy(\034)10 b(;)15 b Fu(\001)p Fz(\))p Fu(k)1263 873 y Fx(M)1373 859 y Fu(\003)20 b Fz(\000)1495 758 y Fn(\021)1550 859 y Fz(\()p Fy(x)p Fz(\))83 b Fu(\024)g Fz(sup)1959 932 y Fw(y)2046 758 y Fn(\020)2100 859 y Fy(\037)p Fz(\()2225 798 y Fy(:)p 2202 838 70 4 v 2202 922 a(R)2282 859 y Fz(\))15 b Fu(k)p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)11 b(;)k Fu(\001)p Fz(\))p Fu(k)2682 873 y Fx(M)2791 859 y Fu(\003)21 b Fz(\000)2914 758 y Fn(\021)2968 859 y Fz(\()p Fy(y)s Fz(\))1755 1091 y Fu(\024)83 b(k)p Fy(\037)p Fu(k)2056 1105 y Fx(1)2162 1091 y Fz(sup)2212 1163 y Fw(y)2299 990 y Fn(\020)2353 1091 y Fu(k)p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)10 b(;)15 b Fu(\001)p Fz(\))p Fu(k)2702 1105 y Fx(M)2812 1091 y Fu(\003)20 b Fz(\000)2934 990 y Fn(\021)2989 1091 y Fz(\()p Fy(y)s Fz(\))p Fy(;)-47 1321 y Fz(as)30 b(claimed.)40 b(T)-8 b(o)31 b(regularize)f(in)f Fy(t)p Fz(,)h(de\014ne)1234 1508 y Fy(V)1287 1522 y Fw(R)1344 1508 y Fz(\()p Fu(\001)p Fy(;)15 b(x)p Fz(\))27 b(:=)e([)p Fy(V)1778 1460 y Fq(\(1\))1757 1537 y Fw(R)1872 1508 y Fz(\()p Fu(\001)p Fy(;)15 b(x)p Fz(\))22 b Fu(\003)e Fy(R)q(\021)s Fz(\()p Fy(R)q Fu(\001)p Fz(\)])p Fy(\021)s Fz(\()2571 1447 y Fy(:)p 2547 1487 V 2547 1570 a(R)2629 1508 y Fz(\))-47 1706 y(where)30 b(the)g(con)m(v)m(olution)h(is)e(in)g (the)i Fy(t)p Fz(-v)-5 b(ariable)29 b(only)-8 b(.)40 b(Again)30 b(one)h(c)m(hec)m(ks)h(that)708 1893 y Fu(k)p Fy(V)806 1907 y Fw(R)864 1893 y Fu(k)909 1907 y Fw(Y)996 1893 y Fu(\024)25 b Fz(\()p Fu(k)p Fy(\021)s Fu(k)1265 1907 y Fx(1)1361 1893 y Fz(+)20 b Fu(k)7 b Fz(^)-52 b Fy(\021)t Fu(k)1591 1907 y Fq(1)1631 1893 y Fz(\))15 b Fu(k)p Fy(V)1800 1845 y Fq(\(1\))1779 1922 y Fw(R)1895 1893 y Fu(k)1940 1907 y Fw(Y)2026 1893 y Fu(\024)25 b Fz(\()p Fu(k)p Fy(\021)s Fu(k)2295 1907 y Fx(1)2392 1893 y Fz(+)19 b Fu(k)7 b Fz(^)-52 b Fy(\021)t Fu(k)2621 1907 y Fq(1)2661 1893 y Fz(\))31 b Fu(k)p Fy(\037)p Fu(k)2874 1907 y Fx(1)2965 1893 y Fu(k)p Fy(V)20 b Fu(k)3128 1907 y Fw(Y)-47 2070 y Fz(for)30 b(an)m(y)h Fy(R)26 b(>)f Fz(0.)41 b(W)-8 b(e)31 b(will)d(use)i(that)h Fy(V)1301 2084 y Fw(R)1384 2070 y Fu(!)25 b Fy(V)51 b Fz(as)30 b Fy(R)c Fu(!)f(1)31 b Fz(in)e(the)h(follo)m(wing)f(sense:)41 b(F)-8 b(or)31 b(a.e.)42 b Fy(t)30 b Fz(one)h(has)1121 2246 y Fu(k)p Fy(V)1219 2260 y Fw(R)1278 2246 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))21 b Fu(\000)f Fy(V)g Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)1844 2297 y Fw(L)1902 2247 y Fr(3)p 1904 2255 31 3 v 1904 2290 a(2)1948 2297 y Fq(\()p Fp(R)2023 2278 y Fr(3)2058 2297 y Fq(\))2115 2246 y Fu(!)25 b Fz(0)61 b(as)31 b Fy(R)26 b Fu(!)f(1)p Fz(.)-2823 b(\(5.13\))-47 2449 y(Firstly)-8 b(,)29 b(it)h(follo)m(ws)g (from)g(standard)f(measure)i(theory)f(that)h(for)f(a.e.)42 b Fy(t)1063 2636 y Fu(k)p Fy(V)1181 2588 y Fq(\(1\))1161 2665 y Fw(R)1276 2636 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))21 b Fu(\000)f Fy(V)g Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)1842 2687 y Fw(L)1900 2637 y Fr(3)p 1902 2645 V 1902 2680 a(2)1946 2687 y Fq(\()p Fp(R)2021 2668 y Fr(3)2056 2687 y Fq(\))2113 2636 y Fu(!)25 b Fz(0)61 b(as)92 b Fy(R)26 b Fu(!)f(1)p Fy(:)-2882 b Fz(\(5.14\))-47 2844 y(Secondly)-8 b(,)30 b(with)f Fy(\021)604 2858 y Fw(R)662 2844 y Fz(\()p Fy(t)p Fz(\))c(:=)h Fy(R)q(\021)s Fz(\()p Fy(R)q(t)p Fz(\),)643 3036 y Fu(k)p Fy(V)741 3050 y Fw(R)799 3036 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))22 b Fu(\000)d Fy(V)1153 2988 y Fq(\(1\))1132 3065 y Fw(R)1247 3036 y Fz(\()p Fy(t;)c Fu(\001)p Fz(\))p Fu(k)1470 3037 y Fr(3)p 1472 3049 V 1472 3090 a(2)479 3221 y Fu(\024)93 b(k)p Fz(\()p Fy(V)797 3173 y Fq(\(1\))776 3250 y Fw(R)912 3221 y Fu(\003)20 b Fy(\021)1022 3235 y Fw(R)1080 3221 y Fz(\)\()p Fy(t;)15 b Fu(\001)p Fz(\))22 b Fu(\000)e Fy(V)1469 3173 y Fq(\(1\))1449 3250 y Fw(R)1563 3221 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)1786 3222 y Fr(3)p 1788 3234 V 1788 3275 a(2)1853 3221 y Fz(+)20 b Fu(k)p Fy(V)2062 3173 y Fq(\(1\))2042 3250 y Fw(R)2157 3221 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)2380 3222 y Fr(3)p 2381 3234 V 2381 3275 a(2)2426 3221 y Fu(j)p Fz(1)21 b Fu(\000)f Fy(\021)s Fz(\()p Fy(t=R)q Fz(\))p Fu(j)479 3434 y(\024)643 3311 y Fn(Z)734 3337 y Fx(1)693 3517 y(\0001)838 3434 y Fy(\021)883 3448 y Fw(R)941 3434 y Fz(\()p Fy(s)p Fz(\))15 b Fu(k)p Fy(V)1188 3387 y Fq(\(1\))1167 3463 y Fw(R)1283 3434 y Fz(\()p Fy(t)20 b Fu(\000)g Fy(s;)15 b Fu(\001)p Fz(\))21 b Fu(\000)f Fy(V)1790 3387 y Fq(\(1\))1770 3463 y Fw(R)1884 3434 y Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)2107 3435 y Fr(3)p 2109 3447 V 2109 3488 a(2)2168 3434 y Fy(ds)21 b Fz(+)f Fu(j)p Fz(1)g Fu(\000)g Fy(\021)s Fz(\()p Fy(t=R)q Fz(\))p Fu(j)15 b(k)p Fy(\021)s Fu(k)2995 3448 y Fx(1)3073 3434 y Fu(k)p Fy(V)21 b Fu(k)3237 3501 y Fw(L)3285 3478 y Ft(1)3285 3521 y Fs(t)3350 3501 y Fq(\()p Fw(L)3435 3435 y Fr(3)p 3435 3443 V 3435 3478 a(2)3425 3512 y Fs(x)3480 3501 y Fq(\))479 3683 y Fu(\024)93 b(k)p Fy(\037)p Fu(k)790 3697 y Fx(1)896 3559 y Fn(Z)987 3586 y Fx(1)946 3766 y(\0001)1091 3683 y Fy(\021)1136 3697 y Fw(R)1194 3683 y Fz(\()p Fy(s)p Fz(\))15 b Fu(k)p Fy(V)21 b Fz(\()p Fy(t)f Fu(\000)g Fy(s;)15 b Fu(\001)p Fz(\))21 b Fu(\000)f Fy(V)g Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)2171 3684 y Fr(3)p 2173 3696 V 2173 3737 a(2)2233 3683 y Fy(ds)20 b Fz(+)g Fy(o)p Fz(\(1\))-2640 b(\(5.15\))469 3864 y Fu(!)83 b Fz(0)-47 4041 y(for)31 b(a.e.)43 b Fy(t)31 b Fz(as)g Fy(R)c Fu(!)g(1)p Fz(.)42 b(The)31 b(conclusion)e(\(5.15\))k(follo)m(ws)d(from)h(the)g(v)m (ector-v)-5 b(alued)32 b(analogue)g(of)f(the)g(Leb)s(esgue)-47 4167 y(di\013eren)m(tiation)44 b(theorem)i(\(in)e(this)h(case)h(\\v)m (ector-v)-5 b(alued")48 b(means)d(with)f(v)-5 b(alues)45 b(in)f Fy(L)3138 4104 y Fr(3)p 3138 4116 V 3138 4157 a(2)3182 4167 y Fz(\).)87 b(In)45 b(com)m(bination)-47 4280 y(with)29 b(\(5.14\))j(this)e(yields)e(\(5.13\))s(.)-47 4393 y(W)-8 b(e)31 b(shall)e(no)m(w)h(pro)m(v)m(e)i(the)e(con)m(v)m (ergence)j(of)d(the)h(appro)m(ximate)f(solutions)f Fy( )2640 4407 y Fw(R)2698 4393 y Fz(\()p Fy(t;)15 b(x)p Fz(\))31 b(satisfying)e(the)i(equation)1388 4569 y Fy(i)15 b(@)1482 4583 y Fw(t)1513 4569 y Fy( )1572 4583 y Fw(R)1650 4569 y Fu(\000)20 b Fz(\001)p Fy( )1876 4583 y Fw(R)1954 4569 y Fz(+)g Fy(V)2098 4583 y Fw(R)2155 4569 y Fz(\()p Fy(t;)15 b(x)p Fz(\))p Fy( )2409 4583 y Fw(R)2493 4569 y Fz(=)25 b(0)p Fy(;)-2706 b Fz(\(5.16\))1388 4707 y Fy( )1447 4721 y Fw(R)1505 4707 y Fu(j)1530 4721 y Fw(t)p Fq(=)p Fw(s)1673 4707 y Fz(=)25 b Fy( )1828 4721 y Fw(s)-47 4884 y Fz(to)32 b(the)g(solution)e Fy( )s Fz(\()p Fy(t;)15 b(x)p Fz(\))33 b(of)f(the)f(original)f(problem)g(corresp)s(onding)g(to) i(the)g(p)s(oten)m(tial)f Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\).)45 b(Note)33 b(that)f(due)-47 4997 y(to)i(the)g(smo)s (othness)f(and)g(b)s(oundedness)e(of)j(the)g(p)s(oten)m(tials)f Fy(V)2174 5011 y Fw(R)2265 4997 y Fz(the)h Fy(L)2487 4964 y Fx(1)2487 5019 y Fw(t)2561 4997 y Fy(L)2623 4964 y Fq(2)2623 5019 y Fw(x)2701 4997 y Fz(function)e Fy( )3119 5011 y Fw(R)3211 4997 y Fz(can)i(b)s(e)f(in)m(terpreted)-47 5110 y(as)38 b(a)h(distributional)34 b(solution)i(of)j(equation)f (\(5.16\))r(.)64 b(In)37 b(fact,)42 b(the)c(left)g(hand-side)e(of)45 b(\(5.16\))c(b)s(elongs)c(to)i(the)-47 5223 y(space)31 b Fy(L)258 5190 y Fx(1)258 5245 y Fw(t)332 5223 y Fy(H)415 5190 y Fx(\000)p Fq(2)510 5223 y Fz(.)40 b(In)30 b(addition,)f Fy( )1132 5237 y Fw(R)1220 5223 y Fz(is)g(also)i(a)f(Duhamel)g (solution)f(as)i(in)e(\(5.3\))r(.)1903 5472 y(29)p eop %%Page: 30 30 30 29 bop -47 274 a Fm(Lemma)32 b(5.4.)47 b Fl(L)-5 b(et)30 b Fy(U)748 288 y Fw(R)805 274 y Fz(\()p Fy(t;)15 b(s)p Fz(\))30 b Fl(b)-5 b(e)30 b(the)g(pr)-5 b(op)g(agator)44 b Fz(\(5.16\))r Fl(,)30 b(i.e.,)f Fy(U)2271 288 y Fw(R)2329 274 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )2574 288 y Fw(s)2637 274 y Fz(=)24 b Fy( )2791 288 y Fw(R)2849 274 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fl(.)42 b(Then)30 b(for)g(any)g Fy(s;)15 b(t)25 b Fu(2)g Fv(R)-47 387 y Fl(such)33 b(that)g Fy(t)25 b Fu(\025)g Fy(s)p Fl(,)32 b(and)i(arbitr)-5 b(ary)35 b(functions)e Fy( )1613 401 y Fw(s)1650 387 y Fy(;)15 b(g)29 b Fu(2)c Fy(L)1910 354 y Fq(2)1949 387 y Fz(\()p Fv(R)2044 354 y Fq(3)2090 387 y Fz(\))p Fl(,)32 b Fu(k)p Fy( )2289 401 y Fw(s)2327 387 y Fu(k)2372 407 y Fw(L)2420 388 y Fr(2)2420 423 y Fs(x)2489 387 y Fz(=)25 b Fu(k)p Fy(g)s Fu(k)2721 407 y Fw(L)2769 388 y Fr(2)2769 423 y Fs(x)2838 387 y Fz(=)g(1)33 b Fl(we)g(have)876 586 y Fy(<)25 b(U)1034 600 y Fw(R)1092 586 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )1337 600 y Fw(s)1374 586 y Fy(;)g(g)29 b(>)c Fu(!)h Fy(<)f(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )2112 600 y Fw(s)2149 586 y Fy(;)g(g)29 b(>)211 b Fl(as)94 b Fy(R)26 b Fu(!)f(1)-3069 b Fz(\(5.17\))-47 774 y Fl(Pr)-5 b(o)g(of.)47 b Fz(First)d(observ)m(e)h(that)g(since)f (the)h(p)s(oten)m(tial)f Fy(V)65 b Fz(satis\014es)44 b(the)h(smallness)e(assumption)g(\(5.5\))r(,)48 b Fy(V)3690 788 y Fw(R)3793 774 y Fz(also)-47 887 y(ob)s(eys)30 b(\(5.5\))i(for)e (all)f Fy(R)d(>)f Fz(0.)41 b(According)30 b(to)i(Lemma)e(5.2,)339 1144 y Fu(h)p Fy(U)436 1158 y Fw(R)494 1144 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )739 1158 y Fw(s)777 1144 y Fy(;)g(g)s Fu(i)26 b Fz(=)1061 1031 y Fx(1)1031 1058 y Fn(X)1020 1252 y Fw(m)p Fq(=0)1279 1144 y Fy(i)1310 1107 y Fw(m)1448 1021 y Fn(Z)1554 1144 y Fu(\001)15 b(\001)g(\001)1675 1021 y Fn(Z)1346 1282 y Fw(s)p Fx(\024)p Fw(s)1467 1290 y Fs(m)1525 1282 y Fx(\024)p Fw(::)p Fx(\024)p Fw(s)1708 1291 y Fr(1)1741 1282 y Fx(\024)p Fw(t)1822 1144 y Fu(h)p Fy(e)1899 1107 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2063 1116 y Fr(1)2098 1107 y Fq(\))p Fw(H)2183 1116 y Fr(0)2223 1144 y Fy(V)2276 1158 y Fw(R)2333 1144 y Fz(\()p Fy(s)2411 1158 y Fq(1)2451 1144 y Fy(;)g Fu(\001)p Fz(\))p Fy(e)2593 1107 y Fw(i)p Fq(\()p Fw(s)2677 1116 y Fr(1)2713 1107 y Fx(\000)p Fw(s)2801 1116 y Fr(2)2835 1107 y Fq(\))p Fw(H)2920 1116 y Fr(0)2959 1144 y Fy(V)3012 1158 y Fw(R)3070 1144 y Fz(\()p Fy(s)3148 1158 y Fq(2)3187 1144 y Fy(;)g Fu(\001)p Fz(\))p Fy(::::)2185 1423 y(V)2238 1437 y Fw(R)2296 1423 y Fz(\()p Fy(s)2374 1437 y Fw(m)2441 1423 y Fy(;)g Fu(\001)p Fz(\))p Fy(e)2583 1385 y Fw(i)p Fq(\()p Fw(s)2667 1393 y Fs(m)2727 1385 y Fx(\000)p Fw(s)p Fq(\))p Fw(H)2900 1394 y Fr(0)2954 1423 y Fy( )3013 1437 y Fw(s)3050 1423 y Fy(;)g(g)s Fu(i)26 b Fy(ds)3287 1437 y Fq(1)3327 1423 y Fy(:::ds)3492 1437 y Fw(m)-47 1423 y Fz(\(5.18\))-47 1611 y(for)k(an)m(y)h Fy( )323 1625 y Fw(s)360 1611 y Fy(;)15 b(g)29 b Fu(2)c Fy(L)620 1578 y Fq(2)659 1611 y Fz(\()p Fv(R)754 1578 y Fq(3)800 1611 y Fz(\).)41 b(Equiv)-5 b(alen)m(tly)d(,)29 b Fy(U)1507 1625 y Fw(R)1564 1611 y Fz(\()p Fy(t;)15 b(s)p Fz(\))31 b(can)g(b)s(e)f(represen)m(ted)g(b)m (y)g(the)h(Neumann)f(series)753 1869 y Fy(U)815 1883 y Fw(R)873 1869 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)1181 1795 y Fn(\002)1219 1869 y Fz(\()p Fy(I)i Fu(\000)19 b(K)1481 1883 y Fw(R)1539 1887 y(s)1576 1869 y Fz(\))1611 1831 y Fx(\000)p Fq(1)1706 1869 y Fy(e)1748 1831 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1992 1840 y Fr(0)2031 1795 y Fn(\003)2069 1869 y Fz(\()p Fy(t)p Fz(\))26 b(=)2335 1755 y Fx(1)2304 1782 y Fn(X)2294 1977 y Fw(m)p Fq(=0)2446 1795 y Fn(\002)2484 1869 y Fu(K)2553 1883 y Fw(R)2611 1831 y(m)2611 1891 y(s)2678 1869 y Fy(e)2720 1831 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2964 1840 y Fr(0)3003 1795 y Fn(\003)3041 1869 y Fz(\()p Fy(t)p Fz(\))-47 2132 y(whic)m(h)j(con)m(v)m(erges)j(in)e(the)g(norm)g(of)h (the)f(space)h Fy(X)38 b Fz(de\014ned)29 b(ab)s(o)m(v)m(e.)43 b(The)30 b(op)s(erators)g Fu(K)3002 2146 y Fw(R)3060 2151 y(s)3122 2132 y Fz(:)c Fy(X)33 b Fu(!)25 b Fy(X)38 b Fz(are)31 b(de\014ned)-47 2245 y(as)983 2392 y(\()p Fu(K)1087 2406 y Fw(R)1145 2410 y(s)1182 2392 y Fy(F)13 b Fz(\)\()p Fy(t;)i Fu(\001)p Fz(\))27 b(=)e Fy(i)1625 2268 y Fn(Z)1716 2294 y Fw(t)1676 2474 y(s)1761 2392 y Fy(e)1803 2354 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)1967 2363 y Fr(1)2002 2354 y Fq(\))p Fw(H)2087 2363 y Fr(0)2126 2392 y Fy(V)2179 2406 y Fw(R)2237 2392 y Fz(\()p Fy(s)2315 2406 y Fq(1)2355 2392 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(F)e Fz(\()p Fy(s)2604 2406 y Fq(1)2644 2392 y Fy(;)i Fu(\001)p Fz(\))g Fy(ds)2849 2406 y Fq(1)2890 2392 y Fy(:)-47 2592 y Fz(T)-8 b(o)38 b(v)m(erify)f(the)h(conclusion)e(of)i(Lemma)g(5.4)h (it)e(su\016ces)h(to)g(sho)m(w)g(that)g(for)g(an)f(arbitrary)g(p)s (ositiv)m(e)g Fy(")g(>)h Fz(0,)i(all)-47 2705 y(p)s(ositiv)m(e)29 b(in)m(tegers)i Fy(m)25 b Fu(\024)g Fy(m)907 2719 y Fq(0)946 2705 y Fz(\()p Fy(")p Fz(\),)32 b(and)d(all)h(su\016cien)m(tly)f(large) h Fy(R)c Fz(=)f Fy(R)2355 2719 y Fq(0)2395 2705 y Fz(\()p Fy(";)15 b(m)2592 2719 y Fq(0)2632 2705 y Fz(\))993 2893 y Fu(jh)1053 2819 y Fn(\000)1095 2893 y Fu(K)1165 2855 y Fw(m)1164 2915 y(s)1252 2893 y Fu(\000)20 b(K)1412 2907 y Fw(R)1470 2855 y(m)1470 2915 y(s)1537 2819 y Fn(\001)1579 2893 y Fy(e)1621 2855 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1865 2864 y Fr(0)1904 2893 y Fy( )1963 2907 y Fw(s)2000 2893 y Fy(;)15 b(g)s Fu(i)p Fz(\()p Fy(t)p Fz(\))p Fu(j)27 b Fy(<)e(")p Fz(\()p Fy(C)2514 2907 y Fq(1)2554 2893 y Fy(c)2593 2907 y Fq(0)2633 2893 y Fz(\))2668 2855 y Fw(m)p Fx(\000)p Fq(1)2825 2893 y Fy(m)-2952 b Fz(\(5.19\))-47 3081 y(The)30 b(p)s(ositiv)m(e)f(in)m (teger)i Fy(m)857 3095 y Fq(0)896 3081 y Fz(\()p Fy(")p Fz(\))g(is)f(c)m(hosen)h(so)f(that)h(2\()p Fy(C)1875 3095 y Fq(1)1916 3081 y Fy(c)1955 3095 y Fq(0)1994 3081 y Fz(\))2029 3048 y Fw(m)2121 3081 y Fu(\024)25 b Fy(")31 b Fz(whic)m(h)e(ensures)h(the)g(smallness)f(of)h(the)h(\\tails")-47 3194 y(of)f(the)h(series)f(for)g Fy(U)10 b Fz(\()p Fy(s;)15 b(t)p Fz(\))31 b(and)e Fy(U)1123 3208 y Fw(R)1181 3194 y Fz(\()p Fy(s;)15 b(t)p Fz(\).)-47 3307 y(F)-8 b(or)31 b(the)f(b)s(ounded)f(op)s(erators)h Fu(K)1115 3321 y Fw(s)1152 3307 y Fy(;)15 b Fu(K)1261 3321 y Fw(R)1320 3325 y(s)1387 3307 y Fz(on)30 b(the)h(space)f Fy(X)38 b Fz(w)m(e)31 b(ha)m(v)m(e)h(the)e(follo)m(wing)f(iden)m(tit)m(y:)1120 3574 y Fu(K)1189 3588 y Fw(R)1247 3536 y(m)1247 3596 y(s)1334 3574 y Fu(\000)20 b(K)1495 3536 y Fw(m)1494 3596 y(s)1587 3574 y Fz(=)1683 3460 y Fw(m)p Fx(\000)p Fq(1)1694 3488 y Fn(X)1700 3685 y Fw(`)p Fq(=0)1851 3574 y Fu(K)1920 3588 y Fw(R)1978 3536 y(`)1978 3596 y(s)2015 3574 y Fz(\()p Fu(K)2119 3588 y Fw(s)2176 3574 y Fu(\000)g(K)2336 3588 y Fw(R)2394 3592 y(s)2431 3574 y Fz(\))p Fu(K)2536 3536 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)2535 3596 y Fw(s)-47 3574 y Fz(\(5.20\))-47 3846 y(W)-8 b(e)31 b(shall)e(pro)m(v)m(e)i(that)g(for)f Fy(`)25 b Fu(2)g Fz([0)p Fy(;)15 b(m)21 b Fu(\000)f Fz(1])832 4034 y Fu(jhK)961 4048 y Fw(R)1020 3997 y(`)1020 4057 y(s)1056 4034 y Fz(\()p Fu(K)1160 4048 y Fw(s)1218 4034 y Fu(\000)g(K)1378 4048 y Fw(R)1436 4052 y(s)1473 4034 y Fz(\))p Fu(K)1578 3997 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)1577 4057 y Fw(s)1819 4034 y Fy(e)1861 3997 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2105 4006 y Fr(0)2144 4034 y Fy( )2203 4048 y Fw(s)2240 4034 y Fy(;)15 b(g)s Fu(i)p Fz(\()p Fy(t)p Fz(\))p Fu(j)28 b Fy(<)d(")p Fz(\()p Fy(C)2755 4048 y Fq(1)2795 4034 y Fy(c)2834 4048 y Fq(0)2873 4034 y Fz(\))2908 3997 y Fw(m)p Fx(\000)p Fq(1)-47 4034 y Fz(\(5.21\))-47 4222 y(whic)m(h)k(immediately)g(implies)e(\(5.19\))r(.) -47 4335 y(In)g(view)f(of)i(Remark)f(5.3)i(the)f(op)s(erator)g Fu(K)1441 4349 y Fw(s)1478 4335 y Fz(,)g(and)f(th)m(us)g(also)g Fu(K)2150 4349 y Fw(R)2208 4353 y(s)2245 4335 y Fz(,)h(maps)f Fy(L)2595 4302 y Fq(2)2595 4358 y Fw(t)2635 4335 y Fy(L)2697 4302 y Fq(6)2697 4358 y Fw(x)2766 4335 y Fu(!)e Fy(w)17 b Fu(\000)d Fy(C)3121 4302 y Fq(0)3114 4358 y Fw(t)3160 4335 y Fz(\()p Fy(L)3257 4302 y Fq(2)3257 4358 y Fw(x)3301 4335 y Fz(\).)40 b(Therefore,)28 b(for)-47 4448 y(an)i(arbitrary)f (\014xed)h Fy(t)25 b Fu(\025)g Fy(s)30 b Fz(w)m(e)h(can)f(de\014ne)g (the)g(op)s(erator)h Fu(K)2068 4462 y Fw(R)2126 4466 y(s;t)2233 4448 y Fz(:)26 b Fy(L)2346 4415 y Fq(2)2346 4471 y Fw(t)2385 4448 y Fy(L)2447 4415 y Fq(6)2447 4471 y Fw(x)2516 4448 y Fu(!)f Fy(L)2694 4415 y Fq(2)2694 4471 y Fw(x)2768 4448 y Fz(via)30 b(the)h(form)m(ula)1573 4636 y Fu(K)1642 4650 y Fw(R)1700 4654 y(s;t)1781 4636 y Fy(F)39 b Fz(=)25 b(\()p Fu(K)2078 4650 y Fw(s)2115 4636 y Fy(F)13 b Fz(\)\()p Fy(t)p Fz(\))-47 4824 y(In)38 b(addition)g(to)i(the)f Fy(L)790 4791 y Fq(2)790 4847 y Fw(x)873 4824 y Fz(pairing)e Fu(h)p Fy(;)15 b Fu(i)41 b Fz(w)m(e)e(de\014ne)g(the)g(space-time)h(pairing)d Fu(h)p Fy(;)15 b Fu(i)2806 4838 y Fw(t;x)2936 4824 y Fz(as)39 b(usual:)57 b(for)39 b(an)m(y)h(pair)e(of)-47 4955 y(functions)29 b Fy(F)38 b Fu(2)25 b Fy(L)589 4910 y Fw(q)589 4979 y(t)627 4955 y Fy(L)689 4910 y Fw(p)689 4966 y(x)763 4955 y Fz(and)30 b Fy(G)25 b Fu(2)g Fy(L)1184 4910 y Fw(q)1218 4887 y Ft(0)1184 4979 y Fw(t)1244 4955 y Fy(L)1306 4910 y Fw(p)1342 4887 y Ft(0)1306 4966 y Fw(x)1399 4955 y Fz(with)k Fy(q)s(;)15 b(p)25 b Fu(2)g Fz([1)p Fy(;)15 b Fu(1)p Fz(])31 b(let)1215 5186 y Fu(h)p Fy(F)s(;)15 b(G)p Fu(i)1457 5200 y Fw(t;x)1573 5186 y Fz(=)1669 5062 y Fn(Z)1719 5268 y Fp(R)1786 5062 y Fn(Z)1837 5268 y Fp(R)1885 5249 y Fr(3)1939 5186 y Fy(F)e Fz(\()p Fy(t;)i(x)p Fz(\))p 2205 5107 268 4 v Fy(G)p Fz(\()p Fy(t;)g(x)p Fz(\))i Fy(dx)e(dt)1903 5472 y Fz(30)p eop %%Page: 31 31 31 30 bop -47 299 a Fz(W)-8 b(e)33 b(no)m(w)g(in)m(tro)s(duce)e(the)h (dual)f(op)s(erator)i Fu(K)1509 313 y Fw(R)1567 263 y Fx(\003)1567 322 y Fw(s;t)1677 299 y Fz(:)c Fy(L)1793 266 y Fq(2)1793 322 y Fw(x)1866 299 y Fu(!)f Fy(L)2047 266 y Fq(2)2047 322 y Fw(t)2086 299 y Fy(L)2158 215 y Fr(6)p 2158 227 31 3 v 2158 268 a(5)2148 311 y Fw(x)2203 299 y Fz(.)46 b(In)32 b(addition,)f(since)h Fu(K)3070 313 y Fw(R)3128 317 y(s)3193 299 y Fz(:)d Fy(L)3309 266 y Fq(2)3309 322 y Fw(t)3348 299 y Fy(L)3410 266 y Fq(6)3410 322 y Fw(x)3483 299 y Fu(!)f Fy(L)3664 266 y Fq(2)3664 322 y Fw(t)3703 299 y Fy(L)3765 266 y Fq(6)3765 322 y Fw(x)3841 299 y Fz(w)m(e)-47 460 y(also)f(de\014ne)g(the)h(dual)e(of)i Fu(K)915 474 y Fw(R)973 478 y(s)1009 460 y Fz(,)h Fu(K)1132 474 y Fw(R)1190 424 y Fx(\003)1190 482 y Fw(s)1254 460 y Fz(:)d Fy(L)1367 427 y Fq(2)1367 482 y Fw(t)1406 460 y Fy(L)1478 375 y Fr(6)p 1478 387 V 1478 428 a(5)1468 471 y Fw(x)1548 460 y Fu(!)f Fy(L)1726 427 y Fq(2)1726 482 y Fw(t)1765 460 y Fy(L)1837 375 y Fr(6)p 1837 387 V 1837 428 a(5)1827 471 y Fw(x)1882 460 y Fz(.)39 b(Therefore)28 b(for)f Fy(`)e Fu(\025)g Fz(1)j(the)g(left)f(hand-side)f(of)35 b(\(5.21\))30 b(can)-47 572 y(b)s(e)f(written)h(as)743 771 y Fy(I)783 785 y Fw(R;s;t)1021 771 y Fz(:=)83 b Fu(hK)1304 785 y Fw(R)1363 789 y(s;t)1444 771 y Fu(K)1513 785 y Fw(R)1571 733 y(`)p Fx(\000)p Fq(1)1571 793 y Fw(s)1695 771 y Fz(\()p Fu(K)1799 785 y Fw(s)1856 771 y Fu(\000)20 b(K)2016 785 y Fw(R)2074 789 y(s)2111 771 y Fz(\))p Fu(K)2216 733 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)2215 793 y Fw(s)2457 771 y Fy(e)2499 733 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2743 742 y Fr(0)2782 771 y Fy( )2841 785 y Fw(s)2878 771 y Fy(;)15 b(g)s Fu(i)1034 924 y Fz(=)95 b Fu(h)p Fz(\()p Fu(K)1339 938 y Fw(s)1397 924 y Fu(\000)20 b(K)1557 938 y Fw(R)1615 942 y(s)1652 924 y Fz(\))p Fu(K)1757 886 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)1756 946 y Fw(s)1998 924 y Fy(e)2040 886 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2284 895 y Fr(0)2323 924 y Fy( )2382 938 y Fw(s)2419 924 y Fy(;)15 b Fu(K)2528 938 y Fw(R)2587 886 y Fx(\003)2587 946 y Fw(s)2626 881 y(`)p Fx(\000)p Fq(1)2749 924 y Fu(K)2818 938 y Fw(R)2876 886 y Fx(\003)2876 946 y Fw(s;t)2958 924 y Fy(g)s Fu(i)3039 938 y Fw(t;x)3129 924 y Fy(:)-47 1122 y Fz(W)-8 b(e)31 b(assume)f(that)h Fy(")g Fz(and)f Fy(m)950 1136 y Fq(0)989 1122 y Fz(\()p Fy(")p Fz(\))i(are)e(no)m(w)h(\014xed)e(and)h(in)m(v)m(ok)m(e)h(Egoro)m (v's)h(theorem.)41 b(According)30 b(to)i(\(5.13\))1042 1320 y Fu(k)p Fy(V)1140 1334 y Fw(R)1199 1320 y Fz(\()p Fy(s)1277 1334 y Fq(1)1316 1320 y Fy(;)15 b Fu(\001)p Fz(\))21 b Fu(\000)f Fy(V)g Fz(\()p Fy(s)1679 1334 y Fq(1)1719 1320 y Fy(;)15 b Fu(\001)p Fz(\))p Fu(k)1864 1387 y Fw(L)1922 1321 y Fr(3)p 1923 1329 V 1923 1364 a(2)1912 1397 y Fs(x)1997 1320 y Fu(!)25 b Fz(0)p Fy(;)198 b Fz(as)91 b Fy(R)26 b Fu(!)f(1)-47 1559 y Fz(for)30 b(a.e.)42 b Fy(s)312 1573 y Fq(1)376 1559 y Fu(2)25 b Fz([)p Fy(s;)15 b(t)p Fz(].)41 b(Therefore,)30 b(for)g(an)m(y)h Fy(\016)e(>)c Fz(0)31 b(there)f(exists)g(a)h(set)g Fu(B)d(\032)d Fz([)p Fy(s;)15 b(t)p Fz(])30 b(suc)m(h)h(that)g Fu(jB)s(j)25 b(\024)g Fy(\016)34 b Fz(and)1012 1758 y Fu(k)p Fy(V)1110 1772 y Fw(R)1168 1758 y Fz(\()p Fy(s)1246 1772 y Fq(1)1286 1758 y Fy(;)15 b Fu(\001)p Fz(\))21 b Fu(\000)f Fy(V)g Fz(\()p Fy(s)1649 1772 y Fq(1)1688 1758 y Fy(;)15 b Fu(\001)p Fz(\))p Fu(k)1833 1824 y Fw(L)1891 1759 y Fr(3)p 1893 1767 V 1893 1802 a(2)1881 1835 y Fs(x)1967 1758 y Fy(<)25 b(";)197 b Fu(8)p Fy(s)2421 1772 y Fq(1)2485 1758 y Fu(2)25 b Fz([)p Fy(s;)15 b(t)p Fz(])21 b Fu(n)f(B)-47 1997 y Fz(and)31 b(all)g(su\016cien)m(tly)f(large)i Fy(R)c Fu(\025)f Fy(R)1203 2011 y Fq(0)1243 1997 y Fz(\()p Fy(";)15 b(\016)s Fz(\).)47 b(Let)32 b Fy(\037)1731 2011 y Fx(B)1815 1997 y Fz(b)s(e)f(the)h(c)m(haracteristic)h(function)d(of)i(the)g(set)g Fu(B)s Fz(.)45 b(W)-8 b(e)33 b(de\014ne)-47 2110 y(op)s(erators)1513 2308 y Fu(Y)1574 2322 y Fw(s)1636 2308 y Fz(=)25 b(\()p Fu(K)1836 2322 y Fw(s)1894 2308 y Fu(\000)20 b(K)2054 2322 y Fw(R)2112 2327 y(s)2148 2308 y Fz(\))p Fy(\037)2240 2322 y Fx(B)2293 2308 y Fy(;)1513 2446 y Fu(Z)1579 2460 y Fw(s)1641 2446 y Fz(=)25 b(\()p Fu(K)1841 2460 y Fw(s)1899 2446 y Fu(\000)20 b(K)2059 2460 y Fw(R)2117 2464 y(s)2154 2446 y Fz(\)\(1)h Fu(\000)f Fy(\037)2438 2460 y Fx(B)2490 2446 y Fz(\))p Fy(;)-47 2645 y Fz(It)28 b(is)f(easy)h(to)h(see)g(that)f Fu(Y)844 2659 y Fw(s)881 2645 y Fy(;)15 b Fu(Z)987 2659 y Fw(s)1049 2645 y Fz(:)26 b Fy(L)1162 2612 y Fq(2)1162 2667 y Fw(t)1201 2645 y Fy(L)1263 2612 y Fq(6)1263 2667 y Fw(x)1332 2645 y Fu(!)f Fy(L)1510 2612 y Fq(2)1510 2667 y Fw(t)1549 2645 y Fy(L)1611 2612 y Fq(6)1611 2667 y Fw(x)1655 2645 y Fz(.)40 b(Moreo)m(v)m(er,)31 b Fu(kZ)2254 2659 y Fw(s)2291 2645 y Fu(k)2336 2668 y Fw(L)2384 2645 y Fr(2)2384 2688 y Fs(t)2419 2668 y Fw(L)2467 2649 y Fr(6)2467 2685 y Fs(x)2506 2668 y Fx(!)p Fw(L)2625 2645 y Fr(2)2625 2688 y Fs(t)2659 2668 y Fw(L)2707 2649 y Fr(6)2707 2685 y Fs(x)2775 2645 y Fu(\024)25 b Fy(C)7 b(")28 b Fz(for)g(all)f Fy(R)f Fu(\025)f Fy(R)3534 2659 y Fq(0)3573 2645 y Fz(\()p Fy(";)15 b(\016)s Fz(\))30 b(and)-47 2775 y Fu(kY)59 2789 y Fw(s)96 2775 y Fu(k)141 2799 y Fw(L)189 2776 y Fr(2)189 2818 y Fs(t)224 2799 y Fw(L)272 2780 y Fr(6)272 2815 y Fs(x)311 2799 y Fx(!)p Fw(L)430 2776 y Fr(2)430 2818 y Fs(t)464 2799 y Fw(L)512 2780 y Fr(6)512 2815 y Fs(x)580 2775 y Fu(\024)25 b Fy(C)741 2789 y Fq(1)781 2775 y Fy(c)820 2789 y Fq(0)885 2775 y Fy(<)990 2739 y Fq(1)p 990 2754 36 4 v 990 2806 a(2)1036 2775 y Fz(,)30 b(see)h(\(5.12\))s(.)40 b(Therefore,)93 3000 y Fy(I)133 3014 y Fw(R;s;t)314 3000 y Fz(=)25 b Fu(hK)515 2963 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)514 3023 y Fw(s)756 3000 y Fy(e)798 2963 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1042 2972 y Fr(0)1081 3000 y Fy( )1140 3014 y Fw(s)1177 3000 y Fy(;)15 b Fu(Z)1290 2963 y Fx(\003)1283 3023 y Fw(s)1330 3000 y Fu(K)1399 3014 y Fw(R)1457 2963 y Fx(\003)1457 3023 y Fw(s)1496 2958 y(`)p Fx(\000)p Fq(1)1620 3000 y Fu(K)1689 3014 y Fw(R)1747 2963 y Fx(\003)1747 3023 y Fw(s;t)1828 3000 y Fy(g)s Fu(i)1909 3014 y Fw(t;x)2020 3000 y Fz(+)k Fu(h)p Fy(\037)2202 3014 y Fx(B)2255 3000 y Fu(K)2325 2963 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)2324 3023 y Fw(s)2566 3000 y Fy(e)2608 2963 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2852 2972 y Fr(0)2891 3000 y Fy( )2950 3014 y Fw(s)2987 3000 y Fy(;)c Fu(Y)3095 2963 y Fx(\003)3088 3023 y Fw(s)3135 3000 y Fu(K)3204 3014 y Fw(R)3262 2963 y Fx(\003)3262 3023 y Fw(s)3301 2958 y(`)p Fx(\000)p Fq(1)3425 3000 y Fu(K)3494 3014 y Fw(R)3551 2963 y Fx(\003)3551 3023 y Fw(s;t)3633 3000 y Fy(g)s Fu(i)3714 3014 y Fw(t;x)-47 3199 y Fz(W)-8 b(e)31 b(can)g(easily)e(estimate)j(the)e(\014rst)g(term) g(b)m(y)198 3397 y Fu(kK)312 3411 y Fw(s)350 3397 y Fu(k)395 3359 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)395 3435 y Fw(L)443 3413 y Fr(2)443 3455 y Fs(t)478 3435 y Fw(L)526 3417 y Fr(6)526 3452 y Fs(x)565 3435 y Fx(!)p Fw(L)684 3413 y Fr(2)684 3455 y Fs(t)718 3435 y Fw(L)766 3417 y Fr(6)766 3452 y Fs(x)809 3397 y Fu(kZ)927 3360 y Fx(\003)920 3420 y Fw(s)967 3397 y Fu(k)1012 3464 y Fw(L)1060 3441 y Fr(2)1060 3483 y Fs(t)1095 3464 y Fw(L)1153 3398 y Fr(6)p 1153 3406 31 3 v 1153 3441 a(5)1143 3474 y Fs(x)1198 3464 y Fx(!)p Fw(L)1317 3441 y Fr(2)1317 3483 y Fs(t)1351 3464 y Fw(L)1409 3398 y Fr(6)p 1409 3406 V 1409 3441 a(5)1399 3474 y Fs(x)1457 3397 y Fu(kK)1572 3360 y Fx(\003)1571 3420 y Fw(s)1613 3397 y Fu(k)1658 3359 y Fw(`)p Fx(\000)p Fq(1)1658 3478 y Fw(L)1706 3455 y Fr(2)1706 3498 y Fs(t)1741 3478 y Fw(L)1799 3413 y Fr(6)p 1799 3421 V 1799 3455 a(5)1789 3489 y Fs(x)1843 3478 y Fx(!)p Fw(L)1962 3455 y Fr(2)1962 3498 y Fs(t)1997 3478 y Fw(L)2055 3413 y Fr(6)p 2054 3421 V 2054 3455 a(5)2045 3489 y Fs(x)2103 3397 y Fu(kK)2218 3360 y Fx(\003)2217 3420 y Fw(s;t)2300 3397 y Fu(k)2345 3464 y Fw(L)2393 3445 y Fr(2)2393 3480 y Fs(x)2432 3464 y Fx(!)p Fw(L)2551 3441 y Fr(2)2551 3483 y Fs(t)2585 3464 y Fw(L)2643 3398 y Fr(6)p 2643 3406 V 2643 3441 a(5)2633 3474 y Fs(x)2692 3397 y Fu(k)p Fy( )2796 3411 y Fw(s)2833 3397 y Fu(k)2878 3417 y Fw(L)2926 3398 y Fr(2)2926 3434 y Fs(x)2970 3397 y Fu(k)p Fy(g)s Fu(k)3106 3417 y Fw(L)3154 3398 y Fr(2)3154 3434 y Fs(x)3224 3397 y Fu(\024)25 b Fy(")p Fz(2)3407 3360 y Fx(\000)p Fq(\()p Fw(m)p Fx(\000)p Fq(1\))3674 3397 y Fy(:)-47 3655 y Fz(F)-8 b(or)31 b(the)f(second)h(term)f(w)m(e)h(ha)m(v)m(e)h (the)e(b)s(ound)211 3836 y Fu(k)p Fy(\037)313 3850 y Fx(B)366 3836 y Fu(K)436 3799 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)435 3859 y Fw(s)677 3836 y Fy(e)719 3799 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)963 3808 y Fr(0)1002 3836 y Fy( )1061 3850 y Fw(s)1098 3836 y Fu(k)1143 3860 y Fw(L)1191 3837 y Fr(2)1191 3880 y Fs(t)1226 3860 y Fw(L)1274 3841 y Fr(6)1274 3876 y Fs(x)1317 3836 y Fu(kY)1430 3799 y Fx(\003)1423 3859 y Fw(s)1470 3836 y Fu(k)1515 3903 y Fw(L)1563 3880 y Fr(2)1563 3922 y Fs(t)1598 3903 y Fw(L)1656 3837 y Fr(6)p 1656 3845 V 1656 3880 a(5)1646 3913 y Fs(x)1701 3903 y Fx(!)p Fw(L)1820 3880 y Fr(2)1820 3922 y Fs(t)1854 3903 y Fw(L)1912 3837 y Fr(6)p 1912 3845 V 1912 3880 a(5)1902 3913 y Fs(x)1960 3836 y Fu(kK)2075 3799 y Fx(\003)2074 3859 y Fw(s)2116 3836 y Fu(k)2161 3798 y Fw(`)p Fx(\000)p Fq(1)2161 3917 y Fw(L)2209 3894 y Fr(2)2209 3937 y Fs(t)2244 3917 y Fw(L)2302 3852 y Fr(6)p 2302 3860 V 2302 3894 a(5)2292 3928 y Fs(x)2346 3917 y Fx(!)p Fw(L)2465 3894 y Fr(2)2465 3937 y Fs(t)2500 3917 y Fw(L)2558 3852 y Fr(6)p 2558 3860 V 2558 3894 a(5)2548 3928 y Fs(x)2606 3836 y Fu(kK)2721 3799 y Fx(\003)2720 3859 y Fw(s;t)2803 3836 y Fu(k)2848 3903 y Fw(L)2896 3884 y Fr(2)2896 3919 y Fs(x)2935 3903 y Fx(!)p Fw(L)3054 3880 y Fr(2)3054 3922 y Fs(t)3088 3903 y Fw(L)3146 3837 y Fr(6)p 3146 3845 V 3146 3880 a(5)3136 3913 y Fs(x)3195 3836 y Fu(k)p Fy( )3299 3850 y Fw(s)3336 3836 y Fu(k)3381 3856 y Fw(L)3429 3837 y Fr(2)3429 3873 y Fs(x)3473 3836 y Fu(k)p Fy(g)s Fu(k)3609 3856 y Fw(L)3657 3837 y Fr(2)3657 3873 y Fs(x)1986 4058 y Fu(\024)25 b(k)p Fy(\037)2184 4072 y Fx(B)2236 4058 y Fu(K)2306 4020 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)2305 4080 y Fw(s)2547 4058 y Fy(e)2589 4020 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2833 4029 y Fr(0)2872 4058 y Fy( )2931 4072 y Fw(s)2968 4058 y Fu(k)3013 4081 y Fw(L)3061 4058 y Fr(2)3061 4101 y Fs(t)3096 4081 y Fw(L)3144 4062 y Fr(6)3144 4098 y Fs(x)3187 4058 y Fz(\()p Fy(C)3287 4072 y Fq(1)3327 4058 y Fy(c)3366 4072 y Fq(0)3406 4058 y Fz(\))3441 4020 y Fw(`)p Fq(+1)-47 4258 y Fz(Observ)m(e)30 b(that)898 4371 y Fu(kK)1013 4333 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)1012 4393 y Fw(s)1254 4371 y Fy(e)1296 4333 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1540 4342 y Fr(0)1579 4371 y Fy( )1638 4385 y Fw(s)1675 4371 y Fu(k)1720 4394 y Fw(L)1768 4372 y Fr(2)1768 4414 y Fs(t)1803 4394 y Fw(L)1851 4376 y Fr(6)1851 4411 y Fs(x)1920 4371 y Fu(\024)25 b Fz(\()p Fy(C)2116 4385 y Fq(1)2155 4371 y Fy(c)2194 4385 y Fq(0)2234 4371 y Fz(\))2269 4333 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)2510 4371 y Fu(k)p Fy( )2614 4385 y Fw(s)2651 4371 y Fu(k)2696 4390 y Fw(L)2744 4372 y Fr(2)2744 4407 y Fs(x)2813 4371 y Fy(<)g Fu(1)-47 4535 y Fz(Therefore,)30 b(w)m(e)h(can)g(c)m(hose)g Fy(\016)e Fz(=)c Fy(\016)s Fz(\()p Fy(m)1254 4549 y Fq(0)1295 4535 y Fz(\))30 b(in)f(Egoro)m(v's)j (theorem)f(in)e(suc)m(h)h(a)h(w)m(a)m(y)g(that)866 4710 y Fw(m)928 4719 y Fr(0)963 4710 y Fq(\()p Fw(")p Fq(\))893 4746 y Fn(X)882 4940 y Fw(m)p Fq(=1)1066 4719 y Fw(m)p Fx(\000)p Fq(1)1076 4746 y Fn(X)1082 4944 y Fw(`)p Fq(=1)1233 4832 y Fu(k)p Fy(\037)1335 4846 y Fx(B)1388 4832 y Fu(K)1458 4795 y Fw(m)p Fx(\000)p Fw(`)p Fx(\000)p Fq(1)1457 4855 y Fw(s)1699 4832 y Fy(e)1741 4795 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1985 4804 y Fr(0)2024 4832 y Fy( )2083 4846 y Fw(s)2120 4832 y Fu(k)2165 4856 y Fw(L)2213 4833 y Fr(2)2213 4876 y Fs(t)2248 4856 y Fw(L)2296 4837 y Fr(6)2296 4873 y Fs(x)2364 4832 y Fu(\024)25 b Fy(")p Fz(\()p Fy(C)2602 4846 y Fq(1)2642 4832 y Fy(c)2681 4846 y Fq(0)2721 4832 y Fz(\))2756 4795 y Fw(m)2818 4804 y Fr(0)2854 4795 y Fx(\000)p Fw(`)p Fx(\000)p Fq(1)-47 5110 y Fz(Hence)31 b(w)m(e)g(ha)m(v)m(e)g(the)g(desired)e(b)s(ound)1539 5223 y Fu(j)p Fy(I)1604 5237 y Fw(R;s;t)1759 5223 y Fu(j)d(\024)f Fy(")p Fz(\()p Fy(C)2048 5237 y Fq(1)2088 5223 y Fy(c)2127 5237 y Fq(0)2167 5223 y Fz(\))2202 5185 y Fw(m)p Fx(\000)p Fq(1)1903 5472 y Fz(31)p eop %%Page: 32 32 32 31 bop -47 274 a Fz(for)30 b(all)f(1)d Fu(\024)f Fy(`)g Fu(\024)g Fy(m)20 b Fu(\000)g Fz(1)31 b(and)e Fy(m)c Fu(\024)g Fy(m)1268 288 y Fq(0)1308 274 y Fz(.)40 b(T)-8 b(o)31 b(settle)g(the)g(remaining)d(case)k(of)e Fy(`)25 b Fz(=)g(0)31 b(w)m(e)g(observ)m(e)g(that)g(for)f Fy(`)25 b Fz(=)g(0)-16 478 y Fy(I)24 492 y Fw(R;s;t)204 478 y Fz(=)g Fu(h)p Fz(\()p Fu(K)439 492 y Fw(s)497 478 y Fu(\000)20 b(K)657 492 y Fw(R)715 496 y(s)752 478 y Fz(\))p Fu(K)857 441 y Fw(m)p Fx(\000)p Fq(1)856 501 y Fw(s)1015 478 y Fy(e)1057 441 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1301 450 y Fr(0)1340 478 y Fy( )1399 492 y Fw(s)1436 478 y Fy(;)15 b(g)s Fu(i)p Fz(\()p Fy(t)p Fz(\))27 b(=)e Fu(hY)1879 492 y Fw(s)1916 478 y Fu(K)1986 441 y Fw(m)p Fx(\000)p Fq(1)1985 501 y Fw(s)2143 478 y Fy(e)2185 441 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)2429 450 y Fr(0)2468 478 y Fy( )2527 492 y Fw(s)2564 478 y Fy(;)15 b(g)s Fu(i)p Fz(\()p Fy(t)p Fz(\))22 b(+)e Fu(hZ)3002 492 y Fw(s)3039 478 y Fu(K)3109 441 y Fw(m)p Fx(\000)p Fq(1)3108 501 y Fw(s)3267 478 y Fy(e)3309 441 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)3553 450 y Fr(0)3592 478 y Fy( )3651 492 y Fw(s)3688 478 y Fy(;)15 b(g)s Fu(i)p Fz(\()p Fy(t)p Fz(\))-47 682 y(Similarly)26 b(to)31 b(the)g(op)s(erator)g Fu(K)1043 696 y Fw(R)1101 701 y(s;t)1213 682 y Fz(w)m(e)g(can)f(de\014ne)g(the)h(op)s(erators)f Fu(Y)2396 696 y Fw(s;t)2478 682 y Fy(;)15 b Fu(Z)2584 696 y Fw(s;t)2691 682 y Fz(:)26 b Fy(L)2804 649 y Fq(2)2804 705 y Fw(t)2843 682 y Fy(L)2905 649 y Fq(6)2905 705 y Fw(x)2974 682 y Fu(!)f Fy(L)3152 649 y Fq(2)3152 705 y Fw(x)3196 682 y Fz(.)40 b(Moreo)m(v)m(er,)988 887 y Fu(kZ)1099 901 y Fw(s;t)1181 887 y Fu(k)1226 910 y Fw(L)1274 887 y Fr(2)1274 930 y Fs(t)1309 910 y Fw(L)1357 891 y Fr(6)1357 927 y Fs(x)1396 910 y Fx(!)p Fw(L)1515 891 y Fr(2)1515 927 y Fs(x)1583 887 y Fu(\024)25 b Fy(C)7 b(";)197 b Fu(kY)2121 901 y Fw(s;t)2203 887 y Fu(k)2248 910 y Fw(L)2296 887 y Fr(2)2296 930 y Fs(t)2331 910 y Fw(L)2379 891 y Fr(6)2379 927 y Fs(x)2418 910 y Fx(!)p Fw(L)2537 891 y Fr(2)2537 927 y Fs(x)2605 887 y Fu(\024)25 b Fy(C)2766 901 y Fq(1)2805 887 y Fy(c)2844 901 y Fq(0)2884 887 y Fy(:)-47 1091 y Fz(Th)m(us)611 1204 y Fu(j)p Fy(I)676 1218 y Fw(R;s;t)832 1204 y Fu(j)g(\024)g Fy(C)1043 1218 y Fq(1)1083 1204 y Fy(c)1122 1218 y Fq(0)1176 1204 y Fu(k)p Fy(\037)1278 1218 y Fx(B)1331 1204 y Fu(K)1401 1166 y Fw(m)p Fx(\000)p Fq(1)1400 1226 y Fw(s)1558 1204 y Fy(e)1600 1166 y Fw(i)p Fq(\()p Fx(\001\000)p Fw(s)p Fq(\))p Fw(H)1844 1175 y Fr(0)1883 1204 y Fu(k)1928 1227 y Fw(L)1976 1205 y Fr(2)1976 1247 y Fs(t)2011 1227 y Fw(L)2059 1209 y Fr(6)2059 1244 y Fs(x)2123 1204 y Fz(+)19 b Fy(")p Fz(\()p Fy(C)2355 1218 y Fq(1)2396 1204 y Fy(c)2435 1218 y Fq(0)2474 1204 y Fz(\))2509 1166 y Fw(m)p Fx(\000)p Fq(1)2692 1204 y Fu(\024)25 b Fz(2)p Fy(")p Fz(\()p Fy(C)2975 1218 y Fq(1)3015 1204 y Fy(c)3054 1218 y Fq(0)3094 1204 y Fz(\))3129 1166 y Fw(m)p Fx(\000)p Fq(1)-47 1371 y Fz(b)m(y)30 b(the)h(c)m(hoice)g(of)f(the)h(constan)m(t)h Fy(\016)i Fz(in)29 b(Egoro)m(v's)i(theorem.)p 3879 1371 4 62 v 3883 1313 55 4 v 3883 1371 V 3937 1371 4 62 v -47 1558 a(Since)26 b(the)i(p)s(oten)m(tials)f Fy(V)811 1572 y Fw(R)869 1558 y Fz(\()p Fy(t;)15 b(x)p Fz(\))29 b(are)f(smo)s(oth)f(in)g(b)s(oth)g(v)-5 b(ariables,)27 b(the)h(solution)e(op)s(erators)i Fy(U)3233 1572 y Fw(R)3291 1558 y Fz(\()p Fy(t;)15 b(s)p Fz(\))28 b(are)g(unitary)-47 1671 y(on)i Fy(L)141 1638 y Fq(2)141 1694 y Fw(x)185 1671 y Fz(.)40 b(T)-8 b(ogether)32 b(with)d(Lemma)i(5.4)g(w)m(e)g(ha)m (v)m(e)g(the)g(follo)m(wing)-47 1859 y Fm(Corollary)k(5.5.)46 b Fl(The)g Fy(L)887 1826 y Fq(2)972 1859 y Fl(norm)g(of)g(the)g (solution)h Fy( )s Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))47 b Fl(of)f(the)g(Schr\177)-46 b(odinger)47 b(e)-5 b(quation)54 b Fz(\(5.2\))47 b Fl(is)f(a)g(non-)-47 1972 y(incr)-5 b(e)g(asing)33 b(function)g(of)g(time,)g(i.e,)1481 2085 y Fu(k)p Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )1843 2099 y Fw(s)1881 2085 y Fu(k)1926 2104 y Fw(L)1974 2085 y Fr(2)1974 2121 y Fs(x)2043 2085 y Fu(\024)25 b(k)p Fy( )2243 2099 y Fw(s)2280 2085 y Fu(k)2325 2104 y Fw(L)2373 2085 y Fr(2)2373 2121 y Fs(x)-47 2252 y Fl(for)33 b(al)5 b(l)33 b Fy(t)25 b Fu(\025)g Fy(s)32 b Fl(and)i(arbitr)-5 b(ary)35 b(functions)e Fy( )1468 2266 y Fw(s)1530 2252 y Fu(2)25 b Fy(L)1678 2219 y Fq(2)1678 2274 y Fw(x)1721 2252 y Fl(.)94 2439 y Fz(Lemma)36 b(5.4)h(also)f(implies)d(that)k(w)m (e)g(can)f(assume)g(henceforth)g(that)g Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))37 b(is)e(a)i(smo)s(oth)f(p)s(oten)m(tial)f(with) -47 2552 y(compact)42 b(supp)s(ort)e(in)g(the)i Fy(x)p Fz(-v)-5 b(ariable)40 b(and)h(the)h(v)-5 b(ariable)45 b(^)-50 b Fy(\034)51 b Fz(of)42 b(the)g(F)-8 b(ourier)41 b(transform)f(relativ)m(e)i(to)g Fy(t)p Fz(.)74 b(W)-8 b(e)-47 2665 y(can)36 b(also)h(assume)f(that)h Fy(V)56 b Fz(satis\014es)36 b(the)h(smallness)d(assumption)h(\(5.5\))r(.)58 b(W)-8 b(e)38 b(shall)d(sho)m(w)h(that)h(the)f(follo)m(wing)-47 2778 y(estimates)d(dep)s(end)e(only)h(on)g(the)h(norm)f(of)g(the)h(p)s (oten)m(tial)f(in)g(the)g(space)h Fy(Y)53 b Fz(de\014ned)31 b(in)h(De\014n)m(tion)g(5.1)i(and)e(the)-47 2891 y(smallness)c(constan) m(t)k Fy(c)759 2905 y Fq(0)799 2891 y Fz(.)-47 3134 y Ff(5.1)112 b(F)-9 b(unctional)37 b(calculus)-47 3306 y Fz(The)26 b(goal)i(of)f(this)f(section)i(is)e(to)i(obtain)e(the)i (explicit)d(represen)m(tation)i(of)g(the)h(in)m(tegral)f(k)m(ernels)f (of)h(the)h(op)s(erators)-47 3419 y(in)m(v)m(olv)m(ed)i(in)f(the)h (Neumann)g(series)g(expansion)f(\(5.6\))j(for)e Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\),)31 b(as)g(some)g(sp)s(ecial)e (oscillatory)g(in)m(tegrals.)94 3532 y(W)-8 b(e)31 b(in)m(tro)s(duce)f (the)g(notation)1444 3664 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)10 b(;)15 b Fu(\001)p Fz(\))27 b(:=)1850 3540 y Fn(Z)1956 3664 y Fy(e)1998 3626 y Fw(it\034)2091 3664 y Fy(V)20 b Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))g Fy(dt:)-47 3880 y Fz(The)25 b Fy(m)p Fz(-th)h(term)g(of)g(the)h(series)e(\(5.6\))r (,)i(whic)m(h)e(w)m(e)h(denote)h(b)m(y)f Fu(I)2146 3894 y Fw(m)2212 3880 y Fz(,)h(can)f(then)g(b)s(e)f(written)g(in)g(the)h (follo)m(wing)f(form)3881 3847 y Fq(2)3920 3880 y Fz(:)405 4126 y Fu(hI)490 4140 y Fw(m)556 4126 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )801 4140 y Fw(s)839 4126 y Fy(;)g(g)s Fu(i)26 b Fz(=)1082 4002 y Fn(Z)1133 4209 y Fp(R)1181 4190 y Fs(m)1259 4126 y Fy(d\034)1346 4140 y Fq(1)1385 4126 y Fy(::d\034)1522 4140 y Fw(m)1660 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Fy(;)31 b(g)s Fu(i)p Fy(:)-47 4609 y Fz(The)f(iden)m(tit)m(y)f(ab)s (o)m(v)m(e)j(is)d(v)m(eri\014ed)h(on)g(arbitrary)f(functions)g Fy( )2107 4623 y Fw(s)2144 4609 y Fy(;)15 b(g)29 b Fu(2)c Fy(L)2404 4576 y Fq(2)2404 4632 y Fw(x)2448 4609 y Fz(.)-47 4722 y(W)-8 b(e)31 b(shall)e(also)h(mak)m(e)i(use)e(of)g(the)h(sp)s (ectral)e(represen)m(tation)i(of)f(the)h(op)s(erator)g Fy(e)2764 4689 y Fw(itH)2871 4698 y Fr(0)2910 4722 y Fz(,)1532 4963 y Fy(e)1574 4926 y Fw(itH)1681 4935 y Fr(0)1746 4963 y Fz(=)1842 4839 y Fn(Z)1892 5045 y Fp(R)1960 4963 y Fy(e)2002 4926 y Fw(it\025)2097 4963 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))p Fy(:)p -47 5061 1597 4 v 56 5115 a Fr(2)91 5147 y Fg(Here)26 b(w)n(e)g(use)f(the)h(fact)g(that)f Fe(V)17 b Fg(\()t(^)-42 b Fe(\034)9 b(;)k Fd(\001)p Fg(\))25 b(has)h(compact)g(supp)r(ort)f(in)30 b(^)-42 b Fe(\034)33 b Fg(to)26 b(in)n(terc)n(hange)g(the)f(in)n(tegrals.)1903 5472 y Fz(32)p eop %%Page: 33 33 33 32 bop -47 274 a Fz(Here,)38 b Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))38 b(is)e(the)g(sp)s(ectral)g(measure)g(asso)s(ciated)h(with)e (the)i(op)s(erator)g Fy(H)2712 288 y Fq(0)2786 274 y Fz(=)e Fu(\000)p Fz(\001.)59 b(In)35 b(dimension)f Fy(n)h Fz(=)g(3,)-47 387 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))31 b(has)g(an)f(explicit)f(represen)m(tation)h(as)h(an)f(in)m(tegral)g(op) s(erator)h(with)e(the)h(k)m(ernel)1199 670 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))28 b(=)1775 515 y Fn(\()1858 564 y Fq(sin)1956 507 y Fx(p)p 2015 507 42 3 v 57 x Fw(\025)p Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p 1858 588 370 4 v 1918 641 a Fq(4)p Fw(\031)r Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)2252 609 y Fy(d\025)92 b(\025)25 b(>)g Fz(0)p Fy(;)1848 753 y Fz(0)551 b Fy(\025)25 b Fu(\024)g Fz(0)-47 961 y(Recall)32 b(also)h(that)g(the)g(resolv)m(en) m(t)h Fy(R)q Fz(\()p Fy(z)t Fz(\))c(=)f(\()p Fy(H)1584 975 y Fq(0)1645 961 y Fu(\000)22 b Fy(z)t Fz(\))1819 928 y Fx(\000)p Fq(1)1947 961 y Fz(is)31 b(an)i(analytic)g(function)e (with)h(v)-5 b(alues)32 b(in)f(the)i(space)h(of)-47 1074 y(b)s(ounded)28 b(op)s(erators)i(in)g Fy(z)f Fu(2)c Fv(C)44 b Fu(n)20 b Fv(R)1201 1088 y Fq(+)1266 1074 y Fz(.)41 b(In)29 b(the)i(ab)s(o)m(v)m(e)h(domain,)1603 1318 y Fy(R)q Fz(\()p Fy(z)t Fz(\))26 b(=)1911 1194 y Fn(Z)1962 1400 y Fp(R)2039 1256 y Fy(dE)5 b Fz(\()p Fy(\026)p Fz(\))p 2039 1297 246 4 v 2056 1380 a Fy(\026)20 b Fu(\000)f Fy(z)-47 1318 y Fz(\(5.23\))-47 1560 y(W)-8 b(e)31 b(shall)e(use)h(the) h(follo)m(wing)e(simpli\014ed)d(v)m(ersion)k(of)h(the)g(limiting)c (absorption)i(principle)e(stating)k(that)g Fy(R)q Fz(\()p Fy(z)t Fz(\))26 b(=)-47 1673 y Fy(R)q Fz(\()p Fy(\025)g Fz(+)h Fy(ib)p Fz(\))40 b(has)g(w)m(ell-de\014ned)e(op)s(erator)i (limits)e Fy(R)1767 1687 y Fq(+)1826 1673 y Fz(\()p Fy(\025)p Fz(\))j(and)e Fy(R)2245 1687 y Fx(\000)2304 1673 y Fz(\()p Fy(\025)p Fz(\),)44 b(for)39 b Fy(\025)j(>)f Fz(0,)i(as)d Fy(b)h Fu(!)g Fz(0)3342 1640 y Fq(+)3442 1673 y Fz(and)e Fy(b)i Fu(!)g Fz(0)3885 1640 y Fx(\000)-47 1786 y Fz(resp)s(ectiv)m (ely)-8 b(.)38 b(The)23 b(op)s(erators)i Fy(R)1120 1800 y Fx(\006)1179 1786 y Fz(\()p Fy(\025)p Fz(\))f(map)g(the)g(space)g(of) g(Sc)m(h)m(w)m(artz)i(functions)c Fu(S)31 b Fz(in)m(to)24 b(the)g(space)g Fy(C)3493 1753 y Fx(1)3575 1786 y Fu(\\)7 b Fy(L)3705 1753 y Fq(4)3744 1786 y Fz(\()p Fv(R)3839 1753 y Fq(3)3884 1786 y Fz(\).)94 1899 y(On)24 b(the)i(real)f(axis,)h (the)g(resolv)m(en)m(t)g Fy(R)q Fz(\()p Fy(\025)p Fz(\))g(can)f(b)s(e)g (then)g(describ)s(ed)e(explicitly)g(as)j(the)g(in)m(tegral)f(op)s (erators)g(with)-47 2012 y(the)30 b(k)m(ernels)458 2261 y Fy(R)527 2275 y Fq(+)586 2261 y Fz(\()p Fy(\025)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)57 b(lim)1042 2326 y Fw(")p Fx(!)p Fq(0)1181 2308 y Fr(+)1246 2261 y Fy(R)q Fz(\()p Fy(\025)21 b Fz(+)f Fy(i")p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)1977 2199 y Fy(e)2019 2166 y Fw(i)2043 2109 y Fx(p)p 2102 2109 42 3 v 57 x Fw(\025)p Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p 1967 2240 362 4 v 1967 2323 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)s Fu(j)2339 2261 y Fy(;)197 b(\025)25 b Fu(\025)g Fz(0)-2827 b(\(5.24\))458 2536 y Fy(R)527 2550 y Fx(\000)586 2536 y Fz(\()p Fy(\025)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)57 b(lim)1042 2602 y Fw(")p Fx(!)p Fq(0)1181 2583 y Fr(+)1246 2536 y Fy(R)q Fz(\()p Fy(\025)21 b Fu(\000)f Fy(i")p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)p 1957 2457 463 4 v 25 w Fy(R)2026 2550 y Fq(+)2085 2536 y Fz(\()p Fy(\025)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)2551 2474 y Fy(e)2593 2441 y Fx(\000)p Fw(i)2672 2385 y Fx(p)p 2731 2385 42 3 v 56 x Fw(\025)p Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p 2551 2515 397 4 v 2568 2598 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)s Fu(j)2957 2536 y Fy(;)197 b(\025)26 b Fu(\025)f Fz(0)p Fy(;)458 2808 y(R)q Fz(\()p Fy(\025)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)p 984 2729 405 4 v 25 w Fy(R)q Fz(\()p Fy(\025)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))26 b(=)1519 2746 y Fy(e)1561 2713 y Fx(\000)1616 2660 y(p)p 1675 2660 97 3 v 53 x(\000)p Fw(\025)p Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p 1519 2787 428 4 v 1552 2870 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)s Fu(j)1957 2808 y Fy(;)197 b(\025)25 b(<)g Fz(0)p Fy(:)-47 3048 y Fz(In)k(particular,)g(w)m(e)i (can)g(write)1624 3244 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))27 b(=)e Fu(=)p Fy(R)q Fz(\()p Fy(\025)p Fz(\))p Fy(:)-2320 b Fz(\(5.25\))-47 3440 y(W)-8 b(e)31 b(shall)e(mak)m(e)i (rep)s(eated)g(use)f(of)g(the)h(follo)m(wing)e(regularization:)756 3570 y Fn(Z)847 3597 y Fw(b)807 3777 y(a)897 3694 y Fy(e)939 3657 y Fw(i\013q)1046 3694 y Fy(dq)g Fz(=)1269 3633 y Fy(e)1311 3600 y Fw(i)p Fq(\()p Fw(\013)p Fq(+)p Fw(i)p Fq(0\))p Fw(b)1604 3633 y Fu(\000)20 b Fy(e)1737 3600 y Fw(i)p Fq(\()p Fw(\013)p Fq(+)p Fw(i)p Fq(0\))p Fw(a)p 1269 3673 749 4 v 1520 3756 a Fy(\013)h Fz(+)f Fy(i)p Fz(0)2053 3694 y(=)56 b(lim)2149 3760 y Fw(")p Fx(!)p Fq(0)2288 3741 y Fr(+)2363 3633 y Fy(e)2405 3600 y Fw(i)p Fq(\()p Fw(\013)p Fq(+)p Fw(i")p Fq(\))p Fw(b)2695 3633 y Fu(\000)20 b Fy(e)2828 3600 y Fw(i)p Fq(\()p Fw(\013)p Fq(+)p Fw(i")p Fq(\))p Fw(a)p 2363 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y Fn(Z)1723 4591 y Fw(\025)1784 4509 y Fy(e)1826 4471 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2079 4395 y(m)p Fq(+1)2089 4422 y Fn(X)2091 4620 y Fw(k)r Fq(=0)2246 4380 y Fn(\034)q(\022)2397 4395 y Fw(k)r Fx(\000)p Fq(1)2403 4422 y Fn(Y)2399 4616 y Fw(r)r Fq(=1)2540 4509 y Fy(R)2609 4523 y Fq(+)2668 4509 y Fz(\()p Fy(\025)c Fz(+)f Fy(\034)2908 4523 y Fw(r)2966 4509 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)3258 4523 y Fw(m)3325 4509 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)3508 4523 y Fw(r)3546 4509 y Fy(;)15 b Fu(\001)p Fz(\))3646 4380 y Fn(\023)963 4828 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)1322 4843 y Fw(k)1385 4828 y Fz(+)g Fy(::)h Fz(+)e Fy(\034)1677 4842 y Fw(m)1744 4828 y Fz(\))1779 4699 y Fn(\022)1894 4714 y Fw(m)p Fq(+1)1912 4741 y Fn(Y)1861 4939 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2094 4828 y Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)2242 4842 y Fw(r)r Fx(\000)p Fq(1)2370 4828 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2539 4842 y Fx(\000)2599 4828 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)2839 4842 y Fw(r)2897 4828 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)3189 4842 y Fw(m)p Fq(+1)3346 4828 y Fz(\))3381 4699 y Fn(\023)3448 4828 y Fy( )3507 4842 y Fw(s)3544 4828 y Fy(;)30 b(g)3645 4699 y Fn(\035)3714 4828 y Fy(;)-3771 b Fz(\(5.26\))-47 5110 y Fl(wher)-5 b(e)41 b(we)g(formal)5 b(ly)42 b(set)f Fy(\034)919 5124 y Fw(m)p Fq(+1)1115 5110 y Fz(=)e(0)p Fl(.)65 b(The)41 b(r)-5 b(epr)g(esentation)43 b(holds)g(true)d(with)h(arbitr)-5 b(ary)43 b(Schwartz)f(functions)-47 5223 y Fy( )12 5237 y Fw(s)49 5223 y Fy(;)15 b(g)29 b Fu(2)c(S)7 b Fl(.)1903 5472 y Fz(33)p eop %%Page: 34 34 34 33 bop -47 274 a Fl(Pr)-5 b(o)g(of.)47 b Fz(W)-8 b(e)26 b(start)h(b)m(y)e(v)m(erifying)f(that)i(the)g(expression)f(on)g(the)h (righ)m(t)f(hand-side)f(of)33 b(\(5.26\))28 b(de\014nes)c(an)i (absolutely)-47 387 y(con)m(v)m(ergen)m(t)39 b(in)m(tegral.)59 b(Recall)36 b(that)i(the)f(p)s(oten)m(tial)f Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)10 b(;)15 b(x)p Fz(\))38 b(is)d(smo)s(oth)i(and)f(has)g(compact)i(supp)s(ort)d(in)h(b)s(oth)-47 500 y(v)-5 b(ariables.)38 b(Therefore,)27 b(the)f(v)-5 b(ariables)25 b Fy(\034)1363 514 y Fq(1)1402 500 y Fy(;)15 b(::;)g(\034)1572 514 y Fw(m)1666 500 y Fz(are)27 b(restricted)f(to)h (a)f(\014nite)f(in)m(terv)-5 b(al)26 b(of)g Fv(R)s Fz(.)45 b(It)27 b(also)f(follo)m(ws,)g(with)-47 613 y(the)34 b(help)e(of)j(our)e(v)m(ersion)g(of)i(the)f(limiting)d(absorption)i (principle,)e(that)k(the)f(op)s(erators)g Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)11 b(;)k Fu(\001)p Fz(\))p Fy(R)3460 627 y Fx(\006)3520 613 y Fz(\()p Fy(\025)p Fz(\))34 b(map)g Fu(S)-47 726 y Fz(in)m(to)c Fu(S)37 b Fz(for)30 b(all)k(^)-49 b Fy(\034)10 b(;)15 b(\025)25 b Fu(2)g Fv(R)s Fz(.)47 b(In)29 b(addition,)g(w)m(e)i(ha)m(v)m(e)g(that)1396 930 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))p Fy(f)36 b Fz(=)25 b Fy(\025)1868 892 y Fx(\000)p Fw(N)1990 930 y Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\)\(\001\))2378 892 y Fw(N)2447 930 y Fy(f)-47 1134 y Fz(for)30 b(an)g(arbitrary)f(Sc)m(h)m(w)m(artz)j (function)d Fy(f)10 b Fz(.)40 b(Hence,)512 1288 y Fn(\014)512 1343 y(\014)512 1398 y(\014)512 1452 y(\014)542 1292 y(\034\022)693 1307 y Fw(k)r Fx(\000)p Fq(1)699 1334 y Fn(Y)695 1528 y Fw(r)r Fq(=1)836 1420 y Fy(R)905 1434 y Fq(+)964 1420 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)1204 1434 y Fw(r)1262 1420 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)1554 1434 y Fw(m)1621 1420 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)1804 1434 y Fw(r)1842 1420 y Fy(;)15 b Fu(\001)p Fz(\))1942 1292 y Fn(\023)-47 1420 y Fz(\(5.27\))876 1736 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)1235 1751 y Fw(k)1297 1736 y Fz(+)g Fy(::)h Fz(+)f Fy(\034)1590 1750 y Fw(m)1656 1736 y Fz(\))1691 1608 y Fn(\022)1806 1622 y Fw(m)p Fq(+1)1824 1650 y Fn(Y)1774 1847 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2006 1736 y Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)2154 1750 y Fw(r)r Fx(\000)p Fq(1)2283 1736 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2452 1750 y Fx(\000)2512 1736 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\034)2751 1750 y Fw(r)2809 1736 y Fz(+)g Fy(::)h Fz(+)f Fy(\034)3102 1750 y Fw(m)3168 1736 y Fz(\))3203 1608 y Fn(\023)3270 1736 y Fy( )3329 1750 y Fw(s)3366 1736 y Fy(;)15 b(g)3452 1608 y Fn(\035)3521 1604 y(\014)3521 1659 y(\014)3521 1713 y(\014)3521 1768 y(\014)512 1979 y Fu(\024)25 b Fy(C)7 b Fz(\(1)20 b(+)g Fu(j)p Fy(\025)p Fu(j)p Fz(\))1009 1942 y Fx(\000)p Fw(N)-47 1979 y Fz(\(5.28\))-47 2184 y(for)k(arbitrary)g(Sc)m(h)m(w)m(artz)i(functions)d Fy( )1294 2198 y Fw(s)1356 2184 y Fz(and)h Fy(g)29 b Fz(with)23 b(a)j(constan)m(t)g Fy(C)31 b Fz(dep)s(ending)22 b(on)j Fy( )2939 2198 y Fw(s)2976 2184 y Fz(,)h Fy(g)s Fz(,)h(and)d Fy(V)45 b Fz(\(in)24 b(particular,)-47 2297 y(on)37 b(the)i(size)e(of)h (the)g(supp)s(ort)e(of)i Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)10 b(;)15 b(x)p Fz(\))39 b(in)i(^)-50 b Fy(\034)10 b Fz(\).)64 b(This)36 b(can)i(b)s(e)f(seen)h(most)g(easily)f(b)m(y)h (mo)m(ving)f(the)i(op)s(erator)-47 2409 y(in)29 b(\(5.27\))j(on)m(to)g Fy(g)s Fz(.)-47 2522 y(In)23 b(what)h(follo)m(ws)f(w)m(e)h(shall)e (manipulate)h(the)h(op)s(erator)g(v)-5 b(alued)23 b(expressions)g(with) f(the)i(tacit)h(understanding)c(that)-47 2635 y(all)31 b(equalities)g(are)h(to)h(b)s(e)f(in)m(terpreted)f(in)g(the)i(w)m(eak)g (sense.)46 b(Ho)m(w)m(ev)m(er,)35 b(for)d(ease)h(of)f(notation)h(w)m(e) f(suppress)f(the)-47 2748 y(pairing)g(with)h(the)i(Sc)m(h)m(w)m(artz)g (functions)e Fy( )1477 2762 y Fw(s)1548 2748 y Fz(and)h Fy(g)s Fz(.)50 b(The)33 b(absolute)g(con)m(v)m(ergence)j(of)d(all)g(of) g(in)m(tegrals)g(in)m(v)m(olv)m(ed)-47 2861 y(\(after)e(silen)m(t)e (pairing)g(with)g Fy( )1018 2875 y Fw(s)1055 2861 y Fy(;)15 b(g)s Fz(\))32 b(will)27 b(also)k(allo)m(w)e(us)h(to)h(freely)f(in)m (terc)m(hange)h(the)g(order)e(of)i(in)m(tegrations.)-47 2974 y(W)-8 b(e)31 b(replace)g(eac)m(h)g(of)g(the)f Fy(e)924 2941 y Fw(i)p Fq(\()p Fw(s)1008 2953 y Fs(k)1047 2941 y Fx(\000)p Fw(s)1135 2953 y Fs(k)q Ft(\000)p Fr(1)1251 2941 y Fq(\))p Fw(H)1336 2950 y Fr(0)1405 2974 y Fz(in)f(\(5.22\))k (with)c(its)h(sp)s(ectral)g(represen)m(tation:)113 3220 y Fu(I)163 3234 y Fw(m)229 3220 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)537 3096 y Fn(Z)588 3303 y Fp(R)636 3284 y Fs(m)713 3220 y Fy(d\034)800 3234 y Fq(1)840 3220 y Fy(::d\034)977 3234 y Fw(m)1059 3096 y Fn(Z)1165 3220 y Fu(\001)15 b(\001)g(\001)1286 3096 y Fn(Z)1030 3360 y Fw(\025)1071 3369 y Fr(1)1106 3360 y Fw(;::;\025)1227 3369 y Fs(m)p Fr(+1)1479 3096 y Fn(Z)1585 3220 y Fu(\001)g(\001)g(\001) 1706 3096 y Fn(Z)1377 3358 y Fw(s)p Fx(\024)p Fw(s)1498 3366 y Fs(m)1556 3358 y Fx(\024)p Fw(::)p Fx(\024)p Fw(s)1739 3367 y Fr(1)1772 3358 y Fx(\024)p Fw(t)1868 3220 y Fy(ds)1958 3234 y Fq(1)1997 3220 y Fy(:::ds)2162 3234 y Fw(m)2255 3220 y Fy(e)2297 3183 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)2461 3192 y Fr(1)2496 3183 y Fq(\))p Fw(\025)2564 3192 y Fr(1)2604 3220 y Fy(e)2646 3183 y Fw(is)2703 3192 y Fr(1)2737 3183 y Fw(\034)2768 3192 y Fr(1)2807 3220 y Fy(dE)5 b Fz(\()p Fy(\025)3014 3234 y Fq(1)3055 3220 y Fz(\))15 b Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)3253 3234 y Fq(1)3293 3220 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)3435 3183 y Fw(i)p Fq(\()p Fw(s)3519 3192 y Fr(1)3555 3183 y Fx(\000)p Fw(s)3643 3192 y Fr(2)3677 3183 y Fq(\))p Fw(\025)3745 3192 y Fr(2)634 3505 y Fy(e)676 3468 y Fw(is)733 3477 y Fr(2)767 3468 y Fw(\034)798 3477 y Fr(2)837 3505 y Fy(dE)5 b Fz(\()p Fy(\025)1044 3519 y Fq(2)1085 3505 y Fz(\))p Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1268 3519 y Fq(2)1308 3505 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(:::e)1525 3468 y Fw(i)p Fq(\()p Fw(s)1609 3477 y Fs(m)p Ft(\000)p Fr(1)1748 3468 y Fx(\000)p Fw(s)1836 3476 y Fs(m)1894 3468 y Fq(\))p Fw(\025)1962 3476 y Fs(m)2026 3505 y Fy(e)2068 3468 y Fw(is)2125 3476 y Fs(m)2183 3468 y Fw(\034)2214 3476 y Fs(m)2277 3505 y Fy(dE)5 b Fz(\()p Fy(\025)2484 3519 y Fw(m)2552 3505 y Fz(\))p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)2736 3519 y Fw(m)2802 3505 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)2944 3468 y Fw(i)p Fq(\()p Fw(s)3028 3476 y Fs(m)3088 3468 y Fx(\000)p Fw(s)p Fq(\))p Fw(\025)3244 3477 y Fs(m)p Fr(+1)3384 3505 y Fy(dE)5 b Fz(\()p Fy(\025)3591 3519 y Fw(m)p Fq(+1)3749 3505 y Fz(\))121 3696 y(=)217 3573 y Fn(Z)267 3779 y Fp(R)315 3760 y Fs(m)393 3696 y Fy(d\034)480 3710 y Fq(1)520 3696 y Fy(::d\034)657 3710 y Fw(m)739 3573 y Fn(Z)845 3696 y Fu(\001)15 b(\001)g(\001)966 3573 y Fn(Z)710 3836 y Fw(\025)751 3845 y Fr(1)786 3836 y Fw(;::;\025)907 3845 y Fs(m)p Fr(+1)1158 3573 y Fn(Z)1264 3696 y Fu(\001)g(\001)g(\001)1386 3573 y Fn(Z)1057 3834 y Fw(s)p Fx(\024)p Fw(s)1178 3842 y Fs(m)1236 3834 y Fx(\024)p Fw(::)p Fx(\024)p Fw(s)1419 3843 y Fr(1)1452 3834 y Fx(\024)p Fw(t)1548 3696 y Fy(ds)1638 3710 y Fq(1)1677 3696 y Fy(:::ds)1842 3710 y Fw(m)1934 3696 y Fy(e)1976 3659 y Fw(it\025)2066 3668 y Fr(1)2106 3696 y Fy(dE)5 b Fz(\()p Fy(\025)2313 3710 y Fq(1)2354 3696 y Fz(\))15 b Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)2552 3710 y Fq(1)2592 3696 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)2734 3659 y Fw(is)2791 3668 y Fr(1)2827 3659 y Fq(\()p Fw(\034)2885 3668 y Fr(1)2920 3659 y Fx(\000)p Fw(\025)3016 3668 y Fr(1)3051 3659 y Fq(+)p Fw(\025)3147 3668 y Fr(2)3181 3659 y Fq(\))3213 3696 y Fy(dE)5 b Fz(\()p Fy(\025)3420 3710 y Fq(2)3460 3696 y Fz(\))p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)3644 3710 y Fq(2)3683 3696 y Fy(;)15 b Fu(\001)p Fz(\))1079 3982 y Fy(e)1121 3944 y Fw(is)1178 3953 y Fr(2)1213 3944 y Fq(\()p Fw(\034)1271 3953 y Fr(2)1306 3944 y Fx(\000)p Fw(\025)1402 3953 y Fr(2)1437 3944 y Fq(+)p Fw(\025)1533 3953 y Fr(3)1568 3944 y Fq(\))1599 3982 y Fy(:::)g(dE)5 b Fz(\()p Fy(\025)1896 3996 y Fw(m)1965 3982 y Fz(\))p Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)2148 3996 y Fw(m)2215 3982 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)2357 3944 y Fw(is)2414 3952 y Fs(m)2474 3944 y Fq(\()p Fw(\034)2532 3952 y Fs(m)2591 3944 y Fx(\000)p Fw(\025)2687 3952 y Fs(m)2746 3944 y Fq(+)p Fw(\025)2842 3953 y Fs(m)p Fr(+1)2977 3944 y Fq(\))3009 3982 y Fy(dE)5 b Fz(\()p Fy(\025)3216 3996 y Fw(m)p Fq(+1)3374 3982 y Fz(\))15 b Fy(e)3466 3944 y Fx(\000)p Fw(is\025)3619 3953 y Fs(m)p Fr(+1)3759 3982 y Fy(:)-3831 b Fz(\(5.29\))-47 4186 y(Consider)28 b(the)j(\014rst)e(term)531 4424 y Fu(I)581 4438 y Fq(1)645 4424 y Fz(=)741 4300 y Fn(Z)791 4506 y Fp(R)859 4424 y Fy(d\034)946 4438 y Fq(1)1000 4300 y Fn(Z)1051 4506 y Fw(\025)1092 4515 y Fr(1)1127 4506 y Fw(;\025)1188 4515 y Fr(2)1241 4300 y Fn(Z)1332 4326 y Fw(t)1292 4506 y(s)1377 4424 y Fy(ds)1467 4438 y Fq(1)1532 4424 y Fy(e)1574 4386 y Fw(it\025)1664 4395 y Fr(1)1703 4424 y Fy(dE)5 b Fz(\()p Fy(\025)1910 4438 y Fq(1)1951 4424 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)2150 4438 y Fq(1)2189 4424 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(e)2331 4386 y Fw(is)2388 4395 y Fr(1)2424 4386 y Fq(\()p Fw(\034)2482 4395 y Fr(1)2517 4386 y Fx(\000)p Fw(\025)2613 4395 y Fr(1)2648 4386 y Fq(+)p Fw(\025)2744 4395 y Fr(2)2778 4386 y Fq(\))2810 4424 y Fy(dE)5 b Fz(\()p Fy(\025)3017 4438 y Fq(2)3057 4424 y Fz(\))p Fy(e)3134 4386 y Fx(\000)p Fw(is\025)3287 4395 y Fr(2)3326 4424 y Fy(:)1903 5472 y Fz(34)p eop %%Page: 35 35 35 34 bop -47 274 a Fz(In)m(tegrating)31 b(explicitly)d(relativ)m(e)i (to)h Fy(s)1294 288 y Fq(1)1364 274 y Fz(w)m(e)f(infer)f(that)150 507 y Fu(I)200 521 y Fq(1)239 507 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)f Fu(\000)p Fy(i)664 383 y Fn(Z)715 589 y Fp(R)782 507 y Fy(d\034)869 521 y Fq(1)924 383 y Fn(Z)974 589 y Fw(\025)1015 598 y Fr(1)1050 589 y Fw(;\025)1111 598 y Fr(2)1165 507 y Fy(e)1207 469 y Fw(it\025)1297 478 y Fr(1)1336 507 y Fy(dE)5 b Fz(\()p Fy(\025)1543 521 y Fq(1)1584 507 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)1783 521 y Fq(1)1822 507 y Fy(;)15 b Fu(\001)p Fz(\))1932 445 y Fy(e)1974 412 y Fw(it)p Fq(\()p Fw(\034)2081 421 y Fr(1)2118 412 y Fx(\000)p Fw(\025)2214 421 y Fr(1)2249 412 y Fq(+)p Fw(\025)2345 421 y Fr(2)2380 412 y Fq(+)p Fw(i)p Fq(0\))2546 445 y Fu(\000)k Fy(e)2678 412 y Fw(is)p Fq(\()p Fw(\034)2793 421 y Fr(1)2829 412 y Fx(\000)p Fw(\025)2925 421 y Fr(1)2959 412 y Fq(+)p Fw(\025)3055 421 y Fr(2)3090 412 y Fq(+)p Fw(i)p Fq(0\))p 1932 486 1303 4 v 2247 569 a Fy(\034)2287 583 y Fq(1)2347 569 y Fu(\000)h Fy(\025)2491 583 y Fq(1)2550 569 y Fz(+)g Fy(\025)2694 583 y Fq(2)2754 569 y Fz(+)g Fy(i)p Fz(0)3246 507 y Fy(dE)5 b Fz(\()p Fy(\025)3453 521 y Fq(2)3493 507 y Fz(\))p Fy(e)3570 469 y Fx(\000)p Fw(is\025)3723 478 y Fr(2)451 780 y Fz(=)25 b Fu(\000)p Fy(i)664 656 y Fn(Z)715 862 y Fp(R)782 780 y Fy(d\034)869 794 y Fq(1)908 780 y Fy(e)950 742 y Fw(it)p Fq(\()p Fw(\034)1057 751 y Fr(1)1094 742 y Fq(+)p Fw(i)p Fq(0\))1255 656 y Fn(Z)1305 862 y Fw(\025)1346 871 y Fr(1)1381 862 y Fw(;\025)1442 871 y Fr(2)1496 780 y Fy(dE)5 b Fz(\()p Fy(\025)1703 794 y Fq(1)1743 780 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)1942 794 y Fq(1)1981 780 y Fy(;)15 b Fu(\001)p Fz(\))2272 718 y Fy(e)2314 685 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2546 694 y Fr(2)p 2092 759 675 4 v 2092 842 a Fy(\034)2132 856 y Fq(1)2192 842 y Fu(\000)20 b Fy(\025)2336 856 y Fq(1)2395 842 y Fz(+)g Fy(\025)2539 856 y Fq(2)2599 842 y Fz(+)g Fy(i)p Fz(0)2776 780 y Fy(dE)5 b Fz(\()p Fy(\025)2983 794 y Fq(2)3024 780 y Fz(\))446 1053 y(+)20 b Fy(i)583 929 y Fn(Z)634 1135 y Fp(R)701 1053 y Fy(d\034)788 1067 y Fq(1)828 1053 y Fy(e)870 1015 y Fw(is)p Fq(\()p Fw(\034)985 1024 y Fr(1)1020 1015 y Fq(+)p Fw(i)p Fq(0\))1181 929 y Fn(Z)1231 1135 y Fw(\025)1272 1144 y Fr(1)1307 1135 y Fw(;\025)1368 1144 y Fr(2)1422 1053 y Fy(dE)5 b Fz(\()p Fy(\025)1629 1067 y Fq(1)1669 1053 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)1868 1067 y Fq(1)1908 1053 y Fy(;)15 b Fu(\001)p Fz(\))2199 991 y Fy(e)2241 958 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2473 967 y Fr(1)p 2019 1032 V 2019 1115 a Fy(\034)2059 1129 y Fq(1)2118 1115 y Fu(\000)20 b Fy(\025)2262 1129 y Fq(1)2322 1115 y Fz(+)f Fy(\025)2465 1129 y Fq(2)2525 1115 y Fz(+)h Fy(i)p Fz(0)2703 1053 y Fy(dE)5 b Fz(\()p Fy(\025)2910 1067 y Fq(2)2950 1053 y Fz(\))451 1305 y(=)25 b Fy(i)593 1181 y Fn(Z)644 1387 y Fp(R)711 1305 y Fy(d\034)798 1319 y Fq(1)838 1305 y Fy(e)880 1267 y Fw(it\034)960 1276 y Fr(1)1015 1181 y Fn(Z)1065 1387 y Fw(\025)1106 1396 y Fr(2)1160 1305 y Fy(e)1202 1267 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1434 1276 y Fr(2)1475 1305 y Fy(R)1544 1319 y Fq(+)1603 1305 y Fz(\()p Fy(\025)1691 1319 y Fq(2)1751 1305 y Fz(+)20 b Fy(\034)1882 1319 y Fq(1)1921 1305 y Fz(\))15 b Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)2119 1319 y Fq(1)2159 1305 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(dE)5 b Fz(\()p Fy(\025)2466 1319 y Fq(2)2507 1305 y Fz(\))446 1553 y(+)20 b Fy(i)583 1430 y Fn(Z)634 1636 y Fp(R)701 1553 y Fy(d\034)788 1567 y Fq(1)828 1553 y Fy(e)870 1516 y Fw(is\034)958 1525 y Fr(1)1012 1430 y Fn(Z)1062 1636 y Fw(\025)1103 1645 y Fr(1)1157 1553 y Fy(e)1199 1516 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1431 1525 y Fr(1)1472 1553 y Fy(dE)5 b Fz(\()p Fy(\025)1679 1567 y Fq(1)1719 1553 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)1918 1567 y Fq(1)1957 1553 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2126 1567 y Fx(\000)2186 1553 y Fz(\()p Fy(\025)2274 1567 y Fq(1)2334 1553 y Fu(\000)20 b Fy(\034)2465 1567 y Fq(1)2505 1553 y Fz(\))451 1810 y(=)25 b Fy(i)593 1686 y Fn(Z)644 1893 y Fp(R)711 1810 y Fy(d\034)798 1824 y Fq(1)838 1810 y Fy(e)880 1773 y Fw(it\034)960 1782 y Fr(1)1015 1686 y Fn(Z)1065 1893 y Fw(\025)1126 1810 y Fy(e)1168 1773 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1405 1682 y Fn(\022)1472 1810 y Fy(R)1541 1824 y Fq(+)1600 1810 y Fz(\()p Fy(\025)c Fz(+)f Fy(\034)1840 1824 y Fq(1)1879 1810 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)2078 1824 y Fq(1)2117 1810 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(dE)5 b Fz(\()p Fy(\025)p Fz(\))23 b(+)d Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)2932 1824 y Fq(1)2971 1810 y Fz(\))15 b Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)3170 1824 y Fq(1)3209 1810 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)3378 1824 y Fx(\000)3439 1810 y Fz(\()p Fy(\025)p Fz(\))3562 1682 y Fn(\023)3629 1810 y Fy(:)-47 2035 y Fz(In)28 b(the)i(ab)s(o)m(v)m(e)h(calculation)d (w)m(e)i(ha)m(v)m(e)h(used)e(the)g(sp)s(ectral)g(represen)m(tation)h (\(5.23\))h(for)f(the)f(resolv)m(en)m(t)h(and)f(\(5.24\))s(.)-47 2148 y(The)h(pro)s(of)f(no)m(w)i(pro)s(ceeds)f(inductiv)m(ely)-8 b(.)39 b(W)-8 b(e)31 b(shall)e(assume)h(that)579 2361 y Fu(I)629 2375 y Fw(m)695 2361 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)f Fy(i)1034 2323 y Fw(m)1116 2237 y Fn(Z)1166 2443 y Fp(R)1214 2424 y Fs(m)1292 2361 y Fy(d\034)1379 2375 y Fq(1)1434 2361 y Fy(:)15 b(:)g(:)h(d\034)1642 2375 y Fw(m)1734 2361 y Fy(e)1776 2323 y Fw(it)p Fq(\()p Fw(\034)1883 2332 y Fr(1)1919 2323 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)2100 2331 y Fs(m)2158 2323 y Fq(\))2205 2237 y Fn(Z)2255 2443 y Fw(\025)2316 2361 y Fy(e)2358 2323 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2595 2361 y Fy(dM)2730 2375 y Fw(m)2797 2361 y Fz(\()p Fy(\025)p Fz(;)f Fy(\034)2965 2375 y Fq(1)3005 2361 y Fy(;)g(::;)g(\034)3175 2375 y Fw(m)3243 2361 y Fz(\))p Fy(;)-3350 b Fz(\(5.30\))-47 2590 y(where)30 b Fy(dM)351 2604 y Fw(m)418 2590 y Fz(\()p Fy(\025)p Fz(;)15 b Fy(\034)586 2604 y Fq(1)626 2590 y Fy(;)g(::;)g(\034)796 2604 y Fw(m)863 2590 y Fz(\))31 b(is)e(the)i(op)s(erator)g(v)-5 b(alued)29 b(measure)2183 2557 y Fq(3)2253 2590 y Fz(de\014ned)g(b)m(y)118 2835 y Fy(dM)253 2849 y Fw(m)320 2835 y Fz(\()p Fy(\025)p Fz(;)15 b Fy(\034)488 2849 y Fq(1)528 2835 y Fy(;)g(::;)g(\034)698 2849 y Fw(m)766 2835 y Fz(\))25 b(=)957 2721 y Fw(m)922 2748 y Fn(X)924 2946 y Fw(k)r Fq(=0)1069 2706 y Fn(\024)1117 2835 y Fy(R)1186 2849 y Fq(+)1245 2835 y Fz(\()p Fy(\025)c Fz(+)e Fy(\034)1484 2849 y Fq(1)1544 2835 y Fz(+)h Fy(::)g Fz(+)g Fy(\034)1836 2849 y Fw(m)1903 2835 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)2086 2849 y Fq(1)2126 2835 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2295 2849 y Fq(+)2355 2835 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\034)2594 2849 y Fq(2)2654 2835 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)2946 2849 y Fw(m)3013 2835 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)3196 2849 y Fq(2)3236 2835 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(:::V)22 b Fz(\()5 b(^)-50 b Fy(\034)3561 2850 y Fw(k)r Fx(\000)p Fq(1)3694 2835 y Fy(;)15 b Fu(\001)p Fz(\))673 3120 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)1032 3135 y Fw(k)1095 3120 y Fz(+)g Fy(::)h Fz(+)f Fy(\034)1388 3134 y Fw(m)1454 3120 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)1637 3135 y Fw(k)1680 3120 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)1849 3134 y Fx(\000)1909 3120 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)2149 3135 y Fw(k)r Fq(+1)2302 3120 y Fz(+)f Fy(::)i Fz(+)f Fy(\034)2594 3134 y Fw(m)2660 3120 y Fz(\))p Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)2844 3135 y Fw(k)r Fq(+1)2977 3120 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(:::V)22 b Fz(\()5 b(^)-50 b Fy(\034)3302 3134 y Fw(m)3369 3120 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)3538 3134 y Fx(\000)3598 3120 y Fz(\()p Fy(\025)p Fz(\))3721 2992 y Fn(\025)3769 3120 y Fy(:)-47 3340 y Fz(F)-8 b(ormally)25 b(setting)h Fy(\034)665 3354 y Fw(m)p Fq(+1)847 3340 y Fz(=)f(0,)i(w)m(e)f(can)g(also)g(write)f(the)h(ab)s(o)m(v)m(e)h (expression)d(in)h(the)h(follo)m(wing)e(more)i(concise)g(form:)363 3601 y Fy(dM)498 3615 y Fw(m)565 3601 y Fz(\()p Fy(\025)p Fz(;)15 b Fy(\034)733 3615 y Fq(1)773 3601 y Fy(;)g(::;)g(\034)943 3615 y Fw(m)1010 3601 y Fz(\))26 b(=)1167 3487 y Fw(m)p Fq(+1)1177 3514 y Fn(X)1179 3712 y Fw(k)r Fq(=0)1319 3472 y Fn(\022)1401 3487 y Fw(k)r Fx(\000)p Fq(1)1408 3514 y Fn(Y)1404 3709 y Fw(r)r Fq(=1)1545 3601 y Fy(R)1614 3615 y Fq(+)1673 3601 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)1913 3615 y Fw(r)1971 3601 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)2263 3615 y Fw(m)2330 3601 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)2513 3615 y Fw(r)2551 3601 y Fy(;)15 b Fu(\001)p Fz(\))2651 3472 y Fn(\023)2719 3601 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)3078 3616 y Fw(k)3141 3601 y Fz(+)g Fy(::)h Fz(+)e Fy(\034)3433 3615 y Fw(m)3500 3601 y Fz(\))1319 3792 y Fn(\022)1434 3806 y Fw(m)p Fq(+1)1452 3833 y Fn(Y)1401 4031 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)1634 3920 y Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)1782 3934 y Fw(r)r Fx(\000)p Fq(1)1910 3920 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2079 3934 y Fx(\000)2140 3920 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\034)2379 3934 y Fw(r)2437 3920 y Fz(+)g Fy(::)h Fz(+)e Fy(\034)2729 3934 y Fw(m)p Fq(+1)2886 3920 y Fz(\))2921 3792 y Fn(\023)2988 3920 y Fy(:)-3060 b Fz(\(5.31\))-47 4186 y(W)-8 b(e)31 b(ha)m(v)m(e)h(already)e(v)m (eri\014ed)f(\(5.30\))k(for)d Fy(m)25 b Fz(=)g(1.)41 b(It)31 b(remains)e(to)i(c)m(hec)m(k)h(that)260 4403 y Fu(I)310 4417 y Fw(m)p Fq(+1)466 4403 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)f Fy(i)805 4366 y Fw(m)p Fq(+1)977 4280 y Fn(Z)1027 4486 y Fp(R)1075 4467 y Fs(m)p Fr(+1)1230 4403 y Fy(d\034)1317 4417 y Fq(1)1357 4403 y Fy(::d\034)1494 4417 y Fw(m)p Fq(+1)1666 4403 y Fy(e)1708 4366 y Fw(it)p Fq(\()p Fw(\034)1815 4375 y Fr(1)1851 4366 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)2032 4374 y Fs(m)2090 4366 y Fq(+)p Fw(\034)2176 4375 y Fs(m)p Fr(+1)2312 4366 y Fq(\))2359 4280 y Fn(Z)2409 4486 y Fw(\025)2470 4403 y Fy(e)2512 4366 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2749 4403 y Fy(dM)2884 4417 y Fw(m)p Fq(+1)3042 4403 y Fz(\()p Fy(\025)p Fz(;)15 b 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Fy(;)g(s)p Fz(\))p Fy(:)p -47 4954 1597 4 v 56 5008 a Fr(3)91 5040 y Fg(Once)29 b(again)h(w)n(e)f(mak)n(e)f(sense)h(of)h Fe(dM)1224 5048 y Fs(m)1283 5040 y Fg(\()p Fe(\025)p Fg(;)12 b Fe(\034)1425 5048 y Fr(1)1460 5040 y Fe(;)h(::;)h(\034)1605 5048 y Fs(m)1663 5040 y Fg(\))29 b(only)g(after)h(pairing)f(it)g(with)h(the)e (Sc)n(h)n(w)n(artz)h(functions)g Fe( )3450 5048 y Fs(s)3513 5040 y Fg(and)f Fe(g)s Fg(.)44 b(Then)-47 5131 y Fd(h)p Fe(dM)97 5139 y Fs(m)155 5131 y Fg(\()p Fe(\025)p Fg(;)13 b Fe(\034)298 5139 y Fr(1)332 5131 y Fe(;)h(::;)f(\034)477 5139 y Fs(m)536 5131 y Fg(\))p Fe( )616 5139 y Fs(s)649 5131 y Fe(;)g(g)s Fd(i)26 b Fg(is)i(a)g(\014nite)e(measure)h(relativ)n (e)h(to)f Fe(\025)g Fg(|)g(in)g(fact,)i(rapidly)e(deca)n(ying)g(in)g Fe(\025)p Fg(,)g(see)h(\(5.28\))g(|)f(whic)n(h)g(dep)r(ends)-47 5223 y(smo)r(othly)e(on)h Fe(\034)427 5231 y Fr(1)461 5223 y Fe(;)13 b(::;)h(\034)606 5231 y Fs(m)690 5223 y Fg(and)25 b(v)l(anishes)h(outside)g(of)g(a)g(compact)g(set)f(in)h (these)g(v)l(ariables.)1903 5472 y Fz(35)p eop %%Page: 36 36 36 35 bop -47 274 a Fz(Substituting)27 b(the)k(expression)e(for)h Fu(I)1257 288 y Fw(m)1354 274 y Fz(from)g(\(5.30\))i(w)m(e)f(obtain)671 487 y Fu(I)721 501 y Fw(m)p Fq(+1)877 487 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)f Fy(i)1216 450 y Fw(m)1298 364 y Fn(Z)1348 570 y Fp(R)1396 551 y Fs(m)p Fr(+1)1551 487 y Fy(d\034)1638 501 y Fq(1)1678 487 y Fy(:::d\034)1840 501 y Fw(m)p Fq(+1)2012 364 y Fn(Z)2063 570 y Fw(\025;\025)2165 579 y Fr(1)2219 364 y Fn(Z)2310 390 y Fw(t)2269 570 y(s)2354 487 y Fy(ds)2444 501 y Fq(1)2484 487 y Fy(e)2526 450 y Fw(it\025)2616 459 y Fr(1)2655 487 y Fy(dE)5 b Fz(\()p Fy(\025)2862 501 y Fq(1)2903 487 y Fz(\))p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)3087 501 y Fq(1)3126 487 y Fy(;)15 b Fu(\001)p Fz(\))1402 703 y Fy(e)1444 666 y Fw(is)1501 675 y Fr(1)1536 666 y Fq(\()p Fw(\034)1594 675 y Fr(1)1629 666 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1810 675 y Fs(m)p Fr(+1)1945 666 y Fx(\000)p Fw(\025)2041 675 y Fr(1)2076 666 y Fq(\))2122 703 y Fy(e)2164 666 y Fw(i)p Fq(\()p Fw(s)2248 675 y Fr(1)2283 666 y Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2444 703 y Fy(dM)2579 717 y Fw(m)2646 703 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)2814 717 y Fq(2)2854 703 y Fy(;)g(::;)g(\034)3024 717 y Fw(m)p Fq(+1)3182 703 y Fz(\))p Fy(:)-47 853 y Fz(In)m(tegrating)31 b(explicitly)d (relativ)m(e)i(to)h Fy(s)1294 867 y Fq(1)1364 853 y Fz(w)m(e)f(infer)f (that)431 1047 y Fu(I)481 1061 y Fw(m)p Fq(+1)637 1047 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)f Fu(\000)p Fy(i)1047 1009 y Fw(m)p Fq(+1)1219 923 y Fn(Z)1269 1129 y Fp(R)1317 1110 y Fs(m)p Fr(+1)1472 1047 y Fy(d\034)1559 1061 y Fq(1)1599 1047 y Fy(:::d\034)1761 1061 y Fw(m)p Fq(+1)1933 923 y Fn(Z)1984 1129 y Fw(\025;\025)2086 1138 y Fr(1)2140 1047 y Fy(e)2182 1009 y Fw(it\025)2272 1018 y Fr(1)2311 1047 y Fy(e)2353 1009 y Fx(\000)p Fw(is\025)2511 1047 y Fy(dE)5 b Fz(\()p Fy(\025)2718 1061 y Fq(1)2758 1047 y Fz(\))p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)2942 1061 y Fq(1)2981 1047 y Fy(;)15 b Fu(\001)p Fz(\))833 1258 y Fy(e)875 1225 y Fw(it)p Fq(\()p Fw(\034)982 1234 y Fr(1)1019 1225 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1200 1234 y Fs(m)p Fr(+1)1335 1225 y Fx(\000)p Fw(\025)1431 1234 y Fr(1)1465 1225 y Fq(+)p Fw(\025)p Fq(+)p Fw(i)p Fq(0\))1727 1258 y Fu(\000)20 b Fy(e)1860 1225 y Fw(is)p Fq(\()p Fw(\034)1975 1234 y Fr(1)2011 1225 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)2192 1234 y Fs(m)p Fr(+1)2326 1225 y Fx(\000)p Fw(\025)2422 1234 y Fr(1)2457 1225 y Fq(+)p Fw(\025)p Fq(+)p Fw(i)p Fq(0\))p 833 1299 1866 4 v 1214 1382 a Fy(\034)1254 1396 y Fq(1)1314 1382 y Fz(+)f Fy(::)i Fz(+)f Fy(\034)1606 1396 y Fw(m)p Fq(+1)1783 1382 y Fu(\000)g Fy(\025)1927 1396 y Fq(1)1986 1382 y Fz(+)g Fy(\025)h Fz(+)e Fy(i)p Fz(0)2709 1320 y Fy(dM)2844 1334 y Fw(m)2911 1320 y Fz(\()p Fy(\025)p Fz(;)c Fy(\034)3079 1334 y Fq(2)3119 1320 y Fy(;)g(::;)g(\034)3289 1334 y Fw(m)p Fq(+1)3447 1320 y Fz(\))849 1559 y(=)25 b Fu(\000)p Fy(i)1047 1522 y Fw(m)p Fq(+1)1219 1436 y Fn(Z)1269 1642 y Fp(R)1317 1623 y Fs(m)p Fr(+1)1472 1559 y Fy(d\034)1559 1573 y Fq(1)1599 1559 y Fy(:::d\034)1761 1573 y Fw(m)p Fq(+1)1943 1559 y Fy(e)1985 1522 y Fw(it)p Fq(\()p Fw(\034)2092 1531 y Fr(1)2129 1522 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)2310 1531 y Fs(m)p Fr(+1)2444 1522 y Fq(+)p Fw(i)p Fq(0\))2605 1436 y Fn(Z)2656 1642 y Fw(\025;\025)2758 1651 y Fr(1)2812 1559 y Fy(e)2854 1522 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1608 1756 y Fy(dE)5 b Fz(\()p Fy(\025)1815 1770 y Fq(1)1855 1756 y Fz(\))p 1197 1797 1104 4 v 1197 1880 a Fy(\034)1237 1894 y Fq(1)1296 1880 y Fz(+)20 b Fy(::)h Fz(+)f Fy(\034)1589 1894 y Fw(m)p Fq(+1)1766 1880 y Fu(\000)g Fy(\025)1910 1894 y Fq(1)1969 1880 y Fz(+)g Fy(\025)g Fz(+)g Fy(i)p Fz(0)2311 1818 y Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)2459 1832 y Fq(1)2499 1818 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(dM)2734 1832 y Fw(m)2802 1818 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)2970 1832 y Fq(2)3010 1818 y Fy(;)g(::;)g(\034)3180 1832 y Fw(m)p Fq(+1)3338 1818 y Fz(\))844 2057 y(+)35 b Fy(i)981 2020 y Fw(m)p Fq(+1)1153 1934 y Fn(Z)1203 2140 y Fp(R)1251 2121 y Fs(m)p Fr(+1)1406 2057 y Fy(d\034)1493 2071 y Fq(1)1533 2057 y Fy(:::d\034)1695 2071 y Fw(m)p Fq(+1)1878 2057 y Fy(e)1920 2020 y Fw(is)p Fq(\()p Fw(\034)2035 2029 y Fr(1)2070 2020 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)2251 2029 y Fs(m)p Fr(+1)2386 2020 y Fq(+)p Fw(i)p Fq(0\))2562 1934 y Fn(Z)2613 2140 y Fw(\025;\025)2715 2149 y Fr(1)2768 2057 y Fy(e)2810 2020 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)3042 2029 y Fr(1)3083 2057 y Fy(dE)5 b Fz(\()p Fy(\025)3290 2071 y Fq(1)3330 2057 y Fz(\))1187 2316 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1335 2330 y Fq(1)1375 2316 y Fy(;)15 b Fu(\001)p Fz(\))1651 2254 y Fy(dM)1786 2268 y Fw(m)1853 2254 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)2021 2268 y Fq(2)2061 2254 y Fy(;)g(::;)g(\034)2231 2268 y Fw(m)p Fq(+1)2389 2254 y Fz(\))p 1485 2295 V 1485 2378 a Fy(\034)1525 2392 y Fq(1)1585 2378 y Fz(+)20 b Fy(::)h Fz(+)f Fy(\034)1878 2392 y Fw(m)p Fq(+1)2054 2378 y Fu(\000)g Fy(\025)2198 2392 y Fq(1)2258 2378 y Fz(+)g Fy(\025)g Fz(+)g Fy(i)p Fz(0)2625 2316 y(=)25 b Fy(J)2771 2330 y Fq(1)2831 2316 y Fz(+)20 b Fy(J)2972 2330 y Fq(2)3012 2316 y Fy(:)-47 2519 y Fz(According)30 b(to)h(\(5.23\))i(and)c(\(5.24\))750 2600 y Fn(Z)800 2807 y Fw(\025)841 2816 y Fr(1)1316 2663 y Fy(dE)5 b Fz(\()p Fy(\025)1523 2677 y Fq(1)1563 2663 y Fz(\))p 905 2703 V 905 2786 a Fy(\034)945 2800 y Fq(1)1005 2786 y Fz(+)20 b Fy(::)g Fz(+)g Fy(\034)1297 2800 y Fw(m)p Fq(+1)1474 2786 y Fu(\000)g Fy(\025)1618 2800 y Fq(1)1678 2786 y Fz(+)f Fy(\025)i Fz(+)f Fy(i)p Fz(0)2045 2724 y(=)25 b Fu(\000)p Fy(R)2281 2738 y Fq(+)2339 2724 y Fz(\()p Fy(\025)c Fz(+)f Fy(\034)2579 2738 y Fq(1)2638 2724 y Fz(+)g Fy(::)h Fz(+)f Fy(\034)2931 2738 y Fw(m)p Fq(+1)3087 2724 y Fz(\))p Fy(:)-47 2931 y Fz(Therefore,)-47 3246 y Fy(J)3 3260 y Fq(1)68 3246 y Fz(=)25 b Fy(i)195 3209 y Fw(m)p Fq(+1)367 3123 y Fn(Z)418 3329 y Fp(R)466 3310 y Fs(m)p Fr(+1)620 3246 y Fy(d\034)707 3260 y Fq(1)747 3246 y Fy(:::d\034)909 3260 y Fw(m)p Fq(+1)1067 3246 y Fy(e)1109 3209 y Fw(it)p Fq(\()p Fw(\034)1216 3218 y Fr(1)1252 3209 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1433 3218 y Fs(m)p Fr(+1)1568 3209 y Fq(\))1614 3123 y Fn(Z)1665 3329 y Fw(\025)1725 3246 y Fy(e)1767 3209 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2005 3246 y Fy(R)2074 3260 y Fq(+)2133 3246 y Fz(\()p Fy(\025)c Fz(+)f Fy(\034)2373 3260 y Fq(1)2432 3246 y Fz(+)g Fy(::)h Fz(+)e Fy(\034)2724 3260 y Fw(m)p Fq(+1)2881 3246 y Fz(\))p Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)3064 3260 y Fq(1)3104 3246 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(dM)3339 3260 y Fw(m)3407 3246 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)3575 3260 y Fq(2)3615 3246 y Fy(;)g(::;)g(\034)3785 3260 y Fw(m)p Fq(+1)3943 3246 y Fz(\))p Fy(:)-47 3080 y Fz(\(5.32\))-47 3444 y(Observ)m(e)30 b(that,)h(with)e(the)i(con)m(v)m(en)m(tion)g(that)g Fy(\034)1579 3458 y Fw(m)p Fq(+2)1761 3444 y Fz(=)25 b(0,)74 3686 y Fy(R)143 3700 y Fq(+)202 3686 y Fz(\()p Fy(\025)c Fz(+)f Fy(\034)442 3700 y Fq(1)501 3686 y Fz(+)g Fy(::)h Fz(+)e Fy(\034)793 3700 y Fw(m)p Fq(+1)950 3686 y Fz(\))p Fy(V)i Fz(\()5 b(^)-50 b Fy(\034)1134 3700 y Fq(1)1173 3686 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(dM)1408 3700 y Fw(m)1476 3686 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)1644 3700 y Fq(2)1684 3686 y Fy(;)g(::;)g(\034)1854 3700 y Fw(m)p Fq(+1)2012 3686 y Fz(\))26 b(=)2168 3573 y Fw(m)p Fq(+2)2179 3600 y Fn(X)2180 3798 y Fw(k)r Fq(=2)2336 3558 y Fn(\024)2384 3613 y(\000)2441 3573 y Fw(k)r Fx(\000)p Fq(1)2447 3600 y Fn(Y)2443 3794 y Fw(r)r Fq(=1)2585 3686 y Fy(R)2654 3700 y Fq(+)2713 3686 y Fz(\()p Fy(\025)21 b Fz(+)e Fy(\034)2952 3700 y Fw(r)3010 3686 y Fz(+)h Fy(::)h Fz(+)f Fy(\034)3303 3700 y Fw(m)p Fq(+1)3459 3686 y Fz(\))p Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)3643 3700 y Fw(r)3681 3686 y Fy(;)15 b Fu(\001)p Fz(\))3781 3613 y Fn(\001)1226 4005 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)1585 4020 y Fw(k)1648 4005 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)1940 4019 y Fw(m)p Fq(+1)2097 4005 y Fz(\))2132 3932 y Fn(\000)2221 3892 y Fw(m)p Fq(+2)2240 3919 y Fn(Y)2189 4117 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2421 4005 y Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)2570 4019 y Fw(r)r Fx(\000)p Fq(1)2698 4005 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2867 4019 y Fx(\000)2927 4005 y Fz(\()p Fy(\025)21 b Fz(+)e Fy(\034)3166 4019 y Fw(r)3224 4005 y Fz(+)h Fy(::)h Fz(+)f Fy(\034)3517 4019 y Fw(m)p Fq(+2)3673 4005 y Fz(\))3708 3932 y Fn(\001)3750 3877 y(\025)3798 4005 y Fy(:)-3870 b Fz(\(5.33\))-47 4248 y(It)30 b(remains)f(to)i (consider)f(the)g(in)m(tegral)g Fy(J)1393 4262 y Fq(2)1433 4248 y Fz(.)-47 4448 y Fy(J)3 4462 y Fq(2)68 4448 y Fz(=)25 b Fy(i)195 4410 y Fw(m)p Fq(+1)413 4324 y Fn(Z)347 4592 y Fp(R)395 4574 y Fs(m)p Fr(+1)566 4448 y Fy(d\034)653 4462 y Fq(1)692 4448 y Fy(:::d\034)854 4462 y Fw(m)p Fq(+1)1012 4448 y Fy(e)1054 4410 y Fw(is)p Fq(\()p Fw(\034)1169 4419 y Fr(1)1204 4410 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1385 4419 y Fs(m)p Fr(+1)1520 4410 y Fq(+)p Fw(i)p Fq(0\))1696 4324 y Fn(Z)1747 4530 y Fw(\025;\025)1849 4539 y Fr(1)1903 4448 y Fy(e)1945 4410 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2177 4419 y Fr(1)2217 4448 y Fy(dE)5 b Fz(\()p Fy(\025)2424 4462 y Fq(1)2464 4448 y Fz(\))p Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)2648 4462 y Fq(1)2687 4448 y Fy(;)15 b Fu(\001)p Fz(\))2964 4387 y Fy(dM)3099 4401 y Fw(m)3166 4387 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)3334 4401 y Fq(2)3374 4387 y Fy(;)g(::;)g(\034)3544 4401 y Fw(m)p Fq(+1)3702 4387 y Fz(\))p 2797 4427 V 2797 4510 a Fy(\034)2837 4524 y Fq(1)2898 4510 y Fz(+)20 b Fy(::)g Fz(+)g Fy(\034)3190 4524 y Fw(m)p Fq(+1)3367 4510 y Fu(\000)g Fy(\025)3511 4524 y Fq(1)3570 4510 y Fz(+)g Fy(\025)h Fz(+)f Fy(i)p Fz(0)68 4886 y(=)25 b Fy(i)195 4848 y Fw(m)p Fq(+1)367 4762 y Fn(Z)418 4968 y Fp(R)466 4949 y Fs(m)p Fr(+1)620 4886 y Fy(d\034)707 4900 y Fq(1)747 4886 y Fy(:::d\034)909 4900 y Fw(m)p Fq(+1)1067 4886 y Fy(e)1109 4848 y Fw(it)p Fq(\()p Fw(\034)1216 4857 y Fr(1)1252 4848 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1433 4857 y Fs(m)p Fr(+1)1568 4848 y Fq(\)+)p Fw(is)p Fq(\()p Fw(i)p Fq(0\))1855 4762 y Fn(Z)1906 4968 y Fw(\025)1947 4977 y Fr(1)2001 4886 y Fy(e)2043 4848 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2275 4857 y Fr(1)2315 4886 y Fy(dE)5 b Fz(\()p Fy(\025)2522 4900 y Fq(1)2583 4886 y Fz(+)20 b Fy(\034)2714 4900 y Fq(1)2773 4886 y Fz(+)g Fy(::)h Fz(+)f Fy(\034)3066 4900 y Fw(m)p Fq(+1)3222 4886 y Fz(\))-130 4714 y(\(5.34\))1238 5140 y Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)1386 5154 y Fq(1)1426 5140 y Fy(;)15 b Fu(\001)p Fz(\))1541 5016 y Fn(Z)1593 5223 y Fw(\025)1663 5079 y Fy(dM)1798 5093 y Fw(m)1865 5079 y Fz(\()p Fy(\025)p Fz(;)g Fy(\034)2033 5093 y Fq(2)2073 5079 y Fy(;)g(::;)g(\034)2243 5093 y Fw(m)p Fq(+1)2401 5079 y Fz(\))p 1663 5119 774 4 v 1827 5203 a Fy(\025)21 b Fu(\000)e Fy(\025)2044 5217 y Fq(1)2104 5203 y Fz(+)h Fy(i)p Fz(0)1903 5472 y(36)p eop %%Page: 37 37 37 36 bop -47 274 a Fz(Insp)s(ection)30 b(of)i(the)f(desired)f (expression)h(for)g Fy(dM)1681 288 y Fw(m)p Fq(+1)1838 274 y Fz(\()p Fy(\025)p Fz(;)15 b Fy(\034)2006 288 y Fq(1)2046 274 y Fy(;)g(::;)g(\034)2216 288 y Fw(m)p Fq(+1)2374 274 y Fz(\))32 b(and)f(equations)g(\(5.32\))s(-\(5.34\))j(su\016ces)d (to)-47 387 y(v)m(erify)f(the)g(follo)m(wing)f(form)m(ula:)494 512 y Fn(Z)544 718 y Fw(\025)615 574 y Fy(dM)750 588 y Fw(m)817 574 y Fz(\()p Fy(\025)p Fz(;)15 b Fy(\034)985 588 y Fq(2)1025 574 y Fy(;)g(::;)g(\034)1195 588 y Fw(m)p Fq(+1)1352 574 y Fz(\))p 615 615 774 4 v 779 698 a Fy(\025)20 b Fu(\000)g Fy(\025)996 712 y Fq(1)1056 698 y Fz(+)g Fy(i)p Fz(0)1423 635 y(=)1519 507 y Fn(\022)1601 522 y Fw(m)p Fq(+1)1619 549 y Fn(Y)1615 743 y Fw(r)r Fq(=2)1769 635 y Fy(R)1838 649 y Fx(\000)1897 635 y Fz(\()p Fy(\025)1985 649 y Fq(1)2045 635 y Fz(+)g Fy(\034)2176 649 y Fw(r)2234 635 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)2526 649 y Fw(m)p Fq(+1)2683 635 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)2866 649 y Fw(r)2904 635 y Fy(;)15 b Fu(\001)p Fz(\))3004 507 y Fn(\023)3072 635 y Fy(R)3141 649 y Fx(\000)3200 635 y Fz(\()p Fy(\025)3288 649 y Fq(1)3328 635 y Fz(\))g Fy(:)-47 881 y Fz(This)28 b(is)i(accomplished)f(in)g(the)h(follo)m (wing)f(t)m(w)m(o)j(lemmas,)e(and)g(w)m(e)h(are)f(done.)p 3879 881 4 62 v 3883 823 55 4 v 3883 881 V 3937 881 4 62 v 94 1063 a(W)-8 b(e)41 b(recall)d(de\014nition)f(\(5.31\))42 b(of)e(the)f(op)s(erator)h(v)-5 b(alued)38 b(measure)i Fy(dM)2643 1077 y Fw(m)2749 1063 y Fz(and)f(pro)m(v)m(e)h(the)g(follo)m (wing)e(more)-47 1176 y(general)30 b(result)-47 1335 y Fm(Lemma)i(5.7.)47 b Fl(L)-5 b(et)30 b Fy(a)734 1349 y Fq(1)773 1335 y Fy(;)15 b(::;)g(a)951 1349 y Fw(m)1044 1335 y Fu(2)25 b Fv(R)39 b Fl(b)-5 b(e)29 b(a)h(se)-5 b(quenc)g(e)30 b(of)g(arbitr)-5 b(ary)32 b(r)-5 b(e)g(al)31 b(numb)-5 b(ers)31 b(and)g(let)e Fy(A)3153 1349 y Fq(1)3193 1335 y Fy(;)15 b(::;)g(A)3391 1349 y Fw(m)3489 1335 y Fl(b)-5 b(e)30 b(arbitr)-5 b(ary)-47 1448 y(op)g(er)g(ators)312 1415 y Fq(4)355 1448 y Fl(.)41 b(Then)427 1576 y Fn(Z)477 1782 y Fw(\025)728 1638 y Fz(1)p 548 1679 407 4 v 548 1762 a Fy(\025)20 b Fu(\000)g Fy(\026)g Fz(+)g Fy(i)p Fz(0)1014 1586 y Fw(m)979 1613 y Fn(X)981 1811 y Fw(k)r Fq(=1)1126 1571 y Fn(\022)1208 1586 y Fw(k)r Fx(\000)p Fq(1)1214 1613 y Fn(Y)1210 1808 y Fw(r)r Fq(=1)1352 1700 y Fy(R)1421 1714 y Fq(+)1480 1700 y Fz(\()p Fy(\025)g Fz(+)g Fy(a)1727 1714 y Fw(r)1765 1700 y Fz(\))p Fy(A)1868 1714 y Fw(r)1907 1571 y Fn(\023)1974 1700 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(a)2341 1715 y Fw(k)2384 1700 y Fz(\))2419 1571 y Fn(\022)2579 1586 y Fw(m)2552 1613 y Fn(Y)2501 1811 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2734 1700 y Fy(A)2802 1714 y Fw(r)r Fx(\000)p Fq(1)2930 1700 y Fy(R)2999 1714 y Fx(\000)3058 1700 y Fz(\()p Fy(\025)h Fz(+)f Fy(a)3306 1714 y Fw(r)3344 1700 y Fz(\))3379 1571 y Fn(\023)1997 2026 y Fz(=)2093 1897 y Fn(\022)2175 1912 y Fw(m)p Fx(\000)p Fq(1)2193 1939 y Fn(Y)2189 2134 y Fw(r)r Fq(=1)2343 2026 y Fy(R)2412 2040 y Fx(\000)2471 2026 y Fz(\()p Fy(\026)g Fz(+)g Fy(a)2720 2040 y Fw(r)2758 2026 y Fz(\))p Fy(A)2861 2040 y Fw(r)2900 1897 y Fn(\023)2967 2026 y Fy(R)3036 2040 y Fx(\000)3095 2026 y Fz(\()p Fy(\026)g Fz(+)g Fy(a)3344 2040 y Fw(m)3410 2026 y Fz(\))q Fy(:)-3518 b Fz(\(5.35\))-47 2271 y Fl(As)41 b(b)-5 b(efor)g(e,)44 b(the)e(identity)g(holds)h(after)f(p)-5 b(airing)43 b(the)f(ab)-5 b(ove)42 b(expr)-5 b(essions)43 b(with)g(a)f(p)-5 b(air)42 b(of)g(Schwartz)h(functions)-47 2384 y Fy( )12 2398 y Fw(s)49 2384 y Fy(;)15 b(g)s Fl(.)-47 2543 y(Pr)-5 b(o)g(of.)47 b Fz(W)-8 b(e)28 b(shall)e(write)h(eac)m(h)h Fy(R)1101 2557 y Fx(\006)1160 2543 y Fz(\()p Fy(\025)14 b Fz(+)g Fy(a)1395 2557 y Fw(r)1434 2543 y Fz(\),)29 b(for)e(all)f(v)-5 b(alues)27 b(of)h Fy(r)f Fz(=)e(1)p Fy(;)15 b(::;)g(m)29 b Fz(di\013eren)m(t)e(from)g Fy(k)k Fz(using)26 b(the)i(sp)s(ectral)-47 2656 y(represen)m(tation)1329 2812 y Fy(R)1398 2826 y Fx(\006)1457 2812 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(a)1705 2826 y Fw(r)1743 2812 y Fz(\))25 b(=)1899 2688 y Fn(Z)1950 2894 y Fw(\025)1991 2902 y Fs(r)2055 2750 y Fy(dE)5 b Fz(\()p Fy(\025)2262 2764 y Fw(r)2321 2750 y Fz(+)20 b Fy(a)2460 2764 y Fw(r)2498 2750 y Fz(\))p 2055 2791 479 4 v 2073 2874 a Fy(\025)2126 2888 y Fw(r)2184 2874 y Fu(\000)g Fy(\025)g Fu(\007)g Fy(i)p Fz(0)2543 2812 y Fy(:)-47 3011 y Fz(W)-8 b(e)41 b(shall)e(also)h(rename)h(the)f(v)-5 b(ariable)39 b(of)i(in)m(tegration)f Fy(\025)h Fz(to)g Fy(\025)2234 3026 y Fw(k)2317 3011 y Fz(in)e(eac)m(h)i(term)g(of)f(the) h(sum)e(in)g Fy(k)s Fz(.)71 b(The)40 b(left)-47 3123 y(hand-side)28 b(of)38 b(\(5.35\))33 b(then)d(tak)m(es)h(the)g(follo)m (wing)e(form:)49 3257 y Fn(Z)155 3380 y Fu(\001)15 b(\001)g(\001)276 3257 y Fn(Z)58 3520 y Fw(\025)99 3529 y Fr(1)134 3520 y Fw(;::;\025)255 3528 y Fs(m)417 3267 y Fw(m)382 3294 y Fn(X)384 3492 y Fw(k)r Fq(=1)741 3319 y Fz(1)p 539 3359 450 4 v 539 3443 a Fy(\025)592 3458 y Fw(k)655 3443 y Fu(\000)k Fy(\026)h Fz(+)g Fy(i)p Fz(0)1013 3267 y Fw(k)r Fx(\000)p Fq(1)1020 3294 y Fn(Y)1016 3488 y Fw(r)r Fq(=1)1387 3319 y Fz(1)p 1167 3359 486 4 v 1167 3443 a Fy(\025)1220 3457 y Fw(r)1278 3443 y Fu(\000)g Fy(\025)1422 3458 y Fw(k)1485 3443 y Fu(\000)g Fy(i)p Fz(0)1755 3267 y Fw(m)1728 3294 y Fn(Y)1678 3492 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2140 3319 y Fz(1)p 1920 3359 V 1920 3443 a Fy(\025)1973 3457 y Fw(r)2032 3443 y Fu(\000)f Fy(\025)2175 3458 y Fw(k)2238 3443 y Fz(+)h Fy(i)p Fz(0)2416 3252 y Fn(\022)2498 3267 y Fw(m)p Fx(\000)p Fq(1)2516 3294 y Fn(Y)2513 3489 y Fw(j)t Fq(=1)2666 3380 y Fy(dE)5 b Fz(\()p Fy(\025)2873 3394 y Fw(j)2931 3380 y Fz(+)19 b Fy(a)3069 3394 y Fw(j)3106 3380 y Fz(\))p Fy(A)3209 3394 y Fw(j)3246 3252 y Fn(\023)3313 3380 y Fy(dE)5 b Fz(\()p Fy(\025)3520 3394 y Fw(m)3608 3380 y Fz(+)20 b Fy(a)3747 3394 y Fw(m)3813 3380 y Fz(\))-47 3670 y(The)28 b(pro)s(of)g(of)h(Lemma)g(5.7)h(is)e(\014nished)e(pro)m (vided)h(that)j(w)m(e)f(can)g(sho)m(w)g(that)g(the)g(follo)m(wing)e (iden)m(tit)m(y)h(holds)g(true:)604 3809 y Fw(m)569 3836 y Fn(X)571 4034 y Fw(k)r Fq(=1)928 3861 y Fz(1)p 726 3901 450 4 v 726 3985 a Fy(\025)779 4000 y Fw(k)842 3985 y Fu(\000)20 b Fy(\026)g Fz(+)g Fy(i)p Fz(0)1200 3809 y Fw(k)r Fx(\000)p Fq(1)1207 3836 y Fn(Y)1203 4030 y Fw(r)r Fq(=1)1574 3861 y Fz(1)p 1354 3901 486 4 v 1354 3985 a Fy(\025)1407 3999 y Fw(r)1465 3985 y Fu(\000)g Fy(\025)1609 4000 y Fw(k)1672 3985 y Fu(\000)g Fy(i)p Fz(0)1943 3809 y Fw(m)1916 3836 y Fn(Y)1865 4034 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2328 3861 y Fz(1)p 2108 3901 V 2108 3985 a Fy(\025)2161 3999 y Fw(r)2219 3985 y Fu(\000)g Fy(\025)2363 4000 y Fw(k)2426 3985 y Fz(+)f Fy(i)p Fz(0)2628 3922 y(=)2755 3809 y Fw(m)2728 3836 y Fn(Y)2724 4030 y Fw(r)r Fq(=1)3073 3861 y Fz(1)p 2873 3901 445 4 v 2873 3985 a Fy(\025)2926 3999 y Fw(r)2985 3985 y Fu(\000)h Fy(\026)g Fz(+)f Fy(i)p Fz(0)-47 4178 y(In)29 b(the)i(distributional)26 b(sense)547 4395 y(lim)496 4461 y Fw(")529 4473 y Fs(k)567 4461 y Fx(!)p Fq(0)673 4442 y Fr(+)788 4395 y Fz(lim)739 4461 y Fw(")772 4470 y Fr(1)806 4461 y Fx(!)p Fq(0)912 4442 y Fr(+)978 4395 y Fy(:::)105 b Fz(lim)1068 4461 y Fw(")1101 4473 y Fs(k)q Ft(\000)p Fr(1)1218 4461 y Fx(!)p Fq(0)1324 4442 y Fr(+)1479 4395 y Fz(lim)1390 4461 y Fw(")1423 4473 y Fs(k)q Fr(+1)1537 4461 y Fx(!)p Fq(0)1643 4442 y Ft(\000)1710 4395 y Fy(:::)78 b Fz(lim)1800 4461 y Fw(")1833 4469 y Fs(m)1893 4461 y Fx(!)p Fq(0)1999 4442 y Ft(\000)2297 4333 y Fz(1)p 2076 4374 490 4 v 2076 4457 a Fy(\025)2129 4472 y Fw(k)2192 4457 y Fu(\000)19 b Fy(\026)h Fz(+)g Fy(i")2521 4472 y Fw(k)2694 4281 y(m)2667 4308 y Fn(Y)2590 4506 y Fw(r)r Fq(=1)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)3123 4333 y Fz(1)p 2886 4374 521 4 v 2886 4457 a Fy(\025)2939 4471 y Fw(r)2997 4457 y Fu(\000)g Fy(\025)3141 4472 y Fw(k)3204 4457 y Fu(\000)g Fy(i")3368 4471 y Fw(r)1520 4712 y Fz(=)57 b(lim)1616 4778 y Fw(")p Fx(!)p Fq(0)1755 4759 y Fr(+)2056 4651 y Fz(1)p 1831 4692 497 4 v 1831 4775 a Fy(\025)1884 4790 y Fw(k)1946 4775 y Fu(\000)20 b Fy(\026)g Fz(+)g Fy(ik)s(")2456 4599 y Fw(m)2430 4626 y Fn(Y)2352 4824 y Fw(r)r Fq(=1)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)3005 4651 y Fz(1)p 2648 4692 759 4 v 2648 4775 a Fy(\025)2701 4789 y Fw(r)2759 4775 y Fu(\000)g Fy(\025)2903 4790 y Fw(k)2966 4775 y Fz(+)g Fy(i)p Fz(\()p Fy(r)j Fu(\000)d Fy(k)s Fz(\))p Fy(")-47 4974 y Fz(Therefore,)34 b(w)m(e)h(can)f(in)m(tro)s(duce)e(the)i(new)f(v)-5 b(ariables)33 b Fy(z)1882 4988 y Fw(r)1951 4974 y Fz(=)e Fy(\025)2106 4988 y Fw(r)2166 4974 y Fu(\000)22 b Fy(\026)g Fz(+)h Fy(ir)s(")p Fz(,)35 b Fy(r)e Fz(=)e(1)p Fy(;)15 b(::;)g(m)35 b Fz(and)e(pro)m(v)m(e)i(instead)e(the)-47 5087 y(follo)m(wing)c (statemen)m(t.)p 3879 5087 4 62 v 3883 5029 55 4 v 3883 5087 V 3937 5087 4 62 v -47 5137 1597 4 v 56 5191 a Fr(4)91 5223 y Fg(It)c(su\016ces)g(to)h(assume)f(that)f(that)h(the)g(op)r (erators)i Fe(A)1654 5232 y Fs(k)1691 5223 y Fg(,)f Fe(k)d Fg(=)e(1)p Fe(;)14 b(::;)g(m)24 b Fg(map)h(the)f(space)i Fe(C)2691 5191 y Ft(1)2756 5223 y Fg(\()p Fc(R)2837 5191 y Fr(3)2875 5223 y Fg(\))16 b Fd(\\)g Fe(L)3040 5191 y Fr(4)3075 5223 y Fg(\()p Fc(R)3156 5191 y Fr(3)3193 5223 y Fg(\))25 b(in)n(to)g(the)g(the)g(space)g Fd(S)6 b Fg(.)1903 5472 y Fz(37)p eop %%Page: 38 38 38 37 bop -47 274 a Fm(Lemma)32 b(5.8.)47 b Fl(F)-7 b(or)33 b(any)g(p)-5 b(airwise)35 b(distinct)e(c)-5 b(omplex)34 b(numb)-5 b(ers)33 b Fy(z)2302 288 y Fq(1)2342 274 y Fy(;)15 b(::;)g(z)2514 288 y Fw(m)2608 274 y Fu(2)24 b Fv(C)18 b Fl(,)1366 429 y Fw(m)1332 456 y Fn(X)1333 654 y Fw(k)r Fq(=1)1508 481 y Fz(1)p 1488 522 85 4 v 1488 605 a Fy(z)1530 620 y Fw(k)1702 429 y(m)1675 456 y Fn(Y)1598 654 y Fw(r)r Fq(=1)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)2009 481 y Fz(1)p 1894 522 277 4 v 1894 605 a Fy(z)1936 619 y Fw(r)1994 605 y Fu(\000)i Fy(z)2127 620 y Fw(k)2206 543 y Fz(=)2332 429 y Fw(m)2305 456 y Fn(Y)2301 651 y Fw(r)r Fq(=1)2468 481 y Fz(1)p 2451 522 81 4 v 2451 605 a Fy(z)2493 619 y Fw(r)2541 543 y Fy(:)-47 839 y Fl(Pr)-5 b(o)g(of.)47 b Fz(The)29 b(k)m(ey)j(iden)m(tit)m(y)d(is)h(the)g (statemen)m(t)j(of)d(the)h(lemma)f(for)g Fy(m)25 b Fz(=)g(2)1495 1015 y(1)p 1304 1055 427 4 v 1304 1138 a Fy(z)1346 1152 y Fq(1)1386 1138 y Fz(\()p Fy(z)1463 1152 y Fq(2)1523 1138 y Fu(\000)20 b Fy(z)1656 1152 y Fq(1)1696 1138 y Fz(\))1761 1076 y(+)2053 1015 y(1)p 1862 1055 V 1862 1138 a Fy(z)1904 1152 y Fq(2)1944 1138 y Fz(\()p Fy(z)2021 1152 y Fq(1)2081 1138 y Fu(\000)g Fy(z)2214 1152 y Fq(2)2254 1138 y Fz(\))2324 1076 y(=)2489 1015 y(1)p 2430 1055 164 4 v 2430 1138 a Fy(z)2472 1152 y Fq(1)2512 1138 y Fy(z)2554 1152 y Fq(2)-47 1324 y Fz(whic)m(h)37 b(follo)m(ws)g (immediately)g(b)m(y)h(insp)s(ection.)63 b(The)38 b(general)g(case)i (then)e(can)g(b)s(e)g(pro)m(v)m(ed)h(b)m(y)f(induction.)63 b(W)-8 b(e)-47 1437 y(shall)26 b(assume)i(that)h(the)f(iden)m(tit)m(y)f (holds)g(true)h(for)g Fy(m)15 b Fu(\000)g Fz(1)29 b(and)e(pro)m(v)m(e)i (the)g(result)e(for)g Fy(m)p Fz(.)40 b(W)-8 b(e)29 b(\014rst)f(note)g (a)h(simple)-47 1550 y(equalit)m(y)1217 1634 y(1)p 1052 1675 376 4 v 1052 1758 a(\()p Fy(z)1129 1772 y Fw(m)1217 1758 y Fu(\000)19 b Fy(z)1349 1773 y Fw(k)1392 1758 y Fz(\))1463 1696 y(=)1732 1634 y(1)p 1569 1675 373 4 v 1569 1758 a(\()p Fy(z)1646 1772 y Fw(m)1733 1758 y Fu(\000)h Fy(z)1866 1772 y Fq(1)1906 1758 y Fz(\))1971 1696 y(+)2307 1634 y Fy(z)2349 1649 y Fw(k)2413 1634 y Fu(\000)f Fy(z)2545 1648 y Fq(1)p 2072 1675 748 4 v 2072 1758 a Fz(\()p Fy(z)2149 1772 y Fw(m)2237 1758 y Fu(\000)h Fy(z)2370 1773 y Fw(k)2413 1758 y Fz(\)\()p Fy(z)2525 1772 y Fw(m)2612 1758 y Fu(\000)g Fy(z)2745 1772 y Fq(1)2785 1758 y Fz(\))2830 1696 y Fy(:)-47 1906 y Fz(Therefore,)244 2075 y Fw(m)209 2103 y Fn(X)211 2300 y Fw(k)r Fq(=1)385 2128 y Fz(1)p 366 2168 85 4 v 366 2251 a Fy(z)408 2266 y Fw(k)580 2075 y(m)553 2103 y Fn(Y)476 2300 y Fw(r)r Fq(=1)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)887 2128 y Fz(1)p 772 2168 277 4 v 772 2251 a Fy(z)814 2265 y Fw(r)872 2251 y Fu(\000)g Fy(z)1005 2266 y Fw(k)1083 2189 y Fz(=)1317 2128 y(1)p 1189 2168 302 4 v 1189 2251 a Fy(z)1231 2265 y Fw(m)1318 2251 y Fu(\000)g Fy(z)1451 2265 y Fq(1)1516 2075 y Fw(m)p Fx(\000)p Fq(1)1526 2103 y Fn(X)1528 2300 y Fw(k)r Fq(=1)1713 2128 y Fz(1)p 1694 2168 85 4 v 1694 2251 a Fy(z)1736 2266 y Fw(k)1863 2075 y(m)p Fx(\000)p Fq(1)1881 2103 y Fn(Y)1804 2300 y Fw(r)r Fq(=1)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)2215 2128 y Fz(1)p 2100 2168 277 4 v 2100 2251 a Fy(z)2142 2265 y Fw(r)2200 2251 y Fu(\000)g Fy(z)2333 2266 y Fw(k)2406 2189 y Fu(\000)2635 2128 y Fz(1)p 2507 2168 302 4 v 2507 2251 a Fy(z)2549 2265 y Fw(m)2636 2251 y Fu(\000)g Fy(z)2769 2265 y Fq(1)2834 2075 y Fw(m)p Fx(\000)p Fq(1)2844 2103 y Fn(X)2846 2300 y Fw(k)r Fq(=2)3031 2128 y Fz(1)p 3011 2168 85 4 v 3011 2251 a Fy(z)3053 2266 y Fw(k)3226 2075 y(m)3199 2103 y Fn(Y)3121 2300 y Fw(r)r Fq(=2)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)3533 2128 y Fz(1)p 3417 2168 277 4 v 3417 2251 a Fy(z)3459 2265 y Fw(r)3518 2251 y Fu(\000)g Fy(z)3651 2266 y Fw(k)1078 2521 y Fz(+)1211 2460 y(1)p 1179 2500 109 4 v 1179 2583 a Fy(z)1221 2597 y Fw(m)1313 2407 y(m)p Fx(\000)p Fq(1)1331 2435 y Fn(Y)1327 2629 y Fw(r)r Fq(=1)1618 2460 y Fz(1)p 1491 2500 301 4 v 1491 2583 a Fy(z)1533 2597 y Fw(r)1591 2583 y Fu(\000)g Fy(z)1724 2597 y Fw(m)1801 2521 y Fy(:)-47 2801 y Fz(According)45 b(to)h(the)f(assumption)f Fy(m)30 b Fu(\000)g Fz(1)46 b(with)e Fy(z)1759 2815 y Fq(1)1798 2801 y Fy(;)15 b(::;)g(z)1970 2815 y Fw(m)p Fx(\000)p Fq(1)2174 2801 y Fz(the)45 b(\014rst)g(term)g(on)h(the)f (righ)m(t)g(hand-side)e(giv)m(es)227 2878 y Fq(1)p -37 2893 565 4 v -37 2946 a(\()p Fw(z)23 2954 y Fs(m)82 2946 y Fx(\000)p Fw(z)170 2955 y Fr(1)204 2946 y Fq(\))p Fw(z)264 2955 y Fr(1)299 2946 y Fx(\001\001\001)o Fw(z)391 2955 y Fs(m)p Ft(\000)p Fr(1)537 2914 y Fz(.)e(W)-8 b(e)32 b(also)e(ha)m(v)m(e)306 3167 y(1)p 177 3207 302 4 v 177 3290 a Fy(z)219 3304 y Fw(m)307 3290 y Fu(\000)19 b Fy(z)439 3304 y Fq(1)504 3114 y Fw(m)p Fx(\000)p Fq(1)515 3142 y Fn(X)516 3339 y Fw(k)r Fq(=2)702 3167 y Fz(1)p 682 3207 85 4 v 682 3290 a Fy(z)724 3305 y Fw(k)896 3114 y(m)869 3142 y Fn(Y)792 3339 y Fw(r)r Fq(=2)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)1203 3167 y Fz(1)p 1088 3207 277 4 v 1088 3290 a Fy(z)1130 3304 y Fw(r)1188 3290 y Fu(\000)h Fy(z)1321 3305 y Fw(k)1399 3228 y Fz(=)1633 3167 y(1)p 1505 3207 302 4 v 1505 3290 a Fy(z)1547 3304 y Fw(m)1634 3290 y Fu(\000)g Fy(z)1767 3304 y Fq(1)1867 3114 y Fw(m)1832 3142 y Fn(X)1833 3339 y Fw(k)r Fq(=2)2008 3167 y Fz(1)p 1989 3207 85 4 v 1989 3290 a Fy(z)2031 3305 y Fw(k)2203 3114 y(m)2176 3142 y Fn(Y)2099 3339 y Fw(r)r Fq(=2)p Fw(;r)r Fx(6)p Fq(=)p Fw(k)2510 3167 y Fz(1)p 2395 3207 277 4 v 2395 3290 a Fy(z)2437 3304 y Fw(r)2495 3290 y Fu(\000)g Fy(z)2628 3305 y Fw(k)2701 3228 y Fu(\000)2930 3167 y Fz(1)p 2802 3207 302 4 v 2802 3290 a Fy(z)2844 3304 y Fw(m)2931 3290 y Fu(\000)g Fy(z)3064 3304 y Fq(1)3155 3167 y Fz(1)p 3123 3207 109 4 v 3123 3290 a Fy(z)3165 3304 y Fw(m)3257 3114 y(m)p Fx(\000)p Fq(1)3276 3142 y Fn(Y)3272 3336 y Fw(r)r Fq(=2)3562 3167 y Fz(1)p 3435 3207 301 4 v 3435 3290 a Fy(z)3477 3304 y Fw(r)3536 3290 y Fu(\000)f Fy(z)3668 3304 y Fw(m)1399 3560 y Fz(=)1832 3499 y(1)p 1505 3539 700 4 v 1505 3622 a(\()p Fy(z)1582 3636 y Fw(m)1670 3622 y Fu(\000)h Fy(z)1803 3636 y Fq(1)1842 3622 y Fz(\))p Fy(z)1919 3636 y Fq(2)1975 3622 y Fu(\001)15 b(\001)g(\001)h Fy(z)2138 3636 y Fw(m)2235 3560 y Fu(\000)2367 3499 y Fz(1)p 2336 3539 109 4 v 2336 3622 a Fy(z)2378 3636 y Fw(m)2470 3446 y(m)p Fx(\000)p Fq(1)2488 3474 y Fn(Y)2484 3668 y Fw(r)r Fq(=1)2775 3499 y Fz(1)p 2647 3539 301 4 v 2647 3622 a Fy(z)2689 3636 y Fw(r)2748 3622 y Fu(\000)k Fy(z)2881 3636 y Fw(m)-47 3840 y Fz(b)m(y)30 b(the)h Fy(m)20 b Fu(\000)g Fz(1)30 b(inductiv)m(e)f(assumption)g(for)h Fy(z)1558 3854 y Fq(2)1598 3840 y Fy(;)15 b(::;)g(z)1770 3854 y Fw(m)1838 3840 y Fz(.)41 b(Finally)-8 b(,)1264 4016 y(1)p 892 4056 790 4 v 892 4140 a(\()p Fy(z)969 4154 y Fw(m)1057 4140 y Fu(\000)20 b Fy(z)1190 4154 y Fq(1)1229 4140 y Fz(\))p Fy(z)1306 4154 y Fq(1)1361 4140 y Fu(\001)15 b(\001)g(\001)i Fy(z)1525 4154 y Fw(m)p Fx(\000)p Fq(1)1712 4077 y Fu(\000)2140 4016 y Fz(1)p 1813 4056 700 4 v 1813 4140 a(\()p Fy(z)1890 4154 y Fw(m)1977 4140 y Fu(\000)j Fy(z)2110 4154 y Fq(1)2150 4140 y Fz(\))p Fy(z)2227 4154 y Fq(2)2282 4140 y Fu(\001)15 b(\001)g(\001)h Fy(z)2445 4154 y Fw(m)2547 4077 y Fz(=)2794 4016 y(1)p 2653 4056 327 4 v 2653 4140 a Fy(z)2695 4154 y Fq(1)2750 4140 y Fu(\001)f(\001)g(\001)h Fy(z)2913 4154 y Fw(m)2990 4077 y Fy(;)-47 4325 y Fz(as)30 b(desired.)p 3879 4325 4 62 v 3883 4267 55 4 v 3883 4325 V 3937 4325 4 62 v -47 4513 a(W)-8 b(e)25 b(shall)e(no)m(w)i(deriv)m(e)f(the)h(explicit)d (represen)m(tation)j(of)g(the)f(in)m(tegral)h(k)m(ernel)f(of)g(the)h (op)s(erator)g Fu(I)3305 4527 y Fw(m)3371 4513 y Fz(\()p Fy(t;)15 b(s)p Fz(\))25 b(acting)g(on)-47 4626 y(the)i(Sc)m(h)m(w)m (artz)i(functions)d Fy( )938 4640 y Fw(s)975 4626 y Fz(.)40 b(W)-8 b(e)28 b(start)g(b)m(y)g(noting)f(the)g(follo)m(wing)f(simple)g (iden)m(tit)m(y)h(whic)m(h)f(holds)g(for)h(arbitrary)-47 4739 y(real)j(n)m(um)m(b)s(ers)f Fy(a)545 4753 y Fq(1)584 4739 y Fy(;)15 b(::;)g(a)762 4753 y Fw(m)p Fq(+1)951 4739 y Fz(with)29 b Fy(m)c Fu(\025)g Fz(1:)940 4911 y Fw(m)p Fq(+1)951 4938 y Fn(X)952 5136 y Fw(k)r Fq(=1)1108 5024 y Fy(e)1150 4987 y Fw(i)p Fq(\()p Fw(a)1238 4996 y Fr(1)1274 4987 y Fq(+)p Fw(::)p Fq(+)p Fw(a)1461 4999 y Fs(k)q Ft(\000)p Fr(1)1577 4987 y Fx(\000)p Fw(a)1669 4999 y Fs(k)q Fr(+1)1784 4987 y Fx(\000)p Fw(::)p Fx(\000)p Fw(a)1971 4996 y Fs(m)p Fr(+1)2106 4987 y Fq(\))2153 5024 y Fz(sin)14 b Fy(a)2328 5039 y Fw(k)2396 5024 y Fz(=)25 b(sin)n(\()2638 4911 y Fw(m)p Fq(+1)2649 4938 y Fn(X)2650 5136 y Fw(k)r Fq(=1)2806 5024 y Fy(a)2854 5039 y Fw(k)2897 5024 y Fz(\))p Fy(:)-3004 b Fz(\(5.36\))1903 5472 y(38)p eop %%Page: 39 39 39 38 bop -47 274 a Fz(This)28 b(iden)m(tit)m(y)i(can)h(b)s(e)f(easily) f(pro)m(v)m(ed)i(b)m(y)f(induction)e(on)i Fy(m)p Fz(.)41 b(Recall)30 b(that)1266 528 y Fy(R)1335 542 y Fx(\006)1394 528 y Fz(\()p Fy(\026)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))26 b(=)1861 466 y Fy(e)1903 433 y Fx(\006)p Fw(i)1982 392 y Fx(p)p 2041 392 43 3 v 41 x Fw(\026)q Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p 1861 507 398 4 v 1879 590 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)s Fu(j)2269 528 y Fy(;)106 b(\026)25 b Fu(2)g Fv(R)-47 773 y Fz(with)165 716 y Fu(p)p 241 716 55 4 v 57 x Fy(\026)35 b Fz(de\014ned)g(in)f(suc)m (h)i(a)g(w)m(a)m(y)g(that)h(Im)1553 716 y Fu(p)p 1629 716 V 57 x Fy(\026)d(>)g Fz(0)i(for)f(Im)p Fy(\026)f(>)f Fz(0.)58 b(W)-8 b(e)36 b(ha)m(v)m(e)h Fy(R)2923 787 y Fq(+)2982 773 y Fz(\()p Fy(\026)p Fz(\))e(=)f Fy(R)3316 787 y Fx(\000)3375 773 y Fz(\()p Fy(\026)p Fz(\))i(for)f Fy(\026)f(<)g Fz(0.)-47 885 y(Also)c(recall)f(that)i(the)g(k)m(ernel)f (of)g(the)h(sp)s(ectral)f(measure)1176 1160 y Fy(dE)5 b Fz(\()p Fy(\026)p Fz(\)\()p Fy(x;)15 b(y)s Fz(\))27 b(=)1753 1004 y Fn(\()1836 1045 y Fq(sin)1934 1003 y Fx(p)p 1993 1003 43 3 v 42 x Fw(\026)q Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)p 1836 1074 371 4 v 1897 1127 a Fq(4)p Fw(\031)r Fx(j)p Fw(x)p Fx(\000)p Fw(y)r Fx(j)2232 1095 y Fy(d\026)91 b(\026)25 b(>)g Fz(0)p Fy(;)1826 1239 y Fz(0)554 b Fy(\026)25 b Fu(\024)g Fz(0)2697 1160 y Fy(:)-47 1439 y Fz(W)-8 b(e)31 b(return)f(to)h(the)f(represen)m(tation) h(\(5.26\))h(for)e(the)h Fu(I)1858 1453 y Fw(m)1924 1439 y Fz(.)41 b(Let)31 b(\(with)e Fy(\034)2435 1453 y Fw(m)p Fq(+1)2617 1439 y Fz(=)c(0\))1175 1635 y Fy(\034)1215 1649 y Fw(j)1272 1635 y Fz(+)20 b Fy(::)g Fz(+)g Fy(\034)1564 1649 y Fw(m)p Fq(+1)1746 1635 y Fz(=)68 b(min)1842 1700 y Fw(r)r Fx(2)p Fq([1)p Fw(;m)p Fq(])2079 1635 y Fz(\()p Fy(\034)2154 1649 y Fw(r)2213 1635 y Fz(+)20 b Fy(::)g Fz(+)g Fy(\034)2505 1649 y Fw(m)p Fq(+1)2662 1635 y Fz(\))p Fy(:)-47 1886 y Fz(T)-8 b(o)31 b(simplify)c(the)j(form)m(ulae)g(w)m(e)h (in)m(tro)s(duce)e(a)i(new)f(op)s(erator)h Fu(J)2191 1900 y Fw(m)2257 1886 y Fz(\()p Fy(t;)15 b(s)p Fz(\),)31 b(implicitly)26 b(dep)s(enden)m(t)k(on)g Fy(\034)3511 1900 y Fq(1)3550 1886 y Fy(;)15 b(::;)g(\034)3720 1900 y Fw(m)3788 1886 y Fz(,)79 2124 y Fu(I)129 2138 y Fw(m)195 2124 y Fz(\()p Fy(t;)g(s)p Fz(\))26 b(=)f Fy(i)534 2087 y Fw(m)616 2000 y Fn(Z)666 2207 y Fp(R)714 2188 y Fs(m)792 2124 y Fy(d\034)879 2138 y Fq(1)919 2124 y Fy(::d\034)1056 2138 y Fw(m)1148 2124 y Fy(e)1190 2087 y Fw(it)p Fq(\()p Fw(\034)1297 2096 y Fr(1)1333 2087 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1514 2095 y Fs(m)1572 2087 y Fq(\))1604 2124 y Fy(e)1646 2087 y Fx(\000)p Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\)\()p Fw(\034)1950 2097 y Fs(j)1984 2087 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)2165 2096 y Fs(m)p Fr(+1)2300 2087 y Fq(\))2332 2124 y Fu(J)2394 2138 y Fw(m)2460 2124 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(\034)2721 2138 y Fq(1)2761 2124 y Fy(;)g(::;)g(\034)2931 2138 y Fw(m)2999 2124 y Fz(\))p Fy(;)79 2530 y Fu(J)141 2544 y Fw(m)207 2530 y Fz(\()p Fy(t;)g(s)p Fz(\))26 b(:=)540 2406 y Fn(Z)591 2612 y Fw(\025)651 2530 y Fy(e)693 2492 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\)\()p Fw(\025)p Fq(+)p Fw(\034)1038 2502 y Fs(j)1073 2492 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1254 2501 y Fs(m)p Fr(+1)1389 2492 y Fq(\))1435 2416 y Fw(m)p Fq(+1)1446 2443 y Fn(X)1447 2641 y Fw(k)r Fq(=1)1603 2402 y Fn(\022)1685 2416 y Fw(k)r Fx(\000)p Fq(1)1691 2443 y Fn(Y)1687 2638 y Fw(r)r Fq(=1)1829 2530 y Fy(R)1898 2544 y Fq(+)1957 2530 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)2197 2544 y Fw(r)2254 2530 y Fz(+)g Fy(::)h Fz(+)f Fy(\034)2547 2544 y Fw(m)2613 2530 y Fz(\))p Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)2797 2544 y Fw(r)2835 2530 y Fy(;)15 b Fu(\001)p Fz(\))2935 2402 y Fn(\023)3003 2530 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)3362 2545 y Fw(k)3425 2530 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)3717 2544 y Fw(m)3783 2530 y Fz(\))-47 2319 y(\(5.37\))1075 2721 y Fn(\022)1190 2735 y Fw(m)p Fq(+1)1208 2763 y Fn(Y)1157 2960 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)1390 2849 y Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)1538 2863 y Fw(r)r Fx(\000)p Fq(1)1666 2849 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)1835 2863 y Fx(\000)1895 2849 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\034)2135 2863 y Fw(r)2193 2849 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)2485 2863 y Fw(m)p Fq(+1)2642 2849 y Fz(\))2677 2721 y Fn(\023)2744 2849 y Fy(:)-47 3131 y Fz(De\014ne)30 b(non-negativ)m(e)i(n)m(um)m(b)s (ers)d Fy(\033)1189 3145 y Fw(r)1226 3131 y Fz(,)i Fy(r)d Fz(=)d(1)p Fy(;)15 b(::;)g(m)22 b Fz(+)e(1)1158 3327 y Fy(\033)1210 3341 y Fw(r)1273 3327 y Fz(=)25 b(\()p Fy(\034)1444 3341 y Fw(r)1503 3327 y Fz(+)19 b Fy(::)i Fz(+)f Fy(\034)1795 3341 y Fw(m)p Fq(+1)1952 3327 y Fz(\))g Fu(\000)g Fz(\()p Fy(\034)2173 3341 y Fw(j)2230 3327 y Fz(+)g Fy(::)g Fz(+)g Fy(\034)2522 3341 y Fw(m)p Fq(+1)2679 3327 y Fz(\))p Fy(:)-47 3523 y Fz(After)30 b(a)h(c)m(hange)h(of)e(v)-5 b(ariables)29 b(w)m(e)i(obtain)f(the)g(expression)85 3802 y Fu(J)147 3816 y Fw(m)213 3802 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)521 3678 y Fn(Z)571 3884 y Fw(\025)631 3802 y Fy(e)673 3764 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)926 3688 y(m)p Fq(+1)937 3715 y Fn(X)938 3913 y Fw(k)r Fq(=1)1094 3673 y Fn(\022)1176 3688 y Fw(k)r Fx(\000)p Fq(1)1182 3715 y Fn(Y)1178 3910 y Fw(r)r Fq(=1)1320 3802 y Fy(R)1389 3816 y Fq(+)1448 3802 y Fz(\()p Fy(\025)21 b Fz(+)e Fy(\033)1699 3816 y Fw(r)1737 3802 y Fz(\))p Fy(V)i Fz(\()5 b(^)-50 b Fy(\034)1921 3816 y Fw(r)1959 3802 y Fy(;)15 b Fu(\001)p Fz(\))2059 3673 y Fn(\023)2127 3802 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\033)2498 3817 y Fw(k)2541 3802 y Fz(\))2576 3673 y Fn(\022)2690 3688 y Fw(m)p Fq(+1)2709 3715 y Fn(Y)2658 3913 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2891 3802 y Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)3039 3816 y Fw(r)r Fx(\000)p Fq(1)3167 3802 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)3336 3816 y Fx(\000)3396 3802 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(\033)3648 3816 y Fw(r)3686 3802 y Fz(\))3721 3673 y Fn(\023)3788 3802 y Fy(:)-47 4100 y Fz(Observ)m(e)g(that)h(due)f(to)i (the)e(presence)h(of)g Fy(dE)5 b Fz(\()p Fy(\025)q Fz(+)q Fy(\033)1670 4115 y Fw(k)1712 4100 y Fz(\))21 b(the)g Fy(k)1965 4067 y Fw(th)2056 4100 y Fz(term)f(in)g(the)g(sum)g(ab)s(o)m (v)m(e)i(v)-5 b(anishes)19 b(for)h Fy(\025)26 b Fu(\024)f(\000)p Fy(\033)3711 4115 y Fw(k)3778 4100 y Fu(\024)g Fz(0.)-47 4213 y(Therefore,)21 4409 y Fu(J)83 4423 y Fw(m)149 4409 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)f Fu(L)520 4423 y Fw(m)586 4409 y Fz(\()p Fy(t;)15 b(s)p Fz(\))21 b(+)f Fu(M)993 4423 y Fw(m)1060 4409 y Fz(\()p Fy(t;)15 b(s)p Fz(\))21 4766 y Fu(L)84 4780 y Fw(m)150 4766 y Fz(\()p Fy(t;)g(s)p Fz(\))26 b(:=)483 4642 y Fn(Z)574 4669 y Fx(1)534 4848 y Fq(0)664 4766 y Fy(e)706 4728 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)959 4652 y(m)p Fq(+1)970 4680 y Fn(X)971 4877 y Fw(k)r Fq(=0)1127 4638 y Fn(\022)1209 4652 y Fw(k)r Fx(\000)p Fq(1)1215 4680 y Fn(Y)1211 4874 y Fw(r)r Fq(=1)1353 4766 y Fy(R)1422 4780 y Fq(+)1481 4766 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\033)1732 4780 y Fw(r)1770 4766 y Fz(\))p Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)1954 4780 y Fw(r)1992 4766 y Fy(;)15 b Fu(\001)p Fz(\))2092 4638 y Fn(\023)2160 4766 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\033)2531 4781 y Fw(k)2573 4766 y Fz(\))2608 4638 y Fn(\022)2723 4652 y Fw(m)p Fq(+1)2741 4680 y Fn(Y)2691 4877 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2923 4766 y Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)3072 4780 y Fw(r)r Fx(\000)p Fq(1)3200 4766 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)3369 4780 y Fx(\000)3429 4766 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\033)3680 4780 y Fw(r)3718 4766 y Fz(\))3753 4638 y Fn(\023)-47 4555 y Fz(\(5.38\))21 5095 y Fu(M)130 5109 y Fw(m)197 5095 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(:=)530 4972 y Fn(Z)621 4998 y Fq(0)580 5178 y Fx(\0001)725 5095 y Fy(e)767 5058 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1020 4982 y(m)p Fq(+1)1031 5009 y Fn(X)1032 5207 y Fw(k)r Fq(=1)1188 4967 y Fn(\022)1270 4982 y Fw(k)r Fx(\000)p Fq(1)1276 5009 y Fn(Y)1272 5203 y Fw(r)r Fq(=1)1414 5095 y Fy(R)1483 5109 y Fq(+)1542 5095 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\033)1793 5109 y Fw(r)1831 5095 y Fz(\))p Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)2015 5109 y Fw(r)2053 5095 y Fy(;)15 b Fu(\001)p Fz(\))2153 4967 y Fn(\023)2221 5095 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(\033)2592 5110 y Fw(k)2634 5095 y Fz(\))2669 4967 y Fn(\022)2784 4982 y Fw(m)p Fq(+1)2802 5009 y Fn(Y)2752 5207 y Fw(r)r Fq(=)p Fw(k)r Fq(+1)2984 5095 y Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)3133 5109 y Fw(r)r Fx(\000)p Fq(1)3261 5095 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)3430 5109 y Fx(\000)3490 5095 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(\033)3741 5109 y Fw(r)3779 5095 y Fz(\))3814 4967 y Fn(\023)3882 5095 y Fy(:)1903 5472 y Fz(39)p eop %%Page: 40 40 40 39 bop -47 274 a Fz(T)-8 b(o)39 b(obtain)f(the)h(explicit)d(form)m (ula)i(for)g(the)h(in)m(tegral)f(k)m(ernel)h(of)f(the)h(op)s(erator)g Fu(L)2863 288 y Fw(m)2929 274 y Fz(\()p Fy(t;)15 b(s)p Fz(\))39 b(w)m(e)g(mak)m(e)h(use)e(of)h(the)-47 387 y(follo)m(wing:)d (the)24 b(parameters)g Fy(\033)1035 402 y Fw(k)1103 387 y Fu(\025)h Fz(0,)h Fy(\025)f Fu(\025)g Fz(0)f(on)g(the)g(in)m(terv)-5 b(al)24 b(of)g(in)m(tegration,)h(and)e(the)i(explicit)d(represen)m (tations)-47 500 y(for)30 b(the)g(k)m(ernels)g(of)h Fy(R)725 514 y Fx(\006)784 500 y Fz(\()p Fy(\026)p Fz(\))g(and)e Fy(dE)5 b Fz(\()p Fy(\026)p Fz(\).)42 b(W)-8 b(e)32 b(ha)m(v)m(e)-32 788 y Fu(L)31 802 y Fw(m)97 788 y Fz(\()p Fy(t;)15 b(s)p Fz(\))q(\()p Fy(x;)g(y)s Fz(\))26 b(=)616 664 y Fn(Z)666 870 y Fp(R)714 851 y Fs(m)792 788 y Fy(dx)891 802 y Fq(1)931 788 y Fy(::dx)1080 802 y Fw(m)1162 664 y Fn(Z)1253 690 y Fx(1)1213 870 y Fq(0)1343 788 y Fy(d\025)15 b(e)1500 750 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1753 674 y(m)p Fq(+1)1764 701 y Fn(X)1765 899 y Fw(k)r Fq(=1)1921 659 y Fn(\024)1969 788 y Fy(e)2011 750 y Fw(i)p Fq(\()2062 698 y Fx(p)p 2122 698 171 3 v 2122 750 a Fw(\025)p Fq(+)p Fw(\033)2258 759 y Fr(1)2293 750 y Fx(j)p Fw(x)p Fx(\000)p Fw(x)2448 759 y Fr(1)2481 750 y Fx(j)p Fq(+)p Fw(::)p Fq(+)2651 691 y Fu(p)p 2726 691 253 4 v 2726 750 a Fw(\025)p Fq(+)p Fw(\033)2862 762 y Fs(k)q Ft(\000)p Fr(1)2979 750 y Fx(j)p Fw(x)3039 762 y Fs(k)q Ft(\000)p Fr(2)3154 750 y Fx(\000)p Fw(x)3249 762 y Fs(k)q Ft(\000)p Fr(1)3366 750 y Fx(j)p Fq(\))284 1093 y Fy(e)326 1055 y Fx(\000)p Fw(i)p Fq(\()432 996 y Fu(p)p 508 996 252 4 v 59 x Fw(\025)p Fq(+)p Fw(\033)644 1067 y Fs(k)q Fr(+1)760 1055 y Fx(j)p Fw(x)820 1067 y Fs(k)857 1055 y Fx(\000)p Fw(x)952 1067 y Fs(k)q Fr(+1)1067 1055 y Fx(j\000)p Fw(::)p Fx(\000)1237 1002 y(p)p 1294 1002 196 3 v 1294 1055 a Fw(\025)p Fq(+)p Fw(\033)1430 1063 y Fs(m)1489 1055 y Fx(j)p Fw(x)1549 1064 y Fs(m)p Ft(\000)p Fr(1)1686 1055 y Fx(\000)p Fw(y)r Fx(j)p Fq(\))1844 1093 y Fz(sin)1971 1019 y Fn(\000)2013 1009 y(p)p 2103 1009 259 4 v 2103 1093 a Fy(\025)21 b Fz(+)f Fy(\033)2320 1108 y Fw(k)2362 1093 y Fu(j)p Fy(x)2439 1108 y Fw(k)r Fx(\000)p Fq(1)2592 1093 y Fu(\000)g Fy(x)2735 1108 y Fw(k)2778 1093 y Fu(j)2803 1019 y Fn(\001)2891 979 y Fw(m)2864 1006 y Fn(Y)2860 1201 y Fw(r)r Fq(=1)3099 1031 y Fy(V)h Fz(\()5 b(^)-50 b Fy(\034)3248 1045 y Fw(r)3286 1031 y Fy(;)15 b(x)3378 1045 y Fw(r)3416 1031 y Fz(\))p 3009 1072 533 4 v 3009 1155 a(4)p Fy(\031)s Fu(j)p Fy(x)3186 1169 y Fw(r)r Fx(\000)p Fq(1)3335 1155 y Fu(\000)20 b Fy(x)3478 1169 y Fw(r)3516 1155 y Fu(j)3778 1031 y Fz(1)p 3587 1072 429 4 v 3587 1155 a(4)p Fy(\031)s Fu(j)p Fy(x)3764 1169 y Fw(m)3851 1155 y Fu(\000)g Fy(y)s Fu(j)4025 964 y Fn(\025)4073 1093 y Fy(;)-47 1377 y Fz(where)30 b(w)m(e)g(set)h Fy(x)543 1391 y Fq(0)608 1377 y Fz(=)25 b Fy(x)p Fz(.)41 b(W)-8 b(e)31 b(no)m(w)g(recall)e(the)i(iden)m(tit)m(y)g(\(5.36\))i(to) e(infer)e(that)-47 1665 y Fu(L)16 1679 y Fw(m)82 1665 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(=)601 1542 y Fn(Z)651 1748 y Fp(R)699 1729 y Fs(m)777 1665 y Fy(dx)876 1679 y Fq(1)916 1665 y Fy(::dx)1065 1679 y Fw(m)1193 1552 y(m)1166 1579 y Fn(Y)1162 1773 y Fw(r)r Fq(=1)1402 1604 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1550 1618 y Fw(r)1588 1604 y Fy(;)15 b(x)1680 1618 y Fw(r)1718 1604 y Fz(\))p 1312 1644 533 4 v 1312 1728 a(4)p Fy(\031)s Fu(j)p Fy(x)1489 1742 y Fw(r)r Fx(\000)p Fq(1)1638 1728 y Fu(\000)20 b Fy(x)1781 1742 y Fw(r)1818 1728 y Fu(j)2055 1604 y Fz(1)p 1864 1644 429 4 v 1864 1728 a(4)p Fy(\031)s Fu(j)p Fy(x)2041 1742 y Fw(m)2128 1728 y Fu(\000)g Fy(y)s Fu(j)2317 1542 y Fn(Z)2408 1568 y Fx(1)2368 1748 y Fq(0)2498 1665 y Fy(d\025)15 b(e)2655 1628 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2908 1665 y Fz(sin)3035 1537 y Fn(\022)3117 1552 y Fw(m)p Fq(+1)3128 1579 y Fn(X)3129 1777 y Fw(k)r Fq(=1)3285 1582 y Fn(p)p 3376 1582 259 4 v 83 x Fy(\025)20 b Fz(+)g Fy(\033)3592 1680 y Fw(k)3635 1665 y Fu(j)p Fy(x)3712 1680 y Fw(k)r Fx(\000)p Fq(1)3865 1665 y Fu(\000)g Fy(x)4008 1680 y Fw(k)4050 1665 y Fu(j)4075 1537 y Fn(\023)4142 1665 y Fy(:)-47 1948 y Fz(Changing)29 b(v)-5 b(ariables)29 b(in)g(the)h Fy(\025)p Fz(-in)m(tegral)h(and)f(in)m(tegrating)g(b)m(y)g (parts)g(yield)48 2093 y Fn(Z)139 2120 y Fx(1)99 2299 y Fq(0)229 2217 y Fy(e)271 2180 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)524 2217 y Fz(sin)650 2089 y Fn(\022)767 2103 y Fw(m)733 2131 y Fn(X)734 2328 y Fw(k)r Fq(=1)879 2133 y Fn(p)p 970 2133 V 84 x Fy(\025)20 b Fz(+)g Fy(\033)1186 2232 y Fw(k)1229 2217 y Fu(j)p Fy(x)1306 2232 y Fw(k)r Fx(\000)p Fq(1)1459 2217 y Fu(\000)g Fy(x)1602 2232 y Fw(k)1644 2217 y Fu(j)1669 2089 y Fn(\023)1762 2217 y Fz(=)25 b(2)1918 2093 y Fn(Z)2009 2120 y Fx(1)1969 2299 y Fq(0)2099 2217 y Fy(d\025)15 b(\025)g(e)2324 2180 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)2556 2156 y Fr(2)2612 2217 y Fz(sin)2739 2089 y Fn(\022)2856 2103 y Fw(m)2821 2131 y Fn(X)2822 2328 y Fw(k)r Fq(=1)2968 2130 y Fn(p)p 3058 2130 299 4 v 3058 2217 a Fy(\025)3111 2191 y Fq(2)3171 2217 y Fz(+)20 b Fy(\033)3314 2232 y Fw(k)3357 2217 y Fu(j)p Fy(x)3434 2232 y Fw(k)r Fx(\000)p Fq(1)3587 2217 y Fu(\000)g Fy(x)3730 2232 y Fw(k)3772 2217 y Fu(j)3797 2089 y Fn(\023)666 2522 y Fz(=)850 2460 y Fy(i)p 772 2501 187 4 v 772 2584 a(t)g Fu(\000)g Fy(s)1018 2408 y Fw(m)984 2436 y Fn(X)990 2633 y Fw(`)p Fq(=1)1130 2398 y Fn(Z)1221 2425 y Fx(1)1181 2604 y Fq(0)1311 2522 y Fy(d\025)15 b(e)1468 2484 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1700 2461 y Fr(2)1756 2522 y Fz(cos)1893 2394 y Fn(\022)2009 2408 y Fw(m)1975 2436 y Fn(X)1976 2633 y Fw(k)r Fq(=1)2121 2435 y Fn(p)p 2212 2435 299 4 v 87 x Fy(\025)2265 2496 y Fq(2)2325 2522 y Fz(+)20 b Fy(\033)2468 2537 y Fw(k)2510 2522 y Fu(j)p Fy(x)2587 2537 y Fw(k)r Fx(\000)p Fq(1)2741 2522 y Fu(\000)g Fy(x)2884 2537 y Fw(k)2926 2522 y Fu(j)2951 2394 y Fn(\023)3191 2460 y Fy(\025)p 3028 2501 380 4 v 3028 2519 a Fn(p)p 3119 2519 289 4 v 82 x Fy(\025)3172 2575 y Fq(2)3232 2601 y Fz(+)g Fy(\033)3375 2616 y Fw(`)3418 2522 y Fu(j)p Fy(x)3495 2537 y Fw(`)p Fx(\000)p Fq(1)3638 2522 y Fu(\000)g Fy(x)3781 2537 y Fw(`)3814 2522 y Fu(j)p Fy(:)-47 2805 y Fz(Therefore,)30 b(\014nally)518 3074 y Fu(L)581 3088 y Fw(m)647 3074 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(=)1253 3012 y Fy(i)p 1176 3053 187 4 v 1176 3136 a(t)20 b Fu(\000)g Fy(s)1422 2960 y Fw(m)1387 2987 y Fn(X)1393 3185 y Fw(`)p Fq(=1)1534 3074 y Fu(L)1597 3036 y Fw(`)1597 3096 y(m)1663 3074 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))p Fy(;)518 3379 y Fu(L)581 3341 y Fw(`)581 3401 y(m)647 3379 y Fz(\()p Fy(t;)g(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(:=)1191 3255 y Fn(Z)1241 3461 y Fp(R)1289 3442 y Fs(m)1367 3379 y Fy(dx)1466 3393 y Fq(1)1521 3379 y Fy(:)15 b(:)g(:)h(dx)1741 3393 y Fw(m)1854 3265 y(m)1827 3292 y Fn(Y)1823 3487 y Fw(r)r Fq(=1)2063 3317 y Fy(V)k Fz(\()5 b(^)-50 b Fy(\034)2211 3331 y Fw(r)2249 3317 y Fy(;)15 b(x)2341 3331 y Fw(r)2379 3317 y Fz(\))p 1972 3358 533 4 v 1972 3441 a(4)p Fy(\031)s Fu(j)p Fy(x)2149 3455 y Fw(r)r Fx(\000)p Fq(1)2299 3441 y Fu(\000)k Fy(x)2441 3455 y Fw(r)2479 3441 y Fu(j)2558 3317 y(j)p Fy(x)2635 3332 y Fw(`)p Fx(\000)p Fq(1)2779 3317 y Fu(\000)h Fy(x)2922 3332 y Fw(`)2955 3317 y Fu(j)p 2555 3358 429 4 v 2555 3441 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)2732 3455 y Fw(m)2819 3441 y Fu(\000)g Fy(y)s Fu(j)-47 3379 y Fz(\(5.39\))1330 3556 y Fn(Z)1421 3583 y Fx(1)1380 3763 y Fq(0)1511 3680 y Fy(d\025)15 b(e)1668 3643 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1900 3619 y Fr(2)1955 3680 y Fz(cos)2092 3552 y Fn(\022)2209 3567 y Fw(m)2174 3594 y Fn(X)2176 3791 y Fw(k)r Fq(=1)2321 3594 y Fn(p)p 2412 3594 299 4 v 86 x Fy(\025)2465 3654 y Fq(2)2524 3680 y Fz(+)20 b Fy(\033)2667 3695 y Fw(k)2710 3680 y Fu(j)p Fy(x)2787 3695 y Fw(k)r Fx(\000)p Fq(1)2940 3680 y Fu(\000)g Fy(x)3083 3695 y Fw(k)3126 3680 y Fu(j)3151 3552 y Fn(\023)3391 3619 y Fy(\025)p 3228 3659 380 4 v 3228 3677 a Fn(p)p 3319 3677 289 4 v 83 x Fy(\025)3372 3733 y Fq(2)3431 3760 y Fz(+)g Fy(\033)3574 3775 y Fw(`)3617 3680 y Fy(:)-47 3979 y Fz(T)-8 b(o)28 b(describ)s(e)d(the)j(in)m(tegral)f(k)m(ernels)g (of)h(the)f(op)s(erators)h Fu(M)1978 3946 y Fw(k)1978 4002 y(m)2045 3979 y Fz(\()p Fy(t;)15 b(s)p Fz(\))28 b(w)m(e)g(shall)d(\014rst)i(order)g(and)g(rename)g(the)h(param-)-47 4092 y(eters)j Fy(\033)223 4107 y Fw(k)265 4092 y Fz(,)g Fy(k)d Fz(=)d(1)p Fy(;)15 b(::;)g(m)22 b Fz(+)e(1.)41 b(In)30 b(fact,)h(de\014ne)f(inductiv)m(ely)1198 4296 y Fy(!)1255 4311 y Fw(d)1320 4296 y Fz(=)25 b(max)p Fu(f)p Fy(\033)1682 4311 y Fw(k)1726 4296 y Fu(g)1771 4315 y Fw(k)r Fx(2)p Fq([1)p Fw(;m)p Fq(+1])2128 4296 y Fu(n)20 b(f)p Fy(!)2295 4311 y Fw(`)2328 4296 y Fu(g)2373 4315 y Fw(`)p Fx(2)p Fq([1)p Fw(;d)p Fx(\000)p Fq(1])2674 4296 y Fy(;)-47 4501 y Fz(and)32 b(set)i Fy(k)e Fz(=)e Fy(k)s Fz(\()p Fy(c)p Fz(\))k(and)e Fy(c)e Fz(=)g Fy(c)p Fz(\()p Fy(k)s Fz(\))k(i\013)e Fy(\033)1355 4516 y Fw(k)1427 4501 y Fz(=)e Fy(!)1585 4515 y Fw(c)1619 4501 y Fz(.)48 b(W)-8 b(e)35 b(shall)c(split)g(the)i(in)m(terv)-5 b(al)32 b(of)h(in)m(tegration)g(in)f Fy(\025)h Fz(in)f(\()p Fu(\0001)p Fy(;)15 b Fz(0])-47 4614 y(in)m(to)30 b(the)h(subin)m(terv)-5 b(als)505 4818 y(\()p Fu(\0001)p Fy(;)15 b Fu(\000)813 4755 y(p)p 888 4755 97 4 v 888 4818 a Fy(!)945 4832 y Fq(1)1010 4818 y Fz(])p Fy(;)g Fz([)p Fu(\000)1171 4759 y(p)p 1247 4759 188 4 v 59 x Fy(!)1304 4833 y Fw(d)p Fx(\000)p Fq(1)1434 4818 y Fy(;)g Fu(\000)1545 4755 y(p)p 1621 4755 98 4 v 63 x Fy(!)1678 4833 y Fw(d)1743 4818 y Fz(])61 b(for)g Fy(d)25 b Fu(2)g Fz([2)p Fy(;)15 b(m)21 b Fz(+)f(1])p Fy(;)77 b Fz(and)90 b([)p Fu(\000)2964 4748 y(p)p 3040 4748 60 4 v 70 x Fy(!)3100 4839 y Fw(m)p Fq(+1)3257 4818 y Fy(;)15 b Fz(0])p Fy(:)-47 5022 y Fz(F)-8 b(or)37 b Fy(\025)f Fu(2)f Fz([)p Fu(\000)403 4957 y(p)p 478 4957 V 478 5022 a Fy(!)538 5048 y Fw(d)p Fx(\000)p Fq(1)669 5022 y Fy(;)15 b Fu(\000)780 4957 y(p)p 856 4957 V 65 x Fy(!)916 5048 y Fw(d)956 5022 y Fz(],)39 b(the)d(sp)s(ectral)g(measures)g Fy(dE)5 b Fz(\()p Fy(\025)26 b Fz(+)e Fy(\033)2331 5041 y Fw(k)r Fq(\()p Fw(c)p Fq(\))2459 5022 y Fz(\))36 b(=)f Fy(dE)5 b Fz(\()p Fy(\025)25 b Fz(+)f Fy(!)3020 5036 y Fw(c)3055 5022 y Fz(\))37 b(v)-5 b(anish)34 b(for)j(all)e Fy(c)h Fu(\025)f Fy(d)p Fz(.)1903 5472 y(40)p eop %%Page: 41 41 41 40 bop -47 274 a Fz(Therefore,)30 b(with)f(the)i(con)m(v)m(en)m (tion)g(that)g Fy(!)1462 288 y Fq(0)1527 274 y Fz(=)25 b Fu(1)30 b Fz(and)f Fy(!)1977 288 y Fw(m)p Fq(+2)2159 274 y Fz(=)c(0,)31 b(w)m(e)g(ha)m(v)m(e)500 555 y Fu(M)609 569 y Fw(m)675 555 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(=)983 441 y Fw(m)p Fq(+2)994 468 y Fn(X)996 666 y Fw(d)p Fq(=1)1151 555 y Fu(M)1260 517 y Fw(d)1260 577 y(m)1327 555 y Fy(;)500 921 y Fu(M)609 883 y Fw(d)609 943 y(m)675 921 y Fz(\()p Fy(t;)15 b(s)p Fz(\))26 b(:=)1068 765 y Fx(\000)1123 723 y(p)p 1181 723 81 3 v 1181 765 a Fw(!)1225 777 y Fs(d)1099 797 y Fn(Z)988 1059 y Fx(\000)1043 1020 y(p)p 1102 1020 159 3 v 39 x Fw(!)1146 1071 y Fs(d)p Ft(\000)p Fr(1)1296 921 y Fy(e)1338 883 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1638 807 y(d)1590 834 y Fn(X)1596 1029 y Fw(c)p Fq(=1)1737 793 y Fn(\022)1819 799 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)1868 834 y Fn(Y)1864 1029 y Fw(r)r Fq(=1)2048 921 y Fy(R)2117 935 y Fq(+)2176 921 y Fz(\()p Fy(\025)21 b Fz(+)f Fy(!)2433 939 y Fw(a)p Fq(\()p Fw(r)r Fq(\))2563 921 y Fz(\))p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)2746 935 y Fw(r)2784 921 y Fy(;)15 b Fu(\001)p Fz(\))2884 793 y Fn(\023)2952 921 y Fy(dE)5 b Fz(\()p Fy(\025)21 b Fz(+)f Fy(!)3328 935 y Fw(c)3363 921 y Fz(\))-3445 b(\(5.40\))1695 1163 y Fn(\022)1852 1178 y Fw(m)p Fq(+1)1871 1205 y Fn(Y)1777 1406 y Fw(r)r Fq(=)p Fw(k)r Fq(\()p Fw(c)p Fq(\)+1)2095 1291 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)2243 1305 y Fw(r)r Fx(\000)p Fq(1)2372 1291 y Fy(;)15 b Fu(\001)p Fz(\))p Fy(R)2541 1305 y Fx(\000)2601 1291 y Fz(\()p Fy(\025)20 b Fz(+)g Fy(!)2857 1310 y Fw(a)p Fq(\()p Fw(r)r Fq(\))2987 1291 y Fz(\))3022 1163 y Fn(\023)3089 1291 y Fy(:)-47 1595 y Fz(The)29 b(in)m(tegral)g(k)m(ernels)f(of)i Fy(R)943 1609 y Fx(\006)1002 1595 y Fz(\()p Fy(\025)18 b Fz(+)g Fy(!)1254 1613 y Fw(a)p Fq(\()p Fw(r)r Fq(\))1384 1595 y Fz(\))29 b(for)h Fy(a)p Fz(\()p Fy(r)s Fz(\))25 b Fu(\024)g Fy(d)18 b Fu(\000)g Fz(1)30 b(con)m(tribute)f(oscillating)f (exp)s(onen)m(tial)g(phases)h(while)-47 1708 y(for)h Fy(a)p Fz(\()p Fy(r)s Fz(\))25 b Fu(\025)g Fy(d)31 b Fz(they)g(pro)s(duce)e(exp)s(onen)m(tially)f(deca)m(ying)j(factors.)42 b(Hence,)336 2038 y Fu(M)445 2000 y Fw(d)445 2060 y(m)512 2038 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(=)1031 1914 y Fn(Z)1081 2120 y Fp(R)1129 2101 y Fs(m)1192 2038 y Fy(dx)1291 2052 y Fq(1)1331 2038 y Fy(::dx)1480 2052 y Fw(m)1593 1924 y(m)1566 1951 y Fn(Y)1562 2146 y Fw(r)r Fq(=1)1801 1976 y Fy(V)21 b Fz(\()5 b(^)-50 b Fy(\034)1950 1990 y Fw(r)1988 1976 y Fy(;)15 b(x)2080 1990 y Fw(r)2118 1976 y Fz(\))p 1711 2017 533 4 v 1711 2100 a(4)p Fy(\031)s Fu(j)p Fy(x)1888 2114 y Fw(r)r Fx(\000)p Fq(1)2037 2100 y Fu(\000)20 b Fy(x)2180 2114 y Fw(r)2218 2100 y Fu(j)2455 1976 y Fz(1)p 2263 2017 429 4 v 2263 2100 a(4)p Fy(\031)s Fu(j)p Fy(x)2440 2114 y Fw(m)2528 2100 y Fu(\000)g Fy(y)s Fu(j)2776 1882 y Fx(\000)2831 1840 y(p)p 2890 1840 81 3 v 42 x Fw(!)2934 1894 y Fs(d)2808 1914 y Fn(Z)2697 2176 y Fx(\000)2752 2137 y(p)p 2811 2137 159 3 v 39 x Fw(!)2855 2188 y Fs(d)p Ft(\000)p Fr(1)3030 2038 y Fy(d\025e)3172 2000 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1210 2298 y(d)p Fx(\000)p Fq(1)1207 2325 y Fn(X)1213 2520 y Fw(c)p Fq(=1)1354 2412 y Fy(e)1396 2374 y Fw(i)p Fq(\()1447 2322 y Fx(p)p 1506 2322 175 3 v 52 x Fw(\025)p Fq(+)p Fw(!)1646 2383 y Fr(1)1681 2374 y Fx(j)p Fw(x)1741 2388 y Fs(k)q Fr(\(1\))p Ft(\000)p Fr(1)1935 2374 y Fx(\000)p Fw(x)2030 2388 y Fs(k)q Fr(\(1\))2146 2374 y Fx(j)p Fq(+)p Fw(::)p Fq(+)2316 2314 y Fu(p)p 2391 2314 251 4 v 2391 2374 a Fw(\025)p Fq(+)p Fw(!)2531 2383 y Fs(c)p Ft(\000)p Fr(1)2641 2374 y Fx(j)p Fw(x)2701 2388 y Fs(k)q Fr(\()p Fs(c)p Fr(\))p Ft(\000)p Fr(2)2893 2374 y Fx(\000)p Fw(x)2988 2388 y Fs(k)q Fr(\()p Fs(c)p Fr(\))p Ft(\000)p Fr(1)3180 2374 y Fx(j)p Fq(\))1192 2667 y Fy(e)1234 2630 y Fx(\000)p Fw(i)p Fq(\()1340 2569 y Fu(p)p 1417 2569 249 4 v 1417 2630 a Fw(\025)p Fq(+)p Fw(!)1557 2639 y Fs(c)p Fr(+1)1665 2630 y Fx(j)p Fw(x)1725 2644 y Fs(k)q Fr(\()p Fs(c)p Fr(+1\))p Ft(\000)p Fr(1)1994 2630 y Fx(\000)p Fw(x)2089 2644 y Fs(k)q Fr(\()p Fs(c)p Fr(+1\))2280 2630 y Fx(j)p Fq(+)p Fw(::)p Fq(+)2450 2570 y Fu(p)p 2524 2570 255 4 v 2524 2630 a Fw(\025)p Fq(+)p Fw(!)2664 2642 y Fs(d)p Ft(\000)p Fr(1)2779 2630 y Fx(j)p Fw(x)2839 2644 y Fs(k)q Fr(\()p Fs(d)p Ft(\000)p Fr(1\))p Ft(\000)p Fr(1)3113 2630 y Fx(\000)p Fw(x)3208 2644 y Fs(k)q Fr(\()p Fs(d)p Ft(\000)p Fr(1\))3404 2630 y Fx(j)p Fq(\))1207 2837 y Fz(sin)1334 2763 y Fn(\000)1376 2753 y(p)p 1467 2753 256 4 v 84 x Fy(\025)g Fz(+)g Fy(!)1688 2851 y Fw(c)1722 2837 y Fu(j)p Fy(x)1799 2855 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)2038 2837 y Fu(\000)g Fy(x)2181 2855 y Fw(k)r Fq(\()p Fw(c)p Fq(\))2309 2837 y Fu(j)2334 2763 y Fn(\001)2376 2837 y Fy(e)2418 2796 y Fx(\000)2485 2747 y Fb(P)2559 2767 y Fs(m)p Fr(+1)2559 2820 y Fs(a)p Fr(=)p Fs(d)p Fr(+1)2764 2744 y Fx(p)p 2823 2744 233 3 v 52 x(\000)p Fw(!)2922 2804 y Fs(a)2960 2796 y Fx(\000)p Fw(\025)p Fx(j)p Fw(x)3116 2810 y Fs(k)q Fr(\()p Fs(a)p Fr(\))p Ft(\000)p Fr(1)3313 2796 y Fx(\000)p Fw(x)3408 2810 y Fs(k)q Fr(\()p Fs(a)p Fr(\))3527 2796 y Fx(j)3551 2837 y Fy(:)-47 3035 y Fz(Once)30 b(again)h(w)m(e)f(recall) g(the)h(iden)m(tit)m(y)g(\(5.36\))h(to)f(infer)e(that)91 3207 y Fw(d)p Fx(\000)p Fq(1)89 3235 y Fn(X)94 3429 y Fw(c)p Fq(=1)235 3321 y Fy(e)277 3280 y Fw(i)301 3206 y Fn(\000)355 3230 y Fb(P)429 3250 y Fs(c)p Ft(\000)p Fr(1)429 3301 y Fs(a)p Fr(=1)556 3227 y Fx(p)p 615 3227 178 3 v 53 x Fw(\025)p Fq(+)p Fw(!)755 3288 y Fs(a)792 3280 y Fx(j)p Fw(x)852 3294 y Fs(k)q Fr(\()p Fs(a)p Fr(\))p Ft(\000)p Fr(1)1050 3280 y Fx(\000)p Fw(x)1145 3294 y Fs(k)q Fr(\()p Fs(a)p Fr(\))1264 3280 y Fx(j\000)1339 3230 y Fb(P)1413 3250 y Fs(d)p Ft(\000)p Fr(1)1413 3303 y Fs(b)p Fr(=)p Fs(c)p Fr(+1)1608 3220 y Fu(p)p 1683 3220 249 4 v 1683 3280 a Fw(\025)p Fq(+)p Fw(!)1823 3292 y Fs(b)p Fr(+1)1932 3280 y Fx(j)p Fw(x)1992 3294 y Fs(k)q Fr(\()p Fs(b)p Fr(+1\))p Ft(\000)p Fr(1)2260 3280 y Fx(\000)p Fw(x)2355 3294 y Fs(k)q Fr(\()p Fs(b)p Fr(+1\))2545 3280 y Fx(j)2565 3206 y Fn(\001)2656 3321 y Fz(sin)2783 3247 y Fn(\000)2824 3237 y(p)p 2915 3237 256 4 v 84 x Fy(\025)20 b Fz(+)g Fy(!)3136 3335 y Fw(c)3171 3321 y Fu(j)p Fy(x)3248 3340 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)3486 3321 y Fu(\000)g Fy(x)3629 3340 y Fw(k)r Fq(\()p Fw(c)p Fq(\))3757 3321 y Fu(j)3782 3247 y Fn(\001)2323 3640 y Fz(=)k(sin)2545 3512 y Fn(\022)2630 3527 y Fw(d)p Fx(\000)p Fq(1)2627 3554 y Fn(X)2633 3748 y Fw(c)p Fq(=1)2774 3556 y Fn(p)p 2865 3556 V 84 x Fy(\025)c Fz(+)g Fy(!)3086 3654 y Fw(c)3120 3640 y Fu(j)p Fy(x)3197 3659 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)3436 3640 y Fu(\000)g Fy(x)3579 3659 y Fw(k)r Fq(\()p Fw(c)p Fq(\))3707 3640 y Fu(j)3732 3512 y Fn(\023)3799 3640 y Fy(:)-47 3915 y Fz(Therefore,)-11 4179 y Fu(M)98 4141 y Fw(d)98 4201 y(m)164 4179 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(=)673 4055 y Fn(Z)723 4261 y Fp(R)771 4242 y Fs(m)849 4179 y Fy(dx)948 4193 y Fq(1)988 4179 y Fy(::dx)1137 4193 y Fw(m)1250 4065 y(m)1223 4093 y Fn(Y)1219 4287 y Fw(r)r Fq(=1)1459 4117 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1607 4131 y Fw(r)1645 4117 y Fy(;)15 b(x)1737 4131 y Fw(r)1775 4117 y Fz(\))p 1369 4158 533 4 v 1369 4241 a(4)p Fy(\031)s Fu(j)p Fy(x)1546 4255 y Fw(r)r Fx(\000)p Fq(1)1695 4241 y Fu(\000)20 b Fy(x)1838 4255 y Fw(r)1875 4241 y Fu(j)2112 4117 y Fz(1)p 1921 4158 429 4 v 1921 4241 a(4)p Fy(\031)s Fu(j)p Fy(x)2098 4255 y Fw(m)2185 4241 y Fu(\000)g Fy(y)s Fu(j)732 4386 y Fx(\000)787 4344 y(p)p 846 4344 81 3 v 42 x Fw(!)890 4398 y Fs(d)763 4418 y Fn(Z)653 4679 y Fx(\000)708 4641 y(p)p 766 4641 159 3 v 766 4679 a Fw(!)810 4691 y Fs(d)p Ft(\000)p Fr(1)960 4541 y Fy(d\025)31 b(e)1133 4504 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1401 4541 y Fz(sin)1527 4413 y Fn(\022)1612 4428 y Fw(d)p Fx(\000)p Fq(1)1610 4455 y Fn(X)1615 4649 y Fw(c)p Fq(=1)1756 4457 y Fn(p)p 1847 4457 256 4 v 84 x Fy(\025)20 b Fz(+)g Fy(!)2068 4555 y Fw(c)2102 4541 y Fu(j)p Fy(x)2179 4560 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)2418 4541 y Fu(\000)g Fy(x)2561 4560 y Fw(k)r Fq(\()p Fw(c)p Fq(\))2689 4541 y Fu(j)2714 4413 y Fn(\023)2781 4541 y Fy(e)2823 4504 y Fx(\000)2890 4454 y Fb(P)2965 4474 y Fs(m)p Fr(+1)2965 4528 y Fs(a)p Fr(=)p Fs(d)3112 4451 y Fx(p)p 3171 4451 233 3 v 53 x(\000)p Fw(!)3270 4512 y Fs(a)3307 4504 y Fx(\000)p Fw(\025)p Fx(j)p Fw(x)3463 4518 y Fs(k)q Fr(\()p Fs(a)p Fr(\))p Ft(\000)p Fr(1)3661 4504 y Fx(\000)p Fw(x)3756 4518 y Fs(k)q Fr(\()p Fs(a)p Fr(\))3875 4504 y Fx(j)3899 4541 y Fy(:)-47 4884 y Fz(W)-8 b(e)38 b(w)m(ould)e(lik)m(e)g(to)h(c)m (hange)h(v)-5 b(ariables)36 b Fy(\025)g Fu(!)g Fy(\025)1638 4851 y Fq(2)1715 4884 y Fz(and)g(in)m(tegrate)i(b)m(y)f(parts)g (relativ)m(e)g(to)g Fy(\025)p Fz(,)i(as)e(w)m(e)h(did)d(for)h Fu(L)3853 4898 y Fw(m)3920 4884 y Fz(.)-47 4997 y(Denote)k(the)g Fy(\025)p Fz(-in)m(tegrand)f Fy(F)996 5012 y Fw(d)1037 4997 y Fz(\()p Fy(\025)p Fz(\))g(in)f(eac)m(h)i(of)g(the)f(k)m(ernels)g Fu(M)2227 4964 y Fw(d)2227 5019 y(m)2293 4997 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\).)69 b(It)39 b(is)f(not)i(di\016cult)d(to)i(see)h(that)-47 5110 y Fy(F)11 5125 y Fw(d)52 5110 y Fz(\()p Fu(\000)p Fy(!)215 5125 y Fw(d)255 5110 y Fz(\))g(=)f Fy(F)498 5125 y Fw(d)p Fq(+1)629 5110 y Fz(\()p Fu(\000)p Fy(!)792 5125 y Fw(d)833 5110 y Fz(\))g(for)g Fy(d)h Fz(=)f(1)p Fy(;)15 b(::;)g(m)28 b Fz(+)e(1.)67 b(Therefore,)41 b(the)e(b)s(oundary)e(terms)i(will)e (cancel)i(eac)m(h)i(other)-47 5223 y(telescopically)-8 b(.)46 b(The)32 b(b)s(oundary)f(terms)h(at)h(the)g(t)m(w)m(o)h(endp)s (oin)m(ts)d Fy(!)2316 5237 y Fw(m)p Fq(+2)2501 5223 y Fz(=)d(0)33 b(and)f Fy(!)2914 5237 y Fq(0)2982 5223 y Fz(=)c Fu(1)33 b Fz(will)c(also)k(disapp)s(ear)1903 5472 y(41)p eop %%Page: 42 42 42 41 bop -47 274 a Fz(as)31 b(in)e Fu(L)234 288 y Fw(m)300 274 y Fz(.)42 b(This)28 b(allo)m(ws)i(us,)g(in)g(what)g(follo)m(ws,)g (to)i(ignore)e(the)h(b)s(oundary)d(terms)j(altogether.)42 b(W)-8 b(e)32 b(no)m(w)f(mak)m(e)g(a)-47 387 y(c)m(hange)g(of)g(v)-5 b(ariables)29 b Fy(\025)c Fu(!)g Fy(\025)981 354 y Fq(2)1041 387 y Fu(\000)20 b Fy(!)1189 402 y Fw(d)p Fx(\000)p Fq(1)1319 387 y Fz(.)41 b(W)-8 b(e)31 b(also)g(re-in)m(tro)s(duce)e(the)i (notation)f Fy(\033)2806 401 y Fw(a)2878 387 y Fz(in)f(the)i(new)f (capacit)m(y:)1154 587 y(0)c Fu(\024)f Fy(\033)1373 601 y Fw(a)1439 587 y Fz(=)g Fy(!)1592 601 y Fw(a)1654 587 y Fu(\000)20 b Fy(!)1802 602 y Fw(d)p Fx(\000)p Fq(1)1932 587 y Fy(;)197 b(a)26 b Fz(=)e(0)p Fy(;)15 b(::;)g(d)23 b Fu(\000)d Fz(1)p Fy(;)1154 725 y Fz(0)26 b Fu(\024)f Fy(\032)1368 739 y Fw(a)1435 725 y Fz(=)f Fy(!)1587 740 y Fw(d)p Fx(\000)p Fq(1)1738 725 y Fu(\000)c Fy(!)1886 739 y Fw(a)1927 725 y Fy(;)197 b(a)26 b Fz(=)f Fy(d;)15 b(::;)g(m)21 b Fz(+)f(2)p Fy(:)-47 922 y Fz(Th)m(us)11 1173 y Fu(M)120 1135 y Fw(d)120 1195 y(m)187 1173 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))26 b(=)695 1049 y Fn(Z)746 1255 y Fp(R)794 1236 y Fs(m)872 1173 y Fy(dx)971 1187 y Fq(1)1010 1173 y Fy(::dx)1159 1187 y Fw(m)1273 1059 y(m)1246 1086 y Fn(Y)1242 1281 y Fw(r)r Fq(=1)1481 1111 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1629 1125 y Fw(r)1667 1111 y Fy(;)15 b(x)1759 1125 y Fw(r)1798 1111 y Fz(\))p 1391 1152 533 4 v 1391 1235 a(4)p Fy(\031)s Fu(j)p Fy(x)1568 1249 y Fw(r)r Fx(\000)p Fq(1)1717 1235 y Fu(\000)20 b Fy(x)1860 1249 y Fw(r)1898 1235 y Fu(j)2135 1111 y Fz(1)p 1943 1152 429 4 v 1943 1235 a(4)p Fy(\031)s Fu(j)p Fy(x)2120 1249 y Fw(m)2208 1235 y Fu(\000)g Fy(y)s Fu(j)716 1338 y Fx(p)p 774 1338 73 3 v 774 1379 a Fw(\032)810 1391 y Fs(d)715 1412 y Fn(Z)723 1672 y Fq(0)842 1535 y Fy(d\025)30 b(\025e)1067 1498 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)1299 1474 y Fr(2)1355 1535 y Fz(sin)1481 1407 y Fn(\022)1566 1422 y Fw(d)p Fx(\000)p Fq(1)1563 1449 y Fn(X)1569 1643 y Fw(c)p Fq(=1)1710 1448 y Fn(p)p 1801 1448 291 4 v 87 x Fy(\025)1854 1509 y Fq(2)1913 1535 y Fz(+)20 b Fy(\033)2056 1549 y Fw(c)2091 1535 y Fu(j)p Fy(x)2168 1554 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)2407 1535 y Fu(\000)g Fy(x)2550 1554 y Fw(k)r Fq(\()p Fw(c)p Fq(\))2678 1535 y Fu(j)2703 1407 y Fn(\023)2770 1535 y Fy(e)2812 1498 y Fx(\000)2879 1448 y Fb(P)2953 1468 y Fs(m)p Fr(+1)2953 1521 y Fs(a)p Fr(=)p Fs(d)3101 1433 y Fu(p)p 3176 1433 205 4 v 3176 1498 a Fw(\032)3212 1506 y Fs(a)3250 1498 y Fx(\000)p Fw(\025)3346 1479 y Fr(2)3381 1498 y Fx(j)p Fw(x)3441 1512 y Fs(k)q Fr(\()p Fs(a)p Fr(\))p Ft(\000)p Fr(1)3638 1498 y Fx(\000)p Fw(x)3733 1512 y Fs(k)q Fr(\()p Fs(a)p Fr(\))3853 1498 y Fx(j)3877 1535 y Fy(:)-47 1843 y Fz(In)m(tegrating)j (b)m(y)f(parts)g(relativ)m(e)g(to)h Fy(\025)f Fz(and)g(canceling)f(the) i(con)m(tribution)e(from)g(the)i(b)s(oundary)d(terms)i(as)h(explained) -47 1956 y(ab)s(o)m(v)m(e,)32 b(w)m(e)e(\014nally)f(obtain)559 2242 y Fu(M)668 2205 y Fw(d)668 2265 y(m)735 2242 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(=)1341 2181 y Fy(i)p 1264 2222 187 4 v 1264 2305 a(t)20 b Fu(\000)f Fy(s)1478 2129 y Fw(d)p Fx(\000)p Fq(1)1475 2156 y Fn(X)1481 2354 y Fw(`)p Fq(=1)1622 2242 y Fu(M)1731 2205 y Fw(d;`)1731 2265 y(m)1820 2242 y Fz(\()p Fy(t;)c(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))21 b(+)2416 2181 y Fy(i)p 2338 2222 V 2338 2305 a(t)f Fu(\000)g Fy(s)2550 2129 y Fw(m)p Fq(+1)2561 2156 y Fn(X)2566 2354 y Fw(`)p Fq(=)p Fw(d)2739 2219 y Fn(f)2718 2242 y Fu(M)2827 2205 y Fw(d;`)2827 2265 y(m)2916 2242 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))p Fy(;)-27 2887 y Fu(M)82 2850 y Fw(d;`)82 2910 y(m)171 2887 y Fz(\()p Fy(t;)g(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))26 b(:=)715 2764 y Fn(Z)765 2970 y Fp(R)813 2951 y Fs(m)891 2887 y Fy(dx)990 2901 y Fq(1)1030 2887 y Fy(::dx)1179 2901 y Fw(m)1317 2774 y(m)1290 2801 y Fn(Y)1286 2995 y Fw(r)r Fq(=1)1526 2826 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1674 2840 y Fw(r)1712 2826 y Fy(;)15 b(x)1804 2840 y Fw(r)1842 2826 y Fz(\))p 1435 2866 533 4 v 1435 2950 a(4)p Fy(\031)s Fu(j)p Fy(x)1612 2964 y Fw(r)r Fx(\000)p Fq(1)1762 2950 y Fu(\000)k Fy(x)1904 2964 y Fw(r)1942 2950 y Fu(j)1988 2817 y(j)p Fy(x)2065 2835 y Fw(k)r Fq(\()p Fw(l)q Fq(\))p Fx(\000)p Fq(1)2295 2817 y Fu(\000)g Fy(x)2437 2835 y Fw(k)r Fq(\()p Fw(l)q Fq(\))2557 2817 y Fu(j)p 1988 2866 595 4 v 2071 2950 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)2248 2964 y Fw(m)2335 2950 y Fu(\000)h Fy(y)s Fu(j)-47 2633 y Fz(\(5.41\))393 3053 y Fx(p)p 451 3053 73 3 v 451 3094 a Fw(\032)487 3106 y Fs(d)392 3126 y Fn(Z)400 3386 y Fq(0)519 3250 y Fy(d\025)30 b(e)691 3212 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)923 3189 y Fr(2)979 3250 y Fz(cos)1115 3122 y Fn(\022)1200 3136 y Fw(d)p Fx(\000)p Fq(1)1197 3164 y Fn(X)1203 3358 y Fw(c)p Fq(=1)1344 3163 y Fn(p)p 1435 3163 291 4 v 87 x Fy(\025)1488 3224 y Fq(2)1547 3250 y Fz(+)20 b Fy(\033)1690 3264 y Fw(c)1725 3250 y Fu(j)p Fy(x)1802 3268 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)2041 3250 y Fu(\000)g Fy(x)2184 3268 y Fw(k)r Fq(\()p Fw(c)p Fq(\))2312 3250 y Fu(j)2337 3122 y Fn(\023)2404 3250 y Fy(e)2446 3212 y Fx(\000)2513 3163 y Fb(P)2587 3183 y Fs(m)p Fr(+1)2587 3236 y Fs(a)p Fr(=)p Fs(d)2735 3148 y Fu(p)p 2811 3148 205 4 v 64 x Fw(\032)2847 3220 y Fs(a)2884 3212 y Fx(\000)p Fw(\025)2980 3193 y Fr(2)3015 3212 y Fx(j)p Fw(x)3075 3226 y Fs(k)q Fr(\()p Fs(a)p Fr(\))p Ft(\000)p Fr(1)3273 3212 y Fx(\000)p Fw(x)3368 3226 y Fs(k)q Fr(\()p Fs(a)p Fr(\))3487 3212 y Fx(j)3699 3188 y Fy(\025)p 3536 3229 380 4 v 3536 3247 a Fn(p)p 3627 3247 289 4 v 82 x Fy(\025)3680 3303 y Fq(2)3739 3329 y Fz(+)g Fy(\033)3882 3344 y Fw(`)-16 3897 y Fn(f)-38 3920 y Fu(M)71 3882 y Fw(d;`)71 3942 y(m)160 3920 y Fz(\()p Fy(t;)15 b(s)p Fz(\))q(\()p Fy(x;)g(y)s Fz(\))26 b(:=)f Fu(\000)790 3796 y Fn(Z)841 4002 y Fp(R)889 3983 y Fs(m)967 3920 y Fy(dx)1066 3934 y Fq(1)1105 3920 y Fy(::dx)1254 3934 y Fw(m)1368 3806 y(m)1341 3833 y Fn(Y)1337 4028 y Fw(r)r Fq(=1)1576 3858 y Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)1724 3872 y Fw(r)1762 3858 y Fy(;)15 b(x)1854 3872 y Fw(r)1893 3858 y Fz(\))p 1486 3899 533 4 v 1486 3982 a(4)p Fy(\031)s Fu(j)p Fy(x)1663 3996 y Fw(r)r Fx(\000)p Fq(1)1812 3982 y Fu(\000)20 b Fy(x)1955 3996 y Fw(r)1993 3982 y Fu(j)2038 3849 y(j)p Fy(x)2115 3868 y Fw(k)r Fq(\()p Fw(l)q Fq(\))p Fx(\000)p Fq(1)2345 3849 y Fu(\000)g Fy(x)2488 3868 y Fw(k)r Fq(\()p Fw(l)q Fq(\))2607 3849 y Fu(j)p 2038 3899 595 4 v 2121 3982 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)2298 3996 y Fw(m)2386 3982 y Fu(\000)f Fy(y)s Fu(j)-47 3665 y Fz(\(5.42\))382 4085 y Fx(p)p 441 4085 73 3 v 41 x Fw(\032)477 4138 y Fs(d)382 4159 y Fn(Z)390 4419 y Fq(0)508 4282 y Fy(d\025)31 b(e)681 4245 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)913 4221 y Fr(2)968 4282 y Fz(cos)1105 4154 y Fn(\022)1190 4169 y Fw(d)p Fx(\000)p Fq(1)1187 4196 y Fn(X)1193 4390 y Fw(c)p Fq(=1)1334 4195 y Fn(p)p 1425 4195 291 4 v 87 x Fy(\025)1478 4256 y Fq(2)1537 4282 y Fz(+)20 b Fy(\033)1680 4296 y Fw(c)1715 4282 y Fu(j)p Fy(x)1792 4301 y Fw(k)r Fq(\()p Fw(c)p Fq(\))p Fx(\000)p Fq(1)2031 4282 y Fu(\000)g Fy(x)2174 4301 y Fw(k)r Fq(\()p Fw(c)p Fq(\))2302 4282 y Fu(j)2327 4154 y Fn(\023)2394 4282 y Fy(e)2436 4245 y Fx(\000)2503 4195 y Fb(P)2577 4215 y Fs(m)p Fr(+1)2577 4268 y Fs(a)p Fr(=)p Fs(d)2725 4180 y Fu(p)p 2800 4180 205 4 v 2800 4245 a Fw(\032)2836 4253 y Fs(a)2874 4245 y Fx(\000)p Fw(\025)2970 4226 y Fr(2)3005 4245 y Fx(j)p Fw(x)3065 4259 y Fs(k)q Fr(\()p Fs(a)p Fr(\))p Ft(\000)p Fr(1)3262 4245 y Fx(\000)p Fw(x)3357 4259 y Fs(k)q Fr(\()p Fs(a)p Fr(\))3477 4245 y Fx(j)3686 4221 y Fy(\025)p 3526 4261 375 4 v 3526 4280 a Fn(p)p 3616 4280 284 4 v 3616 4360 a Fy(\032)3663 4375 y Fw(`)3717 4360 y Fu(\000)g Fy(\025)3861 4334 y Fq(2)3910 4282 y Fy(:)-47 4595 y Fz(Com)m(bining)29 b(\(5.37\))r(-\(5.42\))k(w)m (e)e(can)g(state)h(the)e(follo)m(wing)-47 4783 y Fm(Prop)s(osition)36 b(5.9.)46 b Fl(The)c(inte)-5 b(gr)g(al)43 b(kernel)f(of)g Fu(I)1700 4797 y Fw(m)1766 4783 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fl(,)44 b(the)e Fy(m)p Fl(-th)f(term)i(of)e(the)h(Born)h (series)49 b Fz(\(5.6\))r Fl(,)44 b(c)-5 b(an)42 b(b)-5 b(e)1903 5472 y Fz(42)p eop %%Page: 43 43 43 42 bop -47 274 a Fl(written)33 b(in)g(the)g(fol)5 b(lowing)33 b(form:)-12 529 y Fu(I)38 543 y Fw(m)104 529 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))27 b(=)633 467 y Fy(i)664 434 y Fw(m)p Fq(+1)p 633 508 189 4 v 634 591 a Fy(t)20 b Fu(\000)g Fy(s)846 405 y Fn(Z)897 611 y Fp(R)945 592 y Fs(m)1023 529 y Fy(d\034)1110 543 y Fq(1)1149 529 y Fy(::d\034)1286 543 y Fw(m)1353 529 y Fy(e)1395 491 y Fw(i)p Fq(\()p Fw(\034)1477 500 y Fr(1)1513 491 y Fq(+)p Fw(::)p Fq(+)p Fw(\034)1694 499 y Fs(m)1752 491 y Fq(\))546 685 y Fn(\022)662 699 y Fw(m)628 727 y Fn(X)634 924 y Fw(`)p Fq(=1)774 813 y Fu(L)837 776 y Fw(`)837 836 y(m)903 813 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\)\()p Fy(\033)1386 827 y Fq(1)1428 813 y Fy(;)g(::;)g(\033)1610 827 y Fw(m)1677 813 y Fz(\)+)1950 699 y Fw(m)p Fq(+2)1960 727 y Fn(X)1963 924 y Fw(d)p Fq(=0)2120 699 y Fw(d)p Fx(\000)p Fq(1)2117 727 y Fn(X)2123 924 y Fw(`)p Fq(=1)2264 813 y Fu(M)2373 776 y Fw(d;`)2373 836 y(m)2462 813 y Fz(\()p Fy(t;)g(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\)\()p Fy(\033)2945 827 y Fq(1)2986 813 y Fy(;)g(::;)g(\033)3168 828 y Fw(d)p Fx(\000)p Fq(1)3300 813 y Fy(;)g(\032)3387 828 y Fw(d)3428 813 y Fy(;)g(::;)g(\032)3605 827 y Fw(m)p Fq(+1)3763 813 y Fz(\))20 b(+)1950 1022 y Fw(m)p Fq(+2)1960 1049 y Fn(X)1963 1247 y Fw(d)p Fq(=0)2120 1022 y Fw(d)p Fx(\000)p Fq(1)2117 1049 y Fn(X)2123 1247 y Fw(`)p Fq(=1)2286 1113 y Fn(f)2264 1135 y Fu(M)2373 1098 y Fw(d;`)2373 1158 y(m)2462 1135 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\)\()p Fy(\033)2945 1149 y Fq(1)2986 1135 y Fy(;)g(::;)g(\033)3168 1150 y Fw(d)p Fx(\000)p Fq(1)3300 1135 y Fy(;)g(\032)3387 1150 y Fw(d)3428 1135 y Fy(;)g(::;)g(\032)3605 1149 y Fw(m)p Fq(+1)3763 1135 y Fz(\))3798 1007 y Fn(\023)3865 1135 y Fy(:)-3937 b Fz(\(5.43\))-47 1424 y Fl(We)32 b(interpr)-5 b(et)34 b Fu(I)542 1438 y Fw(m)608 1424 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))34 b Fl(as)f(fol)5 b(lows:)43 b(for)34 b(any)f(p)-5 b(air)34 b(of)f(Schwartz)h(functions)f Fy( )2945 1438 y Fw(s)3015 1424 y Fl(and)g Fy(g)937 1670 y Fu(hI)1022 1684 y Fw(m)1089 1670 y Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )1334 1684 y Fw(s)1371 1670 y Fy(;)31 b(g)s Fu(i)26 b Fz(=)1630 1546 y Fn(Z)1681 1752 y Fp(R)1729 1733 y Fr(6)1783 1670 y Fu(I)1833 1684 y Fw(m)1899 1670 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))g Fy( )2369 1684 y Fw(s)2407 1670 y Fz(\()p Fy(y)s Fz(\))g Fy(g)s Fz(\()p Fy(x)p Fz(\))g Fy(dx)g(dy)s(:)-47 1915 y Fl(The)33 b(functions)957 2119 y Fu(L)1020 2082 y Fw(`)1020 2142 y(m)1086 2119 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))p Fy(;)110 b Fu(M)1726 2082 y Fw(d;`)1726 2142 y(m)1815 2119 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))p Fy(;)2367 2096 y Fn(f)2346 2119 y Fu(M)2455 2082 y Fw(d;`)2455 2142 y(m)2544 2119 y Fz(\()p Fy(t;)g(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))-47 2323 y Fl(ar)-5 b(e)35 b(de\014ne)-5 b(d)35 b(in)40 b Fz(\(5.39\))s Fl(,)34 b Fz(\(5.41\))r Fl(,)g(and)45 b Fz(\(5.42\))36 b Fl(c)-5 b(orr)g(esp)g(ondingly)37 b(with)e(implicit)g(dep)-5 b(endenc)g(e)35 b(on)f(the)h(p)-5 b(ar)g(ameters)-47 2436 y Fy(\033)5 2451 y Fw(k)47 2436 y Fy(;)15 b(\032)134 2451 y Fw(`)168 2436 y Fl(.)41 b(The)33 b(latter)h(ar)-5 b(e)34 b(p)-5 b(ositive)33 b(and)h(dep)-5 b(end)34 b(exclusively)e(and) i(in)e(a)h(line)-5 b(ar)34 b(fashion)g(on)f Fy(\034)3183 2450 y Fq(1)3222 2436 y Fy(;)15 b(::;)g(\034)3392 2450 y Fw(m)3460 2436 y Fl(.)-47 2723 y FA(6)134 b(Estimates)48 b(for)d(oscillatory)h(in)l(tegrals)h(I)-47 2926 y Fz(The)39 b(purp)s(ose)f(of)i(this)e(section)i(is)f(to)h(pro)m(v)m(e)h(the)f (follo)m(wing)e(lemma.)68 b(Up)40 b(to)g(a)g(c)m(hange)h(of)f(v)-5 b(ariables)38 b(in)h Fy(\025)g Fz(it)-47 3039 y(pro)m(vides)27 b(the)h(estimate)h(on)f(the)g(oscillatory)g(in)m(tegral)g(in)e (\(5.39\))31 b(that)e(w)m(e)f(need.)40 b(Strictly)27 b(sp)s(eaking,)g(the)i(second)-47 3152 y(b)s(ound)22 b(in)i(\(6.1\))i(su\016ces)f(for)f(the)h(disp)s(ersiv)m(e)e(estimate,)k (but)d(w)m(e)h(include)d(the)j(\014rst)f(for)h(the)g(sak)m(e)h(of)f (completeness)-47 3264 y(and)30 b(p)s(ossible)d(future)j(applications.) -47 3452 y Fm(Lemma)i(6.1.)47 b Fl(L)-5 b(et)32 b Fy(!)s Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)f Fz(0)33 b Fl(b)-5 b(e)33 b(a)g(twic)-5 b(e)33 b(di\013er)-5 b(entiable,)33 b(monotone)i(function)d(such)h(that)1602 3656 y Fu(j)p Fy(!)1687 3619 y Fq(\()p Fw(j)t Fq(\))1778 3656 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)27 b(\024)e Fy(a)2097 3670 y Fq(0)2151 3656 y Fy(\025)2204 3619 y Fx(\000)p Fw(j)-47 3861 y Fl(for)34 b Fy(j)e Fz(=)27 b(0)p Fy(;)15 b Fz(1)p Fy(;)g Fz(2)36 b Fl(and)e(al)5 b(l)34 b Fy(\025)27 b(>)f Fz(0)p Fl(.)45 b(Then)34 b(ther)-5 b(e)34 b(exists)g(a)g(c)-5 b(onstant)35 b Fy(C)2343 3875 y Fq(0)2416 3861 y Fl(which)g(only)f(dep) -5 b(ends)35 b(on)f(the)g(c)-5 b(onstant)35 b Fy(a)3905 3875 y Fq(0)-47 3973 y Fl(so)e(that)h(for)f(any)g(p)-5 b(ositive)34 b(inte)-5 b(ger)33 b Fy(m)f Fl(and)i(any)f Fz(1)26 b Fu(\024)e Fy(k)29 b Fu(\024)c Fy(m)p Fl(,)211 4091 y Fn(\014)211 4145 y(\014)211 4200 y(\014)211 4254 y(\014)211 4309 y(\014)242 4126 y(Z)332 4152 y Fx(1)292 4332 y Fq(0)422 4250 y Fy(e)474 4185 y Fr(1)p 475 4197 31 3 v 475 4238 a(2)515 4212 y Fw(i\025)580 4189 y Fr(2)634 4250 y Fy(e)676 4210 y Fx(\006)p Fw(i)767 4160 y Fb(P)842 4180 y Fs(m)842 4230 y(j)s Fr(=1)963 4210 y Fw(b)993 4220 y Fs(j)1037 4148 y Fu(p)p 1113 4148 204 4 v 62 x Fw(\025)1154 4191 y Fr(2)1189 4210 y Fq(+)p Fw(\033)1284 4220 y Fs(j)1514 4188 y Fy(\025)p 1346 4229 390 4 v 1346 4247 a Fn(p)p 1437 4247 299 4 v 82 x Fy(\025)1490 4303 y Fq(2)1549 4329 y Fz(+)20 b Fy(\033)1692 4344 y Fw(k)1760 4250 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)2058 4091 y Fn(\014)2058 4145 y(\014)2058 4200 y(\014)2058 4254 y(\014)2058 4309 y(\014)2173 4250 y Fu(\024)82 b Fy(C)2391 4264 y Fq(0)2461 4250 y Fz(min)2628 4176 y Fn(\002)2665 4250 y Fz(\(1)22 b(+)d Fy(\033)2909 4264 y Fq(1)2949 4250 y Fz(\))2994 4185 y Fr(1)p 2994 4197 31 3 v 2994 4238 a(4)3039 4250 y Fy(;)c(m)3169 4185 y Fr(3)p 3169 4197 V 3169 4238 a(2)3229 4250 y Fy(b)3268 4212 y Fx(\000)p Fq(1)3268 4279 y Fw(k)3392 4250 y Fz(max)3462 4311 y Fw(`)3577 4250 y Fy(b)3616 4265 y Fw(`)3649 4176 y Fn(\003)-47 4250 y Fz(\(6.1\))635 4397 y Fn(\014)635 4451 y(\014)635 4506 y(\014)635 4560 y(\014)666 4405 y(Z)757 4431 y Fx(1)716 4611 y Fq(0)846 4528 y Fy(e)898 4464 y Fr(1)p 899 4476 V 899 4517 a(2)939 4491 y Fw(i\025)1004 4468 y Fr(2)1058 4528 y Fy(e)1100 4488 y Fx(\006)p Fw(i)1191 4438 y Fb(P)1266 4459 y Fs(m)1266 4509 y(j)s Fr(=1)1387 4488 y Fw(b)1417 4498 y Fs(j)1462 4427 y Fu(p)p 1537 4427 204 4 v 1537 4488 a Fw(\025)1578 4469 y Fr(2)1613 4488 y Fq(+)p Fw(\033)1708 4498 y Fs(j)1760 4528 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))g Fy(d\025)2058 4397 y Fn(\014)2058 4451 y(\014)2058 4506 y(\014)2058 4560 y(\014)2173 4528 y Fu(\024)82 b Fy(C)2391 4542 y Fq(0)2446 4528 y Fz(\(1)21 b(+)f Fy(\033)2690 4542 y Fq(1)2729 4528 y Fz(\))2774 4464 y Fr(1)p 2774 4476 31 3 v 2774 4517 a(4)-47 4528 y Fz(\(6.2\))-47 4778 y Fl(for)33 b(any)g(choic)-5 b(e)33 b(of)g Fy(\033)705 4792 y Fq(1)770 4778 y Fu(\025)25 b Fy(\033)918 4792 y Fq(2)982 4778 y Fu(\025)g Fy(:)15 b(:)g(:)27 b Fu(\025)e Fy(\033)1358 4792 y Fw(m)1449 4778 y Fu(\025)g Fz(0)33 b Fl(and)h Fy(b)1839 4792 y Fw(j)1901 4778 y Fy(>)25 b Fz(0)p Fl(.)-47 4965 y Fz(The)38 b(pro)s(of)f(of)i(this)e(lemma)h(will)d(b)s(e)j(brok)m(en)g(up)f(in)m (to)h(sev)m(eral)h(sublemmas.)63 b(W)-8 b(e)39 b(start)g(with)e(an)h (elemen)m(tary)-47 5078 y(lemma)29 b(that)i(establishes)e(a)h(basic)g (estimate)h(on)f(the)g(functions)2338 5042 y Fw(\025)p 2244 5057 229 4 v 2244 5067 a Fx(p)p 2303 5067 170 3 v 55 x Fw(\025)2344 5104 y Fr(2)2378 5122 y Fq(+)p Fw(\034)2482 5078 y Fz(.)41 b(W)-8 b(e)31 b(shall)e(use)g(this)g(b)s(ound)f(rep)s (eatedly)-47 5202 y(throughout)i(this)f(section.)1903 5472 y(43)p eop %%Page: 44 44 44 43 bop -47 274 a Fm(Lemma)32 b(6.2.)47 b Fl(F)-7 b(or)33 b(al)5 b(l)33 b(nonne)-5 b(gative)34 b(inte)-5 b(gers)33 b Fy(j)38 b Fl(ther)-5 b(e)34 b(exist)e(c)-5 b(onstants)35 b Fy(C)2647 288 y Fw(j)2716 274 y Fl(so)e(that)1081 407 y Fn(\014)1081 461 y(\014)1081 516 y(\014)1081 570 y(\014)1148 477 y Fy(d)1195 444 y Fw(j)p 1121 518 137 4 v 1121 601 a Fy(d\025)1221 575 y Fw(j)1416 477 y Fy(\025)p 1278 518 330 4 v 1278 536 a Fu(p)p 1354 536 254 4 v 76 x Fy(\025)1407 586 y Fq(2)1467 612 y Fz(+)19 b Fy(\034)1617 407 y Fn(\014)1617 461 y(\014)1617 516 y(\014)1617 570 y(\014)1673 539 y Fu(\024)25 b Fy(C)1834 553 y Fw(j)1901 539 y Fz(min)n(\()p Fy(\034)2137 501 y Fx(\000)2203 470 y Fs(j)p 2203 486 31 3 v 2203 527 a Fr(2)2247 539 y Fy(;)15 b(\025)2340 501 y Fx(\000)p Fw(j)2580 477 y Fy(\025)p 2442 518 330 4 v 2442 536 a Fu(p)p 2518 536 254 4 v 76 x Fy(\025)2571 586 y Fq(2)2630 612 y Fz(+)20 b Fy(\034)2781 539 y Fz(\))-2863 b(\(6.3\))-47 788 y Fl(for)33 b(al)5 b(l)33 b Fy(\034)i(>)25 b Fz(0)33 b Fl(and)h Fy(\025)25 b(>)g Fz(0)p Fl(.)-47 975 y(Pr)-5 b(o)g(of.)47 b Fz(The)29 b(case)i Fy(j)g Fz(=)25 b(0)30 b(in)f(\(6.3\))i(is)e(ob)m(vious.)40 b(Hence)31 b(it)e(su\016ces)h(to)h(consider)d Fy(j)j Fu(\025)25 b Fz(1.)41 b(Let)31 b Fy(f)10 b Fz(\()p Fy(\025)p Fz(\))25 b(=)3604 940 y Fw(\025)p 3512 955 225 4 v 3512 964 a Fx(p)p 3571 964 166 3 v 56 x Fw(\025)3612 1001 y Fr(2)3646 1020 y Fq(+1)3747 975 y Fz(.)40 b(F)-8 b(or)-47 1131 y(an)m(y)30 b Fy(\034)36 b(>)25 b Fz(0)30 b(de\014ne)f Fy(\032)680 1145 y Fw(\034)724 1131 y Fz(\()p Fy(\025)p Fz(\))d(:=)1097 1095 y Fw(\025)p 1004 1110 229 4 v 1004 1120 a Fx(p)p 1063 1120 170 3 v 55 x Fw(\025)1104 1156 y Fr(2)1138 1175 y Fq(+)p Fw(\034)1242 1131 y Fz(.)41 b(Then)29 b Fy(\032)1592 1145 y Fw(\034)1635 1131 y Fz(\()p Fy(\025)p Fz(\))d(=)f Fy(f)10 b Fz(\()p Fy(\025\034)2073 1095 y Fx(\000)2138 1068 y Fr(1)p 2138 1080 31 3 v 2138 1121 a(2)2182 1131 y Fz(\).)41 b(Since)29 b Fu(j)p Fy(f)2600 1098 y Fq(\()p Fw(j)t Fq(\))2691 1131 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)d(\024)f Fy(C)3026 1145 y Fw(j)3078 1131 y Fz(min)o(\(1)p Fy(;)15 b(\025)3403 1098 y Fx(\000)p Fw(j)3495 1131 y Fz(\),)31 b(it)f(follo)m(ws)-47 1254 y(from)g(the)g(c)m(hain)g(rule)f(that)487 1488 y Fu(j)p Fy(\032)559 1451 y Fq(\()p Fw(j)t Fq(\))559 1511 y Fw(\034)651 1488 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)d(\024)f Fy(C)986 1502 y Fw(j)1038 1488 y Fy(\034)1088 1451 y Fx(\000)1154 1420 y Fs(j)p 1153 1436 V 1153 1477 a Fr(2)1212 1488 y Fz(min)1364 1387 y Fn(\020)1418 1488 y Fz(1)p Fy(;)1503 1387 y Fn(\020)1568 1361 y Fu(p)p 1644 1361 51 4 v 66 x Fy(\034)p 1568 1467 126 4 v 1605 1551 a(\025)1704 1387 y Fn(\021)1758 1410 y Fw(j)1795 1387 y Fn(\021)1874 1488 y Fz(=)g Fy(C)2035 1502 y Fw(j)2087 1488 y Fz(min)o(\()p Fy(\034)2324 1451 y Fx(\000)2390 1420 y Fs(j)p 2389 1436 31 3 v 2389 1477 a Fr(2)2433 1488 y Fy(;)15 b(\025)2526 1451 y Fx(\000)p Fw(j)2618 1488 y Fz(\))26 b Fu(\024)f Fy(C)2840 1502 y Fw(j)2891 1488 y Fy(\025)2944 1451 y Fx(\000)p Fw(j)3184 1427 y Fy(\025)p 3046 1467 330 4 v 3046 1486 a Fu(p)p 3122 1486 254 4 v 76 x Fy(\025)3175 1536 y Fq(2)3234 1562 y Fz(+)20 b Fy(\034)3385 1488 y(;)-47 1745 y Fz(as)30 b(desired.)40 b(The)29 b(second)i(inequalit)m(y)e(here) h(follo)m(ws)f(b)m(y)i(considering)d(the)j(cases)g Fy(\025)25 b Fu(\025)2967 1680 y(p)p 3043 1680 51 4 v 65 x Fy(\034)40 b Fz(and)30 b Fy(\025)25 b Fu(\024)3474 1680 y(p)p 3550 1680 V 65 x Fy(\034)10 b Fz(.)p 3879 1745 4 62 v 3883 1687 55 4 v 3883 1745 V 3937 1745 4 62 v -47 1933 a(W)-8 b(e)30 b(no)m(w)f(disp)s(ense)d(with)i(the)h(easy)g(case)h(of)f(the)g (phase)f(with)g(the)h(+)f(sign)g(in)f(\(6.1\))r(.)40 b(By)29 b(our)g(p)s(ositivit)m(y)e(assump-)-47 2046 y(tions)34 b(this)h(phase)g(has)g(no)g(critical)f(p)s(oin)m(t,)i(but)f(some)g (care)h(is)f(needed)g(in)f(terms)h(of)h(upp)s(er)d(b)s(ounds)g(on)i (higher)-47 2159 y(deriv)-5 b(ativ)m(es.)-47 2346 y Fm(Lemma)32 b(6.3.)47 b Fl(L)-5 b(et)37 b Fy(!)s Fz(\()p Fy(\025)p Fz(\))g Fl(b)-5 b(e)37 b(a)g(di\013er)-5 b(entiable)38 b(function)f(such)g(that)i Fu(j)p Fy(!)2508 2313 y Fq(\()p Fw(j)t Fq(\))2599 2346 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)34 b(\024)f Fy(a)2933 2360 y Fq(0)2988 2346 y Fy(\025)3041 2313 y Fx(\000)p Fw(j)3169 2346 y Fl(for)38 b Fy(j)g Fz(=)33 b(0)p Fy(;)15 b Fz(1)38 b Fl(and)g(al)5 b(l)-47 2459 y Fy(\025)25 b(>)g Fz(0)p Fl(.)40 b(Then)28 b(ther)-5 b(e)29 b(exists)e(a)h(c)-5 b(onstant)30 b Fy(C)1434 2473 y Fq(0)1500 2459 y Fl(which)f(only)f(dep)-5 b(ends)29 b(on)f(the)g(c)-5 b(onstant)29 b Fy(a)2953 2473 y Fq(0)2993 2459 y Fl(,)f(so)g(that)h(for)f(any)g(p)-5 b(ositive)-47 2572 y(inte)g(ger)33 b Fy(m)f Fl(and)i(any)f Fz(1)25 b Fu(\024)g Fy(k)k Fu(\024)c Fy(m)p Fl(,)928 2689 y Fn(\014)928 2744 y(\014)928 2798 y(\014)928 2853 y(\014)928 2908 y(\014)958 2725 y(Z)1049 2751 y Fx(1)1009 2931 y Fq(0)1139 2848 y Fy(e)1191 2781 y Fr(1)p 1191 2793 31 3 v 1191 2834 a(2)1231 2808 y Fw(i\025)1296 2785 y Fr(2)1331 2808 y Fq(+)p Fw(i)1422 2758 y Fb(P)1496 2779 y Fs(m)1496 2829 y(j)s Fr(=1)1618 2808 y Fw(b)1648 2818 y Fs(j)1692 2747 y Fu(p)p 1768 2747 204 4 v 61 x Fw(\025)1809 2789 y Fr(2)1844 2808 y Fq(+)p Fw(\033)1939 2818 y Fs(j)2169 2787 y Fy(\025)p 2001 2828 390 4 v 2001 2846 a Fn(p)p 2092 2846 299 4 v 82 x Fy(\025)2145 2902 y Fq(2)2204 2928 y Fz(+)20 b Fy(\033)2347 2943 y Fw(k)2415 2848 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)2713 2689 y Fn(\014)2713 2744 y(\014)2713 2798 y(\014)2713 2853 y(\014)2713 2908 y(\014)2770 2848 y Fu(\024)25 b Fy(C)2931 2862 y Fq(0)-47 3125 y Fl(for)33 b(any)g(choic)-5 b(e)33 b(of)g Fy(\033)705 3139 y Fq(1)770 3125 y Fu(\025)25 b Fy(\033)918 3139 y Fq(2)982 3125 y Fu(\025)g Fy(:)15 b(:)g(:)27 b Fu(\025)e Fy(\033)1358 3139 y Fw(m)1449 3125 y Fu(\025)g Fz(0)33 b Fl(and)h Fy(b)1839 3139 y Fw(j)1901 3125 y Fy(>)25 b Fz(0)p Fl(.)-47 3313 y(Pr)-5 b(o)g(of.)47 b Fz(Let)30 b Fy(\036)p Fz(\()p Fy(\025)p Fz(\))d(=)710 3277 y Fq(1)p 710 3292 36 4 v 710 3344 a(2)755 3313 y Fy(\025)808 3280 y Fq(2)867 3313 y Fz(+)958 3244 y Fn(P)1054 3271 y Fw(m)1054 3339 y(j)t Fq(=1)1196 3313 y Fy(b)1235 3327 y Fw(j)1287 3236 y Fn(p)p 1378 3236 293 4 v 77 x Fy(\025)1431 3286 y Fq(2)1490 3313 y Fz(+)20 b Fy(\033)1633 3327 y Fw(j)1670 3313 y Fz(.)40 b(Then)1317 3603 y Fy(\036)1371 3566 y Fx(0)1394 3603 y Fz(\()p Fy(\025)p Fz(\))84 b(=)f Fy(\025)1808 3502 y Fn(h)1851 3603 y Fz(1)21 b(+)2042 3490 y Fw(m)2007 3517 y Fn(X)2012 3712 y Fw(j)t Fq(=1)2318 3541 y Fy(b)2357 3555 y Fw(j)p 2164 3582 383 4 v 2164 3601 a Fn(p)p 2255 3601 293 4 v 76 x Fy(\025)2308 3651 y Fq(2)2367 3677 y Fz(+)f Fy(\033)2510 3691 y Fw(j)2557 3502 y Fn(i)1298 3899 y Fy(\036)1352 3861 y Fx(00)1394 3899 y Fz(\()p Fy(\025)p Fz(\))84 b(=)1765 3837 y Fy(\036)1819 3804 y Fx(0)1842 3837 y Fz(\()p Fy(\025)p Fz(\))p 1765 3878 202 4 v 1839 3961 a Fy(\025)1996 3899 y Fu(\000)20 b Fy(\021)s Fz(\()p Fy(\025)p Fz(\))-2305 b(\(6.4\))-47 4124 y(where)1162 4375 y Fy(\021)s Fz(\()p Fy(\025)p Fz(\))27 b(=)1490 4261 y Fw(m)1456 4289 y Fn(X)1460 4484 y Fw(j)t Fq(=1)1737 4313 y Fy(b)1776 4327 y Fw(j)1812 4313 y Fy(\025)1865 4280 y Fq(2)p 1612 4354 418 4 v 1612 4460 a Fz(\()p Fy(\025)1700 4434 y Fq(2)1760 4460 y Fz(+)20 b Fy(\033)1903 4474 y Fw(j)1940 4460 y Fz(\))1985 4397 y Fr(3)p 1985 4409 31 3 v 1985 4450 a(2)2065 4375 y Fu(\024)2195 4261 y Fw(m)2161 4289 y Fn(X)2165 4484 y Fw(j)t Fq(=1)2471 4313 y Fy(b)2510 4327 y Fw(j)p 2317 4354 383 4 v 2317 4372 a Fn(p)p 2408 4372 293 4 v 77 x Fy(\025)2461 4422 y Fq(2)2521 4449 y Fz(+)g Fy(\033)2664 4463 y Fw(j)2710 4375 y Fy(:)-47 4700 y Fz(Hence)31 b Fy(\036)277 4667 y Fx(0)300 4700 y Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)f Fy(\025)p Fz(,)694 4655 y Fw(\021)r Fq(\()p Fw(\025)p Fq(\))p 664 4679 195 4 v 664 4734 a Fw(\036)706 4715 y Ft(0)728 4734 y Fq(\()p Fw(\025)p Fq(\))823 4715 y Fr(2)894 4700 y Fu(\024)g Fy(\025)1043 4667 y Fx(\000)p Fq(2)1168 4700 y Fz(and)k(therefore)i(also)1260 4920 y Fu(j)p Fy(\036)1339 4887 y Fx(0)q(0)1382 4920 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)p 1260 4961 271 4 v 1275 5044 a Fy(\036)1329 5018 y Fx(0)1353 5044 y Fz(\()p Fy(\025)p Fz(\))1476 5018 y Fq(2)1566 4982 y Fu(\024)1777 4920 y Fz(1)p 1672 4961 255 4 v 1672 5044 a Fy(\025\036)1779 5018 y Fx(0)1803 5044 y Fz(\()p Fy(\025)p Fz(\))1957 4982 y(+)2154 4920 y Fy(\021)p 2057 4961 241 4 v 2057 5044 a(\036)2111 5018 y Fx(0)2135 5044 y Fz(\()p Fy(\025)p Fz(\))2258 5018 y Fq(2)2333 4982 y Fu(\024)25 b Fz(2)p Fy(\025)2527 4944 y Fx(\000)p Fq(2)2622 4982 y Fy(:)1903 5472 y Fz(44)p eop %%Page: 45 45 45 44 bop -47 274 a Fz(Let)31 b Fy(\037)f Fz(b)s(e)g(a)g(smo)s(oth)h (non-decreasing)e(function)h(with)f Fy(\037)p Fz(\()p Fy(\025)p Fz(\))c(=)g(0)31 b(for)f Fy(\025)c Fu(\024)f Fz(1)30 b(and)g Fy(\037)p Fz(\()p Fy(\025)p Fz(\))c(=)f(1)31 b(for)f Fy(\025)25 b Fu(\025)g Fz(2.)41 b(Then)646 396 y Fn(\014)646 451 y(\014)646 505 y(\014)646 560 y(\014)646 615 y(\014)676 432 y(Z)767 458 y Fx(1)727 638 y Fq(0)857 555 y Fy(e)899 518 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1091 555 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))1452 494 y Fy(\025)p 1284 535 390 4 v 1284 553 a Fn(p)p 1375 553 299 4 v 82 x Fy(\025)1428 609 y Fq(2)1488 635 y Fz(+)20 b Fy(\033)1631 650 y Fw(k)1699 555 y Fy(d\025)1799 396 y Fn(\014)1799 451 y(\014)1799 505 y(\014)1799 560 y(\014)1799 615 y(\014)1855 555 y Fu(\024)25 b Fy(C)h Fz(+)20 b(lim)15 b(sup)2178 634 y Fw(L)p Fx(!1)2427 424 y Fn(\014)2427 478 y(\014)2427 533 y(\014)2427 587 y(\014)2457 432 y(Z)2548 458 y Fx(1)2507 638 y Fq(0)2638 555 y Fy(e)2680 518 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))2871 555 y Fy(g)2914 569 y Fw(L)2967 555 y Fz(\()p Fy(\025)p Fz(\))g Fy(d\025)3205 424 y Fn(\014)3205 478 y(\014)3205 533 y(\014)3205 587 y(\014)-47 555 y Fz(\(6.5\))-47 832 y(where)30 b(w)m(e)g(ha)m(v)m(e)i(set)1102 1056 y Fy(g)1145 1070 y Fw(L)1197 1056 y Fz(\()p Fy(\025)p Fz(\))26 b(:=)f Fy(\037)p Fz(\()p Fy(\025)p Fz(\)\(1)d Fu(\000)e Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))15 b Fy(!)s Fz(\()p Fy(\025)p Fz(\))2540 995 y Fy(\025)p 2370 1035 390 4 v 2370 1053 a Fn(p)p 2463 1053 299 4 v 2463 1135 a Fy(\025)2516 1109 y Fq(2)2575 1135 y Fz(+)20 b Fy(\033)2718 1150 y Fw(k)2771 1056 y Fy(:)-47 1356 y Fz(By)35 b(our)f(assumptions)f(and)h(Lemma)h(6.2,)i Fu(j)p Fy(g)1545 1308 y Fq(\()p Fw(j)t Fq(\))1542 1385 y Fw(L)1637 1356 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)d(\024)e Fy(C)1987 1370 y Fw(j)2038 1356 y Fy(\025)2091 1323 y Fx(\000)p Fw(j)2217 1356 y Fz(for)j Fy(j)j Fz(=)32 b(0)p Fy(;)15 b Fz(1)36 b(uniformly)c(in)h Fy(L)p Fz(.)54 b(In)m(tegrating)35 b(b)m(y)-47 1469 y(parts)30 b(once)h(inside)d(the)j(in)m(tegral)f(on)g (the)g(righ)m(t-hand)g(side)f(of)38 b(\(6.5\))32 b(yields)c(an)j(upp)s (er)d(b)s(ound)g(of)j(the)f(form)484 1599 y Fn(Z)575 1625 y Fx(1)534 1805 y Fq(0)649 1618 y Fn(\014)649 1673 y(\014)649 1727 y(\014)716 1661 y Fy(d)p 690 1702 101 4 v 690 1785 a(d\025)800 1622 y Fn(h)931 1661 y Fz(1)p 853 1702 202 4 v 853 1785 a Fy(\036)907 1759 y Fx(0)930 1785 y Fz(\()p Fy(\025)p Fz(\))1064 1723 y Fy(g)1107 1737 y Fw(L)1159 1723 y Fz(\()p Fy(\025)p Fz(\))1282 1622 y Fn(i)1326 1618 y(\014)1326 1673 y(\014)1326 1727 y(\014)1372 1723 y Fy(d\025)25 b Fu(\024)1593 1599 y Fn(Z)1684 1625 y Fx(1)1644 1805 y Fq(1)1759 1622 y Fn(h)1812 1661 y Fu(j)p Fy(\036)1891 1628 y Fx(00)1934 1661 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)p 1812 1702 271 4 v 1827 1785 a Fy(\036)1881 1759 y Fx(0)1904 1785 y Fz(\()p Fy(\025)p Fz(\))2027 1759 y Fq(2)2113 1723 y Fz(+)2318 1661 y(1)p 2214 1702 255 4 v 2214 1785 a Fy(\025\036)2321 1759 y Fx(0)2344 1785 y Fz(\()p Fy(\025)p Fz(\))2478 1622 y Fn(i)2536 1723 y Fy(d\025)g Fo(.)2757 1599 y Fn(Z)2848 1625 y Fx(1)2808 1805 y Fq(1)2938 1723 y Fy(\025)2991 1685 y Fx(\000)p Fq(2)3101 1723 y Fy(d\025)g Fu(\024)g Fy(C)q(;)-47 1971 y Fz(as)30 b(claimed.)p 3879 1971 4 62 v 3883 1913 55 4 v 3883 1971 V 3937 1971 4 62 v -47 2158 a(W)-8 b(e)34 b(no)m(w)g(turn)e(to)i(the)g(phase)f 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Fq(\(4\))1675 4513 y Fz(\(0\))26 b(=)1959 4451 y(3)p 1922 4492 120 4 v 1922 4575 a Fy(m)2002 4549 y Fq(2)2101 4399 y Fw(m)2066 4426 y Fn(X)2071 4622 y Fw(j)t Fq(=1)2213 4513 y Fy(b)2252 4475 y Fx(\000)p Fq(2)2252 4540 y Fw(j)2346 4513 y Fy(:)-47 4826 y Fl(If)32 b Fy(\036)102 4793 y Fq(\(4\))197 4826 y Fz(\(0\))26 b Fu(\025)f Fz(1)p Fl(,)33 b(then)g(heuristic)-5 b(al)5 b(ly)34 b(sp)-5 b(e)g(aking)966 4970 y Fn(\014)966 5025 y(\014)966 5079 y(\014)996 4951 y(Z)1087 4977 y Fx(1)1047 5157 y Fq(0)1177 5075 y Fy(e)1219 5037 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1386 5075 y Fy(d\025)1486 4970 y Fn(\014)1486 5025 y(\014)1486 5079 y(\014)1542 5075 y Fu(\020)1638 4970 y Fn(\014)1638 5025 y(\014)1638 5079 y(\014)1668 4951 y(Z)1759 4977 y Fx(1)1718 5157 y Fq(0)1849 5075 y Fy(e)1891 5037 y Fw(i\036)1957 5014 y Fr(\(4\))2040 5037 y Fq(\(0\))p Fw(\025)2170 5014 y Fr(4)2210 5075 y Fy(d\025)2310 4970 y Fn(\014)2310 5025 y(\014)2310 5079 y(\014)2366 5075 y Fo(.)24 b Fz(\()p Fy(\036)2550 5037 y Fq(\(4\))2645 5075 y Fz(\(0\)\))2795 5037 y Fx(\000)2860 5010 y Fr(1)p 2862 5022 31 3 v 2862 5063 a(4)2906 5075 y Fy(:)1903 5472 y Fz(45)p eop %%Page: 46 46 46 45 bop -47 299 a Fl(In)31 b(c)-5 b(ase)31 b(al)5 b(l)31 b Fy(b)430 266 y Fx(0)430 325 y Fw(j)467 299 y Fy(s)f Fl(ar)-5 b(e)32 b(c)-5 b(omp)g(ar)g(able,)34 b(this)d(b)-5 b(ound)32 b(agr)-5 b(e)g(es)32 b(with)f(the)h Fy(\033)2297 215 y Fr(1)p 2297 227 31 3 v 2297 268 a(4)2284 325 y Fq(1)2372 299 y Fl(estimate)f(in)38 b Fz(\(6.1\))r Fl(.)j(On)31 b(the)g(other)h(hand,)g(if)-47 429 y Fy(\034)3 396 y Fq(2)42 429 y Fy(\036)96 396 y Fq(\(4\))191 429 y Fz(\(0\))26 b Fu(\025)f Fz(1)p Fl(,)33 b(then)g(again)g(heuristic)-5 b(al)5 b(ly)34 b(sp)-5 b(e)g(aking)89 573 y Fn(\014)89 627 y(\014)89 682 y(\014)119 553 y(Z)210 580 y Fx(1)169 760 y Fq(0)300 677 y Fy(e)342 640 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))656 616 y Fy(\025)p 518 656 330 4 v 518 674 a Fu(p)p 594 674 254 4 v 77 x Fy(\025)647 725 y Fq(2)707 751 y Fz(+)20 b Fy(\034)873 677 y(d\025)973 573 y Fn(\014)973 627 y(\014)973 682 y(\014)1029 677 y Fu(\020)1125 607 y(p)p 1200 607 51 4 v 1200 677 a Fy(\034)1250 573 y Fn(\014)1250 627 y(\014)1250 682 y(\014)1281 553 y(Z)1372 580 y Fx(1)1331 760 y Fq(0)1461 677 y Fy(e)1503 640 y Fw(i\034)1566 616 y Fr(2)1601 640 y Fw(\036)1643 616 y Fr(\(4\))1726 640 y Fq(\(0\))p Fw(\025)1856 616 y Fr(4)156 b Fy(\025)p 1906 656 325 4 v 1906 674 a Fu(p)p 1982 674 250 4 v 77 x Fy(\025)2035 725 y Fq(2)2094 751 y Fz(+)20 b(1)2256 677 y Fy(d\025)2356 573 y Fn(\014)2356 627 y(\014)2356 682 y(\014)2412 677 y Fo(.)2508 607 y Fu(p)p 2583 607 51 4 v 2583 677 a Fy(\034)10 b Fz(\()p Fy(\034)2718 640 y Fq(2)2758 677 y Fy(\036)2812 640 y Fq(\(4\))2907 677 y Fz(\(0\)\))3057 640 y Fx(\000)3122 612 y Fr(1)p 3123 624 31 3 v 3123 666 a(2)3193 677 y Fz(=)25 b(\()p Fy(\034)10 b(\036)3428 640 y Fq(\(4\))3523 677 y Fz(\(0\)\))3673 640 y Fx(\000)3738 612 y Fr(1)p 3739 624 V 3739 666 a(2)3784 677 y Fy(:)-47 931 y Fl(In)32 b(c)-5 b(ase)33 b(al)5 b(l)33 b Fy(b)435 898 y Fx(0)435 957 y Fw(j)471 931 y Fy(s)f Fl(ar)-5 b(e)33 b(c)-5 b(omp)g(ar)g(able)36 b(and)d(one)g(cho)-5 b(oses)34 b Fy(\034)h Fz(=)25 b Fy(\033)2062 946 y Fw(k)2137 931 y Fl(for)33 b(an)g(arbitr)-5 b(ary)34 b Fy(k)s Fl(,)f(then)g(the)f (right-hand)i(side)f(is)-47 1044 y Fy(O)s Fz(\(1\))p Fl(.)42 b(This)34 b(agr)-5 b(e)g(es)33 b(with)h(the)f(se)-5 b(c)g(ond)34 b(term)f(in)39 b Fz(\(6.1\))c Fl(\(ignoring)e(p)-5 b(owers)35 b(of)d Fy(m)p Fl(\).)-47 1232 y Fm(Lemma)g(6.5.)47 b Fl(The)32 b(phase)1362 1500 y Fy(\036)p Fz(\()p Fy(\025)p Fz(\))26 b(=)1671 1439 y(1)p 1671 1479 46 4 v 1671 1563 a(2)1727 1500 y Fy(\025)1780 1463 y Fq(2)1839 1500 y Fu(\000)1965 1387 y Fw(m)1930 1414 y Fn(X)1935 1609 y Fw(j)t Fq(=1)2077 1500 y Fy(b)2116 1514 y Fw(j)2152 1392 y Fn(q)p 2243 1392 293 4 v 108 x Fy(\025)2296 1474 y Fq(2)2356 1500 y Fz(+)20 b Fy(\033)2499 1514 y Fw(j)-47 1794 y Fl(has)33 b(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties)35 b(for)e(any)h(choic)-5 b(e)33 b(of)g(p)-5 b(ar)g(ameters)35 b Fy(b)2274 1808 y Fw(j)2336 1794 y Fy(>)25 b Fz(0)33 b Fl(and)h Fy(\033)2739 1808 y Fw(j)2800 1794 y Fu(\025)25 b Fz(0)p Fl(:)60 1982 y(1.)46 b(Ther)-5 b(e)36 b(exists)g(at)g(most)h (one)f(critic)-5 b(al)36 b(p)-5 b(oint)37 b Fy(\025)1801 1996 y Fq(0)1871 1982 y Fy(>)30 b Fz(0)36 b Fl(of)g Fy(\036)g Fl(on)g Fz(\(0)p Fy(;)15 b Fu(1)p Fz(\))p Fl(,)37 b(with)g Fy(\036)2952 1949 y Fx(0)2975 1982 y Fz(\()p Fy(\025)p Fz(\))31 b Fy(>)g Fz(0)k Fl(for)i Fy(\025)30 b(>)g(\025)3697 1996 y Fq(0)3737 1982 y Fl(,)36 b(and)180 2095 y Fy(\036)234 2062 y Fx(0)257 2095 y Fz(\()p Fy(\025)p Fz(\))30 b Fy(<)e Fz(0)35 b Fl(for)g Fy(\025)29 b(<)f(\025)970 2109 y Fq(0)1010 2095 y Fl(.)47 b(Mor)-5 b(e)g(over,)36 b(if)e Fy(\025)1658 2109 y Fq(0)1732 2095 y Fl(exists,)h(then)g(the)g(se)-5 b(c)g(ond)36 b(derivative)f Fy(\036)3131 2062 y Fx(00)3208 2095 y Fl(satis\014es)g Fy(\036)3605 2062 y Fx(00)3648 2095 y Fz(\()p Fy(\025)p Fz(\))29 b Fy(>)f Fz(0)180 2208 y Fl(for)35 b Fy(\025)29 b(>)g(\025)562 2222 y Fq(0)636 2208 y Fl(and)36 b(ther)-5 b(e)35 b(is)g(at)g(most)h(one)f(zer)-5 b(o)35 b(of)g Fy(\036)2007 2175 y Fx(0)q(0)2085 2208 y Fl(on)g(the)g(interval)g Fz(\(0)p Fy(;)15 b(\025)2878 2222 y Fq(0)2919 2208 y Fz(\))p Fl(,)35 b(which)h(we)f(denote)g(by)g Fy(\025)3878 2222 y Fq(1)3917 2208 y Fl(.)180 2321 y(If)d Fy(\025)328 2335 y Fq(0)400 2321 y Fl(do)-5 b(es)34 b(not)g(exist,)e (then)h Fy(\036)1256 2288 y Fx(0)q(0)1299 2321 y Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)f Fz(0)33 b Fl(for)g(al)5 b(l)33 b Fy(\025)25 b Fu(\025)g Fz(0)p Fl(.)60 2508 y(2.)46 b(One)32 b(has)1783 2638 y Fn(\020)1857 2678 y Fy(\036)1911 2645 y Fx(0)q(0)1954 2678 y Fz(\()p Fy(\025)p Fz(\))p 1847 2718 241 4 v 1847 2809 a Fy(\036)1901 2783 y Fx(0)1925 2772 y Fq(3)1964 2809 y Fz(\()p Fy(\025)p Fz(\))2098 2638 y Fn(\021)2152 2661 y Fx(0)2201 2739 y Fu(\024)25 b Fz(0)-2162 b(\(6.6\))180 2994 y Fl(for)33 b(al)5 b(l)33 b Fy(\025)26 b Fu(6)p Fz(=)f Fy(\025)683 3008 y Fq(0)722 2994 y Fl(.)60 3182 y(3.)46 b(If)32 b Fy(\025)328 3196 y Fq(0)393 3182 y Fy(>)25 b Fz(0)33 b Fl(exists,)g(then)1432 3451 y Fy(\036)1486 3413 y Fx(0)1510 3451 y Fz(\()p Fy(\025)p Fz(\))26 b(=)f Fy(\025)1823 3327 y Fn(Z)1914 3353 y Fw(\025)1873 3533 y(\025)1914 3542 y Fr(0)2009 3337 y Fw(m)1974 3364 y Fn(X)1978 3560 y Fw(j)t Fq(=1)2275 3389 y Fy(b)2314 3403 y Fw(j)2350 3389 y Fy(s)p 2131 3430 407 4 v 2131 3536 a Fz(\()p Fy(s)2209 3509 y Fq(2)2268 3536 y Fz(+)20 b Fy(\033)2411 3550 y Fw(j)2447 3536 y Fz(\))2492 3473 y Fr(3)p 2493 3485 31 3 v 2493 3526 a(2)2562 3451 y Fy(ds;)-2497 b Fz(\(6.7\))180 3744 y Fl(wher)-5 b(e)g(as)35 b(if)d Fy(\025)657 3758 y Fq(0)722 3744 y Fy(>)25 b Fz(0)33 b Fl(do)-5 b(es)33 b(not)h(exist,)e(then)1440 4013 y Fy(\036)1494 3976 y Fx(0)1517 4013 y Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)f Fy(\025)1830 3889 y Fn(Z)1921 3916 y Fw(\025)1881 4096 y Fq(0)2016 3899 y Fw(m)1982 3927 y Fn(X)1986 4122 y Fw(j)t Fq(=1)2282 3951 y Fy(b)2321 3965 y Fw(j)2358 3951 y Fy(s)p 2138 3992 407 4 v 2138 4098 a Fz(\()p Fy(s)2216 4072 y Fq(2)2276 4098 y Fz(+)20 b Fy(\033)2419 4112 y Fw(j)2455 4098 y Fz(\))2500 4035 y Fr(3)p 2500 4047 31 3 v 2500 4088 a(2)2570 4013 y Fy(ds:)-2505 b Fz(\(6.8\))-47 4344 y Fl(Pr)-5 b(o)g(of.)47 b Fz(One)30 b(has)829 4551 y Fy(\036)883 4518 y Fx(0)906 4551 y Fz(\()p Fy(\025)p Fz(\))p 829 4592 202 4 v 903 4675 a Fy(\025)1123 4613 y Fz(=)82 b(1)21 b Fu(\000)1467 4499 y Fw(m)1433 4527 y Fn(X)1437 4722 y Fw(j)t Fq(=1)1743 4551 y Fy(b)1782 4565 y Fw(j)p 1589 4592 383 4 v 1589 4610 a Fn(p)p 1680 4610 293 4 v 76 x Fy(\025)1733 4660 y Fq(2)1793 4686 y Fz(+)f Fy(\033)1936 4700 y Fw(j)-47 4613 y Fz(\(6.9\))819 4928 y Fy(\036)873 4891 y Fx(0)q(0)916 4928 y Fz(\()p Fy(\025)p Fz(\))84 b(=)1286 4867 y Fy(\036)1340 4834 y Fx(0)1364 4867 y Fz(\()p Fy(\025)p Fz(\))p 1286 4908 202 4 v 1361 4991 a Fy(\025)1518 4928 y Fz(+)1643 4815 y Fw(m)1609 4842 y Fn(X)1613 5038 y Fw(j)t Fq(=1)1890 4866 y Fy(b)1929 4880 y Fw(j)1965 4866 y Fy(\025)2018 4833 y Fq(2)p 1765 4908 418 4 v 1765 5014 a Fz(\()p Fy(\025)1853 4987 y Fq(2)1913 5014 y Fz(+)20 b Fy(\033)2056 5028 y Fw(j)2093 5014 y Fz(\))2138 4951 y Fr(3)p 2138 4963 31 3 v 2138 5004 a(2)2218 4928 y Fz(=)25 b(1)20 b Fu(\000)2505 4815 y Fw(m)2470 4842 y Fn(X)2475 5038 y Fw(j)t Fq(=1)2753 4866 y Fy(b)2792 4880 y Fw(j)2829 4866 y Fy(\033)2881 4880 y Fw(j)p 2627 4908 418 4 v 2627 5014 a Fz(\()p Fy(\025)2715 4987 y Fq(2)2775 5014 y Fz(+)f Fy(\033)2917 5028 y Fw(j)2954 5014 y Fz(\))2999 4951 y Fr(3)p 2999 4963 31 3 v 2999 5004 a(2)3054 4928 y Fy(:)-3126 b Fz(\(6.10\))1903 5472 y(46)p eop %%Page: 47 47 47 46 bop -47 274 a Fz(It)30 b(is)g(clear)g(that)h Fy(\025)610 288 y Fq(0)675 274 y Fy(>)25 b Fz(0)30 b(is)g(uniquely)e(giv)m(en)i(b)m (y)1578 543 y(1)c(=)1779 429 y Fw(m)1745 456 y Fn(X)1749 652 y Fw(j)t Fq(=1)2055 481 y Fy(b)2094 495 y Fw(j)p 1901 522 383 4 v 1901 540 a Fn(p)p 1992 540 293 4 v 79 x Fy(\025)2045 587 y Fq(2)2045 645 y(0)2105 619 y Fz(+)20 b Fy(\033)2248 633 y Fw(j)2294 543 y Fy(:)-2366 b Fz(\(6.11\))-47 843 y(Moreo)m(v)m(er,)33 b Fy(\036)432 810 y Fx(00)504 843 y Fz(is)d(non)g(decreasing)g(b)m(y)g(the)h(second)f(equalit)m(y)g (in)f(\(6.10\))r(,)i(and)f(also)1723 1047 y Fy(\036)1777 1010 y Fx(00)1820 1047 y Fz(\()p Fy(\025)1908 1061 y Fq(0)1947 1047 y Fz(\))c Fy(>)f Fz(0)p Fy(:)-47 1252 y Fz(If)38 b(all)g Fy(\033)239 1266 y Fw(j)315 1252 y Fz(=)h(0,)i(then)e Fy(\036)806 1219 y Fx(00)848 1252 y Fz(\()p Fy(\025)p Fz(\))h(=)f(1,)j(whereas)c(if)g(one)h Fy(\033)1904 1266 y Fw(j)1980 1252 y Fu(6)p Fz(=)g(0,)i(then)e Fy(\036)2471 1219 y Fx(00)2552 1252 y Fz(is)f(strictly)g(increasing)f (and)i(can)g(ha)m(v)m(e)-47 1365 y(at)34 b(most)g(one)f(zero)i Fy(\025)711 1379 y Fq(1)784 1365 y Fz(whic)m(h)d(then)h(necessarily)f (falls)g(in)m(to)i(the)f(in)m(terv)-5 b(al)33 b([0)p Fy(;)15 b(\025)2748 1379 y Fq(0)2789 1365 y Fz(].)50 b(T)-8 b(o)34 b(motiv)-5 b(ate)34 b(\(6.6\))r(,)g(observ)m(e)-47 1477 y(that)g(for)g(large)g Fy(\025)f Fz(one)h(has)g Fy(\036)998 1444 y Fx(00)1041 1477 y Fz(\()p Fy(\025)p Fz(\))d Fu(\020)g Fz(1)j(and)f Fy(\036)1610 1444 y Fx(0)1634 1477 y Fz(\()p Fy(\025)p Fz(\))f Fu(\020)e Fy(\025)k Fz(so)g(that)g Fy(\036)2346 1444 y Fx(0)q(0)2389 1477 y Fz(\()p Fy(\025)p Fz(\))p Fy(\036)2566 1444 y Fx(0)2590 1434 y(\000)p Fq(3)2685 1477 y Fz(\()p Fy(\025)p Fz(\))g(is)f (decreasing.)51 b(Ho)m(w)m(ev)m(er,)37 b(the)-47 1590 y(situation)g(is)g(not)i(so)g(clear)f(for)g(smaller)f Fy(\025)h Fz(and)g(w)m(e)h(will)c(need)j(to)i(use)e(the)g(sp)s (eci\014c)f(structure)h(of)h Fy(\036)3577 1557 y Fx(0)3638 1590 y Fz(and)f Fy(\036)3877 1557 y Fx(00)3920 1590 y Fz(.)-47 1703 y(Firstly)-8 b(,)29 b(consider)h(the)g(case)i Fy(\025)25 b(<)g(\025)1196 1717 y Fq(0)1235 1703 y Fz(,)31 b(whic)m(h)e(is)g(precisely)g(the)i(range)g(where)f Fy(\036)2736 1670 y Fx(0)2784 1703 y Fy(<)25 b Fz(0.)41 b(Since)1350 1972 y Fy(\036)1404 1934 y Fx(0)q(00)1466 1972 y Fz(\()p Fy(\025)p Fz(\))26 b(=)f(3)1806 1858 y Fw(m)1772 1886 y Fn(X)1776 2081 y Fw(j)t Fq(=1)2028 1910 y Fy(\025\033)2133 1924 y Fw(j)2170 1910 y Fy(b)2209 1924 y Fw(j)p 1928 1951 418 4 v 1928 2057 a Fz(\()p Fy(\025)2016 2031 y Fq(2)2076 2057 y Fz(+)20 b Fy(\033)2219 2071 y Fw(j)2256 2057 y Fz(\))2301 1994 y Fr(5)p 2301 2006 31 3 v 2301 2047 a(2)2381 1972 y Fu(\025)25 b Fz(0)p Fy(;)-47 2266 y Fz(it)30 b(follo)m(ws)f(that)1260 2441 y Fy(d)p 1233 2482 101 4 v 1233 2565 a(d\025)1343 2402 y Fn(h)1406 2441 y Fy(\036)1460 2408 y Fx(0)q(0)1503 2441 y Fz(\()p Fy(\025)p Fz(\))p 1396 2482 241 4 v 1396 2565 a Fy(\036)1450 2539 y Fx(0)1474 2565 y Fz(\()p Fy(\025)p Fz(\))1597 2539 y Fq(3)1647 2402 y Fn(i)1715 2503 y Fz(=)1822 2441 y Fy(\036)1876 2408 y Fx(000)1937 2441 y Fz(\()p Fy(\025)p Fz(\))p 1821 2482 V 1821 2565 a Fy(\036)1875 2539 y Fx(0)1898 2565 y Fz(\()p Fy(\025)p Fz(\))2021 2539 y Fq(3)2092 2503 y Fu(\000)20 b Fz(3)2238 2441 y Fy(\036)2292 2408 y Fx(00)2335 2441 y Fz(\()p Fy(\025)p Fz(\))2458 2408 y Fq(2)p 2238 2482 260 4 v 2248 2565 a Fy(\036)2302 2539 y Fx(0)2325 2565 y Fz(\()p Fy(\025)p Fz(\))2448 2539 y Fq(4)2533 2503 y Fu(\024)25 b Fz(0)-47 2751 y(in)k(that)i(range.)41 b(T)-8 b(o)31 b(deal)f(with)f(the)h(range)h Fy(\025)25 b Fu(\025)g Fy(\025)1703 2765 y Fq(0)1743 2751 y Fz(,)30 b(w)m(e)h(use)f(the)h(iden)m(tit)m(y)1519 2999 y Fy(\036)1573 2962 y Fx(00)1615 2999 y Fz(\()p Fy(\025)p Fz(\))26 b(=)1870 2938 y Fy(\036)1924 2905 y Fx(0)1948 2938 y Fz(\()p Fy(\025)p Fz(\))p 1870 2978 202 4 v 1944 3062 a Fy(\025)2102 2999 y Fz(+)20 b Fy( )s Fz(\()p Fy(\025)p Fz(\))-2425 b(\(6.12\))-47 3224 y(where)1491 3475 y Fy( )s Fz(\()p Fy(\025)p Fz(\))26 b(=)1833 3362 y Fw(m)1798 3389 y Fn(X)1802 3584 y Fw(j)t Fq(=1)2079 3413 y Fy(\025)2132 3380 y Fq(2)2172 3413 y Fy(b)2211 3427 y Fw(j)p 1955 3454 418 4 v 1955 3560 a Fz(\()p Fy(\025)2043 3534 y Fq(2)2103 3560 y Fz(+)19 b Fy(\033)2245 3574 y Fw(j)2282 3560 y Fz(\))2327 3497 y Fr(3)p 2327 3509 31 3 v 2327 3551 a(2)2382 3475 y Fy(:)-47 3769 y Fz(One)30 b(therefore)g(has)702 3939 y Fy(d)p 675 3979 101 4 v 675 4062 a(d\025)786 3899 y Fn(h)849 3939 y Fy(\036)903 3906 y Fx(00)945 3939 y Fz(\()p Fy(\025)p Fz(\))p 839 3979 241 4 v 839 4062 a Fy(\036)893 4036 y Fx(0)916 4062 y Fz(\()p Fy(\025)p Fz(\))1039 4036 y Fq(3)1089 3899 y Fn(i)1215 4000 y Fz(=)1405 3939 y Fy(d)p 1379 3979 101 4 v 1379 4062 a(d\025)1489 3899 y Fn(h)1640 3939 y Fz(1)p 1542 3979 241 4 v 1542 4062 a Fy(\036)1596 4036 y Fx(0)1620 4062 y Fz(\()p Fy(\025)p Fz(\))1743 4036 y Fq(2)1818 3939 y Fy(\036)1872 3906 y Fx(00)1914 3939 y Fz(\()p Fy(\025)p Fz(\))p 1818 3979 221 4 v 1828 4062 a Fy(\036)1882 4036 y Fx(0)1905 4062 y Fz(\()p Fy(\025)p Fz(\))2048 3899 y Fn(i)2116 4000 y Fz(=)2249 3939 y Fy(d)p 2222 3979 101 4 v 2222 4062 a(d\025)2333 3899 y Fn(h)2483 3939 y Fz(1)p 2385 3979 241 4 v 2385 4062 a Fy(\036)2439 4036 y Fx(0)2463 4062 y Fz(\()p Fy(\025)p Fz(\))2586 4036 y Fq(2)2636 3899 y Fn(\020)2704 3939 y Fz(1)p 2700 3979 54 4 v 2700 4062 a Fy(\025)2783 4000 y Fz(+)2912 3939 y Fy( )s Fz(\()p Fy(\025)p Fz(\))p 2884 3979 241 4 v 2884 4062 a Fy(\036)2938 4036 y Fx(0)2962 4062 y Fz(\()p 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Fx(0)763 4510 y Fz(:)1252 4779 y Fy( )1314 4741 y Fx(0)1338 4779 y Fz(\()p Fy(\025)p Fz(\))84 b(=)f(2)1754 4717 y Fy( )s Fz(\()p Fy(\025)p Fz(\))p 1754 4758 187 4 v 1821 4841 a Fy(\025)1970 4779 y Fu(\000)2096 4665 y Fw(m)2061 4693 y Fn(X)2066 4888 y Fw(j)t Fq(=1)2320 4717 y Fz(3)p Fy(\025)2418 4684 y Fq(3)2458 4717 y Fy(b)2497 4731 y Fw(j)p 2218 4758 418 4 v 2218 4864 a Fz(\()p Fy(\025)2306 4838 y Fq(2)2366 4864 y Fz(+)20 b Fy(\033)2509 4878 y Fw(j)2545 4864 y Fz(\))2590 4801 y Fr(5)p 2590 4813 31 3 v 2590 4854 a(2)1545 5075 y Fz(=)1709 5013 y(2)p Fy(\036)1808 4980 y Fx(00)1851 5013 y Fz(\()p Fy(\025)p Fz(\))p 1709 5054 266 4 v 1815 5137 a Fy(\025)2005 5075 y Fu(\000)2105 5013 y Fz(2)p Fy(\036)2204 4980 y Fx(0)2228 5013 y Fz(\()p Fy(\025)p Fz(\))p 2105 5054 247 4 v 2183 5137 a Fy(\025)2236 5111 y Fq(2)2382 5075 y Fu(\000)g Fy(\021)s Fz(\()p Fy(\025)p Fz(\))-2691 b(\(6.14\))1903 5472 y(47)p eop %%Page: 48 48 48 47 bop -47 274 a Fz(where)25 b Fy(\021)j Fu(\025)d Fz(0)h(in)e(\(6.14\))k(is)d(de\014ned)f(to)i(b)s(e)f(the)h(sum)f(in)f (the)h(preceding)g(line,)g(and)g(w)m(e)h(ha)m(v)m(e)h(used)d(\(6.12\))k (to)e(obtain)-47 387 y(the)k(\014rst)g(expression)f(in)g(\(6.14\))r(.) 41 b(Com)m(bining)30 b(\(6.13\))i(with)d(\(6.14\))k(leads)d(to)572 559 y Fy(d)p 545 600 101 4 v 545 683 a(d\025)655 520 y Fn(h)718 559 y Fy(\036)772 526 y Fx(0)q(0)815 559 y Fz(\()p Fy(\025)p Fz(\))p 708 600 241 4 v 708 683 a Fy(\036)762 657 y Fx(0)786 683 y Fz(\()p Fy(\025)p Fz(\))909 657 y Fq(3)959 520 y Fn(i)1027 621 y Fz(=)25 b Fu(\000)p Fz(2)1249 559 y Fy(\036)1303 526 y Fx(00)1346 559 y Fz(\()p Fy(\025)p Fz(\))1469 526 y Fq(2)p 1249 600 260 4 v 1259 683 a Fy(\036)1313 657 y Fx(0)1336 683 y Fz(\()p Fy(\025)p Fz(\))1459 657 y Fq(4)1539 621 y Fu(\000)1784 559 y Fz(3)p 1640 600 334 4 v 1640 683 a Fy(\025)1693 657 y Fq(2)1732 683 y Fy(\036)1786 657 y Fx(0)1810 683 y Fz(\()p Fy(\025)p Fz(\))1933 657 y Fq(2)2003 621 y Fz(+)2118 559 y(2)p Fy(\036)2217 526 y Fx(0)q(0)2260 559 y Fz(\()p Fy(\025)p Fz(\))p 2104 600 294 4 v 2104 683 a Fy(\025\036)2211 657 y Fx(0)2235 683 y Fz(\()p Fy(\025)p Fz(\))2358 657 y Fq(3)2428 621 y Fu(\000)2563 559 y Fy(\021)s Fz(\()p Fy(\025)p Fz(\))p 2529 600 241 4 v 2529 683 a Fy(\036)2583 657 y Fx(0)2606 683 y Fz(\()p Fy(\025)p Fz(\))2729 657 y Fq(3)2800 621 y Fu(\000)2911 559 y Fy(\036)2965 526 y Fx(00)3007 559 y Fz(\()p Fy(\025)p Fz(\))p 2900 600 V 2900 683 a Fy(\036)2954 657 y Fx(0)2978 683 y Fz(\()p Fy(\025)p Fz(\))3101 657 y Fq(4)3151 621 y Fy( )s Fz(\()p Fy(\025)p Fz(\))p Fy(:)-3408 b Fz(\(6.15\))-47 850 y(Since)27 b(b)s(oth)g Fy(\036)454 817 y Fx(0)477 850 y Fz(\()p Fy(\025)p Fz(\))f Fu(\025)f Fz(0)j(and)g Fy(\036)1024 817 y Fx(00)1066 850 y Fz(\()p Fy(\025)p Fz(\))e Fu(\025)f Fz(0)j(in)f(the)h(range)g Fy(\025)e Fu(\025)e Fy(\025)2114 864 y Fq(0)2154 850 y Fz(,)29 b(all)d(but)h(the)h(third)e(term)i(are)h (negativ)m(e.)41 b(W)-8 b(e)29 b(no)m(w)-47 963 y(sho)m(w)h(that)h(the) g(\014rst)e(and)h(third)e(terms)j(in)e(\(6.15\))j(result)e(in)f(a)i (negativ)m(e)g(expression:)660 1196 y Fu(\000)p Fz(2)786 1135 y Fy(\036)840 1102 y Fx(00)883 1135 y Fz(\()p Fy(\025)p Fz(\))1006 1102 y Fq(2)p 786 1176 260 4 v 796 1259 a Fy(\036)850 1233 y Fx(0)873 1259 y Fz(\()p Fy(\025)p Fz(\))996 1233 y Fq(4)1076 1196 y Fz(+)1191 1135 y(2)p Fy(\036)1290 1102 y Fx(0)q(0)1333 1135 y Fz(\()p Fy(\025)p Fz(\))p 1177 1176 294 4 v 1177 1259 a Fy(\025\036)1284 1233 y Fx(0)1307 1259 y Fz(\()p Fy(\025)p Fz(\))1430 1233 y Fq(3)1506 1196 y Fz(=)1612 1135 y(2)p Fy(\036)1711 1102 y Fx(00)1754 1135 y Fz(\()p Fy(\025)p Fz(\))p 1612 1176 266 4 v 1625 1259 a Fy(\036)1679 1233 y Fx(0)1702 1259 y Fz(\()p Fy(\025)p Fz(\))1825 1233 y Fq(4)1887 1096 y Fn(\020)1952 1135 y Fy(\036)2006 1102 y Fx(0)2029 1135 y Fz(\()p Fy(\025)p Fz(\))p 1952 1176 202 4 v 2026 1259 a Fy(\025)2183 1196 y Fu(\000)20 b Fy(\036)2328 1159 y Fx(00)2371 1196 y Fz(\()p Fy(\025)p Fz(\))2494 1096 y Fn(\021)2574 1196 y Fz(=)25 b Fu(\000)2751 1135 y Fz(2)p Fy(\036)2850 1102 y Fx(00)2893 1135 y Fz(\()p Fy(\025)p Fz(\))p Fy( )s Fz(\()p Fy(\025)p Fz(\))p 2751 1176 453 4 v 2856 1259 a Fy(\036)2910 1233 y Fx(0)2934 1259 y Fz(\()p Fy(\025)p Fz(\))3057 1233 y Fq(4)3212 1196 y Fy(;)-47 1424 y Fz(and)30 b(\(6.6\))i(has)e(b)s(een)f (established.)-47 1536 y(As)h(for)g(the)h(\014nal)e(assertion,)h(if)f Fy(\025)1126 1550 y Fq(0)1196 1536 y Fz(exists,)h(then)h(\(6.7\))h (follo)m(ws)d(b)m(y)h(in)m(tegrating)1418 1684 y Fn(\020)1482 1723 y Fy(\036)1536 1690 y Fx(0)1559 1723 y Fz(\()p Fy(s)p Fz(\))p 1482 1764 191 4 v 1556 1847 a Fy(s)1683 1684 y Fn(\021)1737 1706 y Fx(0)1786 1784 y Fz(=)1916 1671 y Fw(m)1881 1698 y Fn(X)1886 1893 y Fw(j)t Fq(=1)2182 1722 y Fy(b)2221 1736 y Fw(j)2258 1722 y Fy(s)p 2038 1764 407 4 v 2038 1870 a Fz(\()p Fy(s)2116 1843 y Fq(2)2176 1870 y Fz(+)19 b Fy(\033)2318 1884 y Fw(j)2355 1870 y Fz(\))2400 1807 y Fr(3)p 2400 1819 31 3 v 2400 1860 a(2)2455 1784 y Fy(:)-47 2052 y Fz(If)30 b Fy(\025)97 2066 y Fq(0)161 2052 y Fy(>)25 b Fz(0)31 b(do)s(es)f(not)h(exist,)f(then)g(necessarily) 1680 2295 y(1)c Fu(\025)1881 2182 y Fw(m)1847 2209 y Fn(X)1851 2404 y Fw(j)t Fq(=1)2047 2233 y Fy(b)2086 2247 y Fw(j)p 2003 2274 165 4 v 2003 2306 a Fu(p)p 2079 2306 89 4 v 52 x Fy(\033)2131 2372 y Fw(j)2177 2295 y Fy(;)-2249 b Fz(\(6.16\))-47 2576 y(so)30 b(that)h Fy(\036)315 2543 y Fx(0)339 2576 y Fz(\()p Fy(s)p Fz(\))p Fy(s)495 2543 y Fx(\000)p Fq(1)614 2576 y Fu(\025)25 b Fz(0)31 b(for)f(all)f Fy(s)c Fu(\025)g Fz(0.)41 b(In)m(tegrating)31 b Fy(s)f Fz(from)g(0)h(to)g Fy(\025)f Fz(therefore)h(pro)m(v)m(es)g(\(6.8\))r(.) p 3879 2576 4 62 v 3883 2518 55 4 v 3883 2576 V 3937 2576 4 62 v -47 2759 a(In)d(the)h(pro)s(of)e(of)i(Lemma)g(6.1)h(w)m(e)f (consider)e(the)i(in)m(terv)-5 b(als)28 b([)p Fy(\025)2105 2773 y Fq(0)2145 2759 y Fy(;)15 b Fu(1)p Fz(\))29 b(and)f([0)p Fy(;)15 b(\025)2678 2773 y Fq(0)2719 2759 y Fz(\))28 b(separately)-8 b(.)41 b(W)-8 b(e)30 b(start)f(with)f(the)-47 2872 y(former)k(case,)j(and)d(presen)m(t)h(a)g(lemma)g(that)g(reduces)g (matters)g(to)h(establishing)c(suitable)i(lo)m(w)m(er)h(b)s(ounds)d(on) j Fy(\036)3896 2839 y Fx(0)3920 2872 y Fz(.)-47 2985 y(The)c(exact)i(form)e(of)g(the)h(assumed)f(lo)m(w)m(er)g(b)s(ounds)f (migh)m(t)h(b)s(e)g(unmotiv)-5 b(ated)28 b(at)j(this)d(p)s(oin)m(t,)h (but)g(it)g(will)d(b)s(ecome)-47 3098 y(clearer)k(later,)h(see)g(Lemma) f(6.8.)-47 3264 y Fm(Lemma)i(6.6.)47 b Fl(L)-5 b(et)1362 3490 y Fy(\036)p Fz(\()p Fy(\025)p Fz(\))26 b(=)1671 3428 y(1)p 1671 3469 46 4 v 1671 3552 a(2)1727 3490 y Fy(\025)1780 3452 y Fq(2)1839 3490 y Fu(\000)1965 3376 y Fw(m)1930 3403 y Fn(X)1935 3599 y Fw(j)t Fq(=1)2077 3490 y Fy(b)2116 3504 y Fw(j)2152 3382 y Fn(q)p 2243 3382 293 4 v 108 x Fy(\025)2296 3463 y Fq(2)2356 3490 y Fz(+)20 b Fy(\033)2499 3504 y Fw(j)-47 3763 y Fl(with)35 b(arbitr)-5 b(ary)36 b Fy(b)572 3777 y Fw(j)636 3763 y Fy(>)28 b Fz(0)34 b Fl(and)h Fy(\033)1044 3777 y Fw(j)1108 3763 y Fu(\025)28 b Fz(0)p Fl(,)34 b(and)h(supp)-5 b(ose)36 b Fy(!)s Fz(\()p Fy(\025)p Fz(\))28 b Fu(\025)g Fz(0)34 b Fl(is)g(a)g(twic)-5 b(e)35 b(di\013er)-5 b(entiable,)35 b(monotone)h(function)-47 3876 y(with)1603 4054 y Fu(j)p Fy(!)1688 4017 y Fq(\()p Fw(j)t Fq(\))1780 4054 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)26 b(\024)f Fy(a)2098 4068 y Fw(j)2150 4054 y Fy(\025)2203 4017 y Fx(\000)p Fw(j)-47 4233 y Fl(for)33 b Fy(j)e Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fy(;)g Fz(2)p Fl(.)60 4399 y(1.)46 b(Supp)-5 b(ose)34 b(a)f(critic)-5 b(al)34 b(p)-5 b(oint)34 b Fy(\025)1194 4413 y Fq(0)1258 4399 y Fy(>)25 b Fz(0)33 b Fl(of)g Fy(\036)g Fl(exists,)g(and)g(assume) h(that)1239 4587 y Fy(\036)1293 4550 y Fx(0)1317 4587 y Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)f Fy(A)p Fz(\()p Fy(\025)c Fu(\000)e Fy(\025)1882 4601 y Fq(0)1922 4587 y Fz(\))65 b Fl(for)34 b(al)5 b(l)65 b Fy(\025)25 b(>)g(\025)2557 4601 y Fq(0)2617 4587 y Fz(+)20 b Fy(A)2776 4550 y Fx(\000)2841 4522 y Fr(1)p 2841 4534 31 3 v 2841 4576 a(2)180 4587 y Fz(\(6.17\))180 4766 y Fl(for)33 b(some)h Fy(A)25 b(>)g Fz(0)p Fl(.)42 b(Then)1466 4857 y Fn(\014)1466 4912 y(\014)1466 4966 y(\014)1466 5021 y(\014)1496 4865 y(Z)1587 4891 y Fx(1)1547 5071 y Fw(\025)1588 5080 y Fr(0)1677 4989 y Fy(e)1719 4951 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1886 4989 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)2184 4857 y Fn(\014)2184 4912 y(\014)2184 4966 y(\014)2184 5021 y(\014)2240 4989 y Fu(\024)25 b Fy(C)2401 5003 y Fq(1)2456 4989 y Fy(A)2524 4951 y Fx(\000)2589 4924 y Fr(1)p 2589 4936 V 2589 4977 a(2)2634 4989 y Fy(:)-2479 b Fz(\(6.18\))180 5223 y Fl(If)32 b Fy(\025)328 5237 y Fq(0)393 5223 y Fy(>)25 b Fz(0)33 b Fl(do)-5 b(es)34 b(not)f(exist,)g(then)40 b Fz(\(6.17\))35 b Fl(with)e Fy(\025)1891 5237 y Fq(0)1956 5223 y Fz(=)25 b(0)33 b Fl(implies)41 b Fz(\(6.18\))35 b Fl(with)e Fy(\025)2964 5237 y Fq(0)3029 5223 y Fz(=)25 b(0)p Fl(.)1903 5472 y Fz(48)p eop %%Page: 49 49 49 48 bop 60 274 a Fl(2.)46 b(L)-5 b(et)33 b Fy(\034)i(>)25 b Fz(0)32 b Fl(and)i(supp)-5 b(ose)33 b(the)g(critic)-5 b(al)33 b(p)-5 b(oint)34 b Fy(\025)1832 288 y Fq(0)1897 274 y Fy(>)24 b Fz(0)33 b Fl(exists)g(and)g(satis\014es)g Fy(\025)2889 288 y Fq(0)2954 274 y Fy(<)3050 208 y Fu(p)p 3125 208 51 4 v 3125 274 a Fy(\034)10 b Fl(.)42 b(Assume)32 b(the)h(lower)180 387 y(b)-5 b(ounds)1088 625 y Fy(\036)1142 587 y Fx(0)1165 625 y Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)1410 497 y Fn(\032)1520 568 y Fy(c)15 b(R)1644 535 y Fx(\000)p Fq(1)1739 568 y Fy(\025)1792 535 y Fq(2)1831 568 y Fz(\()p Fy(\025)21 b Fu(\000)e Fy(\025)2083 582 y Fq(0)2123 568 y Fz(\))83 b Fl(for)h Fy(\025)2490 582 y Fq(0)2555 568 y Fu(\024)25 b Fy(\025)g Fu(\024)2825 491 y(p)p 2901 491 70 4 v 77 x Fy(R)1520 686 y(c)p Fz(\()p Fy(\025)c Fu(\000)f Fy(\025)1812 700 y Fq(0)1851 686 y Fz(\))355 b Fl(for)84 b Fy(\025)25 b Fu(\025)2611 609 y(p)p 2687 609 V 77 x Fy(R)180 625 y Fz(\(6.19\))180 893 y Fl(with)34 b(c)-5 b(onstants)34 b Fz(1)26 b Fy(>)f(c)g(>)g Fz(0)33 b Fl(and)h Fy(R)26 b Fu(\025)f Fy(\025)1603 860 y Fq(2)1603 918 y(0)1642 893 y Fl(.)42 b(Then)1100 1035 y Fn(\014)1100 1089 y(\014)1100 1144 y(\014)1100 1198 y(\014)1130 1043 y(Z)1221 1069 y Fx(1)1181 1249 y Fw(\025)1222 1258 y Fr(0)1311 1167 y Fy(e)1353 1129 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1667 1105 y Fy(\025)p 1529 1146 330 4 v 1529 1164 a Fu(p)p 1605 1164 254 4 v 77 x Fy(\025)1658 1214 y Fq(2)1718 1241 y Fz(+)20 b Fy(\034)1884 1167 y(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)2182 1035 y Fn(\014)2182 1089 y(\014)2182 1144 y(\014)2182 1198 y(\014)2238 1167 y Fu(\024)25 b Fy(C)2399 1181 y Fq(1)2454 1066 y Fn(h)2497 1167 y Fy(c)2536 1129 y Fx(\000)2601 1102 y Fr(3)p 2601 1114 31 3 v 2601 1155 a(2)2666 1167 y Fz(+)2757 1022 y Fn(r)p 2848 1022 110 4 v 2867 1105 a Fy(R)p 2858 1146 90 4 v 2858 1229 a(c\034)2957 1066 y Fn(i)3000 1167 y Fy(:)-2845 b Fz(\(6.20\))180 1424 y Fl(If)32 b Fy(\025)328 1438 y Fq(0)393 1424 y Fy(>)25 b Fz(0)33 b Fl(do)-5 b(es)34 b(not)f(exist,)g(then)g(the)g(assumption)42 b Fz(\(6.19\))34 b Fl(with)g Fy(\025)2521 1438 y Fq(0)2586 1424 y Fz(=)25 b(0)33 b Fl(implies)40 b Fz(\(6.20\))35 b Fl(with)f Fy(\025)3594 1438 y Fq(0)3658 1424 y Fz(=)25 b(0)p Fl(.)-47 1609 y(In)32 b(al)5 b(l)34 b(c)-5 b(ases)33 b Fy(C)499 1623 y Fq(1)571 1609 y Fl(only)g(dep)-5 b(ends)34 b(on)g(the)f(c)-5 b(onstants)34 b Fy(a)1836 1623 y Fq(0)1876 1609 y Fy(;)15 b(a)1964 1623 y Fq(1)2004 1609 y Fy(;)g(a)2092 1623 y Fq(2)2131 1609 y Fl(.)-47 1795 y(Pr)-5 b(o)g(of.)47 b Fz(Setting)31 b Fy(\025)605 1809 y Fq(0)672 1795 y Fz(=)26 b(0)32 b(if)f(a)h(p)s (ositiv)m(e)e(critical)h(p)s(oin)m(t)f(do)s(es)h(not)h(exist)f(allo)m (ws)g(us)g(to)h(ignore)f(the)h(issue)e(whether)-47 1908 y(or)i(not)h(suc)m(h)f(a)g(p)s(oin)m(t)g(exists.)46 b(Let)33 b Fy(\037)f Fz(b)s(e)g(a)g(smo)s(oth)h(non-decreasing)f(function)f (with)g Fy(\037)p Fz(\()p Fy(\025)p Fz(\))e(=)f(0)33 b(for)f Fy(\025)c Fu(\024)h Fz(1)j(and)-47 2021 y Fy(\037)p Fz(\()p Fy(\025)p Fz(\))26 b(=)f(1)30 b(for)g Fy(\025)c Fu(\025)f Fz(2.)41 b(With)30 b(an)g(arbitrary)f(parameter)i Fy(L)25 b(>)g Fz(0)31 b(de\014ne)1105 2222 y Fy(g)1148 2236 y Fw(L)1200 2222 y Fz(\()p Fy(\025)p Fz(\))26 b(:=)f Fy(\037)p Fz(\()1562 2141 y Fu(p)p 1638 2141 69 4 v 81 x Fy(A)p Fz(\()p Fy(\025)c Fu(\000)f Fy(\025)1959 2236 y Fq(0)1998 2222 y Fz(\)\)\(1)i Fu(\000)e Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))p Fy(!)s Fz(\()p Fy(\025)p Fz(\))p Fy(:)-47 2424 y Fz(Then)29 b(for)h(an)m(y)h Fy(j)g Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fy(;)g Fz(2,)1365 2640 y Fu(j)p Fy(g)1436 2592 y Fq(\()p Fw(j)t Fq(\))1433 2669 y Fw(L)1528 2640 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)84 b(\024)f Fy(C)22 b(A)2080 2572 y Fs(j)p 2079 2588 31 3 v 2079 2629 a Fr(2)2184 2640 y Fz(for)30 b(all)60 b Fy(\025)-2580 b Fz(\(6.21\))-47 2861 y(Clearly)-8 b(,)29 b Fy(\037)p Fz(\()378 2784 y Fu(p)p 454 2784 69 4 v 77 x Fy(A)q Fz(\()p Fy(\025)20 b Fu(\000)g Fy(\025)775 2875 y Fq(0)814 2861 y Fz(\)\))31 b(pro)m(vides)f(a)h(cut-o\013)g(to)g(an)f(in)m(terv)-5 b(al)30 b([)p Fy(\025)2280 2875 y Fq(0)2320 2861 y Fy(;)15 b(\025)2413 2875 y Fq(0)2473 2861 y Fz(+)20 b(2)p Fy(A)2677 2825 y Fx(\000)2742 2798 y Fr(1)p 2742 2810 31 3 v 2742 2851 a(2)2787 2861 y Fz(].)41 b(Th)m(us)648 2981 y Fn(\014)648 3036 y(\014)648 3090 y(\014)648 3145 y(\014)679 2989 y(Z)770 3015 y Fx(1)729 3195 y Fw(\025)770 3204 y Fr(0)859 3113 y Fy(e)901 3075 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1068 3113 y Fz(\(1)21 b Fu(\000)f Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))15 b Fy(!)s Fz(\()p Fy(\025)p Fz(\))g Fy(d\025)1895 2981 y Fn(\014)1895 3036 y(\014)1895 3090 y(\014)1895 3145 y(\014)1953 3113 y Fu(\024)25 b Fy(C)d(A)2204 3075 y Fx(\000)2269 3048 y Fr(1)p 2269 3060 V 2269 3101 a(2)2333 3113 y Fz(+)2424 2981 y Fn(\014)2424 3036 y(\014)2424 3090 y(\014)2424 3145 y(\014)2455 2989 y(Z)2545 3015 y Fx(1)2505 3195 y Fw(\025)2546 3204 y Fr(0)2635 3113 y Fy(e)2677 3075 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))2844 3113 y Fy(g)2887 3127 y Fw(L)2939 3113 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)3177 2981 y Fn(\014)3177 3036 y(\014)3177 3090 y(\014)3177 3145 y(\014)3224 3113 y Fy(:)-3296 b Fz(\(6.22\))-47 3364 y(Since)834 3497 y Fn(\020)925 3537 y Fy(d)p 899 3577 101 4 v 899 3661 a(d\025)1097 3537 y Fz(1)p 1019 3577 202 4 v 1019 3661 a Fy(\036)1073 3634 y Fx(0)1096 3661 y Fz(\()p Fy(\025)p Fz(\))1230 3497 y Fn(\021)1284 3520 y Fq(2)1349 3598 y Fz(=)25 b Fy(\036)1499 3561 y Fx(0)1522 3598 y Fz(\()p Fy(\025)p Fz(\))1645 3561 y Fx(\000)p Fq(2)1777 3537 y Fy(d)1824 3504 y Fq(2)p 1750 3577 140 4 v 1750 3661 a Fy(d\025)1850 3634 y Fq(2)1920 3598 y Fu(\000)20 b Fz(3)2077 3537 y Fy(\036)2131 3504 y Fx(00)2173 3537 y Fz(\()p Fy(\025)p Fz(\))p 2066 3577 241 4 v 2066 3661 a Fy(\036)2120 3634 y Fx(0)2144 3661 y Fz(\()p Fy(\025)p Fz(\))2267 3634 y Fq(3)2354 3537 y Fy(d)p 2327 3577 101 4 v 2327 3661 a(d\025)2458 3598 y Fu(\000)2585 3537 y Fy(d)p 2559 3577 V 2559 3661 a(d\025)2669 3497 y Fn(\020)2743 3537 y Fy(\036)2797 3504 y Fx(0)q(0)2840 3537 y Fz(\()p Fy(\025)p Fz(\))p 2733 3577 241 4 v 2733 3661 a Fy(\036)2787 3634 y Fx(0)2811 3661 y Fz(\()p Fy(\025)p Fz(\))2934 3634 y Fq(3)2984 3497 y Fn(\021)3038 3598 y Fy(;)-47 3849 y Fz(in)m(tegrating)30 b(b)m(y)g(parts)g(t)m(wice)h (sho)m(ws)f(that)h(the)g(in)m(tegral)f(on)g(the)h(righ)m(t-hand)e(side) g(of)38 b(\(6.22\))33 b(is)c(no)h(larger)g(than)457 3976 y Fn(Z)548 4003 y Fx(1)508 4182 y Fq(0)638 4100 y Fu(j)p Fy(\036)717 4062 y Fx(0)741 4100 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)889 4062 y Fx(\000)p Fq(2)999 4100 y Fu(j)p Fy(g)1070 4062 y Fx(0)q(0)1067 4122 y Fw(L)1120 4100 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)21 b Fz(+)f(3)1555 3976 y Fn(Z)1647 4003 y Fx(1)1607 4182 y Fq(0)1722 3995 y Fn(\014)1722 4050 y(\014)1722 4105 y(\014)1772 4038 y Fy(\036)1826 4005 y Fx(00)1869 4038 y Fz(\()p Fy(\025)p Fz(\))p 1762 4079 V 1762 4162 a Fy(\036)1816 4136 y Fx(0)1839 4162 y Fz(\()p Fy(\025)p Fz(\))1962 4136 y Fq(3)2012 3995 y Fn(\014)2012 4050 y(\014)2012 4105 y(\014)2058 4100 y Fu(j)p Fy(g)2129 4062 y Fx(0)2126 4122 y Fw(L)2179 4100 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)21 b Fz(+)2554 3976 y Fn(Z)2645 4003 y Fx(1)2605 4182 y Fq(0)2720 3995 y Fn(\014)2720 4050 y(\014)2720 4105 y(\014)2787 4038 y Fy(d)p 2760 4079 101 4 v 2760 4162 a(d\025)2871 3999 y Fn(h)2934 4038 y Fy(\036)2988 4005 y Fx(00)3030 4038 y Fz(\()p Fy(\025)p Fz(\))p 2923 4079 241 4 v 2923 4162 a Fy(\036)2977 4136 y Fx(0)3001 4162 y Fz(\()p Fy(\025)p Fz(\))3124 4136 y Fq(3)3174 3999 y Fn(i)3217 3995 y(\014)3217 4050 y(\014)3217 4105 y(\014)3262 4100 y Fu(j)p Fy(g)3330 4114 y Fw(L)3383 4100 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025:)-3718 b Fz(\(6.23\))-47 4377 y(Setting)32 b Fy(\015)i Fz(=)28 b Fy(A)516 4341 y Fx(\000)581 4314 y Fr(1)p 581 4326 31 3 v 581 4368 a(2)658 4377 y Fz(for)k(con)m(v)m (enience,)j(\(6.21\))f(and)e(\(6.17\))j(imply)30 b(that)j(the)f (\014rst)g(term)h(on)f(the)h(righ)m(t-hand)e(side)-47 4490 y(of)37 b(\(6.23\))c(do)s(es)d(not)h(exceed)969 4612 y Fn(Z)1060 4639 y Fx(1)1019 4819 y Fw(\025)1060 4828 y Fr(0)1095 4819 y Fq(+)p Fw(\015)1209 4736 y Fu(j)p Fy(A)p Fz(\()p Fy(\025)21 b Fu(\000)f Fy(\025)1555 4750 y Fq(0)1595 4736 y Fz(\))p Fu(j)1655 4699 y Fx(\000)p Fq(2)1765 4736 y Fy(\015)1817 4699 y Fx(\000)p Fq(2)1926 4736 y Fy(d\025)26 b Fu(\024)f Fy(C)d(A)2303 4699 y Fx(\000)p Fq(2)2397 4736 y Fy(\015)2449 4699 y Fx(\000)p Fq(3)2569 4736 y Fz(=)j Fy(C)c(A)2819 4699 y Fx(\000)2884 4671 y Fr(1)p 2884 4683 V 2884 4725 a(2)-47 4997 y Fz(uniformly)27 b(in)h Fy(L)p Fz(.)40 b(In)29 b(view)g(of)37 b(\(6.6\))32 b(in)d(Lemma)g(6.5)i(the)f(absolute)g(v)-5 b(alues)29 b(in)f(the)i(third)e(term)i(of)37 b(\(6.23\))32 b(can)e(b)s(e)-47 5110 y(tak)m(en)35 b(outside)e(the)i(in)m(tegral.)51 b(In)m(tegrating)35 b(b)m(y)f(parts)g(therefore)g(reduces)g(the)g (third)f(term)h(to)h(the)f(second.)52 b(As)-47 5223 y(for)32 b(the)i(latter,)g(Lemma)f(6.5)h(sho)m(ws)f(that)g Fy(\036)1523 5190 y Fx(0)q(0)1566 5223 y Fz(\()p Fy(\025)p Fz(\))p Fy(\036)1743 5190 y Fx(0)1767 5179 y(\000)p Fq(3)1861 5223 y Fz(\()p Fy(\025)p Fz(\))e Fy(>)e Fz(0)k(for)g(all)f Fy(\025)d(>)h(\025)2700 5237 y Fq(0)2739 5223 y Fz(.)48 b(Whereas)34 b Fy(g)3233 5190 y Fx(0)3230 5250 y Fw(L)3315 5223 y Fz(do)s(es)f(not)g(ha)m(v)m(e)h(a)1903 5472 y(49)p eop %%Page: 50 50 50 49 bop -47 274 a Fz(de\014nite)25 b(sign,)h(it)g(is)f(the)h(sum)g (of)g(three)h(terms)f(eac)m(h)h(of)f(whic)m(h)f(do)s(es)h(ha)m(v)m(e)i (a)e(de\014nite)f(sign,)i(sa)m(y)f Fy(g)3333 241 y Fx(0)3330 301 y Fw(L)3408 274 y Fz(=)f Fy(f)3549 288 y Fq(1)3600 274 y Fz(+)12 b Fy(f)3728 288 y Fq(2)3778 274 y Fz(+)g Fy(f)3906 288 y Fq(3)-47 387 y Fz(where)26 b Fy(f)257 402 y Fw(k)299 387 y Fz(\()p Fy(\025)p Fz(\))g Fu(\025)f Fz(0)i(for)g(all)f Fy(\025)f(>)g(\025)1102 401 y Fq(0)1168 387 y Fz(or)i Fy(f)1321 402 y Fw(k)1363 387 y Fz(\()p Fy(\025)p Fz(\))f Fu(\024)f Fz(0)j(for)e(all)g Fy(\025)f(>)g(\025)2166 401 y Fq(0)2206 387 y Fz(.)39 b(W)-8 b(e)28 b(can)f(therefore)h(mo)m(v) m(e)g(the)f(absolute)f(v)-5 b(alues)-47 500 y(outside)29 b(the)i(second)f(in)m(tegral)g(pro)m(vided)f(w)m(e)i(consider)f(eac)m (h)h(of)g(the)f Fy(f)2419 515 y Fw(k)2492 500 y Fz(separately)-8 b(.)41 b(Since)524 624 y Fn(Z)615 650 y Fx(1)574 830 y Fq(0)725 686 y Fy(\036)779 653 y Fx(00)821 686 y Fz(\()p Fy(\025)p Fz(\))p 715 727 241 4 v 715 810 a Fy(\036)769 784 y Fx(0)792 810 y Fz(\()p Fy(\025)p Fz(\))915 784 y Fq(3)980 748 y Fy(f)1025 762 y Fw(j)1061 748 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)85 b Fz(=)d Fu(\000)1623 624 y Fn(Z)1714 650 y Fx(1)1674 830 y Fq(0)1804 748 y Fy(\036)1858 710 y Fx(0)1882 748 y Fz(\()p Fy(\025)p Fz(\))2020 647 y Fn(\020)2103 686 y Fy(f)2148 700 y Fw(j)2184 686 y Fz(\()p Fy(\025)p Fz(\))p 2085 727 V 2085 810 a Fy(\036)2139 784 y Fx(0)2162 810 y Fz(\()p Fy(\025)p Fz(\))2285 784 y Fq(3)2335 647 y Fn(\021)2390 670 y Fx(0)2428 748 y Fy(d\025)1384 998 y Fz(=)g Fu(\000)1623 874 y Fn(Z)1714 900 y Fx(1)1674 1080 y Fq(0)1912 936 y Fz(1)p 1814 977 V 1814 1060 a Fy(\036)1868 1034 y Fx(0)1891 1060 y Fz(\()p Fy(\025)p Fz(\))2014 1034 y Fq(2)2080 998 y Fy(f)2135 960 y Fx(0)2125 1020 y Fw(j)2161 998 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)21 b Fz(+)f(3)2571 874 y Fn(Z)2663 900 y Fx(1)2622 1080 y Fq(0)2773 936 y Fy(\036)2827 903 y Fx(00)2869 936 y Fz(\()p Fy(\025)p Fz(\))p 2762 977 V 2762 1060 a Fy(\036)2816 1034 y Fx(0)2840 1060 y Fz(\()p Fy(\025)p Fz(\))2963 1034 y Fq(3)3028 998 y Fy(f)3073 1012 y Fw(j)3109 998 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025;)-47 1257 y Fz(and)32 b Fu(j)p Fy(f)212 1224 y Fx(0)202 1285 y Fw(k)245 1257 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)e(\024)g Fy(C)7 b(\015)648 1224 y Fx(\000)p Fq(2)775 1257 y Fz(for)33 b(eac)m(h)h Fy(k)i Fz(and)d(all)f Fy(\025)h Fz(\(see)h(\(6.21\))h(with) d Fy(j)k Fz(=)29 b(2\),)35 b(the)e(second)g(term)h(satis\014es)e(the)h (same)-47 1370 y(estimates)e(as)f(the)h(\014rst.)40 b(P)m(assing)30 b(to)h(the)g(limit)d Fy(L)d Fu(!)g(1)30 b Fz(in)f(\(6.22\))k(pro)m(v)m (es)e(\(6.18\))r(.)94 1483 y(As)f(for)g(the)h(second)f(part)g(of)g(the) h(lemma,)f(w)m(e)h(\014rst)e(consider)g(the)i(case)g Fy(R)26 b Fu(\025)f Fz(1.)41 b(With)30 b Fy(L)25 b(>)g Fz(0)30 b(arbitrary)f(and)-47 1596 y(some)i Fy(\015)f(>)25 b Fz(0)31 b(that)g(will)c(b)s(e)j(sp)s(eci\014ed)f(b)s(elo)m(w,)h(set) 1008 1837 y Fy(g)1051 1851 y Fw(L)1104 1837 y Fz(\()p Fy(\025)p Fz(\))c(:=)f Fy(\037)1431 1736 y Fn(\020)1495 1776 y Fy(\025)20 b Fu(\000)g Fy(\025)1712 1790 y Fq(0)p 1495 1816 257 4 v 1597 1900 a Fy(\015)1762 1736 y Fn(\021)1816 1837 y Fz(\(1)h Fu(\000)f Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))p Fy(!)s Fz(\()p Fy(\025)p Fz(\))2663 1776 y Fy(\025)p 2523 1816 330 4 v 2523 1835 a Fu(p)p 2601 1835 254 4 v 2601 1911 a Fy(\025)2654 1885 y Fq(2)2713 1911 y Fz(+)g Fy(\034)2864 1837 y(:)-47 2084 y Fz(Since)k Fy(\025)i Fu(\025)f Fy(\015)15 b Fz(+)c Fy(\025)557 2098 y Fq(0)622 2084 y Fu(\025)25 b Fy(\015)30 b Fz(on)c(the)g(supp)s(ort)e (of)h Fy(g)1542 2098 y Fw(L)1595 2084 y Fz(,)i(our)e(assumptions)f(on)h Fy(!)k Fz(and)c(Lemma)g(6.2)i(imply)c(the)j(deriv)-5 b(ativ)m(e)-47 2197 y(b)s(ounds)1435 2420 y Fu(j)p Fy(g)1506 2373 y Fq(\()p Fw(j)t Fq(\))1503 2449 y Fw(L)1598 2420 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)26 b(\024)f Fy(C)d(\015)2007 2383 y Fx(\000)p Fw(j)2262 2359 y Fy(\025)p 2124 2400 330 4 v 2124 2418 a Fu(p)p 2199 2418 254 4 v 2199 2494 a Fy(\025)2252 2468 y Fq(2)2312 2494 y Fz(+)e Fy(\034)-47 2420 y Fz(\(6.24\))-47 2667 y(for)30 b Fy(j)h Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fy(;)g Fz(2)32 b(and)e(uniformly)d(in)i Fy(L)d(>)f Fz(0.)41 b(By)30 b(de\014nition)e(of)j Fy(g)2188 2681 y Fw(L)2240 2667 y Fz(,)265 2795 y Fn(\014)265 2850 y(\014)265 2905 y(\014)265 2959 y(\014)295 2804 y(Z)386 2830 y Fx(1)346 3010 y Fw(\025)387 3019 y Fr(0)476 2927 y Fy(e)518 2890 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))848 2866 y Fy(\025)p 710 2906 330 4 v 710 2925 a Fu(p)p 785 2925 254 4 v 785 3001 a Fy(\025)838 2975 y Fq(2)898 3001 y Fz(+)20 b Fy(\034)1049 2927 y Fz(\(1)h Fu(\000)f Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))p Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)1861 2795 y Fn(\014)1861 2850 y(\014)1861 2905 y(\014)1861 2959 y(\014)1919 2927 y Fu(\024)25 b Fy(a)2063 2941 y Fq(0)2117 2804 y Fn(Z)2208 2830 y Fw(\025)2249 2839 y Fr(0)2284 2830 y Fq(+2)p Fw(\015)2168 3010 y(\025)2209 3019 y Fr(0)2582 2866 y Fy(\025)p 2444 2906 330 4 v 2444 2925 a Fu(p)p 2520 2925 254 4 v 76 x Fy(\025)2573 2975 y Fq(2)2632 3001 y Fz(+)20 b Fy(\034)2798 2927 y(d\025)h Fz(+)3010 2795 y Fn(\014)3010 2850 y(\014)3010 2905 y(\014)3010 2959 y(\014)3040 2804 y(Z)3131 2830 y Fx(1)3091 3010 y Fw(\025)3132 3019 y Fr(0)3221 2927 y Fy(e)3263 2890 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))3460 2927 y Fy(g)3503 2941 y Fw(L)3555 2927 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)3793 2795 y Fn(\014)3793 2850 y(\014)3793 2905 y(\014)3793 2959 y(\014)3840 2927 y Fy(:)-3912 b Fz(\(6.25\))-47 3186 y(The)30 b(\014rst)f(in)m(tegral)h(on)h(the)f(righ)m(t-hand)f (side)h(of)37 b(\(6.25\))c(is)1074 3355 y Fn(p)p 1164 3355 573 4 v 1164 3437 a Fz(\()p Fy(\025)1252 3451 y Fq(0)1312 3437 y Fz(+)20 b(2)p Fy(\015)5 b Fz(\))1535 3411 y Fq(2)1596 3437 y Fz(+)20 b Fy(\034)30 b Fu(\000)1848 3327 y Fn(q)p 1939 3327 254 4 v 110 x Fy(\025)1992 3406 y Fq(2)1992 3464 y(0)2052 3437 y Fz(+)20 b Fy(\034)35 b Fu(\024)25 b Fy(C)2410 3376 y(\015)5 b Fz(\()p Fy(\015)26 b Fz(+)20 b Fy(\025)2714 3390 y Fq(0)2754 3376 y Fz(\))p 2410 3417 379 4 v 2537 3435 a Fu(p)p 2612 3435 51 4 v 2612 3500 a Fy(\034)2799 3437 y(:)-2871 b Fz(\(6.26\))-47 3689 y(This)30 b(b)s(ound)g(is)h(w)m(asteful)h(if)f Fy(\015)5 b Fz(\()p Fy(\015)27 b Fz(+)21 b Fy(\025)1284 3703 y Fq(0)1324 3689 y Fz(\))28 b Fy(>)g(\034)42 b Fz(but)32 b(that)g(do)s(es)g(not)g(concern)h(us.)45 b(In)m(tegrating)33 b(b)m(y)f(parts)g(t)m(wice)g(in)-47 3802 y(the)f(in)m(tegral)f(on)h (the)g(righ)m(t-hand)e(side)h(of)38 b(\(6.25\))33 b(and)d(rep)s(eating) g(the)h(argumen)m(ts)g(that)g(reduced)f(\(6.23\))j(to)f(the)-47 3915 y(\014rst)d(term,)i(allo)m(ws)f(us)f(to)j(b)s(ound)c(the)i (\014nal)f(term)i(in)e(\(6.25\))k(b)m(y)512 4073 y Fn(Z)603 4043 y Fx(p)p 662 4043 54 3 v 56 x Fw(R)p Fx(_)p Fq(\()p Fw(\025)830 4108 y Fr(0)866 4099 y Fq(+)p Fw(\015)t Fq(\))563 4279 y Fw(\025)604 4288 y Fr(0)638 4279 y Fq(+)p Fw(\015)1118 4135 y Fy(c)1157 4102 y Fx(\000)p Fq(2)1267 4135 y Fy(R)1337 4102 y Fq(2)p 1017 4175 460 4 v 1017 4259 a Fy(\025)1070 4232 y Fq(4)1110 4259 y Fz(\()p Fy(\025)21 b Fu(\000)e Fy(\025)1362 4273 y Fq(0)1402 4259 y Fz(\))1437 4232 y Fq(2)1487 4196 y Fy(\015)1539 4159 y Fx(\000)p Fq(2)1679 4135 y Fy(\025)p 1643 4175 126 4 v 1643 4194 a Fu(p)p 1719 4194 51 4 v 65 x Fy(\034)1794 4196 y(d\025)i Fz(+)e Fy(c)2044 4159 y Fx(\000)p Fq(2)2154 4073 y Fn(Z)2245 4099 y Fx(1)2205 4223 y(p)p 2264 4223 54 3 v 56 x Fw(R)p Fx(_)p Fq(\()p Fw(\025)2432 4288 y Fr(0)2467 4279 y Fq(+)p Fw(\015)t Fq(\))2780 4135 y Fz(1)p 2619 4175 367 4 v 2619 4259 a(\()p Fy(\025)i Fu(\000)f Fy(\025)2872 4273 y Fq(0)2911 4259 y Fz(\))2946 4232 y Fq(2)3011 4196 y Fy(\015)3063 4159 y Fx(\000)p Fq(2)3306 4135 y Fy(\025)p 3167 4175 330 4 v 3167 4194 a Fu(p)p 3243 4194 254 4 v 76 x Fy(\025)3296 4244 y Fq(2)3356 4270 y Fz(+)g Fy(\034)3522 4196 y(d\025)358 4466 y Fo(.)83 b Fy(c)551 4428 y Fx(\000)p Fq(2)774 4404 y Fy(R)844 4371 y Fq(2)883 4404 y Fy(\015)935 4371 y Fx(\000)p Fq(3)p 656 4445 492 4 v 656 4529 a Fz(\()p Fy(\025)744 4543 y Fq(0)804 4529 y Fz(+)20 b Fy(\015)5 b Fz(\))982 4502 y Fq(3)1022 4463 y Fu(p)p 1097 4463 51 4 v 1097 4529 a Fy(\034)1177 4466 y Fz(+)20 b Fy(c)1307 4428 y Fx(\000)p Fq(2)1402 4466 y Fy(R)1472 4428 y Fx(\000)1537 4401 y Fr(1)p 1537 4413 31 3 v 1537 4454 a(2)1581 4466 y Fy(\015)1633 4428 y Fx(\000)p Fq(2)1728 4466 y Fy(:)-1800 b Fz(\(6.27\))-47 4719 y(One)34 b(no)m(w)g(c)m(ho)s(oses)h Fy(\015)40 b Fz(in)33 b(suc)m(h)h(a)g(w)m(a)m(y)i(that)f(the)f(\014rst) g(term)g(in)f(\(6.27\))k(equals)d(the)g(righ)m(t-hand)f(side)h(of)41 b(\(6.26\))s(.)-47 4831 y(This)28 b(leads)i(to)1263 5099 y Fy(\015)g Fz(=)1436 4944 y Fn(\()1551 5025 y Fz(\()p Fy(R)q(=c)p Fz(\))1785 4962 y Fr(1)p 1786 4974 V 1786 5016 a(4)1914 5025 y Fz(if)107 b(\()p Fy(R)q(=c)p Fz(\))2309 4962 y Fr(1)p 2310 4974 V 2310 5016 a(4)2380 5025 y Fy(>)25 b(\025)2529 5039 y Fq(0)1561 5067 y Fu(p)p 1637 5067 120 4 v 62 x Fw(R=c)p 1561 5161 196 4 v 1620 5213 a(\025)1661 5222 y Fr(0)1914 5182 y Fz(if)82 b(\()p Fy(R)q(=c)p Fz(\))2284 5119 y Fr(1)p 2285 5131 31 3 v 2285 5172 a(4)2355 5182 y Fu(\024)25 b Fy(\025)2504 5196 y Fq(0)2543 5182 y Fy(:)-47 5099 y Fz(\(6.28\))1903 5472 y(50)p eop %%Page: 51 51 51 50 bop -47 278 a Fz(Note)33 b(that)f Fy(R)c Fu(\025)f Fz(1)32 b(implies)d(that)j Fy(\015)g Fu(\025)27 b Fy(c)1368 242 y Fx(\000)1433 215 y Fr(1)p 1433 227 31 3 v 1433 268 a(4)1509 278 y Fz(in)k(the)g(\014rst)g(case,)i(whereas)f(in)e(the)i (second)g(case)g Fy(\015)h Fu(\025)27 b Fy(c)3498 242 y Fx(\000)3563 215 y Fr(1)p 3563 227 V 3563 268 a(2)3639 278 y Fz(b)s(ecause)-47 391 y(of)66 314 y Fu(p)p 141 314 70 4 v 141 391 a Fy(R)42 b Fu(\025)e Fy(\025)416 405 y Fq(0)455 391 y Fz(.)69 b(Therefore,)42 b(the)d(\014nal)g(term)g (in)g(\(6.27\))i(is)e(at)h(most)g Fy(c)2480 355 y Fx(\000)2545 328 y Fr(3)p 2546 340 31 3 v 2546 381 a(2)2590 391 y Fz(.)68 b(Inserting)39 b(the)g(preceding)g(b)s(ounds)-47 504 y(in)m(to)30 b(\(6.25\))j(sho)m(ws)d(that)758 646 y Fn(\014)758 701 y(\014)758 755 y(\014)758 810 y(\014)788 654 y(Z)879 680 y Fx(1)838 860 y Fw(\025)879 869 y Fr(0)969 778 y Fy(e)1011 740 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1340 716 y Fy(\025)p 1202 757 330 4 v 1202 775 a Fu(p)p 1278 775 254 4 v 77 x Fy(\025)1331 825 y Fq(2)1391 852 y Fz(+)20 b Fy(\034)1542 778 y Fz(\(1)h Fu(\000)f Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))p Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)2354 646 y Fn(\014)2354 701 y(\014)2354 755 y(\014)2354 810 y(\014)2412 778 y Fu(\024)24 b Fy(C)2594 677 y Fn(h)2637 633 y(r)p 2728 633 110 4 v 2748 716 a Fy(R)p 2738 757 90 4 v 2738 840 a(c\034)2857 778 y Fz(+)c Fy(c)2987 740 y Fx(\000)3052 713 y Fr(3)p 3053 725 31 3 v 3053 766 a(2)3097 677 y Fn(i)-47 778 y Fz(\(6.29\))-47 1037 y(for)30 b(b)s(oth)g(c)m(hoices)h(of)f Fy(\015)36 b Fz(in)29 b(\(6.28\))r(,)i(pro)m(vided)d Fy(R)e Fu(\025)f Fz(1.)42 b(Finally)-8 b(,)29 b(if)g Fy(R)d Fu(\024)f Fz(1,)31 b(then)f(also)g Fy(\025)3013 1051 y Fq(0)3078 1037 y Fu(\024)25 b Fz(1.)41 b(Hence)369 1160 y Fn(\014)369 1214 y(\014)369 1269 y(\014)369 1323 y(\014)399 1168 y(Z)490 1194 y Fx(1)450 1374 y Fw(\025)491 1383 y Fr(0)580 1291 y Fy(e)622 1254 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))952 1230 y Fy(\025)p 814 1271 330 4 v 814 1289 a Fu(p)p 889 1289 254 4 v 889 1365 a Fy(\025)942 1339 y Fq(2)1002 1365 y Fz(+)20 b Fy(\034)1153 1291 y Fz(\(1)h Fu(\000)f Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))p Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)1965 1160 y Fn(\014)1965 1214 y(\014)1965 1269 y(\014)1965 1323 y(\014)2023 1291 y Fu(\024)2119 1160 y Fn(\014)2119 1214 y(\014)2119 1269 y(\014)2119 1323 y(\014)2149 1168 y(Z)2240 1194 y Fx(1)2200 1374 y Fw(\025)2241 1383 y Fr(0)2330 1291 y Fy(e)2372 1254 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))2702 1230 y Fy(\025)p 2564 1271 330 4 v 2564 1289 a Fu(p)p 2639 1289 254 4 v 2639 1365 a Fy(\025)2692 1339 y Fq(2)2752 1365 y Fz(+)20 b Fy(\034)2903 1291 y Fz(\(1)h Fu(\000)f Fy(\037)p Fz(\()p 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Fq(2)1463 1799 y Fu(j)p Fy(g)1534 1762 y Fx(0)q(0)1531 1822 y Fw(L)1584 1799 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)-1894 b Fz(\(6.31\))-47 2063 y(where)41 b Fy(g)270 2077 y Fw(L)322 2063 y Fz(\()p Fy(\025)p Fz(\))j(:=)733 2027 y Fw(\025)p 639 2042 229 4 v 639 2052 a Fx(p)p 698 2052 170 3 v 55 x Fw(\025)739 2088 y Fr(2)774 2107 y Fq(+)p Fw(\034)877 2063 y Fy(\037)p Fz(\()p Fy(c\025)p Fz(\)\(1)30 b Fu(\000)d Fy(\037)p Fz(\()p Fy(\025=L)p Fz(\)\))p Fy(!)s Fz(\()p Fy(\025)p Fz(\).)76 b(T)-8 b(o)41 b(pass)g(to)i(\(6.31\))g(w)m(e)f (used)f(the)g(lo)m(w)m(er)h(b)s(ound)d(on)i Fy(\036)3921 2030 y Fx(0)-47 2193 y Fz(from)36 b(\(6.19\))j(as)e(w)m(ell)f(as)h(the) g(usual)e(reduction)h(argumen)m(ts)h(in)m(v)m(olving)f(\(6.23\))s(.)60 b(Since)35 b Fu(j)p Fy(g)3165 2160 y Fx(0)q(0)3162 2220 y Fw(L)3215 2193 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)i Fo(.)f Fz(1,)j(sa)m(y)-8 b(,)39 b(and)-47 2306 y Fy(\025)6 2320 y Fq(0)70 2306 y Fu(\024)25 b Fz(1,)k(\(6.31\))g(is)d(at)i(most)f Fy(c)978 2273 y Fx(\000)p Fq(1)1100 2306 y Fz(uniformly)d(in)i Fy(L)p Fz(.)39 b(P)m(assing)27 b(to)h(the)f(limit)e Fy(L)g Fu(!)g(1)i Fz(in)f(\(6.29\))j(and)e(\(6.30\))i(\014nishes)-47 2419 y(the)h(pro)s(of.)p 3879 2419 4 62 v 3883 2361 55 4 v 3883 2419 V 3937 2419 4 62 v -47 2607 a(W)-8 b(e)31 b(no)m(w)g(turn)e(to)i(the)g(con)m(tribution)e(of)h(the)h(in)m(terv)-5 b(al)29 b([0)p Fy(;)15 b(\025)2038 2621 y Fq(0)2079 2607 y Fz(])30 b(to)h(the)g(oscillatory)f(in)m(tegral.)-47 2794 y Fm(Lemma)i(6.7.)47 b Fl(L)-5 b(et)36 b Fy(\036)p Fz(\()p Fy(\025)p Fz(\))p Fy(;)15 b(!)s Fz(\()p Fy(\025)p Fz(\))39 b Fl(b)-5 b(e)36 b(as)h(in)g(the)g(pr)-5 b(evious)37 b(lemma)h(and)f(supp)-5 b(ose)39 b(that)e Fy(\036)g Fl(has)g(a)g(p)-5 b(ositive)38 b(critic)-5 b(al)-47 2907 y(p)g(oint)34 b Fy(\025)236 2921 y Fq(0)275 2907 y Fl(.)42 b(If)962 3121 y Fu(j)p Fy(\036)1041 3084 y Fx(0)1064 3121 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)27 b(\025)e Fy(B)5 b(\025)p Fz(\()p Fy(\025)1550 3135 y Fq(0)1609 3121 y Fu(\000)20 b Fy(\025)p Fz(\))66 b Fl(for)33 b(al)5 b(l)65 b Fz(\()p Fy(B)5 b(\025)2323 3135 y Fq(0)2363 3121 y Fz(\))2398 3084 y Fx(\000)2463 3057 y Fr(1)p 2463 3069 31 3 v 2463 3110 a(2)2533 3121 y Fy(<)25 b(\025)g(<)g(\025)2856 3135 y Fq(0)2895 3121 y Fy(;)-2967 b Fz(\(6.32\))-47 3326 y Fl(then)894 3437 y Fn(\014)894 3492 y(\014)894 3546 y(\014)894 3601 y(\014)924 3445 y(Z)1015 3471 y Fw(\025)1056 3480 y Fr(0)974 3651 y Fq(0)1110 3569 y Fy(e)1152 3531 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1467 3507 y Fy(\025)p 1328 3548 330 4 v 1328 3566 a Fu(p)p 1404 3566 254 4 v 77 x Fy(\025)1457 3617 y Fq(2)1517 3643 y Fz(+)20 b Fy(\034)1683 3569 y(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)1981 3437 y Fn(\014)1981 3492 y(\014)1981 3546 y(\014)1981 3601 y(\014)2038 3569 y Fu(\024)24 b Fy(C)2198 3583 y Fq(1)2253 3569 y Fz(\()p Fy(B)5 b(\025)2415 3583 y Fq(0)2454 3569 y Fz(\))2489 3531 y Fx(\000)2554 3504 y Fr(1)p 2555 3516 31 3 v 2555 3557 a(2)2750 3507 y Fy(\025)2803 3521 y Fq(0)p 2624 3548 345 4 v 2624 3566 a Fn(p)p 2715 3566 254 4 v 79 x Fy(\025)2768 3614 y Fq(2)2768 3671 y(0)2828 3645 y Fz(+)20 b Fy(\034)2979 3569 y(:)-3051 b Fz(\(6.33\))-47 3834 y Fl(The)33 b(c)-5 b(onstant)34 b Fy(C)566 3848 y Fq(1)638 3834 y Fl(only)f(dep)-5 b(ends)35 b(on)e Fy(!)s Fl(.)-47 4022 y(Pr)-5 b(o)g(of.)47 b Fz(As)30 b(in)g(the)h(previous)e (lemma,)i(w)m(e)g(will)e(cut)i(o\013)g(in)m(terv)-5 b(als)30 b(of)h(size)f Fy(\015)36 b Fz(from)31 b(the)g(endp)s(oin)m(ts)e(0)i (and)f Fy(\025)3728 4036 y Fq(0)3798 4022 y Fz(and)-47 4135 y(in)m(tegrate)35 b(b)m(y)f(parts)g(b)s(et)m(w)m(een)g Fy(\015)39 b Fz(and)34 b Fy(\025)1379 4149 y Fq(0)1441 4135 y Fu(\000)22 b Fy(\015)5 b Fz(.)52 b(More)35 b(precisely)-8 b(,)34 b(w)m(e)g(assume)g(that)h(\()p Fy(B)5 b(\025)3114 4149 y Fq(0)3153 4135 y Fz(\))3188 4099 y Fx(\000)3253 4072 y Fr(1)p 3253 4084 31 3 v 3253 4125 a(2)3329 4135 y Fu(\024)31 b Fy(\015)37 b Fu(\024)3627 4099 y Fq(1)p 3627 4114 36 4 v 3627 4166 a(2)3672 4135 y Fy(\025)3725 4149 y Fq(0)3798 4135 y Fz(and)-47 4248 y(de\014ne)1086 4472 y Fy(h)p Fz(\()p Fy(\025)p Fz(\))26 b(:=)f Fy(\037)p Fz(\()p Fy(\025=\015)5 b Fz(\))p Fy(\037)p Fz(\(\()p Fy(\025)1865 4486 y Fq(0)1927 4472 y Fu(\000)19 b Fy(\025)p Fz(\))p Fy(=\015)5 b Fz(\))2387 4411 y Fy(\025)p 2247 4451 330 4 v 2247 4469 a Fu(p)p 2325 4469 254 4 v 2325 4546 a Fy(\025)2378 4520 y Fq(2)2437 4546 y Fz(+)20 b Fy(\034)2603 4472 y(!)s Fz(\()p Fy(\025)p Fz(\))p Fy(:)-47 4738 y Fz(Since)289 4702 y Fw(\025)p 195 4717 229 4 v 195 4727 a Fx(p)p 254 4727 170 3 v 55 x Fw(\025)295 4764 y Fr(2)330 4782 y Fq(+)p Fw(\034)459 4738 y Fz(is)k(an)i(increasing)e (function)g(and)h Fy(\025)g Fu(\025)g Fy(\015)30 b Fz(on)c(the)f(supp)s (ort)f(of)i Fy(h)p Fz(,)h(Lemma)e(6.2)i(and)e(our)g(assumptions)-47 4862 y(on)30 b Fy(!)j Fz(imply)1432 5104 y Fu(j)p Fy(h)1509 5066 y Fq(\()p Fw(j)t Fq(\))1601 5104 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)26 b(\024)f Fy(C)d(\015)2010 5066 y Fx(\000)p Fw(j)2237 5042 y Fy(\025)2290 5056 y Fq(0)p 2111 5083 345 4 v 2111 5101 a Fn(p)p 2202 5101 254 4 v 79 x Fy(\025)2255 5149 y Fq(2)2255 5206 y(0)2315 5180 y Fz(+)e Fy(\034)-47 5104 y Fz(\(6.34\))1903 5472 y(51)p eop %%Page: 52 52 52 51 bop -47 274 a Fz(for)30 b Fy(j)h Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fy(;)g Fz(2.)43 b(By)30 b(the)h(de\014nition)d(of)i Fy(h)p Fz(,)580 386 y Fn(\014)580 440 y(\014)580 495 y(\014)580 549 y(\014)610 394 y(Z)701 420 y Fw(\025)742 429 y Fr(0)661 600 y Fq(0)796 517 y Fy(e)838 480 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1178 456 y Fy(\025)p 1040 496 330 4 v 1040 515 a Fu(p)p 1116 515 254 4 v 76 x Fy(\025)1169 565 y Fq(2)1228 591 y Fz(+)20 b Fy(\034)1394 517 y(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)1692 386 y Fn(\014)1692 440 y(\014)1692 495 y(\014)1692 549 y(\014)1749 517 y Fu(\024)25 b Fz(2)p Fy(a)1938 531 y Fq(0)1978 517 y Fy(\015)2166 456 y(\025)2219 470 y Fq(0)p 2040 496 345 4 v 2040 515 a Fn(p)p 2131 515 254 4 v 79 x Fy(\025)2184 562 y Fq(2)2184 620 y(0)2243 594 y Fz(+)20 b Fy(\034)2415 517 y Fz(+)2505 386 y Fn(\014)2505 440 y(\014)2505 495 y(\014)2505 549 y(\014)2536 394 y(Z)2627 420 y Fw(\025)2668 429 y Fr(0)2586 600 y Fq(0)2722 517 y Fy(e)2764 480 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))2956 517 y Fy(h)p Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)3246 386 y Fn(\014)3246 440 y(\014)3246 495 y(\014)3246 549 y(\014)3293 517 y Fy(:)-3365 b Fz(\(6.35\))-47 765 y(In)m(tegrating)31 b(b)m(y)f(parts)g(t)m(wice)h(in)e(the)i(last)f (terms)g(yields)f(the)h(upp)s(er)f(b)s(ound)481 873 y Fn(Z)572 899 y Fx(1)532 1079 y Fq(0)662 997 y Fu(j)p Fy(\036)741 959 y Fx(0)765 997 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)913 959 y Fx(\000)p Fq(2)1023 997 y Fu(j)p Fy(h)1100 959 y Fx(0)q(0)1143 997 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)22 b Fz(+)e(3)1579 873 y Fn(Z)1670 899 y Fx(1)1630 1079 y Fq(0)1745 892 y Fn(\014)1745 947 y(\014)1745 1001 y(\014)1796 935 y Fy(\036)1850 902 y Fx(00)1892 935 y Fz(\()p Fy(\025)p Fz(\))p 1785 976 241 4 v 1785 1059 a Fy(\036)1839 1033 y Fx(0)1863 1059 y Fz(\()p Fy(\025)p Fz(\))1986 1033 y Fq(3)2036 892 y Fn(\014)2036 947 y(\014)2036 1001 y(\014)2082 997 y Fu(j)p Fy(h)2159 959 y Fx(0)2182 997 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)22 b Fz(+)2558 873 y Fn(Z)2649 899 y Fx(1)2609 1079 y Fq(0)2724 892 y Fn(\014)2724 947 y(\014)2724 1001 y(\014)2790 935 y Fy(d)p 2764 976 101 4 v 2764 1059 a(d\025)2874 896 y Fn(h)2937 935 y Fy(\036)2991 902 y Fx(0)q(0)3034 935 y Fz(\()p Fy(\025)p Fz(\))p 2927 976 241 4 v 2927 1059 a Fy(\036)2981 1033 y Fx(0)3005 1059 y Fz(\()p Fy(\025)p Fz(\))3128 1033 y Fq(3)3178 896 y Fn(i)3221 892 y(\014)3221 947 y(\014)3221 1001 y(\014)3266 997 y Fu(j)p Fy(h)p Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025;)-3678 b Fz(\(6.36\))-47 1232 y(see)30 b(\(6.23\))r(.)40 b(By)31 b(\(6.6\))h(in)c(Lemma)h(6.5,)i(the)f(absolute)f(v)-5 b(alues)28 b(in)h(the)g(third)f(term)h(can)h(b)s(e)e(tak)m(en)j (outside,)e(whic)m(h)-47 1345 y(reduces)k(it)h(to)h(the)f(second)g (term)g(in)f(\(6.36\))r(.)52 b(T)-8 b(o)34 b(deal)g(with)f(the)h (second)g(term,)h(recall)e(from)h(Lemma)g(6.5)h(that)-47 1458 y(there)d(can)g(b)s(e)f(at)i(most)f(one)g(zero)h(of)f Fy(\036)1338 1425 y Fx(00)1413 1458 y Fz(on)f(the)h(supp)s(ort)f(of)h Fy(h)p Fz(,)h(whic)m(h)d(w)m(e)j(denote)f(b)m(y)g Fy(\025)3124 1472 y Fq(1)3195 1458 y Fz(\(if)f(there)h(is)f(no)h(suc)m(h)-47 1571 y(zero,)38 b(then)e(apply)e(the)j(follo)m(wing)d(argumen)m(t)i (with)f Fy(\025)1875 1585 y Fq(1)1949 1571 y Fz(=)f(0\).)59 b(Since)34 b Fy(\036)2514 1538 y Fx(0)q(0)2557 1571 y Fz(\()p Fy(\025)p Fz(\))h Fu(\025)g Fz(0)h(for)g Fy(\025)e Fu(\025)h Fy(\025)3293 1585 y Fq(1)3332 1571 y Fz(,)j Fy(\036)3449 1538 y Fx(00)3491 1571 y Fz(\()p Fy(\025)p Fz(\))d Fu(\024)g Fz(0)h(for)-47 1684 y Fy(\025)25 b Fu(\024)g Fy(\025)180 1698 y Fq(1)219 1684 y Fz(,)31 b(and)f Fy(\036)506 1651 y Fx(0)529 1684 y Fz(\()p Fy(\025)p Fz(\))c Fu(\024)f Fz(0)31 b(on)f(the)h(supp)s(ort)d(of)j Fy(h)p Fz(,)532 1808 y Fn(Z)622 1835 y Fx(1)582 2014 y Fq(0)697 1827 y Fn(\014)697 1882 y(\014)697 1936 y(\014)748 1870 y Fy(\036)802 1837 y Fx(00)844 1870 y Fz(\()p Fy(\025)p Fz(\))p 737 1911 V 737 1994 a Fy(\036)791 1968 y Fx(0)815 1994 y Fz(\()p Fy(\025)p Fz(\))938 1968 y Fq(3)988 1827 y Fn(\014)988 1882 y(\014)988 1936 y(\014)1033 1932 y Fu(j)p Fy(h)1110 1894 y Fx(0)1134 1932 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)27 b Fz(=)e Fu(\000)1606 1808 y Fn(Z)1697 1835 y Fx(1)1656 2014 y Fw(\025)1697 2023 y Fr(1)1807 1870 y Fy(\036)1861 1837 y Fx(00)1903 1870 y Fz(\()p Fy(\025)p Fz(\))p 1797 1911 V 1797 1994 a Fy(\036)1851 1968 y Fx(0)1874 1994 y Fz(\()p Fy(\025)p Fz(\))1997 1968 y Fq(3)2062 1932 y Fu(j)p Fy(h)2139 1894 y Fx(0)2163 1932 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)22 b Fz(+)2539 1808 y Fn(Z)2630 1835 y Fw(\025)2671 1844 y Fr(1)2589 2014 y Fq(0)2745 1870 y Fy(\036)2799 1837 y Fx(00)2842 1870 y Fz(\()p Fy(\025)p Fz(\))p 2735 1911 V 2735 1994 a Fy(\036)2789 1968 y Fx(0)2812 1994 y Fz(\()p Fy(\025)p Fz(\))2935 1968 y Fq(3)3001 1932 y Fu(j)p Fy(h)3078 1894 y Fx(0)3102 1932 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)-3412 b Fz(\(6.37\))-47 2190 y(In)33 b(view)h(of)g(the)h (de\014nition)d(of)j Fy(h)f Fz(and)g(the)h(assumed)e(monotonicit)m(y)i (of)f Fy(!)j Fz(one)e(has)f Fy(h)3009 2157 y Fx(0)3065 2190 y Fz(=)3167 2122 y Fn(P)3263 2149 y Fq(4)3263 2217 y Fw(i)p Fq(=1)3397 2190 y Fy(f)3442 2204 y Fw(i)3504 2190 y Fz(where)g(eac)m(h)-47 2303 y(function)29 b Fy(f)354 2317 y Fw(i)412 2303 y Fz(has)h(a)h(de\014nite)e(sign.)40 b(In)m(tegrating)31 b(b)m(y)f(parts)g(once)h(one)g(obtains)533 2535 y Fy(J)j Fz(:=)738 2411 y Fn(Z)829 2437 y Fx(1)789 2617 y Fw(\025)830 2626 y Fr(1)939 2473 y Fy(\036)993 2440 y Fx(0)q(0)1036 2473 y Fz(\()p Fy(\025)p Fz(\))p 929 2514 V 929 2597 a Fy(\036)983 2571 y Fx(0)1007 2597 y Fz(\()p Fy(\025)p Fz(\))1130 2571 y Fq(3)1195 2535 y Fy(f)1240 2549 y Fw(i)1267 2535 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)27 b Fz(=)e Fu(\000)1714 2411 y Fn(Z)1805 2437 y Fx(1)1764 2617 y Fw(\025)1805 2626 y Fr(1)2002 2473 y Fz(1)p 1905 2514 V 1905 2597 a Fy(\036)1959 2571 y Fx(0)1982 2597 y Fz(\()p Fy(\025)p Fz(\))2105 2571 y Fq(2)2170 2535 y Fy(f)2225 2497 y Fx(0)2215 2557 y Fw(i)2248 2535 y Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)21 b Fz(+)f(3)p Fy(J)30 b Fu(\000)2941 2473 y Fz(1)p 2824 2514 281 4 v 2824 2597 a Fy(\036)2878 2571 y Fx(0)2901 2597 y Fz(\()p Fy(\025)2989 2611 y Fq(1)3029 2597 y Fz(\))3064 2571 y Fq(2)3129 2535 y Fy(f)3174 2549 y Fw(i)3202 2535 y Fz(\()p Fy(\025)3290 2549 y Fq(1)3330 2535 y Fz(\))-47 2776 y(and)g(similarly)c(for)31 b(the)f(second)h(in)m(tegral)f(in)f (\(6.37\))r(.)41 b(Hence)31 b(the)g(second)f(term)g(in)f(\(6.37\))k(is) c(no)i(larger)f(than)933 3047 y Fy(C)1068 2933 y Fq(4)1020 2961 y Fn(X)1028 3156 y Fw(i)p Fq(=1)1166 2919 y Fn(\032)1234 2923 y(Z)1325 2949 y Fx(1)1285 3129 y Fq(0)1415 3047 y Fu(j)p Fy(\036)1494 3009 y Fx(0)1518 3047 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)1666 3009 y Fx(\000)p Fq(2)1776 3047 y Fu(j)p Fy(f)1856 3009 y Fx(0)1846 3069 y Fw(i)1879 3047 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)15 b Fy(d\025)22 b Fz(+)2382 2985 y(1)p 2265 3026 V 2265 3109 a Fy(\036)2319 3083 y Fx(0)2342 3109 y Fz(\()p Fy(\025)2430 3123 y Fq(1)2470 3109 y Fz(\))2505 3083 y Fq(2)2570 3047 y Fu(j)p Fy(f)2640 3061 y Fw(i)2668 3047 y Fz(\()p Fy(\025)2756 3061 y Fq(1)2796 3047 y Fz(\))p Fu(j)2856 2919 y Fn(\033)2939 3047 y Fy(:)-3011 b Fz(\(6.38\))-47 3315 y(As)30 b Fy(\015)g Fu(\024)25 b Fy(\025)313 3329 y Fq(1)378 3315 y Fu(\024)g Fy(\025)527 3329 y Fq(0)587 3315 y Fu(\000)20 b Fy(\015)5 b Fz(,)30 b(one)h(concludes)f(from)g(\(6.32\))i(and)e(\(6.34\))j(with)c Fy(j)i Fz(=)24 b(1)31 b(that)578 3472 y Fq(4)530 3499 y Fn(X)539 3695 y Fw(i)p Fq(=1)804 3524 y Fz(1)p 687 3565 V 687 3648 a Fy(\036)741 3622 y Fx(0)764 3648 y Fz(\()p Fy(\025)852 3662 y Fq(1)892 3648 y Fz(\))927 3622 y Fq(2)992 3586 y Fu(j)p Fy(f)1062 3600 y Fw(i)1090 3586 y Fz(\()p Fy(\025)1178 3600 y Fq(1)1218 3586 y Fz(\))p Fu(j)26 b Fo(.)f Fy(B)1474 3548 y Fx(\000)p Fq(2)1804 3524 y Fz(1)p 1577 3565 499 4 v 1577 3650 a Fy(\025)1630 3618 y Fq(2)1630 3676 y(1)1670 3650 y Fz(\()p Fy(\025)1758 3664 y Fq(0)1818 3650 y Fu(\000)20 b Fy(\025)1962 3664 y Fq(1)2001 3650 y Fz(\))2036 3624 y Fq(2)2101 3586 y Fy(\015)2153 3548 y Fx(\000)p Fq(1)2383 3524 y Fy(\025)2436 3538 y Fq(0)p 2257 3565 345 4 v 2257 3583 a Fn(p)p 2348 3583 254 4 v 79 x Fy(\025)2401 3631 y Fq(2)2401 3688 y(0)2461 3662 y Fz(+)g Fy(\034)2637 3586 y Fo(.)25 b Fy(B)2807 3548 y Fx(\000)p Fq(2)2901 3586 y Fy(\025)2954 3547 y Fx(\000)p Fq(2)2954 3612 y(0)3048 3586 y Fy(\015)3100 3548 y Fx(\000)p Fq(3)3346 3524 y Fy(\025)3399 3538 y Fq(0)p 3220 3565 345 4 v 3220 3583 a Fn(p)p 3311 3583 254 4 v 79 x Fy(\025)3364 3631 y Fq(2)3364 3688 y(0)3423 3662 y Fz(+)20 b Fy(\034)3574 3586 y(:)-3646 b Fz(\(6.39\))-47 3856 y(On)30 b(the)i(other)f(hand,)g(since)g Fy(f)1034 3823 y Fx(0)1024 3881 y Fw(i)1087 3856 y Fz(and)g Fy(h)1317 3823 y Fx(0)q(0)1391 3856 y Fz(b)s(oth)g(satisfy)h(\(6.34\))r(,)g(the)f (\014rst)g(term)g(in)f(\(6.36\))k(as)d(w)m(ell)f(as)i(the)g(con)m(tri-) -47 3969 y(bution)d(b)m(y)h(the)g(in)m(tegrals)g(in)f(\(6.38\))k(do)s (es)d(not)h(exceed)655 4217 y Fy(B)729 4180 y Fx(\000)p Fq(2)838 4093 y Fn(Z)929 4120 y Fw(\025)970 4129 y Fr(0)1005 4120 y Fx(\000)p Fw(\015)888 4300 y(\015)1336 4156 y Fz(1)p 1129 4196 460 4 v 1129 4280 a Fy(\025)1182 4253 y Fq(2)1222 4280 y Fz(\()p Fy(\025)1310 4294 y Fq(0)1370 4280 y Fu(\000)19 b Fy(\025)p Fz(\))1548 4253 y Fq(2)1613 4217 y Fy(\015)1665 4180 y Fx(\000)p Fq(2)1911 4156 y Fy(\025)1964 4170 y Fq(0)p 1785 4196 345 4 v 1785 4214 a Fn(p)p 1876 4214 254 4 v 79 x Fy(\025)1929 4262 y Fq(2)1929 4320 y(0)1989 4293 y Fz(+)g Fy(\034)2155 4217 y(d\025)25 b Fo(.)g Fy(B)2450 4180 y Fx(\000)p Fq(2)2544 4217 y Fy(\025)2597 4179 y Fx(\000)p Fq(2)2597 4243 y(0)2691 4217 y Fy(\015)2743 4180 y Fx(\000)p Fq(3)2989 4156 y Fy(\025)3042 4170 y Fq(0)p 2863 4196 345 4 v 2863 4214 a Fn(p)p 2954 4214 254 4 v 79 x Fy(\025)3007 4262 y Fq(2)3007 4320 y(0)3066 4293 y Fz(+)20 b Fy(\034)3217 4217 y(;)-47 4497 y Fz(whic)m(h)31 b(is)h(the)h(same)g(as)f(\(6.39\))s(.)47 b(Inserting)31 b(this)h(b)s(ound)e(with)i(the)h(c)m(hoice)g(of)g Fy(\015)h Fz(=)29 b(\()p Fy(B)5 b(\025)3052 4511 y Fq(0)3091 4497 y Fz(\))3126 4461 y Fx(\000)3191 4434 y Fr(1)p 3191 4446 31 3 v 3191 4487 a(2)3268 4497 y Fz(in)m(to)33 b(\(6.35\))i (yields)-47 4610 y(the)e(desired)f(estimate)i(\(6.33\))s(.)49 b(Recall)33 b(that)h(this)e(argumen)m(t)i(assumed)f(that)h Fy(\015)h Fu(\024)2926 4574 y Fq(1)p 2926 4589 36 4 v 2926 4641 a(2)2971 4610 y Fy(\025)3024 4624 y Fq(0)3064 4610 y Fz(,)f(i.e.,)h Fy(B)f Fo(&)c Fy(\025)3556 4572 y Fx(\000)p Fq(3)3556 4636 y(0)3684 4610 y Fz(b)m(y)j(our)-47 4723 y(c)m(hoice)e(of)f Fy(\015)5 b Fz(.)41 b(Ho)m(w)m(ev)m(er,)33 b(if)c Fy(B)h Fo(.)25 b Fy(\025)1167 4685 y Fx(\000)p Fq(3)1167 4749 y(0)1261 4723 y Fz(,)31 b(then)f(one)h(has)698 4843 y Fn(\014)698 4897 y(\014)698 4952 y(\014)698 5006 y(\014)728 4851 y(Z)819 4877 y Fw(\025)860 4886 y Fr(0)779 5057 y Fq(0)914 4975 y Fy(e)956 4937 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1296 4913 y Fy(\025)p 1158 4954 330 4 v 1158 4972 a Fu(p)p 1234 4972 254 4 v 77 x Fy(\025)1287 5022 y Fq(2)1346 5049 y Fz(+)20 b Fy(\034)1512 4975 y(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)1810 4843 y Fn(\014)1810 4897 y(\014)1810 4952 y(\014)1810 5006 y(\014)1867 4975 y Fu(\024)2099 4913 y Fy(\025)2152 4880 y Fq(2)2152 4938 y(0)p 1973 4954 345 4 v 1973 4972 a Fn(p)p 2064 4972 254 4 v 79 x Fy(\025)2117 5020 y Fq(2)2117 5077 y(0)2177 5051 y Fz(+)20 b Fy(\034)2353 4975 y Fo(.)25 b Fz(\()p Fy(B)5 b(\025)2611 4989 y Fq(0)2650 4975 y Fz(\))2685 4937 y Fx(\000)2750 4910 y Fr(1)p 2750 4922 31 3 v 2750 4963 a(2)2946 4913 y Fy(\025)2999 4927 y Fq(0)p 2820 4954 345 4 v 2820 4972 a Fn(p)p 2911 4972 254 4 v 79 x Fy(\025)2964 5020 y Fq(2)2964 5077 y(0)3023 5051 y Fz(+)20 b Fy(\034)3174 4975 y(;)-47 5223 y Fz(and)30 b(w)m(e)g(are)h(done.)p 3879 5223 4 62 v 3883 5165 55 4 v 3883 5223 V 3937 5223 4 62 v 1903 5472 a(52)p eop %%Page: 53 53 53 52 bop -47 274 a Fz(In)29 b(the)i(next)g(lemma)e(w)m(e)i(obtain)f (general)g(lo)m(w)m(er)h(b)s(ounds)d(on)j Fy(\036)2185 241 y Fx(0)2238 274 y Fz(and)f Fy(\036)2469 241 y Fx(00)2512 274 y Fz(.)-47 436 y Fm(Lemma)i(6.8.)47 b Fl(L)-5 b(et)32 b Fy(\036)h Fl(and)h Fy(\025)1005 450 y Fq(0)1077 436 y Fl(b)-5 b(e)32 b(as)h(in)g(L)-5 b(emma)34 b(6.5.)60 598 y(1.)46 b(If)32 b Fy(\025)328 612 y Fq(0)393 598 y Fy(>)25 b Fz(0)33 b Fl(exists,)g(then)902 805 y Fy(\036)956 767 y Fx(0)980 805 y Fz(\()p Fy(\025)p Fz(\))83 b Fu(\025)1350 743 y Fz(1)p 1350 784 46 4 v 1350 867 a(2)1406 805 y(\(1)21 b(+)1597 741 y Fu(p)p 1673 741 92 4 v 64 x Fy(\033)1725 819 y Fq(1)1765 805 y Fz(\))1800 767 y Fx(\000)p Fq(1)1894 805 y Fz(\()p Fy(\025)g Fu(\000)f Fy(\025)2147 819 y Fq(0)2186 805 y Fz(\))66 b Fl(for)33 b(al)5 b(l)65 b Fy(\025)26 b Fu(\025)f Fy(\025)2822 819 y Fq(0)2881 805 y Fz(+)20 b(2)p Fy(\033)3082 720 y Fr(1)p 3083 732 31 3 v 3083 773 a(4)3069 831 y Fq(1)180 805 y Fz(\(6.40\))852 1017 y Fu(j)p Fy(\036)931 979 y Fx(0)954 1017 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)84 b(\025)1340 943 y Fn(\002)1378 1017 y Fy(\025)1431 1031 y Fq(0)1470 1017 y Fz(\(1)21 b(+)1662 954 y Fu(p)p 1738 954 92 4 v 63 x Fy(\033)1790 1031 y Fq(1)1830 1017 y Fz(\))1865 943 y Fn(\003)1903 966 y Fx(\000)p Fq(1)1997 1017 y Fy(\025)p Fz(\()p Fy(\025)g Fu(\000)f Fy(\025)2303 1031 y Fq(0)2342 1017 y Fz(\))65 b Fl(for)34 b(al)5 b(l)65 b Fy(\033)2815 932 y Fr(1)p 2815 944 31 3 v 2815 985 a(4)2802 1043 y Fq(1)2885 1017 y Fu(\024)25 b Fy(\025)h(<)f(\025)3209 1031 y Fq(0)3248 1017 y Fy(:)-3093 b Fz(\(6.41\))180 1190 y Fl(If)32 b Fy(\025)328 1204 y Fq(0)393 1190 y Fy(>)25 b Fz(0)33 b Fl(do)-5 b(es)34 b(not)f(exist,)g(then)40 b Fz(\(6.40\))35 b Fl(holds)f(for)f(al)5 b(l)33 b Fy(\025)26 b Fu(\025)f Fz(0)p Fl(.)60 1368 y(2.)46 b(Now)33 b(supp)-5 b(ose)34 b Fy(\025)772 1382 y Fq(0)844 1368 y Fl(exists.)42 b(Then)1071 1541 y Fy(\036)1125 1503 y Fx(0)1148 1541 y Fz(\()p Fy(\025)p Fz(\))84 b Fu(\025)f Fy(\036)1563 1503 y Fx(00)1605 1541 y Fz(\()p Fy(\025)1693 1555 y Fq(0)1733 1541 y Fz(\)\()p Fy(\025)21 b Fu(\000)f Fy(\025)2021 1555 y Fq(0)2060 1541 y Fz(\))66 b Fl(for)33 b(al)5 b(l)65 b Fy(\025)26 b Fu(\025)f Fy(\025)2696 1555 y Fq(0)180 1541 y Fz(\(6.42\))1020 1722 y Fu(j)p Fy(\036)1099 1684 y Fx(0)1123 1722 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)84 b(\025)1518 1660 y Fz(1)p 1518 1701 46 4 v 1518 1784 a(2)1574 1722 y Fy(\036)1628 1684 y Fx(00)1670 1722 y Fz(\()p Fy(\025)1758 1736 y Fq(0)1798 1722 y Fz(\))1863 1660 y Fy(\025)p 1843 1701 93 4 v 1843 1784 a(\025)1896 1798 y Fq(0)1946 1722 y Fz(\()p Fy(\025)2034 1736 y Fq(0)2094 1722 y Fu(\000)20 b Fy(\025)p Fz(\))65 b Fl(for)34 b(al)5 b(l)65 b Fz(0)26 b Fu(\024)f Fy(\025)g Fu(\024)g Fy(\025)3040 1736 y Fq(0)3080 1722 y Fy(:)-2925 b Fz(\(6.43\))180 1952 y Fl(If)32 b Fy(\025)328 1966 y Fq(0)393 1952 y Fu(\025)489 1894 y(p)p 565 1894 95 4 v 58 x Fy(\033)617 1967 y Fw(k)660 1952 y Fl(,)g(then)h Fy(\036)976 1919 y Fx(00)1019 1952 y Fz(\()p Fy(\025)1107 1966 y Fq(0)1146 1952 y Fz(\))26 b Fu(\025)1450 1912 y Fw(b)1480 1924 y Fs(k)p 1313 1931 343 4 v 1313 1984 a Fq(4)p Fw(m)12 b Fq(max)1554 1996 y Fs(`)1596 1984 y Fw(b)1626 1996 y Fs(`)1666 1952 y Fl(.)-47 2114 y(Pr)-5 b(o)g(of.)47 b Fz(By)32 b(\(6.10\))r(,)e(for)h(all)e Fy(\025)c Fu(\025)g Fy(\025)1162 2128 y Fq(0)1232 2114 y Fz(\(or)31 b Fy(\025)25 b Fu(\025)g Fz(0)31 b(if)e Fy(\025)1765 2128 y Fq(0)1835 2114 y Fz(do)s(es)h(not)g(exist\),)675 2357 y Fy(\036)729 2320 y Fx(0)q(0)772 2357 y Fz(\()p Fy(\025)p Fz(\))84 b Fu(\025)e Fz(1)21 b Fu(\000)1323 2244 y Fw(m)1289 2271 y Fn(X)1293 2466 y Fw(j)t Fq(=1)1599 2295 y Fy(b)1638 2309 y Fw(j)p 1445 2336 383 4 v 1445 2355 a Fn(p)p 1536 2355 293 4 v 76 x Fy(\025)1589 2405 y Fq(2)1649 2431 y Fz(+)f Fy(\033)1792 2445 y Fw(j)1858 2357 y Fz(+)2061 2296 y Fy(\025)2114 2263 y Fq(2)p 1959 2336 295 4 v 1959 2420 a Fy(\025)2012 2393 y Fq(2)2072 2420 y Fz(+)g Fy(\033)2215 2434 y Fq(1)2314 2244 y Fw(m)2279 2271 y Fn(X)2284 2466 y Fw(j)t Fq(=1)2590 2295 y Fy(b)2629 2309 y Fw(j)p 2436 2336 383 4 v 2436 2355 a Fn(p)p 2527 2355 293 4 v 76 x Fy(\025)2580 2405 y Fq(2)2639 2431 y Fz(+)g Fy(\033)2782 2445 y Fw(j)979 2673 y Fu(\025)82 b Fz(1)21 b Fu(\000)1401 2611 y Fy(\033)1453 2625 y Fq(1)p 1299 2652 295 4 v 1299 2735 a Fy(\025)1352 2709 y Fq(2)1412 2735 y Fz(+)e Fy(\033)1554 2749 y Fq(1)1653 2559 y Fw(m)1619 2586 y Fn(X)1623 2782 y Fw(j)t Fq(=1)1929 2611 y Fy(b)1968 2625 y Fw(j)p 1775 2652 383 4 v 1775 2670 a Fn(p)p 1866 2670 293 4 v 76 x Fy(\025)1919 2720 y Fq(2)1979 2746 y Fz(+)h Fy(\033)2122 2760 y Fw(j)2194 2673 y Fu(\025)25 b Fz(1)20 b Fu(\000)2558 2611 y Fy(\033)2610 2625 y Fq(1)p 2456 2652 295 4 v 2456 2735 a Fy(\025)2509 2709 y Fq(2)2569 2735 y Fz(+)g Fy(\033)2712 2749 y Fq(1)2786 2673 y Fz(=)2993 2611 y Fy(\025)3046 2578 y Fq(2)p 2892 2652 V 2892 2735 a Fy(\025)2945 2709 y Fq(2)3005 2735 y Fz(+)g Fy(\033)3148 2749 y Fq(1)3197 2673 y Fy(:)-3269 b Fz(\(6.44\))-47 2941 y(F)-8 b(or)31 b(the)f(second)h(inequalit)m(y)e(sign)g(in)g (\(6.44\))k(w)m(e)e(used)e(the)i(fact)g(that)g(if)e Fy(\025)2554 2955 y Fq(0)2624 2941 y Fz(exists,)h(then)1301 3070 y Fw(m)1267 3097 y Fn(X)1271 3293 y Fw(j)t Fq(=1)1577 3122 y Fy(b)1616 3136 y Fw(j)p 1423 3163 383 4 v 1423 3181 a Fn(p)p 1514 3181 293 4 v 76 x Fy(\025)1567 3231 y Fq(2)1627 3257 y Fz(+)20 b Fy(\033)1770 3271 y Fw(j)1841 3184 y Fu(\024)25 b Fz(1)61 b(for)30 b(all)60 b Fy(\025)25 b Fu(\025)g Fy(\025)2566 3198 y Fq(0)2606 3184 y Fy(;)-47 3460 y Fz(and)33 b(for)h(all)f Fy(\025)f Fu(\025)f Fz(0)k(if)e Fy(\025)813 3474 y Fq(0)887 3460 y Fz(do)s(es)h(not)g(exist.)52 b(See)34 b(\(6.11\))j(and)c(\(6.16\))s(.)52 b(Hence)35 b Fy(\036)2773 3427 y Fx(00)2816 3460 y Fz(\()p Fy(\025)p Fz(\))d Fu(\025)f Fz(\(1)24 b(+)3270 3401 y Fu(p)p 3346 3401 92 4 v 59 x Fy(\033)3398 3474 y Fq(1)3437 3460 y Fz(\))3472 3427 y Fx(\000)p Fq(1)3601 3460 y Fz(pro)m(vided)-47 3613 y Fy(\025)33 b Fu(\025)h Fy(\033)209 3529 y Fr(1)p 209 3541 31 3 v 209 3582 a(4)196 3639 y Fq(1)254 3613 y Fz(.)55 b(Recall)35 b(from)g(Lemma)h(6.5)g(that)h Fy(\036)1571 3580 y Fx(00)1613 3613 y Fz(\()p Fy(\025)p Fz(\))e Fu(\025)e Fz(0)j(for)f(all)f Fy(\025)g Fu(\025)f Fy(\025)2475 3627 y Fq(0)2550 3613 y Fz(\(or)j(for)f(all)f Fy(\025)g Fu(\025)g Fz(0)h(in)g(case)h(a)g(p)s(ositiv)m(e)-47 3726 y(critical)29 b(p)s(oin)m(t)g(do)s(es)h(not)g(exist\).)41 b(Setting)30 b Fy(\025)1511 3740 y Fq(0)1576 3726 y Fz(=)25 b(0)30 b(if)g(a)g(p)s(ositiv)m(e)f(critical)g(p)s(oin)m(t)h(do)s(es)f(not)i (exist,)f(it)g(follo)m(ws)f(that)710 3961 y Fy(\036)764 3924 y Fx(0)788 3961 y Fz(\()p Fy(\025)p Fz(\))83 b Fu(\025)1148 3837 y Fn(Z)1239 3864 y Fw(\025)1199 4044 y(\025)1240 4053 y Fr(0)1299 3961 y Fy(\036)1353 3924 y Fx(0)q(0)1396 3961 y Fz(\()p Fy(s)p Fz(\))15 b Fy(ds)26 b Fu(\025)1736 3837 y Fn(Z)1827 3864 y Fw(\025)1786 4044 y(\025)1827 4053 y Fr(0)1862 4044 y Fq(+)p Fw(\033)1969 3978 y Fr(1)p 1969 3986 V 1969 4021 a(4)1957 4065 y(1)2033 3961 y Fy(\036)2087 3924 y Fx(0)q(0)2130 3961 y Fz(\()p Fy(s)p Fz(\))15 b Fy(ds)994 4218 y Fu(\025)83 b Fz(\(1)21 b(+)1340 4155 y Fu(p)p 1416 4155 92 4 v 63 x Fy(\033)1468 4232 y Fq(1)1507 4218 y Fz(\))1542 4180 y Fx(\000)p Fq(1)1637 4218 y Fz(\()p Fy(\025)f Fu(\000)g Fy(\025)1889 4232 y Fq(0)1949 4218 y Fu(\000)g Fy(\033)2105 4134 y Fr(1)p 2105 4146 31 3 v 2105 4187 a(4)2092 4244 y Fq(1)2150 4218 y Fz(\))25 b Fu(\025)2316 4157 y Fz(1)p 2316 4197 46 4 v 2316 4280 a(2)2372 4218 y(\(1)c(+)2563 4155 y Fu(p)p 2639 4155 92 4 v 63 x Fy(\033)2691 4232 y Fq(1)2731 4218 y Fz(\))2766 4180 y Fx(\000)p Fq(1)2860 4218 y Fz(\()p Fy(\025)g Fu(\000)f Fy(\025)3113 4232 y Fq(0)3152 4218 y Fz(\))-47 4466 y(for)30 b(all)f Fy(\025)c Fu(\025)g Fy(\025)445 4480 y Fq(0)505 4466 y Fz(+)20 b(2)p Fy(\033)706 4381 y Fr(1)p 707 4393 31 3 v 707 4435 a(4)693 4492 y Fq(1)751 4466 y Fz(,)31 b(as)f(claimed.)-47 4579 y(No)m(w)h(assume)f(that)h Fy(\025)724 4593 y Fq(0)789 4579 y Fy(>)25 b Fz(0)30 b(exists.)41 b(If)30 b Fy(\025)25 b(<)g(\025)1565 4593 y Fq(0)1604 4579 y Fz(,)31 b(then)f(\(6.7\))i(implies)c(that)559 4822 y Fu(j)p Fy(\036)638 4784 y Fx(0)662 4822 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)84 b Fz(=)e Fy(\025)1115 4698 y Fn(Z)1206 4724 y Fw(\025)1247 4733 y Fr(0)1166 4904 y Fw(\025)1336 4708 y(m)1302 4735 y Fn(X)1306 4931 y Fw(j)t Fq(=1)1602 4760 y Fy(b)1641 4774 y Fw(j)1678 4760 y Fy(s)p 1458 4801 407 4 v 1458 4907 a Fz(\()p Fy(s)1536 4880 y Fq(2)1596 4907 y Fz(+)19 b Fy(\033)1738 4921 y Fw(j)1775 4907 y Fz(\))1820 4844 y Fr(3)p 1820 4856 31 3 v 1820 4897 a(2)1890 4822 y Fy(ds)25 b Fu(\025)g Fy(\025)2169 4698 y Fn(Z)2260 4724 y Fw(\025)2301 4733 y Fr(0)2220 4904 y Fw(\025)2390 4708 y(m)2355 4735 y Fn(X)2360 4931 y Fw(j)t Fq(=1)2660 4760 y Fy(b)2699 4774 y Fw(j)p 2512 4801 373 4 v 2512 4819 a Fn(p)p 2603 4819 282 4 v 76 x Fy(s)2646 4869 y Fq(2)2705 4895 y Fz(+)20 b Fy(\033)2848 4909 y Fw(j)3020 4760 y Fy(s)3063 4727 y Fq(2)p 2919 4801 285 4 v 2919 4884 a Fy(s)2962 4858 y Fq(2)3021 4884 y Fz(+)g Fy(\033)3164 4898 y Fq(1)3239 4760 y Fy(ds)p 3239 4801 90 4 v 3263 4884 a(s)894 5137 y Fu(\025)1159 5076 y Fy(\025)1212 5043 y Fq(3)p 1057 5116 295 4 v 1057 5200 a Fy(\025)1110 5173 y Fq(2)1170 5200 y Fz(+)g Fy(\033)1313 5214 y Fq(1)1377 5014 y Fn(Z)1468 5040 y Fw(\025)1509 5049 y Fr(0)1428 5220 y Fw(\025)1598 5024 y(m)1563 5051 y Fn(X)1568 5246 y Fw(j)t Fq(=1)1873 5075 y Fy(b)1912 5089 y Fw(j)p 1720 5116 383 4 v 1720 5135 a Fn(p)p 1811 5135 293 4 v 78 x Fy(\025)1864 5182 y Fq(2)1864 5239 y(0)1923 5213 y Fz(+)g Fy(\033)2066 5227 y Fw(j)2139 5076 y Fy(ds)p 2138 5116 93 4 v 2138 5200 a(\025)2191 5214 y Fq(0)2265 5137 y Fz(=)2473 5076 y Fy(\025)2526 5043 y Fq(2)p 2371 5116 295 4 v 2371 5200 a Fy(\025)2424 5173 y Fq(2)2484 5200 y Fz(+)g Fy(\033)2627 5214 y Fq(1)2706 5076 y Fy(\025)p 2686 5116 93 4 v 2686 5200 a(\025)2739 5214 y Fq(0)2789 5137 y Fz(\()p Fy(\025)2877 5151 y Fq(0)2937 5137 y Fu(\000)g Fy(\025)p Fz(\))p Fy(;)-3188 b Fz(\(6.45\))1903 5472 y(53)p eop %%Page: 54 54 54 53 bop -47 299 a Fz(whic)m(h)26 b(is)g(\(6.41\))j(pro)m(vided)d Fy(\025)g Fu(\025)e Fy(\033)1167 215 y Fr(1)p 1168 227 31 3 v 1168 268 a(4)1154 325 y Fq(1)1212 299 y Fz(.)40 b(T)-8 b(o)28 b(pass)e(to)i(\(6.45\))i(w)m(e)d(used)g(that)2574 263 y Fw(s)2607 240 y Fr(2)p 2509 278 198 4 v 2509 333 a Fw(s)2542 314 y Fr(2)2576 333 y Fq(+)p Fw(\033)2671 342 y Fr(1)2743 299 y Fz(is)f(increasing,)h(whereas)g(the)g(\014nal)-47 412 y(expression)i(uses)h(\(6.11\))r(.)-47 525 y(The)h(pro)s(of)f(of)39 b(\(6.42\))33 b(is)e(an)g(immediate)f(consequence)i(of)g(the)f(fact)i (that)f Fy(\036)2628 492 y Fx(00)2670 525 y Fz(\()p Fy(\025)p Fz(\))g(is)f(increasing.)42 b(F)-8 b(or)32 b(\(6.43\))i(one)-47 638 y(again)c(uses)g(\(6.7\))r(.)41 b(Indeed,)29 b(for)h(0)c Fu(\024)f Fy(\025)g Fu(\024)g Fy(\025)1494 652 y Fq(0)1534 638 y Fz(,)901 899 y Fu(j)p Fy(\036)980 862 y Fx(0)1003 899 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)84 b(\025)f Fy(\025)1457 776 y Fn(Z)1548 802 y Fw(\025)1589 811 y Fr(0)1508 982 y Fw(\025)1678 786 y(m)1643 813 y Fn(X)1648 1009 y Fw(j)t Fq(=1)1949 837 y Fy(b)1988 851 y Fw(j)2025 837 y Fy(s)p 1800 879 418 4 v 1800 985 a Fz(\()p Fy(\025)1888 953 y Fq(2)1888 1011 y(0)1948 985 y Fz(+)20 b Fy(\033)2091 999 y Fw(j)2127 985 y Fz(\))2172 922 y Fr(3)p 2172 934 31 3 v 2172 975 a(2)2242 899 y Fy(ds)1235 1207 y Fu(\025)83 b Fy(\025)1457 1084 y Fn(Z)1548 1110 y Fw(\025)1589 1119 y Fr(0)1508 1290 y Fw(\025)1653 1146 y Fy(\036)1707 1113 y Fx(00)1750 1146 y Fz(\()p Fy(\025)1838 1160 y Fq(0)1878 1146 y Fz(\))p 1653 1186 260 4 v 1737 1271 a Fy(\025)1790 1240 y Fq(2)1790 1297 y(0)1938 1207 y Fy(s)15 b(ds)25 b Fu(\025)2217 1146 y Fz(1)p 2217 1186 46 4 v 2217 1270 a(2)2282 1146 y Fy(\036)2336 1113 y Fx(0)q(0)2379 1146 y Fz(\()p Fy(\025)2467 1160 y Fq(0)2507 1146 y Fz(\))p 2282 1186 260 4 v 2366 1271 a Fy(\025)2419 1240 y Fq(2)2419 1297 y(0)2552 1207 y Fy(\025)p Fz(\()p Fy(\025)2693 1170 y Fq(2)2693 1230 y(0)2753 1207 y Fu(\000)20 b Fy(\025)2897 1170 y Fq(2)2936 1207 y Fz(\))p Fy(;)-47 1453 y Fz(as)30 b(claimed.)40 b(Assume)30 b Fy(\025)823 1467 y Fq(0)888 1453 y Fu(\025)984 1395 y(p)p 1059 1395 95 4 v 1059 1453 a Fy(\033)1111 1468 y Fw(k)1154 1453 y Fz(.)41 b(By)32 b(\(6.11\))g(one)f(has)939 1694 y(max)905 1756 y Fq(1)p Fx(\024)p Fw(`)p Fx(\024)p Fw(m)1321 1633 y Fy(b)1360 1648 y Fw(`)p 1167 1673 380 4 v 1167 1692 a Fn(p)p 1258 1692 289 4 v 79 x Fy(\025)1311 1739 y Fq(2)1311 1797 y(0)1370 1771 y Fz(+)20 b Fy(\033)1513 1786 y Fw(`)1581 1694 y Fu(\025)1705 1633 y Fz(1)p 1687 1673 80 4 v 1687 1757 a Fy(m)1838 1694 y Fz(and)30 b(th)m(us)g(also)60 b Fy(m)15 b Fz(max)2592 1756 y Fw(`)2707 1694 y Fy(b)2746 1709 y Fw(`)2804 1694 y Fu(\025)25 b Fy(\025)2953 1708 y Fq(0)-47 1953 y Fz(F)-8 b(urthermore,)30 b(\(6.10\))j(implies)27 b(that)1192 2187 y Fy(\036)1246 2150 y Fx(0)q(0)1289 2187 y Fz(\()p Fy(\025)1377 2201 y Fq(0)1417 2187 y Fz(\))e Fu(\025)g Fy(\025)1626 2150 y Fq(2)1626 2210 y(0)1847 2126 y Fy(b)1886 2141 y Fw(k)p 1676 2167 424 4 v 1676 2273 a Fz(\()p Fy(\025)1764 2241 y Fq(2)1764 2299 y(0)1824 2273 y Fz(+)20 b Fy(\033)1967 2288 y Fw(k)2009 2273 y Fz(\))2054 2210 y Fr(3)p 2055 2222 31 3 v 2055 2263 a(2)2134 2187 y Fu(\025)2414 2126 y Fy(b)2453 2141 y Fw(k)p 2240 2167 430 4 v 2240 2250 a Fz(4)p Fy(m)15 b Fz(max)2550 2265 y Fw(`)2598 2250 y Fy(b)2637 2265 y Fw(`)2680 2187 y Fy(;)-47 2449 y Fz(and)30 b(the)g(lemma)g(follo)m(ws.)p 3879 2449 4 62 v 3883 2391 55 4 v 3883 2449 V 3937 2449 4 62 v -47 2635 a Fl(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(6.1.)46 b Fz(The)27 b(b)s(ound)e(\(1)15 b(+)g Fy(\033)1514 2649 y Fq(1)1553 2635 y Fz(\))1598 2572 y Fr(1)p 1599 2584 31 3 v 1599 2625 a(4)1671 2635 y Fz(on)27 b(the)h(righ)m(t-hand)e (side)h(of)35 b(\(6.1\))29 b(follo)m(ws)e(from)g(\(6.41\))r(,)h (\(6.40\))s(,)-47 2761 y(\(6.18\))37 b(with)c Fy(A)g Fu(\020)f Fz(\(1)24 b(+)e Fy(\033)884 2775 y Fq(1)924 2761 y Fz(\))959 2725 y Fx(\000)1024 2698 y Fr(1)p 1024 2710 V 1024 2751 a(2)1068 2761 y Fz(,)36 b(and)e(\(6.33\))j(with)d Fy(B)j Fu(\020)32 b Fz([)p Fy(\025)2076 2728 y Fq(2)2076 2785 y(0)2115 2761 y Fz(\(1)24 b(+)f Fy(\033)2365 2775 y Fq(1)2404 2761 y Fz(\)])2464 2725 y Fx(\000)2529 2698 y Fr(1)p 2530 2710 V 2530 2751 a(2)2574 2761 y Fz(.)54 b(As)35 b(for)f(the)h(other)g(b)s(ound)d(in)i(\(6.1\))r(,)-47 2874 y(\014x)d(some)i(1)c Fu(\024)f Fy(k)k Fu(\024)c Fy(m)p Fz(.)46 b(If)32 b Fy(\025)961 2888 y Fq(0)1029 2874 y Fy(>)c Fz(0)33 b(exists)f(and)g Fy(\025)1691 2888 y Fq(0)1759 2874 y Fy(>)1858 2816 y Fu(p)p 1934 2816 95 4 v 58 x Fy(\033)1986 2889 y Fw(k)2028 2874 y Fz(,)h(then)f(one)h (applies)d(\(6.18\))35 b(with)c Fy(A)e Fu(\020)3568 2834 y Fw(b)3598 2846 y Fs(k)p 3449 2853 308 4 v 3449 2905 a Fw(m)12 b Fq(max)3654 2917 y Fs(`)3696 2905 y Fw(b)3726 2917 y Fs(`)3798 2874 y Fz(and)-47 3006 y(\(6.33\))32 b(with)d Fy(B)h Fu(\020)25 b Fy(\025)670 2968 y Fx(\000)p Fq(1)670 3032 y(0)765 3006 y Fy(A)p Fz(,)30 b(see)h(the)g(second)f (part)g(of)h(Lemma)f(6.8.)42 b(Observ)m(e)31 b(that)g(this)e(leads)h (to)h(the)f(b)s(ound)1669 3292 y Fy(m)1759 3227 y Fr(1)p 1759 3239 31 3 v 1759 3280 a(2)1803 3127 y Fn(s)p 1894 3127 310 4 v 1904 3230 a Fz(max)2073 3245 y Fw(`)2121 3230 y Fy(b)2160 3245 y Fw(`)p 1904 3271 290 4 v 2008 3354 a Fy(b)2047 3369 y Fw(k)2203 3292 y Fy(;)-47 3552 y Fz(whic)m(h)e(is)h(smaller)g(than)h(the)g(one)g(in)f(\(6.1\))q(.)41 b(It)30 b(therefore)g(remains)f(to)i(consider)e(the)h(case)h Fy(\025)3193 3566 y Fq(0)3257 3552 y Fu(\024)3353 3493 y(p)p 3429 3493 95 4 v 59 x Fy(\033)3481 3567 y Fw(k)3524 3552 y Fz(,)f(where)f(w)m(e)-47 3664 y(set)i Fy(\025)148 3678 y Fq(0)213 3664 y Fz(=)26 b(0)31 b(if)e(a)i(p)s(ositiv)m(e)f (critical)f(p)s(oin)m(t)h(do)s(es)g(not)h(exist.)41 b(Cho)s(ose)31 b(a)g(maximal)e Fy(n)i Fz(so)f(that)i Fy(k)d Fu(\024)c Fy(n)g Fu(\024)h Fy(m)k Fz(and)g(for)-47 3777 y(whic)m(h)f Fy(\033)265 3791 y Fw(n)p Fq(+1)427 3777 y Fu(\024)c Fy(\025)576 3744 y Fq(2)576 3802 y(0)641 3777 y Fu(\024)g Fy(\033)789 3791 y Fw(n)836 3777 y Fz(.)40 b(Here)31 b(w)m(e)g(ha)m(v)m(e)h(set)f Fy(\033)1654 3791 y Fw(m)p Fq(+1)1836 3777 y Fz(:=)25 b(0.)41 b(If)30 b Fy(\025)2212 3791 y Fq(0)2277 3777 y Fy(>)25 b Fz(0)30 b(exists,)h(then)f(\(6.11\))i (implies)c(that)1122 4040 y(1)e(=)1323 3927 y Fw(m)1289 3954 y Fn(X)1293 4150 y Fw(j)t Fq(=1)1599 3978 y Fy(b)1638 3992 y Fw(j)p 1445 4020 383 4 v 1445 4038 a Fn(p)p 1536 4038 293 4 v 79 x Fy(\025)1589 4085 y Fq(2)1589 4143 y(0)1649 4117 y Fz(+)20 b Fy(\033)1792 4131 y Fw(j)1863 4040 y Fu(\020)1993 3979 y Fz(1)p 1969 4020 93 4 v 1969 4103 a Fy(\025)2022 4117 y Fq(0)2087 3954 y Fn(X)2087 4150 y Fw(j)t(>n)2233 4040 y Fy(b)2272 4054 y Fw(j)2329 4040 y Fz(+)2420 3954 y Fn(X)2420 4150 y Fw(j)t Fx(\024)p Fw(n)2621 3978 y Fy(b)2660 3992 y Fw(j)p 2576 4020 165 4 v 2576 4051 a Fu(p)p 2652 4051 89 4 v 52 x Fy(\033)2704 4117 y Fw(j)2750 4040 y Fy(:)-2822 b Fz(\(6.46\))-47 4328 y(If)30 b Fy(\025)97 4342 y Fq(0)161 4328 y Fz(=)25 b(0)31 b(\(whic)m(h)f(includes)e(the)i(case)i(that)f(the)f(p)s(ositiv)m (e)f(critical)h(p)s(oin)m(t)f(do)s(es)h(not)h(exist\),)g(then)1680 4589 y(1)26 b Fu(\025)1881 4475 y Fw(m)1847 4503 y Fn(X)1851 4698 y Fw(j)t Fq(=1)2047 4527 y Fy(b)2086 4541 y Fw(j)p 2003 4568 165 4 v 2003 4599 a Fu(p)p 2079 4599 89 4 v 52 x Fy(\033)2131 4665 y Fw(j)2177 4589 y Fy(;)-47 4876 y Fz(see)31 b(\(6.16\))r(.)41 b(If)30 b(in)f(fact)1712 5024 y Fw(m)1678 5051 y Fn(X)1682 5246 y Fw(j)t Fq(=1)1878 5075 y Fy(b)1917 5089 y Fw(j)p 1834 5116 165 4 v 1834 5147 a Fu(p)p 1910 5147 89 4 v 53 x Fy(\033)1962 5214 y Fw(j)2033 5137 y Fu(\024)2139 5076 y Fz(1)p 2139 5116 46 4 v 2139 5200 a(2)2195 5137 y Fy(;)1903 5472 y Fz(54)p eop %%Page: 55 55 55 54 bop -47 274 a Fz(then)33 b(\(6.9\))j(implies)31 b(that)j Fy(\036)950 241 y Fx(0)974 274 y Fz(\()p Fy(\025)p Fz(\))e Fu(\025)1240 238 y Fq(1)p 1240 253 36 4 v 1240 305 a(2)1286 274 y Fy(\025)h Fz(for)h(all)f Fy(\025)e Fu(\025)g Fz(0.)51 b(Hence)35 b(\(6.18\))h(with)c Fy(A)g Fz(=)2913 238 y Fq(1)p 2913 253 V 2913 305 a(2)2992 274 y Fz(yields)g(a)i Fy(O)s Fz(\(1\))h(b)s(ound)d(for)-47 387 y(the)e(en)m(tire)h(oscillatory)e(in)m(tegral.)41 b(Hence)31 b(w)m(e)g(can)f(assume)h(without)e(loss)h(of)g(generalit)m (y)h(that)1594 656 y(1)26 b Fu(\025)1795 542 y Fw(m)1761 569 y Fn(X)1765 765 y Fw(j)t Fq(=1)1962 594 y Fy(b)2001 608 y Fw(j)p 1917 635 165 4 v 1917 666 a Fu(p)p 1993 666 89 4 v 52 x Fy(\033)2045 732 y Fw(j)2117 656 y Fu(\025)2223 594 y Fz(1)p 2223 635 46 4 v 2223 718 a(2)2278 656 y Fy(:)-2350 b Fz(\(6.47\))-47 954 y(This)29 b(again)j(implies)c(that)k (the)g(righ)m(t-hand)e(side)g(of)39 b(\(6.46\))33 b(is)e(comparable)g (to)h(1.)44 b(Supp)s(ose)30 b(that)i(the)f(\014rst)g(sum)-47 1067 y(on)f(the)h(righ)m(t-hand)e(side)g(of)38 b(\(6.46\))32 b(dominates.)41 b(Then)1110 1341 y Fy(\036)1164 1303 y Fx(00)1207 1341 y Fz(\()p Fy(\025)1295 1355 y Fq(0)1335 1341 y Fz(\))25 b(=)1526 1227 y Fw(m)1491 1255 y Fn(X)1495 1450 y Fw(j)t Fq(=1)1772 1279 y Fy(b)1811 1293 y Fw(j)1848 1279 y Fy(\025)1901 1246 y Fq(2)1901 1303 y(0)p 1648 1320 418 4 v 1648 1426 a Fz(\()p Fy(\025)1736 1395 y Fq(2)1736 1452 y(0)1796 1426 y Fz(+)19 b Fy(\033)1938 1440 y Fw(j)1975 1426 y Fz(\))2020 1363 y Fr(3)p 2020 1375 31 3 v 2020 1416 a(2)2100 1341 y Fo(&)25 b Fy(\025)2249 1303 y Fx(\000)p Fq(1)2249 1367 y(0)2374 1255 y Fn(X)2374 1450 y Fw(j)t(>n)2520 1341 y Fy(b)2559 1355 y Fw(j)2621 1341 y Fo(&)g Fz(1)p Fy(:)-47 1640 y Fz(In)35 b(view)g(of)h(the)h (second)f(part)g(of)g(Lemma)g(6.8)h(one)f(can)h(apply)f(\(6.18\))i(and) d(\(6.33\))k(with)34 b Fy(A)h Fu(\020)f Fy(B)40 b Fu(\020)34 b Fz(1,)k(whic)m(h)-47 1753 y(leads)28 b(to)i(a)f Fy(O)s Fz(\(1\))h(b)s(ound)d(in)g(this)h(case.)41 b(No)m(w)30 b(supp)s(ose)d(that)j(the)f(second)g(sum)f(in)f(\(6.46\))k(is)d(the)h (larger)g(one,)g(i.e.,)1688 2000 y(1)c Fu(\020)1854 1914 y Fn(X)1855 2110 y Fw(j)t Fx(\024)p Fw(n)2055 1938 y Fy(b)2094 1952 y Fw(j)p 2011 1979 165 4 v 2011 2010 a Fu(p)p 2086 2010 89 4 v 2086 2063 a Fy(\033)2138 2077 y Fw(j)2185 2000 y Fy(:)-47 2295 y Fz(Then)k(there)i(is)e(some)i Fy(p)25 b Fu(\024)g Fy(n)30 b Fz(for)g(whic)m(h)1743 2542 y Fy(b)1782 2556 y Fw(p)1847 2542 y Fo(&)1952 2419 y Fu(p)p 2028 2419 92 4 v 52 x Fy(\033)2080 2485 y Fw(p)p 1952 2521 168 4 v 1996 2604 a Fy(m)2130 2542 y(:)-2202 b Fz(\(6.48\))-47 2772 y(By)32 b(\(6.10\))g(therefore)1151 3032 y Fy(\036)1205 2994 y Fx(0)q(0)1248 3032 y Fz(\()p Fy(\025)1336 3046 y Fq(0)1376 3032 y Fz(\))25 b Fu(\025)g Fy(\025)1585 2994 y Fq(2)1585 3054 y(0)1806 2970 y Fy(b)1845 2984 y Fw(p)p 1635 3011 421 4 v 1635 3117 a Fz(\()p Fy(\025)1723 3085 y Fq(2)1723 3143 y(0)1783 3117 y Fz(+)20 b Fy(\033)1926 3131 y Fw(p)1965 3117 y Fz(\))2010 3054 y Fr(3)p 2011 3066 31 3 v 2011 3107 a(2)2090 3032 y Fo(&)25 b Fy(\025)2239 2994 y Fq(2)2239 3054 y(0)2304 2970 y Fy(b)2343 2984 y Fw(p)p 2289 3011 110 4 v 2289 3138 a Fy(\033)2354 3054 y Fr(3)p 2354 3066 31 3 v 2354 3107 a(2)2341 3150 y Fw(p)2434 3032 y Fo(&)2579 2970 y Fy(\025)2632 2937 y Fq(2)2632 2995 y(0)p 2540 3011 172 4 v 2540 3094 a Fy(m\033)2672 3108 y Fw(p)2721 3032 y Fy(:)-2793 b Fz(\(6.49\))-47 3344 y(Com)m(bining)29 b(\(6.49\))r(,)i(\(6.43\))r(,)g(and)f(\(6.33\))i (with)d Fy(B)h Fu(\020)25 b Fy(\025)1886 3305 y Fx(\000)p Fq(1)1886 3370 y(0)1980 3344 y Fy(\036)2034 3311 y Fx(0)q(0)2077 3344 y Fz(\()p 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y Fw(k)2998 3628 y Fy(;)-3070 b Fz(\(6.50\))-47 3910 y(whic)m(h)28 b(is)g(precisely)g (the)h(second)g(b)s(ound)e(in)h(\(6.1\))r(.)40 b(T)-8 b(o)30 b(pass)f(to)h(the)f(\014nal)f(expression)g(in)g(\(6.50\))j(one)f (uses)f(\(6.48\))-47 4023 y(as)39 b(w)m(ell)g(as)g Fy(b)426 4038 y Fw(k)509 4023 y Fu(\024)620 3944 y Fn(p)p 710 3944 299 4 v 710 4023 a Fy(\033)762 4038 y Fw(k)825 4023 y Fz(+)20 b Fy(\025)969 3991 y Fq(2)969 4049 y(0)1049 4023 y Fo(.)1160 3965 y Fu(p)p 1235 3965 95 4 v 1235 4023 a Fy(\033)1287 4038 y Fw(k)1330 4023 y Fz(,)42 b(whic)m(h)c(follo) m(ws)g(immediately)f(from)i(\(6.11\))j(or)d(\(6.47\))r(.)68 b(W)-8 b(e)40 b(will)d(no)m(w)-47 4136 y(estimate)30 b(the)f(oscillatory)g(in)m(tegral)g(on)g(the)h(in)m(terv)-5 b(al)28 b([)p Fy(\025)1924 4150 y Fq(0)1964 4136 y Fy(;)15 b Fu(1)p Fz(\))30 b(b)m(y)f(means)g(of)36 b(\(6.20\))s(.)k(Recall)29 b(that)h Fy(\025)3491 4150 y Fq(0)3556 4136 y Fu(\024)3652 4084 y(p)p 3727 4084 92 4 v 3727 4136 a Fy(\033)3779 4150 y Fw(p)3819 4136 y Fz(.)40 b(If)-47 4249 y Fy(\025)6 4263 y Fq(0)70 4249 y Fu(\024)25 b Fy(\025)h Fu(\024)f Fz(2)386 4196 y Fu(p)p 462 4196 V 53 x Fy(\033)514 4263 y Fw(p)554 4249 y Fz(,)30 b(then)g(b)m(y)i(\(6.7\))777 4527 y Fy(\036)831 4489 y Fx(0)854 4527 y Fz(\()p Fy(\025)p Fz(\))26 b Fu(\025)f Fy(\025)1167 4403 y Fn(Z)1258 4429 y Fw(\025)1218 4609 y(\025)1259 4618 y Fr(0)1473 4465 y Fy(b)1512 4479 y Fw(p)1552 4465 y Fy(s)p 1329 4506 411 4 v 1329 4612 a Fz(\()p Fy(s)1407 4586 y Fq(2)1466 4612 y Fz(+)20 b Fy(\033)1609 4626 y Fw(p)1649 4612 y Fz(\))1694 4549 y Fr(3)p 1694 4561 31 3 v 1694 4602 a(2)1764 4527 y Fy(ds)25 b Fo(&)g Fy(\025)2043 4403 y Fn(Z)2134 4429 y Fw(\025)2094 4609 y(\025)2135 4618 y Fr(0)2204 4465 y Fy(b)2243 4479 y Fw(p)2283 4465 y Fy(s)p 2204 4506 122 4 v 2210 4634 a(\033)2275 4549 y Fr(3)p 2275 4561 31 3 v 2275 4602 a(2)2262 4645 y Fw(p)2351 4527 y Fy(ds)g Fu(\025)2611 4465 y Fy(\025)2664 4432 y Fq(2)p 2572 4506 172 4 v 2572 4589 a Fy(m\033)2704 4603 y Fw(p)2753 4527 y Fz(\()p Fy(\025)c Fu(\000)f Fy(\025)3006 4541 y Fq(0)3045 4527 y Fz(\))p Fy(;)-3152 b Fz(\(6.51\))-47 4821 y(whereas)30 b(for)g(2)481 4769 y Fu(p)p 557 4769 92 4 v 52 x Fy(\033)609 4835 y Fw(p)674 4821 y Fu(\024)25 b Fy(\025)p Fz(,)1223 5098 y Fy(\036)1277 5060 y Fx(0)1300 5098 y Fz(\()p Fy(\025)p Fz(\))h Fu(\025)f Fy(\025)1613 4974 y Fn(Z)1704 5001 y Fq(2)1739 4962 y Fx(p)p 1798 4962 77 3 v 39 x Fw(\033)1838 5009 y Fs(p)1664 5180 y Fw(\025)1705 5189 y Fr(0)2048 5036 y Fy(b)2087 5050 y Fw(p)2127 5036 y Fy(s)p 1904 5077 411 4 v 1904 5183 a Fz(\()p Fy(s)1982 5157 y Fq(2)2041 5183 y Fz(+)20 b Fy(\033)2184 5197 y Fw(p)2224 5183 y Fz(\))2269 5120 y Fr(3)p 2269 5132 31 3 v 2269 5173 a(2)2339 5098 y Fy(ds)25 b Fo(&)2573 5037 y Fy(\025)p 2560 5077 80 4 v 2560 5160 a(m)2650 5098 y(:)-2722 b Fz(\(6.52\))1903 5472 y(55)p eop %%Page: 56 56 56 55 bop -47 274 a Fz(It)40 b(follo)m(ws)f(from)g(\(6.51\))j(and)d (\(6.52\))j(that)e(the)g(condition)f(\(6.19\))j(in)c(Lemma)i(6.6)h (holds)d(with)h Fy(R)i Fz(=)g Fy(\033)3719 288 y Fw(p)3798 274 y Fz(and)-47 387 y Fy(c)25 b Fu(\020)g Fy(m)193 354 y Fx(\000)p Fq(1)287 387 y Fz(.)41 b(Hence)31 b(\(6.20\))i(implies)28 b(that)826 509 y Fn(\014)826 564 y(\014)826 618 y(\014)826 673 y(\014)826 727 y(\014)857 545 y(Z)948 571 y Fx(1)907 751 y Fw(\025)948 760 y Fr(0)1037 668 y Fy(e)1079 631 y Fw(i\036)p Fq(\()p Fw(\025)p Fq(\))1449 607 y Fy(\025)p 1281 647 390 4 v 1281 666 a Fn(p)p 1372 666 299 4 v 82 x Fy(\025)1425 721 y Fq(2)1485 748 y Fz(+)20 b Fy(\033)1628 763 y Fw(k)1695 668 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)1993 509 y Fn(\014)1993 564 y(\014)1993 618 y(\014)1993 673 y(\014)1993 727 y(\014)2050 668 y Fo(.)25 b Fy(m)2236 604 y Fr(3)p 2236 616 31 3 v 2236 657 a(2)2301 668 y Fz(+)2391 542 y Fn(r)p 2482 542 192 4 v 2492 606 a Fy(m\033)2624 620 y Fw(p)p 2492 647 172 4 v 2531 731 a Fy(\033)2583 746 y Fw(k)2699 668 y Fo(.)g Fy(m)2885 604 y Fr(3)p 2885 616 31 3 v 2885 657 a(2)2941 606 y Fy(b)2980 620 y Fw(p)p 2939 647 82 4 v 2939 731 a Fy(b)2978 746 y Fw(k)3031 668 y Fy(;)-3103 b Fz(\(6.53\))-47 968 y(as)26 b(desired.)37 b(T)-8 b(o)27 b(pass)e(to)h(the)g(\014nal)f (inequalit)m(y)f(w)m(e)i(used)f(\(6.48\))j(and)d(that)h Fy(b)2591 983 y Fw(k)2659 968 y Fu(\024)2755 889 y Fn(p)p 2846 889 299 4 v 79 x Fy(\033)2898 983 y Fw(k)2961 968 y Fz(+)20 b Fy(\025)3105 937 y Fq(2)3105 994 y(0)3170 968 y Fo(.)3266 910 y Fu(p)p 3341 910 95 4 v 3341 968 a Fy(\033)3393 983 y Fw(k)3436 968 y Fz(.)39 b(The)25 b(lemma)-47 1081 y(is)k(pro)m(v)m(en)i(in)e(all)g(cases.)p 3879 1081 4 62 v 3883 1023 55 4 v 3883 1081 V 3937 1081 4 62 v -47 1268 a Fl(R)-5 b(emark)34 b(6.9.)46 b Fz(It)23 b(is)f(p)s(ossible)f(to)j(obtain)e(the)i(same)f(b)s(ounds)e(as)j(in)e (Lemma)h(6.1)h(b)m(y)f(means)g(of)g(a)h(single)e(in)m(tegration)-47 1381 y(b)m(y)33 b(parts.)49 b(Whereas)34 b(this)e(w)m(ould)g(lead)g(to) i(some)g(minor)e(simpli\014cations,)f(as)i(a)m(v)m(oiding)h(\(6.6\))r (,)g(w)m(e)g(b)s(eliev)m(e)e(that)-47 1494 y(the)d(previous)g(argumen)m (t)h(in)m(v)m(olving)e(t)m(w)m(o)i(in)m(tegrations)g(b)m(y)f(parts)h (migh)m(t)f(ha)m(v)m(e)i(some)f(in)m(terest)f(in)f(its)h(o)m(wn)h(righ) m(t.)-47 1607 y(F)-8 b(or)32 b(example,)h(it)e(applies)f(to)j(p)s(oten) m(tials)e(whose)h(\014rst)f(deriv)-5 b(ativ)m(e)32 b(do)s(es)g(not)g (deca)m(y)h(\(but)f(is)f(b)s(ounded,)f(sa)m(y\).)47 b(In)-47 1720 y(the)30 b(next)h(section)f(w)m(e)h(sho)m(w)f(ho)m(w)h(to)g (implemen)m(t)e(an)h(argumen)m(t)h(based)f(on)g(a)h(single)e(in)m (tegration)i(b)m(y)f(parts.)-47 2007 y FA(7)134 b(Estimates)48 b(for)d(oscillatory)h(in)l(tegrals)h(I)t(I)-47 2209 y Fz(The)39 b(purp)s(ose)f(of)i(this)f(section)h(is)f(to)i(pro)m(v)m(e)g (Lemmas)e(7.5)i(and)f(7.6)h(b)s(elo)m(w.)68 b(These)40 b(lemmas)f(are)i(needed)e(to)-47 2322 y(con)m(trol)c(the)f(oscillatory) g(in)m(tegrals)g(arising)e(in)h(\(5.41\))k(and)d(\(5.42\))r(.)53 b(The)34 b(argumen)m(ts)g(are)h(similar)d(to)j(the)g(ones)-47 2435 y(from)e(the)h(previous)f(section,)i(the)f(main)f(di\013erence)g (b)s(eing)g(the)h(singularit)m(y)d(at)k(the)f(p)s(oin)m(t)3209 2379 y Fu(p)p 3285 2379 80 4 v 56 x Fy(\032)3332 2450 y Fw(`)3365 2435 y Fz(.)51 b(T)-8 b(o)34 b(o)m(v)m(ercome)-47 2562 y(it,)29 b(w)m(e)h(c)m(hange)h(v)-5 b(ariables)28 b Fy(u)d Fz(=)1050 2481 y Fn(p)p 1141 2481 284 4 v 81 x Fy(\032)1188 2577 y Fw(`)1241 2562 y Fu(\000)20 b Fy(\025)1385 2536 y Fq(2)1425 2562 y Fz(.)40 b(This)28 b(leads)g(to)j(a)f(new)f (class)g(of)g(phase)h(functions)e Fy( )s Fz(\()p Fy(u)p Fz(\).)41 b(W)-8 b(e)31 b(presen)m(t)-47 2675 y(some)g(useful)d(prop)s (erties)h(of)h(this)g(class)g(in)f(the)h(follo)m(wing)f(lemma.)-47 2863 y Fm(Lemma)j(7.1.)47 b Fl(L)-5 b(et)39 b Fy( )s Fz(\()p Fy(u)p Fz(\))g(=)1036 2827 y Fq(1)p 1036 2842 36 4 v 1036 2894 a(2)1081 2863 y Fy(u)1133 2830 y Fq(2)1198 2863 y Fz(+)1294 2795 y Fn(P)1390 2821 y Fw(m)1390 2890 y(j)t Fq(=1)1532 2863 y Fy(b)1571 2877 y Fw(j)1622 2786 y Fn(p)p 1713 2786 279 4 v 77 x Fy(\034)1753 2877 y Fw(j)1810 2863 y Fu(\000)20 b Fy(u)1953 2836 y Fq(2)2031 2863 y Fl(wher)-5 b(e)41 b Fy(\034)2335 2877 y Fq(1)2412 2863 y Fu(\025)c Fy(\034)2560 2877 y Fq(2)2637 2863 y Fu(\025)g Fy(:)15 b(:)g(:)38 b Fu(\025)g Fy(\034)3037 2877 y Fw(m)3140 2863 y Fy(>)g Fz(0)h Fl(and)i Fy(b)3556 2877 y Fw(j)3630 2863 y Fy(>)c Fz(0)j Fl(ar)-5 b(e)-47 2976 y(arbitr)g(ary.)44 b(F)-7 b(or)33 b(any)h(choic)-5 b(e)33 b(of)g(these)g(p)-5 b(ar)g(ameters)35 b(the)e(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties)35 b(hold:)60 3163 y(1.)46 b Fy( )s Fz(\()p Fy(u)p Fz(\))38 b Fl(has)f(at)g(most)g(one)g(critic)-5 b(al)37 b(p)-5 b(oint)38 b Fy(u)32 b Fu(2)f Fz(\(0)p Fy(;)1920 3105 y Fu(p)p 1997 3105 107 4 v 1997 3163 a Fy(\034)2037 3177 y Fw(m)2104 3163 y Fz(\))p Fl(.)53 b(This)37 b(p)-5 b(oint)37 b(wil)5 b(l)37 b(always)h(b)-5 b(e)36 b(denote)-5 b(d)38 b(by)f Fy(u)3762 3177 y Fq(0)3801 3163 y Fl(.)53 b(If)180 3276 y Fy(u)232 3290 y Fq(0)299 3276 y Fy(<)28 b(u)g(<)577 3218 y Fu(p)p 652 3218 V 652 3276 a Fy(\034)692 3290 y Fw(m)759 3276 y Fl(,)34 b(then)g Fy( )1086 3243 y Fx(0)1110 3276 y Fz(\()p Fy(u)p Fz(\))29 b Fy(<)f Fz(0)34 b Fl(and)h Fy( )1679 3243 y Fx(0)q(0)1722 3276 y Fz(\()p Fy(u)p Fz(\))29 b Fy(<)e Fz(0)p Fl(,)35 b(wher)-5 b(e)g(as)36 b Fy( )2478 3243 y Fx(0)2502 3276 y Fz(\()p Fy(u)p Fz(\))28 b Fy(>)g Fz(0)34 b Fl(on)h Fz(\(0)p Fy(;)15 b(u)3134 3290 y Fq(0)3175 3276 y Fz(\))p Fl(.)46 b(Ther)-5 b(e)35 b(is)f(at)h(most)180 3389 y(one)e(p)-5 b(oint)34 b Fy(u)634 3403 y Fq(1)699 3389 y Fu(2)25 b Fz(\(0)p Fy(;)15 b(u)957 3403 y Fq(0)997 3389 y Fz(\))33 b Fl(such)g(that)h Fy( )1520 3356 y Fx(00)1563 3389 y Fz(\()p Fy(u)1650 3403 y Fq(1)1689 3389 y Fz(\))26 b(=)f(0)p Fl(.)60 3577 y(2.)46 b(If)32 b Fy(u)327 3591 y Fq(0)392 3577 y Fy(>)25 b Fz(0)33 b Fl(exists,)g(then)1420 3845 y Fy( )1482 3808 y Fx(0)1506 3845 y Fz(\()p Fy(u)p Fz(\))26 b(=)f Fu(\000)p Fy(u)1888 3722 y Fn(Z)1979 3748 y Fw(u)1938 3928 y(u)1979 3937 y Fr(0)2073 3732 y Fw(m)2039 3759 y Fn(X)2043 3954 y Fw(j)t Fq(=1)2326 3783 y Fy(b)2365 3797 y Fw(j)2417 3783 y Fy(s)p 2195 3824 395 4 v 2195 3930 a Fz(\()p Fy(\034)2270 3944 y Fw(j)2327 3930 y Fu(\000)20 b Fy(s)2461 3904 y Fq(2)2500 3930 y Fz(\))2545 3867 y Fr(3)p 2545 3879 31 3 v 2545 3921 a(2)2615 3845 y Fy(ds)-2525 b Fz(\(7.1\))180 4142 y Fl(for)33 b(any)g Fz(0)26 b Fy(<)f(u)g(<)839 4083 y Fu(p)p 915 4083 107 4 v 59 x Fy(\034)955 4156 y Fw(m)1021 4142 y Fl(,)33 b(wher)-5 b(e)g(as)35 b(if)d Fy(u)1558 4156 y Fq(0)1623 4142 y Fy(>)25 b Fz(0)33 b Fl(do)-5 b(es)33 b(not)h(exist,)e(then)1430 4422 y Fu(j)p Fy( )1517 4384 y Fx(0)1541 4422 y Fz(\()p Fy(u)p Fz(\))p Fu(j)26 b(\025)f Fy(u)1877 4298 y Fn(Z)1968 4324 y Fw(u)1928 4504 y Fq(0)2063 4308 y Fw(m)2029 4335 y Fn(X)2033 4531 y Fw(j)t Fq(=1)2316 4360 y Fy(b)2355 4374 y Fw(j)2406 4360 y Fy(s)p 2185 4401 395 4 v 2185 4507 a Fz(\()p Fy(\034)2260 4521 y Fw(j)2317 4507 y Fu(\000)20 b Fy(s)2451 4480 y Fq(2)2490 4507 y Fz(\))2535 4444 y Fr(3)p 2535 4456 31 3 v 2535 4497 a(2)2605 4422 y Fy(ds)-2515 b Fz(\(7.2\))180 4718 y Fl(for)33 b(al)5 b(l)33 b Fz(0)26 b Fy(<)f(u)g(<)795 4659 y Fu(p)p 871 4659 107 4 v 59 x Fy(\034)911 4732 y Fw(m)977 4718 y Fl(.)60 4924 y(3.)46 b(If)278 4856 y Fn(P)374 4882 y Fw(m)374 4951 y(j)t Fq(=1)556 4876 y Fw(b)586 4886 y Fs(j)p 526 4903 123 4 v 526 4917 a Fx(p)p 585 4917 64 3 v 38 x Fw(\034)616 4965 y Fs(j)689 4924 y Fu(\024)801 4888 y Fq(1)p 801 4903 36 4 v 801 4955 a(2)846 4924 y Fl(,)36 b(then)g Fy( )1177 4891 y Fx(0)1201 4924 y Fz(\()p Fy(u)p Fz(\))31 b Fu(\025)1465 4888 y Fq(1)p 1465 4903 V 1465 4955 a(4)1511 4924 y Fy(u)k Fl(for)h(al)5 b(l)36 b Fz(0)31 b Fy(<)g(u)f(<)2251 4888 y Fq(1)p 2251 4903 V 2251 4955 a(2)2296 4865 y Fu(p)p 2372 4865 107 4 v 59 x Fy(\034)2412 4938 y Fw(m)2478 4924 y Fl(.)50 b(If)2655 4856 y Fn(P)2751 4882 y Fw(m)2751 4951 y(j)t Fq(=1)2933 4876 y Fw(b)2963 4886 y Fs(j)p 2903 4903 123 4 v 2903 4917 a Fx(p)p 2961 4917 64 3 v 2961 4955 a Fw(\034)2992 4965 y Fs(j)3066 4924 y Fu(\025)30 b Fz(2)p Fl(,)37 b(then)f Fy( )3544 4891 y Fx(0)3567 4924 y Fz(\()p Fy(u)p Fz(\))c Fu(\024)e(\000)p Fy(u)180 5055 y Fl(for)j(al)5 b(l)33 b Fz(0)26 b Fy(<)f(u)g(<)795 4996 y Fu(p)p 871 4996 107 4 v 59 x Fy(\034)911 5069 y Fw(m)977 5055 y Fl(.)1903 5472 y Fz(56)p eop %%Page: 57 57 57 56 bop -47 274 a Fl(Pr)-5 b(o)g(of.)47 b Fz(The)29 b(\014rst)h(assertion)g(follo)m(ws)g(from)g(the)g(explicit)f (expressions)834 481 y Fy( )896 448 y Fx(0)920 481 y Fz(\()p Fy(u)p Fz(\))p 834 522 209 4 v 913 605 a Fy(u)1136 543 y Fz(=)83 b(1)20 b Fu(\000)1481 429 y Fw(m)1446 456 y Fn(X)1451 652 y Fw(j)t Fq(=1)1750 481 y Fy(b)1789 495 y Fw(j)p 1603 522 370 4 v 1603 540 a Fn(p)p 1694 540 279 4 v 76 x Fy(\034)1734 630 y Fw(j)1790 616 y Fu(\000)g Fy(u)1933 590 y Fq(2)825 858 y Fy( )887 821 y Fx(0)q(0)930 858 y Fz(\()p Fy(u)p Fz(\))84 b(=)1300 797 y Fy( )1362 764 y Fx(0)1385 797 y Fz(\()p Fy(u)p Fz(\))p 1300 837 209 4 v 1378 921 a Fy(u)1538 858 y Fu(\000)1664 745 y Fw(m)1629 772 y Fn(X)1634 967 y Fw(j)t Fq(=1)1897 796 y Fy(b)1936 810 y Fw(j)1987 796 y Fy(u)2039 763 y Fq(2)p 1786 837 405 4 v 1786 943 a Fz(\()p Fy(\034)1861 957 y Fw(j)1917 943 y Fu(\000)20 b Fy(u)2060 917 y Fq(2)2100 943 y Fz(\))2145 880 y Fr(3)p 2145 892 31 3 v 2145 933 a(2)2225 858 y Fz(=)25 b(1)c Fu(\000)2512 745 y Fw(m)2477 772 y Fn(X)2482 967 y Fw(j)t Fq(=1)2760 796 y Fy(b)2799 810 y Fw(j)2836 796 y Fy(\034)2876 810 y Fw(j)p 2634 837 405 4 v 2634 943 a Fz(\()p Fy(\034)2709 957 y Fw(j)2766 943 y Fu(\000)f Fy(u)2909 917 y Fq(2)2948 943 y Fz(\))2993 880 y Fr(3)p 2993 892 31 3 v 2993 933 a(2)3048 858 y Fy(:)-47 1157 y Fz(F)-8 b(or)31 b(the)f(second)h(assertion,)f(viz.)g (\(7.1\))j(and)c(\(7.2\))r(,)i(in)m(tegrate)g(out)1384 1330 y Fn(\020)1449 1369 y Fy( )1511 1336 y Fx(0)1534 1369 y Fz(\()p Fy(u)p Fz(\))p 1449 1410 209 4 v 1527 1493 a Fy(u)1667 1330 y Fn(\021)1721 1352 y Fx(0)1770 1431 y Fz(=)25 b Fu(\000)1986 1317 y Fw(m)1952 1344 y Fn(X)1956 1540 y Fw(j)t Fq(=1)2222 1369 y Fy(b)2261 1383 y Fw(j)2312 1369 y Fy(u)p 2108 1410 370 4 v 2108 1428 a Fn(p)p 2199 1428 279 4 v 76 x Fy(\034)2239 1518 y Fw(j)2296 1504 y Fu(\000)20 b Fy(u)2439 1478 y Fq(2)2488 1431 y Fy(:)-47 1757 y Fz(If)30 b Fy(u)96 1771 y Fq(0)162 1757 y Fy(>)c Fz(0)31 b(do)s(es)g(not)g(exist,)g(then)g(use)f(that)1517 1713 y Fw( )1565 1689 y Ft(0)1588 1713 y Fq(\()p Fw(u)p Fq(\))p 1517 1736 167 4 v 1580 1789 a Fw(u)1719 1757 y Fu(\024)c Fz(0)32 b(for)e(all)g Fy(u)c Fu(\025)g Fz(0.)43 b(If)2538 1689 y Fn(P)2634 1715 y Fw(m)2634 1784 y(j)t Fq(=1)2816 1709 y Fw(b)2846 1719 y Fs(j)p 2786 1736 123 4 v 2786 1751 a Fx(p)p 2845 1751 64 3 v 38 x Fw(\034)2876 1799 y Fs(j)2945 1757 y Fu(\024)3052 1722 y Fq(1)p 3052 1737 36 4 v 3052 1789 a(2)3097 1757 y Fz(,)31 b(then)g(for)g(all)e(0)e Fy(<)f(u)g(<)-37 1865 y Fq(1)p -37 1880 V -37 1932 a(2)8 1842 y Fu(p)p 84 1842 107 4 v 59 x Fy(\034)124 1915 y Fw(m)1278 2070 y(m)1244 2097 y Fn(X)1248 2292 y Fw(j)t Fq(=1)1548 2121 y Fy(b)1587 2135 y Fw(j)p 1400 2162 370 4 v 1400 2181 a Fn(p)p 1491 2181 279 4 v 76 x Fy(\034)1531 2271 y Fw(j)1588 2257 y Fu(\000)20 b Fy(u)1731 2231 y Fq(2)1805 2183 y Fu(\024)1901 2040 y Fn(r)p 1992 2040 66 4 v 2002 2122 a Fz(4)p 2002 2162 46 4 v 2002 2246 a(3)2107 2070 y Fw(m)2073 2097 y Fn(X)2077 2292 y Fw(j)t Fq(=1)2268 2121 y Fy(b)2307 2135 y Fw(j)p 2229 2162 153 4 v 2229 2193 a Fu(p)p 2305 2193 77 4 v 53 x Fy(\034)2345 2260 y Fw(j)2416 2183 y Fu(\024)2560 2122 y Fz(1)p 2522 2162 122 4 v 2522 2181 a Fu(p)p 2598 2181 46 4 v 75 x Fz(3)-47 2508 y(whic)m(h)29 b(implies)e(that)k Fy( )780 2475 y Fx(0)804 2508 y Fz(\()p Fy(u)p Fz(\))26 b Fu(\025)f Fy(u)p Fz(\(1)c Fu(\000)e Fz(1)p Fy(=)1381 2433 y Fu(p)p 1458 2433 V 1458 2508 a Fz(3\))26 b Fu(\025)f Fy(u=)p Fz(4)31 b(for)f(those)h Fy(u)p Fz(.)41 b(If)29 b(on)h(the)h(other)f (hand)3166 2440 y Fn(P)3262 2466 y Fw(m)3262 2535 y(j)t Fq(=1)3443 2460 y Fw(b)3473 2470 y Fs(j)p 3413 2487 123 4 v 3413 2502 a Fx(p)p 3472 2502 64 3 v 37 x Fw(\034)3503 2549 y Fs(j)3571 2508 y Fu(\025)25 b Fz(2,)31 b(then)1380 2816 y Fu(\000)p Fy( )1513 2778 y Fx(0)1537 2816 y Fz(\()p Fy(u)p Fz(\))26 b Fu(\025)e Fy(u)1897 2702 y Fw(m)1863 2730 y Fn(X)1867 2925 y Fw(j)t Fq(=1)2057 2754 y Fy(b)2096 2768 y Fw(j)p 2019 2795 153 4 v 2019 2826 a Fu(p)p 2095 2826 77 4 v 52 x Fy(\034)2135 2892 y Fw(j)2201 2816 y Fu(\000)c Fy(u)26 b Fu(\025)f Fy(u)-47 3113 y Fz(for)30 b(all)f(0)d Fy(<)f(u)g(<)558 3054 y Fu(p)p 634 3054 107 4 v 59 x Fy(\034)674 3127 y Fw(m)740 3113 y Fz(.)p 3879 3113 4 62 v 3883 3054 55 4 v 3883 3113 V 3937 3113 4 62 v -47 3300 a(In)k(what)h(follo)m(ws)f(let)h Fy(\037)g Fz(b)s(e)f(a)h(smo)s(oth)g(non-decreasing)g(function)e(with)h Fy(\037)p Fz(\()p Fy(\025)p Fz(\))d(=)f(0)30 b(for)g Fy(\025)25 b Fu(\024)3207 3264 y Fq(1)p 3207 3279 36 4 v 3207 3332 a(4)3282 3300 y Fz(and)30 b Fy(\037)p Fz(\()p Fy(\025)p Fz(\))c(=)f(1)30 b(for)-47 3413 y Fy(\025)25 b Fu(\025)137 3377 y Fq(1)p 137 3392 V 137 3444 a(2)182 3413 y Fz(.)41 b(F)-8 b(urthermore,)30 b(w)m(e)h(shall)e(use)h(the)g (notation)h Fy(\037)1881 3427 y Fw(E)1971 3413 y Fz(for)f(the)h (indicator)e(of)h(a)h(set)g Fy(E)5 b Fz(.)-47 3611 y Fm(Lemma)32 b(7.2.)47 b Fl(L)-5 b(et)30 b Fy( )s Fz(\()p Fy(u)p Fz(\))c(=)1002 3575 y Fq(1)p 1002 3590 V 1002 3642 a(2)1048 3611 y Fy(u)1100 3578 y Fq(2)1154 3611 y Fz(+)1240 3543 y Fn(P)1335 3569 y Fw(m)1335 3638 y(j)t Fq(=1)1477 3611 y Fy(b)1516 3625 y Fw(j)1568 3535 y Fn(p)p 1659 3535 279 4 v 76 x Fy(\034)1699 3625 y Fw(j)1755 3611 y Fu(\000)20 b Fy(u)1898 3585 y Fq(2)1968 3611 y Fl(wher)-5 b(e)31 b Fy(\034)2262 3625 y Fq(1)2326 3611 y Fu(\025)25 b Fy(\034)2462 3625 y Fq(2)2527 3611 y Fu(\025)g Fy(:)15 b(:)g(:)26 b Fu(\025)f Fy(\034)2890 3625 y Fw(m)2981 3611 y Fy(>)g Fz(0)31 b Fl(and)g Fy(b)3366 3625 y Fw(j)3428 3611 y Fy(>)25 b Fz(0)p Fl(.)41 b(Assume)-47 3724 y(that)34 b Fy(!)h Fl(is)e(a)g(di\013er)-5 b(entiable)33 b(function)g(such)g (that)1603 3928 y Fu(j)p Fy(!)1688 3891 y Fq(\()p Fw(j)t Fq(\))1779 3928 y Fz(\()p Fy(u)p Fz(\))p Fu(j)27 b(\024)e Fy(a)2097 3942 y Fq(0)2151 3928 y Fy(u)2203 3891 y Fx(\000)p Fw(j)-47 4132 y Fl(for)33 b(al)5 b(l)33 b Fy(u)25 b(>)g Fz(0)33 b Fl(and)h Fy(j)d Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fl(.)60 4320 y(1.)46 b(Supp)-5 b(ose)34 b(that)g Fy(A)26 b(>)f Fz(0)p Fy(;)15 b(u)1039 4334 y Fq(0)1104 4320 y Fu(\025)25 b Fz(0)33 b Fl(\(wher)-5 b(e)34 b Fy(u)1624 4334 y Fq(0)1696 4320 y Fl(is)f(not)g(ne)-5 b(c)g(essarily)34 b(a)f(critic)-5 b(al)34 b(p)-5 b(oint)34 b(of)f Fy( )s Fl(\))g(have)g(the)g(pr)-5 b(op)g(erty)180 4433 y(that)34 b Fu(j)p Fy( )453 4400 y Fx(0)477 4433 y Fz(\()p Fy(u)p Fz(\))p Fu(j)26 b(\025)f Fy(Au)866 4400 y Fq(2)906 4433 y Fz(\()p Fy(u)20 b Fu(\000)g Fy(u)1156 4447 y Fq(0)1196 4433 y Fz(\))33 b Fl(and)g Fy( )1502 4400 y Fx(0)q(0)1545 4433 y Fz(\()p Fy(u)p Fz(\))26 b Fu(\024)f Fz(0)33 b Fl(for)g(al)5 b(l)2142 4374 y Fu(p)p 2218 4374 107 4 v 59 x Fy(\034)2258 4447 y Fw(m)2350 4433 y Fy(>)25 b(u)g Fu(\025)g Fy(u)2671 4447 y Fq(0)2710 4433 y Fl(.)42 b(L)-5 b(et)33 b Fy(\034)i Fu(\025)25 b Fy(\034)3148 4447 y Fw(m)3215 4433 y Fl(.)41 b(Then)721 4561 y Fn(\014)721 4616 y(\014)721 4670 y(\014)721 4725 y(\014)752 4569 y(Z)843 4595 y Fx(1)802 4775 y Fw(u)843 4784 y Fr(0)933 4693 y Fy(e)975 4655 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1173 4592 y Fn(\020)1227 4693 y Fz(1)21 b Fu(\000)e Fy(\037)1440 4592 y Fn(\020)1570 4631 y Fy(u)p 1505 4672 183 4 v 1505 4697 a Fu(p)p 1580 4697 107 4 v 1580 4755 a Fy(\034)1620 4769 y Fw(m)1697 4592 y Fn(\021\021)1805 4693 y Fy(!)s Fz(\()p Fy(u)p Fz(\))2035 4631 y Fy(u)p 1997 4672 126 4 v 1997 4690 a Fu(p)p 2073 4690 51 4 v 66 x Fy(\034)2149 4693 y(du)2248 4561 y Fn(\014)2248 4616 y(\014)2248 4670 y(\014)2248 4725 y(\014)2361 4693 y Fu(\024)98 b Fy(C)2595 4707 y Fq(0)2665 4693 y Fz(min)n(\()p Fy(A)2919 4655 y Fx(\000)2984 4628 y Fr(1)p 2985 4640 31 3 v 2985 4681 a(4)3029 4693 y Fy(;)15 b Fz(\()p Fy(A\034)10 b Fz(\))3257 4655 y Fx(\000)3322 4628 y Fr(1)p 3324 4640 V 3324 4681 a(2)3368 4693 y Fz(\))-3223 b(\(7.3\))867 4818 y Fn(\014)867 4872 y(\014)867 4927 y(\014)867 4981 y(\014)897 4826 y(Z)988 4852 y Fx(1)948 5032 y Fw(u)989 5041 y Fr(0)1078 4950 y Fy(e)1120 4912 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1318 4849 y Fn(\020)1373 4950 y Fz(1)20 b Fu(\000)g Fy(\037)1586 4849 y Fn(\020)1715 4888 y Fy(u)p 1650 4929 183 4 v 1650 4953 a Fu(p)p 1726 4953 107 4 v 59 x Fy(\034)1766 5026 y Fw(m)1842 4849 y Fn(\021)q(\021)1951 4950 y Fy(!)s Fz(\()p Fy(u)p Fz(\))15 b Fy(du)2247 4818 y Fn(\014)2247 4872 y(\014)2247 4927 y(\014)2247 4981 y(\014)2361 4950 y Fu(\024)98 b Fy(C)2595 4964 y Fq(0)2650 4950 y Fy(A)2718 4912 y Fx(\000)2783 4885 y Fr(1)p 2783 4897 31 3 v 2783 4938 a(4)2827 4950 y Fy(:)-2672 b Fz(\(7.4\))1903 5472 y(57)p eop %%Page: 58 58 58 57 bop 60 274 a Fl(2.)46 b(Supp)-5 b(ose)34 b(that)g Fu(j)p Fy( )799 241 y Fx(0)823 274 y Fz(\()p Fy(u)p Fz(\))p Fu(j)26 b(\025)1102 238 y Fq(1)p 1102 253 36 4 v 1102 305 a(2)1147 274 y Fy(u)33 b Fl(and)h Fy( )1471 241 y Fx(00)1514 274 y Fz(\()p Fy(u)p Fz(\))26 b Fu(\024)e Fz(0)33 b Fl(for)h(al)5 b(l)2121 238 y Fq(1)p 2121 253 V 2121 305 a(2)2166 215 y Fu(p)p 2242 215 107 4 v 59 x Fy(\034)2282 288 y Fw(m)2373 274 y Fy(>)25 b(u)h Fu(\025)f Fz(0)p Fl(.)42 b(Then)1224 390 y Fn(\014)1224 445 y(\014)1224 499 y(\014)1224 554 y(\014)1254 398 y(Z)1345 425 y Fx(1)1305 605 y Fq(0)1435 522 y Fy(e)1477 485 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1675 421 y Fn(\020)1729 522 y Fz(1)21 b Fu(\000)f Fy(\037)1943 421 y Fn(\020)2072 461 y Fy(u)p 2007 501 183 4 v 2007 526 a Fu(p)p 2083 526 107 4 v 59 x Fy(\034)2123 599 y Fw(m)2199 421 y Fn(\021\021)2308 522 y Fy(!)s Fz(\()p Fy(u)p Fz(\))15 b Fy(du)2604 390 y Fn(\014)2604 445 y(\014)2604 499 y(\014)2604 554 y(\014)2660 522 y Fu(\024)40 b Fy(C)2836 536 y Fq(0)2876 522 y Fy(:)-2721 b Fz(\(7.5\))-47 796 y Fl(In)32 b(b)-5 b(oth)34 b(c)-5 b(ases)34 b(the)f(c)-5 b(onstant)34 b Fy(C)1075 810 y Fq(0)1147 796 y Fl(only)f(dep)-5 b(ends)35 b(on)e Fy(a)1862 810 y Fq(0)1901 796 y Fl(.)-47 972 y(Pr)-5 b(o)g(of.)47 b Fz(By)30 b(our)g(assumptions)f(the)h(function)1308 1179 y Fy(h)p Fz(\()p Fy(u)p Fz(\))c(:=)1629 1078 y Fn(\020)1683 1179 y Fz(1)21 b Fu(\000)f Fy(\037)1897 1078 y Fn(\020)2026 1117 y Fy(u)p 1961 1158 183 4 v 1961 1183 a Fu(p)p 2037 1183 107 4 v 58 x Fy(\034)2077 1255 y Fw(m)2153 1078 y Fn(\021\021)2262 1179 y Fy(!)s Fz(\()p Fy(u)p Fz(\))2491 1117 y Fy(u)p 2454 1158 126 4 v 2454 1176 a Fu(p)p 2530 1176 51 4 v 66 x Fy(\034)-47 1418 y Fz(satis\014es)30 b(the)g(deriv)-5 b(ativ)m(e)30 b(estimates)1034 1608 y Fu(j)p Fy(h)1111 1570 y Fq(\()p Fw(j)t Fq(\))1203 1608 y Fz(\()p Fy(u)p Fz(\))p Fu(j)c Fo(.)f Fy(u)1524 1570 y Fx(\000)p Fw(j)1662 1546 y Fy(u)p 1626 1587 126 4 v 1626 1605 a Fu(p)p 1701 1605 51 4 v 1701 1670 a Fy(\034)1822 1608 y Fz(for)30 b(all)60 b Fy(u)25 b(>)g Fz(0)31 b(and)e Fy(j)i Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fy(:)-47 1840 y Fz(Th)m(us)358 1925 y Fn(\014)358 1980 y(\014)358 2034 y(\014)358 2089 y(\014)388 1933 y(Z)479 1960 y Fx(1)439 2140 y Fw(u)480 2149 y Fr(0)569 2057 y Fy(e)611 2020 y Fw(i )r Fq(\()p Fw(u)p Fq(\))809 1956 y Fn(\020)863 2057 y Fz(1)21 b Fu(\000)f Fy(\037)1077 1956 y Fn(\020)1206 1996 y Fy(u)p 1141 2036 183 4 v 1141 2061 a Fu(p)p 1217 2061 107 4 v 59 x Fy(\034)1257 2134 y Fw(m)1333 1956 y Fn(\021\021)1442 2057 y Fy(!)s Fz(\()p Fy(u)p Fz(\))1671 1996 y Fy(u)p 1634 2036 126 4 v 1634 2054 a Fu(p)p 1710 2054 51 4 v 66 x Fy(\034)1785 2057 y(du)1884 1925 y Fn(\014)1884 1980 y(\014)1884 2034 y(\014)1884 2089 y(\014)204 2314 y Fo(.)368 2252 y Fy(\015)5 b Fz(\()p Fy(u)507 2266 y Fq(0)567 2252 y Fz(+)20 b Fy(\015)5 b Fz(\))p 368 2293 378 4 v 494 2311 a Fu(p)p 570 2311 51 4 v 66 x Fy(\034)776 2314 y Fz(+)867 2182 y Fn(\014)867 2237 y(\014)867 2291 y(\014)867 2346 y(\014)897 2190 y(Z)988 2216 y Fx(1)947 2396 y Fw(u)988 2405 y Fr(0)1078 2314 y Fy(e)1120 2276 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1318 2314 y Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)22 b Fu(\000)e Fy(u)1836 2328 y Fq(0)1875 2314 y Fz(\))p Fy(=\015)5 b Fz(\))15 b Fy(du)2156 2182 y Fn(\014)2156 2237 y(\014)2156 2291 y(\014)2156 2346 y(\014)204 2570 y Fo(.)368 2509 y Fy(\015)5 b Fz(\()p Fy(u)507 2523 y Fq(0)567 2509 y Fz(+)20 b Fy(\015)5 b Fz(\))p 368 2549 378 4 v 494 2567 a Fu(p)p 570 2567 51 4 v 66 x Fy(\034)776 2570 y Fz(+)867 2446 y Fn(Z)957 2473 y Fx(1)917 2653 y Fw(u)958 2662 y Fr(0)1057 2509 y Fu(j)p Fy( )1144 2476 y Fx(0)q(0)1187 2509 y Fz(\()p Fy(u)p Fz(\))p Fu(j)p 1057 2549 279 4 v 1072 2633 a Fy( )1134 2606 y Fx(0)1158 2633 y Fz(\()p Fy(u)p Fz(\))1280 2606 y Fq(2)1361 2570 y Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)22 b Fu(\000)d Fy(u)1878 2584 y Fq(0)1918 2570 y Fz(\))p Fy(=\015)5 b Fz(\))15 b Fy(du)22 b Fz(+)2312 2446 y Fn(Z)2403 2473 y Fx(1)2362 2653 y Fw(u)2403 2662 y Fr(0)2609 2509 y Fz(1)p 2503 2549 260 4 v 2503 2633 a Fu(j)p Fy( )2590 2606 y Fx(0)2614 2633 y Fz(\()p Fy(u)p Fz(\))p Fu(j)2787 2570 y(j)p Fz([)p Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)g Fu(\000)e Fy(u)3355 2584 y Fq(0)3395 2570 y Fz(\))p Fy(=\015)5 b Fz(\)])3587 2533 y Fx(0)3611 2570 y Fu(j)15 b Fy(du:)-3822 b Fz(\(7.6\))-47 2810 y(The)30 b(second)g(term)g(here)h(can)f(b)s(e)g(reduced)g(to)h(the)f(third)f(b)m (y)h(means)g(of)h(a)g(further)e(in)m(tegration)h(b)m(y)g(parts:)466 2920 y Fn(Z)556 2947 y Fx(1)516 3126 y Fw(u)557 3135 y Fr(0)666 2983 y Fy( )728 2950 y Fx(0)q(0)771 2983 y Fz(\()p Fy(u)p Fz(\))p 656 3023 248 4 v 656 3106 a Fy( )718 3080 y Fx(0)742 3106 y Fz(\()p Fy(u)p Fz(\))864 3080 y Fq(2)929 3044 y Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)22 b Fu(\000)e Fy(u)1447 3058 y Fq(0)1487 3044 y Fz(\))p Fy(=\015)5 b Fz(\))15 b Fy(du)27 b Fz(=)e Fu(\000)1977 2920 y Fn(Z)2067 2947 y Fx(1)2027 3126 y Fw(u)2068 3135 y Fr(0)2157 3044 y Fy( )2219 3007 y Fx(0)2243 3044 y Fz(\()p Fy(u)p Fz(\))2365 2943 y Fn(h)2520 2983 y Fz(1)p 2419 3023 V 2419 3106 a Fy( )2481 3080 y Fx(0)2505 3106 y Fz(\()p Fy(u)p Fz(\))2627 3080 y Fq(2)2692 3044 y Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)d Fu(\000)e Fy(u)3210 3058 y Fq(0)3249 3044 y Fz(\))p Fy(=\015)5 b Fz(\))3416 2943 y Fn(i)3460 2966 y Fx(0)3499 3044 y Fy(du)466 3300 y Fz(=)24 b(2)621 3177 y Fn(Z)713 3203 y Fx(1)673 3383 y Fw(u)714 3392 y Fr(0)823 3239 y Fy( )885 3206 y Fx(0)q(0)928 3239 y Fz(\()p Fy(u)p Fz(\))p 813 3279 V 813 3363 a Fy( )875 3336 y Fx(0)899 3363 y Fz(\()p Fy(u)p Fz(\))1021 3336 y Fq(2)1086 3300 y Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)e Fu(\000)e Fy(u)1604 3314 y Fq(0)1643 3300 y Fz(\))p Fy(=\015)5 b Fz(\))15 b Fy(du)22 b Fu(\000)2037 3177 y Fn(Z)2128 3203 y Fx(1)2088 3383 y Fw(u)2129 3392 y Fr(0)2309 3239 y Fz(1)p 2228 3279 209 4 v 2228 3363 a Fy( )2290 3336 y Fx(0)2314 3363 y Fz(\()p Fy(u)p Fz(\))2462 3300 y([)p Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)g Fu(\000)e Fy(u)3005 3314 y Fq(0)3044 3300 y Fz(\))p Fy(=\015)5 b Fz(\)])3236 3263 y Fx(0)3276 3300 y Fy(du:)-47 3545 y Fz(The)30 b(\014nal)f(in)m (tegral)h(in)f(\(7.6\))j(is)d(no)i(larger)f(than)462 3655 y Fn(Z)553 3682 y Fx(1)512 3861 y Fw(u)553 3870 y Fr(0)588 3861 y Fq(+)p Fw(\015)939 3717 y Fz(1)p 712 3758 501 4 v 712 3841 a Fy(A)15 b(u)847 3815 y Fq(2)887 3841 y Fz(\()p Fy(u)21 b Fu(\000)e Fy(u)1137 3855 y Fq(0)1177 3841 y Fz(\))1222 3678 y Fn(h)1265 3779 y Fy(\015)1317 3741 y Fx(\000)p Fq(1)1422 3717 y Fy(u)1474 3731 y Fq(0)1533 3717 y Fz(+)h Fy(\015)p 1422 3758 255 4 v 1486 3776 a Fu(p)p 1562 3776 51 4 v 66 x Fy(\034)1686 3779 y(\037)1743 3807 y Fq([)1773 3780 y Fr(1)p 1773 3792 31 3 v 1773 3833 a(4)1813 3807 y Fw(\015)t Fx(\024)p Fw(u)p Fx(\000)p Fw(u)2045 3816 y Fr(0)2079 3807 y Fx(\024)2144 3780 y Fr(1)p 2144 3792 V 2144 3833 a(2)2185 3807 y Fw(\015)t Fq(])2269 3779 y Fz(+)2410 3717 y(1)p 2370 3758 126 4 v 2370 3776 a Fu(p)p 2446 3776 51 4 v 66 x Fy(\034)2506 3678 y Fn(i)2564 3779 y Fy(du)25 b Fo(.)2994 3717 y Fz(1)p 2794 3758 446 4 v 2794 3841 a Fy(A)p Fz(\()p Fy(u)2949 3855 y Fq(0)3009 3841 y Fz(+)20 b Fy(\015)5 b Fz(\))p Fy(\015)3300 3717 y Fz(1)p 3260 3758 126 4 v 3260 3776 a Fu(p)p 3336 3776 51 4 v 66 x Fy(\034)3396 3779 y(;)-3468 b Fz(\(7.7\))-47 4022 y(where)35 b Fy(\037)g Fz(refers)g(to)h(a)f (usual)f(indicator)g(function.)55 b(As)35 b(alw)m(a)m(ys,)i(w)m(e)f(no) m(w)f(c)m(ho)s(ose)i Fy(\015)j Fz(so)c(that)g(the)f(\014rst)g(term)g (in)-47 4135 y(\(7.6\))d(equals)e(the)g(third.)39 b(This)29 b(leads)g(to)j(the)e(estimate)1633 4340 y(\(7.6\))d Fo(.)e Fz(\()p Fy(A\034)10 b Fz(\))2129 4302 y Fx(\000)2194 4275 y Fr(1)p 2195 4287 31 3 v 2195 4328 a(2)2239 4340 y Fy(:)-47 4543 y Fz(T)-8 b(o)31 b(obtain)g(the)g Fy(A)596 4507 y Fx(\000)661 4480 y Fr(1)p 661 4492 V 661 4533 a(4)737 4543 y Fz(b)s(ound)e(in)g(\(7.3\))r(,)j(one)f(includes)e(the)i (factor)2342 4507 y Fw(u)p 2313 4522 98 4 v 2313 4531 a Fx(p)p 2372 4531 39 3 v 48 x Fw(\034)2452 4543 y Fz(in)m(to)g(the)g (function)f Fy(!)s Fz(.)42 b(Since)31 b Fy(u)26 b Fu(\024)3692 4477 y(p)p 3768 4477 51 4 v 66 x Fy(\034)41 b Fz(on)-47 4666 y(the)32 b(supp)s(ort)d(of)j Fy(h)g Fz(this)f(can)h(b)s(e)f(done)g (without)g(violating)f(the)i(conditions)e(on)i Fy(!)s Fz(.)44 b(The)31 b(same)h(argumen)m(ts)g(as)g(in)-47 4779 y(the)e(previous)f(case)j(lead)e(to)460 4882 y Fn(\014)460 4936 y(\014)460 4991 y(\014)460 5046 y(\014)490 4890 y(Z)581 4916 y Fx(1)541 5096 y Fw(u)582 5105 y Fr(0)671 5014 y Fy(e)713 4976 y Fw(i )r Fq(\()p Fw(u)p Fq(\))911 4913 y Fn(\020)965 5014 y Fz(1)21 b Fu(\000)f Fy(\037)1179 4913 y Fn(\020)1308 4952 y Fy(u)p 1243 4993 183 4 v 1243 5017 a Fu(p)p 1319 5017 107 4 v 59 x Fy(\034)1359 5090 y Fw(m)1435 4913 y Fn(\021\021)1543 5014 y Fy(!)s Fz(\()p Fy(u)p Fz(\))1773 4952 y Fy(u)p 1735 4993 126 4 v 1735 5011 a Fu(p)p 1812 5011 51 4 v 1812 5076 a Fy(\034)1887 5014 y(du)1986 4882 y Fn(\014)1986 4936 y(\014)1986 4991 y(\014)1986 5046 y(\014)2100 5014 y Fo(.)82 b Fy(\015)26 b Fz(+)2417 4890 y Fn(Z)2507 4916 y Fx(1)2467 5096 y Fw(u)2508 5105 y Fr(0)2542 5096 y Fq(+)p Fw(\015)2887 4952 y Fz(1)p 2667 4993 485 4 v 2667 5076 a Fy(Au)2787 5050 y Fq(2)2827 5076 y Fz(\()p Fy(u)20 b Fu(\000)g Fy(u)3077 5090 y Fq(0)3117 5076 y Fz(\))3177 5014 y Fy(\015)3229 4976 y Fx(\000)p Fq(1)3339 5014 y Fy(du)2100 5223 y Fo(.)82 b Fy(\015)26 b Fz(+)20 b Fy(A)2485 5185 y Fx(\000)p Fq(1)2579 5223 y Fy(\015)2631 5185 y Fx(\000)p Fq(3)2725 5223 y Fy(:)1903 5472 y Fz(58)p eop %%Page: 59 59 59 58 bop -47 278 a Fz(Setting)40 b Fy(\015)47 b Fz(=)42 b Fy(A)551 242 y Fx(\000)616 215 y Fr(1)p 616 227 31 3 v 616 268 a(4)701 278 y Fz(\014nishes)d(the)h(pro)s(of)g(of)h(b)s (oth)f(\(7.3\))i(and)e(\(7.4\))r(.)71 b(The)40 b(pro)s(of)g(of)48 b(\(7.5\))42 b(is)d(similar.)69 b(More)-47 391 y(precisely)-8 b(,)29 b(b)m(y)i(the)f(same)h(argumen)m(ts)g(in)m(v)m(olving)f(\(7.6\)) i(one)f(obtains)740 493 y Fn(\014)740 548 y(\014)740 603 y(\014)740 657 y(\014)770 502 y(Z)861 528 y Fx(1)821 708 y Fq(0)951 625 y Fy(e)993 588 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1191 524 y Fn(\020)1245 625 y Fz(1)21 b Fu(\000)f Fy(\037)1459 524 y Fn(\020)1588 564 y Fy(u)p 1523 604 183 4 v 1523 629 a Fu(p)p 1599 629 107 4 v 59 x Fy(\034)1639 702 y Fw(m)1715 524 y Fn(\021\021)1824 625 y Fy(!)s Fz(\()p Fy(u)p Fz(\))15 b Fy(du)2120 493 y Fn(\014)2120 548 y(\014)2120 603 y(\014)2120 657 y(\014)2176 625 y Fo(.)25 b Fz(1)c(+)2429 493 y Fn(\014)2429 548 y(\014)2429 603 y(\014)2429 657 y(\014)2459 502 y(Z)2550 528 y Fx(1)2510 708 y Fq(0)2640 625 y Fy(e)2682 588 y Fw(i )r Fq(\()p Fw(u)p Fq(\))2880 625 y Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\()p Fy(u)p Fz(\))15 b Fy(du)3347 493 y Fn(\014)3347 548 y(\014)3347 603 y(\014)3347 657 y(\014)586 876 y Fo(.)83 b Fz(1)20 b(+)896 752 y Fn(Z)987 778 y Fx(1)947 958 y Fq(0)1169 814 y Fz(1)p 1087 855 209 4 v 1087 938 a Fy( )1149 912 y Fx(0)1173 938 y Fz(\()p Fy(u)p Fz(\))1306 876 y Fu(j)p Fz([)p Fy(h)p Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\()p Fy(u)p Fz(\)])1734 838 y Fx(0)1759 876 y Fu(j)15 b Fy(du)26 b Fo(.)f Fz(1)c(+)2177 752 y Fn(Z)2268 778 y Fx(1)2237 931 y Fr(1)p 2237 943 31 3 v 2237 984 a(4)2371 814 y Fz(1)p 2368 855 53 4 v 2368 938 a Fy(u)2430 775 y Fn(h)2473 876 y Fy(\037)2530 903 y Fq([)2560 876 y Fr(1)p 2559 888 31 3 v 2559 929 a(4)2600 903 y Fw(;)2630 876 y Fr(1)p 2629 888 V 2629 929 a(2)2669 903 y Fq(])2693 876 y Fz(\()p Fy(u)p Fz(\))g(+)2940 814 y(1)p 2937 855 53 4 v 2937 938 a Fy(u)2999 775 y Fn(i)3057 876 y Fy(du)26 b Fo(.)f Fz(1)-47 1127 y(and)30 b(w)m(e)g(are)h(done.)p 3879 1127 4 62 v 3883 1069 55 4 v 3883 1127 V 3937 1127 4 62 v -47 1312 a Fm(Lemma)h(7.3.)47 b Fl(L)-5 b(et)32 b Fy( )k Fl(and)e Fy(!)h Fl(b)-5 b(e)33 b(as)g(ab)-5 b(ove.)43 b(Supp)-5 b(ose)34 b(that)g(ther)-5 b(e)33 b(ar)-5 b(e)34 b(c)-5 b(onstants)34 b Fy(A)f Fl(and)g Fy(u)3205 1326 y Fq(0)3277 1312 y Fl(so)h(that)1077 1496 y Fy( )1139 1459 y Fx(0)1163 1496 y Fz(\()p Fy(u)p Fz(\))26 b Fo(&)f Fy(Au)1527 1510 y Fq(0)1581 1496 y Fy(u)p Fz(\()p Fy(u)c Fu(\000)f Fy(u)1884 1510 y Fq(0)1923 1496 y Fz(\))66 b Fl(for)33 b(al)5 b(l)98 b Fz(0)26 b Fu(\024)f Fy(u)g Fu(\024)g Fy(u)2756 1510 y Fq(0)2796 1496 y Fy(:)-47 1681 y Fl(Then)33 b(for)g(any)g Fy(\034)43 b Fl(so)33 b(that)h Fy(\034)h Fu(\025)25 b Fy(\034)1105 1695 y Fw(m)1171 1681 y Fl(,)613 1779 y Fn(\014)613 1833 y(\014)613 1888 y(\014)613 1942 y(\014)643 1787 y(Z)734 1813 y Fw(u)775 1822 y Fr(0)694 1993 y Fq(0)829 1911 y Fy(e)871 1873 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1069 1810 y Fn(\020)1123 1911 y Fz(1)c Fu(\000)f 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Fz(\))p Fy(h)p Fz(\()p Fy(u)3484 5191 y Fq(1)3526 5177 y Fz(\))-3608 b(\(7.11\))1903 5472 y(59)p eop %%Page: 60 60 60 59 bop -47 274 a Fz(and)27 b(similarly)d(for)j(the)h(in)m(tegral)g (o)m(v)m(er)g(the)g(in)m(terv)-5 b(al)27 b([0)p Fy(;)15 b(u)1948 288 y Fq(1)1989 274 y Fz(].)40 b(It)27 b(follo)m(ws)g(that)h (the)g(estimate)h(in)d(\(7.10\))k(reduces)d(to)320 522 y Fy(\015)399 461 y(u)451 475 y Fq(0)p 382 501 126 4 v 382 520 a Fu(p)p 458 520 51 4 v 65 x Fy(\034)538 522 y Fz(+)629 399 y Fn(Z)720 425 y Fw(u)761 434 y Fr(0)679 605 y Fq(0)906 461 y Fz(1)p 825 501 209 4 v 825 585 a Fy( )887 558 y Fx(0)910 585 y Fz(\()p Fy(u)p Fz(\))1043 522 y Fu(j)p Fz([)p Fy(\037)p Fz(\()p Fy(u=\015)5 b Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)1548 536 y Fq(0)1610 522 y Fu(\000)20 b Fy(u)p Fz(\))p Fy(=\015)5 b Fz(\))p Fy(h)p Fz(\()p Fy(u)p Fz(\)])2119 485 y Fx(0)2145 522 y Fu(j)15 b Fy(du)21 b Fz(+)2507 461 y(1)p 2406 501 248 4 v 2406 585 a Fy( )2468 558 y Fx(0)2491 585 y Fz(\()p Fy(u)2578 599 y Fq(1)2618 585 y 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b(On)29 b(the)i(other)g(hand,)e(the)i(in)m(tegral)f(in)f (\(7.12\))j(is)e(b)s(ounded)e(b)m(y)468 1395 y Fn(Z)559 1422 y Fw(u)600 1431 y Fr(0)518 1601 y Fq(0)745 1457 y Fz(1)p 664 1498 209 4 v 664 1581 a Fy( )726 1555 y Fx(0)749 1581 y Fz(\()p Fy(u)p Fz(\))882 1519 y Fu(j)p Fz([)p Fy(\037)p Fz(\()p Fy(u=\015)5 b Fz(\))p Fy(\037)p Fz(\(\()p Fy(u)1387 1533 y Fq(0)1449 1519 y Fu(\000)20 b Fy(u)p Fz(\))p Fy(=\015)5 b Fz(\))p Fy(h)p Fz(\()p Fy(u)p Fz(\)])1958 1481 y Fx(0)1984 1519 y Fu(j)15 b Fy(du)468 1785 y Fo(.)564 1661 y Fn(Z)655 1687 y Fw(u)696 1696 y Fr(0)730 1687 y Fx(\000)p Fw(\015)t(=)p Fq(4)614 1867 y Fw(\015)t(=)p Fq(4)1179 1723 y Fz(1)p 925 1764 553 4 v 925 1847 a Fy(Au)1045 1861 y Fq(0)1100 1847 y Fy(u)p Fz(\()p Fy(u)1239 1861 y Fq(0)1299 1847 y Fu(\000)20 b Fy(u)p Fz(\))1503 1684 y Fn(h)1545 1785 y Fy(\015)1597 1747 y Fx(\000)p Fq(1)1692 1785 y Fy(\037)1749 1805 y Fq([)1779 1775 y Fs(\015)p 1778 1790 36 3 v 1781 1832 a Fr(4)1824 1805 y Fw(;)1854 1775 y Fs(\015)p 1854 1790 V 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1654 y Fu(\025)f Fy(\034)2682 1668 y Fq(2)2763 1654 y Fu(\025)f Fy(:)15 b(:)g(:)43 b Fu(\025)e Fy(\034)3175 1668 y Fw(p)3257 1654 y Fy(>)g Fz(0)h Fl(and)h Fy(b)3681 1668 y Fw(j)3759 1654 y Fy(>)f Fz(0)p Fl(.)-47 1767 y(Assume)32 b(that)i Fy(!)h Fl(is)e(a)g(di\013er) -5 b(entiable)33 b(function)g(such)g(that)1603 1955 y Fu(j)p Fy(!)1688 1917 y Fq(\()p Fw(j)t Fq(\))1779 1955 y Fz(\()p Fy(u)p Fz(\))p Fu(j)27 b(\024)e Fy(a)2097 1969 y Fq(0)2151 1955 y Fy(u)2203 1917 y Fx(\000)p Fw(j)-47 2143 y Fl(for)33 b(al)5 b(l)33 b Fy(u)25 b(>)g Fz(0)33 b Fl(and)h Fy(j)d Fz(=)25 b(0)p Fy(;)15 b Fz(1)p Fl(.)43 b(Then)33 b(for)g(any)g Fz(1)26 b Fu(\024)f Fy(k)j Fu(\024)d Fy(m)p Fl(,)552 2243 y Fn(\014)552 2298 y(\014)552 2352 y(\014)552 2407 y(\014)582 2251 y(Z)673 2278 y Fx(1)632 2458 y Fq(0)763 2375 y Fy(e)805 2338 y Fw(i )r Fq(\()p Fw(u)p Fq(\))1003 2274 y Fn(\020)1057 2375 y Fz(1)c Fu(\000)f Fy(\037)1271 2274 y Fn(\020)1400 2314 y Fy(u)p 1335 2354 183 4 v 1335 2379 a Fu(p)p 1410 2379 107 4 v 1410 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3330 y Fw(k)2855 3315 y Fz(for)30 b(all)g Fy(k)s Fz(.)40 b(Let)1653 3579 y Fy(A)25 b Fz(:=)1902 3465 y Fw(m)1867 3492 y Fn(X)1872 3688 y Fw(j)t Fq(=1)2079 3517 y Fy(b)2118 3531 y Fw(j)p 2024 3558 187 4 v 2024 3645 a Fy(\034)2064 3659 y Fw(j)2100 3618 y Fq(3)p Fw(=)p Fq(2)2220 3579 y Fy(:)-47 3856 y Fz(Then)1453 4044 y Fy(A)h Fo(&)f Fz(max)1827 3970 y Fn(\000)1869 4044 y Fy(\034)1919 4005 y Fx(\000)p Fq(1)1909 4070 y(1)2013 4044 y Fy(;)15 b Fz(\()p Fy(m\034)2208 4058 y Fw(p)2248 4044 y Fz(\))2283 4006 y Fx(\000)p Fq(1)2378 3970 y Fn(\001)2419 4044 y Fy(:)-2491 b Fz(\(7.18\))-47 4231 y(If)30 b Fy(u)96 4245 y Fq(0)160 4231 y Fy(>)25 b Fz(0)31 b(exists,)f(then)g(b)m(y)i(\(7.1\))703 4463 y Fu(j)p Fy( )790 4426 y Fx(0)814 4463 y Fz(\()p Fy(u)p Fz(\))p Fu(j)84 b(\025)f Fy(u)1266 4339 y Fn(Z)1357 4366 y Fw(u)1317 4546 y(u)1358 4555 y Fr(0)1417 4463 y Fy(A)15 b(s)g(ds)26 b Fo(&)f Fy(A)15 b(u)1905 4426 y Fq(2)1945 4463 y Fz(\()p Fy(u)20 b Fu(\000)g Fy(u)2195 4477 y Fq(0)2235 4463 y Fz(\))60 b(for)31 b(all)2626 4400 y Fu(p)p 2702 4400 107 4 v 63 x Fy(\034)2742 4477 y Fw(m)2834 4463 y Fy(>)24 b(u)i(>)f(u)3155 4477 y Fq(0)703 4714 y Fu(j)p Fy( )790 4677 y Fx(0)814 4714 y Fz(\()p Fy(u)p Fz(\))p Fu(j)84 b(\025)f Fy(u)1266 4590 y Fn(Z)1357 4617 y Fw(u)1398 4626 y Fr(0)1317 4797 y Fw(u)1452 4714 y Fy(A)15 b(s)g(ds)25 b Fo(&)g Fy(A)15 b(u)1939 4728 y Fq(0)1994 4714 y Fy(u)p Fz(\()p Fy(u)2133 4728 y Fq(0)2193 4714 y Fu(\000)20 b Fy(u)p Fz(\))61 b(for)30 b(all)60 b(0)26 b Fy(<)f(u)g(<)g(u)3120 4728 y Fq(0)3159 4714 y Fy(:)-47 4948 y Fz(By)30 b(Lemma)h(7.2)g(and)f (7.3)i(the)e(oscillatory)g(in)m(tegral)g(in)f(\(7.17\))k(is)c (therefore)i(no)f(larger)g(than)1174 5191 y(min)1325 5090 y Fn(\020)1380 5191 y Fy(\034)1440 5107 y Fr(1)p 1440 5119 31 3 v 1440 5160 a(4)1420 5217 y Fq(1)1484 5191 y Fy(;)1524 5090 y Fn(\020)1589 5129 y Fy(m\034)1709 5143 y Fw(p)p 1589 5170 160 4 v 1627 5253 a Fy(\034)1667 5268 y Fw(k)1758 5090 y Fn(\021)1822 5086 y Fr(1)p 1822 5098 31 3 v 1822 5139 a(2)1867 5090 y Fn(\021)1946 5191 y Fo(.)25 b Fz(min)2194 5090 y Fn(\020)2248 5191 y Fy(\034)2308 5107 y Fr(1)p 2308 5119 V 2308 5160 a(4)2288 5217 y Fq(1)2353 5191 y Fy(;)15 b(m)2483 5126 y Fr(3)p 2483 5138 V 2483 5180 a(2)2554 5129 y Fy(b)2593 5143 y Fw(p)p 2553 5170 82 4 v 2553 5253 a Fy(b)2592 5268 y Fw(k)2644 5090 y Fn(\021)2699 5191 y Fy(;)1903 5472 y Fz(61)p eop %%Page: 62 62 62 61 bop -47 274 a Fz(as)30 b(claimed.)40 b(If)30 b Fy(u)576 288 y Fq(0)641 274 y Fy(>)25 b Fz(0)30 b(do)s(es)g(not)h (exist,)f(then)h(\(7.2\))h(implies)27 b(that)1220 455 y Fu(j)p Fy( )1307 418 y Fx(0)1331 455 y Fz(\()p Fy(u)p Fz(\))p Fu(j)f Fo(&)f Fy(A)15 b(u)1735 418 y Fq(3)1835 455 y Fz(for)30 b(all)60 b(0)26 b Fy(<)f(u)g(<)2471 392 y Fu(p)p 2547 392 107 4 v 63 x Fy(\034)2587 469 y Fw(m)2653 455 y Fy(:)-47 637 y Fz(No)m(w)31 b(apply)f(\(7.3\))i(of)f(Lemma)f(7.2) i(with)d Fy(u)1462 651 y Fq(0)1527 637 y Fz(=)c(0.)p 3879 637 4 62 v 3883 579 55 4 v 3883 637 V 3937 637 4 62 v -47 820 a(The)30 b(previous)e(estimates)j(allo)m(w)f(us)g(to)h (deal)f(with)f(the)i(singularit)m(y)d(at)j Fy(\025)25 b Fz(=)2655 764 y Fu(p)p 2731 764 80 4 v 56 x Fy(\032)2778 835 y Fw(`)2841 820 y Fz(in)k(the)i(follo)m(wing)e(situation.)-47 989 y Fm(Lemma)j(7.5.)47 b Fl(Ther)-5 b(e)38 b(exists)g(a)g(c)-5 b(onstant)39 b Fy(C)1568 1003 y Fq(0)1645 989 y Fl(so)f(that)h(for)f (any)g(choic)-5 b(e)38 b(of)g Fy(\033)2729 1003 y Fq(1)2803 989 y Fu(\025)c Fy(\033)2960 1003 y Fq(2)3033 989 y Fu(\025)g Fy(:)15 b(:)g(:)36 b Fu(\025)e Fy(\033)3436 1003 y Fw(m)3536 989 y Fu(\025)g Fz(0)p Fl(,)39 b Fy(\032)3800 1003 y Fq(1)3874 989 y Fu(\025)-47 1102 y Fy(\032)0 1116 y Fq(2)64 1102 y Fu(\025)25 b Fy(:)15 b(:)g(:)27 b Fu(\025)e Fy(\032)435 1117 y Fw(`)493 1102 y Fy(>)g Fz(0)p Fl(,)33 b Fy(b)734 1116 y Fw(j)796 1102 y Fy(>)24 b Fz(0)p Fl(,)33 b(and)h Fy(c)1213 1116 y Fw(i)1267 1102 y Fy(>)25 b Fz(0)p Fl(,)32 b(one)h(has)684 1215 y Fn(\014)684 1269 y(\014)684 1324 y(\014)684 1378 y(\014)684 1433 y(\014)714 1250 y(Z)805 1235 y Fx(p)p 864 1235 67 3 v 41 x Fw(\032)900 1288 y Fs(`)764 1456 y Fq(0)949 1374 y Fy(e)1001 1309 y Fr(1)p 1001 1321 31 3 v 1001 1362 a(2)1042 1336 y Fw(i\025)1107 1313 y Fr(2)1161 1374 y Fy(e)1203 1334 y Fx(\006)p Fw(i)1294 1284 y Fb(P)1368 1304 y Fs(m)1368 1354 y(j)s Fr(=1)1490 1334 y Fw(b)1520 1344 y Fs(j)1564 1273 y Fu(p)p 1640 1273 204 4 v 61 x Fw(\025)1681 1315 y Fr(2)1716 1334 y Fq(+)p Fw(\033)1811 1344 y Fs(j)1878 1374 y Fz(exp)2017 1273 y Fn(\020)2071 1374 y Fu(\000)2208 1260 y Fw(`)2157 1288 y Fn(X)2165 1483 y Fw(i)p Fq(=1)2303 1374 y Fy(c)2342 1388 y Fw(i)2371 1289 y Fn(p)p 2462 1289 279 4 v 85 x Fy(\032)2509 1388 y Fw(i)2557 1374 y Fu(\000)20 b Fy(\025)2701 1348 y Fq(2)2741 1273 y Fn(\021)2988 1312 y Fy(\025)p 2820 1353 390 4 v 2820 1371 a Fn(p)p 2911 1371 299 4 v 82 x Fy(\025)2964 1427 y Fq(2)3024 1453 y Fz(+)f Fy(\033)3166 1468 y Fw(k)3234 1374 y Fy(d\025)3334 1215 y Fn(\014)3334 1269 y(\014)3334 1324 y(\014)3334 1378 y(\014)3334 1433 y(\014)1055 1630 y Fu(\024)25 b Fy(C)1216 1644 y Fq(0)1271 1630 y Fz(min)1422 1529 y Fn(h)1465 1630 y Fz(\(1)c(+)f Fy(\033)1709 1644 y Fq(1)1769 1630 y Fz(+)g Fy(\032)1907 1645 y Fw(`)1940 1630 y Fz(\))1985 1565 y Fr(1)p 1985 1577 31 3 v 1985 1618 a(4)2030 1630 y Fy(;)15 b(m)2160 1565 y Fr(3)p 2160 1577 V 2160 1618 a(2)2204 1630 y Fy(b)2243 1592 y Fx(\000)p Fq(1)2243 1659 y Fw(k)2388 1630 y Fz(max)2353 1689 y Fq(1)p Fx(\024)p Fw(j)t Fx(\024)p Fw(m)2608 1630 y Fy(b)2647 1644 y Fw(j)2683 1529 y Fn(i)-47 1630 y Fz(\(7.19\))-47 1851 y Fl(for)33 b(any)g Fz(1)26 b Fu(\024)f Fy(k)j Fu(\024)d Fy(m)p Fl(.)-47 2020 y(Pr)-5 b(o)g(of.)47 b Fz(F)-8 b(or)31 b(the)f(purp)s(oses)f(of)h(this)g(pro)s(of,)g(set)g Fy(w)1685 2035 y Fw(k)1728 2020 y Fz(\()p Fy(\025)p Fz(\))c(=)2105 1984 y Fw(\025)p 1983 1999 285 4 v 1983 2010 a Fu(p)p 2059 2010 210 4 v 65 x Fw(\025)2100 2056 y Fr(2)2135 2075 y Fq(+)p Fw(\033)2230 2087 y Fs(k)2278 2020 y Fz(.)41 b(W)-8 b(e)31 b(claim)f(that)721 2238 y Fn(\014)721 2293 y(\014)721 2347 y(\014)788 2281 y Fy(d)835 2248 y Fw(j)p 761 2322 137 4 v 761 2405 a Fy(d\025)861 2379 y Fw(j)908 2242 y Fn(h)951 2343 y Fz(exp)1090 2242 y Fn(\020)1144 2343 y Fu(\000)1281 2229 y Fw(`)1230 2256 y Fn(X)1239 2452 y Fw(i)p Fq(=1)1377 2343 y Fy(c)1416 2357 y Fw(i)1444 2257 y Fn(p)p 1535 2257 279 4 v 86 x Fy(\032)1582 2357 y Fw(i)1630 2343 y Fu(\000)20 b Fy(\025)1774 2316 y Fq(2)1814 2242 y Fn(\021)1883 2343 y Fy(w)1948 2358 y Fw(k)1991 2343 y Fz(\()p Fy(\025)p Fz(\))2114 2242 y Fn(\020)2169 2343 y Fz(1)h Fu(\000)f Fy(\037)2383 2242 y Fn(\020)2498 2281 y Fy(\025)p 2447 2322 156 4 v 2447 2348 a Fu(p)p 2522 2348 80 4 v 2522 2405 a Fy(\032)2569 2420 y Fw(`)2612 2242 y Fn(\021)q(\021i)2764 2238 y(\014)2764 2293 y(\014)2764 2347 y(\014)2820 2343 y Fu(\024)k Fy(C)2980 2357 y Fw(j)3032 2343 y Fy(\025)3085 2305 y Fx(\000)p Fw(j)-47 2343 y Fz(\(7.20\))-47 2601 y(for)30 b(ev)m(ery)h Fy(j)g Fu(\025)25 b Fz(0)31 b(and)e Fy(\025)d(>)f Fz(0.)41 b(In)29 b(view)h(of)h(Lemma)f (6.5)i(it)d(su\016ces)i(to)g(sho)m(w)f(that)1264 2729 y Fn(\014)1264 2783 y(\014)1264 2838 y(\014)1330 2772 y Fy(d)1377 2739 y Fw(j)p 1304 2812 137 4 v 1304 2896 a Fy(d\025)1404 2869 y Fw(j)1466 2833 y Fz(exp)1605 2732 y Fn(\020)1659 2833 y Fu(\000)p Fy(c)1769 2847 y Fw(i)1797 2748 y Fn(p)p 1888 2748 279 4 v 85 x Fy(\032)1935 2847 y Fw(i)1984 2833 y Fu(\000)20 b Fy(\025)2128 2807 y Fq(2)2167 2732 y Fn(\021)2221 2729 y(\014)2221 2783 y(\014)2221 2838 y(\014)2277 2833 y Fu(\024)25 b Fy(C)2438 2847 y Fw(j)2489 2833 y Fy(\025)2542 2796 y Fx(\000)p Fw(j)-47 3060 y Fz(for)35 b(ev)m(ery)g Fy(j)k Fu(\025)33 b Fz(0,)k(1)c Fu(\024)g Fy(i)h Fu(\024)f Fy(`)p Fz(,)j(and)e(0)g Fy(<)f(\025)g(<)1642 3025 y Fq(1)p 1642 3040 36 4 v 1642 3092 a(2)1687 3004 y Fu(p)p 1763 3004 80 4 v 56 x Fy(\032)1810 3075 y Fw(`)1843 3060 y Fz(.)54 b(Fix)35 b(some)g(suc)m(h)g Fy(i)h Fz(and)e(note)i(that) f Fy(\025)f Fu(7!)3399 2980 y Fn(p)p 3490 2980 279 4 v 80 x Fy(\032)3537 3074 y Fw(i)3585 3060 y Fu(\000)20 b Fy(\025)3729 3034 y Fq(2)3803 3060 y Fz(is)34 b(a)-47 3173 y(di\013eomorphism)27 b(on)k(the)f(in)m(terv)-5 b(al)30 b(0)25 b Fy(<)g(\025)h(<)1555 3137 y Fq(1)p 1555 3152 36 4 v 1555 3205 a(2)1600 3117 y Fu(p)p 1676 3117 80 4 v 56 x Fy(\032)1723 3188 y Fw(`)1756 3173 y Fz(.)40 b(Since)959 3315 y Fn(\014)959 3370 y(\014)959 3424 y(\014)1025 3358 y Fy(d)1072 3325 y Fw(j)p 999 3399 136 4 v 999 3482 a Fy(du)1098 3456 y Fw(j)1145 3420 y Fy(e)1187 3382 y Fx(\000)p Fw(cu)1318 3315 y Fn(\014)1318 3370 y(\014)1318 3424 y(\014)1373 3420 y Fz(=)1469 3315 y Fn(\014)1469 3370 y(\014)1469 3424 y(\014)1499 3420 y Fy(c)1538 3382 y Fw(j)1575 3420 y Fy(e)1617 3382 y Fx(\000)p Fw(cu)1748 3315 y Fn(\014)1748 3370 y(\014)1748 3424 y(\014)1804 3420 y Fu(\024)25 b Fy(u)1952 3382 y Fx(\000)p Fw(j)2058 3420 y Fz(sup)2063 3495 y Fw(v)r(>)p Fq(0)2195 3420 y Fz(\()p Fy(cv)s Fz(\))2351 3382 y Fw(j)2389 3420 y Fy(e)2431 3382 y Fx(\000)p Fw(cv)2583 3420 y Fz(=)g Fy(C)2744 3434 y Fw(j)2795 3420 y Fy(u)2847 3382 y Fx(\000)p Fw(j)-47 3653 y Fz(irresp)s(ectiv)m(e)37 b(of)i(the)g(c)m(hoice)h(of)f Fy(c)p Fz(,)i(\(7.20\))g(follo)m(ws.)65 b(Lemma)39 b(6.1,)j(with)37 b Fy(!)42 b Fz(equal)c(to)i(the)e(function)g(in)f(\(7.20\))s(,)-47 3766 y(therefore)31 b(implies)c(that)469 3870 y Fn(\014)469 3925 y(\014)469 3980 y(\014)469 4034 y(\014)469 4089 y(\014)500 3906 y(Z)591 3891 y Fx(p)p 649 3891 67 3 v 649 3932 a Fw(\032)685 3944 y Fs(`)550 4112 y Fq(0)735 4030 y Fy(e)787 3965 y Fr(1)p 787 3977 31 3 v 787 4018 a(2)827 3992 y Fw(i\025)892 3969 y Fr(2)947 4030 y Fy(e)989 3989 y Fx(\006)p Fw(i)1080 3940 y Fb(P)1154 3960 y Fs(m)1154 4010 y(j)s Fr(=1)1275 3989 y Fw(b)1305 3999 y Fs(j)1350 3928 y Fu(p)p 1426 3928 204 4 v 61 x Fw(\025)1467 3970 y Fr(2)1501 3989 y Fq(+)p Fw(\033)1596 3999 y Fs(j)1664 4030 y Fz(exp)1802 3929 y Fn(\020)1857 4030 y Fu(\000)1994 3916 y Fw(`)1943 3943 y Fn(X)1951 4139 y Fw(i)p Fq(=1)2089 4030 y Fy(c)2128 4044 y Fw(i)2157 3944 y Fn(p)p 2247 3944 279 4 v 2247 4030 a Fy(\032)2294 4044 y Fw(i)2343 4030 y Fu(\000)20 b Fy(\025)2487 4003 y Fq(2)2526 3929 y Fn(\021)2596 4030 y Fy(w)2661 4045 y Fw(k)2703 4030 y Fz(\()p Fy(\025)p Fz(\))2826 3929 y Fn(\020)2881 4030 y Fz(1)h Fu(\000)f Fy(\037)3095 3929 y Fn(\020)3210 3968 y Fy(\025)p 3159 4009 156 4 v 3159 4035 a Fu(p)p 3235 4035 80 4 v 57 x Fy(\032)3282 4107 y Fw(`)3325 3929 y Fn(\021\021)3449 4030 y Fy(d\025)3549 3870 y Fn(\014)3549 3925 y(\014)3549 3980 y(\014)3549 4034 y(\014)3549 4089 y(\014)833 4285 y Fu(\024)25 b Fy(C)994 4299 y Fq(0)1064 4285 y Fz(min)1215 4185 y Fn(h)1258 4285 y Fz(\(1)c(+)f Fy(\033)1502 4299 y Fq(1)1541 4285 y Fz(\))1586 4221 y Fr(1)p 1587 4233 31 3 v 1587 4274 a(4)1631 4285 y Fy(;)15 b(m)1761 4221 y Fr(3)p 1761 4233 V 1761 4274 a(2)1821 4285 y Fy(b)1860 4247 y Fx(\000)p Fq(1)1860 4315 y Fw(k)2020 4285 y Fz(max)1985 4345 y Fq(1)p Fx(\024)p Fw(j)t Fx(\024)p Fw(m)2240 4285 y Fy(b)2279 4299 y Fw(j)2315 4185 y Fn(i)2358 4285 y Fy(:)-2430 b Fz(\(7.21\))-47 4508 y(It)32 b(remains)f(to)i(deal) 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4818 y Fn(X)2127 5014 y Fw(i)p Fq(=1)2265 4905 y Fy(c)2304 4919 y Fw(i)2332 4820 y Fn(p)p 2423 4820 279 4 v 85 x Fy(\032)2470 4919 y Fw(i)2518 4905 y Fu(\000)20 b Fy(\025)2662 4879 y Fq(2)2702 4804 y Fn(\021)2771 4905 y Fy(w)2836 4920 y Fw(k)2879 4905 y Fz(\()p Fy(\025)p Fz(\))p Fy(\037)3059 4804 y Fn(\020)3175 4843 y Fy(\025)p 3124 4884 156 4 v 3124 4911 a Fu(p)p 3200 4911 80 4 v 56 x Fy(\032)3247 4982 y Fw(`)3289 4804 y Fn(\021)3359 4905 y Fy(d\025)521 5182 y Fz(=)83 b Fy(e)717 5145 y Fw(i\032)777 5157 y Fs(`)808 5145 y Fw(=)p Fq(2)897 5059 y Fn(Z)988 5085 y Fx(1)948 5265 y Fq(0)1078 5182 y Fy(e)1120 5145 y Fx(\000)p Fw(i )1245 5154 y Ft(\007)1298 5145 y Fq(\()p Fw(u)p Fq(\))1423 5182 y Fy(!)s Fz(\()p Fy(u)p Fz(\))1785 5121 y Fy(u)p 1631 5161 362 4 v 1631 5181 a Fu(p)p 1706 5181 286 4 v 1706 5245 a Fy(\033)1758 5260 y Fw(k)1821 5245 y Fz(+)20 b Fy(\032)1959 5260 y Fw(`)2017 5182 y Fy(du;)-2188 b Fz(\(7.22\))1903 5472 y(62)p eop %%Page: 63 63 63 62 bop -47 274 a Fz(where)1236 525 y Fy( )1295 539 y Fx(\007)1354 525 y Fz(\()p Fy(u)p Fz(\))26 b(=)1608 463 y(1)p 1608 504 46 4 v 1608 587 a(2)1664 525 y Fy(u)1716 487 y Fq(2)1775 525 y Fu(\007)1901 411 y Fw(m)1866 439 y Fn(X)1871 634 y Fw(j)t Fq(=1)2013 525 y Fy(b)2052 539 y Fw(j)2088 417 y Fn(q)p 2179 417 483 4 v 108 x Fy(\033)2231 539 y Fw(j)2288 525 y Fz(+)20 b Fy(\032)2426 540 y Fw(`)2479 525 y Fu(\000)g Fy(u)2622 499 y Fq(2)-47 525 y Fz(\(7.23\))-47 819 y(and)616 1087 y Fy(!)s Fz(\()p Fy(u)p Fz(\))26 b(=)f(exp)1059 986 y Fn(\020)1113 1087 y Fu(\000)1250 974 y Fw(`)1199 1001 y Fn(X)1208 1196 y Fw(i)p Fq(=1)1346 1087 y Fy(c)1385 1101 y Fw(i)1413 1002 y Fn(p)p 1504 1002 469 4 v 85 x Fy(\032)1551 1101 y Fw(i)1599 1087 y Fu(\000)20 b Fy(\032)1737 1102 y Fw(`)1790 1087 y Fz(+)g Fy(u)1933 1061 y Fq(2)1973 986 y Fn(\021)2027 1087 y Fy(\037)2084 986 y Fn(\020)2138 1005 y(p)p 2229 1005 374 4 v 82 x Fz(1)h Fu(\000)f Fy(u)2438 1061 y Fq(2)2477 1087 y Fy(=\032)2569 1102 y Fw(`)2603 986 y Fn(\021)2657 956 y(r)p 2748 956 509 4 v 2859 1026 a Fy(\033)2911 1041 y Fw(k)2974 1026 y Fz(+)g Fy(\032)3112 1041 y Fw(`)p 2758 1066 489 4 v 2758 1150 a Fy(\033)2810 1165 y Fw(k)2873 1150 y Fz(+)g Fy(\032)3011 1165 y Fw(`)3064 1150 y Fu(\000)g Fy(u)3207 1123 y Fq(2)3256 1087 y Fy(:)-47 1388 y Fz(By)25 b(the)h(same)g(argumen)m(ts)g(that)f(lead)g(to)i (\(7.20\))g(one)f(sees)g(that)g Fu(j)p Fy(!)2262 1355 y Fq(\()p Fw(j)t Fq(\))2353 1388 y Fz(\()p Fy(u)p Fz(\))p Fu(j)g(\024)f Fy(C)2687 1402 y Fw(j)2739 1388 y Fy(u)2791 1355 y Fx(\000)p Fw(j)2908 1388 y Fz(for)g(ev)m(ery)h Fy(j)31 b Fu(\025)25 b Fz(0.)39 b(Moreo)m(v)m(er,)-47 1536 y Fy(!)s Fz(\()p Fy(u)p Fz(\))26 b(=)f(0)c(for)g(all)g Fy(u)k Fu(\025)744 1430 y Fn(q)p 835 1430 91 4 v 845 1500 a Fq(15)p 845 1515 71 4 v 845 1567 a(16)925 1479 y Fu(p)p 1001 1479 80 4 v 57 x Fy(\032)1048 1551 y Fw(`)1081 1536 y Fz(.)38 b(In)20 b(case)j(of)e(the)h(more)f(di\016cult)f(phase)h Fy( )2520 1550 y Fq(+)2579 1536 y Fz(\()p Fy(u)p Fz(\),)j(Corollary)c (7.4)j(with)d Fy(\034)3519 1550 y Fw(j)3580 1536 y Fz(:=)25 b Fy(\033)3753 1550 y Fw(j)3792 1536 y Fz(+)r Fy(\032)3912 1551 y Fw(`)-47 1666 y Fz(therefore)31 b(yields)d(the)j(desired)e(b)s (ound)f(\(7.19\))r(.)41 b(If,)30 b(on)g(the)h(other)f(hand,)g Fy( )s Fz(\()p Fy(u)p Fz(\))c(=)f Fy( )2888 1680 y Fx(\000)2948 1666 y Fz(\()p Fy(u)p Fz(\),)31 b(then)715 1940 y Fy( )777 1903 y Fx(0)800 1940 y Fz(\()p Fy(u)p Fz(\))26 b(=)f Fy(u)c Fz(+)1242 1826 y Fw(m)1207 1854 y Fn(X)1212 2049 y Fw(j)t Fq(=1)1485 1878 y Fy(b)1524 1892 y Fw(j)1560 1878 y Fy(u)p 1364 1919 370 4 v 1364 1937 a Fn(p)p 1455 1937 279 4 v 77 x Fy(\034)1495 2028 y Fw(j)1551 2014 y Fu(\000)f Fy(u)1694 1987 y Fq(2)2107 1940 y Fy( )2169 1903 y Fx(0)q(0)2212 1940 y Fz(\()p Fy(u)p Fz(\))26 b(=)f(1)c(+)2647 1826 y Fw(m)2613 1854 y Fn(X)2617 2049 y Fw(j)t Fq(=1)2895 1878 y Fy(b)2934 1892 y Fw(j)2971 1878 y Fy(\034)3011 1892 y Fw(j)p 2769 1919 405 4 v 2769 2025 a Fz(\()p Fy(\034)2844 2039 y Fw(j)2901 2025 y Fu(\000)f Fy(u)3044 1999 y Fq(2)3083 2025 y Fz(\))3128 1962 y Fr(3)p 3129 1974 31 3 v 3129 2015 a(2)-47 2241 y Fz(sho)m(ws)30 b(that)h Fy( )473 2208 y Fx(0)497 2241 y Fz(\()p Fy(u)p Fz(\))26 b Fu(\025)f Fy(u)30 b Fz(and)g Fy( )1062 2208 y Fx(00)1105 2241 y Fz(\()p Fy(u)p Fz(\))c Fu(\025)f Fz(0)30 b(on)h(the)f(supp)s(ort)f(of)h Fy(!)s Fz(.)41 b(Th)m(us)29 b(in)m(tegrating)h(b)m(y)h(parts)f(once)h (leads)f(to)435 2363 y Fn(\014)435 2418 y(\014)435 2472 y(\014)435 2527 y(\014)465 2371 y(Z)556 2397 y Fx(1)516 2577 y Fq(0)646 2495 y Fy(e)688 2457 y Fx(\000)p Fw(i )r Fq(\()p Fw(u)p Fq(\))941 2495 y Fy(!)s Fz(\()p Fy(u)p Fz(\))1304 2433 y Fy(u)p 1149 2474 362 4 v 1149 2494 a Fu(p)p 1225 2494 286 4 v 63 x Fy(\033)1277 2572 y Fw(k)1339 2557 y Fz(+)20 b Fy(\032)1477 2572 y Fw(`)1535 2495 y Fy(du)1634 2363 y Fn(\014)1634 2418 y(\014)1634 2472 y(\014)1634 2527 y(\014)435 2746 y Fo(.)25 b Fz(1)c(+)688 2622 y Fn(Z)779 2648 y Fx(1)738 2828 y Fq(0)889 2684 y Fy( )951 2651 y Fx(00)993 2684 y Fz(\()p Fy(u)p Fz(\))p 878 2725 248 4 v 878 2808 a Fy( )940 2782 y Fx(0)964 2808 y Fz(\()p Fy(u)p Fz(\))1086 2782 y Fq(2)1151 2746 y Fy(!)s Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\()p Fy(u)p Fz(\))1694 2684 y Fy(u)p 1539 2725 362 4 v 1539 2745 a Fu(p)p 1615 2745 286 4 v 63 x Fy(\033)1667 2823 y Fw(k)1729 2808 y Fz(+)f Fy(\032)1867 2823 y Fw(`)1925 2746 y Fy(du)h Fz(+)2136 2622 y Fn(Z)2227 2648 y Fx(1)2186 2828 y Fq(0)2408 2684 y Fz(1)p 2327 2725 209 4 v 2327 2808 a Fy( )2389 2782 y Fx(0)2412 2808 y Fz(\()p Fy(u)p Fz(\))2560 2641 y Fn(\014)2560 2696 y(\014)2560 2750 y(\014)2591 2645 y(h)2633 2746 y Fy(!)s Fz(\()p Fy(u)p Fz(\))p Fy(\037)p Fz(\()p Fy(u)p Fz(\))3176 2684 y Fy(u)p 3021 2725 362 4 v 3021 2745 a Fu(p)p 3097 2745 286 4 v 63 x Fy(\033)3149 2823 y Fw(k)3211 2808 y Fz(+)f Fy(\032)3349 2823 y Fw(`)3392 2645 y Fn(i)3435 2668 y Fx(0)3458 2641 y Fn(\014)3458 2696 y(\014)3458 2750 y(\014)3504 2746 y Fy(du:)-3675 b Fz(\(7.24\))-47 3000 y(In)m(tegrating)31 b(b)m(y)f(parts)g(once)h (more)g(in)e(the)h(\014rst)g(in)m(tegral,)g(see)h(\(7.11\))s(,)f (reduces)g(it)g(to)h(the)g(second.)40 b(Hence)1388 3248 y(\(7.24\))28 b Fo(.)1742 3124 y Fn(Z)1833 3151 y Fx(1)1792 3330 y Fq(1)2040 3187 y Fz(1)p 1932 3227 261 4 v 1932 3310 a Fy(u )2046 3284 y Fx(0)2070 3310 y Fz(\()p Fy(u)p Fz(\))2218 3248 y Fy(du)e Fo(.)f Fz(1)p Fy(;)-47 3496 y Fz(and)30 b(the)g(lemma)g(follo)m(ws.)p 3879 3496 4 62 v 3883 3438 55 4 v 3883 3496 V 3937 3496 4 62 v -47 3684 a(T)-8 b(o)31 b(conclude)e(this)h(section,)g(w)m(e)h(turn)f(to)h (oscillatory)e(in)m(tegrals)h(with)f(singular)f(w)m(eigh)m(ts.)-47 3871 y Fm(Lemma)k(7.6.)47 b Fl(Ther)-5 b(e)38 b(exists)g(a)g(c)-5 b(onstant)39 b Fy(C)1568 3885 y Fq(0)1645 3871 y Fl(so)f(that)h(for)f (any)g(choic)-5 b(e)38 b(of)g Fy(\033)2729 3885 y Fq(1)2803 3871 y Fu(\025)c Fy(\033)2960 3885 y Fq(2)3033 3871 y Fu(\025)g Fy(:)15 b(:)g(:)36 b Fu(\025)e Fy(\033)3436 3885 y Fw(m)3536 3871 y Fu(\025)g Fz(0)p Fl(,)39 b Fy(\032)3800 3885 y Fq(1)3874 3871 y Fu(\025)-47 3984 y Fy(\032)0 3998 y Fq(2)64 3984 y Fu(\025)25 b Fy(:)15 b(:)g(:)27 b Fu(\025)e Fy(\032)435 3999 y Fw(`)493 3984 y Fy(>)g Fz(0)p Fl(,)33 b Fy(b)734 3998 y Fw(j)796 3984 y Fy(>)24 b Fz(0)p Fl(,)33 b(and)h Fy(c)1213 3998 y Fw(i)1267 3984 y Fy(>)25 b Fz(0)p Fl(,)32 b(one)h(has)686 4120 y Fn(\014)686 4175 y(\014)686 4229 y(\014)686 4284 y(\014)686 4338 y(\014)716 4155 y(Z)807 4140 y Fx(p)p 866 4140 67 3 v 42 x Fw(\032)902 4194 y Fs(`)767 4362 y Fq(0)952 4279 y Fy(e)1004 4214 y Fr(1)p 1004 4226 31 3 v 1004 4268 a(2)1044 4242 y Fw(i\025)1109 4218 y Fr(2)1163 4279 y Fy(e)1205 4239 y Fx(\006)p Fw(i)1296 4189 y Fb(P)1371 4210 y Fs(m)1371 4260 y(j)s Fr(=1)1492 4239 y Fw(b)1522 4249 y Fs(j)1567 4178 y Fu(p)p 1642 4178 204 4 v 1642 4239 a Fw(\025)1683 4220 y Fr(2)1718 4239 y Fq(+)p Fw(\033)1813 4249 y Fs(j)1880 4279 y Fz(exp)2019 4178 y Fn(\020)2074 4279 y Fu(\000)2211 4165 y Fw(`)2159 4193 y Fn(X)2168 4388 y Fw(i)p Fq(=1)2306 4279 y Fy(c)2345 4293 y Fw(i)2373 4194 y Fn(p)p 2464 4194 279 4 v 85 x Fy(\032)2511 4293 y Fw(i)2560 4279 y Fu(\000)20 b Fy(\025)2704 4253 y Fq(2)2743 4178 y Fn(\021)2988 4218 y Fy(\025)p 2822 4258 385 4 v 2822 4276 a Fn(p)p 2913 4276 294 4 v 81 x Fy(\032)2960 4372 y Fw(k)3023 4357 y Fu(\000)g Fy(\025)3167 4331 y Fq(2)3232 4279 y Fy(d\025)3332 4120 y Fn(\014)3332 4175 y(\014)3332 4229 y(\014)3332 4284 y(\014)3332 4338 y(\014)1058 4535 y Fu(\024)25 b Fy(C)1219 4549 y Fq(0)1273 4535 y Fz(min)1425 4434 y Fn(h)1468 4535 y Fz(\(1)c(+)f Fy(\033)1712 4549 y Fq(1)1771 4535 y Fz(+)g Fy(\032)1909 4550 y Fw(`)1942 4535 y Fz(\))1987 4470 y Fr(1)p 1988 4482 31 3 v 1988 4524 a(4)2032 4535 y Fy(;)15 b(m)2162 4470 y Fr(3)p 2162 4482 V 2162 4524 a(2)2207 4535 y Fy(c)2246 4497 y Fx(\000)p Fq(1)2246 4564 y Fw(k)2356 4535 y Fz(max)2404 4594 y Fw(j;i)2525 4535 y Fz(\()p Fy(b)2599 4549 y Fw(j)2656 4535 y Fz(+)20 b Fy(c)2786 4549 y Fw(i)2814 4535 y Fz(\))2849 4434 y Fn(i)-47 4535 y Fz(\(7.25\))-47 4779 y Fl(for)33 b(any)g Fz(1)26 b Fu(\024)f Fy(k)j Fu(\024)d Fy(m)p Fl(.)-47 4966 y(Pr)-5 b(o)g(of.)47 b Fz(W)-8 b(e)31 b(start)g(with)e(the)i (elemen)m(tary)g(commen)m(t)g(that)g(w)m(e)g(can)g(assume)f(that)1813 5171 y Fy(\032)1860 5186 y Fw(`)1918 5171 y Fu(\025)25 b Fz(1)p Fy(:)-2131 b Fz(\(7.26\))1903 5472 y(63)p eop %%Page: 64 64 64 63 bop -47 274 a Fz(Indeed,)29 b(if)37 b(\(7.26\))32 b(fails,)e(then)g(the)g(oscillatory)g(in)m(tegral)g(in)f(\(7.25\))k(is) 856 541 y Fu(\024)952 417 y Fn(Z)1043 402 y Fx(p)p 1102 402 67 3 v 42 x Fw(\032)1138 456 y Fs(`)1002 624 y Fq(0)1363 480 y Fy(\025)p 1197 520 385 4 v 1197 538 a Fn(p)p 1288 538 294 4 v 81 x Fy(\032)1335 634 y Fw(k)1398 619 y Fu(\000)20 b Fy(\025)1542 593 y Fq(2)1606 541 y Fy(d\025)26 b Fu(\024)1828 417 y Fn(Z)1919 402 y Fx(p)p 1978 402 67 3 v 42 x Fw(\032)2014 456 y Fs(`)1878 624 y Fq(0)2234 480 y Fy(\025)p 2073 520 375 4 v 2073 538 a Fn(p)p 2164 538 284 4 v 81 x Fy(\032)2211 634 y Fw(`)2264 619 y Fu(\000)20 b Fy(\025)2408 593 y Fq(2)2472 541 y Fy(d\025)26 b Fz(=)2694 480 y Fu(p)p 2770 480 80 4 v 61 x Fy(\032)2817 556 y Fw(`)2875 541 y Fu(\024)f Fz(1)p Fy(:)-47 825 y Fz(As)i(in)f(the)i(previous)e(pro)s (of,)i(w)m(e)g(\014rst)f(consider)f(the)i(case)g(0)e Fu(\024)f Fy(\025)g Fu(\024)2322 790 y Fq(1)p 2322 805 36 4 v 2322 857 a(2)2367 769 y Fu(p)p 2443 769 80 4 v 56 x Fy(\032)2490 840 y Fw(`)2523 825 y Fz(.)40 b(Set)28 b Fy(w)2807 840 y Fw(k)2849 825 y Fz(\()p Fy(\025)p Fz(\))e(:=)3249 790 y Fw(\025)p 3129 805 281 4 v 3129 815 a Fu(p)p 3205 815 205 4 v 65 x Fw(\032)3241 892 y Fs(k)3279 880 y Fx(\000)p Fw(\025)3375 861 y Fr(2)3448 825 y Fz(and)g(\014x)h(some)-47 974 y(1)e Fu(\024)g Fy(k)k Fu(\024)c Fy(`)p Fz(.)40 b(Then)899 1255 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))26 b(:=)f(exp)1368 1154 y Fn(\020)1422 1255 y Fu(\000)1559 1141 y Fw(`)1508 1169 y Fn(X)1516 1364 y Fw(i)p Fq(=1)1654 1255 y Fy(c)1693 1269 y Fw(i)1722 1170 y Fn(p)p 1813 1170 279 4 v 85 x Fy(\032)1860 1269 y Fw(i)1908 1255 y Fu(\000)20 b Fy(\025)2052 1229 y Fq(2)2092 1154 y Fn(\021)2161 1255 y Fy(w)2226 1270 y Fw(k)2269 1255 y Fz(\()p Fy(\025)p Fz(\))2392 1154 y Fn(\020)2447 1255 y Fz(1)g Fu(\000)g Fy(\037)2660 1154 y Fn(\020)2776 1193 y Fy(\025)p 2725 1234 156 4 v 2725 1261 a Fu(p)p 2800 1261 80 4 v 2800 1317 a Fy(\032)2847 1332 y Fw(`)2890 1154 y Fn(\021)q(\021)-47 1536 y Fz(satsi\014es)30 b(the)g(deriv)-5 b(ativ)m(e)30 b(b)s(ounds)1595 1740 y Fu(j)p Fy(!)1680 1703 y Fq(\()p Fw(j)t Fq(\))1771 1740 y Fz(\()p Fy(\025)p Fz(\))p Fu(j)d(\024)e Fy(C)2107 1754 y Fw(j)2158 1740 y Fy(\025)2211 1703 y Fx(\000)p Fw(j)-47 1740 y Fz(\(7.27\))-47 1944 y(for)30 b(all)f Fy(\025)c(>)g Fz(0)31 b(and)f Fy(j)h Fu(\025)25 b Fz(0.)41 b(Hence)31 b(\(6.2\))h(of)f(Lemma)f(6.1)i(implies)27 b(that)935 2067 y Fn(\014)935 2121 y(\014)935 2176 y(\014)935 2230 y(\014)935 2285 y(\014)965 2102 y(Z)1056 2087 y Fx(p)p 1115 2087 67 3 v 41 x Fw(\032)1151 2140 y Fs(`)1016 2308 y Fq(0)1200 2226 y Fy(e)1252 2161 y Fr(1)p 1253 2173 31 3 v 1253 2214 a(2)1293 2188 y Fw(i\025)1358 2165 y Fr(2)1412 2226 y Fy(e)1454 2186 y Fx(\006)p Fw(i)1545 2136 y Fb(P)1619 2156 y Fs(m)1619 2206 y(j)s Fr(=1)1741 2186 y Fw(b)1771 2196 y Fs(j)1815 2124 y Fu(p)p 1891 2124 204 4 v 62 x Fw(\025)1932 2167 y Fr(2)1967 2186 y Fq(+)p Fw(\033)2062 2196 y Fs(j)2114 2226 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))15 b Fy(d\025)2412 2067 y Fn(\014)2412 2121 y(\014)2412 2176 y(\014)2412 2230 y(\014)2412 2285 y(\014)2468 2226 y Fo(.)25 b Fz(\(1)c(+)f Fy(\033)2808 2240 y Fq(1)2848 2226 y Fz(\))2893 2161 y Fr(1)p 2893 2173 31 3 v 2893 2214 a(4)2938 2226 y Fy(:)-3010 b Fz(\(7.28\))-47 2527 y(F)-8 b(or)31 b(the)f(part)h(close)f(to)798 2470 y Fu(p)p 874 2470 80 4 v 57 x Fy(\032)921 2542 y Fw(`)985 2527 y Fz(w)m(e)g(again)h(use)f(the)g(c)m(hange)i(of)e(v)-5 b(ariables)29 b Fy(u)d Fz(=)2629 2446 y Fn(p)p 2720 2446 284 4 v 81 x Fy(\032)2767 2542 y Fw(`)2820 2527 y Fu(\000)20 b Fy(\025)2964 2501 y Fq(2)3003 2527 y Fz(,)31 b(whic)m(h)e(yields)675 2702 y Fn(Z)766 2687 y Fx(p)p 825 2687 67 3 v 42 x Fw(\032)861 2741 y Fs(`)726 2909 y Fq(0)910 2826 y Fy(e)962 2761 y Fr(1)p 963 2773 31 3 v 963 2815 a(2)1003 2789 y Fw(i\025)1068 2765 y Fr(2)1122 2826 y Fy(e)1164 2786 y Fx(\006)p Fw(i)1255 2736 y Fb(P)1330 2757 y Fs(m)1330 2807 y(j)s Fr(=1)1451 2786 y Fw(b)1481 2796 y Fs(j)1525 2725 y Fu(p)p 1601 2725 204 4 v 61 x Fw(\025)1642 2767 y Fr(2)1677 2786 y Fq(+)p Fw(\033)1772 2796 y Fs(j)1839 2826 y Fz(exp)1978 2725 y Fn(\020)2032 2826 y Fu(\000)2169 2713 y Fw(`)2118 2740 y Fn(X)2127 2935 y Fw(i)p Fq(=1)2265 2826 y Fy(c)2304 2840 y Fw(i)2332 2741 y Fn(p)p 2423 2741 279 4 v 85 x Fy(\032)2470 2840 y Fw(i)2518 2826 y Fu(\000)20 b Fy(\025)2662 2800 y Fq(2)2702 2725 y Fn(\021)2771 2826 y Fy(w)2836 2841 y Fw(k)2879 2826 y Fz(\()p Fy(\025)p Fz(\))p Fy(\037)3059 2725 y Fn(\020)3175 2765 y Fy(\025)p 3124 2805 156 4 v 3124 2832 a Fu(p)p 3200 2832 80 4 v 57 x Fy(\032)3247 2904 y Fw(`)3289 2725 y Fn(\021)3359 2826 y Fy(d\025)521 3104 y Fz(=)83 b Fy(e)717 3066 y Fw(i\032)777 3078 y Fs(`)808 3066 y Fw(=)p Fq(2)897 2980 y Fn(Z)988 3006 y Fx(1)948 3186 y Fq(0)1078 3104 y Fy(e)1120 3066 y Fx(\000)p Fw(i )1245 3075 y Ft(\007)1298 3066 y Fq(\()p Fw(u)p Fq(\))1423 3104 y Fy(!)s Fz(\()p Fy(u)p Fz(\))15 b Fy(du:)-1791 b Fz(\(7.29\))-47 3354 y(Here)31 b Fy( )227 3368 y Fx(\007)286 3354 y Fz(\()p Fy(u)p Fz(\))g(are)g(as)f(in)f(\(7.23\))s(,)h(whereas) 619 3645 y Fy(!)s Fz(\()p Fy(u)p Fz(\))c(=)f(exp)1061 3544 y Fn(\020)1116 3645 y Fu(\000)1253 3531 y Fw(`)1202 3559 y Fn(X)1210 3754 y Fw(i)p Fq(=1)1348 3645 y Fy(c)1387 3659 y Fw(i)1416 3560 y Fn(p)p 1506 3560 469 4 v 1506 3645 a Fy(\032)1553 3659 y Fw(i)1602 3645 y Fu(\000)20 b Fy(\032)1740 3660 y Fw(`)1793 3645 y Fz(+)g Fy(u)1936 3619 y Fq(2)1975 3544 y Fn(\021)2030 3645 y Fy(\037)2087 3544 y Fn(\020)2141 3562 y(p)p 2232 3562 374 4 v 83 x Fz(1)g Fu(\000)g Fy(u)2440 3619 y Fq(2)2480 3645 y Fy(=\032)2572 3660 y Fw(`)2605 3544 y Fn(\021)2931 3584 y Fy(u)p 2669 3624 575 4 v 2669 3642 a Fn(p)p 2760 3642 484 4 v 81 x Fy(\032)2807 3738 y Fw(k)2870 3723 y Fu(\000)g Fy(\032)3008 3738 y Fw(`)3061 3723 y Fz(+)g Fy(u)3204 3697 y Fq(2)3254 3645 y Fy(:)-3326 b Fz(\(7.30\))-47 3976 y(Observ)m(e)27 b(that)h Fy(!)s Fz(\()p Fy(u)p Fz(\))e(=)f(0)i(if)g Fy(u)e Fu(\025)1123 3870 y Fn(q)p 1214 3870 91 4 v 1224 3940 a Fq(15)p 1224 3955 71 4 v 1224 4007 a(16)1304 3919 y Fu(p)p 1380 3919 80 4 v 57 x Fy(\032)1427 3991 y Fw(`)1460 3976 y Fz(.)40 b(Moreo)m(v)m(er,)30 b(the)e(reasoning)e(that)i(lead)f (to)h(\(7.20\))i(yields)25 b(in)i(this)f(case)-47 4101 y(that)1551 4235 y Fn(\014)1551 4290 y(\014)1551 4344 y(\014)1617 4278 y Fy(d)1664 4245 y Fw(j)p 1591 4319 136 4 v 1591 4402 a Fy(du)1690 4376 y Fw(j)1752 4340 y Fy(!)s Fz(\()p Fy(u)p Fz(\))1934 4235 y Fn(\014)1934 4290 y(\014)1934 4344 y(\014)1990 4340 y Fu(\024)f Fy(C)2151 4354 y Fw(j)2203 4340 y Fy(u)2255 4302 y Fx(\000)p Fw(j)-47 4570 y Fz(for)30 b(all)f Fy(j)i Fu(\025)25 b Fz(0)31 b(and)f Fy(u)25 b(>)g Fz(0.)41 b(Hence)31 b(\(7.4\))h(of)f(Lemma)f(7.2) i(and)e(\(7.9\))i(of)e(Lemma)h(7.3)g(imply)d(that)1153 4692 y Fn(\014)1153 4747 y(\014)1153 4801 y(\014)1153 4856 y(\014)1183 4700 y(Z)1274 4726 y Fx(1)1233 4906 y Fq(0)1364 4824 y Fy(e)1406 4786 y Fx(\000)p Fw(i )1531 4795 y Fr(+)1582 4786 y Fq(\()p Fw(u)p Fq(\))1707 4824 y Fy(!)s Fz(\()p Fy(u)p Fz(\))15 b Fy(du)2003 4692 y Fn(\014)2003 4747 y(\014)2003 4801 y(\014)2003 4856 y(\014)2060 4824 y Fo(.)25 b Fz(\(1)c(+)e Fy(\033)2399 4838 y Fq(1)2459 4824 y Fz(+)h Fy(\032)2597 4839 y Fw(`)2630 4824 y Fz(\))2675 4759 y Fr(1)p 2675 4771 31 3 v 2675 4812 a(4)2720 4824 y Fy(;)-47 5078 y Fz(whereas)37 b(the)h(case)h(of)f Fy( )838 5092 y Fx(\000)935 5078 y Fz(is)e(the)i(dealt)g(with)e(in)g(the)i(same) g(w)m(a)m(y)h(as)f(in)e(the)i(previous)e(pro)s(of,)j(see)g(\(7.24\))r (.)63 b(In)-47 5191 y(conjunction)27 b(with)g(\(7.28\))k(this)c(yields) g(the)h(\014rst)g(b)s(ound)e(in)h(\(7.25\))s(.)40 b(T)-8 b(o)29 b(obtain)e(the)i(second)g(estimate)g(of)35 b(\(7.25\))s(,)1903 5472 y(64)p eop %%Page: 65 65 65 64 bop -47 274 a Fz(w)m(e)32 b(\014rst)f(consider)f(the)i(case)h(of) f Fy(\025)27 b Fu(\024)1277 238 y Fq(1)p 1277 253 36 4 v 1277 305 a(2)1323 217 y Fu(p)p 1398 217 80 4 v 1398 274 a Fy(\032)1445 289 y Fw(`)1478 274 y Fz(,)32 b(cf.)h(\(7.28\))r(.) 44 b(In)31 b(con)m(trast)i(to)g(the)f(latter)g(form)m(ula,)f(though,)h (w)m(e)g(seek)-47 387 y(b)s(ounds)c(on)1291 475 y Fn(\014)1291 530 y(\014)1291 584 y(\014)1291 639 y(\014)1291 694 y(\014)1322 511 y(Z)1412 496 y Fx(p)p 1471 496 67 3 v 41 x Fw(\032)1507 549 y Fs(`)1372 717 y Fq(0)1557 634 y Fy(e)1599 597 y Fw(i\036)1665 606 y Ft(\006)1717 597 y Fq(\()p Fw(\025)p Fq(\))1843 634 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))2217 573 y Fy(\025)p 2051 614 385 4 v 2051 632 a Fn(p)p 2142 632 294 4 v 80 x Fy(\032)2189 727 y Fw(k)2252 712 y Fz(+)20 b Fy(\025)2396 686 y Fq(2)2461 634 y Fy(d\025)2561 475 y Fn(\014)2561 530 y(\014)2561 584 y(\014)2561 639 y(\014)2561 694 y(\014)-47 634 y Fz(\(7.31\))-47 920 y(with)29 b Fy(\036)214 934 y Fx(\006)298 920 y Fz(=)404 884 y Fq(1)p 404 899 36 4 v 404 952 a(2)449 920 y Fy(\025)502 887 y Fq(2)562 920 y Fu(\006)653 852 y Fn(P)749 878 y Fw(m)749 947 y(j)t Fq(=1)891 920 y Fy(b)930 934 y Fw(j)982 844 y Fn(p)p 1072 844 293 4 v 1072 920 a Fy(\025)1125 894 y Fq(2)1185 920 y Fz(+)20 b Fy(\033)1328 934 y Fw(j)1364 920 y Fz(,)31 b(and)792 1222 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))26 b(:=)f(exp)1261 1121 y Fn(\020)1316 1222 y Fu(\000)1453 1109 y Fw(`)1401 1136 y Fn(X)1410 1331 y Fw(i)p Fq(=1)1548 1222 y Fy(c)1587 1236 y Fw(i)1615 1137 y Fn(p)p 1706 1137 279 4 v 85 x Fy(\032)1753 1236 y Fw(i)1802 1222 y Fu(\000)20 b Fy(\025)1946 1196 y Fq(2)1985 1121 y Fn(\021)c(\020)2109 1222 y Fz(1)k Fu(\000)g Fy(\037)2322 1121 y Fn(\020)2438 1161 y Fy(\025)p 2387 1201 156 4 v 2387 1228 a Fu(p)p 2462 1228 80 4 v 2462 1285 a Fy(\032)2509 1300 y Fw(`)2552 1121 y Fn(\021)q(\021)2676 1056 y(s)p 2767 1056 314 4 v 2777 1161 a Fy(\032)2824 1176 y Fw(k)2887 1161 y Fz(+)g Fy(\025)3031 1134 y Fq(2)p 2777 1201 294 4 v 2777 1285 a Fy(\032)2824 1300 y Fw(k)2887 1285 y Fu(\000)g Fy(\025)3031 1258 y Fq(2)3080 1222 y Fy(:)-47 1508 y Fz(In)31 b(view)g(of)i(these)f (c)m(hoices)h(\(7.31\))h(is)d(equal)h(to)h(\(7.28\))r(.)46 b(Since)31 b Fy(\025)d Fu(\024)2370 1472 y Fq(1)p 2370 1487 36 4 v 2370 1539 a(2)2416 1451 y Fu(p)p 2491 1451 80 4 v 2491 1508 a Fy(\032)2538 1523 y Fw(`)2599 1508 y Fu(\024)2708 1472 y Fq(1)p 2708 1487 36 4 v 2708 1539 a(2)2753 1451 y Fu(p)p 2829 1451 90 4 v 57 x Fy(\032)2876 1523 y Fw(k)2951 1508 y Fz(on)k(the)g(supp)s(ort)e(of)i Fy(!)s Fz(,)h(this)-47 1620 y(function)c(satis\014es)h(the)h(deriv)-5 b(ativ)m(e)31 b(b)s(ounds)d(\(7.27\))s(.)41 b(Supp)s(ose)29 b(that)j(a)f(critical)e(p)s(oin)m(t)h Fy(\025)3059 1634 y Fq(0)3124 1620 y Fy(>)c Fz(0)31 b(of)g Fy(\036)3455 1634 y Fx(\000)3545 1620 y Fz(exists.)42 b(By)-47 1733 y(de\014nition)28 b(of)i Fy(\025)513 1747 y Fq(0)553 1733 y Fz(,)g(see)h(\(6.11\))s(,)f(there)h(exists)f(1)25 b Fu(\024)g Fy(p)g Fu(\024)g Fy(m)31 b Fz(suc)m(h)f(that)1596 1962 y Fy(m)15 b(b)1730 1976 y Fw(p)1795 1962 y Fu(\025)1891 1852 y Fn(q)p 1981 1852 296 4 v 1981 1962 a Fy(\033)2033 1976 y Fw(p)2093 1962 y Fz(+)20 b Fy(\025)2237 1931 y Fq(2)2237 1988 y(0)2277 1962 y Fy(:)-2349 b Fz(\(7.32\))-47 2169 y(Hence,)31 b(see)g(\(6.10\))s(,)934 2431 y Fy(\036)988 2394 y Fx(00)1030 2431 y Fz(\()p Fy(\025)1118 2445 y Fq(0)1158 2431 y Fz(\))84 b(=)1465 2317 y Fw(m)1430 2345 y Fn(X)1435 2540 y Fw(j)t Fq(=1)1711 2369 y Fy(b)1750 2383 y Fw(j)1787 2369 y Fy(\025)1840 2336 y Fq(2)1840 2393 y(0)p 1587 2410 418 4 v 1587 2516 a Fz(\()p Fy(\033)1674 2530 y Fw(j)1731 2516 y Fz(+)20 b Fy(\025)1875 2485 y Fq(2)1875 2542 y(0)1914 2516 y Fz(\))1959 2453 y Fr(3)p 1959 2465 31 3 v 1959 2506 a(2)2039 2431 y Fu(\025)25 b Fy(\025)2188 2394 y Fq(2)2188 2454 y(0)2391 2369 y Fy(b)2430 2383 y Fw(p)p 2238 2410 387 4 v 2238 2428 a Fn(p)p 2328 2428 296 4 v 2328 2507 a Fy(\025)2381 2476 y Fq(2)2381 2533 y(0)2441 2507 y Fz(+)20 b Fy(\033)2584 2521 y Fw(p)2784 2370 y Fz(1)p 2659 2410 V 2659 2495 a Fy(\025)2712 2464 y Fq(2)2712 2521 y(0)2771 2495 y Fz(+)g Fy(\033)2914 2509 y Fw(p)1277 2733 y Fu(\025)1617 2671 y Fy(\025)1670 2638 y Fq(2)1670 2695 y(0)p 1440 2712 446 4 v 1440 2797 a Fy(m)p Fz(\()p Fy(\025)1608 2765 y Fq(2)1608 2823 y(0)1668 2797 y Fz(+)g Fy(\033)1811 2811 y Fw(p)1851 2797 y Fz(\))1896 2733 y Fy(:)-1968 b Fz(\(7.33\))-47 2980 y(In)29 b(view)h(of)38 b(\(6.43\))32 b(one)f(can)g(apply)e(Lemma)h(6.7)i(with)1699 3223 y Fy(B)e Fz(=)1904 3161 y Fy(\036)1958 3128 y Fx(00)2000 3161 y Fz(\()p Fy(\025)2088 3175 y Fq(0)2128 3161 y Fz(\))p 1904 3202 260 4 v 1987 3285 a Fy(\025)2040 3299 y Fq(0)2173 3223 y Fy(:)-47 3455 y Fz(By)i(\(7.33\))g(this)e(leads)f(to)i(the)g(b)s (ound)418 3579 y Fn(\014)418 3634 y(\014)418 3688 y(\014)418 3743 y(\014)418 3797 y(\014)448 3614 y(Z)539 3641 y Fw(\025)580 3650 y Fr(0)499 3820 y Fq(0)634 3738 y Fy(e)676 3701 y Fw(i\036)742 3710 y Ft(\000)795 3701 y Fq(\()p Fw(\025)p Fq(\))920 3738 y Fy(!)s Fz(\()p Fy(\025)p Fz(\))1294 3677 y Fy(\025)p 1129 3717 385 4 v 1129 3735 a Fn(p)p 1220 3735 294 4 v 81 x Fy(\032)1267 3831 y Fw(k)1330 3816 y Fz(+)19 b Fy(\025)1473 3790 y Fq(2)1538 3738 y Fy(d\025)1638 3579 y Fn(\014)1638 3634 y(\014)1638 3688 y(\014)1638 3743 y(\014)1638 3797 y(\014)1694 3738 y Fo(.)25 b Fy(\036)1844 3701 y Fx(00)1886 3738 y Fz(\()p Fy(\025)1974 3752 y Fq(0)2014 3738 y Fz(\))2049 3701 y Fx(\000)2114 3673 y Fr(1)p 2114 3685 31 3 v 2114 3727 a(2)2315 3677 y Fy(\025)2368 3691 y Fq(0)p 2169 3717 385 4 v 2169 3735 a Fn(p)p 2260 3735 294 4 v 79 x Fy(\025)2313 3783 y Fq(2)2313 3840 y(0)2373 3814 y Fz(+)19 b Fy(\032)2510 3829 y Fw(k)2588 3738 y Fo(.)2684 3668 y Fu(p)p 2760 3668 80 4 v 70 x Fy(m)2840 3574 y Fn(s)p 2931 3574 316 4 v 2941 3676 a Fy(\033)2993 3690 y Fw(p)3053 3676 y Fz(+)h Fy(\025)3197 3645 y Fq(2)3197 3702 y(0)p 2941 3717 296 4 v 2942 3802 a Fy(\032)2989 3817 y Fw(k)3052 3802 y Fz(+)g Fy(\025)3196 3771 y Fq(2)3196 3828 y(0)3271 3738 y Fo(.)25 b Fy(m)3457 3673 y Fr(3)p 3457 3685 31 3 v 3457 3727 a(2)3555 3676 y Fy(b)3594 3690 y Fw(p)p 3511 3717 166 4 v 3511 3744 a Fu(p)p 3587 3744 90 4 v 56 x Fy(\032)3634 3815 y Fw(k)3687 3738 y Fy(:)-3759 b Fz(\(7.34\))-47 4008 y(W)-8 b(e)41 b(will)d(no)m(w)j(estimate)g(the)f (con)m(tribution)f(b)m(y)i(the)f(in)m(terv)-5 b(al)40 b([)p Fy(\025)2289 4022 y Fq(0)2329 4008 y Fy(;)15 b Fu(1)p Fz(\))41 b(b)m(y)f(means)g(of)h(Lemma)f(6.6)i(with)d Fy(\034)52 b Fz(=)-47 4121 y Fy(\032)0 4136 y Fw(k)42 4121 y Fz(.)61 b(The)37 b(condition)e Fy(\025)780 4135 y Fq(0)856 4121 y Fu(\024)h Fy(\034)48 b Fz(in)35 b(that)j(lemma)f(can) g(b)s(e)f(assumed)h(to)h(hold,)f(as)h(otherwise)e Fy(!)k Fz(v)-5 b(anishes)36 b(on)h(the)-47 4234 y(in)m(terv)-5 b(al)31 b([)p Fy(\025)361 4248 y Fq(0)401 4234 y Fy(;)15 b Fu(1)p Fz(\).)48 b(If)32 b(the)g(p)s(ositiv)m(e)g(critical)f(p)s(oin) m(t)h Fy(\025)1828 4248 y Fq(0)1899 4234 y Fz(do)s(es)h(not)f(exist,)h (then)f(w)m(e)h(formally)e(set)i Fy(\025)3416 4248 y Fq(0)3484 4234 y Fz(=)c(0.)47 b(Recall)-47 4347 y(that)1700 4587 y(1)26 b Fu(\025)1901 4473 y Fw(m)1867 4501 y Fn(X)1871 4696 y Fw(j)t Fq(=1)2068 4525 y Fy(b)2107 4539 y Fw(j)p 2023 4566 165 4 v 2023 4597 a Fu(p)p 2099 4597 89 4 v 52 x Fy(\033)2151 4663 y Fw(j)-47 4875 y Fz(if)j Fy(\025)89 4889 y Fq(0)154 4875 y Fy(>)c Fz(0)31 b(do)s(es)f(not)g(exist,)h(see)g (\(6.16\))r(.)41 b(If)1688 5076 y(1)p 1688 5116 46 4 v 1688 5200 a(2)1768 5137 y Fu(\025)1899 5024 y Fw(m)1864 5051 y Fn(X)1869 5246 y Fw(j)t Fq(=1)2065 5075 y Fy(b)2104 5089 y Fw(j)p 2021 5116 165 4 v 2021 5147 a Fu(p)p 2096 5147 89 4 v 2096 5200 a Fy(\033)2148 5214 y Fw(j)2195 5137 y Fy(;)1903 5472 y Fz(65)p eop %%Page: 66 66 66 65 bop -47 274 a Fz(then)29 b Fy(\036)213 241 y Fx(0)236 274 y Fz(\()p Fy(\025)p Fz(\))d Fu(\025)491 238 y Fq(1)p 491 253 36 4 v 491 305 a(2)537 274 y Fy(\025)p Fz(.)40 b(Th)m(us)28 b(\(6.17\))k(holds)c(with)h Fy(A)c Fz(=)1789 238 y Fq(1)p 1789 253 V 1789 305 a(2)1834 274 y Fz(,)30 b(and)f(therefore)h(\(6.18\))i(leads)d(to)h(a)g Fy(O)s Fz(\(1\))g(b)s(ound.)39 b(In)29 b(view)-47 387 y(of)37 b(\(6.11\))c(and)d(the)g(preceding)g(w)m(e)h(can)f(th)m(us)g(assume)g (that)1591 661 y(1)c Fu(\020)1792 547 y Fw(m)1757 574 y Fn(X)1762 770 y Fw(j)t Fq(=1)2068 599 y Fy(b)2107 613 y Fw(j)p 1914 640 383 4 v 1914 658 a Fn(p)p 2005 658 293 4 v 79 x Fy(\033)2057 751 y Fw(j)2113 737 y Fz(+)20 b Fy(\025)2257 705 y Fq(2)2257 763 y(0)-47 954 y Fz(regardless)38 b(of)h(the)g(v)-5 b(alue)38 b(of)h Fy(\025)1067 968 y Fq(0)1106 954 y Fz(.)66 b(In)38 b(particular,)h(there)g(exists)f(some)h (1)h Fu(\024)f Fy(p)g Fu(\024)f Fy(m)h Fz(with)e(the)i(prop)s(ert)m(y)f (that)-47 1067 y Fy(m)15 b(b)87 1081 y Fw(p)152 1067 y Fo(&)248 989 y Fn(p)p 339 989 296 4 v 78 x Fy(\033)391 1081 y Fw(p)450 1067 y Fz(+)20 b Fy(\025)594 1036 y Fq(2)594 1093 y(0)634 1067 y Fz(.)40 b(W)-8 b(e)31 b(no)m(w)f(pro)s(ceed)g(to)g (v)m(erify)g(the)g(conditions)e(\(6.19\))k(with)c Fy(R)e Fu(\020)f Fy(\033)3044 1081 y Fw(p)3103 1067 y Fz(+)19 b Fy(\025)3246 1034 y Fq(2)3246 1092 y(0)3285 1067 y Fz(.)41 b(More)31 b(precisely)-8 b(,)-47 1180 y(let)30 b Fy(\025)137 1194 y Fq(0)202 1180 y Fu(\024)25 b Fy(\025)g Fu(\024)g Fz(2)517 1101 y Fn(p)p 608 1101 V 79 x Fy(\033)660 1194 y Fw(p)720 1180 y Fz(+)20 b Fy(\025)864 1149 y Fq(2)864 1206 y(0)904 1180 y Fz(.)40 b(Then)30 b(\(6.7\))i(or)e(\(6.8\))i(imply) c(that)867 1450 y Fy(\036)921 1413 y Fx(0)945 1450 y Fz(\()p Fy(\025)p Fz(\))83 b Fo(&)g Fy(\025)1373 1326 y Fn(Z)1464 1353 y Fw(\025)1424 1533 y(\025)1465 1542 y Fr(0)1665 1388 y Fy(b)1704 1402 y Fw(p)1759 1388 y Fy(s)p 1535 1429 398 4 v 1535 1535 a Fz(\()p Fy(b)1609 1549 y Fw(p)1669 1535 y Fz(+)20 b Fy(s)1803 1509 y Fq(2)1842 1535 y Fz(\))1887 1472 y Fr(3)p 1887 1484 31 3 v 1887 1525 a(2)1957 1450 y Fy(ds)25 b Fo(&)g Fy(\025)2221 1413 y Fq(2)2435 1388 y Fy(b)2474 1402 y Fw(p)p 2270 1429 408 4 v 2270 1535 a Fz(\()p Fy(b)2344 1549 y Fw(p)2405 1535 y Fz(+)20 b Fy(\025)2549 1504 y Fq(2)2549 1561 y(0)2588 1535 y Fz(\))2633 1472 y Fr(3)p 2633 1484 31 3 v 2633 1525 a(2)2703 1450 y Fz(\()p Fy(\025)h Fu(\000)f Fy(\025)2956 1464 y Fq(0)2995 1450 y Fz(\))1151 1732 y Fo(&)1492 1671 y Fy(\025)1545 1638 y Fq(2)p 1315 1711 446 4 v 1315 1796 a Fy(m)p Fz(\()p Fy(\033)1482 1810 y Fw(p)1542 1796 y Fz(+)g Fy(\025)1686 1765 y Fq(2)1686 1822 y(0)1726 1796 y Fz(\))1786 1732 y(\()p Fy(\025)h Fu(\000)e Fy(\025)2038 1746 y Fq(0)2078 1732 y Fz(\))p Fy(;)-47 1998 y Fz(whic)m(h)38 b(is)g(the)h(\014rst)f(condition)g(in)g(\(6.19\))j(with)d Fy(R)i Fu(\020)g Fy(\033)1965 2012 y Fw(p)2030 1998 y Fz(+)26 b Fy(\025)2180 1965 y Fq(2)2180 2023 y(0)2258 1998 y Fz(and)39 b Fy(c)h Fz(=)f Fy(m)2713 1965 y Fx(\000)p Fq(1)2808 1998 y Fz(.)66 b(If)39 b(on)g(the)g(other)g(hand)f Fy(\025)i Fu(\025)-47 2111 y Fz(2)-2 2032 y Fn(p)p 89 2032 296 4 v 79 x Fy(\033)141 2125 y Fw(p)201 2111 y Fz(+)20 b Fy(\025)345 2080 y Fq(2)345 2137 y(0)384 2111 y Fz(.)41 b(Then)877 2381 y Fy(\036)931 2344 y Fx(0)955 2381 y Fz(\()p Fy(\025)p Fz(\))83 b Fo(&)g Fy(\025)1383 2257 y Fn(Z)1474 2284 y Fw(\025)1434 2401 y Fu(p)p 1510 2401 208 4 v 63 x Fw(\033)1550 2472 y Fs(p)1586 2464 y Fq(+)p Fw(\025)1682 2441 y Fr(2)1682 2485 y(0)1876 2319 y Fy(b)1915 2333 y Fw(p)1970 2319 y Fy(s)p 1746 2360 398 4 v 1746 2466 a Fz(\()p Fy(b)1820 2480 y Fw(p)1880 2466 y Fz(+)20 b Fy(s)2014 2440 y Fq(2)2053 2466 y Fz(\))2098 2403 y Fr(3)p 2099 2415 31 3 v 2099 2456 a(2)2168 2381 y Fy(ds)25 b Fo(&)g Fy(\025)2447 2257 y Fn(Z)2538 2284 y Fw(\025)2498 2401 y Fu(p)p 2574 2401 208 4 v 63 x Fw(\033)2614 2472 y Fs(p)2650 2464 y Fq(+)p Fw(\025)2746 2441 y Fr(2)2746 2485 y(0)2812 2319 y Fy(b)2851 2333 y Fw(p)p 2810 2360 83 4 v 2810 2443 a Fy(s)2853 2417 y Fq(2)2917 2381 y Fy(ds)1161 2651 y Fo(&)1479 2589 y Fy(b)1518 2603 y Fw(p)p 1325 2630 387 4 v 1325 2648 a Fn(p)p 1416 2648 296 4 v 79 x Fy(\033)1468 2741 y Fw(p)1528 2727 y Fz(+)20 b Fy(\025)1672 2696 y Fq(2)1672 2753 y(0)1736 2651 y Fz(\()p Fy(\025)h Fu(\000)1936 2541 y Fn(q)p 2027 2541 V 110 x Fy(\033)2079 2665 y Fw(p)2139 2651 y Fz(+)f Fy(\025)2283 2620 y Fq(2)2283 2677 y(0)2322 2651 y Fz(\))26 b Fo(&)f Fy(m)2559 2614 y Fx(\000)p Fq(1)2668 2651 y Fz(\()p Fy(\025)20 b Fu(\000)g Fy(\025)2920 2665 y Fq(0)2960 2651 y Fz(\))p Fy(:)-3067 b Fz(\(7.35\))-47 2939 y(T)-8 b(o)31 b(pass)f(to)h(the)f (last)g(line)f(w)m(e)i(used)f(that)h Fy(b)1474 2953 y Fw(p)1539 2939 y Fo(&)24 b Fy(m)1714 2906 y Fx(\000)p Fq(1)1809 2861 y Fn(p)p 1900 2861 V 78 x Fy(\033)1952 2953 y Fw(p)2011 2939 y Fz(+)c Fy(\025)2155 2908 y Fq(2)2155 2966 y(0)2225 2939 y Fz(as)31 b(w)m(ell)e(as)1414 3184 y(2\()p Fy(\025)21 b Fu(\000)1659 3073 y Fn(q)p 1750 3073 V 111 x Fy(\033)1802 3198 y Fw(p)1862 3184 y Fz(+)f Fy(\025)2006 3152 y Fq(2)2006 3210 y(0)2045 3184 y Fz(\))26 b Fu(\025)f Fy(\025)20 b Fu(\000)g Fy(\025)2419 3198 y Fq(0)2458 3184 y Fy(:)-47 3401 y Fz(Since)29 b(\(7.35\))k(v)m (eri\014es)d(the)g(second)h(condition)e(in)g(\(6.19\))j(w)m(e)f (conclude)f(from)g(\(6.20\))i(that)506 3541 y Fn(\014)506 3595 y(\014)506 3650 y(\014)506 3704 y(\014)506 3759 y(\014)536 3576 y(Z)627 3602 y Fx(1)587 3782 y Fw(\025)628 3791 y Fr(0)717 3700 y Fy(e)759 3662 y Fw(i\036)825 3671 y Ft(\000)878 3662 y Fq(\()p Fw(\025)p Fq(\))1003 3700 y Fy(!)s Fz(\()p Fy(\025)p 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4513 y Fx(0)1166 4568 y Fq(+)1225 4546 y Fz(\()p Fy(\025)p Fz(\))h Fu(\025)e Fy(\025)p Fz(,)j(the)e(in)m(tegral)g(in)f(\(7.31\))k(with)35 b(phase)i Fy(\036)3021 4560 y Fq(+)3118 4546 y Fz(is)f Fy(O)s Fz(\(1\))i(b)m(y)f(the)g(usual)-47 4659 y(argumen)m(t)27 b(in)m(v)m(olving)e(a)h(single)g(in)m(tegration)g(b)m(y)g(parts.)40 b(Th)m(us)25 b(\(7.31\))k(satis\014es)d(the)g(estimate)h(\(7.36\))i(in) c(all)h(cases.)-47 4772 y(Clearly)-8 b(,)29 b(b)s(oth)h(\(7.31\))j(and) c(the)i(in)m(tegral)f(in)f(\(7.25\))k(are)d(no)h(larger)f(than)364 5039 y(min)341 5100 y Fq(1)p Fx(\024)p Fw(i)p Fx(\024)p Fw(`)554 4915 y Fn(Z)645 4900 y Fx(p)p 703 4900 67 3 v 703 4941 a Fw(\032)739 4953 y Fs(`)604 5121 y Fq(0)789 5039 y Fy(e)831 5001 y Fx(\000)p Fw(c)917 5011 y Fs(i)943 4937 y Fu(p)p 1019 4937 194 4 v 64 x Fw(\032)1055 5011 y Fs(i)1081 5001 y Fx(\000)p Fw(\025)1177 4982 y Fr(2)1417 4977 y Fy(\025)p 1251 5018 385 4 v 1251 5036 a Fn(p)p 1342 5036 294 4 v 81 x Fy(\032)1389 5132 y Fw(k)1452 5117 y Fu(\000)20 b Fy(\025)1596 5091 y Fq(2)1660 5039 y Fy(d\025)26 b Fu(\024)48 b Fz(min)1882 5100 y Fq(1)p Fx(\024)p Fw(i)p Fx(\024)p Fw(`)2095 4915 y Fn(Z)2186 4900 y Fx(p)p 2245 4900 67 3 v 41 x Fw(\032)2281 4953 y Fs(`)2146 5121 y Fq(0)2330 5039 y Fy(e)2372 5001 y Fx(\000)p Fw(c)2458 5011 y Fs(i)2484 4937 y Fu(p)p 2560 4937 197 4 v 64 x Fw(\032)2596 5013 y Fs(`)2626 5001 y Fx(\000)p Fw(\025)2722 4982 y Fr(2)2957 4977 y Fy(\025)p 2796 5018 375 4 v 2796 5036 a Fn(p)p 2887 5036 284 4 v 81 x Fy(\032)2934 5132 y Fw(`)2987 5117 y Fu(\000)20 b Fy(\025)3131 5091 y Fq(2)3196 5039 y Fy(d\025)25 b Fz(=)49 b(min)3417 5100 y Fq(1)p Fx(\024)p Fw(i)p Fx(\024)p Fw(`)3630 5039 y Fy(c)3669 5001 y Fx(\000)p Fq(1)3669 5066 y Fw(i)3764 5039 y Fy(:)-3836 b Fz(\(7.37\))1903 5472 y(66)p eop %%Page: 67 67 67 66 bop -47 274 a Fz(The)29 b(desired)g(b)s(ound)f(\(7.25\))k(no)m(w) e(follo)m(ws)f(easily)g(from)h(\(7.36\))i(and)e(\(7.37\))r(.)40 b(Indeed,)30 b(if)f Fy(c)3121 289 y Fw(k)3189 274 y Fu(\024)3285 217 y(p)p 3361 217 90 4 v 57 x Fy(\032)3408 289 y Fw(k)3451 274 y Fz(,)h(then)g(\(7.36\))-47 387 y(implies)e(\(7.25\))r(.)43 b(On)30 b(the)h(other)g(hand,)f(if)g Fy(c)1485 402 y Fw(k)1554 387 y Fu(\025)1651 330 y(p)p 1727 330 V 57 x Fy(\032)1774 402 y Fw(k)1817 387 y Fz(,)h(then)g(in)e(view)h(of)39 b(\(7.26\))33 b(one)e(has)g Fy(c)3141 402 y Fw(k)3210 387 y Fu(\025)3307 330 y(p)p 3383 330 V 57 x Fy(\032)3430 402 y Fw(k)3498 387 y Fu(\025)3595 330 y(p)p 3671 330 80 4 v 57 x Fy(\032)3718 402 y Fw(`)3777 387 y Fu(\025)26 b Fz(1.)-47 500 y(In)j(that)i(case)h(\(7.37\))g(yields)d(a)i Fy(O)s Fz(\(1\))g(b)s(ound.)-47 613 y(It)f(remains)g(to)h(consider)f (the)g(case)i(of)f Fy(\025)f Fz(close)h(to)1723 556 y Fu(p)p 1798 556 V 1798 613 a Fy(\032)1845 628 y Fw(`)1878 613 y Fz(.)42 b(In)29 b(that)j(case)f(the)g(c)m(hange)h(of)f(v)-5 b(ariables)29 b Fy(u)c Fz(=)3561 532 y Fn(p)p 3652 532 294 4 v 81 x Fy(\032)3699 628 y Fw(k)3762 613 y Fu(\000)19 b Fy(\025)3905 586 y Fq(2)-47 726 y Fz(reduces)30 b(matters)h(to)g (\(7.29\))i(with)c Fy(!)s Fz(\()p Fy(u)p Fz(\))i(as)f(in)f(\(7.30\))s (.)40 b(Supp)s(ose)29 b(\014rst)g(that)1616 880 y Fw(m)1582 908 y Fn(X)1586 1103 y Fw(j)t Fq(=1)1878 932 y Fy(b)1917 946 y Fw(j)p 1738 973 356 4 v 1738 997 a Fu(p)p 1814 997 280 4 v 59 x Fy(\033)1866 1070 y Fw(j)1923 1056 y Fz(+)20 b Fy(\032)2061 1071 y Fw(`)2129 994 y Fu(\024)2235 933 y Fz(1)p 2235 973 46 4 v 2235 1056 a(2)2290 994 y Fy(:)-47 1298 y Fz(Then)32 b(Lemma)h(7.1)i(implies)30 b(that)k Fy( )1243 1265 y Fx(0)1240 1321 y Fq(+)1299 1298 y Fz(\()p Fy(u)p Fz(\))d Fu(\025)1563 1262 y Fq(1)p 1563 1277 36 4 v 1563 1329 a(2)1608 1298 y Fy(u)i Fz(on)h(the)f(supp)s (ort)f(of)h Fy(!)s Fz(,)h(whereas)f(in)f(all)g(cases)i Fy( )3425 1265 y Fx(0)3422 1321 y(\000)3482 1298 y Fz(\()p Fy(u)p Fz(\))c Fu(\025)g Fy(u)p Fz(.)50 b(In)-47 1411 y(case)34 b(of)g Fy( )314 1425 y Fq(+)373 1411 y Fz(\()p Fy(u)p Fz(\))h(a)f Fy(O)s Fz(\(1\))g(b)s(ound)e(is)g(obtained)h(via)g (\(7.5\))r(,)i(whereas)e(for)h Fy( )2533 1425 y Fx(\000)2592 1411 y Fz(\()p Fy(u)p Fz(\))g(this)f(b)s(ound)e(follo)m(ws)i(b)m(y)g(a) h(single)-47 1524 y(in)m(tegration)c(b)m(y)g(parts)g(in)g(the)g(usual)f (w)m(a)m(y)-8 b(.)42 b(Hence)31 b(it)f(su\016ces)g(to)h(assume)g(that) 1616 1674 y Fw(m)1582 1701 y Fn(X)1586 1896 y Fw(j)t Fq(=1)1878 1725 y Fy(b)1917 1739 y Fw(j)p 1738 1766 356 4 v 1738 1790 a Fu(p)p 1814 1790 280 4 v 60 x Fy(\033)1866 1864 y Fw(j)1923 1850 y Fz(+)20 b Fy(\032)2061 1865 y Fw(`)2129 1787 y Fu(\025)2235 1726 y Fz(1)p 2235 1766 46 4 v 2235 1850 a(2)2290 1787 y Fy(:)-47 2088 y Fz(There)31 b(exists)g(some)i(1)28 b Fu(\024)f Fy(p)g Fu(\024)h Fy(m)j Fz(so)h(that)h Fy(m)15 b(b)1599 2102 y Fw(p)1666 2088 y Fu(\025)1764 2029 y(p)p 1840 2029 283 4 v 59 x Fy(\033)1892 2102 y Fw(p)1952 2088 y Fz(+)20 b Fy(\032)2090 2103 y Fw(`)2123 2088 y Fz(.)45 b(By)33 b(\(7.26\))h(this)d(implies)e(that)j Fy(b)3328 2102 y Fw(p)3395 2088 y Fu(\025)c Fy(m)3574 2055 y Fx(\000)p Fq(1)3668 2088 y Fz(.)45 b(Since)-47 2201 y(the)24 b(estimate)i(\(7.37\))g(also)f(applies)d(to)k(the)e (oscillatory)g(in)m(tegrals)g(after)h(c)m(hanging)g(v)-5 b(ariables)23 b(to)i Fy(u)p Fz(,)h(one)f(concludes)-47 2314 y(from)30 b(the)g(preceding)g(that)h(there)f(is)g(an)g(upp)s(er)e (b)s(ound)g(of)j(the)g(form)1622 2551 y Fy(m)1737 2489 y Fz(max)1906 2503 y Fq(1)p Fx(\024)p Fw(j)t Fx(\024)p Fw(m)2165 2489 y Fy(b)2204 2503 y Fw(j)p 1737 2531 504 4 v 1948 2614 a Fy(c)1987 2629 y Fw(k)2251 2551 y Fy(;)-47 2786 y Fz(and)f(w)m(e)g(are)h(done.)p 3879 2786 4 62 v 3883 2728 55 4 v 3883 2786 V 3937 2786 4 62 v -47 3072 a FA(8)134 b(Putting)46 b(it)f(all)h(together)-47 3275 y Fz(By)30 b(com)m(bining)f(the)i(results)e(of)i(the)f(previous)f (three)i(sections)f(w)m(e)h(are)g(no)m(w)f(able)g(to)h(pro)m(v)m(e)g (our)f(main)f(result.)-47 3458 y Fm(Theorem)34 b(8.1.)46 b Fl(L)-5 b(et)33 b Fy(V)20 b Fz(\()p Fy(t;)15 b(x)p Fz(\))33 b Fl(b)-5 b(e)33 b(a)g(r)-5 b(e)g(al-value)g(d)34 b(me)-5 b(asur)g(able)35 b(function)e(on)g Fv(R)2714 3425 y Fq(4)2792 3458 y Fl(such)g(that)762 3705 y Fz(sup)817 3778 y Fw(t)914 3705 y Fu(k)p Fy(V)20 b Fz(\()p Fy(t;)15 b Fu(\001)p Fz(\))p Fu(k)1245 3755 y Fw(L)1303 3706 y Fr(3)p 1305 3714 31 3 v 1305 3748 a(2)1349 3755 y Fq(\()p Fp(R)1424 3737 y Fr(3)1459 3755 y Fq(\))1516 3705 y Fy(<)25 b(c)1651 3719 y Fq(0)1756 3705 y Fl(and)96 b Fz(sup)1980 3789 y Fw(y)r Fx(2)p Fp(R)2112 3770 y Fr(3)2162 3581 y Fn(Z)2213 3787 y Fp(R)2261 3768 y Fr(3)2325 3643 y Fu(k)p Fy(V)20 b Fz(\()5 b(^)-50 b Fy(\034)11 b(;)k(x)p Fz(\))p Fu(k)2701 3657 y Fx(M)p 2325 3684 466 4 v 2427 3767 a Fu(j)p Fy(x)20 b Fu(\000)g Fy(y)s Fu(j)2815 3705 y Fy(dx)26 b(<)f Fz(4)p Fy(\031)-3180 b Fz(\(8.1\))-47 3968 y Fl(for)33 b(some)g(smal)5 b(l)34 b(c)-5 b(onstant)34 b Fy(c)973 3982 y Fq(0)1038 3968 y Fy(>)25 b Fz(0)p Fl(,)33 b(se)-5 b(e)33 b(De\014nition)g(5.1.)42 b(Then)582 4177 y Fu(k)p Fy(U)10 b Fz(\()p Fy(t;)15 b(s)p Fz(\))p Fy( )944 4191 y Fw(s)982 4177 y Fu(k)1027 4191 y Fx(1)1128 4177 y Fu(\024)25 b Fy(C)7 b Fu(j)p Fy(t)19 b Fu(\000)h 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5110 y Fz(\).)53 b(F)-8 b(urthermore,)36 b(Prop)s(osition)c(5.9)k(pro)m(vides)e(a)g(represen)m (tation)h(of)g(the)g(k)m(ernel)f(of)-47 5223 y Fu(I)3 5237 y Fw(m)69 5223 y Fz(\()p Fy(t;)15 b(s)p Fz(\))27 b(in)f(terms)h(of)g(three)h(kinds)d(of)i(oscillatory)f(in)m(tegrals,)i (whic)m(h)d(are)j(de\014ned)e(in)f(\(5.39\))s(,)j(\(5.41\))r(,)g(and)e (\(5.42\))s(.)1903 5472 y(67)p eop %%Page: 68 68 68 67 bop -47 274 a Fz(Supp)s(ose)30 b Fy(t)d(>)g(s)p Fz(.)44 b(Changing)31 b(v)-5 b(ariables)30 b Fy(\025)d Fu(7!)1644 238 y Fw(\025)p 1579 253 172 4 v 1579 263 a Fx(p)p 1638 263 114 3 v 50 x Fw(t)p Fx(\000)p Fw(s)1792 274 y Fz(in)k(eac)m(h)h(of)g(these)g(in)m(tegrals)g(brings)d(out)j(one) g(factor)h(of)f(\()p Fy(t)21 b Fu(\000)-47 429 y Fy(s)p Fz(\))31 394 y Fx(\000)96 367 y Fr(1)p 96 379 31 3 v 96 420 a(2)140 429 y Fz(,)29 b(whereas)g(\(5.43\))h(already)f(con)m (tains)f(the)h(factor)h(\()p Fy(t)16 b Fu(\000)h Fy(s)p Fz(\))2135 397 y Fx(\000)p Fq(1)2229 429 y Fz(.)40 b(This)27 b(leads)h(to)h(the)g(desired)e(p)s(o)m(w)m(er)h(\()p Fy(t)17 b Fu(\000)f Fy(s)p Fz(\))3810 394 y Fx(\000)3875 367 y Fr(3)p 3875 379 V 3875 420 a(2)3920 429 y Fz(.)-47 542 y(More)31 b(precisely)-8 b(,)30 b(for)g(the)g(oscillatory)g(in)m (tegrals)g(from)g(\(5.39\))i(this)e(pro)s(cess)f(leads)h(to)529 680 y Fn(Z)620 706 y Fx(1)580 886 y Fq(0)710 803 y Fy(e)752 766 y Fw(i)p Fq(\()p Fw(t)p Fx(\000)p Fw(s)p Fq(\))p Fw(\025)984 742 y Fr(2)1039 803 y Fz(cos)1176 675 y Fn(\022)1293 690 y Fw(m)1258 717 y Fn(X)1259 915 y Fw(k)r Fq(=1)1405 717 y Fn(p)p 1496 717 299 4 v 86 x Fy(\025)1549 777 y Fq(2)1608 803 y Fz(+)20 b Fy(\033)1751 818 y Fw(k)1794 803 y Fu(j)p Fy(x)1871 818 y Fw(k)r Fx(\000)p Fq(1)2024 803 y Fu(\000)g Fy(x)2167 818 y Fw(k)2209 803 y Fu(j)2234 675 y Fn(\023)2475 742 y Fy(\025)p 2312 782 380 4 v 2312 801 a Fn(p)p 2403 801 289 4 v 82 x Fy(\025)2456 856 y Fq(2)2515 883 y Fz(+)g Fy(\033)2658 898 y Fw(`)2716 803 y Fy(d\025)375 1108 y Fz(=)83 b(\()p Fy(t)21 b Fu(\000)e Fy(s)p Fz(\))786 1071 y Fx(\000)851 1044 y Fr(1)p 851 1056 31 3 v 851 1097 a(2)926 984 y Fn(Z)1017 1011 y Fx(1)977 1191 y Fq(0)1107 1108 y Fy(e)1149 1071 y Fw(i\025)1214 1047 y Fr(2)1268 1108 y Fz(cos)1405 980 y Fn(\022)1522 995 y Fw(m)1487 1022 y Fn(X)1489 1219 y Fw(k)r Fq(=1)1634 1026 y Fn(p)p 1725 1026 556 4 v 82 x Fy(\025)1778 1082 y Fq(2)1837 1108 y Fz(+)h Fy(\033)1980 1123 y Fw(k)2023 1108 y Fz(\()p Fy(t)g Fu(\000)g Fy(s)p Fz(\))2290 1047 y Fu(j)p Fy(x)2367 1062 y Fw(k)r Fx(\000)p Fq(1)2520 1047 y Fu(\000)g Fy(x)2663 1062 y Fw(k)2706 1047 y Fu(j)p 2290 1087 442 4 v 2379 1105 a(p)p 2455 1105 187 4 v 71 x Fy(t)g Fu(\000)g Fy(s)2741 980 y Fn(\023)3110 1047 y Fy(\025)p 2818 1087 637 4 v 2818 1105 a Fn(p)p 2909 1105 546 4 v 79 x Fy(\025)2962 1157 y Fq(2)3021 1184 y Fz(+)g Fy(\033)3164 1199 y Fw(`)3197 1184 y Fz(\()p Fy(t)h Fu(\000)f Fy(s)p Fz(\))3480 1108 y Fy(d\025;)-3652 b Fz(\(8.2\))-47 1384 y(and)24 b(similarly)d(for)k(\(5.41\))i(and)d (\(5.42\))r(.)39 b(Th)m(us)24 b(the)h(parameters)g Fy(\033)2241 1398 y Fw(j)2302 1384 y Fz(and)f Fy(\032)2520 1399 y Fw(k)2588 1384 y Fz(and)g Fu(j)p Fy(x)2836 1398 y Fw(i)p Fq(+1)2964 1384 y Fu(\000)9 b Fy(x)3096 1398 y Fw(i)3123 1384 y Fu(j)25 b Fz(in)f(these)h(expressions)-47 1513 y(are)44 b(rescaled)f(to)h Fy(\033)651 1527 y Fw(j)703 1513 y Fz(\()p Fy(t)29 b Fu(\000)g Fy(s)p Fz(\),)47 b Fy(\032)1097 1528 y Fw(k)1154 1513 y Fz(\()p Fy(t)29 b Fu(\000)g Fy(s)p Fz(\),)47 b(and)1701 1468 y Fx(j)p Fw(x)1761 1478 y Fs(i)p Fr(+1)1864 1468 y Fx(\000)p Fw(x)1959 1478 y Fs(i)1985 1468 y Fx(j)p 1701 1493 304 4 v 1767 1502 a(p)p 1825 1502 114 3 v 1825 1553 a Fw(t)p Fx(\000)p Fw(s)2014 1513 y Fz(,)g(resp)s(ectiv)m(ely)-8 b(.)80 b(W)-8 b(e)45 b(no)m(w)e(estimate)h(\(8.2\))i(and)d(the)-47 1642 y(analogous)31 b(in)m(tegrals)g(from)f(\(5.41\))r(,)i(and)e (\(5.42\))j(b)m(y)e(means)g(of)g(Lemma)g(6.1,)h(7.5,)h(7.6,)f(resp)s (ectiv)m(ely)-8 b(.)42 b(Using)31 b(the)-47 1755 y(second)h(b)s(ound)e (in)h(eac)m(h)j(of)e(these)h(lemmas,)f(whic)m(h)f(is)h(in)m(v)-5 b(arian)m(t)31 b(under)g(the)h(aforemen)m(tioned)h(rescaling)e(of)i (the)-47 1868 y(parameters,)e(one)f(arriv)m(es)g(at)h(the)g(upp)s(er)d (b)s(ound)h(\(setting)h Fy(x)c Fz(=)f Fy(x)2277 1882 y Fq(0)2346 1868 y Fz(and)30 b Fy(y)e Fz(=)d Fy(x)2744 1882 y Fw(m)p Fq(+1)2901 1868 y Fz(\))1472 2163 y Fy(m)1562 2098 y Fr(3)p 1562 2110 31 3 v 1562 2151 a(2)1722 2046 y Fz(max)1642 2105 y Fq(0)p Fx(\024)p Fw(j)t Fx(\024)p Fw(m)p Fq(+1)1987 2046 y Fu(j)p Fy(x)2064 2060 y Fw(j)t Fq(+1)2211 2046 y Fu(\000)20 b Fy(x)2354 2060 y Fw(j)2390 2046 y Fu(j)p 1642 2142 774 4 v 1818 2225 a(j)p Fy(x)1895 2240 y Fw(`)1948 2225 y Fu(\000)g Fy(x)2091 2240 y Fw(`)p Fx(\000)p Fq(1)2214 2225 y Fu(j)-47 2414 y Fz(in)29 b(case)i(of)g Fu(L)418 2381 y Fw(`)418 2436 y(m)484 2414 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\),)32 b(and)1472 2709 y Fy(m)1562 2644 y Fr(3)p 1562 2656 31 3 v 1562 2697 a(2)1722 2592 y Fz(max)1642 2651 y Fq(0)p Fx(\024)p Fw(j)t Fx(\024)p Fw(m)p Fq(+1)1987 2592 y Fu(j)p Fy(x)2064 2606 y Fw(j)t Fq(+1)2211 2592 y Fu(\000)20 b Fy(x)2354 2606 y Fw(j)2390 2592 y Fu(j)p 1642 2688 774 4 v 1724 2771 a(j)p Fy(x)1801 2789 y Fw(k)r Fq(\()p Fw(`)p Fq(\))1948 2771 y Fu(\000)g Fy(x)2091 2789 y Fw(k)r Fq(\()p Fw(`)p Fq(\))p Fx(\000)p Fq(1)2308 2771 y Fu(j)-47 2984 y Fz(in)f(case)i(of)f Fu(M)433 2940 y Fw(d;`)433 2996 y(m)523 2984 y Fz(\()p Fy(t;)15 b(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\))p Fy(;)1007 2961 y Fn(f)986 2984 y Fu(M)1095 2940 y Fw(d;`)1095 2996 y(m)1184 2984 y Fz(\()p Fy(t;)g(s)p Fz(\)\()p Fy(x;)g(y)s Fz(\).)38 b(Inserting)19 b(these)i(b)s(ounds)d(in)m(to)i(the)h (\(rescaled\))g(de\014nitions)d(\(5.39\))r(,)-47 3097 y(\(5.41\))r(,)31 b(and)e(\(5.42\))k(\014nally)28 b(leads)i(to)h(the)g (estimate)360 3288 y Fu(jhI)470 3302 y Fw(m)537 3288 y Fy( )596 3302 y Fw(s)633 3288 y Fy(;)15 b(g)s Fu(ij)206 3530 y(\024)83 b Fy(C)22 b(m)537 3465 y Fr(7)p 537 3477 31 3 v 537 3518 a(2)606 3530 y Fu(j)p Fy(t)f Fu(\000)f Fy(s)p Fu(j)844 3492 y Fx(\000)909 3465 y Fr(3)p 908 3477 V 908 3518 a(2)983 3406 y Fn(Z)1034 3612 y Fp(R)1082 3593 y Fs(m)1159 3406 y Fn(Z)1210 3612 y Fp(R)1258 3593 y Fs(m)1367 3416 y Fw(m)1340 3443 y Fn(Y)1336 3637 y Fw(r)r Fq(=1)1550 3468 y Fu(j)p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)1723 3482 y Fw(r)1761 3468 y Fy(;)15 b(x)1853 3482 y Fw(r)1892 3468 y Fz(\))p Fu(j)p 1485 3509 533 4 v 1485 3592 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)1662 3606 y Fw(r)r Fx(\000)p Fq(1)1811 3592 y Fu(\000)20 b Fy(x)1954 3606 y Fw(r)1992 3592 y Fu(j)2148 3413 y Fz(max)2067 3472 y Fq(0)p Fx(\024)p Fw(j)t Fx(\024)p Fw(m)p Fq(+1)2413 3413 y Fu(j)p Fy(x)2490 3427 y Fw(j)t Fq(+1)2637 3413 y Fu(\000)g Fy(x)2780 3427 y Fw(j)2816 3413 y Fu(j)p 2067 3509 774 4 v 2290 3592 a(j)p Fy(x)2367 3606 y Fw(m)2454 3592 y Fu(\000)g Fy(y)s Fu(j)2876 3530 y Fy(dx)2975 3544 y Fq(1)3030 3530 y Fy(:)15 b(:)g(:)i(dx)3251 3544 y Fw(m)3333 3530 y Fy(d\034)3420 3544 y Fq(1)3474 3530 y Fy(:)e(:)g(:)h(d\034)3682 3544 y Fw(m)3749 3530 y Fy(:)-47 3797 y Fz(In)29 b(view)h(of)h(Lemma)f (2.5)h(this)f(is)f(no)h(larger)h(than)980 4062 y Fy(C)21 b(m)1156 3998 y Fr(9)p 1156 4010 31 3 v 1156 4051 a(2)1226 4062 y Fu(j)p Fy(t)f Fu(\000)g Fy(s)p Fu(j)1463 4025 y Fx(\000)1528 3998 y Fr(3)p 1528 4010 V 1528 4051 a(2)1618 3907 y Fn( )1705 4062 y Fz(sup)1690 4146 y Fw(y)r Fx(2)p Fp(R)1822 4127 y Fr(3)1872 3939 y Fn(Z)1922 4145 y Fp(R)1970 4126 y Fr(3)2024 3939 y Fn(Z)2153 4001 y Fu(j)p Fy(V)g Fz(\()5 b(^)-50 b Fy(\034)11 b(;)k(x)p Fz(\))p Fu(j)p 2140 4042 362 4 v 2140 4125 a Fz(4)p Fy(\031)s Fu(j)p Fy(x)21 b Fu(\000)f Fy(y)s Fu(j)2527 4062 y Fy(d\034)26 b(dx)2739 3907 y Fn(!)2811 3930 y Fw(m)2893 4062 y Fy(;)-47 4326 y Fz(and)k(w)m(e)g(are)h(done.)p 3879 4326 4 62 v 3883 4268 55 4 v 3883 4326 V 3937 4326 4 62 v -47 4611 a FA(References)-47 4813 y Fz([Ag])46 b(Agmon,)25 b(S.)e Fl(Sp)-5 b(e)g(ctr)g(al)29 b(pr)-5 b(op)g(erties)28 b(of)e(Schr\177)-46 b(odinger)28 b(op)-5 b(er)g(ators)29 b(and)e(sc)-5 b(attering)27 b(the)-5 b(ory.)25 b Fz(Ann.)d(Scuola)h(Norm.)140 4926 y(Sup.)29 b(Pisa)g(Cl.)h(Sci.)g(\(4\))h(2)g(\(1975\),)i(no.)d(2,)h (151{218.)-47 5110 y([AS])45 b(Aizenman,)33 b(M.,)h(Simon,)d(B.)i Fl(Br)-5 b(ownian)36 b(motion)g(and)g(Harnack)f(ine)-5 b(quality)35 b(for)g(Schr\177)-46 b(odinger)36 b(op)-5 b(er)g(ators.)140 5223 y Fz(Comm.)30 b(Pure)f(Appl.)g(Math.)j(35)f (\(1982\),)i(no.)e(2,)g(209{273.)1903 5472 y(68)p eop %%Page: 69 69 69 68 bop -47 274 a Fz([A)-8 b(Y])47 b(Artbazar,)28 b(G.,)h(Y)-8 b(a)5 b(jima,)29 b(K.)e Fl(The)j Fy(L)1444 241 y Fw(p)1484 274 y Fl(-c)-5 b(ontinuity)30 b(of)g(wave)h(op)-5 b(er)g(ators)33 b(for)d(one)h(dimensional)g(Schr\177)-46 b(odinger)140 387 y(op)-5 b(er)g(ators.)33 b Fz(J.)e(Math.)g(Sci.)e(Univ.)h(T)-8 b(oky)m(o)32 b(7)e(\(2000\),)j(no.)e(2,)g(221{240.)-47 575 y([Bo1])47 b(Bourgain,)33 b(J.)g Fl(Gr)-5 b(owth)37 b(of)e(Sob)-5 b(olev)36 b(norms)g(in)f(line)-5 b(ar)36 b(Schr\177)-46 b(odinger)36 b(e)-5 b(quations)36 b(with)g(quasi-p)-5 b(erio)g(dic)36 b(p)-5 b(o-)140 687 y(tential.)31 b Fz(Comm.)f(Math.)h (Ph)m(ys.)f(204)i(\(1999\),)h(no.)e(1,)g(207{247.)-47 875 y([Bo2])47 b(Bourgain,)42 b(J.)d Fl(On)i(gr)-5 b(owth)44 b(of)d(Sob)-5 b(olev)42 b(norms)h(in)e(line)-5 b(ar)42 b(Schr\177)-46 b(odinger)43 b(e)-5 b(quations)42 b(with)g(smo)-5 b(oth)44 b(time)140 988 y(dep)-5 b(endent)34 b(p)-5 b(otential.)32 b Fz(J.)e(Anal.)g(Math.)i(77)f(\(1999\),)i(315{348.)-47 1176 y([Bo3])47 b(Bourgain,)36 b(J.)f Fl(F)-7 b(ourier)37 b(tr)-5 b(ansform)40 b(r)-5 b(estriction)38 b(phenomena)h(for)e(c)-5 b(ertain)38 b(lattic)-5 b(e)38 b(subsets)e(and)i(applic)-5 b(a-)140 1289 y(tions)37 b(to)g(nonline)-5 b(ar)39 b(evolution)e(e)-5 b(quations.)38 b(I.)f(Schr\177)-46 b(odinger)38 b(e)-5 b(quations.)36 b Fz(Geom.)g(F)-8 b(unct.)35 b(Anal.)g(3)g(\(1993\),)140 1401 y(no.)30 b(2,)h(107{156.)-47 1589 y([BGT])46 b(Burq,)26 b(N.,)i(Gerard,)e(P)-8 b(.,)28 b(Tzv)m(etk)m(o)m(v,)h(N.)d Fl(Strichartz)k(ine)-5 b(qualities)29 b(and)g(the)g(nonline)-5 b(ar)30 b(Schr)-5 b(o)g(dinger)30 b(e)-5 b(qua-)140 1702 y(tion)33 b(on)g(c)-5 b(omp)g(act)35 b(manifolds.)d Fz(preprin)m(t)-47 1890 y([CK])45 b(Christ,)24 b(M.,)i(Kiselev,)e(A.)g Fl(Maximal)k (functions)f(asso)-5 b(ciate)g(d)29 b(with)e(\014ltr)-5 b(ations,)28 b Fz(J.)c(F)-8 b(unct.)24 b(Anal.)g(179)h(\(2001\),)140 2002 y(409-425)-47 2190 y([D])68 b(Da)m(vies,)31 b(E.)f(B.)h Fl(Time-dep)-5 b(endent)34 b(sc)-5 b(attering)34 b(the)-5 b(ory.)32 b Fz(Math.)f(Ann.)f(210)h(\(1974\),)j(149-162)-47 2378 y([GM])46 b(Ginibre,)30 b(J.,)h(Moulin,)f(M.)h Fl(Hilb)-5 b(ert)34 b(sp)-5 b(ac)g(e)34 b(appr)-5 b(o)g(ach)37 b(to)d(the)g (quantum)f(me)-5 b(chanic)g(al)36 b(thr)-5 b(e)g(e-b)g(o)g(dy)35 b(pr)-5 b(oblem.)140 2491 y Fz(Ann.)29 b(Inst.)h(H.)h(P)m(oincar)m (\023)-43 b(e)32 b(Sect.)f(A)g(\(N.S.\))g(21)g(\(1974\),)i(97{145.)-47 2678 y([Ik])56 b(Ik)m(eb)s(e,)29 b(T.)f Fl(Eigenfunction)i(exp)-5 b(ansions)32 b(asso)-5 b(ciate)g(d)33 b(with)f(the)f(Schr)-5 b(o)g(e)g(dinger)33 b(op)-5 b(er)g(ators)34 b(and)d(their)g(applic)-5 b(a-)140 2791 y(tions)33 b(to)g(sc)-5 b(attering)34 b(the)-5 b(ory.)32 b Fz(Arc)m(h.)f(Rational)e(Mec)m(h.)j(Anal.)e(5)h(1960)h (1{34)g(\(1960\).)-47 2979 y([J1])46 b(Jensen,)e(A.)e Fl(Sp)-5 b(e)g(ctr)g(al)45 b(pr)-5 b(op)g(erties)45 b(of)e(Schr\177)-46 b(odinger)45 b(op)-5 b(er)g(ators)46 b(and)e(time-de)-5 b(c)g(ay)44 b(of)f(the)g(wave)g(functions)140 3092 y(r)-5 b(esults)33 b(in)g Fy(L)598 3059 y Fq(2)637 3092 y Fz(\()p Fy(R)742 3059 y Fw(m)809 3092 y Fz(\))p Fl(,)f Fy(m)26 b Fu(\025)e Fz(5)p Fl(.)31 b Fz(Duk)m(e)g(Math.)g(J.)g(47)g(\(1980\),)i (no.)e(1,)g(57{80.)-47 3279 y([J2])46 b(Jensen,)41 b(A.)e Fl(Sp)-5 b(e)g(ctr)g(al)43 b(pr)-5 b(op)g(erties)43 b(of)e(Schr\177)-46 b(odinger)42 b(op)-5 b(er)g(ators)44 b(and)e(time-de)-5 b(c)g(ay)42 b(of)f(the)g(wave)g(functions.)140 3392 y(R)-5 b(esults)33 b(in)g Fy(L)626 3359 y Fq(2)665 3392 y Fz(\()p Fy(R)770 3359 y Fq(4)810 3392 y Fz(\))p Fl(.)d Fz(J.)g(Math.)i(Anal.)d (Appl.)g(101)j(\(1984\),)h(no.)e(2,)g(397{422.)-47 3580 y([JK])45 b(Jensen,)40 b(A.,)g(Kato,)h(T.)d Fl(Sp)-5 b(e)g(ctr)g(al)42 b(pr)-5 b(op)g(erties)42 b(of)e(Schr\177)-46 b(odinger)41 b(op)-5 b(er)g(ators)42 b(and)f(time-de)-5 b(c)g(ay)41 b(of)e(the)h(wave)140 3693 y(functions.)30 b Fz(Duk)m(e)h(Math.)h(J.)e(46)h(\(1979\),)i(no.)e(3,)g(583{611.)-47 3880 y([JN])46 b(Jensen,)27 b(A.,)i(Nak)-5 b(am)m(ura,)29 b(S.)e Fy(L)1253 3847 y Fw(p)1323 3880 y Fl(and)k(Besov)f(estimates)h (for)g(Schr\177)-46 b(odinger)31 b(Op)-5 b(er)g(ators.)30 b Fz(Adv)-5 b(anced)27 b(Studies)140 3993 y(in)i(Pure)g(Math.)j(23,)f (Sp)s(ectral)e(and)h(Scattering)h(Theory)f(and)f(Applications)g (\(1994\),)k(187{209.)-47 4181 y([JSS])44 b(Journ)m(\023)-43 b(e,)30 b(J.-L.,)g(So\013er,)f(A.,)h(Sogge,)h(C.)d(D.)i Fl(De)-5 b(c)g(ay)32 b(estimates)g(for)g(Schr\177)-46 b(odinger)33 b(op)-5 b(er)g(ators.)32 b Fz(Comm.)c(Pure)140 4294 y(Appl.)h(Math.)i(44)g(\(1991\),)i(no.)e(5,)g(573{604.)-47 4482 y([H1])46 b(Ho)m(wland,)39 b(J.)e(S.)g Fl(Born)i(series)g(and)h (sc)-5 b(attering)40 b(by)e(time-dep)-5 b(endent)40 b(p)-5 b(otentaials.)40 b Fz(Ro)s(c)m(ky)e(Moun)m(t.)g(J.)f(of)140 4594 y(Math.)31 b(10)g(\(1980\),)i(no.3,)e(521-531)-47 4782 y([H2])46 b(Ho)m(wland,)35 b(J.)f(S.)f Fl(Stationary)39 b(sc)-5 b(attering)36 b(the)-5 b(ory)38 b(for)f(time-dep)-5 b(endent)37 b(Hamiltonians.)e Fz(Math.)g(Ann.)f(207)140 4895 y(\(1974\),)f(315{335.)-47 5083 y([Ka])46 b(Kato,)e(T.)d Fl(Wave)h(op)-5 b(er)g(ators)46 b(and)d(similarity)g(for)g(some)g (non-selfadjoint)h(op)-5 b(er)g(ators.)44 b Fz(Math.)d(Ann.)g(162)140 5196 y(\(1965/1966\),)35 b(258{279.)1903 5472 y(69)p eop %%Page: 70 70 70 69 bop -47 274 a Fz([KT])45 b(Keel,)30 b(M.,)i(T)-8 b(ao,)31 b(T.)f Fl(Endp)-5 b(oint)35 b(Strichartz)f(estimates.)d Fz(Amer.)g(J.)f(Math.)h(120)h(\(1998\),)h(no.)e(5,)g(955{980.)-47 462 y([Ku1])45 b(Kuro)s(da,)40 b(S.)e(T.)g Fl(Sc)-5 b(attering)41 b(the)-5 b(ory)41 b(for)g(di\013er)-5 b(ential)41 b(op)-5 b(er)g(ators.)43 b(I.)d(Op)-5 b(er)g(ator)42 b(the)-5 b(ory.)40 b Fz(J.)f(Math.)g(So)s(c.)140 575 y(Japan)30 b(25)h(\(1973\),)i(75{104.)-47 762 y([Ku2])45 b(Kuro)s(da,)30 b(S.)h(T.)g Fl(Sc)-5 b(attering)34 b(the)-5 b(ory)35 b(for)f(di\013er)-5 b(ential)34 b(op)-5 b(er)g(ators.)36 b(II.)d(Self-adjoint)h(el)5 b(liptic)34 b(op)-5 b(er)g(ators.)34 b Fz(J.)140 875 y(Math.)d(So)s(c.)f(Japan)g(25)h(\(1973\),)j(222{234.) -47 1063 y([R])70 b(Rauc)m(h,)33 b(J.)e Fl(L)-5 b(o)g(c)g(al)37 b(de)-5 b(c)g(ay)35 b(of)f(sc)-5 b(attering)36 b(solutions)f(to)g (Schr\177)-46 b(odinger's)36 b(e)-5 b(quation.)32 b Fz(Comm.)g(Math.)h (Ph)m(ys.)f(61)140 1176 y(\(1978\),)h(no.)d(2,)h(149{168.)-47 1363 y([RS])45 b(Reed,)32 b(M.,)g(Simon,)d(B.)j Fl(Metho)-5 b(ds)34 b(of)g(mo)-5 b(dern)35 b(mathematic)-5 b(al)36 b(physics.)e(IV.)e(A)n(nalysis)i(of)f(op)-5 b(er)g(ators.)35 b Fz(Aca-)140 1476 y(demic)29 b(Press)h([Harcourt)h(Brace)h(Jo)m(v)-5 b(ano)m(vic)m(h,)31 b(Publishers],)d(New)i(Y)-8 b(ork-London,)31 b(1978.)-47 1664 y([Si1])45 b(Simon,)36 b(B.)h Fl(Quantum)h(me)-5 b(chanics)39 b(for)f(Hamiltonians)i(de\014ne)-5 b(d)39 b(as)f(quadr)-5 b(atic)39 b(forms.)f Fz(Princeton)d(Series)140 1777 y(in)29 b(Ph)m(ysics.)g(Princeton)h(Univ)m(ersit)m(y)g(Press,)g (Princeton,)g(N.)h(J.,)f(1971.)-47 1964 y([Si2])45 b(Simon,)29 b(B.)i Fl(Schr\177)-46 b(odinger)34 b(semigr)-5 b(oups.)32 b Fz(Bull.)d(AMS.)h(v)m(ol.)h(7,)g(no.)g(3,)g(\(1982\),)i(447{526.)-47 2152 y([SoSm])45 b(Smith,)63 b(H.,)i(Sogge,)h(C.)57 b Fl(Glob)-5 b(al)60 b(Strichartz)f(estimates)g(for)f(nontr)-5 b(apping)60 b(p)-5 b(erturb)g(ations)61 b(of)c(the)140 2265 y(L)-5 b(aplac)g(e)g(an.)33 b Fz(Comm.)d(PDE,)h(to)g(app)s(ear.) -47 2452 y([ST])45 b(Sta\016lani,)28 b(G.,)h(T)-8 b(ataru,)30 b(D.)g Fl(Strichartz)i(estimates)h(for)e(a)h(Schr)-5 b(o)g(dinger)33 b(op)-5 b(er)g(ator)34 b(with)e(nonsmo)-5 b(oth)34 b(c)-5 b(o)g(e\016-)140 2565 y(cients.)30 b Fz(preprin)m(t)e(2000.)-47 2753 y([St])51 b(Stein,)29 b(E.)i Fl(Bejing)h(le)-5 b(ctur)g(es)33 b(in)g(harmonic)h(analysis.)e Fz(Princeton)d(Univ)m(ersit)m(y)h(Press,)g(1986.)-47 2941 y([Str])45 b(Stric)m(hartz,)29 b(R.)g Fl(R)-5 b(estrictions)32 b(of)f(F)-7 b(ourier)32 b(tr)-5 b(ansforms)33 b(to)f(quadr)-5 b(atic)32 b(surfac)-5 b(es)31 b(and)h(de)-5 b(c)g(ay)32 b(of)f(solutions)i(of)140 3054 y(wave)g(e)-5 b(quations.)31 b Fz(Duk)m(e)g(Math.)g(J.)f(44)i(\(1977\),)h(no.)d(3,)h(705{714.)-47 3241 y([Y1])46 b(Y)-8 b(a)5 b(jima,)30 b(K.)f Fl(Existenc)-5 b(e)31 b(of)h(solutions)h(for)f(Schr\177)-46 b(odinger)33 b(evolution)f(e)-5 b(quations,)30 b Fz(Comm.)f(Math.)h(Ph)m(ys.)f(110) 140 3354 y(\(1987\),)k(no.)d(3,)h(415{426.)-47 3542 y([Y2])46 b(Y)-8 b(a)5 b(jima,)46 b(K.)d Fl(The)h Fy(W)953 3509 y Fw(k)r(;p)1050 3542 y Fl(-c)-5 b(ontinuity)44 b(of)g(wave)g(op)-5 b(er)g(ators)48 b(for)c(Schr\177)-46 b(odinger)45 b(op)-5 b(er)g(ators.)46 b Fz(J.)c(Math.)i(So)s(c.)140 3655 y(Japan)30 b(47)h(\(1995\),)i(no.)d(3,)h(551{581.)-47 3842 y([Y3])46 b(Y)-8 b(a)5 b(jima,)23 b(K.)e Fy(L)674 3809 y Fw(p)713 3842 y Fl(-b)-5 b(ounde)g(dness)25 b(of)f(wave)g(op)-5 b(er)g(ators)27 b(for)d(two-dimensional)i(Schr\177)-46 b(odinger)25 b(op)-5 b(er)g(ators.)24 b Fz(Comm.)140 3955 y(Math.)31 b(Ph)m(ys.)f(208)i(\(1999\),)h(no.)d(1,)h(125{152.)-47 4168 y Fa(R)m(odnianski:)41 b(Dep)-6 b(ar)g(tment)25 b(of)i(Ma)-6 b(thema)g(tics,)27 b(Princeton)g(University,)h(Fine)g (Hall,)g(Princeton)-47 4281 y(N.J.)34 b(08544,)h(U.S.A.)-47 4394 y Fl(email:)47 b Fm(iro)s(d@math.princeton.edu)-47 4556 y Fa(Schla)n(g:)42 b(Division)32 b(of)f(Astr)n(onomy,)f(Ma)-6 b(thema)g(tics,)30 b(and)h(Physics,)g(253-37)i(Cal)-6 b(tech,)31 b(P)-8 b(asadena,)-47 4669 y(CA)35 b(91125,)g(U.S.A.)-47 4782 y Fl(curr)-5 b(ent)33 b(addr)-5 b(ess:)84 b Fa(Institute)33 b(f)n(or)h(Ad)n(v)-8 b(anced)32 b(Stud)n(y,)h(Olden)g(Lane,)h (Princeton,)f(NJ)h(08540)-47 4895 y Fl(email:)47 b Fm(sc)m (hlag@ias.edu)1903 5472 y Fz(70)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0110121019673--