Content-Type: multipart/mixed; boundary="-------------9910191739585" This is a multi-part message in MIME format. ---------------9910191739585 Content-Type: text/plain; name="99-398.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-398.keywords" intermittency, fractal, Schr\"odinger, quantum dynamics, Hausdorff, packing, renyi ---------------9910191739585 Content-Type: application/postscript; name="interm.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="interm.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: interm.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips interm.dvi -o %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.10.19:1459 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 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(y)g(separable)g(Hilb)s(ert)e(space)i Fn(H)q Fp(.)67 b(If)38 b(w)m(e)i(fo)s(cus)e(on)h(the)-24 4594 y(particular)29 b(case)i Fn(H)26 b Fp(=)f Fo(`)825 4561 y Fh(2)865 4594 y Fp(\()p Fg(Z)965 4561 y Fm(d)1001 4594 y Fp(\),)31 b(our)f(main)f(result)g(reads)i(as)f(follo)m(ws)1145 4848 y Fo(\013)1203 4810 y Fl(\000)1263 4848 y Fp(\()p Fo( )s(;)15 b(p;)g(d)p Fp(\))27 b Fn(\025)e Fo(D)1769 4810 y Fl(\000)1766 4870 y Fm(\026)1808 4882 y Fi( )1875 4720 y Fj(\022)2077 4787 y Fp(1)p 1952 4827 296 4 v 1952 4910 a(1)20 b(+)g Fo(p=d)2257 4720 y Fj(\023)2354 4848 y Fo(p=d)-24 5097 y Fp(and)1133 5237 y Fo(\013)1191 5200 y Fh(+)1250 5237 y Fp(\()p Fo( )s(;)15 b(p;)g(d)p Fp(\))27 b Fn(\025)e Fo(D)1756 5200 y Fh(+)1753 5260 y Fm(\026)1795 5272 y Fi( )1862 5109 y Fj(\022)2064 5176 y Fp(1)p 1939 5216 V 1939 5300 a(1)c(+)f Fo(p=d)2244 5109 y Fj(\023)2341 5237 y Fo(p=d:)1796 5611 y Fp(2)p eop %%Page: 3 3 3 2 bop -24 375 a Fp(where)31 b Fo(D)318 342 y Fl(\000)315 397 y Fm(\026)357 409 y Fi( )408 375 y Fp(\()p Fo(q)s Fp(\))h(and)e Fo(D)809 342 y Fh(+)806 397 y Fm(\026)848 409 y Fi( )900 375 y Fp(\()p Fo(q)s Fp(\))h(are)h(the)f(lo)m(w)m(er)g (and)f(upp)s(er)f(\\)p Fo(q)s Fp(-th)j(generalized)e(R)m(\023)-43 b(en)m(yi)32 b(dimensions")c(\(see)-24 488 y(De\014nition)33 b(2.2\),)38 b(and)c Fo(X)41 b Fp(the)35 b(usual)e("p)s(osition\\)h(op)s (erator.)53 b(T)-8 b(o)35 b(ac)m(hiev)m(e)h(this)d(w)m(e)i(\014rst)f (deriv)m(e)g(a)h Fo(p=d)p Fp(-)-24 601 y(dep)s(enden)m(t)h(lo)m(w)m(er) h(b)s(ound)e Fo(L)1015 616 y Fm( )1067 601 y Fp(\()p Fo(T)13 b Fp(\))37 b(for)g Fn(hhj)p Fo(X)7 b Fn(jii)1658 616 y Fm( )r(;T)1820 601 y Fp(\(Theorem)37 b(3.1\),)i(and)e(then)f (establish)f Ff(via)75 b Fp(Theo-)-24 714 y(rem)30 b(4.2)i(the)e (connection)h(b)s(et)m(w)m(een)g Fo(L)1329 729 y Fm( )1382 714 y Fp(\()p Fo(T)13 b Fp(\))30 b(and)g(the)h(generalized)f(R)m(\023) -43 b(en)m(yi)30 b(dimensions.)118 827 y(Moreo)m(v)m(er,)d(w)m(e)d (obtain)f(a)h(kind)e(of)i(optimalit)m(y)-8 b(,)24 b(in)f(the)h(sense)f (that)h(the)g(quan)m(tit)m(y)g Fo(L)2990 842 y Fm( )3043 827 y Fp(\()p Fo(T)13 b Fp(\))24 b(that)g(minors)-24 939 y Fn(hhj)p Fo(X)7 b Fn(j)178 906 y Fm(p)220 939 y Fn(ii)290 954 y Fm( )r(;T)444 939 y Fp(has)30 b(its)g(gro)m(wth)h(exp)s (onen)m(ts)f(exactly)h(equal)f(to)h Fo(D)2208 906 y Fl(\006)2205 962 y Fm(\026)2247 974 y Fi( )2299 939 y Fp(\(1)p Fo(=)p Fp(\(1)22 b(+)e Fo(p=d)p Fp(\)\))15 b Fo(p=d)p Fp(.)118 1065 y(W)-8 b(e)34 b(p)s(oin)m(t)e(out)i(that)g Fo(D)960 1032 y Fl(\006)957 1087 y Fm(\026)999 1099 y Fi( )1051 1065 y Fp(\(1)p Fo(=)p Fp(\(1)24 b(+)e Fo(p=d)p Fp(\)\))34 b(are)g(increasing)e(function)g(of)i Fo(p=d)p Fp(,)g(and)f(are)h(resp)s (ectiv)m(ely)-24 1178 y(larger)d(than)f Fo(d)494 1192 y Fm(H)561 1178 y Fp(\()p Fo(\026)651 1193 y Fm( )704 1178 y Fp(\))h(and)f Fo(d)994 1192 y Fm(P)1053 1178 y Fp(\()p Fo(\026)1143 1193 y Fm( )1196 1178 y Fp(\))g(\(for)h(all)e Fo(p=d)d(>)f Fp(0\).)42 b(Therefore,)31 b(our)f(result)f(impro)m(v)m (es)h(notably)g(the)-24 1291 y(ab)s(o)m(v)m(e)i(\(2\))f(and)f(\(3\))i (b)s(ounds.)118 1403 y(These)f(new)g(b)s(ounds)e(are)j(a)g(consequence) h(of)e(a)h(double)e(impro)m(v)m(emen)m(t)i(of)g(Guarneri-Com)m(b)s (es-Last)-24 1516 y(\(G-C-L\))c([7,)f(22])g(and)e(Barbaroux-Tc)m (heremc)m(han)m(tsev)j(\(BT\))f([5)q(])f(approac)m(hes.)40 b(The)26 b(\014rst)f(impro)m(v)m(emen)m(t)-24 1629 y(is)30 b(due,)g(after)h(a)f(decomp)s(ostion)g Fo( )f Fp(=)c Fo(')c Fp(+)e Fo(\037)p Fp(,)31 b Fo(')26 b Fn(?)f Fo(\037)p Fp(,)30 b(to)h(a)g(b)s(etter)f(con)m(trol)h(of)g(the)f(k)m(ey)h(quan)m (tit)m(y)373 1951 y Fo(B)442 1966 y Fm( )494 1951 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))54 b Fn(\021)945 1889 y Fp(1)p 935 1930 66 4 v 935 2013 a Fo(T)1026 1827 y Fj(Z)1117 1854 y Fl(1)1077 2033 y Fh(0)1222 1865 y Fj(X)1207 2062 y Fm(n)p Fl(\024)p Fm(N)1383 1951 y Fn(jh)p Fp(e)1483 1913 y Fl(\000)p Fm(itH)1655 1951 y Fo( )s(;)15 b(e)1799 1965 y Fm(n)1848 1951 y Fn(ij)1908 1913 y Fh(2)1948 1951 y Fo(h)p Fp(\()p Fo(t=T)e Fp(\)d)p Fo(t)802 2241 y Fn(\024)52 b Fo(B)994 2255 y Fm(\037)1042 2241 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))21 b(+)1428 2180 y(2)p 1418 2220 V 1418 2303 a Fo(T)1509 2241 y Fp(Re)1616 2117 y Fj(Z)1707 2144 y Fl(1)1667 2323 y Fh(0)1797 2155 y Fj(X)1782 2352 y Fm(n)p Fl(\024)p Fm(N)1943 2241 y Fn(h)p Fp(e)2018 2204 y Fl(\000)p Fm(itH)2190 2241 y Fo(';)15 b(e)2331 2255 y Fm(n)2379 2241 y Fn(i)p 2414 2159 475 4 v(h)p Fp(e)2489 2215 y Fl(\000)p Fm(itH)2662 2241 y Fo( )s(;)g(e)2806 2255 y Fm(n)2854 2241 y Fn(i)p Fo(h)p Fp(\()p Fo(t=T)e Fp(\)d)p Fo(t;)281 b Fp(\(4\))-24 2523 y(and)28 b(more)h(particulary)e (of)h(the)h(last)f(term)h(\(the)g(crossed)f(term\).)41 b(Here)29 b Fo(h)p Fp(\()p Fo(z)t Fp(\))h(is)d(some)i(p)s(ositiv)m(e)f (function)-24 2636 y(in)33 b Fo(C)158 2603 y Fl(1)151 2660 y Fh(0)232 2636 y Fp(\([0)p Fo(;)15 b Fp(1]\))36 b(suc)m(h)d(that)926 2562 y Fj(R)987 2589 y Fh(1)969 2668 y(0)1041 2636 y Fo(h)p Fp(\()p Fo(z)t Fp(\)d)p Fo(z)j Fp(=)30 b(1.)50 b(Note)35 b(that)f(the)g(consideration)e(of)i(this)e (function)g Fo(h)i Fp(seems)-24 2749 y(to)29 b(b)s(e)d(a)i(necessary)g (tric)m(k)g(if)e(one)i(w)m(an)m(ts)g(in)e(whole)h(generalit)m(y)g(to)h (get)h(rid)d(of)h(the)h(negativ)m(e)h(v)-5 b(alues)27 b(of)g(the)-24 2862 y(time)h Fo(t)p Fp(.)39 b(T)-8 b(aking)27 b Fo(h)h Fp(as)g(the)f(usual)f(gaussian)h Fo(e)1559 2829 y Fl(\000)p Fm(z)1650 2805 y Fe(2)1684 2829 y Fm(=)p Fh(4)1786 2862 y Fp(will)e(only)i(yield)e(informations)h(ab)s(out)h (the)h(momen)m(ts)-24 2974 y Fn(j)p Fo(X)7 b Fn(j)108 2930 y Fm(p)108 3004 y( )162 2974 y Fp(\()p Fo(t)p Fp(\))37 b(a)m(v)m(eraged)j(o)m(v)m(er)e([)p Fn(\000)p Fo(T)8 b(;)15 b(T)e Fp(],)39 b(and)d(not)i(o)m(v)m(er)g([0)p Fo(;)15 b(T)e Fp(].)62 b(This)35 b(should)g(b)s(e)h(compared)h(to)h (the)f(recen)m(t)-24 3101 y([15)r(])30 b(where)g(related)g(results)f (ha)m(v)m(e)j(b)s(een)e(sim)m(ultaneously)e(obtained)i(on)g Fn(H)c Fp(=)f Fo(`)2782 3068 y Fh(2)2821 3101 y Fp(\()p Fg(N)7 b Fp(\).)118 3214 y(Afterw)m(ards,)29 b(and)f(this)g(is)g (Theorem)h(3.1,)i(one)e(is)f(able)h(to)h(obtain)e(a)i(constan)m(t)g Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\))27 b Fo(>)e Fp(0,)30 b(suc)m(h)-24 3327 y(that)h(for)f(all)g Fo(T)38 b(>)25 b Fp(0,)306 3586 y Fn(hhj)p Fo(X)7 b Fn(j)508 3549 y Fm(p)549 3586 y Fn(ii)619 3601 y Fm( )r(;T)769 3586 y Fn(\025)25 b Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\))p Fo(L)1309 3601 y Fm( )1363 3586 y Fp(\()p Fo(T)e Fp(\))p Fo(;)106 b Fp(with)90 b Fo(L)1960 3601 y Fm( )2013 3586 y Fp(\()p Fo(T)13 b Fp(\))25 b(=)57 b(sup)2270 3664 y Fm(')p Fl(2H)2423 3676 y Fi( )2485 3431 y Fj(\()2660 3525 y Fn(jh)p Fo(';)15 b( )s Fn(ij)2941 3492 y Fh(2+2)p Fm(\014)p 2568 3565 640 4 v 2568 3664 a Fn(k)p Fo(')p Fn(k)2717 3638 y Fh(2)2773 3664 y Fp(\()q Fo(U)2871 3679 y Fm('; )2989 3664 y Fp(\()p Fo(T)e Fp(\)\))3161 3622 y Fm(\014)3218 3431 y Fj(\))3306 3586 y Fo(;)-24 3846 y Fp(where)30 b Fn(H)316 3861 y Fm( )399 3846 y Fp(is)f(the)i(cyclic)f (subspace)g(spanned)f(b)m(y)h Fo( )k Fp(and)c Fo(H)7 b Fp(,)990 4080 y Fo(U)1052 4095 y Fm('; )1170 4080 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1428 3956 y Fj(Z)1478 4162 y Fk(R)1526 4143 y Fe(2)1580 4080 y Fp(d)p Fo(\026)1686 4095 y Fm( )1738 4080 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)1966 4094 y Fm(')2016 4080 y Fp(\()p Fo(y)s Fp(\))p Fo(R)q Fp(\()p Fo(T)13 b Fp(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\)\))p Fo(;)-24 4313 y Fp(and)29 b Fo(R)h Fp(is)e(some)i(b)s(ounded) d(fast)j(deca)m(ying)f(function)f(de\014ned)g(b)m(y)h(\(11\))i(in)d (Section)h(3)h(\(one)g(should)d(think)-24 4426 y(to)41 b Fo(R)g Fp(as)f(to)g(the)g(gaussian)g Fo(e)1031 4393 y Fl(\000)p Fm(w)1139 4370 y Fe(2)1173 4393 y Fm(=)p Fh(4)1248 4426 y Fp(\).)69 b(Afterw)m(ards,)43 b(as)d(in)e([5)q(,)i(14) q(],)j(one)d(can)g(\\c)m(ho)s(ose",)45 b(for)39 b(eac)m(h)i Fo(T)13 b Fp(,)-24 4539 y(a)44 b Fo(T)13 b Fp(-dep)s(enden)m(t)42 b(v)m(ector)j Fo(')f Fp(in)e(the)h(decomp)s(osition)f Fo( )50 b Fp(=)d Fo(')29 b Fp(+)f Fo(\037)p Fp(,)47 b(that)d(con)m (tains)f(enough)g(sp)s(ectral)-24 4652 y(informations)25 b(to)i(appro)m(ximate)g(the)g(suprem)m(um)e(in)g Fo(L)1886 4667 y Fm( )1938 4652 y Fp(\()p Fo(T)13 b Fp(\).)40 b(The)26 b(second)h(impro)m(v)m(emen)m(t)g(then)f(consists)-24 4765 y(in)38 b(the)h(w)m(a)m(y)h(one)f(c)m(ho)s(oses)h(this)e (particular)f(v)m(ector)j Fo(')p Fp(.)67 b(W)-8 b(e)40 b(sho)m(w)f(\(Theorem)g(4.1\))h(that)g(a)f(judicious)-24 4878 y(c)m(hoice)32 b(enables)d(one)i(to)g(connect,)h(up)d(to)i(a)g (logarithmic)e(factor,)i(the)g(quan)m(tit)m(y)g Fo(L)2868 4893 y Fm( )2920 4878 y Fp(\()p Fo(T)13 b Fp(\))31 b(to)g(the)g(in)m (tegral)537 5120 y Fo(I)577 5134 y Fm(\026)619 5146 y Fi( )686 4992 y Fj(\022)888 5059 y Fp(1)p 763 5099 296 4 v 763 5182 a(1)20 b(+)g Fo(p=d)1068 5120 y(;)15 b(T)1174 5083 y Fl(\000)p Fh(1)1268 4992 y Fj(\023)1361 5120 y Fp(=)1456 4996 y Fj(Z)1507 5203 y Fk(R)1574 5120 y Fp(d)p Fo(\026)1680 5135 y Fm( )1732 5120 y Fp(\()p Fo(x)p Fp(\))26 b Fo(\026)1935 5135 y Fm( )2003 5046 y Fj(\000)o(\002)2082 5120 y Fo(x)20 b Fn(\000)g Fo(T)2311 5083 y Fl(\000)p Fh(1)2405 5120 y Fo(;)15 b(x)21 b Fp(+)f Fo(T)2675 5083 y Fl(\000)p Fh(1)2769 5046 y Fj(\003\001)2848 5066 y Fl(\000)2952 5032 y Fi(p=d)p 2913 5051 173 3 v 2913 5092 a Fe(1+)p Fi(p=d)-24 5362 y Fp(that)31 b(de\014nes)f(the)g(generalized) g(fractal)h(exp)s(onen)m(ts)f Fo(D)1889 5329 y Fl(\006)1886 5385 y Fm(\026)1928 5397 y Fi( )1980 5362 y Fp(\(1)p Fo(=)p Fp(\(1)22 b(+)e Fo(p=d)p Fp(\)\).)1796 5611 y(3)p eop %%Page: 4 4 4 3 bop 118 375 a Fp(Our)25 b(metho)s(d)h(also)g(applies)f(to)i(the)g (previous)e(approac)m(hes)i(\(G-C-L\))h(and)e(\(BT\),)h(where)g(the)f (crossed)-24 488 y(term)36 b(in)d(\(4\))k(w)m(asn't)e(w)m(ell)f (treated.)56 b(It)35 b(resp)s(ectiv)m(ely)g(yields,)g(as)g(stated)h(in) e(App)s(endix)e(B,)k(the)f(fractal)-24 601 y(dimensions)c Fo(D)523 568 y Fl(\006)520 623 y Fm(\026)562 635 y Fi( )613 601 y Fp(\(\(1)23 b(+)f Fo(\014)5 b Fp(\))p Fo(=)p Fp(\(1)24 b(+)e(2)p Fo(\014)5 b Fp(\)\))34 b(and)e Fo(D)1639 568 y Fl(\006)1636 623 y Fm(\026)1678 635 y Fi( )1730 601 y Fp(\(\(1)23 b(+)f(2)p Fo(\014)5 b Fp(\))p Fo(=)p Fp(\(1)24 b(+)e(3)p Fo(\014)5 b Fp(\)\).)49 b(Since)32 b(the)i(functions)d Fo(D)3456 568 y Fl(\006)3453 623 y Fm(\026)3495 635 y Fi( )3547 601 y Fp(\()p Fo(q)s Fp(\))-24 714 y(are)h(nonincreasing)d (functions)g(of)i Fo(q)s Fp(,)g(our)g(Theorem)f(2.1)i(giv)m(es)f(a)h(b) s(etter)f(lo)m(w)m(er)g(b)s(ound.)40 b(It)31 b(means)g(that)-24 827 y(a)g(b)s(etter)g(estimating)f(of)g(the)h(crossed)f(term)g(in)g (\(4\))h(do)s(es)f(pro)m(vide)g(an)g(impro)m(v)m(emen)m(t.)118 985 y(Ev)m(en)m(tually)-8 b(,)29 b(with)e(extra)j(assumptions)d(on)h (the)h(deca)m(y)h(of)f(the)g(generalized)f(eigenfunctions)g Fo(u)p Fp(\()p Fo(n;)15 b(x)p Fp(\))-24 1097 y(of)39 b Fo(H)45 b Fp(\(in)37 b(the)h(spirit)e(of)i([18)q(])h([20)q(]\),)i(it) c(is)h(p)s(ossible)d(to)k(impro)m(v)m(e)f(the)g(ab)s(o)m(v)m(e)i(b)s (ounds.)61 b(In)38 b(particular,)-24 1210 y(supp)s(ose)28 b(that)h(there)g(exists)g(a)g(constan)m(t)h Fo(C)35 b Fp(suc)m(h)29 b(that)g(for)g Fo(\026)2124 1225 y Fm( )2205 1210 y Fp(a.e.)p Fo(x)p Fp(,)2448 1142 y Fj(P)2544 1237 y Fm(n)f Fp(0)p Ff(,)33 b Fo(I)2447 2242 y Fm(\026)2526 2228 y Ff(takes)g(the)g(value)f Fp(+)p Fn(1)p Ff(.)-24 2386 y Fd(Remark)j(2.2.)-24 2499 y Fp(i\))c(F)-8 b(or)31 b(our)f(purp)s(ose,)f(it)h(is)g(su\016cien)m(t)g(to)i(discuss)c(the)j (case)h Fo(q)c Fn(2)e Fp(\()p Fn(\0001)p Fo(;)15 b Fp(1\))31 b(\(see)h(e.g.)43 b([8])31 b(for)g(the)f(general)-24 2612 y(case\).)43 b(W)-8 b(e)31 b(shall)e(ev)m(en)i(only)e(need)h(in)f (this)h(pap)s(er)f Fo(q)f Fn(2)d Fp(\(0)p Fo(;)15 b Fp(1\).)-24 2725 y(ii\))38 b(There)h(exists)g(actually)g(a)h(wide)e(n)m(um)m(b)s (er)g(of)h(\\generalized)h(fractal)f(dimensions".)65 b(F)-8 b(or)40 b(example,)-24 2838 y(they)34 b(can)f(b)s(e)g(de\014ned) e(with)h(the)i(help)d(of)j(the)f(so-called)g(\\singularit)m(y)e(sp)s (ectrum)h(function")h Fo(f)3350 2852 y Fm(\026)3429 2838 y Fp(of)g(the)-24 2951 y(measure)e Fo(\026)f Fp(\([16)q(],)h([24)q (]\),)g(or)g(as)f(a)h(solution)e(of)i(an)f(implicit)d(equation)j(\([16) r(,)g(F)-8 b(orm)m(ula)31 b(2.8],)h([24)q(]\).)118 3064 y(The)g(resulting)g(dimensions)e(coincide)i(with)g(eac)m(h)j(other)e (in)f(certain)h(v)m(ery)h(sp)s(eci\014c)e(cases,)j(lik)m(e)d(e.g.)-24 3177 y(Co)s(okie)e(Cutter)h(measures)f(in)f Fg(R)39 b Fp([24)q(].)118 3335 y(In)25 b(order)h(to)i(state)g(our)e(results)f(w)m (e)i(also)g(de\014ne)f(the)g(follo)m(wing)g(in)m(tegrals)g(that)h (could)f(b)s(e)g(considered)-24 3448 y(as)k(appro)m(ximations)d(of)i (the)g(quan)m(tities)g Fo(I)1443 3462 y Fm(\026)1489 3448 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\).)42 b(The)28 b(function)g Fo(R)h Fp(is)f(a)i(b)s(ounded)c(ev)m(en)k(function)e(with)-24 3561 y(fast)j(deca)m(y)h(prop)s(erties)c(at)k Fn(\0061)d Fp(de\014ned)h(b)m(y)g(\(11\).)770 3797 y Fo(K)847 3811 y Fm(\026)894 3797 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))27 b(=)1213 3673 y Fj(Z)1263 3879 y Fh(supp)q Fm(\026)1470 3797 y Fp(d)p Fo(\026)p Fp(\()p Fo(x)p Fp(\))1713 3668 y Fj(\022)1780 3673 y(Z)1831 3879 y Fk(R)1898 3797 y Fp(d)p Fo(\026)p Fp(\()p Fo(y)s Fp(\))p Fo(R)q Fp(\(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\))p Fo(=")p Fp(\))2631 3668 y Fj(\023)2699 3691 y Fm(q)r Fl(\000)p Fh(1)2842 3797 y Fo(;)-24 4025 y Fp(In)28 b(Lemma)h(4.3)g(w)m(e)g(shall)e(pro)m (v)m(e)i(that)g(for)f(an)m(y)h(measure)g Fo(\026)f Fp(with)f(a)i(b)s (ounded)d(supp)s(ort)h(taking)h Fo(K)3417 4039 y Fm(\026)3464 4025 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))-24 4138 y(in)27 b(\(6\))i(instead)e(of)h Fo(I)676 4152 y Fm(\026)723 4138 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))29 b(leads)e(to)i(the)f(same)g (v)-5 b(alues)27 b(of)h(dimensions.)37 b(Less)28 b(restrictiv)m(e)g (assumptions,)-24 4250 y(in)h(order)h(to)h(get)h(the)f(same)f(prop)s (ert)m(y)-8 b(,)31 b(are)f(studied)f(in)g(App)s(endix)f(C.)118 4401 y(W)-8 b(e)29 b(review)f(b)s(elo)m(w)g(some)g(prop)s(erties)f(of)i (the)g(fractal)f(dimension)e(n)m(um)m(b)s(ers)h Fo(D)2874 4368 y Fl(\006)2871 4424 y Fm(\026)2933 4401 y Fp(\()p Fo(q)s 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Fo(q)s Fp(\))h(=)f Fo(d)769 5376 y Fm(P)828 5362 y Fp(\()p Fo(\026)p Fp(\))g(=)g Fo(d)1121 5376 y Fm(H)1189 5362 y Fp(\()p Fo(\026)p Fp(\))h(=)f Fo(\013)p Ff(.)1796 5611 y Fp(5)p eop %%Page: 6 6 6 5 bop -24 375 a Fd(Pro)s(of)36 b(of)f(Prop)s(osition)h(2.1.)-24 488 y Fp(Statemen)m(t)f(i\))e(is)g(already)g(kno)m(wn)g(\(see)h Ff(e.g.)49 b Fp([8)q(]\).)h(This)32 b(a)i(straigh)m(tforw)m(ard)f (consequence)h(of)g(conca)m(v)m(e)-24 601 y(and)42 b(con)m(v)m(ex)i (Jensen)d(inequalities.)74 b(F)-8 b(or)43 b(the)f(pro)s(of)g(of)g (ii\),)j(iii\))40 b(see)j(App)s(endix)c(A.)k(The)e(statemen)m(t)-24 714 y(iv\))c(is)e(pro)m(v)m(ed)i(in)f(Lemma)h(4.1.)60 b(Note)38 b(that)g(iv\))e(do)s(esn't)g(hold)g(if)f(one)i(lets)g Fo(q)i Fp(v)-5 b(ary)37 b(in)e(\()p Fn(\0001)p Fo(;)15 b Fp(1\))38 b(-)f(See)-24 827 y(App)s(endix)28 b(D.)j(Finally)-8 b(,)29 b(the)i(pro)s(of)e(of)i(v\))f(can)h(b)s(e)f(found)f(e.g.)42 b(in)29 b([25)q(].)1140 b Fc(\003)118 989 y Fp(F)-8 b(rom)24 b(no)m(w)g(on,)h(eac)m(h)h(time)d(w)m(e)i(will)c(refer)j(to)g(the)h (exp)s(onen)m(t)f Fo(\014)29 b Fp(it)23 b(should)f(b)s(e)i(understo)s (o)s(d)e(that)i Fo(\014)31 b Fp(=)25 b Fo(p)-24 1102 y Fp(if)k Fn(H)h Fp(is)e(a)i(general)g(separable)e(Hilb)s(ert)g(space)i (with)e(basis)g Fn(f)p Fo(e)2112 1116 y Fm(n)2160 1102 y Fn(g)2205 1116 y Fm(n)p Fl(2)p Fk(N)2347 1102 y Fp(,)i(and)f Fo(\014)h Fp(=)25 b Fo(p=d)30 b Fp(in)e(the)i(sp)s(eci\014c)e(case)-24 1215 y Fn(H)e Fp(=)f Fo(`)213 1182 y Fh(2)253 1215 y Fp(\()p Fg(Z)353 1182 y Fm(d)389 1215 y Fp(\))31 b(equipp)s(ed)d(with)h (its)g(canonical)h(basis)g Fn(f)p Fo(\016)1885 1229 y Fm(n)1932 1215 y Fn(g)1977 1237 y Fm(n)p Fl(2)p Fk(Z)2118 1218 y Fi(d)2152 1215 y Fp(.)118 1378 y(Our)f(main)g(result)g(is)h(the) g(follo)m(wing)f(one:)-24 1615 y Fd(Theorem)34 b(2.1.)46 b Ff(L)-5 b(et)33 b Fo(H)40 b Ff(b)-5 b(e)32 b(a)h(Schr\177)-46 b(odinger)34 b(op)-5 b(er)g(ator)36 b(on)d(a)g(sep)-5 b(ar)g(able)35 b(Hilb)-5 b(ert)33 b(sp)-5 b(ac)g(e)34 b Fn(H)q 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b(Theorem)d(2.1)i(still)d(holds)g(under)g(the)i(more)g(general)g (assumptions)-24 3470 y(\(H2\))44 b(and)d(\(H3\))i(of)f(App)s(endix)d (C.)j(See)g(also)g([3])g(for)g(further)f(results.)74 b(But)42 b(at)g(the)g(momen)m(t,)k(it)41 b(is)-24 3583 y(not)h(clear)f(to)h(us)f(whether)f(Theorem)h(2.1)i(w)m(ould)d(remain)g (true)h(without)f(an)m(y)i(assumptions)e(on)h Fo(\026)3584 3598 y Fm( )3636 3583 y Fp(.)-24 3696 y(Concerning)29 b(this)h(tec)m(hnical)g(p)s(oin)m(t,)g(w)m(e)g(refer)h(the)f(reader)g (to)i(Remark)e(4.3.)-24 3809 y(iii\))39 b(In)g([15)q(],)k(a)e(similar)c (but)j(w)m(eak)m(er)h(v)m(ersion)f(of)g(Theorem)g(2.1)h(has)f(b)s(een)g (obtained)f(indep)s(enden)m(tly)-24 3922 y(and)j(sim)m(ultaneously)-8 b(.)75 b(There,)45 b(they)e(assume)f(in)f(addition)g(that)i(for)f(all)f Fo(q)48 b Fn(2)d Fp(\()p Fn(\0001)p Fo(;)15 b Fp(+)p Fn(1)p Fp(\))43 b(one)g(has)-24 4035 y Fo(D)54 4002 y Fh(+)51 4057 y Fm(\026)113 4035 y Fp(\()p Fo(q)s Fp(\))33 b(=)f Fo(D)441 4002 y Fl(\000)438 4057 y Fm(\026)500 4035 y Fp(\()p Fo(q)s Fp(\).)53 b(F)-8 b(urthermore,)36 b(they)e(consider)g(the)h(momen)m(ts)f(a)m(v)m(eraged)j(o)m(v)m(er)f([) p Fn(\000)p Fo(T)8 b(;)15 b(T)e Fp(])34 b(instead)g(of)-24 4148 y([0)p Fo(;)15 b(T)e Fp(].)-24 4335 y Fd(Pro)s(of)36 b(of)f(Theorem)f(2.1.)-24 4448 y Fp(The)39 b(pro)s(of)g(of)h(Theorem)f (2.1)i(is)e(the)g(com)m(bination)g(of)h(Theorem)f(3.1)i(and)e(Theorem)g (4.2)i(whic)m(h)d(are)-24 4561 y(pro)m(v)m(ed)31 b(resp)s(ectiv)m(ely)f (in)f(Sections)h(3)h(and)e(4.)42 b(And)29 b(so)i(is)e(the)i(pro)s(of.) 1207 b Fc(\003)-24 4897 y Fq(3)135 b(A)45 b(general)h(lo)l(w)l(er)g(b)t (ound)-24 5100 y Fp(This)24 b(section)i(can)f(b)s(e)g(regarded)g(as)h (the)g(\014rst)e(part)i(of)f(the)h(pro)s(of)e(of)i(our)f(main)f(result) h(Theorem)g(2.1.)40 b(Let)-24 5213 y Fo(H)35 b Fp(b)s(e)28 b(a)h(self-adjoin)m(t)e(op)s(erator)h(in)f(Hilb)s(ert)g(space)h Fn(H)q Fp(,)h(and)e Fn(f)p Fo(e)2161 5227 y Fm(n)2209 5213 y Fn(g)i Fp(an)f(orthonormal)f(basis)g(in)g Fn(H)i Fp(lab)s(elled)-24 5326 y(b)m(y)j Fo(n)27 b Fn(2)g Fg(N)45 b Fp(or)31 b(b)m(y)h Fo(n)27 b Fn(2)g Fg(Z)847 5293 y Fm(d)883 5326 y Fp(.)45 b(Let)32 b Fo( )j Fp(b)s(e)c(some)i(v)m(ector)g (in)d Fn(H)j Fp(suc)m(h)e(that)h Fn(k)p Fo( )s Fn(k)d Fp(=)f(1.)45 b(W)-8 b(e)33 b(are)f(in)m(terested)g(in)1796 5611 y(6)p eop %%Page: 7 7 7 6 bop -24 375 a Fp(lo)m(w)m(er)30 b(b)s(ounds)e(for)i(the)g(momen)m (ts)g(of)g(the)g(abstract)h(p)s(osition)c(op)s(erator)k(asso)s(ciated)f (to)g(the)g(basis)f Fn(f)p Fo(e)3543 389 y Fm(n)3591 375 y Fn(g)p Fp(,)-24 488 y(de\014ned)h(as)1173 601 y Fn(j)p Fo(X)7 b Fn(j)1305 556 y Fm(p)1305 630 y( )1359 601 y Fp(\()p Fo(t)p Fp(\))25 b(=)1583 514 y Fj(X)1628 705 y Fm(n)1730 601 y Fn(j)p Fo(n)p Fn(j)1835 563 y Fm(p)1875 601 y Fn(jh)p Fp(e)1975 563 y Fl(\000)p Fm(itH)2147 601 y Fo( )s(;)15 b(e)2291 615 y Fm(n)2339 601 y Fn(ij)2399 563 y Fh(2)2439 601 y Fo(:)1081 b Fp(\(7\))-24 850 y(In)39 b(particular,)g(if)f Fn(H)j Fp(=)e Fo(`)912 817 y Fh(2)952 850 y Fp(\()p Fg(Z)1052 817 y Fm(d)1088 850 y Fp(\),)j(one)d(can)g(tak) m(e)i(the)e(canonical)g(basis)f Fo(e)2593 864 y Fm(n)2640 850 y Fp(\()p Fo(k)s Fp(\))j(=)e Fo(\016)2951 865 y Fm(nk)3037 850 y Fo(;)93 b(n;)15 b(k)43 b Fn(2)d Fg(Z)3506 817 y Fm(d)3581 850 y Fp(to)-24 963 y(obtain)33 b(the)f(momen)m(ts)i(of)e (the)h(usual)f(p)s(osition)e(op)s(erator.)48 b(Our)32 b(results)f(can)i(also)g(b)s(e)f(extended)h(to)g(the)-24 1076 y(case)f Fn(H)26 b Fp(=)f(L)425 1043 y Fh(2)464 1076 y Fp(\()p Fg(R)559 1043 y Fm(d)605 1076 y Fp(\))31 b(and)740 1310 y Fn(j)p Fo(X)7 b Fn(j)872 1265 y Fm(p)872 1339 y( )925 1310 y Fp(\()p Fo(t)p Fp(\))26 b(=)1150 1186 y Fj(Z)1201 1392 y Fk(R)1249 1373 y Fi(d)1304 1310 y Fn(j)p Fo(x)p Fn(j)1406 1272 y Fm(p)1446 1310 y Fn(j)p Fo( )s Fp(\()p Fo(t;)15 b(x)p Fp(\))p Fn(j)1753 1272 y Fh(2)1795 1310 y Fo(dx;)106 b( )s Fp(\()p Fo(t;)15 b(x)p Fp(\))27 b(=)e(\(e)2480 1272 y Fl(\000)p Fm(itH)2652 1310 y Fo( )s Fp(\)\()p Fo(x)p Fp(\))p Fo(:)-24 1548 y Fp(It)37 b(can)g(b)s(e)f(done)h(using)e(the)i(Theorem)f(3.2)i(in)d (the)i(same)g(manner)f(as)h(in)f([5])h(\(see)h(Corollary)d(2.4)j(and) -24 1661 y(Theorem)27 b(2.5)i(of)f([5],)h(see)f(also)f([22)q(]\).)40 b(Ho)m(w)m(ev)m(er,)31 b(for)c(the)h(sak)m(e)g(of)g(simplicit)m(y)-8 b(,)26 b(w)m(e)h(shall)f(consider)g(in)g(this)-24 1774 y(pap)s(er)k(only)f Fn(j)p Fo(X)7 b Fn(j)563 1741 y Fm(p)634 1774 y Fp(giv)m(en)30 b(b)m(y)h(\(7\).)118 1886 y(W)-8 b(e)34 b(deriv)m(e)f(a)h(lo)m(w)m(er)g(b)s(ound)d(for)i(the)h(time)f(a) m(v)m(eraged)j(momen)m(ts)e(of)f(p)s(osition)f(op)s(erator)i(in)e(term) h(of)-24 1999 y(an)e(abstract)g(quan)m(tit)m(y)g(in)e(the)h(spirit)e (of)j([5)q(],)f(namely)304 2250 y Fo(L)366 2265 y Fm( )419 2250 y Fp(\()p Fo(T)13 b Fp(\))25 b(=)g(sup)14 b Fn(f)q Fo(L)936 2265 y Fm( )988 2250 y Fp(\()p Fo(';)h(T)e Fp(\))p Fo(;)47 b(')26 b Fn(2)f(H)1543 2265 y Fm( )1595 2250 y Fo(;)76 b Fn(h)p Fo( )s(;)15 b(')p Fn(i)28 b(6)p Fp(=)d(0)p Fn(g)16 b Fo(;)106 b(L)2350 2265 y Fm( )2403 2250 y Fp(\()p 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Fh(\()p Fm(R)p Fh(\))1062 3940 y Fp(\()p Fo(x;)i(")p Fp(\))26 b Fn(\021)1388 3817 y Fj(Z)1439 4023 y Fk(R)1506 3940 y Fp(d)p Fo(\026)1612 3955 y Fm( )1664 3940 y Fp(\()p Fo(y)s Fp(\))p Fo(R)1868 3812 y Fj(\022)1944 3879 y Fo(x)21 b Fn(\000)f Fo(y)p 1944 3919 211 4 v 2029 4003 a(")2165 3812 y Fj(\023)2278 3940 y Fo(;)76 b(")25 b Fp(=)g Fo(T)2608 3903 y Fl(\000)p Fh(1)2702 3940 y Fo(:)-24 4197 y Fp(So)41 b(that)h Fo(U)383 4212 y Fm('; )501 4197 y Fp(\()p Fo(T)13 b Fp(\))43 b(=)793 4124 y Fj(R)836 4229 y Fk(R)904 4197 y Fp(d)p Fo(\026)1010 4211 y Fm(')1059 4197 y Fp(\()p Fo(x)p Fp(\))p Fo(b)1220 4164 y Fh(\()p Fm(R)p Fh(\))1333 4197 y Fp(\()p Fo(x;)15 b Fp(1)p Fo(=T)e Fp(\).)74 b(The)40 b(quan)m(tit)m(y)h Fo(U)2383 4212 y Fm('; )2502 4197 y Fp(\()p Fo(T)13 b Fp(\))41 b(is)f(crucial)f(since)i(one)g(ma)m(y)-24 4310 y(consider)34 b(that)h(it)f(co)s(des)g(the)h(determinating)e(part)h(of) h(the)f(sp)s(ectral)g(informations)f(that)i(are)g(in)m(v)m(olv)m(ed)-24 4423 y(in)d(the)g(dynamical)f(b)s(eha)m(viour)g(of)i(the)f(considered)g (quan)m(tum)g(system.)47 b(The)32 b(quan)m(tit)m(y)g(in)f(\(10\))j (should)-24 4536 y(b)s(e)c(compared)h(to)g(other)f(quan)m(tities)g(suc) m(h)g(as)525 4641 y Fj(Z)576 4847 y Fl(j)p Fm(x)p Fl(\000)p Fm(y)r Fl(j\024)p Fh(1)p Fm(=T)942 4765 y Fp(d)p Fo(\026)1048 4780 y Fm( )1100 4765 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)1328 4780 y Fm( )1381 4765 y Fp(\()p Fo(y)s Fp(\))p Fo(;)107 b Fp(or)1818 4641 y Fj(Z)1868 4847 y Fk(R)1936 4641 y Fj(Z)1986 4847 y Fk(R)2053 4765 y Fp(d)p Fo(\026)2159 4780 y Fm( )2211 4765 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)2439 4780 y Fm( )2492 4765 y Fp(\()p Fo(y)s Fp(\)e)2650 4727 y Fl(\000)p Fh(\()p Fm(x)p Fl(\000)p Fm(y)r Fh(\))2891 4704 y Fe(2)2927 4727 y Fm(T)2978 4704 y Fe(2)3013 4727 y Fm(=)p Fh(4)3087 4765 y Fo(;)-24 5027 y Fp(whic)m(h)34 b(already)h(app)s(ear,)h(in)e(the)i(limit)d Fo(")g Fp(=)g Fo(T)1639 4994 y Fl(\000)p Fh(1)1767 5027 y Fn(!)g Fp(0,)k(as)f(k)m(ey) g(quan)m(tities)e(to)i(discuss)e(the)h(nature)g(of)-24 5140 y(the)c(sp)s(ectrum)e([21)q(,)i(22)q(,)f(10)q(,)h(5].)118 5299 y(The)e(main)h(result)f(of)h(this)g(section)g(is)g(the)g(follo)m (wing.)1796 5611 y(7)p eop %%Page: 8 8 8 7 bop -24 375 a Fd(Theorem)34 b(3.1.)46 b Ff(L)-5 b(et)34 b Fo(H)40 b Ff(b)-5 b(e)33 b(a)h(Schr\177)-46 b(odinger)35 b(op)-5 b(er)g(ator)36 b(on)e Fn(H)f Ff(and)i(let)e Fo( )k Ff(b)-5 b(e)33 b(a)g(ve)-5 b(ctor)34 b(in)f Fn(H)q Ff(,)g Fn(k)p Fo( )s Fn(k)28 b Fp(=)e(1)p Ff(.)-24 488 y(L)-5 b(et)33 b Fn(f)p Fo(e)220 502 y Fm(n)268 488 y Fn(g)g Ff(b)-5 b(e)33 b(some)g(orthonormal)j(b)-5 b(asis)34 b(in)e Fn(H)h Ff(and)923 734 y Fn(hhj)p Fo(X)7 b Fn(j)1125 696 y Fm(p)1167 734 y Fn(ii)1237 749 y Fm( )1290 734 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1568 672 y(1)p 1558 713 66 4 v 1558 796 a Fo(T)1649 610 y Fj(Z)1740 636 y Fm(T)1699 816 y Fh(0)1810 647 y Fj(X)1854 838 y Fm(n)1956 734 y Fn(j)p Fo(n)p Fn(j)2061 696 y Fm(p)2101 734 y Fn(jh)p Fp(e)2201 696 y Fl(\000)p Fm(itH)2374 734 y Fo(;)15 b(e)2456 748 y Fm(n)2503 734 y Fn(ij)2563 696 y Fh(2)2603 734 y Fp(dt)p Fo(:)-24 1018 y Ff(L)-5 b(et)33 b Fo(h)26 b Fn(2)f Fo(C)369 985 y Fl(1)362 1042 y Fh(0)443 1018 y Fp(\([0)p Fo(;)15 b Fp(1]\))35 b Ff(b)-5 b(e)33 b(any)g(p)-5 b(ositive)33 b(function)g(such)g(that)2092 944 y Fj(R)2152 971 y Fh(1)2135 1049 y(0)2207 1018 y Fo(h)p Fp(\()p Fo(z)t Fp(\))p Fo(dz)e Fp(=)25 b(1)33 b Ff(and)h(de\014ne)1174 1270 y Fo(R)q Fp(\()p Fo(w)r Fp(\))27 b(=)1504 1141 y Fj(\032)1613 1209 y Fp(1)351 b(if)39 b Fn(j)p Fo(w)r Fn(j)26 b(\024)f Fp(1)1613 1330 y Fn(j)1639 1306 y Fp(^)1638 1330 y Fo(h)q Fp(\()p Fo(w)r Fp(\))p Fn(j)1853 1297 y Fh(2)2009 1330 y Fp(if)39 b Fn(j)p Fo(w)r Fn(j)26 b Fo(>)f Fp(1)p Fo(;)3500 1270 y Fp(\(11\))-24 1531 y Ff(wher)-5 b(e)237 1507 y Fp(^)237 1531 y Fo(h)36 b Ff(stands)h(for)g(the)g(F)-7 b(ourier)37 b(tr)-5 b(ansform)38 b(of)f Fo(h)p Ff(.)52 b(Then,)37 b(for)g(al)5 b(l)36 b Fo(p)c(>)f Fp(0)36 b Ff(and)h(with)g Fo(L)3162 1546 y Fm( )3215 1531 y Fp(\()p Fo(T)13 b Fp(\))36 b Ff(de\014ne)-5 b(d)-24 1644 y(Line)33 b(\(8\),)g(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\))34 b Ff(such)f(that)h(for)f(al)5 b(l)33 b Fo(T)38 b(>)25 b Fp(0)1161 1829 y Fn(hhj)p Fo(X)7 b Fn(j)1363 1792 y Fm(p)1404 1829 y Fn(ii)1474 1844 y Fm( )1527 1829 y Fp(\()p Fo(T)13 b Fp(\))26 b Fn(\025)f Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\))p Fo(L)2229 1844 y Fm( )2283 1829 y Fp(\()p Fo(T)e Fp(\))33 b Fo(:)-24 2014 y Fd(Remark)i(3.1.)46 b Fp(As)23 b(already)f(p)s(oin)m(ted)g(out)h(in)f(the)h(in)m(tro)s(duction,)g Fo(h)g Fp(seems)g(to)h(b)s(e)e(the)h(necessary)h(tric)m(k)f(to)-24 2127 y(tak)m(e)28 b(in)m(to)f(accoun)m(t)g(a)m(v)m(eraging)h(on)f(time) f(b)s(et)m(w)m(een)h([0)p Fo(;)15 b(T)e Fp(])27 b(only)e(\(and)i(not)f ([)p Fn(\000)p Fo(T)8 b(;)15 b(T)e Fp(]\).)40 b(This)24 b(tric)m(k)j(actually)-24 2240 y(allo)m(ws)34 b(one)g(to)g(deal)g(with) e(the)i(crossed)g(term)g(in)e(\(4\))j(and)e(to)i(reco)m(v)m(er)g(a)g (function)d Fo(R)j Fp(with)d(fast)i(deca)m(y)-24 2353 y(at)e Fn(\0061)p Fp(.)42 b(Replacing)30 b Fo(h)h Fp(b)m(y)g(the)g (usual)f(gaussian)g Fo(e)1751 2320 y Fl(\000)p Fm(z)1842 2297 y Fe(2)1876 2320 y Fm(=)p Fh(4)1982 2353 y Fp(in)f(\(21\))k(b)s (elo)m(w)d(and)g(th)m(us)h(in)e(\(12\))k(will)28 b(lead)j(to)-24 2466 y(the)g(same)g(result)e(but)h(with)f Fn(j)p Fo(X)7 b Fn(j)1120 2422 y Fm(p)1120 2495 y( )1173 2466 y Fp(\()p Fo(t)p Fp(\))31 b(a)m(v)m(eraged)i(o)m(v)m(er)e([)p Fn(\000)p Fo(T)8 b(;)15 b(T)e Fp(].)118 2638 y(As)32 b(it)g(is)g(no)m(w)g(w)m (ell-kno)m(wn)f([13)r(,)h(7)q(,)g(22)q(],)i(a)f(k)m(ey)g(p)s(oin)m(t)f (in)f(the)h(pro)s(of)g(of)h(lo)m(w)m(er)g(b)s(ound)d(for)i Fn(j)p Fo(X)7 b Fn(j)3449 2605 y Fm(p)3522 2638 y Fp(is)32 b(a)-24 2750 y(go)s(o)s(d)f(con)m(trol)f(of)h(the)g(b)s(eha)m(viour)d (of)j(the)g(w)m(a)m(v)m(e-pac)m(k)m(et)j(inside)28 b(a)j(ball)e(of)h (radius)f Fo(N)10 b Fp(,)30 b(namely)851 2975 y Fo(B)920 2990 y Fm( )973 2975 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))26 b(=)1369 2913 y(1)p 1359 2954 V 1359 3037 a Fo(T)1450 2851 y Fj(Z)1541 2877 y Fl(1)1500 3057 y Fh(0)1665 2888 y Fj(X)1631 3090 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)1846 2975 y Fn(jh)p Fp(e)1946 2937 y Fl(\000)p Fm(itH)2118 2975 y Fo( )s(;)15 b(e)2262 2989 y Fm(n)2311 2975 y Fn(ij)2371 2937 y Fh(2)2411 2975 y Fo(h)p Fp(\()p Fo(t=T)e Fp(\)d)p Fo(t:)714 b Fp(\(12\))-24 3259 y(T)-8 b(o)35 b(that)g(end,)g(w)m(e)g (shall)e(need)h(Theorem)g(3.2)i(b)s(elo)m(w,)f(whic)m(h)e(is)g(a)i (generalization)f(of)h(Theorem)f(2.1)i(in)-24 3372 y([5)q(].)56 b(In)34 b(order)h(to)h(state)g(it)f(w)m(e)h(\014rst)e(recall)h(w)m (ell-kno)m(wn)f(facts)i(in)m(v)m(olving)e(generalized)h(eigenfunctions) -24 3485 y(and)30 b(the)h(sp)s(ectral)e(measure)i Fo(\026)1061 3500 y Fm( )1143 3485 y Fp(\()p Ff(se)-5 b(e)33 b(e.g.)40 b Fp([20)q(])31 b([27)q(])g(and)e(ref.)41 b(therein\).)-24 3598 y(Let)35 b Fo(W)229 3613 y Fm( )316 3598 y Fp(b)s(e)e(the)i (unitary)d(map)i(from)g(the)g(cyclic)g(subspace)g Fn(H)2228 3613 y Fm( )2314 3598 y Fp(in)m(to)h(the)f(space)h(L)2967 3565 y Fh(2)3006 3598 y Fp(\()p Fg(R)s Fo(;)15 b Fp(d)p Fo(\026)3247 3613 y Fm( )3305 3598 y Fp(\).)53 b(Recall)-24 3710 y(that)27 b Fo(W)255 3725 y Fm( )308 3710 y Fp(\()p Fo( )s Fp(\))f(=)f(1)i(and)e Fo(W)892 3725 y Fm( )945 3710 y Fp(\(e)1020 3677 y Fl(\000)p Fm(itH)1192 3710 y Fo( )s Fp(\))h(=)f(e)1451 3677 y Fl(\000)p Fm(itx)1626 3710 y Fp(for)h(all)f Fo(x)g Fn(2)g Fg(R)s Fp(.)45 b(W)-8 b(e)27 b(shall)e(denote)h(b)m(y)g Fo(P)3005 3725 y Fm( )3085 3710 y Fp(the)g(orthogonal)-24 3823 y(pro)5 b(jection)30 b(on)h Fn(H)612 3838 y Fm( )664 3823 y Fp(.)41 b(The)30 b(map)g Fo(W)1205 3838 y Fm( )1288 3823 y Fp(has)g(a)g(k)m(ernel)g Fo(u)p Fp(\()p Fo(n;)15 b(x)p Fp(\))31 b(de\014ned)e(b)m(y)i Fo(u)p Fp(\()p Fo(n)p Fp(\))25 b(=)g Fo(W)2917 3838 y Fm( )2970 3823 y Fp(\()p Fo(P)3063 3838 y Fm( )3116 3823 y Fo(e)3158 3837 y Fm(n)3205 3823 y Fp(\),)31 b(so)g(that)1084 4055 y Fn(h)p Fo(e)1161 4018 y Fl(\000)p Fm(iH)5 b(t)1333 4055 y Fo( )s(;)15 b(e)1477 4069 y Fm(n)1525 4055 y Fn(i)26 b Fp(=)1682 3931 y Fj(Z)1788 4055 y Fo(e)1830 4018 y Fl(\000)p Fm(ixt)p 1978 3976 270 4 v 1978 4055 a Fo(u)p Fp(\()p Fo(n;)15 b(x)p Fp(\))q(d)p Fo(\026)2354 4070 y Fm( )2406 4055 y Fp(\()p Fo(x)p Fp(\))p Fo(;)947 b Fp(\(13\))-24 4277 y(and)30 b(more)h(generally)-8 b(,)30 b(for)g(an)m(y)h(v)m(ector)h Fo(\030)d Fn(2)c(H)q Fp(,)30 b(one)h(has)981 4499 y Fn(h)p Fo(P)1074 4514 y Fm( )1128 4499 y Fo(\030)t(;)15 b(e)1254 4513 y Fm(n)1301 4499 y Fn(i)26 b Fp(=)1458 4375 y Fj(Z)1564 4499 y Fo(W)1650 4514 y Fm( )1718 4499 y Fp(\()p Fo(P)1811 4514 y Fm( )1864 4499 y Fo(\030)t Fp(\))15 b(\()p Fo(x)p Fp(\))p 2080 4420 V Fo(u)p Fp(\()p Fo(n;)g(x)p Fp(\))r(d)p Fo(\026)2457 4514 y Fm( )2509 4499 y Fp(\()p Fo(x)p Fp(\))p Fo(:)844 b Fp(\(14\))-24 4737 y(In)35 b(the)h(case)g Fn(H)e Fp(=)f Fo(`)707 4704 y Fh(2)747 4737 y Fp(\()p Fg(Z)847 4704 y Fm(d)883 4737 y Fp(\))j(and)e Fo(e)1177 4751 y Fm(n)1225 4737 y Fp(\()p Fo(k)s Fp(\))g(=)f Fo(\016)1523 4752 y Fm(nk)1609 4737 y Fp(,)k(for)e(eac)m(h)i(\014xed)d Fo(x)g Fn(2)f Fg(R)44 b Fp(the)36 b(v)m(ector)g(\()p Fo(u)p Fp(\()p Fo(n;)15 b(x)p Fp(\))3273 4759 y Fm(n)p Fl(2)p Fk(Z)3415 4740 y Fi(d)3449 4737 y Fp(\))36 b(is)e(a)-24 4850 y(generalized)c(eigenfunction)f(\()p Ff(i.e.)41 b Fp(in)29 b(a)h(distributional)d(sense\).)118 4962 y(The)37 b(expansions)f(\(13\)-\(14\))42 b(are)c(of)g(course)g(also)g(p)s (ossible)d(with)i(an)m(y)h(v)m(ector)h Fo(')g Fn(2)e(H)h Fp(and)g(corre-)-24 5075 y(sp)s(onding)h(generalized)h(eigenfunctions)g Fo(v)s Fp(\()p Fo(n;)15 b(y)s Fp(\))43 b(=)f Fo(W)1961 5089 y Fm(')2011 5075 y Fp(\()p Fo(P)2104 5089 y Fm(')2155 5075 y Fo(e)2197 5089 y Fm(n)2245 5075 y Fp(\)\()p Fo(\025)p Fp(\))f(\(actually)g(if)f Fo(')j Fp(=)g Fo(f)10 b Fp(\()p Fo(H)d Fp(\))p Fo( )s Fp(,)44 b(with)-24 5188 y Fo(f)35 b Fn(2)25 b Fp(L)199 5155 y Fh(2)238 5188 y Fp(\()p Fg(R)s Fo(;)15 b Fp(d)p Fo(\026)479 5203 y Fm( )537 5188 y Fp(\),)31 b(then)f(one)h(c)m(hec)m(ks)h(that)f Fo(v)s Fp(\()p Fo(n;)15 b(y)s Fp(\))26 b(=)f Fo(f)10 b Fp(\()p Fo(y)s Fp(\))p Fo(u)p Fp(\()p Fo(n;)15 b(y)s Fp(\)\).)-24 5301 y(W)-8 b(e)32 b(are)f(no)m(w)f(ready)g(to)h(form)m(ulate)g(Theorem)f(3.2.)1796 5611 y(8)p eop %%Page: 9 9 9 8 bop -24 375 a Fd(Theorem)34 b(3.2.)46 b Ff(L)-5 b(et)39 b Fo(h)p Fp(\()p Fo(z)t Fp(\))g Ff(b)-5 b(e)38 b(some)h(function)f(fr) -5 b(om)39 b Fo(L)1998 342 y Fh(1)2038 375 y Fp(\()p Fg(R)s Fp(\))p Ff(,)45 b Fo(H)g Ff(a)38 b(self-adjoint)h(op)-5 b(er)g(ator)42 b(acting)c(on)-24 488 y Fn(H)c Ff(and)h Fo(A)e Ff(a)h(Hilb)-5 b(ert-Schmidt)35 b(op)-5 b(er)g(ator)36 b(in)e Fn(H)q Ff(.)44 b(F)-7 b(or)34 b(any)g(c)-5 b(ouple)35 b(of)e(ve)-5 b(ctors)35 b Fo( )s(;)15 b(')34 b Ff(fr)-5 b(om)35 b Fn(H)f Ff(de\014ne)g(the)-24 601 y(quantity)881 764 y Fo(D)959 716 y Fh(\()p Fm(h)p Fh(\))956 794 y Fm('; )1074 764 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1352 703 y(1)p 1342 743 66 4 v 1342 827 a Fo(T)1433 641 y Fj(Z)1524 667 y Fh(+)p Fl(1)1483 847 y(\0001)1653 764 y Fn(h)p Fo(A)p Fp(e)1796 727 y Fl(\000)p Fm(itH)1969 764 y Fo(';)15 b Fp(e)2108 727 y Fl(\000)p Fm(itH)2281 764 y Fo( )s Fn(i)p Fo(h)p Fp(\()p Fo(t=T)e Fp(\)dt)r Fo(:)-24 972 y Ff(L)-5 b(et)912 1104 y Fo(U)984 1056 y Fh(\()p Fm(h)p Fh(\))974 1134 y Fm('; )1092 1104 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1350 981 y Fj(Z)1401 1187 y Fk(R)1468 981 y Fj(Z)1518 1187 y Fk(R)1586 1104 y Fp(d)p Fo(\026)1692 1118 y Fm(')1741 1104 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)1969 1119 y Fm( )2022 1104 y Fp(\()p Fo(y)s Fp(\))p Fn(j)2166 1080 y Fp(^)2165 1104 y Fo(h)q Fp(\(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\))p Fo(T)13 b Fp(\))p Fn(j)2661 1067 y Fh(2)2701 1104 y Fo(:)118 1307 y Ff(The)32 b(fol)5 b(lowing)34 b(estimate)f(holds:)1201 1550 y Fn(j)p Fo(D)1304 1502 y Fh(\()p Fm(h)p Fh(\))1301 1580 y Fm('; )1420 1550 y Fp(\()p Fo(T)13 b Fp(\))p Fn(j)26 b(\024)f(k)p Fo(A)p Fn(k)1861 1564 y Fh(2)1916 1449 y Fj(\020)1971 1550 y Fo(U)2043 1502 y Fh(\()p Fm(h)p Fh(\))2033 1580 y Fm('; )2151 1550 y Fp(\()p Fo(T)13 b Fp(\))2287 1449 y Fj(\021)2352 1445 y Fe(1)p 2352 1457 31 3 v 2352 1498 a(2)2411 1550 y Fo(;)-24 1766 y Ff(wher)-5 b(e)34 b Fn(k)p Fo(A)p Fn(k)391 1780 y Fh(2)464 1766 y Ff(is)f(the)g(Hilb)-5 b(ert-Schmidt)33 b(norm)h(of)f Fo(A)p Ff(.)118 1879 y(In)f(the)h(sp)-5 b(e)g(cial)34 b(c)-5 b(ase)33 b Fo(A)26 b Fp(=)1060 1811 y Fj(P)1156 1906 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)1360 1879 y Fn(h\001)p Fo(;)15 b(e)1502 1893 y Fm(n)1550 1879 y Fn(i)p Fo(e)1627 1893 y Fm(n)1675 1879 y Ff(,)32 b(one)h(has)h Fn(k)p Fo(A)p Fn(k)2228 1893 y Fh(2)2294 1879 y Fn(\024)25 b Fo(C)7 b(N)2545 1846 y Fm(d=)p Fh(2)2655 1879 y Ff(,)32 b(and)i(ther)-5 b(efor)g(e)101 2197 y Fn(j)p Fo(D)204 2149 y Fh(\()p Fm(h)p Fh(\))201 2226 y Fm('; )319 2197 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))p Fn(j)27 b Fp(=)721 2011 y Fj(\014)721 2065 y(\014)721 2120 y(\014)721 2174 y(\014)721 2229 y(\014)721 2283 y(\014)771 2135 y Fp(1)p 761 2176 66 4 v 761 2259 a Fo(T)852 2073 y Fj(Z)943 2099 y Fh(+)p Fl(1)902 2279 y(\0001)1122 2111 y Fj(X)1087 2312 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)1287 2197 y Fn(h)p Fp(e)1362 2159 y Fl(\000)p Fm(itH)1535 2197 y Fo(';)15 b(e)1676 2211 y Fm(n)1724 2197 y Fn(ih)p Fo(e)1836 2211 y Fm(n)1884 2197 y Fo(;)g Fp(e)1964 2159 y Fl(\000)p Fm(itH)2136 2197 y Fo( )s Fn(i)p Fo(h)p Fp(\()p Fo(t=T)e Fp(\)d)p Fo(t)2583 2011 y Fj(\014)2583 2065 y(\014)2583 2120 y(\014)2583 2174 y(\014)2583 2229 y(\014)2583 2283 y(\014)2640 2197 y Fn(\024)25 b Fo(C)7 b(N)2891 2159 y Fm(d=)p Fh(2)3017 2096 y Fj(\020)3071 2197 y Fo(U)3143 2149 y Fh(\()p Fm(h)p Fh(\))3133 2226 y Fm('; )3251 2197 y Fp(\()p Fo(T)13 b Fp(\))3387 2096 y Fj(\021)3452 2092 y Fe(1)p 3452 2104 31 3 v 3452 2145 a(2)3512 2197 y Fo(:)-24 2494 y Ff(Her)-5 b(e)32 b Fo(d)25 b Fp(=)g(1)32 b Ff(for)g(abstr)-5 b(act)33 b(p)-5 b(osition)34 b(op)-5 b(er)g(ator)34 b(asso)-5 b(ciate)g(d)34 b(with)f(the)f(b)-5 b(ase)32 b Fn(f)p Fo(e)2670 2508 y Fm(n)2717 2494 y Fn(g)g Ff(lab)-5 b(el)5 b(le)-5 b(d)33 b(by)f Fo(n)24 b Fn(2)h Fg(N)7 b Ff(,)37 b(and)-24 2607 y Fo(d)26 b Fn(\025)f Fp(1)33 b Ff(in)f(the)h(c)-5 b(ase)34 b Fn(H)25 b 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Fo(D)892 4431 y Fh(\()p Fm(h)p Fh(\))889 4508 y Fm('; )1008 4479 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1266 4355 y Fj(Z)1316 4561 y Fk(R)1383 4355 y Fj(Z)1434 4561 y Fk(R)1501 4479 y Fp(d)p Fo(\026)1607 4493 y Fm(')1657 4479 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)1885 4494 y Fm( )1937 4479 y Fp(\()p Fo(y)s Fp(\))2056 4455 y(^)2055 4479 y Fo(h)q Fp(\(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\))p Fo(T)13 b Fp(\))p Fo(S)5 b Fp(\()p Fo(x;)15 b(y)s Fp(\))p Fo(;)678 b Fp(\(17\))-24 4713 y(where)1213 4872 y Fo(S)5 b Fp(\()p Fo(x;)15 b(y)s Fp(\))26 b(=)1638 4759 y Fl(1)1607 4786 y Fj(X)1606 4980 y Fm(n)p Fh(=1)1755 4872 y Fo(E)1822 4886 y Fm(n)p 1869 4793 V 1869 4872 a Fo(u)p Fp(\()p Fo(n;)15 b(x)p Fp(\))p Fo(v)s Fp(\()p Fo(n;)g(y)s Fp(\))p Fo(:)-24 5122 y Fp(The)29 b(sum)e(con)m(v)m(erges)k(in)d(L)920 5089 y Fh(2)959 5122 y Fp(\()p Fg(R)1054 5089 y Fh(2)1099 5122 y Fo(;)15 b(\026)1194 5136 y Fm(')1261 5122 y Fn(\002)i Fo(\026)1404 5137 y Fm( )1457 5122 y Fp(\).)40 b(Applying)26 b(the)j(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)i(inequalit)m(y)c(to)i(\(17\),)i (one)-24 5235 y(gets)1030 5359 y Fn(j)p Fo(D)1133 5311 y Fh(\()p Fm(h)p Fh(\))1130 5388 y Fm('; )1248 5359 y Fp(\()p Fo(T)13 b Fp(\))p Fn(j)1409 5321 y Fh(2)1475 5359 y Fn(\024)25 b Fo(U)1643 5311 y Fh(\()p Fm(h)p Fh(\))1633 5388 y Fm('; )1751 5359 y Fp(\()p Fo(T)13 b Fp(\))p Fn(k)p Fo(S)5 b Fn(k)2038 5321 y Fh(2)2038 5387 y(L)2082 5368 y Fe(2)2118 5387 y Fh(\()p Fk(R)2193 5368 y Fe(2)2228 5387 y Fm(;d\026)2326 5395 y Fi(')2371 5387 y Fl(\002)p Fm(d\026)2504 5399 y Fi( )2551 5387 y Fh(\))2583 5359 y Fo(:)892 b Fp(\(18\))1796 5611 y(9)p eop %%Page: 10 10 10 9 bop -24 375 a Fp(One)30 b(can)h(easily)e(see)i(that)1054 603 y Fn(k)p Fo(S)5 b Fn(k)1205 566 y Fh(2)1205 631 y(L)1249 612 y Fe(2)1284 631 y Fh(\()p Fk(R)1359 612 y Fe(2)1394 631 y Fm(;d\026)1492 639 y Fi(')1537 631 y Fl(\002)p Fm(d\026)1670 643 y Fi( )1718 631 y Fh(\))1775 603 y Fp(=)1931 490 y Fl(1)1900 517 y Fj(X)1871 715 y Fm(n;k)r Fh(=1)2077 603 y Fo(E)2144 617 y Fm(n)2191 603 y Fo(E)2258 618 y Fm(k)2301 603 y Fo(a)2349 618 y Fm(nk)p 2434 529 125 4 v 2434 603 a Fo(b)2473 618 y Fm(nk)2559 603 y Fo(;)-24 859 y Fp(where)139 1042 y Fo(a)187 1057 y Fm(nk)298 1042 y Fp(=)394 919 y Fj(Z)444 1125 y Fk(R)512 1042 y Fp(d)p Fo(\026)618 1056 y Fm(')667 1042 y Fp(\()p Fo(x)p Fp(\))p Fo(u)p Fp(\()p Fo(k)s(;)15 b(x)p Fp(\))p 1053 963 270 4 v Fo(u)p Fp(\()p Fo(n;)g(x)p Fp(\))28 b(=)d Fn(h)p Fo(W)1567 1056 y Fm(')1617 1042 y Fp(\()p Fo(P)1710 1056 y Fm(')1761 1042 y Fo(f)1806 1057 y Fm(k)1848 1042 y Fp(\))p Fo(;)15 b(W)2009 1056 y Fm(')2060 1042 y Fp(\()p Fo(P)2153 1056 y Fm(')2204 1042 y Fo(f)2249 1056 y Fm(n)2295 1042 y Fp(\))p Fn(i)2365 1062 y Fh(L)2409 1043 y Fe(2)2445 1062 y Fh(\()p Fk(R)p Fm(;)p Fh(d)p Fm(\026)2621 1070 y Fi(')2666 1062 y Fh(\))2723 1042 y Fp(=)25 b Fn(h)p Fo(P)2912 1056 y Fm(')2963 1042 y Fo(f)3008 1057 y Fm(k)3050 1042 y Fo(;)15 b(f)3135 1056 y Fm(n)3182 1042 y Fn(i)3217 1056 y Fl(H)3312 1042 y Fo(;)163 b Fp(\(19\))-24 1252 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Fh(2)1758 1871 y Fo(;)1890 1757 y Fl(1)1859 1784 y Fj(X)1861 1982 y Fm(k)r Fh(=1)2006 1871 y Fn(j)p Fo(b)2070 1886 y Fm(nk)2156 1871 y Fn(j)2181 1833 y Fh(2)2245 1871 y Fp(=)g Fn(k)p Fo(P)2444 1886 y Fm( )2498 1871 y Fo(g)2541 1885 y Fm(n)2588 1871 y Fn(k)2633 1833 y Fh(2)2673 1871 y Fo(:)-24 2101 y Fp(Therefore,)31 b(as)f Fn(k)p Fo(f)615 2116 y Fm(k)658 2101 y Fn(k)c Fp(=)f Fn(k)p Fo(g)913 2115 y Fm(n)960 2101 y Fn(k)h Fp(=)f(1)31 b(for)f(all)f Fo(k)s(;)15 b(n)p Fp(,)358 2359 y Fn(k)p Fo(S)5 b Fn(k)509 2322 y Fh(4)509 2387 y(L)553 2368 y Fe(2)589 2387 y Fh(\()p Fk(R)664 2368 y Fe(2)699 2387 y Fm(;d\026)797 2395 y Fi(')841 2387 y Fl(\002)p Fm(d\026)974 2399 y Fi( )1022 2387 y Fh(\))1079 2359 y Fn(\024)1205 2245 y Fl(1)1175 2273 y Fj(X)1176 2470 y Fm(k)r Fh(=1)1321 2359 y Fo(E)1393 2322 y Fh(2)1388 2382 y Fm(k)1433 2359 y Fn(k)p Fo(P)1536 2373 y Fm(')1587 2359 y Fo(f)1632 2374 y Fm(k)1674 2359 y Fn(k)1719 2322 y Fh(2)1805 2245 y Fl(1)1775 2273 y Fj(X)1774 2467 y Fm(n)p Fh(=1)1922 2359 y Fo(E)1994 2322 y Fh(2)1989 2382 y Fm(n)2036 2359 y Fn(k)p Fo(P)2139 2374 y Fm( )2193 2359 y Fo(g)2236 2373 y Fm(n)2283 2359 y Fn(k)2328 2322 y Fh(2)2393 2359 y Fn(\024)2489 2204 y Fj( )2592 2245 y Fl(1)2562 2273 y Fj(X)2561 2467 y Fm(n)p Fh(=1)2709 2359 y Fo(E)2781 2322 y Fh(2)2776 2382 y Fm(n)2823 2204 y Fj(!)2895 2226 y Fh(2)2960 2359 y Fp(=)25 b Fn(k)p Fo(A)p Fn(k)3214 2322 y Fh(4)3214 2382 y(2)3255 2359 y Fo(:)220 b Fp(\(20\))-24 2607 y(The)30 b(\014rst)f(statemen)m(t)i(of)f(the)g(Theorem)f(follo)m (ws)g(from)g(\(18\))j(and)d(\(20\).)42 b(The)29 b(pro)s(of)g(of)h(the)g (second)f(part)-24 2720 y(is)f(essen)m(tially)f(the)i(same.)40 b(The)28 b(only)g(di\013erence)g(is)f(that)i(in)e(the)i(case)g Fn(H)d Fp(=)f Fo(`)2674 2687 y Fh(2)2713 2720 y Fp(\()p Fg(Z)2813 2687 y Fm(d)2850 2720 y Fp(\))k(the)f(sums)f(are)i(tak)m(en) -24 2833 y(o)m(v)m(er)j Fo(n)25 b Fn(2)g Fg(Z)402 2800 y Fm(d)463 2833 y Fp(:)56 b Fn(j)p Fo(n)p Fn(j)25 b(\024)g Fo(N)10 b Fp(.)41 b(In)29 b(particular,)g(the)i(estimate)g(\(20\))h (reads)e(as)667 3011 y Fn(k)p Fo(S)5 b Fn(k)818 2973 y Fh(4)818 3039 y(L)862 3020 y Fe(2)897 3039 y Fh(\()p Fk(R)972 3020 y Fe(2)1007 3039 y Fm(;d\026)1105 3047 y Fi(')1150 3039 y Fl(\002)p Fm(d\026)1283 3051 y Fi( )1331 3039 y Fh(\))1388 3011 y Fn(\024)1516 2924 y Fj(X)1484 3126 y Fl(j)p Fm(k)r Fl(j\024)p Fm(N)1694 3011 y Fn(k)p Fo(P)1797 3025 y Fm(')1848 3011 y Fo(e)1890 3026 y Fm(k)1933 3011 y Fn(k)1978 2973 y Fh(2)2068 2924 y Fj(X)2033 3126 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)2249 3011 y Fn(k)p Fo(P)2352 3026 y Fm( )2405 3011 y Fo(e)2447 3025 y Fm(n)2494 3011 y Fn(k)2539 2973 y Fh(2)2604 3011 y Fn(\024)25 b Fo(C)d(N)2870 2973 y Fh(2)p Fm(d)2946 3011 y Fo(:)-24 3274 y Fp(The)30 b(pro)s(of)g(is)f(completed.)2670 b Fc(\003)118 3423 y Fp(One)27 b(should)e(stress)j(that)g(the)g(pro)s(of)f(w)m(e)h(presen) m(ted)f(here)h(is)e(simpler)f(than)j(that)g(of)g(Theorem)f(2.1)i(in)-24 3536 y([5)q(],)d(b)s(ecause)e(w)m(e)g(do)g(not)h(use)e(the)i(pro)s (duct)d(space)j Fn(H)8 b(\012)g(H)q Fp(.)38 b(In)23 b(the)h(case)h Fo( )k Fp(=)c Fo(')f Fp(and)g Fo(h)p Fp(\()p Fo(z)t Fp(\))j(=)d(exp\()p Fn(\000)p Fo(z)3495 3503 y Fh(2)3535 3536 y Fo(=)p Fp(4\))-24 3649 y(the)31 b(result)e(of)i(Theorem)f(3.2)h(is)f(equiv)-5 b(alen)m(t)30 b(to)h(that)g(of)f(Theorem)g(2.1)i(in)d([5].)-24 3798 y Fd(Pro)s(of)k(of)g(Theorem)e(3.1.)40 b Fp(Pic)m(k)28 b(a)g(p)s(ositiv)m(e)f(function)g Fo(h)p Fp(\()p Fo(z)t Fp(\))f Fn(2)f Fo(C)2336 3765 y Fl(1)2329 3823 y Fh(0)2411 3798 y Fp(\([0)p Fo(;)15 b Fp(1]\))30 b(suc)m(h)e(that)3088 3725 y Fj(R)3149 3751 y Fh(1)3131 3830 y(0)3203 3798 y Fo(h)p Fp(\()p Fo(z)t Fp(\)d)p Fo(z)j Fp(=)25 b(1.)-24 3926 y(The)35 b(role)g(of)g Fo(h)h Fp(is)e(to)i(supply)c(a)k(fast)f (deca)m(ying)h(function)e Fn(j)2079 3902 y Fp(^)2078 3926 y Fo(h)p Fp(\()p Fo(w)r Fp(\))p Fn(j)2292 3893 y Fh(2)2333 3926 y Fp(.)55 b(Note)37 b(that)e(one)h(trivially)c(has,)37 b(for)-24 4038 y(an)m(y)31 b Fo(z)f Fn(2)24 b Fp([0)p Fo(;)15 b Fp(1],)33 b Fo(h)p Fp(\()p Fo(z)t Fp(\))26 b Fn(\024)f(k)p Fo(h)p Fn(k)975 4052 y Fl(1)1051 4038 y Fo(\037)1108 4057 y Fh([0)p Fm(;)p Fh(1])1242 4038 y Fp(\()p Fo(z)t Fp(\).)41 b(Then)30 b(one)h(v)m(eri\014es)e(that)22 4292 y Fn(hh)p Fo(X)174 4255 y Fm(p)215 4292 y Fn(ii)285 4307 y Fm( )338 4292 y Fp(\()p Fo(T)13 b Fp(\))26 b Fn(\025)692 4231 y Fp(1)p 606 4271 219 4 v 606 4354 a Fn(k)p Fo(h)p Fn(k)748 4368 y Fl(1)849 4168 y Fj(Z)940 4195 y Fl(1)899 4375 y Fh(0)1030 4206 y Fj(X)1074 4397 y Fm(n)1176 4292 y Fn(j)p Fo(n)p Fn(j)1281 4255 y Fm(p)1321 4292 y Fn(jh)p Fp(e)1421 4255 y Fl(\000)p Fm(itH)1593 4292 y Fo( )s(;)15 b(e)1737 4306 y Fm(n)1786 4292 y Fn(ij)1846 4255 y Fh(2)1886 4292 y Fo(h)p Fp(\()p Fo(t=T)e Fp(\))2162 4231 y(d)p Fo(t)p 2162 4271 84 4 v 2171 4354 a(T)2281 4292 y Fn(\025)2435 4231 y Fo(N)2518 4198 y Fm(p)p 2387 4271 219 4 v 2387 4354 a Fn(k)p Fo(h)p Fn(k)2529 4368 y Fl(1)2615 4137 y Fj( )2687 4292 y Fn(k)p Fo( )s Fn(k)2839 4255 y Fh(2)2895 4292 y Fn(\000)i Fo(B)3050 4307 y Fm( )3102 4292 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))3356 4137 y Fj(!)3429 4292 y Fo(;)46 b Fp(\(21\))-24 4533 y(with)38 b Fo(B)261 4548 y Fm( )314 4533 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))40 b(de\014ned)e(line)g(\(12\).)69 b(As)39 b(usual,)h(one)g(needs)f(a)g (con)m(trol)h(of)f(this)f(quan)m(tit)m(y)i Fo(B)3354 4548 y Fm( )3407 4533 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))-24 4646 y(whic)m(h)40 b(represen)m(ts)g(the)h(b)s(eha)m(viour) e(of)i(the)g(w)m(a)m(v)m(e-pac)m(k)m(et)j(in)c(a)h(ball)e(of)h(radius)f Fo(N)10 b Fp(.)71 b(Decomp)s(ose)42 b(the)-24 4759 y(v)m(ector)32 b Fo( )i Fp(as)d Fo(')20 b Fp(+)g Fo(\037)p Fp(,)31 b(with)e Fn(h)p Fo(';)15 b(\037)p Fn(i)27 b Fp(=)e(0)30 b(and)g Fo(')c Fn(6)p Fp(=)f(0)30 b(\(one)i(should)c(think)h(to)i Fo(')26 b Fp(=)f Fo(\037)2881 4773 y Fh(\012)2936 4759 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )s Fp(\).)42 b(Th)m(us)30 4967 y Fo(B)99 4982 y Fm( )151 4967 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))84 b(=)f Fo(B)712 4981 y Fm(')762 4967 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))21 b(+)f Fo(B)1197 4981 y Fm(\037)1245 4967 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))21 b(+)1631 4906 y(2)p 1621 4946 66 4 v 1621 5030 a Fo(T)1712 4967 y Fp(Re)1834 4844 y Fj(Z)1925 4870 y Fl(1)1884 5050 y Fh(0)2049 4881 y Fj(X)2015 5082 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)2215 4967 y Fn(h)p Fp(e)2290 4930 y Fl(\000)p Fm(itH)2462 4967 y Fo(';)15 b(e)2603 4981 y Fm(n)2651 4967 y Fn(i)p 2686 4885 470 4 v(h)p Fp(e)2761 4941 y Fl(\000)p Fm(itH)2934 4967 y Fo(\037;)g(e)3073 4981 y Fm(n)3121 4967 y Fn(i)p Fo(h)p Fp(\()p Fo(t=T)e Fp(\)d)p Fo(t;)489 5267 y Fp(=)83 b Fn(\000)p Fo(B)783 5281 y Fm(')833 5267 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))21 b(+)f Fo(B)1268 5281 y Fm(\037)1316 5267 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))20 b(+)1701 5206 y(2)p 1691 5246 66 4 v 1691 5330 a Fo(T)1782 5267 y Fp(Re)1905 5144 y Fj(Z)1996 5170 y Fl(1)1955 5350 y Fh(0)2120 5181 y Fj(X)2085 5382 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)2285 5267 y Fn(h)p Fp(e)2360 5230 y Fl(\000)p Fm(itH)2533 5267 y Fo(';)15 b(e)2674 5281 y Fm(n)2722 5267 y Fn(i)p 2757 5185 475 4 v(h)p Fp(e)2832 5241 y Fl(\000)p Fm(itH)3005 5267 y Fo( )s(;)g(e)3149 5281 y Fm(n)3197 5267 y Fn(i)p Fo(h)p Fp(\()p Fo(t=T)e Fp(\)d)p Fo(t:)1773 5611 y Fp(10)p eop %%Page: 11 11 11 10 bop -24 375 a Fp(Then,)42 b(taking)e(in)m(to)g(accoun)m(t)h(that) f(1)p Fo(=T)1458 302 y Fj(R)1519 328 y Fl(1)1501 407 y Fh(0)1609 375 y Fo(h)p Fp(\()p Fo(t=T)13 b Fp(\))p Fo(dt)42 b Fp(=)f(1)f(and)f Fo(h)p Fp(\()p Fo(z)t Fp(\))k Fn(\025)e Fp(0,)h(w)m(e)f(ha)m(v)m(e)g Fo(B)3247 389 y Fm(\037)3295 375 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))42 b Fn(\024)-24 488 y(k)p Fo(\037)p Fn(k)123 455 y Fh(2)189 488 y Fp(=)25 b Fn(k)p Fo( )s Fn(k)437 455 y Fh(2)498 488 y Fn(\000)20 b(k)p Fo(')p Fn(k)738 455 y Fh(2)779 488 y Fp(.)40 b(Let)31 b Fo(A)26 b Fp(=)1196 420 y Fj(P)1292 515 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)1497 488 y Fn(h\001)p Fo(;)15 b(e)1639 502 y Fm(n)1687 488 y Fn(i)p Fo(e)1764 502 y Fm(n)1812 488 y Fp(.)40 b(Then)955 729 y Fo(B)1024 744 y Fm( )1076 729 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))26 b Fn(\024)f(k)p Fo( )s Fn(k)1604 691 y Fh(2)1665 729 y Fn(\000)20 b(k)p Fo(')p Fn(k)1905 691 y Fh(2)1966 729 y Fp(+)g(2Re)q Fo(D)2288 681 y Fh(\()p Fm(h)p Fh(\))2285 758 y Fm('; )2403 729 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))p Fo(;)-24 968 y Fp(where)36 b Fo(D)323 920 y Fh(\()p Fm(h)p Fh(\))320 998 y Fm('; )439 968 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))37 b(w)m(as)g(de\014ned)e(in)g(the)h (Theorem)g(3.2.)60 b(The)36 b(second)h(statemen)m(t)h(of)e(this)g (Theorem)-24 1081 y(giv)m(es)31 b(immediately)895 1245 y Fo(B)964 1260 y Fm( )1016 1245 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))26 b Fn(\024)f(k)p Fo( )s Fn(k)1544 1207 y Fh(2)1605 1245 y Fn(\000)20 b(k)p Fo(')p Fn(k)1845 1207 y Fh(2)1906 1245 y Fp(+)g Fo(C)7 b(N)2162 1180 y Fi(d)p 2162 1192 33 3 v 2163 1233 a Fe(2)2223 1144 y Fj(\020)2277 1245 y Fo(U)2349 1197 y Fh(\()p Fm(h)p Fh(\))2339 1274 y Fm('; )2457 1245 y Fp(\()p Fo(T)13 b Fp(\))2593 1144 y Fj(\021)2658 1139 y Fe(1)p 2658 1151 31 3 v 2658 1192 a(2)2718 1245 y Fo(;)757 b Fp(\(22\))-24 1429 y(where,)31 b(as)f(in)f(Theorem)h(3.2,)912 1670 y Fo(U)984 1622 y Fh(\()p Fm(h)p Fh(\))974 1700 y Fm('; )1092 1670 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1350 1546 y Fj(Z)1401 1753 y Fk(R)1468 1546 y Fj(Z)1518 1753 y Fk(R)1586 1670 y Fp(d)p Fo(\026)1692 1684 y Fm(')1741 1670 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)1969 1685 y Fm( )2022 1670 y Fp(\()p Fo(y)s Fp(\))p Fn(j)2166 1646 y Fp(^)2165 1670 y Fo(h)q Fp(\(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\))p Fo(T)13 b Fp(\))p Fn(j)2661 1633 y Fh(2)2701 1670 y Fo(:)-24 1939 y Fp(As)30 b Fn(j)136 1915 y Fp(^)135 1939 y Fo(h)q Fp(\()p Fo(w)r Fp(\))p Fn(j)c(\024)f Fp(1)31 b(for)e(all)g Fo(w)j Fp(and)d(b)m(y)h(de\014nition)d(\(11\))k(of)f Fo(R)q Fp(,)g(w)m(e)g(clearly)f(ha)m(v)m(e)i Fo(R)q Fp(\()p Fo(w)r Fp(\))26 b Fn(\025)f(j)3020 1915 y Fp(^)3019 1939 y Fo(h)q Fp(\()p Fo(w)r Fp(\))p Fn(j)3234 1906 y Fh(2)3305 1939 y Fp(for)k(all)g Fo(w)r Fp(.)-24 2068 y(Therefore)36 b Fo(U)466 2020 y Fh(\()p Fm(h)p Fh(\))456 2097 y Fm('; )574 2068 y Fp(\()p Fo(T)13 b Fp(\))35 b Fn(\024)e Fo(U)911 2083 y Fm('; )1030 2068 y Fp(\()p Fo(T)13 b Fp(\),)37 b(and)e(the)h(estimate)g(\(22\))h(is)e(v)-5 b(alid)34 b(with)g Fo(U)2734 2083 y Fm('; )2853 2068 y Fp(\()p Fo(T)13 b Fp(\))36 b(de\014ned)e(b)m(y)i(\(10\),)-24 2206 y(that)31 b(is)f(with)f(the)h(function)f Fo(R)i Fp(instead)f(of)h Fn(j)1528 2182 y Fp(^)1527 2206 y Fo(h)p Fn(j)1604 2173 y Fh(2)1644 2206 y Fp(.)118 2319 y(W)-8 b(e)27 b(are)g(no)m(w)g(in)f(p)s(osition)e(to)k(\014nish)c(the)j(pro)s (of)f(of)h(Theorem)f(3.1.)41 b(The)26 b(basic)g(strategy)i(is)e (standard:)-24 2432 y(let)31 b Fo(N)40 b Fp(b)s(e)30 b(the)g(largest)h(in)m(teger)g(suc)m(h)f(that)h Fo(C)7 b(N)1653 2399 y Fm(d=)p Fh(2)1763 2432 y Fo(U)1825 2447 y Fm('; )1943 2432 y Fp(\()p Fo(T)13 b Fp(\))2089 2369 y Fe(1)p 2090 2381 V 2090 2422 a(2)2160 2432 y Fn(\024)25 b(k)p Fo(')p Fn(k)2405 2399 y Fh(2)2445 2432 y Fo(=)p Fp(2,)32 b(it)e(yields:)1301 2696 y Fo(B)1370 2711 y Fm( )1422 2696 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))26 b Fn(\024)f(k)p Fo( )s Fn(k)1950 2659 y Fh(2)2011 2696 y Fn(\000)2112 2635 y(k)p Fo(')p Fn(k)2261 2602 y Fh(2)p 2112 2675 190 4 v 2184 2759 a Fp(2)2312 2696 y Fo(:)1163 b Fp(\(23\))-24 2926 y(The)30 b(inequalities)e(\(23\))k(and)e(\(21\))i (yield)c(with)h(some)i(p)s(ositiv)m(e)f(constan)m(t)h Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\):)528 3189 y Fn(hhj)p Fo(X)7 b Fn(j)730 3152 y Fm(p)771 3189 y Fn(ii)841 3204 y Fm( )r(;T)990 3189 y Fn(\025)25 b Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\))1500 3128 y Fn(k)p Fo(')p Fn(k)1649 3095 y Fh(2+4)p Fm(\014)p 1478 3169 364 4 v 1478 3252 a Fo(U)1540 3267 y Fm('; )1660 3252 y Fp(\()p Fo(T)e Fp(\))1796 3226 y Fm(\014)1879 3189 y Fp(=)25 b Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p;)g(h)p Fp(\))p Fo(L)2419 3204 y Fm( )2473 3189 y Fp(\()p Fo(';)g(T)e Fp(\))p Fo(;)107 b(\014)31 b Fp(=)3029 3128 y Fo(p)p 3028 3169 48 4 v 3028 3252 a(d)3085 3189 y(:)390 b Fp(\(24\))-24 3447 y(One)28 b(reco)m(v)m(ers)i Fo(L)573 3462 y Fm( )626 3447 y Fp(\()p Fo(';)15 b(T)e Fp(\))29 b(as)g(giv)m(en)g(Line)e(\(9\).) 41 b(It)29 b(is)e(this)g(latter)i(lo)m(w)m(er)g(b)s(ound)d(that)j(will) d(b)s(e)h(used)h(in)f(the)-24 3560 y(pro)s(of)k(of)h(the)g(lo)m(w)m(er) g(b)s(ound)d(in)i(Theorem)g(4.1.)46 b(Ho)m(w)m(ev)m(er,)34 b(in)c(the)i(more)g(general)f(case)i(where)e Fo(')h Fp(is)f(an)m(y)-24 3673 y(function)e(of)i Fn(H)g Fp(with)e Fn(h)p Fo( )s(;)15 b(')p Fn(i)27 b(6)p Fp(=)e(0,)31 b(one)g(gets)g(the)g(b)s(ound)d(with) 1283 3936 y Fo(L)1345 3951 y Fm( )1398 3936 y Fp(\()p Fo(';)15 b(T)e Fp(\))27 b Fn(\021)1814 3875 y(jh)p Fo(';)15 b( )s Fn(ij)2095 3842 y Fh(2+2)p Fm(\014)p 1765 3915 554 4 v 1765 3999 a Fn(k)p Fo(')p Fn(k)1914 3972 y Fh(2)1955 3999 y Fo(U)2017 4014 y Fm('; )2136 3999 y Fp(\()p Fo(T)e Fp(\))2272 3972 y Fm(\014)2329 3936 y Fo(:)1146 b Fp(\(25\))-24 4202 y(Indeed)36 b(tak)m(e)i(suc)m(h)e(a)g Fo(')p Fp(,)j(and)d(then)g (de\014ne)f(as)i(in)e([5])52 b(~)-60 b Fo(')36 b Fp(=)e(\()p Fn(h)p Fo(';)15 b( )s Fn(ik)p Fo(')p Fn(k)2537 4169 y Fl(\000)p Fh(2)2636 4202 y Fp(\))p Fo(')p Fp(.)59 b(So)36 b(one)h(c)m(hec)m(ks)g(that)g(if)-24 4315 y Fo(\037)31 b Fp(=)g Fo( )26 b Fn(\000)36 b Fp(~)-59 b Fo(')p Fp(,)35 b(then)f Fn(h)15 b Fp(~)-60 b Fo(';)15 b(\037)p Fn(i)32 b Fp(=)f(0)j(and)f(one)h(is)f(able)g(to)i(apply)d(the)i(result)f(line)f (\(9\))j(to)f Fo( )j Fp(and)48 b(~)-60 b Fo(')q Fp(.)51 b(T)-8 b(aking)-24 4427 y(in)m(to)33 b(accoun)m(t)h(that)f Fn(k)15 b Fp(~)-60 b Fo(')q Fn(k)29 b Fp(=)g Fn(jh)p Fo(';)15 b( )s Fn(ijk)p Fo(')p Fn(k)1409 4394 y Fl(\000)p Fh(1)1539 4427 y Fp(and)32 b(that)h Fo(U)1990 4442 y Fh(~)-46 b Fm('; )2098 4427 y Fp(\()p Fo(T)13 b Fp(\))29 b(=)g Fn(jh)p Fo(';)15 b( )s Fn(ij)2644 4394 y Fh(2)2686 4427 y Fn(k)p Fo(')p Fn(k)2835 4394 y Fl(\000)p Fh(4)2930 4427 y Fo(U)2992 4442 y Fm('; )3111 4427 y Fp(\()p Fo(T)e Fp(\),)33 b(one)g(\014nds)-24 4540 y(the)f(announced)f(expression)g (\(25\).)47 b(T)-8 b(o)32 b(optimise)f(the)h(lo)m(w)m(er)g(b)s(ound,)e (w)m(e)j(should)d(tak)m(e)j(the)f(suprem)m(um)-24 4653 y(of)37 b Fo(L)148 4668 y Fm( )200 4653 y Fp(\()p Fo(';)15 b(T)e Fp(\))38 b(for)e(a)g(giv)m(en)g Fo(T)49 b Fp(o)m(v)m(er)38 b(all)d(p)s(ossible)f Fo(')p Fp(.)58 b(One)36 b(can)h(sho)m(w)f(in)f (the)h(same)h(manner)e(as)h(in)f([5)q(],)-24 4766 y(Lemma)i(3.1,)i (that)e(it)f(is)g(su\016cien)m(t)g(to)h(tak)m(e)h Fo(')f Fp(only)f(from)g(the)g(cyclic)g(subspace)g Fn(H)2996 4781 y Fm( )3049 4766 y Fp(.)59 b(This)34 b(giv)m(es)j(us)-24 4879 y Fo(L)38 4894 y Fm( )91 4879 y Fp(\()p Fo(T)13 b Fp(\))31 b(de\014ned)e(line)g(\(8\).)1773 5611 y(11)p eop %%Page: 12 12 12 11 bop -24 375 a Fq(4)135 b(T)-11 b(o)l(w)l(ards)45 b(the)g(fractal)i(dimensions)-24 578 y Fp(This)36 b(section)h(deals)f (with)g(the)h(connection)g(b)s(et)m(w)m(een)g(the)h(dynamic)d(quan)m (tit)m(y)i Fo(L)2898 593 y Fm( )2951 578 y Fp(\()p Fo(T)13 b Fp(\))37 b(in)m(tro)s(duced)f(in)-24 691 y(the)f(previous)f(section,) 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378 4 v 602 1669 a Fn(j)15 b Fp(log)i Fo(")p Fn(j)842 1642 y Fh(1+)p Fm(\014)990 1606 y Fo(K)1067 1620 y Fm(\026)1109 1632 y Fi( )1175 1478 y Fj(\022)1336 1545 y Fp(1)p 1252 1585 213 4 v 1252 1669 a(1)k(+)f Fo(\014)1475 1606 y(;)15 b(")1557 1478 y Fj(\023)1625 1501 y Fh(1+)p Fm(\014)1787 1606 y Fn(\024)25 b Fo(L)1945 1621 y Fm( )1998 1606 y Fp(\()p Fo(")2075 1569 y Fl(\000)p Fh(1)2170 1606 y Fp(\))g Fn(\024)g Fo(K)2403 1620 y Fm(\026)2445 1632 y Fi( )2512 1478 y Fj(\022)2673 1545 y Fp(1)p 2589 1585 V 2589 1669 a(1)c(+)e Fo(\014)2812 1606 y(;)c(")2894 1478 y Fj(\023)2961 1501 y Fh(1+)p Fm(\014)3114 1606 y Fo(;)361 b Fp(\(26\))-24 1836 y Ff(wher)-5 b(e)534 2015 y Fo(K)611 2029 y Fm(\026)653 2041 y Fi( )720 1887 y Fj(\022)880 1953 y Fp(1)p 796 1994 V 796 2077 a(1)21 b(+)f Fo(\014)1019 2015 y(;)15 b(")1101 1887 y Fj(\023)1194 2015 y Fp(=)1290 1891 y Fj(Z)1341 2097 y Fh(supp)p Fm(\026)1528 2109 y Fi( )1595 2015 y Fp(d)p Fo(\026)1701 2030 y Fm( )1753 2015 y Fp(\()p Fo(x)p Fp(\))1890 1887 y Fj(\022)1957 1891 y(Z)2008 2097 y Fk(R)2075 2015 y Fo(d\026)2177 2030 y Fm( )2230 2015 y Fp(\()p Fo(y)s Fp(\))p Fo(R)q Fp(\(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\))p Fo(=")p Fp(\))2857 1887 y Fj(\023)2950 1879 y Fb(\000)p Fi(\014)p 2935 1894 115 3 v 2935 1935 a Fe(1+)p Fi(\014)3079 2015 y Fo(:)-24 2249 y Fp(This)29 b(will)f(actually)i(imply)-24 2420 y Fd(Theorem)k(4.2.)46 b Ff(Under)33 b(the)g(same)h(hyp)-5 b(othesis)35 b(\(H1\))e(as)g(pr)-5 b(eviously,)34 b(one)f(has)313 2654 y Fp(lim)15 b(inf)343 2715 y Fm(T)10 b Fl(!1)590 2591 y Fp(log)17 b Fo(L)785 2606 y Fm( )837 2591 y Fp(\()p Fo(T)c Fp(\))p 590 2634 384 4 v 683 2717 a(log)j Fo(T)1009 2654 y Fp(=)25 b Fo(\014)5 b(D)1239 2617 y Fl(\000)1236 2677 y Fm(\026)1278 2689 y Fi( )1345 2526 y Fj(\022)1505 2593 y Fp(1)p 1422 2634 213 4 v 1422 2717 a(1)20 b(+)g Fo(\014)1644 2526 y Fj(\023)1726 2654 y Fo(;)109 b Fp(lim)15 b(sup)1903 2733 y Fm(T)10 b Fl(!1)2163 2591 y Fp(log)17 b Fo(L)2358 2606 y Fm( )2410 2591 y Fp(\()p Fo(T)c Fp(\))p 2163 2634 384 4 v 2256 2717 a(log)j Fo(T)2582 2654 y Fp(=)25 b Fo(\014)5 b(D)2812 2617 y Fh(+)2809 2677 y Fm(\026)2851 2689 y Fi( )2918 2526 y Fj(\022)3078 2593 y Fp(1)p 2995 2634 213 4 v 2995 2717 a(1)20 b(+)g Fo(\014)3218 2526 y Fj(\023)3300 2654 y Fo(:)-24 2885 y Fd(Remark)35 b(4.1.)-24 2998 y Fp(i\))45 b(W)-8 b(e)46 b(p)s(oin)m(t)e(out)h(that)g(the)g(left)g(inequalit)m(y)e (in)g(\(26\),)51 b(that)45 b(is)f(the)h(lo)m(w)m(er)g(b)s(ound)e(on)i Fo(L)3267 3013 y Fm( )3319 2998 y Fp(\()p Fo(")3396 2965 y Fl(\000)p Fh(1)3491 2998 y Fp(\),)k(is)-24 3111 y(su\016cien)m(t)34 b(to)g(pro)m(v)m(e)h(our)e(main)g(result)g(Theorem)g(2.1.)52 b(Ho)m(w)m(ev)m(er)36 b(the)e(righ)m(t)g(part)f(of)h(\(26\),)j(that)d (is)f(the)-24 3224 y(upp)s(er)f(b)s(ound,)g(is)g(of)i(in)m(terest)f(to) s(o.)50 b(In)33 b(particular)e(it)i(sa)m(ys)h(that)g(once)g(one)f (deriv)m(ed)f(the)i(lo)m(w)m(er)f(b)s(ound)-24 3337 y Fo(L)38 3352 y Fm( )91 3337 y Fp(\()p Fo(T)13 b Fp(\),)31 b(one)g(cannot)g(hop)s(e)e(a)i(b)s(etter)f(result)g(than)g(the)g(one)h (w)m(e)g(stated.)-24 3449 y(ii\))38 b(F)-8 b(or)39 b(clarit)m(y)-8 b(,)41 b(the)e(ab)s(o)m(v)m(e)h(t)m(w)m(o)f(theorems)g(are)g(stated)h (under)d(the)h(v)m(ery)h(simple)e(assumption)g(\(H1\),)-24 3562 y(namely)32 b(the)g(b)s(oundedness)e(of)i(the)h(supp)s(ort)d(of)j (the)f(sp)s(ectral)f(measure)i Fo(\026)2609 3577 y Fm( )2661 3562 y Fp(.)46 b(Since)31 b(the)i(pro)s(of)e(of)i(The-)-24 3675 y(orems)f(4.1)h(and)62 b(4.2)33 b(are)f(based)f(on)h(the)f(tec)m (hnical)h(but)f(cen)m(tral)h(Lemma)g(4.1,)h(it)e(is)g(a)h(straigh)m (tforw)m(ard)-24 3788 y(consequence)k(of)f(App)s(endix)d(C)i(that)i (Theorem)e(4.1)i(and)e(Theorem)h(4.2)g(remain)f(true)h(if)e(one)j (replaces)-24 3901 y(assumption)23 b(\(H1\))i(b)m(y)f(the)g(w)m(eak)m (er)i(assumption)c(\(H2\))j(of)f(Prop)s(osition)e(5.1)k(or)e(\(H3\))h (of)f(Prop)s(osition)e(5.2.)-24 4014 y(See)31 b(also)f(Remark)h(2.3)g (P)m(oin)m(t)g(ii\).)-24 4127 y(iii\))e(Note)k(also)e(that)g(the)g (righ)m(t-hand-side)f(inequalit)m(y)f(in)h(\(26\),)j(that)e(is)f(the)h (upp)s(er)e(b)s(ound,)h(is)g(true)h(for)-24 4240 y(an)m(y)g(measures)f Fo(\026)593 4255 y Fm( )646 4240 y Fp(,)g(not)h(only)e(those)i(with)e (b)s(ounded)g(supp)s(ort.)118 4411 y(W)-8 b(e)38 b(start)g(b)m(y)g(pro) m(viding)d(a)j(pro)s(of)f(for)g(Theorem)g(4.1.)64 b(Throughout)36 b(this)h(section,)i(ev)m(en)g(if)d(there)-24 4524 y(is)d(no)g(men)m (tion,)h(in)m(tegrations)f(m)m(ust)g(b)s(e)f(understo)s(o)s(d)g(on)h (the)g(supp)s(ort)f(of)h(the)h(measure)f Fo(\026)3220 4539 y Fm( )3306 4524 y Fp(or)g Fo(\026)3475 4538 y Fm(')3558 4524 y Fp(w)m(e)-24 4637 y(consider.)-24 4793 y Fd(Pro)s(of)j(of)f(the) g(upp)s(er)g(b)s(ound)g(in)g(Theorem)f(4.1.)118 4949 y Fp(The)43 b(upp)s(er)g(b)s(ound)f(of)i(\(26\))i(in)d(Theorem)h(4.1)i (will)c(b)s(e)h(pro)m(v)m(ed,)49 b(if)43 b(taking)h(an)m(y)h(function)e Fo(f)58 b Fn(2)-24 5062 y Fp(L)33 5029 y Fh(2)73 5062 y Fp(\()p Fg(R)s Fo(;)15 b Fp(d)p Fo(\026)314 5077 y Fm( )372 5062 y Fp(\),)31 b(one)g(sho)m(ws)400 5322 y Fo(L)462 5337 y Fm( )514 5322 y Fp(\()p Fo(f)10 b Fp(\()p Fo(H)d Fp(\))p Fo( )s(;)15 b(")901 5284 y Fl(\000)p Fh(1)997 5322 y Fp(\))26 b Fn(\024)f Fo(K)1231 5336 y Fm(\026)1273 5348 y Fi( )1340 5193 y Fj(\022)1500 5260 y Fp(1)p 1416 5301 V 1416 5384 a(1)c(+)f Fo(\014)1639 5322 y(;)15 b(")1721 5193 y Fj(\023)1789 5216 y Fh(1+)p Fm(\014)1951 5322 y Fp(=)2047 5193 y Fj(\022)2114 5198 y(Z)2220 5322 y Fo(d\026)2322 5337 y Fm( )2375 5322 y Fp(\()p Fo(x)p Fp(\))p Fo(b)2536 5284 y Fh(\()p Fm(R)p Fh(\))2649 5322 y Fp(\()p Fo(x;)g(")p Fp(\))2879 5245 y Fb(\000)p Fi(\014)p 2865 5261 115 3 v 2865 5302 a Fe(1+)p Fi(\014)2993 5193 y Fj(\023)3060 5216 y Fh(1+)p Fm(\014)3213 5322 y Fo(;)262 b Fp(\(27\))1773 5611 y(12)p eop %%Page: 13 13 13 12 bop -24 375 a Fp(where)38 b Fo(b)286 342 y Fh(\()p Fm(R)p Fh(\))398 375 y Fp(\()p Fo(x;)15 b(")p Fp(\))39 b(=)749 302 y Fj(R)825 375 y Fp(d)p Fo(\026)931 390 y Fm( )983 375 y Fp(\()p Fo(y)s Fp(\))p Fo(R)q Fp(\(\()p Fo(x)26 b Fn(\000)f Fo(y)s Fp(\))p Fo(=")p Fp(\))p Fo(:)39 b Fp(This)d(result)g(will)f(follo)m(w)i(from)h(Cauc)m(h)m(y-Sc)m(h)m(w) m(arz)h(and)-24 488 y(H\177)-45 b(older)30 b(Inequalities.)39 b(Pic)m(k)30 b Fo(')c Fp(=)f Fo(f)10 b Fp(\()p Fo(H)d Fp(\))p Fo( )s Fp(,)31 b(with)e Fo(f)34 b Fn(2)25 b Fp(L)1932 455 y Fh(2)1971 488 y Fp(\()p Fg(R)s Fo(;)16 b Fp(d)p Fo(\026)2213 503 y Fm( )2271 488 y Fp(\).)41 b(Therefore)770 739 y(d)p Fo(\026)876 753 y Fm(')926 739 y Fp(\()p Fo(x)p Fp(\))26 b(=)f Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)1397 701 y Fh(2)1437 739 y Fp(d)p Fo(\026)1543 754 y Fm( )1595 739 y Fp(\()p Fo(x)p Fp(\))30 b(and)g Fn(h)p Fo(';)15 b( )s Fn(i)28 b Fp(=)2279 615 y Fj(Z)2385 739 y Fp(d)p Fo(\026)2491 754 y Fm( )2543 739 y Fp(\()p Fo(x)p Fp(\))p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fo(:)-24 1000 y Fp(Remem)m(b)s(er)26 b(also)f(that)h Fo(U)859 1015 y Fm('; )978 1000 y Fp(\()p Fo(T)13 b Fp(\))25 b(=)1235 926 y Fj(R)1311 1000 y Fp(d)p Fo(\026)1417 1014 y Fm(')1467 1000 y Fp(\()p Fo(x)p Fp(\))p Fo(b)1628 967 y Fh(\()p Fm(R)p Fh(\))1741 1000 y Fp(\()p Fo(x;)15 b(")p Fp(\).)40 b(Then,)26 b(rewriting)e Fo(L)2716 1015 y Fm( )2768 1000 y Fp(\(\()p Fo(f)10 b Fp(\()p Fo(H)d Fp(\))p Fo( )s(;)15 b(")3190 967 y Fl(\000)p Fh(1)3287 1000 y Fp(\))25 b(as)h(giv)m(en)-24 1113 y(in)j(\(8\))j(one)f(gets)g(:)506 1399 y Fo(L)568 1414 y Fm( )621 1399 y Fp(\()p Fo(f)10 b Fp(\()p Fo(H)d Fp(\))p Fo( )s(;)15 b(")1008 1362 y Fl(\000)p Fh(1)1104 1399 y Fp(\))25 b(=)1802 1255 y Fj(\014)1802 1309 y(\014)1832 1259 y(R)1908 1332 y Fo(d\026)2010 1347 y Fm( )2062 1332 y Fp(\()p Fo(x)p Fp(\))p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))2361 1255 y Fj(\014)2361 1309 y(\014)2392 1281 y Fh(2+2)p Fm(\014)p 1270 1378 1827 4 v 1270 1413 a Fj(\000R)1388 1486 y Fo(d\026)1490 1501 y Fm( )1542 1486 y Fp(\()p Fo(x)p Fp(\))p Fn(j)p Fo(f)g Fp(\()p Fo(x)p Fp(\))p Fn(j)1891 1460 y Fh(2)1932 1413 y Fj(\001)15 b(\000)o(R)2106 1486 y Fo(d\026)2208 1501 y Fm( )2261 1486 y Fp(\()p Fo(x)p Fp(\))p Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)2610 1460 y Fh(2)2650 1486 y Fo(b)2689 1460 y Fh(\()p Fm(R)p Fh(\))2802 1486 y Fp(\()p Fo(x;)15 b(")p Fp(\))3006 1413 y Fj(\001)3049 1435 y Fm(\014)3106 1399 y Fo(:)369 b Fp(\(28\))-24 1681 y(One)30 b(starts)h(with)e(a)h(Cauc)m(h)m(y-Sc)m(h)m (w)m(arz)i(inequalit)m(y)d(applied)f(to)j(the)f(n)m(umerator,)h(and)e (to)i(the)g(functions)-24 1821 y Fo(b)15 1788 y Fh(\()p Fm(R)p Fh(\))128 1821 y Fp(\()p Fo(x;)15 b(")p Fp(\))397 1741 y Fb(\000)p Fi(\014)p 343 1757 193 3 v 343 1798 a Fe(2\(1+)p Fi(\014)s Fe(\))581 1821 y Fp(and)29 b Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fo(b)973 1788 y Fh(\()p Fm(R)p Fh(\))1086 1821 y Fp(\()p Fo(x;)15 b(")p Fp(\))1379 1741 y Fi(\014)p 1301 1757 V 1301 1798 a Fe(2\(1+)p Fi(\014)s Fe(\))1508 1821 y Fp(.)41 b(It)30 b(yields:)43 1963 y Fj(\014)43 2017 y(\014)43 2072 y(\014)43 2126 y(\014)73 1971 y(Z)179 2095 y Fo(d\026)281 2110 y Fm( )334 2095 y Fp(\()p Fo(x)p Fp(\))p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))633 1963 y Fj(\014)633 2017 y(\014)633 2072 y(\014)633 2126 y(\014)664 1989 y Fh(2+2)p Fm(\014)889 2095 y Fn(\024)1013 1966 y Fj(\022)1080 1971 y(Z)1186 2095 y Fo(d\026)1288 2110 y Fm( )1340 2095 y Fp(\()p Fo(x)p Fp(\))p Fo(b)1501 2057 y Fh(\()p Fm(R)p Fh(\))1615 2095 y Fp(\()p Fo(x;)15 b(")p Fp(\))1845 2018 y Fb(\000)p Fi(\014)p 1830 2034 115 3 v 1830 2075 a Fe(1+)p Fi(\014)1959 1966 y Fj(\023)2025 1989 y Fh(1+)p Fm(\014)2163 1966 y Fj(\022)2230 1971 y(Z)2336 2095 y Fo(d\026)2438 2110 y Fm( )2491 2095 y Fp(\()p Fo(x)p Fp(\))p Fo(b)2652 2057 y Fh(\()p Fm(R)p Fh(\))2765 2095 y Fp(\()p Fo(x;)g(")p Fp(\))3019 2018 y Fi(\014)p 2980 2034 V 2980 2075 a Fe(1+)p Fi(\014)3109 2095 y Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)3336 2057 y Fh(2)3376 1966 y Fj(\023)3442 1989 y Fh(1+)p Fm(\014)3500 2095 y Fp(\(29\))889 2366 y(=)53 b Fo(K)1090 2380 y Fm(\026)1132 2392 y Fi( )1198 2238 y Fj(\022)1359 2304 y Fp(1)p 1275 2345 213 4 v 1275 2428 a(1)21 b(+)f Fo(\014)1498 2366 y(;)15 b(")1580 2238 y Fj(\023)1648 2260 y Fh(1+)p Fm(\014)1800 2238 y Fj(\022)1867 2242 y(Z)1973 2366 y Fo(d\026)2075 2381 y Fm( )2128 2366 y Fp(\()p Fo(x)p Fp(\))p Fo(b)2289 2328 y Fh(\()p Fm(R)p Fh(\))2402 2366 y Fp(\()p Fo(x;)g(")p Fp(\))2656 2289 y Fi(\014)p 2618 2305 115 3 v 2618 2346 a Fe(1+)p Fi(\014)2746 2366 y Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)2973 2328 y Fh(2)3013 2238 y Fj(\023)3080 2260 y Fh(1+)p Fm(\014)3232 2366 y Fo(:)-24 2615 y Fp(An)45 b(H\177)-45 b(older)44 b(inequalit)m(y)f(applied)g(to)j(the)f(last)g (term,)k(and)44 b(with)g(the)h(co)s(e\016cien)m(ts)g Fo(p)50 b Fp(=)f(1)30 b(+)g Fo(\014)50 b Fp(and)-24 2728 y Fo(p)22 2695 y Fl(0)71 2728 y Fp(=)25 b(\(1)c(+)f Fo(\014)5 b Fp(\))p Fo(=\014)g Fp(,)31 b(leads)f(to:)685 2874 y Fj(\022)751 2878 y(Z)858 3002 y Fo(d\026)960 3017 y Fm( )1012 3002 y Fp(\()p Fo(x)p Fp(\))p Fo(b)1173 2964 y Fh(\()p Fm(R)p Fh(\))1287 3002 y Fp(\()p Fo(x;)15 b(")p Fp(\))1540 2926 y Fi(\014)p 1502 2941 V 1502 2982 a Fe(1+)p Fi(\014)1630 3002 y Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)1857 2964 y Fh(2)1897 2874 y Fj(\023)1964 2896 y Fh(1+)p Fm(\014)768 3285 y Fp(=)921 3157 y Fj(\022)988 3161 y(Z)1094 3285 y Fo(d\026)1196 3300 y Fm( )1249 3285 y Fp(\()p Fo(x)p Fp(\)\()p Fn(j)p Fo(f)g Fp(\()p Fo(x)p Fp(\))p Fn(j)1633 3247 y Fh(2)1674 3285 y Fp(\))1761 3212 y Fe(1)p 1719 3224 V 1719 3265 a(1+)p Fi(\014)1863 3184 y Fj(\020)1917 3285 y Fn(j)p Fo(f)g Fp(\()p Fo(x)p Fp(\))p Fn(j)2144 3247 y Fh(2)2184 3285 y Fo(b)2223 3247 y Fh(\()p Fm(R)p Fh(\))2336 3285 y Fp(\()p Fo(x;)15 b(")p Fp(\))2540 3184 y Fj(\021)2644 3176 y Fi(\014)p 2605 3191 V 2605 3233 a Fe(1+)p Fi(\014)2734 3157 y Fj(\023)2801 3168 y Fh(1+)p Fm(\014)768 3556 y Fn(\024)921 3428 y Fj(\022)988 3432 y(Z)1094 3556 y Fo(d\026)1196 3571 y Fm( )1249 3556 y Fp(\()p Fo(x)p Fp(\))p Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)1598 3519 y Fh(2)1639 3428 y Fj(\023)15 b(\022)1788 3432 y(Z)1894 3556 y Fo(d\026)1996 3571 y Fm( )2048 3556 y Fp(\()p Fo(x)p Fp(\))p Fn(j)p Fo(f)10 b Fp(\()p Fo(x)p Fp(\))p Fn(j)2397 3519 y Fh(2)2438 3556 y Fo(b)2477 3519 y Fh(\()p Fm(R)p Fh(\))2590 3556 y Fp(\()p Fo(x;)15 b(")p Fp(\))2794 3428 y Fj(\023)2862 3451 y Fm(\014)2924 3556 y Fo(:)551 b Fp(\(30\))-24 3810 y(One)30 b(th)m(us)g(reco)m(v)m(ers)i(exactly)f(the)g(denominator)f(term,)g(and) g(\(27\))i(holds.)1030 b Fc(\003)-24 4048 y Fd(Remark)35 b(4.2.)-24 4161 y Fp(i\))44 b(W)-8 b(e)46 b(stress)e(out)g(the)h(v)m (ery)f(natural)g(link)e(that)j(comes)g(out)f(b)s(et)m(w)m(een)h Fo(L)2695 4176 y Fm( )2748 4161 y Fp(\()p Fo(")2825 4128 y Fl(\000)p Fh(1)2920 4161 y Fp(\))g(and)e(the)i(in)m(tegral)-24 4274 y Fo(K)53 4288 y Fm(\026)95 4300 y Fi( )147 4274 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))44 b(with)e(the)h(particular)f(v)-5 b(alue)43 b Fo(q)49 b Fp(=)d(1)p Fo(=)p Fp(\(1)31 b(+)d Fo(\014)5 b Fp(\).)79 b(The)43 b(same)h(deep)e(link)f(app)s(ears)i(to)h (hold)-24 4386 y(with)36 b(the)h(other)g(lo)m(w)m(er)g(b)s(ounds)d Fo(L)1225 4400 y Fh(1)1265 4386 y Fp(\()p Fo(T)13 b Fp(\))37 b(and)f Fo(L)1683 4400 y Fh(2)1722 4386 y Fp(\()p Fo(T)13 b Fp(\))37 b(de\014ned)f(in)f(App)s(endix)g(B)i(and)f(that)h(come)h (from)-24 4499 y(the)d(former)g(approac)m(hes)g(\(G-C-L)g(and)f(BT\).)i (This)d(sho)m(ws)h(ho)m(w)h(relev)-5 b(an)m(t)35 b(are)g(the)g(fractal) g(dimension)-24 4612 y(n)m(um)m(b)s(ers)c Fo(D)423 4579 y Fl(\006)420 4635 y Fm(\026)482 4612 y Fp(\()p Fo(q)s Fp(\))h(with)f(regards)h(to)h(the)f(time)g(b)s(eha)m(viour)f(of)h(the)g (quan)m(tities)g(that)g(ha)m(v)m(e)i(b)s(een)d(studied)-24 4725 y(for)g(man)m(y)f(y)m(ears)h(in)e(quan)m(tum)h(dynamics.)-24 4838 y(ii\))24 b(W)-8 b(e)26 b(also)e(p)s(oin)m(t)g(out)h(that)g(this)e (is)h(the)h(\014rst)f(time)g(that)h(suc)m(h)f(upp)s(er)f(b)s(ounds)f (are)j(obtained,)h(supplying)-24 4951 y(thereb)m(y)35 b(the)f(theoretic)h(b)s(est)e(result)g(one)i(can)f(hop)s(e)g(while)e (studying)g(the)j(quan)m(tit)m(y)f Fo(L)3078 4966 y Fm( )3131 4951 y Fp(\()p Fo(T)13 b Fp(\))34 b(\(whic)m(h)f(is)-24 5064 y(actually)h(a)g(more)h(re\014ned)e(lo)m(w)m(er)h(b)s(ound)e(than) i(what)g(one)g(used)f(to)i(get)g([2)q(]\).)53 b(It)34 b(is)f(remarquable)g(that)-24 5177 y(one)e(happ)s(ens)e(to)i(pro)m(v)m (e)g(that)g(these)g(theoretic)g(limits)d(are)j(actually)f(reac)m(hed.) 1773 5611 y(13)p eop %%Page: 14 14 14 13 bop 118 375 a Fp(W)-8 b(e)32 b(no)m(w)f(turn)g(to)h(the)f(second) h(and)e(main)h(part)g(of)g(Theorem)h(4.1,)g(that)g(is)f(the)g(lo)m(w)m (er)h(b)s(ound.)42 b(Our)-24 488 y(basic)37 b(strategy)i(to)f(get)g (the)f(lo)m(w)m(er)h(b)s(ound)d(in)h(\(26\))j(is)d(to)i(estimate)g(the) f(quan)m(tit)m(y)h Fo(L)3036 503 y Fm( )3088 488 y Fp(\()p Fo(';)15 b(T)e Fp(\))39 b(in)d(\(25\),)-24 601 y(with)42 b(a)h(v)m(ector)h Fo(')j Fp(=)e Fo(\037)848 619 y Fh(\012\()p Fm(r)r Fh(\))992 601 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )s Fp(.)79 b(And)42 b(\012\()p Fo(r)s Fp(\))g(will)e(b)s(e)i(a)i(\\thin")e (set)h(of)g(the)g(form)f(\012\()p Fo(r)s Fp(\))k(=)g Fn(f)p Fo(x)g Fn(2)-24 727 y Fp(supp)o Fo(\026)219 742 y Fm( )271 727 y Fo(;)e(")382 694 y Fm(r)r Fh(+)p Fm(A=)l(N)651 727 y Fo(<)28 b(b)789 694 y Fh(\()p Fm(R)p Fh(\))901 727 y Fp(\()p Fo(x;)15 b(")p Fp(\))30 b Fn(\024)e Fo(")1276 694 y Fm(r)1314 727 y Fn(g)p Fp(,)34 b(but)d(whic)m(h)g(supp)s(orts,)g (roughly)g(sp)s(eaking,)h(a)g(signi\014can)m(t)f(part)-24 840 y(of)i(the)f(mass)g(of)h(the)f(in)m(tegral)g Fo(K)1139 854 y Fm(\026)1181 866 y Fi( )1232 840 y Fp(\(1)p Fo(=)p Fp(\(1)24 b(+)d Fo(\014)5 b Fp(\))p Fo(;)15 b(")p Fp(\).)48 b(The)32 b(constan)m(t)h Fo(A)f Fp(is)g(\014xed)f(b)m(y)h(Lemma)h(4.1)g (b)s(elo)m(w,)-24 953 y(and)27 b(has)f(the)h(v)-5 b(alue)27 b Fo(A)e Fp(=)g(2\(1)13 b(+)g Fo(\014)5 b Fp(\).)42 b(The)26 b(in)m(teger)h Fo(N)37 b Fp(will)24 b(stand)j(for)f(the)i(in)m(teger)f (part)g(of)g(log)16 b Fo(T)d Fp(.)39 b(Before)-24 1066 y(going)31 b(on)f(with)f(the)i(pro)s(of)e(of)i(Theorem)f(4.1,)i(w)m(e)f (need)f(the)g(t)m(w)m(o)i(follo)m(wing)d(lemmas:)-24 1216 y Fd(Lemma)k(4.1.)46 b Ff(L)-5 b(et)36 b Fo(q)e Fn(2)c Fp(\(0)p Fo(;)15 b Fp(1\))p Ff(,)38 b(and)e(supp)-5 b(ose)38 b(that)f Fo(\026)1906 1231 y Fm( )1993 1216 y Ff(has)g(a)f(b)-5 b(ounde)g(d)37 b(supp)-5 b(ort)38 b(with)e(length)h Fo(D)h Ff(\(i.e.)-24 1329 y(that)g(\(H1\))f(holds)h (for)f Fo(\026)831 1344 y Fm( )884 1329 y Ff(\).)53 b(De\014ne)36 b Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))34 b(=)1672 1256 y Fj(R)1748 1329 y Fp(d)p Fo(\026)1854 1344 y Fm( )1906 1329 y Fp(\()p Fo(x)p Fp(\))p Fo(g)s Fp(\(\()p Fo(x)25 b Fn(\000)e Fo(y)s Fp(\))p Fo(")2440 1296 y Fl(\000)p Fh(1)2535 1329 y Fp(\))p Ff(,)37 b(with)h Fo(g)s Fp(\()p Fo(w)r Fp(\))c(=)e Fo(\037)3214 1348 y Fh([)p Fl(\000)p Fh(1)p Fm(;)p Fh(1])3402 1329 y Fp(\()p Fo(w)r Fp(\))38 b Ff(or)-24 1442 y Fo(R)q Fp(\()p Fo(w)r Fp(\))p Ff(.)43 b(Then)33 b(ther)-5 b(e)34 b(exists)e Fo(A)h Ff(\(actual)5 b(ly)34 b Fo(A)25 b Fp(=)g(2)p Fo(=q)s Ff(\))34 b(such)e(that)1144 1520 y Fj(Z)1194 1726 y Fm(b)p Fh(\()p Fm(x;")p Fh(\))p Fl(\024)p Fm(")1459 1707 y Fi(A)1528 1644 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1771 1606 y Fm(q)r Fl(\000)p Fh(1)1901 1644 y Fp(d)p Fo(\026)2007 1659 y Fm( )2059 1644 y Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)f Fp(2)p Fo(D)s(":)1007 b Fp(\(31\))-24 1871 y Ff(Mor)-5 b(e)g(over,)34 b Fo(D)482 1838 y Fl(\006)479 1894 y Fm(\026)521 1906 y Fi( )598 1871 y Fn(2)25 b Fp([0)p Fo(;)15 b Fp(1])34 b Ff(for)f(any)h Fo(q)28 b Fn(2)d Fp(\(0)p Fo(;)15 b Fp(1\))p Ff(.)-24 2022 y Fd(Lemma)33 b(4.2.)46 b Ff(L)-5 b(et)41 b Fo(q)j Ff(and)e Fo(\026)1045 2037 y Fm( )1138 2022 y Ff(b)-5 b(e)40 b(as)i(in)e(the)i(pr)-5 b(evious)41 b(lemma)h(and)g Fo(A)e Fp(=)g(2)p Fo(=q)s Ff(.)67 b(L)-5 b(et)41 b Fo(N)50 b(>)40 b Fp(0)h Ff(b)-5 b(e)41 b(an)-24 2135 y(inte)-5 b(ger.)42 b(Then,)32 b(for)g Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))33 b Ff(de\014ne)-5 b(d)33 b(as)f(in)g(L)-5 b(emma)33 b(4.1,)f(ther)-5 b(e)33 b(exists)f(an)g Fo(r)2675 2149 y Fh(0)2746 2135 y Ff(and)g(a)g(set)g Fp(\012\()p Fo(r)3282 2149 y Fh(0)3322 2135 y Fp(\))25 b(=)g Fn(f)p Fo(x)h Fn(2)-24 2247 y Fp(supp)o Fo(\026)219 2262 y Fm( )271 2247 y Fo(;)41 b(")379 2214 y Fm(r)411 2223 y Fe(0)446 2214 y Fh(+)p Fm(A=)l(N)677 2247 y Fo(<)25 b(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))27 b Fn(\024)e Fo(")1181 2214 y Fm(r)1213 2223 y Fe(0)1252 2247 y Fn(g)33 b Ff(such)g(that,)g (for)h(al)5 b(l)33 b Fo(")g Ff(smal)5 b(l)33 b(enough)837 2330 y Fj(Z)887 2537 y Fh(\012\()p Fm(r)997 2546 y Fe(0)1032 2537 y Fh(\))1079 2454 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1322 2417 y Fm(q)r Fl(\000)p Fh(1)1451 2454 y Fp(d)p Fo(\026)1557 2469 y Fm( )1609 2454 y Fp(\()p Fo(x)p Fp(\))26 b Fn(\025)1905 2393 y Fp(1)p 1863 2433 129 4 v 1863 2516 a(2)p Fo(N)2017 2330 y Fj(Z)2123 2454 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))2366 2417 y Fm(q)r Fl(\000)p Fh(1)2495 2454 y Fp(d)p Fo(\026)2601 2469 y Fm( )2653 2454 y Fp(\()p Fo(x)p Fp(\))p Fo(:)118 2671 y Fp(Lemma)30 b(4.2)i(is)e(the)h(k)m(ey)g (lemma)g(to)g(get)h(Theorem)f(4.1,)h(since)e(it)g(is)g(this)g(lemma)g (that)h(supplies)d(the)-24 2784 y(set)i(\012\()p Fo(r)s Fp(\),)f(and)g(so)g(the)h(v)m(ector)g Fo(')c Fp(=)f Fo(\037)1302 2803 y Fh(\012\()p Fm(r)r Fh(\))1446 2784 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )s Fp(,)30 b(that)g(is)e(needed)h(to)h(pro)m(v)m(e)g (Theorem)f(4.1.)41 b(Lemma)29 b(4.1)-24 2897 y(will)g(not)i(en)m(ter)h (explicitely)d(in)h(the)h(pro)s(of)f(of)i(Theorem)e(4.1)j(but)d(is)g (an)h(imp)s(ortan)m(t)f(ingredien)m(t)g(that)i(w)m(e)-24 3010 y(shall)d(use)h(t)m(wice:)41 b(once)31 b(while)e(pro)m(ving)g (Lemma)i(4.2)g(and)f(then)g(in)f(Lemma)i(4.3.)118 3123 y(W)-8 b(e)29 b(stress)f(out)g(that)h(Lemma)f(4.1)h(is)e(the)h(only)-8 b(,)28 b(but)g(crucial,)f(tec)m(hnical)h(p)s(oin)m(t)f(where)h(w)m(e)g (shall)f(tak)m(e)-24 3236 y(adv)-5 b(an)m(tage)36 b(of)e(the)g(b)s (oundedness)d(assumption)h(of)i(the)g(supp)s(ort)d(of)j(the)g(measure)g Fo(\026)2983 3251 y Fm( )3069 3236 y Fp(\(H1\).)52 b(Prop)s(osi-)-24 3349 y(tions)28 b(5.1)h(and)f(5.2)h(in)e(App)s(endix)f(C)i(are)g (analog)h(of)f(Lemma)h(4.1,)h(but)d(under)g(the)h(w)m(eak)m(er)i (assumptions)-24 3462 y(\(H2\),)40 b(resp.)58 b(\(H3\).)j(A)36 b(consequence)h(is)f(that,)j(as)d(already)g(men)m(tionned)g(in)f(the)i (Remark)f(4.1)i(ab)s(o)m(v)m(e,)-24 3575 y(the)31 b(conclusion)e(of)i (Lemma)f(4.2,)i(as)f(w)m(ell)e(as)i(the)f(one)h(of)g(the)f(forthcoming) g(Lemma)h(4.3,)h(remain)d(v)-5 b(alid)-24 3688 y(under)29 b(Hyp)s(otheses)i(\(H2\))g(and)f(\(H3\).)-24 3836 y Fd(Pro)s(of)36 b(of)f(the)g(lo)m(w)m(er)g(b)s(ound)g(in)g(Theorem)f(4.1.)118 3983 y Fp(Let)j Fo(N)46 b Fp(b)s(e)37 b(the)g(in)m(teger)g(part)f(of)h Fn(\000)15 b Fp(log)i Fo(")p Fp(.)60 b(F)-8 b(or)37 b(a)h(sak)m(e)f(of) g(simplicit)m(y)d(w)m(e)k(shall)d(use)h Fo(N)46 b Fp(=)36 b Fn(\000)15 b Fp(log)h Fo(")-24 4096 y Fp(\(rather)41 b(than)g(the)g(in)m(teger)g(part\).)73 b(T)-8 b(ak)m(e)42 b Fo(A)f Fp(as)g(in)f(Lemma)h(4.2.)73 b(Here)41 b Fo(q)46 b Fp(=)c(1)p Fo(=)p Fp(\(1)29 b(+)e Fo(\014)5 b Fp(\),)44 b(and)d(th)m(us)-24 4209 y Fo(q)27 b Fn(\000)d Fp(1)35 b(=)f Fn(\000)p Fo(\014)5 b(=)p Fp(\(1)25 b(+)f Fo(\014)5 b Fp(\).)58 b(F)-8 b(or)36 b(\014xed)f Fo(")p Fp(,)j(w)m(e)f(c)m(ho)s (ose)g Fo(')e Fp(=)f Fo(\037)2058 4228 y Fh(\012\()p Fm(r)2168 4237 y Fe(0)2203 4228 y Fh(\))2234 4209 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )40 b Fp(with)35 b Fo(r)2740 4223 y Fh(0)2815 4209 y Fp(giv)m(en)h(b)m(y)g(Lemma)g(4.2.)-24 4322 y(Th)m(us,)30 b(from)g(the)g(de\014nition)e(of)j(\012\()p Fo(r)1253 4336 y Fh(0)1292 4322 y Fp(\))g(w)m(e)g(obtain)782 4524 y Fo(U)844 4539 y Fm('; )962 4524 y Fp(\()p Fo(")1039 4486 y Fl(\000)p Fh(1)1135 4524 y Fp(\))25 b(=)1291 4400 y Fj(Z)1342 4606 y Fh(\012\()p Fm(r)1452 4615 y Fe(0)1486 4606 y Fh(\))1533 4524 y Fo(b)1572 4486 y Fh(\()p Fm(R)p Fh(\))1685 4524 y Fp(\()p Fo(x;)15 b(")p Fp(\)d)p Fo(\026)1995 4539 y Fm( )2048 4524 y Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)f Fo(")2334 4486 y Fm(r)2366 4495 y Fe(0)2405 4524 y Fo(\026)2460 4539 y Fm( )2513 4524 y Fp(\(\012\()p Fo(r)2690 4538 y Fh(0)2729 4524 y Fp(\)\))31 b Fo(;)-24 4754 y Fp(Therefore,)g(using)e (the)h(expression)f(\(28\))j(of)f Fo(L)1604 4769 y Fm( )1656 4754 y Fp(\()p Fo(f)10 b Fp(\()p Fo(H)d Fp(\))p Fo( )s(;)15 b(")2043 4721 y Fl(\000)p Fh(1)2139 4754 y Fp(\))31 b(with)e Fo(f)34 b Fp(=)25 b Fo(\037)2644 4772 y Fh(\012\()p Fm(r)2754 4781 y Fe(0)2789 4772 y Fh(\))2821 4754 y Fp(,)30 b(one)h(has)45 5023 y Fo(L)107 5038 y Fm( )159 5023 y Fp(\()p Fo(")236 4986 y Fl(\000)p Fh(1)332 5023 y Fp(\))83 b Fn(\025)629 4960 y Fo(\026)684 4975 y Fm( )736 4960 y Fp(\(\012\()p Fo(r)913 4974 y Fh(0)953 4960 y Fp(\)\))1023 4927 y Fh(1+2)p Fm(\014)p 614 5002 599 4 v 614 5085 a Fo(")656 5059 y Fm(r)688 5068 y Fe(0)722 5059 y Fm(\014)770 5085 y Fo(\026)825 5100 y Fm( )877 5085 y Fp(\(\012\()p Fo(r)1054 5099 y Fh(0)1094 5085 y Fp(\)\))1164 5059 y Fm(\014)1247 5023 y Fn(\025)25 b Fo(")1395 4955 y Fi(A\014)p 1395 4970 84 3 v 1410 5012 a(N)1508 4868 y Fj( )1580 4899 y(Z)1630 5106 y Fh(\012\()p Fm(r)1740 5115 y Fe(0)1775 5106 y Fh(\))1822 5023 y Fo(b)1861 4986 y Fh(\()p Fm(R)p Fh(\))1974 5023 y Fp(\()p Fo(x;)15 b(")p Fp(\))2204 4947 y Fb(\000)p Fi(\014)p 2189 4962 115 3 v 2189 5004 a Fe(1+)p Fi(\014)2317 5023 y Fp(d)p Fo(\026)2423 5038 y Fm( )2475 5023 y Fp(\()p Fo(x)p Fp(\))2597 4868 y Fj(!)2670 4890 y Fh(1+)p Fm(\014)3500 5023 y Fp(\(32\))450 5322 y Fn(\025)772 5260 y Fo(e)814 5227 y Fl(\000)p Fm(A\014)p 614 5301 515 4 v 614 5384 a Fp(\()p Fn(\000)p Fp(2)g(log)i Fo(")p Fp(\))990 5358 y Fh(1+)p Fm(\014)1153 5193 y Fj(\022)1220 5198 y(Z)1326 5322 y Fo(b)1365 5284 y Fh(\()p Fm(R)p Fh(\))1478 5322 y Fp(\()p Fo(x;)e(")p Fp(\))1708 5245 y Fb(\000)p Fi(\014)p 1693 5261 115 3 v 1693 5302 a Fe(1+)p Fi(\014)1822 5322 y Fp(d)p Fo(\026)1928 5337 y Fm( )1979 5322 y Fp(\()p Fo(x)p Fp(\))2101 5193 y Fj(\023)2169 5216 y Fh(1+)p Fm(\014)2332 5322 y Fp(=)2574 5260 y Fo(C)2639 5274 y Fh(1)p 2438 5301 378 4 v 2438 5384 a Fn(j)g Fp(log)i Fo(")p Fn(j)2678 5358 y Fh(1+)p Fm(\014)2825 5322 y Fo(K)2902 5336 y Fm(\026)2944 5348 y Fi( )3011 5193 y Fj(\022)3172 5260 y Fp(1)p 3088 5301 213 4 v 3088 5384 a(1)k(+)f Fo(\014)3311 5322 y(;)15 b(")3393 5193 y Fj(\023)3461 5216 y Fh(1+)p Fm(\014)3568 5322 y Fo(;)1773 5611 y Fp(14)p eop %%Page: 15 15 15 14 bop -24 375 a Fp(where)39 b(in)f(the)i(last)f(inequalit)m(y)e(w)m (e)j(used)f(Lemma)g(4.2)h(and)f Fo(N)50 b Fp(=)40 b Fn(\000)15 b Fp(log)h Fo(")p Fp(.)68 b(This)37 b(holds)h(for)h Fo(")h Fp(small)-24 488 y(enough.)3307 b Fc(\003)118 601 y Fp(Remark)39 b(that)h(the)g(fact)g(that)g Fo(r)1276 615 y Fh(0)1355 601 y Fp(disapp)s(ears)d(in)h(the)i(relation)e(\(32\))j(is)e(crucial.) 67 b(It)39 b(is)g(of)g(course)-24 714 y(related)27 b(to)g(the)g (particular)e(v)-5 b(alue)26 b Fo(q)i Fp(=)d(1)p Fo(=)p Fp(\(1)13 b(+)g Fo(\014)5 b Fp(\))28 b(of)f(the)g(fractal)g(dimension)d (that)j(en)m(ters)g(in)m(to)g(accoun)m(t.)-24 827 y(This)i(is)g(the)i (place)f(where)g(the)h(deep)f(link)e(b)s(et)m(w)m(een)j Fo(L)1888 842 y Fm( )1941 827 y Fp(\()p Fo(T)13 b Fp(\))30 b(and)g Fo(D)2362 794 y Fl(\006)2359 849 y Fm(\026)2401 861 y Fi( )2453 827 y Fp(\(1)p Fo(=)p Fp(\(1)22 b(+)e Fo(\014)5 b Fp(\)\))31 b(comes)g(out.)118 977 y(W)-8 b(e)31 b(are)g(left)f(with)f(the)i(pro)s(ofs)e(of)i(Lemmas)f(4.1)h(and) f(4.2.)-24 1127 y Fd(Pro)s(of)36 b(of)f(Lemma)e(4.1.)118 1277 y Fp(Let)26 b(us)f(\014rst)g(co)m(v)m(er)i(supp)p Fo(\026)1045 1292 y Fm( )1122 1277 y Fp(with)d(disjoin)m(t)h(in)m(terv) -5 b(als)24 b Fo(I)2043 1291 y Fm(j)2105 1277 y Fp(=)h([)p Fo(x)2278 1291 y Fm(j)2326 1277 y Fn(\000)11 b Fo("=)p Fp(2)p Fo(;)k(x)2632 1291 y Fm(j)2681 1277 y Fp(+)c Fo("=)p Fp(2[.)40 b(If)26 b Fo(D)i Fp(is)d(the)h(length)-24 1390 y(of)e(the)g(supp)s(ort)e(of)i Fo(\026)705 1405 y Fm( )758 1390 y Fp(,)h(then)f(note)g(that)h(one)f(needs)f(at)i(most)f(2)p Fo(D)s(=")h Fp(suc)m(h)e(in)m(terv)-5 b(als)23 b Fo(I)2948 1404 y Fm(j)3008 1390 y Fp(to)i(co)m(v)m(er)g(supp)p Fo(\026)3585 1405 y Fm( )3636 1390 y Fp(.)-24 1503 y(Then)30 b(de\014ne)f Fo(B)545 1517 y Fm(A)628 1503 y Fp(=)c Fn(f)p Fo(x)g Fn(2)g Fp(supp)n Fo(\026)1174 1518 y Fm( )1227 1503 y Fo(;)41 b(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))26 b Fn(\024)f Fo(")1700 1470 y Fm(A)1758 1503 y Fn(g)30 b Fp(and)g(for)g(an)m(y)h Fo(j)36 b Fp(suc)m(h)30 b(that)h Fo(\026)2851 1518 y Fm( )2903 1503 y Fp(\()p Fo(I)2978 1517 y Fm(j)3035 1503 y Fn(\\)20 b Fo(B)3185 1517 y Fm(A)3242 1503 y Fp(\))26 b Fn(6)p Fp(=)f(0,)1219 1670 y Fo(b)p 1219 1685 40 4 v 22 x Fm(j)1320 1670 y Fp(=)g(inf)5 b Fn(f)p Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))p Fo(;)43 b(x)25 b Fn(2)g Fo(I)2085 1684 y Fm(j)2142 1670 y Fn(\\)19 b Fo(B)2291 1684 y Fm(A)2348 1670 y Fn(g)p Fo(:)-24 1845 y Fp(Remark)31 b(that)g Fo(b)p 516 1860 V 23 x Fm(j)617 1845 y Fn(\024)25 b Fo(")755 1812 y Fm(A)812 1845 y Fp(.)41 b(W)-8 b(e)31 b(\014rst)f(pro)m(v)m(e)h(that)1467 2012 y Fo(\026)1522 2027 y Fm( )1575 2012 y Fp(\()p Fo(I)1650 2026 y Fm(j)1707 2012 y Fn(\\)19 b Fo(B)1856 2026 y Fm(A)1913 2012 y Fp(\))26 b Fn(\024)f Fo(b)p 2070 2027 V 22 x Fm(j)2145 2012 y Fo(:)1330 b Fp(\(33\))-24 2179 y(T)-8 b(o)40 b(that)g(end)e(w)m (e)i(sho)m(w)e(that)i Fo(\026)1142 2194 y Fm( )1195 2179 y Fp(\()p Fo(I)1270 2193 y Fm(j)1306 2179 y Fp(\))g Fn(\024)g Fo(b)p 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a(")2347 2384 y Fj(\023)2440 2512 y Fn(\024)2536 2388 y Fj(Z)2642 2512 y Fp(d)p Fo(\026)2748 2527 y Fm( )2800 2512 y Fp(\()p Fo(y)s Fp(\))p Fo(R)3003 2384 y Fj(\022)3080 2451 y Fo(x)g Fn(\000)g Fo(y)p 3080 2491 V 3164 2574 a(")3301 2384 y Fj(\023)3383 2512 y Fo(:)-24 2729 y Fp(The)44 b(latter)g(means)g(that)g Fo(\026)995 2744 y Fm( )1048 2729 y Fp(\()p Fo(I)1123 2743 y Fm(j)1159 2729 y Fp(\))49 b Fn(\024)e Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\),)49 b(whenev)m(er)44 b Fo(g)s Fp(\()p Fo(w)r Fp(\))h(is)e (de\014ned)g(as)h Fo(\037)2933 2747 y Fh([)p Fl(\000)p Fh(1)p Fm(;)p Fh(1])3121 2729 y Fp(\()p Fo(w)r Fp(\))h(or)f Fo(R)q Fp(\()p Fo(w)r Fp(\).)-24 2842 y(Th)m(us)34 b(\(33\))i(holds.)52 b(As)35 b Fo(q)g Fn(2)d Fp(\(0)p Fo(;)15 b Fp(1\),)38 b(one)d(has,)h(taking)e(in)m(to)h(accoun)m(t)h(only)e(the)h(in)m(terv) -5 b(als)33 b Fo(I)3214 2856 y Fm(j)3285 2842 y Fp(suc)m(h)i(that)-24 2955 y Fo(\026)31 2970 y Fm( )84 2955 y Fp(\()p Fo(I)159 2969 y Fm(j)216 2955 y Fn(\\)20 b Fo(B)366 2969 y Fm(A)423 2955 y Fp(\))25 b Fn(6)p Fp(=)g(0)31 b(\(so)g(that)g Fo(b)p 999 2970 40 4 v 22 x Fm(j)1100 2955 y Fo(>)25 b Fp(0\):)564 3052 y Fj(Z)614 3258 y Fm(B)667 3269 y Fi(A)737 3176 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))980 3138 y Fm(q)r Fl(\000)p Fh(1)1109 3176 y Fp(d)p Fo(\026)1215 3191 y Fm( )1267 3176 y Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)1511 3089 y Fj(X)1561 3285 y Fm(j)1658 3176 y Fo(b)p 1658 3191 V -45 x Fm(q)r Fl(\000)p Fh(1)1697 3203 y Fm(j)1825 3176 y Fo(\026)1880 3191 y Fm( )1932 3176 y Fp(\()p Fo(I)2007 3190 y Fm(j)2064 3176 y Fn(\\)20 b Fo(B)2214 3190 y Fm(A)2271 3176 y Fp(\))26 b Fn(\024)2428 3089 y Fj(X)2477 3285 y Fm(j)2574 3176 y Fo(b)p 2574 3191 V -45 x Fm(q)2613 3203 y(j)2677 3176 y Fn(\024)2782 3114 y Fp(2)p Fo(D)p 2782 3155 124 4 v 2823 3238 a(")2916 3176 y(")2958 3138 y Fm(q)r(A)3049 3176 y Fo(;)426 b Fp(\(34\))-24 3437 y(where)28 b(w)m(e)g(used)f(\(33\).)42 b(No)m(w)28 b(w)m(e)h(are)f(in)f (p)s(osition)f(to)i(\014nish)e(the)i(pro)s(of.)39 b(First,)28 b(taking)g Fo(A)e Fp(=)e(2)p Fo(=q)s Fp(,)30 b(w)m(e)e(get)-24 3550 y(\(31\).)43 b(Next,)31 b(ob)m(viously)1167 3581 y Fj(Z)1217 3787 y Fm(b)p Fh(\()p Fm(x;")p Fh(\))p Fm(>")1501 3705 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1744 3667 y Fm(q)r Fl(\000)p Fh(1)1874 3705 y Fp(d)p Fo(\026)1980 3720 y Fm( )2031 3705 y Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)f Fo(")2317 3667 y Fm(q)r Fl(\000)p Fh(1)2446 3705 y Fo(:)1029 b Fp(\(35\))-24 3915 y(This)29 b(b)s(ound)f(together)k(with)d(\(34\))j (for)e Fo(A)c Fp(=)f(1)30 b(giv)m(es)1006 4000 y Fj(Z)1057 4206 y Fh(supp)p Fm(\026)1244 4218 y Fi( )1311 4124 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1554 4086 y Fm(q)r Fl(\000)p Fh(1)1683 4124 y Fp(d)p Fo(\026)1789 4139 y Fm( )1841 4124 y Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)f Fp(\(2)p Fo(D)f Fp(+)c(1\))p Fo(")2477 4086 y Fm(q)r Fl(\000)p Fh(1)2606 4124 y Fo(;)-24 4363 y Fp(so)31 b Fo(D)166 4330 y Fl(\006)163 4385 y Fm(\026)205 4397 y Fi( )257 4363 y Fp(\()p Fo(q)s Fp(\))25 b Fn(\024)g Fp(1.)41 b(This)29 b(ends)h(the)g(pro)s(of)g(of)g(Lemma)h(4.1.)1610 b Fc(\003)-24 4513 y Fd(Pro)s(of)36 b(of)f(Lemma)e(4.2.)118 4663 y Fp(T)-8 b(o)29 b(alleviate)g(the)h(notations,)g(denote)f(as)h (in)e(the)h(previous)f(lemma)h Fo(B)2574 4677 y Fm(A)2656 4663 y Fp(=)c Fn(f)p Fo(x)h Fn(2)f Fp(supp)n Fo(\026)3203 4678 y Fm( )3255 4663 y Fo(;)41 b(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))27 b Fn(\024)-24 4776 y Fo(")18 4743 y Fm(A)76 4776 y Fn(g)34 b Fp(and)f(also)g Fo(B)594 4743 y Fm(A)681 4776 y Fp(=)e Fn(f)p Fo(x)g Fn(2)f Fp(supp)n Fo(\026)1244 4791 y Fm( )1297 4776 y Fo(;)46 b(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))32 b Fo(>)e(")1786 4743 y Fm(A)1843 4776 y Fn(g)p Fp(.)51 b(Then,)34 b(using)e(the)h(b)s(ound)f(of)i(Lemma)f (4.1,)j(one)-24 4889 y(has)437 4951 y Fj(Z)543 5075 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))786 5037 y Fm(q)r Fl(\000)p Fh(1)916 5075 y Fp(d)p Fo(\026)1022 5090 y Fm( )1074 5075 y Fp(\()p Fo(x)p Fp(\))84 b(=)1433 4951 y Fj(Z)1484 5157 y Fm(B)1540 5139 y Fi(A)1610 5075 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1853 5037 y Fm(q)r Fl(\000)p Fh(1)1982 5075 y Fp(d)p Fo(\026)2088 5090 y Fm( )2140 5075 y Fp(\()p Fo(x)p Fp(\))21 b(+)2374 4951 y Fj(Z)2425 5157 y Fm(B)2478 5168 y Fi(A)2547 5075 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))2790 5037 y Fm(q)r Fl(\000)p Fh(1)2920 5075 y Fp(d)p Fo(\026)3026 5090 y Fm( )3078 5075 y Fp(\()p Fo(x)p Fp(\))1280 5326 y Fn(\024)1433 5202 y Fj(Z)1484 5408 y Fm(B)1540 5389 y Fi(A)1610 5326 y Fo(b)p Fp(\()p Fo(x;)g(")p Fp(\))1853 5288 y Fm(q)r Fl(\000)p Fh(1)1982 5326 y Fp(d)p Fo(\026)2088 5341 y Fm( )2140 5326 y Fp(\()p Fo(x)p Fp(\))21 b(+)f(2)p Fo(D)s(":)1773 5611 y Fp(15)p eop %%Page: 16 16 16 15 bop -24 375 a Fp(Remark)28 b(that)g(one)g(alw)m(a)m(ys)g(has)g Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))26 b Fn(\024)f Fp(1)j(\(since)g Fo(R)q Fp(\()p Fo(w)r Fp(\))e Fn(\024)f Fp(1\).)41 b(Th)m(us)26 b(one)i(can)g(divide)d(the)j(remaining)-24 488 y(set)41 b Fo(B)202 455 y Fm(A)298 488 y Fp(in)m(to)f Fo(N)50 b Fp(parts:)60 b(\012\()p Fo(k)s(A=)-5 b(N)10 b Fp(\))42 b(=)f Fn(f)p Fo(x)g Fn(2)g Fp(supp)n Fo(\026)1916 503 y Fm( )1969 488 y Fo(;)15 b(")2051 455 y Fh(\()p Fm(k)r Fh(+1\))p Fm(A=)l(N)2428 488 y Fo(<)41 b(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))42 b Fn(\024)f Fo(")2979 455 y Fm(k)r(A=)l(N)3170 488 y Fn(g)p Fp(,)i(with)38 b Fo(k)45 b Fp(=)-24 601 y(0)p Fo(;)15 b Fp(1)p Fo(;)g Fn(\001)g(\001)g(\001)j Fo(;)d(N)34 b Fn(\000)23 b Fp(1.)55 b(A)m(t)36 b(least)g(one)f(of)g (these)h Fo(N)45 b Fp(sets)36 b(\012\()p Fo(k)s(A=)-5 b(N)10 b Fp(\))36 b(giv)m(es)f(rise)f(to)i(an)f(in)m(tegral)g(bigger)g (than)-24 714 y(1)p Fo(=)-5 b(N)43 b Fp(times)32 b(the)h(in)m(tegral)f (o)m(v)m(er)h(the)g(whole)e(set)i Fo(B)1746 681 y Fm(A)1802 714 y Fp(.)47 b(And)31 b(so)h(are)h(pic)m(k)m(ed)f Fo(k)2671 728 y Fh(0)2711 714 y Fp(,)h Fo(r)2810 728 y Fh(0)2878 714 y Fp(=)28 b Fo(Ak)3092 728 y Fh(0)3132 714 y Fo(=)-5 b(N)10 b Fp(,)33 b(and)f(th)m(us)-24 827 y(the)f(set)g(\012\()p Fo(r)417 841 y Fh(0)456 827 y Fp(\))g(of)g(the)f(Lemma.)118 939 y(T)-8 b(o)37 b(end)f(the)h(pro)s(of,)i(remark)d(that)1392 866 y Fj(R)1468 939 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1711 906 y Fm(q)r Fl(\000)p Fh(1)1840 939 y Fp(d)p Fo(\026)1946 954 y Fm( )1998 939 y Fp(\()p Fo(x)p Fp(\))37 b Fn(\025)f Fp(1,)j(since)e Fo(q)i(<)d Fp(1.)61 b(Therefore,)38 b(for)f Fo(")g Fn(\024)-24 1052 y Fp(1)p Fo(=)p Fp(\(4)p Fo(D)s Fp(\),)c(one)d(gets)601 1183 y Fj(Z)652 1389 y Fh(\012\()p Fm(r)762 1398 y Fe(0)797 1389 y Fh(\))843 1307 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1086 1269 y Fm(q)r Fl(\000)p Fh(1)1216 1307 y Fp(d)p Fo(\026)1322 1322 y Fm( )1374 1307 y Fp(\()p Fo(x)p Fp(\))84 b Fn(\025)1762 1245 y Fp(1)p 1743 1286 83 4 v 1743 1369 a Fo(N)1851 1178 y Fj(\022)1918 1183 y(Z)2024 1307 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))2267 1269 y Fm(q)r Fl(\000)p Fh(1)2397 1307 y Fp(d)p Fo(\026)2503 1322 y Fm( )2555 1307 y Fp(\()p Fo(x)p Fp(\))21 b Fn(\000)f Fp(2)p Fo(D)s(")2954 1178 y Fj(\023)1580 1563 y Fn(\025)1785 1501 y Fp(1)p 1743 1542 129 4 v 1743 1625 a(2)p Fo(N)1897 1439 y Fj(Z)2003 1563 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))2246 1525 y Fm(q)r Fl(\000)p Fh(1)2376 1563 y Fp(d)p Fo(\026)2482 1578 y Fm( )2534 1563 y Fp(\()p Fo(x)p Fp(\))p Fo(:)3591 1802 y Fc(\003)118 1964 y Fp(W)-8 b(e)26 b(no)m(w)f(turn)f(to)i(the)f (pro)s(of)f(of)h(Theorem)50 b(4.2,)28 b(whic)m(h)23 b(is)h(actually)h (a)g(consequence)h(of)f(Theorem)g(4.1)-24 2077 y(and)32 b(of)g(the)g(follo)m(wing)e(Lemma)i(4.3)h(that)f(relates)h(the)f(in)m (tegrals)f Fo(K)2369 2091 y Fm(\026)2411 2103 y Fi( )2463 2077 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))33 b(to)f(the)g(in)m(tegrals)g Fo(I)3371 2091 y Fm(\026)3413 2103 y Fi( )3464 2077 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))-24 2190 y(that)38 b(en)m(ter)g(in)m (to)f(accoun)m(t)h(in)e(the)h(de\014nition)d(of)k(the)f(generalized)f (R)m(\023)-43 b(en)m(yi)38 b(dimension)c Fo(D)3166 2157 y Fl(\006)3163 2213 y Fm(\026)3205 2225 y Fi( )3257 2190 y Fp(\()p Fo(q)s Fp(\).)61 b(More)-24 2303 y(precisely)-8 b(,)33 b(Lemma)g(4.3)h(sa)m(ys)f(that)h(b)s(oth)e Fo(K)1534 2317 y Fm(\026)1576 2329 y Fi( )1627 2303 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))34 b(and)f Fo(I)2077 2317 y Fm(\026)2119 2329 y Fi( )2170 2303 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))34 b(giv)m(e)f(rise)f(to)i(the)f(same)g(dimension)-24 2416 y(n)m(um)m(b)s(ers.)k(This)22 b(is)g(of)h(course,)j(as)d(already)g (said,)h(b)s(ecause)f(of)h(the)f(fast)h(deca)m(y)g(prop)s(erties)e(of)h (the)h(function)-24 2529 y Fo(R)31 b Fp(w)m(e)g(ha)m(v)m(e)h(c)m(hosen) f(\()p Ff(via)g Fp(the)f(c)m(hoice)h(of)g Fo(h)26 b Fn(2)e Fo(C)1658 2496 y Fl(1)1651 2553 y Fh(0)1733 2529 y Fp(\([0)p Fo(;)15 b Fp(1]\)\).)-24 2766 y Fd(Lemma)33 b(4.3.)46 b Ff(L)-5 b(et)43 b Fo(q)k Fn(2)d Fp(\(0)p Fo(;)15 b Fp(1\))p Ff(.)75 b(Supp)-5 b(ose)44 b(that)g Fo(\026)1825 2781 y Fm( )1921 2766 y Ff(has)g(a)f(b)-5 b(ounde)g(d)45 b(supp)-5 b(ort)45 b(\(H1\).)73 b(Then)43 b(for)h(al)5 b(l)-24 2879 y Fo(\027)31 b Fn(2)25 b Fp(\(0)p Fo(;)15 b Fp(1\))1081 2969 y(1)p 1017 3010 174 4 v 1017 3093 a(2)1062 3067 y Fh(2)p Fl(\000)p Fm(q)1201 3031 y Fo(I)1241 3045 y Fm(\026)1283 3057 y Fi( )1334 3031 y Fp(\()p Fo(q)s(;)g(")1495 2993 y Fh(1)p Fl(\000)p Fm(\027)1629 3031 y Fp(\))26 b Fn(\024)f Fo(K)1863 3045 y Fm(\026)1905 3057 y Fi( )1956 3031 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))27 b Fn(\024)e Fo(I)2315 3045 y Fm(\026)2357 3057 y Fi( )2408 3031 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))p Fo(;)871 b Fp(\(36\))-24 3223 y Ff(wher)-5 b(e)34 b(the)f(left)g(ine)-5 b(quality)33 b(holds)h(for)f Fo(")g Ff(smal)5 b(l)34 b(enough)f(\()p Fo(")26 b Fn(\024)f Fo(")p Fp(\()p Fo(\027)6 b Fp(\))p Ff(\).)42 b(As)33 b(a)g(c)-5 b(onse)g(quenc)g(e)206 3423 y Fp(1)p 129 3463 201 4 v 129 3547 a(1)20 b Fn(\000)g Fo(q)354 3484 y Fp(lim)15 b(inf)411 3542 y Fm(")p Fl(!)p Fh(0)631 3414 y Fp(log)i Fo(K)841 3428 y Fm(\026)883 3440 y Fi( )934 3414 y Fp(\()p Fo(q)s(;)e(")p Fp(\))p 631 3463 501 4 v 751 3547 a Fn(\000)g Fp(log)i Fo(")1167 3484 y Fp(=)25 b Fo(D)1341 3447 y Fl(\000)1338 3507 y Fm(\026)1380 3519 y Fi( )1432 3484 y Fp(\()p Fo(q)s Fp(\))93 b(and)1966 3423 y(1)p 1888 3463 201 4 v 1888 3547 a(1)21 b Fn(\000)f Fo(q)2114 3484 y Fp(lim)15 b(sup)2184 3560 y Fm(")p Fl(!)p Fh(0)2417 3414 y Fp(log)i Fo(K)2627 3428 y Fm(\026)2669 3440 y Fi( )2721 3414 y Fp(\()p Fo(q)s(;)e(")p Fp(\))p 2417 3463 501 4 v 2537 3547 a Fn(\000)g Fp(log)i Fo(")2953 3484 y Fp(=)25 b Fo(D)3127 3447 y Fh(+)3124 3507 y Fm(\026)3166 3519 y Fi( )3218 3484 y Fp(\()p Fo(q)s Fp(\))p Fo(:)143 b Fp(\(37\))-24 3732 y Fd(Remark)35 b(4.3.)46 b Fp(This)37 b(lemma)i(remains)f(true)h(if)g(w)m(e)h(replace) f(the)g(b)s(oundedness)e(assumption)h(on)h(the)-24 3845 y(supp)s(ort)c(of)h Fo(\026)483 3860 y Fm( )572 3845 y Fp(\(H1\))h(b)m(y)f(assumption)f(\(H2\))i(or)g(\(H3\))g(giv)m(en)f (in)f(App)s(endix)f(B.)i(In)g([3)q(])g(w)m(e)h(sho)m(w)f(that)-24 3958 y Fo(\026)31 3973 y Fm( )118 3958 y Fn(2)f Fo(P)272 3972 y Fh(1)311 3958 y Fp(\()p Fg(R)s Fp(\))43 b(\(see)37 b([24)q(])f(for)g(a)g(de\014nition\))e(is)h(also)h(a)g(su\016cien)m(t)g (condition)e(for)i(\(37\))i(to)e(b)s(e)g(true.)57 b(But)-24 4071 y(it)37 b(is)f(not)h(clear)g(for)g(us)f(whether)h(or)g(not)g (Lemma)g(4.3)h(\(and)f(more)g(precisely)f(\(37\)\))j(is)d(true)g(in)g (whole)-24 4183 y(generalit)m(y)k(as)f(it)g(is)f(stated)i(in)e(the)h (app)s(endix)e(of)i([15)q(].)67 b(Ho)m(w)m(ev)m(er,)44 b(it)38 b(comes)i(out)g(from)e(Hyp)s(otheses)-24 4296 y(\(H1\),)g(\(H2\),)g(\(H3\))e(and)f(also)g(from)g([3])h(that)f(the)h (in)m(tegrals)e Fo(K)2218 4310 y Fm(\026)2260 4322 y Fi( )2312 4296 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))36 b(and)f Fo(I)2766 4310 y Fm(\026)2808 4322 y Fi( )2859 4296 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))37 b(do)e(lead)f(to)i(the)-24 4409 y(same)31 b(dimension)d(n)m(um)m(b)s(ers)h Fo(D)1079 4376 y Fl(\006)1076 4432 y Fm(\026)1118 4444 y Fi( )1169 4409 y Fp(\()p Fo(q)s Fp(\))i(for)f(a)h(v)m(ery)g(large)f(class)h(of)f (measure.)41 b(See)30 b(Remark)h(5.1.)-24 4647 y Fd(Pro)s(of)36 b(of)f(Lemma)e(4.3.)118 4809 y Fp(Throughout)c(this)g(pro)s(of)h(w)m(e) h(shall)d(denote)j Fo(B)5 b Fp(\()p Fo(x;)15 b(")p Fp(\))26 b(=)f([)p Fo(x)c Fn(\000)f Fo(";)15 b(x)21 b Fp(+)e Fo(")p Fp(].)42 b(One)30 b(has)98 5080 y Fo(I)138 5094 y Fm(\026)180 5106 y Fi( )231 5080 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))27 b(=)550 4956 y Fj(Z)656 5080 y Fo(\026)711 5095 y Fm( )763 5080 y Fp(\()p Fo(B)5 b Fp(\()p Fo(x;)15 b(")p Fp(\)\))1111 5043 y Fm(q)r Fl(\000)p Fh(1)1241 5080 y Fp(d)p Fo(\026)1347 5095 y Fm( )1399 5080 y Fp(\()p Fo(x)p Fp(\))26 b(=)1643 4956 y Fj(Z)1749 5080 y Fp(d)p Fo(\026)1855 5095 y Fm( )1907 5080 y Fp(\()p Fo(x)p Fp(\))2044 4952 y Fj(\022)2112 4956 y(Z)2218 5080 y Fo(\037)2275 5099 y Fh([)p Fl(\000)p Fh(1)p Fm(;)p Fh(1])2478 4952 y Fj(\022)2555 5019 y Fo(x)20 b Fn(\000)g Fo(y)p 2555 5059 211 4 v 2639 5143 a(")2776 4952 y Fj(\023)2858 5080 y Fp(d)p Fo(\026)2964 5095 y Fm( )3016 5080 y Fp(\()p Fo(x)p Fp(\))3138 4952 y Fj(\023)3205 4975 y Fm(q)r Fl(\000)p Fh(1)3349 5080 y Fo(:)1773 5611 y Fp(16)p eop %%Page: 17 17 17 16 bop -24 375 a Fp(Since,)33 b(b)m(y)f(\(11\),)j Fo(\037)647 393 y Fh([)p Fl(\000)p Fh(1)p Fm(;)p Fh(1])835 375 y Fp(\()p Fo(w)r Fp(\))30 b Fn(\024)f Fo(R)q Fp(\()p Fo(w)r Fp(\),)34 b(and)e Fo(q)25 b Fn(\000)c Fp(1)29 b Fo(<)g Fp(0,)34 b(one)f(has)f Fo(K)2393 389 y Fm(\026)2435 401 y Fi( )2486 375 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))31 b Fn(\024)d Fo(I)2852 389 y Fm(\026)2894 401 y Fi( )2946 375 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\).)48 b(So)33 b(w)m(e)g(need)-24 488 y(to)e(get)h(the)f(lo)m(w)m(er)f(b)s(ound)e(of)j(\(36\).)42 b(First)30 b(notice)h(that)303 742 y Fo(b)342 704 y Fh(\()p Fm(R)p Fh(\))454 742 y Fp(\()p Fo(x;)15 b(")p Fp(\))85 b(=)896 618 y Fj(Z)1002 742 y Fp(d)p Fo(\026)1108 757 y Fm( )1160 742 y Fp(\()p Fo(y)s Fp(\))p Fo(R)1364 614 y Fj(\022)1441 680 y Fo(x)20 b Fn(\000)g Fo(y)p 1441 721 211 4 v 1525 804 a(")1662 614 y Fj(\023)743 993 y Fp(=)896 870 y Fj(Z)947 1082 y Fn(j)982 1045 y Fi(x)p Fb(\000)p Fi(y)p 982 1061 117 3 v 1026 1102 a(")1108 1082 y Fn(j)1133 1076 y Fm(<")1221 1057 y Fb(\000)p Fi(\027)1327 993 y Fp(d)p Fo(\026)1433 1008 y Fm( )1485 993 y Fp(\()p Fo(y)s Fp(\))p Fo(R)1688 865 y Fj(\022)1765 932 y Fo(x)g Fn(\000)g Fo(y)p 1765 972 211 4 v 1849 1056 a(")1986 865 y Fj(\023)2073 993 y Fp(+)2164 870 y Fj(Z)2214 1082 y Fn(j)2250 1045 y Fi(x)p Fb(\000)p Fi(y)p 2250 1061 117 3 v 2293 1102 a(")2376 1082 y Fn(j)2401 1076 y Fl(\025)p Fm(")2489 1057 y Fb(\000)p Fi(\027)2594 993 y Fp(d)p Fo(\026)2700 1008 y Fm( )2752 993 y Fp(\()p Fo(y)s Fp(\))p Fo(R)2955 865 y Fj(\022)3032 932 y Fo(x)g Fn(\000)g Fo(y)p 3032 972 211 4 v 3117 1056 a(")3253 865 y Fj(\023)743 1218 y Fn(\024)82 b Fo(\026)951 1233 y Fm( )1004 1218 y Fp(\()p Fo(B)5 b Fp(\()p Fo(x;)15 b(")1282 1181 y Fh(1)p Fl(\000)p Fm(\027)1416 1218 y Fp(\)\))21 b(+)84 b(sup)1598 1302 y Fl(j)p Fm(w)r Fl(j\025)p Fm(")1779 1283 y Fb(\000)p Fi(\027)1879 1218 y Fo(R)q Fp(\()p Fo(w)r Fp(\))p Fo(;)-24 1507 y Fp(where)30 b(w)m(e)h(used)f Fo(R)q Fp(\()p Fo(w)r Fp(\))c Fn(\024)f Fp(1.)41 b(Let)p 1184 1434 70 4 v 31 w Fo(R)q Fp(\()p Fo(z)t Fp(\))26 b(=)f(sup)1629 1529 y Fl(j)p Fm(w)r Fl(j\025)p Fm(z)1830 1507 y Fo(R)q 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Fl(\000)p Fm(\027)551 3157 y Fp(\))25 b Fn(\024)g Fo(")749 3124 y Fm(A)807 3157 y Fp(.)40 b(Hence,)32 b(with)1399 3134 y(~)1375 3157 y Fo(A)1443 3171 y Fm(\027)1486 3157 y Fp(\()p Fo(")p Fp(\))27 b(=)e Fn(f)p Fo(x)g Fn(2)g Fp(supp)n Fo(\026)2171 3172 y Fm( )2224 3157 y Fn(j)p Fo(\026)2304 3172 y Fm( )2357 3157 y Fp(\()p Fo(B)5 b Fp(\()p Fo(x;)15 b(")2635 3124 y Fh(1)p Fl(\000)p Fm(\027)2769 3157 y Fp(\)\))26 b Fn(\024)f Fo(")3003 3124 y Fm(A)3060 3157 y Fn(g)p Fp(,)518 3408 y(2)563 3370 y Fh(1)p Fl(\000)p Fm(q)692 3408 y Fo(K)769 3422 y Fm(\026)811 3434 y Fi( )862 3408 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))85 b Fn(\025)e Fo(I)1337 3422 y Fm(\026)1379 3434 y Fi( )1430 3408 y Fp(\()p Fo(q)s(;)15 b(")1591 3370 y Fh(1)p Fl(\000)p Fm(\027)1725 3408 y Fp(\))21 b Fn(\000)1872 3284 y Fj(Z)1941 3474 y Fh(~)1922 3490 y Fm(A)1975 3498 y Fi(\027)2014 3490 y Fh(\()p Fm(")p Fh(\))2120 3408 y Fp(d)p Fo(\026)2226 3423 y Fm( )2278 3408 y Fp(\()p Fo(x)p Fp(\))p Fo(\026)2455 3423 y Fm( )2508 3408 y Fp(\()p 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b(lim)15 b(sup)29 b(and)h(notice)i(that)f(the)g(result)-24 4750 y(is)f(v)-5 b(alid)28 b(for)j(all)e Fo(\027)i(>)25 b Fp(0)31 b(as)f(small)f(as)i(one)f(w)m(an)m(ts.)1911 b Fc(\003)-24 4912 y Fd(Pro)s(of)36 b(of)f(Theorem)69 b(4.2.)-24 5025 y Fp(Theorem)61 b(4.2)31 b(is)f(a)g(direct)g (consequence)h(of)g(Theorem)f(4.1)h(and)f(of)h(Lemma)f(4.3.)737 b Fc(\003)118 5188 y Fp(As)33 b(already)g(men)m(tioned)g(at)h(the)g (end)f(of)g(Section)h(2,)h(our)d(main)h(Theorem,)h(namely)f(Theorem)g (2.1,)-24 5301 y(is)d(a)h(direct)e(consequence)j(of)e(Theorem)g(3.1)i (\(Section)e(3\))h(and)f(of)h(Theorem)f(4.2.)1773 5611 y(17)p eop %%Page: 18 18 18 17 bop 118 375 a Fp(W)-8 b(e)44 b(end)e(this)f(section)i(with)f(a)h (few)g(w)m(ords)f(concerning)g(the)h(case)h(where,)i(as)d(in)e([18)r(]) i(and)f([20)q(],)-24 488 y(one)g(assumes)f(some)h(further)e(prop)s (erties)g(on)h(the)g(spatial)g(b)s(eha)m(viour)f(of)h(the)h (eigenfunctions)e(of)h Fo(H)7 b Fp(.)-24 601 y(Supp)s(ose)35 b(that)j(for)e(some)i(constan)m(t)g Fo(C)7 b Fp(,)38 b(indep)s(enden)m(t)c(of)j(the)g(energy)g Fo(x)p Fp(,)i(and)d(for)h Fo(\015)k Fn(\024)35 b Fo(d)i Fp(one)g(kno)m(ws)-24 714 y(that:)219 645 y Fj(P)315 740 y Fl(j)p Fm(n)p Fl(j\024)p Fm(N)535 714 y Fn(j)p Fo(u)p Fp(\()p Fo(n;)15 b(x)p Fp(\))p Fn(j)854 681 y Fh(2)928 714 y Fn(\024)34 b Fo(C)7 b(N)1188 681 y Fm(\015)1232 714 y Fp(,)37 b(for)f Fo(\026)1494 729 y Fm( )1580 714 y Fo(a:e)f(x)h Fp(\(Hyp)s(othesis)e(\(HS\)\),)j (then)e(one)h(can)g(not)g(only)f(get)-24 827 y(a)44 b(b)s(etter)g (result)f(taking)g Fo(\014)53 b Fp(=)47 b Fo(p=\015)i Fp(instead)43 b(of)g Fo(\014)53 b Fp(=)47 b Fo(p=d)p Fp(,)h(but)43 b(it)g(turns)f(out)i(that)g(one)g(can)g(push)-24 939 y(one)35 b(step)f(further)e(our)i(approac)m(h)g(to)g(reac)m(h)h (the)f(dimension)d(n)m(um)m(b)s(ers)i Fo(D)2613 906 y Fl(\006)2610 962 y Fm(\026)2652 974 y Fi( )2704 939 y Fp(\(1)23 b Fn(\000)f Fo(\014)5 b Fp(\).)52 b(The)33 b(gain)h(tak)m(es)-24 1052 y(place)29 b(in)e(Inequalit)m(y)g(\(18\),)k (where)d(a)h(direct)f(estimate)h(is)f(made)g(p)s(ossible)e(thanks)i(to) h(Hyp)s(othesis)f(\(HS\).)-24 1165 y(This)39 b(thereb)m(y)h(enable)g (us)g(to)h(a)m(v)m(oid)g(the)f(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)i (inequalit)m(y)d(w)m(e)i(made)f(in)f(order)h(to)h(split)-24 1278 y(the)34 b(in)m(tegral)e(in)m(to)h(a)h(spatial)e(part)h(and)f(a)h (sp)s(ectral)g(part.)48 b(This)31 b(therefore)j(leads)e(to)i(the)f(lo)m (w)m(er)g(b)s(ound)-24 1407 y Fn(hhj)p Fo(X)7 b Fn(j)178 1374 y Fm(p)220 1407 y Fn(ii)290 1422 y Fm( )r(;T)463 1407 y Fn(\025)49 b Fo(C)7 b(L)717 1374 y Fl(0)717 1435 y Fm( )769 1407 y Fp(\()p Fo(T)h(;)15 b(')p Fp(\))50 b Fn(\021)f Fo(C)1268 1362 y Fl(k)p Fm(')p Fl(k)1384 1339 y Fe(2+2)p Fi(\014)p 1250 1386 302 4 v 1250 1443 a Fm(U)1298 1455 y Fi('; )1404 1443 y Fh(\()p Fm(T)10 b Fh(\))1509 1424 y Fi(\014)1562 1407 y Fo(:)44 b Fp(Then)g(the)h(same) g(tec)m(hnic)g(as)g(the)g(one)g(used)f(to)i(pro)m(v)m(e)-24 1536 y(Theorem)30 b(4.1)i(supplies)27 b(the)k(follo)m(wing)e(result.) -24 1714 y Fd(Theorem)34 b(4.3.)46 b Ff(Supp)-5 b(ose)30 b(in)e(addition)i(to)e(the)g(hyp)-5 b(otheses)31 b(of)d(The)-5 b(or)g(em)30 b(2.1)f(that)g(the)f(sp)-5 b(atial)30 b(Hyp)-5 b(oth-)-24 1827 y(esis)33 b(\(HS\))f(ab)-5 b(ove)34 b(holds)g(for)f (some)h Fo(\015)j Ff(and)d(c)-5 b(onstant)34 b Fo(C)7 b Ff(.)41 b(Then,)33 b(if)g Fo(\014)d Fp(=)25 b Fo(p=\015)31 b(<)24 b Fp(1)p Ff(,)33 b(one)g(has)675 2083 y Fp(lim)15 b(inf)705 2144 y Fm(T)10 b Fl(!1)952 2020 y Fp(log)q Fn(hhj)p Fo(X)d Fn(j)1271 1987 y Fm(p)1313 2020 y Fn(ii)1383 2035 y Fm( )r(;T)p 952 2062 555 4 v 1130 2146 a Fp(log)17 b Fo(T)1542 2083 y Fn(\025)25 b Fp(lim)15 b(inf)1668 2144 y Fm(T)10 b Fl(!1)1915 2007 y Fp(log)16 b Fo(L)2109 1974 y Fl(0)2109 2035 y Fm( )2162 2007 y Fp(\()p Fo(T)d Fp(\))p 1915 2062 384 4 v 2007 2146 a(log)k Fo(T)2334 2083 y Fp(=)24 b Fo(\014)5 b(D)2563 2046 y Fl(\000)2560 2106 y Fm(\026)2602 2118 y Fi( )2654 2083 y Fp(\(1)21 b Fn(\000)f Fo(\014)5 b Fp(\))p Fo(;)-24 2314 y Ff(and)648 2468 y Fp(lim)15 b(sup)692 2547 y Fm(T)10 b Fl(!1)952 2405 y Fp(log)q Fn(hhj)p Fo(X)d Fn(j)1271 2372 y Fm(p)1313 2405 y Fn(ii)1383 2420 y Fm( )r(;T)p 952 2447 555 4 v 1130 2531 a Fp(log)17 b Fo(T)1542 2468 y Fn(\025)25 b Fp(lim)15 b(sup)1681 2547 y Fm(T)10 b Fl(!1)1941 2392 y Fp(log)17 b Fo(L)2136 2359 y Fl(0)2136 2420 y Fm( )2188 2392 y Fp(\()p Fo(T)c Fp(\))p 1941 2447 384 4 v 2034 2531 a(log)j Fo(T)2360 2468 y Fp(=)25 b Fo(\014)5 b(D)2590 2431 y Fh(+)2587 2491 y Fm(\026)2629 2503 y Fi( )2681 2468 y Fp(\(1)21 b Fn(\000)f Fo(\014)5 b Fp(\))p Fo(:)118 2697 y Fp(F)-8 b(urther)36 b(results)f(concerning)i(Theorem)f(4.3)i (will)c(b)s(e)i(found)f(in)h(the)h(forthcoming)f(article)g([3)q(].)60 b(In)-24 2810 y(particular)29 b(w)m(e)i(shall)d(discuss)g(ho)m(w)j(far) e(Hyp)s(othesis)g(\(HS\))i(is)e(relev)-5 b(an)m(t,)31 b(and)f(also)g(pro)m(vide)f(a)i(v)m(ersion)e(of)-24 2923 y(this)h(theorem)g(that)h(holds)e(for)h(all)g Fo(\014)35 b Fp(\(and)30 b(not)h(only)e Fo(\014)i(<)25 b 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y Fi(H)2495 4998 y Fh(\()p Fm(\026)p Fh(\))p Fl(\000)p Fm(\027)2691 5036 y Fn(g)p Fo(:)-24 5228 y Fp(W)-8 b(e)32 b(ha)m(v)m(e)732 5341 y Fo(A)800 5303 y Fh(\()p Fm(H)5 b Fh(\))800 5363 y Fm(\027)948 5341 y Fn(\021)25 b Fp(lim)15 b(inf)1100 5399 y Fm(")p Fl(!)p Fh(0)1311 5341 y Fo(A)1379 5303 y Fh(\()p Fm(H)5 b Fh(\))1379 5363 y Fm(\027)1501 5341 y Fp(\()p Fo(")p Fp(\))26 b Fn(\033)f(f)p Fo(x)h Fn(2)f Fg(R)39 b Fn(j)30 b Fo(d)p 2095 5356 48 4 v 2143 5363 a Fm(\026)2189 5341 y Fp(\()p Fo(x)p Fp(\))c Fo(>)f(d)2480 5355 y Fm(H)2548 5341 y Fp(\()p Fo(\026)p Fp(\))20 b Fn(\000)g Fo(\027)6 b Fn(g)p Fo(:)1773 5611 y Fp(18)p eop %%Page: 19 19 19 18 bop -24 375 a Fp(By)43 b(the)f(de\014nition)e(of)i Fo(d)877 389 y Fm(H)945 375 y Fp(\()p Fo(\026)p Fp(\))g(giv)m(en)g (Line)f(\(5\))i(w)m(e)g(get)g Fo(\026)p Fp(\()p Fn(f)p Fo(x)i Fn(2)g Fg(R)50 b Fn(j)42 b Fo(d)p 2555 390 48 4 v 2603 397 a Fm(\026)2649 375 y Fp(\()p Fo(x)p Fp(\))k Fo(>)e(d)2979 389 y Fm(H)3047 375 y Fp(\()p Fo(\026)p Fp(\))28 b 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4961 y(Since)23 b(w)m(e)h(shall)f(fo)s(cus)h (on)g(the)g(relation)g(b)s(et)m(w)m(een)g(those)h(lo)m(w)m(er)g(b)s (ounds)d(and)h(the)i(fractal)f(dimensions)-24 5074 y Fo(D)54 5041 y Fl(\006)51 5097 y Fm(\026)93 5109 y Fi( )145 5074 y Fp(\()p Fo(q)s Fp(\),)40 b(and)d(for)h(a)g(sak)m(e)g(of)g (simplicit)m(y)-8 b(,)37 b(w)m(e)h(shall)e(only)h(consider)g(v)m (ectors)i Fo(')f Fp(of)f(the)h(form)f Fo(\037)3390 5088 y Fh(\012)3445 5074 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )-24 5187 y Fp(instead)30 b(of)h(general)f Fo(')c Fn(2)f(H)955 5202 y Fm( )1037 5187 y Fp(as)31 b(in)e(\(8\).)1773 5611 y(19)p eop %%Page: 20 20 20 19 bop 118 375 a Fp(W)-8 b(e)33 b(\014rst)f(consider)f(the)h(lo)m(w) m(er)h(b)s(ound)d(that)j(app)s(ears)f(in)f([5)q(].)46 b(F)-8 b(or)33 b(some)g(constan)m(t)h Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p)p Fp(\))29 b Fo(>)f Fp(0,)34 b(it)-24 488 y(is)c(sho)m(wn)g(that)h(for)f(all)f Fo(T)38 b(>)25 b Fp(0,)31 739 y Fn(hhj)p Fo(X)7 b Fn(j)233 702 y Fm(p)275 739 y Fn(ii)345 754 y Fm( )r(;T)494 739 y Fn(\025)25 b Fo(C)7 b Fp(\()p Fo( )s(;)15 b(p)p Fp(\))p Fo(L)942 753 y Fh(2)982 739 y Fp(\()p Fo(T)e Fp(\))p Fo(;)107 b Fp(with)89 b Fo(L)1579 753 y Fh(2)1619 739 y Fp(\()p Fo(T)13 b Fp(\))25 b(=)g(sup)2028 611 y Fj(\032)2096 739 y Fo(L)2158 753 y Fh(2)2198 739 y Fp(\()p Fo(';)15 b(T)e Fp(\))26 b(:=)2590 678 y Fn(k)p Fo(')p Fn(k)2739 645 y Fh(2+8)p Fm(\014)p 2590 719 323 4 v 2607 802 a Fo(S)2663 816 y Fm(')2713 802 y Fp(\()p Fo(T)13 b Fp(\))2849 776 y Fm(\014)2923 739 y Fo(;)41 b(')26 b Fp(=)f Fo(\037)3227 753 y Fh(\012)3282 739 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )3497 611 y Fj(\033)3581 739 y Fo(;)-24 985 y Fp(where)1028 1117 y Fo(S)1084 1131 y Fm(')1134 1117 y Fp(\()p Fo(T)13 b Fp(\))26 b(=)1392 993 y Fj(Z)1442 1199 y Fk(R)1490 1180 y Fe(2)1544 1117 y Fp(d)p Fo(\026)1650 1131 y Fm(')1700 1117 y Fp(\()p Fo(x)p Fp(\)d)p Fo(\026)1928 1131 y Fm(')1978 1117 y Fp(\()p Fo(y)s Fp(\))p Fo(R)q Fp(\()p Fo(T)13 b Fp(\()p Fo(x)21 b Fn(\000)f Fo(y)s Fp(\)\))p Fo(:)-24 1347 y Fp(Here)27 b Fo(R)q 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b Fp(\))31 b(and)e(using)f(a)j(set)f(\012\()p Fo(r)s Fp(\))g(in)f(the)h(same)g (spirit)e(as)i(previously)e(yields)g(the)i(lo)m(w)m(er)g(b)s(ound,)e (again)-24 1799 y(up)i(to)h(a)g(logarithmic)e(factor.)41 b(The)30 b(follo)m(wing)f(theorem)i(then)f(holds:)-24 1981 y Fd(Theorem)k(5.1.)46 b Ff(Under)33 b(the)g(same)h(hyp)-5 b(otheses)35 b(as)e(in)f(The)-5 b(or)g(em)35 b(2.1,)e(one)g(has)153 2224 y Fp(lim)15 b(inf)183 2284 y Fm(T)10 b Fl(!1)431 2162 y Fp(log)16 b Fo(L)625 2176 y Fh(2)664 2162 y Fp(\()p Fo(T)d Fp(\))p 431 2203 371 4 v 517 2286 a(log)j Fo(T)836 2224 y Fp(=)25 b Fo(\014)5 b(D)1066 2186 y Fl(\000)1063 2246 y Fm(\026)1105 2258 y Fi( )1172 2096 y Fj(\022)1249 2162 y Fp(1)21 b(+)e(2)p Fo(\014)p 1249 2203 259 4 v 1249 2286 a Fp(1)i(+)e(3)p Fo(\014)1517 2096 y Fj(\023)1599 2224 y Fo(;)109 b Fp(and)e(lim)15 b(sup)2030 2302 y Fm(T)10 b Fl(!1)2290 2162 y Fp(log)17 b Fo(L)2485 2176 y Fh(2)2524 2162 y Fp(\()p Fo(T)c Fp(\))p 2290 2203 371 4 v 2376 2286 a(log)k Fo(T)2696 2224 y Fp(=)25 b 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Fj(R)1788 2811 y Fp(d)p Fo(\026)1894 2825 y Fm(')1943 2811 y Fp(\()p Fo(y)s Fp(\))p Fo(R)q Fp(\()p Fo(T)e Fp(\()p Fo(x)22 b Fn(\000)f Fo(y)s Fp(\)\))33 b(where)f Fo(R)q Fp(\()p Fo(w)r Fp(\))c(=)g(exp\()p Fn(\000)p Fo(w)3495 2778 y Fh(2)3535 2811 y Fo(=)p Fp(4\))-24 2924 y(\(notice)38 b(that)g Fo(G)562 2938 y Fm(')612 2924 y Fp(\()p Fo(T)13 b Fp(\))37 b(will)e(b)s(e)h(smaller)g(than)h Fo(C)7 b(T)1772 2891 y Fl(\000)p Fm(\013)1913 2924 y Fp(if)36 b Fo(\026)2058 2938 y Fm(')2145 2924 y Fp(is)g(uniformly)f Fo(\013)p Fp(-H\177)-45 b(older)37 b(con)m(tin)m(uous,)i(see)-24 3037 y(\(3.10\))44 b(in)c([22)q(]\).)74 b(One)40 b(then)h(uses)g Fo(G)1338 3051 y Fm(')1388 3037 y Fp(\()p Fo(T)13 b Fp(\))41 b(in)f(order)h(to)h(b)s(ound)d Fo(B)2413 3051 y Fm(')2463 3037 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))42 b(de\014ned)e(as)h(in)f (\(12\))j(but)-24 3164 y(with)29 b(1)p Fo(=T)355 3091 y Fj(R)415 3117 y Fm(T)398 3196 y Fh(0)500 3164 y Fp(rather)h(than)g(1) p Fo(=T)1157 3091 y Fj(R)1217 3117 y Fh(+)p 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b(2.1,)e(one)g(has)153 5275 y Fp(lim)15 b(inf)183 5336 y Fm(T)10 b Fl(!1)431 5213 y Fp(log)16 b Fo(L)625 5227 y Fh(1)664 5213 y Fp(\()p Fo(T)d Fp(\))p 431 5254 371 4 v 517 5337 a(log)j Fo(T)836 5275 y Fp(=)25 b Fo(\014)5 b(D)1066 5237 y Fl(\000)1063 5297 y Fm(\026)1105 5309 y Fi( )1172 5147 y Fj(\022)1272 5213 y Fp(1)20 b(+)g Fo(\014)p 1249 5254 259 4 v 1249 5337 a Fp(1)h(+)e(2)p Fo(\014)1517 5147 y Fj(\023)1599 5275 y Fo(;)109 b Fp(and)e(lim)15 b(sup)2030 5353 y Fm(T)10 b Fl(!1)2290 5213 y Fp(log)17 b Fo(L)2485 5227 y Fh(1)2524 5213 y Fp(\()p Fo(T)c Fp(\))p 2290 5254 371 4 v 2376 5337 a(log)k Fo(T)2696 5275 y Fp(=)25 b Fo(\014)5 b(D)2926 5237 y Fh(+)2923 5297 y Fm(\026)2965 5309 y Fi( )3032 5147 y Fj(\022)3132 5213 y Fp(1)20 b(+)g Fo(\014)p 3109 5254 259 4 v 3109 5337 a Fp(1)h(+)e(2)p Fo(\014)3377 5147 y Fj(\023)3459 5275 y Fo(:)1773 5611 y Fp(20)p eop %%Page: 21 21 21 20 bop 118 375 a Fp(One)33 b(should)e(also)j(compare)g(the)g (expression)e(of)i Fo(L)1927 389 y Fh(1)1967 375 y Fp(\()p Fo(T)13 b Fp(\))34 b(and)f Fo(L)2379 389 y Fh(2)2418 375 y Fp(\()p Fo(T)13 b Fp(\))34 b(to)g(the)g(expression)f(of)g Fo(L)3472 390 y Fm( )3525 375 y Fp(\()p Fo(T)13 b Fp(\))-24 488 y(giv)m(en)39 b(Line)e(\(9\),)42 b(that)d(is)f(where)g(as)g(ab)s(o) m(v)m(e)i(the)f(suprem)m(um)d(is)i(tak)m(en)h(o)m(v)m(er)h(the)f(set)g (of)g(v)m(ectors)g Fo(')h Fp(=)-24 601 y Fo(\037)33 615 y Fh(\012)88 601 y Fp(\()p Fo(H)7 b Fp(\))p Fo( )s Fp(,)42 b(namely)c(the)g(suprem)m(um)f(of)h Fn(k)p Fo(')p Fn(k)1569 568 y Fh(2+4)p Fm(\014)1743 601 y Fo(=U)1850 616 y Fm('; )1969 601 y Fp(\()p Fo(T)13 b Fp(\))2105 568 y Fm(\014)2153 601 y Fp(.)64 b(One)38 b(ma)m(y)h(ha)m(v)m(e)g(the)g(righ)m(t)f(to)h(w) m(onder)-24 714 y(whether)31 b(there)g(are)g(some)g(links)e(b)s(et)m(w) m(een)i(these)g(quan)m(tities,)g(and)f(also)h(whether)f Fo(L)2956 729 y Fm( )3008 714 y Fp(\()p Fo(T)13 b Fp(\))32 b(is)d(e\013ectiv)m(ely)-24 827 y(a)i(b)s(etter)f(lo)m(w)m(er)h(b)s (ound)d(than)i Fo(L)1114 841 y Fh(1)1154 827 y Fp(\()p Fo(T)13 b Fp(\))30 b(and)g Fo(L)1559 841 y Fh(2)1598 827 y Fp(\()p Fo(T)13 b Fp(\).)41 b(Theorem)30 b(5.3)i(b)s(elo)m(w)d (answ)m(ers)h(to)h(these)g(questions.)-24 939 y(This)24 b(theorem)i(is)f(actually)g(a)h(consequence)g(of)g(Theorems)f(4.1,)j (5.2)e(and)f(5.1,)j(and)d(of)h(the)f(nonincreasing)-24 1052 y(prop)s(ert)m(y)30 b(of)h(the)f(functions)f Fo(D)1078 1019 y Fl(\006)1075 1075 y Fm(\026)1117 1087 y Fi( )1169 1052 y Fp(\()p Fo(q)s Fp(\))i(\(p)s(oin)m(t)e(i\))h(of)h(Prop)s (osition)d(2.1\).)-24 1240 y Fd(Theorem)34 b(5.3.)46 b Ff(Under)33 b(the)g(same)h(hyp)-5 b(otheses)35 b(as)e(pr)-5 b(eviously,)34 b(one)f(has)692 1489 y Fp(lim)15 b(inf)722 1549 y Fm(T)10 b Fl(!1)969 1425 y Fp(log)17 b Fo(L)1164 1440 y Fm( )1217 1425 y Fp(\()p Fo(T)c Fp(\))p 969 1468 384 4 v 1062 1551 a(log)k Fo(T)1388 1489 y Fn(\025)25 b Fp(lim)15 b(inf)1514 1549 y Fm(T)10 b Fl(!1)1761 1427 y Fp(log)17 b Fo(L)1956 1441 y Fh(1)1995 1427 y Fp(\()p Fo(T)c Fp(\))p 1761 1468 371 4 v 1847 1551 a(log)k Fo(T)2167 1489 y Fn(\025)25 b Fp(lim)15 b(inf)2293 1549 y Fm(T)10 b Fl(!1)2540 1427 y Fp(log)17 b Fo(L)2735 1441 y Fh(2)2774 1427 y Fp(\()p Fo(T)c Fp(\))p 2540 1468 V 2626 1551 a(log)k Fo(T)2920 1489 y(;)-24 1731 y Ff(and)653 1871 y Fp(lim)e(sup)696 1949 y Fm(T)10 b Fl(!1)956 1808 y Fp(log)17 b Fo(L)1151 1823 y Fm( )1203 1808 y Fp(\()p Fo(T)c Fp(\))p 956 1850 384 4 v 1049 1933 a(log)j Fo(T)1375 1871 y Fn(\025)25 b Fp(lim)15 b(sup)1514 1949 y Fm(T)10 b Fl(!1)1774 1810 y Fp(log)17 b Fo(L)1969 1824 y Fh(1)2008 1810 y Fp(\()p Fo(T)c Fp(\))p 1774 1850 371 4 v 1860 1933 a(log)k Fo(T)2180 1871 y Fn(\025)25 b Fp(lim)15 b(sup)2319 1949 y Fm(T)10 b Fl(!1)2579 1810 y Fp(log)17 b Fo(L)2774 1824 y Fh(2)2813 1810 y Fp(\()p Fo(T)c Fp(\))p 2579 1850 V 2665 1933 a(log)k Fo(T)2960 1871 y(:)-24 2154 y Fd(Remarks.)-24 2267 y Fp(1\))24 b(It)f(is)g(w)m(orth)g(to)h(p)s(oin)m(t)e(out)h(that)h(suc)m (h)f(a)g(comparison)g(is)f(made)h(p)s(ossible)e(thanks)h(to)i(the)f (upp)s(er)f(b)s(ounds)-24 2380 y(obtained)35 b(for)h Fo(L)562 2395 y Fm( )615 2380 y Fp(\()p Fo(T)13 b Fp(\),)37 b Fo(L)875 2394 y Fh(1)915 2380 y Fp(\()p Fo(T)13 b Fp(\))36 b(and)f Fo(L)1331 2394 y Fh(2)1370 2380 y Fp(\()p Fo(T)13 b Fp(\);)39 b(upp)s(er)34 b(b)s(ounds)f(that)k(are)f(for)f(the)h (\014rst)f(time)h(deriv)m(ed)f(for)-24 2493 y(suc)m(h)30 b(quan)m(tities.)-24 2606 y(2\))24 b(Theorem)e(5.3)i(tells)e(us)g(that) h(it)g(is)e(w)m(orth)i(to)h(deal)e(with)f(the)i(crossed)g(term)g(in)e (\(4\))j(while)d(one)i(dev)m(elopps)-24 2719 y Fo(B)45 2734 y Fm( )98 2719 y Fp(\()p Fo(T)8 b(;)15 b(N)10 b Fp(\))30 b(with)e Fo( )h Fp(=)c Fo(')19 b Fp(+)f Fo(\037)p Fp(.)40 b(Dealing)29 b(with)f(the)i(crossed)f(term)h(do)s(es)f(lead)g (to)h(an)f(impro)m(v)m(emen)m(t,)i(with)-24 2832 y(regards)g(to)g(the)f (former)g(approac)m(hes)h(\(G-C-L\))h(and)d(\(BT\))i(where)f(this)f (term)i(w)m(asn't)g(w)m(ell)e(treated.)-24 2945 y(3\))40 b(One)f(can)g(sho)m(w)g(that)h Fo(L)974 2960 y Fm( )1027 2945 y Fp(\()p Fo(';)15 b(T)e Fp(\))41 b Fn(\025)f Fo(c)1453 2959 y Fh(1)1492 2945 y Fo(L)1554 2959 y Fh(1)1594 2945 y Fp(\()p Fo(';)15 b(T)e Fp(\))40 b(with)e(some)i(constan)m(t)g Fo(c)2736 2959 y Fh(1)2816 2945 y Fo(>)f Fp(0)h(uniform)d(in)h Fo(T)8 b(;)15 b(')p Fp(.)-24 3058 y(The)35 b(latter)g(is)f(not)h(true)f (as)h(one)g(compares)h Fo(L)1632 3073 y Fm( )1684 3058 y Fp(\()p Fo(';)15 b(T)e Fp(\))36 b(and)e Fo(L)2198 3072 y Fh(2)2238 3058 y Fp(\()p Fo(';)15 b(T)e Fp(\))36 b(for)e(some)h Fo(')p Fp(.)55 b(W)-8 b(e)36 b(confess)f(that)-24 3171 y(the)40 b(inequalities)d(in)m(v)m(olving)i Fo(L)1090 3185 y Fh(2)1129 3171 y Fp(\()p Fo(T)13 b Fp(\))40 b(in)f(Theorem)g (5.3)i(are)f(quite)f(a)h(surprise)d(for)j(us,)i(since)d(w)m(e)h(w)m (ere)-24 3283 y(exp)s(ecting)30 b(the)h(exp)s(onen)m(ts)f(of)h Fo(L)1135 3297 y Fh(2)1174 3283 y Fp(\()p Fo(T)13 b Fp(\))31 b(to)g(b)s(e)f(bigger)g(than)g(those)h(of)f Fo(L)2464 3297 y Fh(1)2504 3283 y Fp(\()p Fo(T)13 b Fp(\).)-24 3396 y(4\))40 b(W)-8 b(e)39 b(b)s(eliev)m(e)f(a)h(stronger)f(v)m (ersion)g(of)h(Theorem)f(5.3)i(to)f(b)s(e)f(true,)i(namely:)56 b Fo(L)2898 3411 y Fm( )2951 3396 y Fp(\()p Fo(T)13 b Fp(\))39 b Fn(\025)f Fo(c)3274 3410 y Fh(1)3314 3396 y Fo(L)3376 3410 y Fh(1)3416 3396 y Fp(\()p Fo(T)13 b Fp(\))39 b Fn(\025)-24 3509 y Fo(c)15 3523 y Fh(2)55 3509 y Fo(L)117 3523 y Fh(2)156 3509 y Fp(\()p Fo(T)13 b Fp(\).)-24 3722 y Fy(App)s(endix)38 b(C:)f(Alternativ)m(e)f(h)m(yp)s (otheses)i(concerning)f(the)h(measure.)118 3885 y Fp(Lemma)c(4.1)h(is)e (the)h(only)f(tec)m(hnical)h(lemma)f(where)h(w)m(e)g(used)g(the)g (assumption)e(\(H1\),)37 b(that)d(is)f(the)-24 3997 y(b)s(oundedness)d (of)j(the)g(supp)s(ort)e(of)h(the)h(measure)f Fo(\026)1791 4012 y Fm( )1844 3997 y Fp(.)47 b(W)-8 b(e)34 b(shall)d(b)s(elo)m(w)g (prop)s(ose)h(t)m(w)m(o)i(v)m(ersions)e(of)g(this)-24 4110 y(Lemma)27 b(extended)f(to)h(more)f(general)g(h)m(yp)s(otheses,)h (whic)m(h)e(in)g(particular)f(enable)i(one)g(to)h(deal)f(with)f(non)-24 4223 y(compactly)i(supp)s(orted)d(sp)s(ectrum.)38 b(Note)27 b(that)f(since)g(Lemma)g(4.2)h(and)e(4.3)i(are)f(pro)m(v)m(en)g(indep)s (enden)m(tly)-24 4336 y(of)g(the)g(assumption)e(on)h Fo(\026)879 4351 y Fm( )931 4336 y Fp(,)i(pro)m(vided)d(Equation)h (\(31\))i(is)e(ful\014lled,)e(they)i(remain)g(v)-5 b(alid)24 b(if)g(one)i(supp)s(oses)-24 4449 y(for)38 b Fo(\026)178 4464 y Fm( )268 4449 y Fp(\(H2\))g(of)g(Prop)s(osition)d(5.1)k(or)f (\(H3\))g(of)g(Prop)s(osition)e(5.2)i(b)s(elo)m(w)f(instead)g(of)g(the) h(assumption)-24 4562 y(\(H1\))31 b(of)f(Theorem)f(2.1.)42 b(W)-8 b(e)31 b(also)e(refer)h(the)g(reader)f(to)i(a)f(forthcoming)f (article)g([3)q(])h(for)f(further)f(results.)118 4675 y(Prop)s(osition)i(5.1)j(deals)e(with)g(v)-5 b(alues)32 b Fo(q)j Fp(in)30 b(\(0)p Fo(;)15 b Fp(1\),)35 b(and)d(Prop)s(osition)e (5.2)j(with)e(v)-5 b(alues)31 b(in)g(\()p Fn(\0001)p Fo(;)15 b Fp(1\).)-24 4788 y(In)30 b(the)g(next)g(t)m(w)m(o)i(statemen) m(ts)g Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))31 b(is)e(either)1703 4715 y Fj(R)1778 4788 y Fp(d)p Fo(\026)p Fp(\()p 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Fo(>)e Fp(0)33 b Ff(such)f(that)1094 497 y Fj(Z)1144 703 y Fm(b)p Fh(\()p Fm(x;")p Fh(\))p Fl(\024)p Fm(")1409 684 y Fi(A)1478 621 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1721 583 y Fm(q)r Fl(\000)p Fh(1)1851 621 y Fo(d\026)p Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)f Fo(C)7 b Fp(\()p Fo(q)s(;)15 b(\025)p Fp(\))p Fo(";)-24 893 y Ff(Mor)-5 b(e)g(over,)34 b Fo(D)482 860 y Fl(\006)479 915 y Fm(\026)541 893 y Fp(\()p Fo(q)s Fp(\))26 b Fn(\024)f Fp(1)p Ff(.)42 b(As)32 b(a)h(c)-5 b(onse)g(quenc)g(e)33 b(the)g(c)-5 b(onclusions)34 b(of)f(L)-5 b(emmas)34 b(4.2)f(and)h(4.3)f (hold.)-24 1205 y Fd(Prop)s(osition)k(5.2.)46 b Ff(L)-5 b(et)33 b Fo(q)i Ff(b)-5 b(e)32 b(in)h Fp(\()p Fn(\0001)p Fo(;)15 b Fp(1\))34 b Ff(and)f Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))34 b Ff(de\014ne)-5 b(d)34 b(as)f(in)g(L)-5 b(emma)33 b(4.1.)43 b(Assume)893 1409 y Fp(\()p Fd(H3)p Fp(\))93 b(F)-8 b(or)31 b(some)g Fo(q)1625 1372 y Fl(0)1674 1409 y Fo(<)25 b(q)s(;)40 b Fp(one)31 b(has)55 b Fo(D)2311 1372 y Fh(+)2308 1432 y Fm(\026)2370 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Fp(=)e(inf)5 b Fn(f)p Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))p Fo(;)47 b(x)30 b Fn(2)g Fo(I)2026 3771 y Fm(j)2085 3757 y Fn(\\)21 b Fp(\012\()p Fo(")p Fp(\))p Fn(g)p Fp(.)51 b(As)33 b(in)f(the)h(pro)s (of)g(of)g(Lemma)g(4.1)-24 3879 y(note)e(that)g Fo(\026)p Fp(\()p Fo(I)505 3893 y Fm(j)562 3879 y Fn(\\)20 b Fp(\012\()p Fo(")p Fp(\)\))26 b Fn(\024)f Fo(b)p 978 3894 V 22 x Fm(j)1079 3879 y Fn(\024)g Fo(")1217 3846 y Fm(A)1305 3879 y Fp(and)30 b(one)g(has)1229 4014 y Fj(Z)1280 4220 y Fh(\012\()p Fm(")p Fh(\))1438 4137 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1681 4100 y Fm(q)r Fl(\000)p Fh(1)1810 4137 y Fp(d)p Fo(\026)p Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)2160 4051 y Fj(X)2209 4246 y Fm(j)2306 4137 y Fo(b)p 2306 4152 V -44 x Fm(q)2345 4164 y(j)2383 4137 y Fo(:)-24 4431 y Fp(But)31 b(no)m(w)f(one)h(can)g(rewrite)e(this)h(summation)f (as)h(follo)m(ws:)1212 4607 y Fj(X)1261 4803 y Fm(j)1358 4694 y Fo(b)p 1358 4709 V -45 x Fm(q)1397 4721 y(j)1461 4694 y Fp(=)1560 4580 y Fh(+)p Fl(1)1557 4607 y Fj(X)1558 4805 y Fm(k)r Fh(=0)1703 4694 y Fo(S)1759 4709 y Fm(k)1801 4694 y Fo(;)107 b(S)1989 4709 y Fm(k)2056 4694 y Fn(\021)2165 4607 y Fj(X)2152 4804 y Fm(j)t Fl(2)p Fm(J)2271 4816 y Fi(k)2324 4694 y Fo(b)p 2324 4709 V -45 x Fm(q)2363 4721 y(j)2401 4694 y Fo(;)1074 b Fp(\(42\))-24 5005 y(where)37 b Fo(J)296 5020 y Fm(k)375 5005 y Fp(=)f Fn(f)p Fo(j)5 b Fn(j)p Fo(I)634 5019 y Fm(j)708 5005 y Fn(\032)36 b Fp([)p Fo(e)882 4972 y Fm(k)925 5005 y Fo(;)15 b(e)1007 4972 y Fm(k)r Fh(+1)1140 5005 y Fp(\))25 b Fn([)g Fp([)p Fn(\000)p Fo(e)1424 4972 y Fm(k)r Fh(+1)1557 5005 y Fo(;)15 b Fn(\000)p Fo(e)1710 4972 y Fm(k)1753 5005 y Fp(\))p Fn(g)37 b Fp(for)g Fo(k)i(>)d Fp(0)i(and)e Fo(J)2525 5019 y Fh(0)2601 5005 y Fp(=)g Fn(f)p Fo(j)5 b Fn(j)p Fo(I)2860 5019 y Fm(j)2934 5005 y Fn(\032)36 b Fp([)p Fn(\000)p Fo(e;)15 b(e)p Fp(\))p Fn(g)p Fp(.)62 b(T)-8 b(o)37 b(b)s(e)-24 5118 y(rigourous)c(Equalit)m(y)g(\(42\))j(holds)d (if)g(one)h(replaces)g(in)f(the)h(de\014nition)e(of)j(the)f(sets)g Fo(J)2971 5133 y Fm(k)3049 5118 y Fp(the)g(quan)m(tit)m(y)g Fo(e)3618 5085 y Fm(k)-24 5231 y Fp(b)m(y)28 b([)p Fo(e)167 5198 y Fm(k)211 5231 y Fo(=")p Fp(])p Fo(")16 b Fp(+)g Fo("=)p Fp(2,)30 b(where)d([)p Fn(\001)p Fp(])i(stands)e(for)h(the)g (in)m(teger)g(part,)h(but)e(for)h(a)g(sak)m(e)i(of)e(simplicit)m(y)d(w) m(e)j(shall)f(use)-24 5344 y(the)k(de\014nition)d(of)j Fo(J)691 5359 y Fm(k)764 5344 y Fp(giv)m(en)f(ab)s(o)m(v)m(e.)1773 5611 y(22)p eop %%Page: 23 23 23 22 bop 118 375 a Fp(Let)22 b Fo(\015)30 b Fn(2)25 b Fp(\(0)p Fo(;)15 b Fp(1\).)40 b(F)-8 b(or)23 b Fo(k)28 b Fn(\025)d Fp(0,)g(b)m(y)c(a)i(H\177)-45 b(older)21 b(inequalit)m(y)-8 b(,)23 b(and)e(using)g(that)h(Card)p Fo(J)2860 390 y Fm(k)2928 375 y Fn(\024)j Fp(2\()p Fo(e)3146 342 y Fm(k)r Fh(+1)3283 375 y Fn(\000)s Fo(e)3399 342 y Fm(k)3442 375 y Fp(\))p Fo(=")i Fn(\024)-24 488 y Fp(4)p Fo(e)63 455 y Fm(k)107 488 y Fo(=")p Fp(,)227 813 y Fo(S)283 828 y Fm(k)408 813 y Fp(=)575 727 y Fj(X)562 924 y Fm(j)t Fl(2)p Fm(J)681 936 y Fi(k)733 813 y Fo(b)p 733 828 40 4 v -44 x Fm(q)772 840 y(j)836 813 y Fp(=)944 727 y Fj(X)932 924 y Fm(j)t Fl(2)p Fm(J)1051 936 y Fi(k)1103 813 y Fo(b)p 1103 828 V -44 x Fm(\015)t(q)1142 840 y(j)1220 813 y Fo(b)p 1220 828 V -48 x Fh(\(1)p Fl(\000)p Fm(\015)t Fh(\))p Fm(q)1259 840 y(j)1508 813 y Fn(\024)1604 630 y Fj(0)1604 794 y(@)1696 727 y(X)1683 924 y Fm(j)t Fl(2)p Fm(J)1802 936 y Fi(k)1855 813 y Fo(b)p 1855 828 V 22 x Fm(j)1930 630 y Fj(1)1930 794 y(A)2010 653 y Fm(\015)t(q)2103 630 y Fj(0)2103 794 y(@)2195 727 y(X)2183 924 y Fm(j)t Fl(2)p Fm(J)2302 936 y Fi(k)2354 813 y Fo(b)p 2354 828 V 2403 712 a Fe(\(1)p Fb(\000)p Fi(\015)s Fe(\))p Fi(q)p 2403 731 193 3 v 2427 772 a Fe(1)p Fb(\000)p Fi(\015)s(q)2393 840 y Fm(j)2609 630 y Fj(1)2609 794 y(A)2689 653 y Fh(1)p Fl(\000)p Fm(\015)t(q)408 1176 y Fn(\024)83 b Fo(")604 1138 y Fm(A)p Fh(\(1)p Fl(\000)p Fm(\015)t Fh(\))p Fm(q)896 1048 y Fj(\022)973 1114 y Fp(4)p Fo(e)1060 1081 y Fm(k)p 973 1155 131 4 v 1017 1238 a Fo(")1113 1048 y Fj(\023)1180 1061 y Fh(1)p Fl(\000)p Fm(\015)t(q)1364 993 y Fj(0)1364 1157 y(@)1456 1089 y(X)1443 1286 y Fm(j)t Fl(2)p Fm(J)1562 1298 y Fi(k)1615 1176 y Fo(b)p 1615 1191 40 4 v 22 x Fm(j)1690 993 y Fj(1)1690 1157 y(A)1770 1016 y Fm(\015)t(q)1873 1176 y Fp(=)25 b(4)p Fo(")2056 1138 y Fm(A)p Fh(\(1)p Fl(\000)p Fm(\015)t Fh(\))p Fm(q)r Fl(\000)p Fh(\(1)p Fl(\000)p Fm(\015)t(q)r Fh(\))2607 1176 y Fo(e)2649 1138 y Fm(k)r Fh(\(1)p Fl(\000)p Fm(\015)t(q)r Fh(\))2927 993 y Fj(0)2927 1157 y(@)3019 1089 y(X)3006 1286 y Fm(j)t Fl(2)p Fm(J)3125 1298 y Fi(k)3177 1176 y Fo(b)p 3177 1191 V 22 x Fm(j)3253 993 y Fj(1)3253 1157 y(A)3333 1016 y Fm(\015)t(q)-24 1479 y Fp(Assume)30 b(that)1447 1506 y Fj(X)1434 1703 y Fm(j)t Fl(2)p Fm(J)1553 1715 y Fi(k)1605 1592 y Fo(b)p 1605 1607 V 23 x Fm(j)1706 1592 y Fn(\024)25 b Fo(C)7 b Fp(\()p Fo(\025)p Fp(\))p Fo(e)2039 1555 y Fl(\000)p Fm(k)r(\025)2178 1592 y Fo(:)1297 b Fp(\(43\))-24 1850 y(Then)30 b(the)g(pro)s(of)g(can)h(b)s(e)e(completed)i(as)f(follo) m(ws:)623 2044 y Fj(X)672 2240 y Fm(j)769 2130 y Fo(b)p 769 2145 V -44 x Fm(q)808 2158 y(j)872 2130 y Fn(\024)24 b Fp(4)p Fo(C)7 b Fp(\()p Fo(\025)p Fp(\))p Fo(")1249 2093 y Fm(A)p Fh(\(1)p Fl(\000)p Fm(\015)t Fh(\))p Fm(q)r Fl(\000)p Fh(\(1)p Fl(\000)p Fm(\015)t(q)r Fh(\))1819 2017 y(+)p Fl(1)1816 2044 y Fj(X)1817 2242 y Fm(k)r Fh(=0)1963 2130 y Fo(e)2005 2093 y Fl(\000)p Fm(k)r Fh(\()p Fm(\025\015)t(q)r Fl(\000)p Fh(\(1)p Fl(\000)p Fm(\015)t(q)r Fh(\)\))2572 2130 y Fn(\024)25 b Fo(C)7 b Fp(\()p Fo(q)s(;)15 b(\025)p Fp(\))p Fo(";)486 b Fp(\(44\))-24 2429 y(pro)m(vided)34 b Fo(\025)e(>)h Fp(\(1)24 b Fn(\000)e Fo(q)s Fp(\))p Fo(=q)s Fp(,)37 b Fo(\015)j Fp(close)35 b(enough)f(to)i(1,)h(and)d (\014nally)e Fo(A)h Fp(=)f(\(2)24 b Fn(\000)f Fo(\015)5 b(q)s Fp(\))p Fo(=)p Fp(\(1)25 b Fn(\000)e Fo(\015)5 b Fp(\))p Fo(q)s Fp(.)54 b(Moreo)m(v)m(er,)-24 2542 y(taking)29 b Fo(A)d Fp(=)f(1)k(in)f(\(44\),)j(one)f(gets)g(that)1387 2469 y Fj(R)1430 2574 y Fm(b)p Fh(\()p Fm(x;")p Fh(\))p Fl(\024)p Fm(")1714 2542 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1957 2509 y Fm(q)r Fl(\000)p Fh(1)2087 2542 y Fp(d)p Fo(\026)p Fp(\()p Fo(x)p Fp(\))25 b Fn(\024)g Fo(C)7 b Fp(\()p Fo(q)s(;)15 b(\025)p Fp(\))p Fo(")2757 2509 y Fm(q)r Fl(\000)p Fh(1)2886 2542 y Fp(.)40 b(The)29 b(latter)g(implies)-24 2672 y(\(see)j(the)e(pro)s(of)g(of)g(Lemma)h (4.1)g(\))g(that)g Fo(D)1468 2639 y Fl(\006)1465 2695 y Fm(\026)1527 2672 y Fp(\()p Fo(q)s Fp(\))26 b Fn(\024)f Fp(1.)118 2785 y(W)-8 b(e)26 b(are)f(then)g(left)g(with)e(the)j(pro)s (of)e(of)h(Estimate)h(\(43\).)40 b(This)23 b(will)g(b)s(e)h(a)i (consequence)f(of)h(Hyp)s(othesis)-24 2898 y(\(H2\))i(and)e(of)h(the)f (Cheb)m(yc)m(hev)h(Inequalit)m(y)-8 b(.)39 b(Indeed)25 b(from)h(these)h(t)m(w)m(o)h(p)s(oin)m(ts)e(one)g(deriv)m(es)g(the)h (follo)m(wing)-24 3011 y(estimate:)1346 3124 y Fo(\026)p Fp(\()p Fn(j)p Fo(x)p Fn(j)e(\025)g Fo(D)s Fp(\))h Fn(\024)f Fo(C)7 b Fp(\()p Fo(\025)p Fp(\))p Fo(D)2167 3086 y Fl(\000)p Fm(\025)2267 3124 y Fo(;)1208 b Fp(\(45\))-24 3291 y(for)41 b(some)g Fo(C)7 b Fp(\()p Fo(\025)p Fp(\))43 b Fn(\024)f Fp(+)p Fn(1)p Fp(.)71 b(Moreo)m(v)m(er,)46 b(observ)m(e)41 b(that)g Fo(b)p 1952 3306 V 22 x Fm(j)2070 3291 y Fn(\024)h Fo(b)p Fp(\()p Fo(x)2309 3305 y Fm(j)2346 3291 y Fo(;)15 b(")p Fp(\))42 b(\(where)f Fo(x)2866 3305 y Fm(j)2945 3291 y Fp(=)h Fo(j)5 b(")p Fp(\).)73 b(Indeed,)42 b(if)-24 3404 y Fo(x)28 3418 y Fm(j)90 3404 y Fn(2)25 b Fp(\012\()p Fo(")p Fp(\),)31 b(clearly)-8 b(,)989 3517 y Fo(b)p 989 3532 V 22 x Fm(j)1090 3517 y Fp(=)25 b(inf)5 b Fn(f)p Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))p Fo(;)48 b(x)25 b Fn(2)g Fo(I)1860 3531 y Fm(j)1917 3517 y Fn([)19 b Fp(\012\()p Fo(")p Fp(\))p Fn(g)27 b(\024)e Fo(b)p Fp(\()p Fo(x)2469 3531 y Fm(j)2506 3517 y Fo(;)15 b(")p Fp(\))p Fo(:)-24 3692 y Fp(On)30 b(the)g(other)h(hand,)f(if)f Fo(x)910 3706 y Fm(j)982 3692 y Fo(=)-55 b Fn(2)25 b Fp(\012\()p Fo(")p Fp(\),)31 b(then)f Fo(b)p Fp(\()p Fo(x)1625 3706 y Fm(j)1662 3692 y Fo(;)15 b(")p Fp(\))26 b Fo(>)f(")1943 3659 y Fm(A)2026 3692 y Fn(\025)g Fo(b)p 2122 3707 V 23 x Fm(j)2197 3692 y Fp(.)41 b(Therefore,)30 b(in)f(an)m(y)i(cases,)1421 3836 y Fj(X)1409 4033 y Fm(j)t Fl(2)p Fm(J)1528 4045 y Fi(k)1580 3923 y Fo(b)p 1580 3938 V 22 x Fm(j)1681 3923 y Fn(\024)1789 3836 y Fj(X)1777 4033 y Fm(j)t Fl(2)p Fm(J)1896 4045 y Fi(k)1948 3923 y Fo(b)p Fp(\()p Fo(x)2074 3937 y Fm(j)2111 3923 y Fo(;)15 b(")p Fp(\))1272 b(\(46\))-24 4223 y(First)38 b(consider)f(the)i(case)g (where)f Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))40 b(=)f Fo(\026)p Fp(\()p Fo(x)25 b Fn(\000)h Fo(";)15 b(x)26 b Fp(+)f Fo(")p Fp(\).)66 b(Since)37 b Fn([)2591 4237 y Fm(j)t Fl(2)p Fm(J)2710 4249 y Fi(k)2751 4223 y Fp([)p Fo(x)2828 4237 y Fm(j)2890 4223 y Fn(\000)25 b Fo(";)15 b(x)3120 4237 y Fm(j)3183 4223 y Fp(+)25 b Fo(")p Fp(\))40 b(co)m(v)m(ers,)-24 4336 y(roughly)28 b(sp)s(eaking,)g(t)m(w)m(o)j (times)d(the)h(set)h Fn(f)p Fo(e)1499 4303 y Fm(k)r Fh(+1)1658 4336 y Fn(\025)25 b(j)p Fo(x)p Fn(j)g(\025)g Fo(e)2019 4303 y Fm(k)2062 4336 y Fn(g)p Fp(,)30 b(the)f(r.h.s)g(of)g(\(46\))h (is)e(then)h(b)s(ounded)e(from)-24 4449 y(ab)s(o)m(v)m(e)36 b(b)m(y)f(2)p Fo(\026)p Fp(\()p Fo(e)546 4416 y Fm(k)r Fh(+1)712 4449 y Fn(\025)d(j)p Fo(x)p Fn(j)h(\025)f Fo(e)1095 4416 y Fm(k)1138 4449 y Fp(\))h Fn(\024)f Fp(2)p Fo(\026)p Fp(\()p Fn(j)p Fo(x)p Fn(j)h(\025)f Fo(e)1724 4416 y Fm(k)1767 4449 y Fp(\).)54 b(Th)m(us,)35 b(com)m(bining)e(this)h(with)f (\(45\))k(one)d(gets)i(the)-24 4562 y(desired)29 b(result.)118 4675 y(Estimate)h(\(43\))j(is)c(a)i(little)f(bit)f(more)i(tric)m(ky)g (in)e(the)i(second)f(case,)i(that)g(is)d(when)h Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))32 b(is)e(de\014ned)-24 4788 y(with)37 b(the)h(function)e Fo(R)q Fp(.)63 b(In)37 b(this)g(case,)k(the)d(summation)e(other)i(the)g Fo(x)2511 4802 y Fm(j)2585 4788 y Fp(cannot)h(b)s(e)e(in)m(terpreted)g(as)h(a)-24 4901 y(summation)33 b(of)h(balls)e([)p Fo(x)850 4915 y Fm(j)909 4901 y Fn(\000)22 b Fo(";)15 b(x)1136 4915 y Fm(j)1196 4901 y Fp(+)22 b Fo(")p Fp(\),)35 b(whic)m(h)d(led)h (previously)f(to)i Fo(\026)p Fp(\()p Fo(e)2524 4868 y Fm(k)r Fh(+1)2688 4901 y Fn(\025)c(j)p Fo(x)p Fn(j)i(\025)e Fo(e)3066 4868 y Fm(k)3109 4901 y Fp(\).)51 b(One)33 b(has)h(to)-24 5014 y(handle)29 b(with)g(the)i(remainder)e(part)h(of)g Fo(R)i Fp(outside)d([)p Fn(\000)p Fp(1)p Fo(;)15 b Fp(1].)118 5126 y(T)-8 b(o)31 b(that)h(aim)f(w)m(e)h(shall)d(consider)i(ann)m (ulars)e(cen)m(tered)k(at)f(p)s(oin)m(ts)e Fo(x)2500 5140 y Fm(j)2563 5126 y Fp(=)d Fo(j)5 b(")p Fp(,)33 b Fo(j)f Fn(2)27 b Fo(J)3010 5141 y Fm(k)3053 5126 y Fp(.)43 b(As)32 b(previously)-24 5239 y(w)m(e)c(shall)d(denote)p 606 5166 70 4 v 28 w Fo(R)q Fp(\()p Fo(x)p Fp(\))h(=)f(sup)1056 5261 y Fl(j)p Fm(w)r Fl(j\025)p Fm(x)1262 5239 y Fo(R)q Fp(\()p Fo(w)r Fp(\).)41 b(Note)28 b(that)g(as)f(the)g(function)f Fo(R)q Fp(,)p 2682 5166 V 28 w Fo(R)i Fp(deca)m(ys)g(faster)f(than)g (an)m(y)1773 5611 y(23)p eop %%Page: 24 24 24 23 bop -24 375 a Fp(in)m(v)m(erse)31 b(p)s(o)m(w)m(er.)40 b(With)30 b(some)h(abuse)f(of)h(notation,)f(one)h(has,)g(using)d (\014rst)i(\(46\),)-2 520 y Fj(X)-14 717 y Fm(j)t Fl(2)p Fm(J)105 729 y Fi(k)157 606 y Fo(b)p 157 621 40 4 v 23 x Fm(j)315 606 y Fn(\024)482 520 y Fj(X)469 717 y Fm(j)t Fl(2)p Fm(J)588 729 y Fi(k)640 606 y Fo(b)p Fp(\()p Fo(x)766 620 y Fm(j)803 606 y Fo(;)15 b(")p Fp(\))27 b Fn(\024)1055 520 y Fj(X)1043 717 y Fm(j)t Fl(2)p Fm(J)1162 729 y Fi(k)1214 483 y Fj(Z)1320 606 y Fp(d)p Fo(\026)p Fp(\()p Fo(y)s Fp(\))p Fo(R)1629 478 y Fj(\022)1705 544 y Fo(x)1757 558 y Fm(j)1814 544 y Fn(\000)20 b Fo(y)p 1705 585 248 4 v 1808 669 a(")1963 478 y Fj(\023)315 961 y Fn(\024)482 875 y Fj(X)469 1072 y Fm(j)t Fl(2)p Fm(J)588 1084 y Fi(k)640 779 y Fj(0)640 942 y(@)720 839 y Fh(\()p Fm(e)780 816 y Fi(k)q Fb(\000)p Fe(2)897 839 y Fm(=")p Fh(\))p Fl(\000)p Fh(1)835 875 y Fj(X)835 1069 y Fm(n)p Fh(=0)1097 838 y Fj(Z)1148 1044 y Fm(n")p Fl(\024j)p Fm(x)1339 1054 y Fi(j)1370 1044 y Fl(\000)p Fm(y)r Fl(j)p Fm(<)p Fh(\()p Fm(n)p Fh(+1\))p Fm(")1701 961 y Fp(d)p Fo(\026)p Fp(\()p Fo(y)s Fp(\))p Fo(R)1980 833 y Fj(\022)2056 899 y Fo(x)2108 913 y Fm(j)2165 899 y Fn(\000)f Fo(y)p 2056 940 V 2158 1024 a(")2313 833 y Fj(\023)2400 961 y Fp(+)2491 838 y Fj(Z)2542 1044 y Fm(e)2575 1025 y Fi(k)q Fb(\000)p Fe(2)2691 1044 y Fl(\024j)p Fm(x)2806 1054 y Fi(j)2838 1044 y Fl(\000)p Fm(y)r Fl(j)2908 961 y Fp(d)p Fo(\026)p Fp(\()p Fo(y)s Fp(\))p Fo(R)3187 833 y Fj(\022)3263 899 y Fo(x)3315 913 y Fm(j)3372 899 y Fn(\000)h Fo(y)p 3263 940 V 3366 1024 a(")3521 833 y Fj(\023)3557 779 y(1)3557 942 y(A)315 1325 y Fn(\024)469 1203 y Fh(\()p Fm(e)529 1179 y Fi(k)q Fb(\000)p Fe(2)646 1203 y Fm(=")p Fh(\))p Fl(\000)p Fh(1)585 1238 y Fj(X)584 1433 y Fm(n)p Fh(=0)846 1142 y Fj(0)846 1306 y(@)p 926 1252 70 4 v 19 x Fo(R)q Fp(\()p Fo(n)p Fp(\))1149 1238 y Fj(X)1136 1435 y Fm(j)t Fl(2)p Fm(J)1255 1447 y Fi(k)1307 1325 y Fo(\026)p Fp(\()p Fo(n")26 b Fn(\024)f(j)p Fo(x)1693 1339 y Fm(j)1750 1325 y Fn(\000)20 b Fo(y)s Fn(j)25 b Fo(<)g Fp(\()p Fo(n)20 b Fp(+)g(1\))p 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Fi(k)729 2136 y Fo(b)p 729 2151 40 4 v 23 x Fm(j)887 2136 y Fn(\024)83 b Fp(2)1101 2014 y Fh(\()p Fm(e)1161 1991 y Fi(k)1200 2014 y Fm(=")p Fh(\))p Fl(\000)p Fh(1)1178 2050 y Fj(X)1177 2244 y Fm(n)p Fh(=0)p 1400 2063 70 4 v 1400 2136 a Fo(R)q Fp(\()p Fo(n)p Fp(\))p Fo(\026)p Fp(\()p Fn(j)p Fo(y)s Fn(j)26 b(\025)f Fo(e)1947 2099 y Fm(k)r Fl(\000)p Fh(1)2080 2136 y Fp(\))c(+)2237 2075 y(4)p Fo(e)2324 2042 y Fm(k)p 2237 2116 131 4 v 2281 2199 a Fo(")p 2377 2063 70 4 v 2377 2136 a(R)q Fp(\()p Fo(e)2524 2099 y Fm(k)r Fl(\000)p Fh(2)2657 2136 y Fo(=")p Fp(\))887 2443 y Fn(\024)83 b Fo(C)7 b(\026)p Fp(\()p Fn(j)p Fo(x)p Fn(j)25 b(\025)g Fo(e)1468 2405 y Fm(k)r Fl(\000)p Fh(1)1601 2443 y Fp(\))c(+)f(4)p Fo(e)1835 2405 y Fh(2)1885 2381 y Fo(e)1927 2348 y Fm(k)r Fl(\000)p Fh(2)p 1885 2422 176 4 v 1951 2505 a Fo(")p 2070 2370 70 4 v 2070 2443 a(R)q Fp(\()p Fo(e)2217 2405 y Fm(k)r Fl(\000)p Fh(2)2350 2443 y Fo(=")p Fp(\))27 b Fn(\024)e Fo(C)7 b Fp(\()p Fo(q)s(;)15 b(\025)p Fp(\))p Fo(e)2916 2405 y 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Fj(Z)178 4832 y Fm(b)p Fh(\()p Fm(x;")p Fh(\))p Fl(\024)p Fm(")443 4813 y Fi(A)512 4749 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))755 4712 y Fm(q)r Fl(\000)p Fh(1)885 4749 y Fp(d)p Fo(\026)p Fp(\()p Fo(x)p Fp(\))25 b Fn(\024)1235 4663 y Fj(X)1234 4861 y Fm(k)r Fl(2)p Fk(N)1383 4626 y Fj(Z)1433 4832 y Fh(\012)1484 4844 y Fi(k)1522 4832 y Fh(\()p Fm(")p Fh(\))1629 4749 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1872 4712 y Fm(q)r Fl(\000)p Fh(1)2002 4749 y Fp(d)p Fo(\026)p Fp(\()p Fo(x)p Fp(\))25 b Fn(\024)2352 4663 y Fj(X)2351 4861 y Fm(k)r Fl(2)p Fk(N)2500 4621 y Fj(\022)2576 4688 y Fo(e)2618 4655 y Fm(k)r Fh(+1)p 2576 4728 176 4 v 2614 4812 a Fo(")2656 4785 y Fm(A)2762 4621 y Fj(\023)2829 4635 y Fh(1)p Fl(\000)p Fm(q)2972 4749 y Fo(\026)p Fp(\(\012)3128 4764 y Fm(k)3170 4749 y Fp(\()p Fo(")p Fp(\)\))p Fo(:)-24 5019 y Fp(Using)30 b(no)m(w)g(\(48\),)132 5160 y Fj(Z)183 5366 y Fm(b)p Fh(\()p Fm(x;")p Fh(\))p Fl(\024)p Fm(")448 5347 y Fi(A)517 5283 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))760 5246 y 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Fm(")1512 939 y Fi(A)1581 875 y Fo(b)p Fp(\()p Fo(x;)15 b(")p Fp(\))1824 838 y Fm(q)r Fl(\000)p Fh(1)1953 875 y Fp(d)p Fo(\026)p Fp(\()p Fo(x)p Fp(\))26 b Fn(\024)e Fo(C)7 b(":)3591 1126 y Fc(\003)-24 1332 y Fy(App)s(endix)39 b(D:)f(An)h(example)f(of)g(\014nite)g(pure)h(p)s(oin)m(t)e(measure)i (with)e(non)m(trivial)g(gen-)-24 1445 y(eralized)g(fractal)g (dimensions.)118 1605 y Fp(F)-8 b(or)30 b Fo(\025)25 b(>)g Fp(1)30 b(and)g Fo(\013)25 b(>)g Fp(0,)31 b(let)f Fo(a)1165 1619 y Fm(n)1237 1605 y Fp(=)25 b Fo(a=n)1481 1572 y Fm(\025)1526 1605 y Fp(,)30 b Fo(x)1633 1619 y Fm(n)1706 1605 y Fp(=)25 b(1)p Fo(=n)1947 1572 y Fm(\013)1996 1605 y Fp(,)31 b(where)e Fo(a)c(>)g Fp(0)30 b(is)f(a)h(normalization)f (constan)m(t.)-24 1717 y(W)-8 b(e)34 b(de\014ne)e(the)h(\014nite)e (pure)g(p)s(oin)m(t)h(probabilit)m(y)e(measure)i Fo(\026)d Fp(=)2249 1649 y Fj(P)2345 1676 y Fl(1)2345 1744 y Fm(n)p Fh(=1)2497 1717 y Fo(a)2545 1731 y Fm(n)2592 1717 y Fo(\016)2632 1731 y Fm(x)2672 1739 y Fi(n)2719 1717 y Fp(.)47 b(As)32 b(the)h(measure)f Fo(\026)h Fp(has)-24 1830 y(a)h(b)s(ounded)d(supp)s (ort,)h(0)e Fn(\024)g Fo(D)1048 1797 y Fl(\006)1045 1853 y Fm(\026)1107 1830 y Fp(\()p Fo(q)s Fp(\))g Fn(\024)g Fp(1)j(for)g(an)m(y)g Fo(q)g Fn(2)d Fp(\(0)p Fo(;)15 b Fp(1\))35 b(-)e(see)h(P)m(oin)m(t)f(iv\))g(of)g(Prop)s(osition)e (2.1.)50 b(W)-8 b(e)-24 1943 y(denote)31 b(b)m(y)e Fo(B)5 b Fp(\()p Fo(x;)15 b(")p Fp(\))31 b(the)f(closed)g(ball)e(of)i(cen)m (ter)g Fo(x)g Fp(and)f(radius)f Fo(")p Fp(.)41 b(F)-8 b(or)30 b(an)m(y)h(giv)m(en)e Fo(")d(>)f Fp(0,)30 b(tak)m(e)i Fo(N)39 b Fp(to)31 b(b)s(e)-24 2056 y(the)h(in)m(teger)g(part)f(of)h Fo(")781 2023 y Fl(\000)p Fh(1)p Fm(=)p Fh(\(1+)p Fm(\013)p Fh(\))1102 2056 y Fp(.)44 b(Th)m(us,)31 b(there)h(exists)f(a)h(constan) m(t)h Fo(c)27 b(>)g Fp(0)32 b(uniform)d(in)i Fo(N)41 b Fp(\(and)31 b Fo(")p Fp(\))i(suc)m(h)-24 2169 y(that)e(for)f(all)g Fo(n)25 b Fn(\024)g Fo(cN)10 b Fp(,)30 b Fo(\026)p Fp(\()p Fo(B)5 b Fp(\()p Fo(x)1043 2183 y Fm(n)1090 2169 y Fo(;)15 b(")p Fp(\)\))27 b(=)e Fo(\026)p Fp(\()p Fn(f)p Fo(x)1552 2183 y Fm(n)1600 2169 y Fn(g)p Fp(\))h(=)f Fo(a)1850 2183 y Fm(n)1897 2169 y Fp(.)40 b(Let)31 b Fo(q)d(<)d Fp(1)p Fo(=\025)p Fp(.)42 b(Then)29 b(w)m(e)i(ha)m(v)m(e)206 2450 y Fo(I)246 2464 y Fm(\026)292 2450 y Fp(\()p Fo(q)s(;)15 b(")p Fp(\))27 b(=)611 2364 y Fj(X)655 2555 y Fm(n)757 2450 y Fo(a)805 2464 y Fm(n)852 2450 y Fo(\026)p Fp(\()p Fo(B)5 b Fp(\()p Fo(x)1103 2464 y Fm(n)1150 2450 y Fo(;)15 b(")p Fp(\)\))1302 2413 y Fm(q)r Fl(\000)p Fh(1)1457 2450 y Fn(\025)1573 2336 y Fm(cN)1554 2364 y Fj(X)1553 2558 y Fm(n)p Fh(=1)1701 2450 y Fo(\026)p Fp(\()p Fo(B)5 b Fp(\()p Fo(x)1952 2464 y Fm(n)1999 2450 y Fo(;)15 b(")p Fp(\)\))2151 2413 y Fm(q)r Fl(\000)p Fh(1)2281 2450 y Fo(a)2329 2464 y Fm(n)2401 2450 y Fp(=)2517 2336 y Fm(cN)2498 2364 y Fj(X)2497 2558 y Fm(n)p Fh(=1)2645 2450 y Fo(a)2693 2413 y Fm(q)2693 2473 y(n)2766 2450 y Fn(\030)25 b Fo(C)c(")2990 2407 y Fl(\000)3055 2377 y Fe(1)p Fb(\000)p Fi(q)r(\025)p 3055 2392 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Fp(,)h(pap)s(er)e(1)i (\(1999\).)-24 4252 y([15])47 b(Guarneri)31 b(I.,)j(Sc)m(h)m (ulz-Baldes)e(H.,)h Ff(Intermittent)j(lower)g(b)-5 b(ound)35 b(on)g(quantum)g(di\013usion)p Fp(,)f(Preprin)m(t)163 4365 y(\(1999\),)f(or)e(mp-arc)f(99-342.)-24 4547 y([16])47 b(Halsey)c(T.C.,)i(Jensen)d(M.,)k(Kaddano)m(v)d(L.,)i(Pro)s(caccia)e (I.,)j(Sc)m(hraiman)41 b(B.,)47 b Ff(F)-7 b(r)i(actal)46 b(Me)-5 b(asur)g(es)163 4660 y(and)40 b(their)g(Singularities:)56 b(the)40 b(Char)-5 b(acterization)42 b(of)e(Str)-5 b(ange)40 b(stes)p Fp(,)g(Ph)m(ys.)e(Rev.)g(A)g Fd(33)g Fp(\(1986\),)163 4773 y(1141-1151.)-24 4955 y([17])47 b(Hen)m(tsc)m(hel)29 b(H.G.E.,)g(Pro)s(caccia)g(I.,)f Ff(The)j(In\014nite)f(Numb)-5 b(er)30 b(of)g(Gener)-5 b(alize)g(d)33 b(Dimensions)d(of)h(F)-7 b(r)i(ac-)163 5067 y(tals)34 b(and)f(Str)-5 b(ange)34 b(A)n(ttr)-5 b(actors)p Fp(,)32 b(Ph)m(ys.)f(D)f Fd(8)h Fp(\(1983\),)i(435-444.)-24 5249 y([18])47 b(Ketzmeric)m(k)26 b(R.,)g(Kruze)f(K.,)h(Kraut)f(S.,)h(Geisel)e(T.,)i Ff(What)j (Determines)f(the)g(Spr)-5 b(e)g(ading)30 b(of)d(a)i(Wave)163 5362 y(Packet)p Fp(,)i(Ph)m(ys.)f(Rev.)h(A)f Fd(79)h Fp(\(1997\),)i(1959-1963.)1773 5611 y(26)p eop %%Page: 27 27 27 26 bop -24 375 a Fp([19])47 b(Ketzmeric)m(k)39 b(R.,)h(P)m(etsc)m (hel)f(G.,)i(Geisel)d(T,)f Ff(Slow)k(de)-5 b(c)g(ay)41 b(of)f(temp)-5 b(or)g(al)43 b(c)-5 b(orr)g(elations)42 b(in)e(quantum)163 488 y(systems)34 b(with)f(Cantor)h(sp)-5 b(e)g(ctr)g(a)p Fp(,)33 b(Ph)m(ys.)e(Rev.)f(Lett.)i Fd(69)f Fp(\(1992\),)i(695-698.)-24 675 y([20])47 b(Kiselev)41 b(A.,)j(Last)e(Y.,)j Ff(Solutions,)i(sp)-5 b(e)g(ctrum,)47 b(and)d(dynamics)g(for)f(Schr\177)-46 b(odinger)45 b(op)-5 b(er)g(ators)46 b(on)163 788 y(in\014nte)33 b(domains)p Fp(,)f(Preprin)m(t)d(\(1999\),)k(or)e(mp-arc)f(99-206.)-24 976 y([21])47 b(Kunz)42 b(H.,)47 b(Souillard)40 b(B.,)48 b Ff(Sur)c(le)h(sp)-5 b(e)g(ctr)g(e)46 b(des)e(op)n(\023)-44 b(er)-5 b(ateurs)47 b(aux)e(di\013)n(\023)-44 b(er)-5 b(enc)g(es)45 b(\014nies)f(al)n(\023)-44 b(eatoir)-5 b(es)p Fp(,)163 1089 y(Comm)m(un.)30 b(Math.)h(Ph)m(ys.)f Fd(78)p Fp(,)h(201-246)j(\(1980\).)-24 1277 y([22])47 b(Last)40 b(Y.,)j Ff(Quantum)f(Dynamics)f(and)h(De)-5 b(c)g(omp)g(osition)44 b(of)e(Singular)g(Continuous)g(Sp)-5 b(e)g(ctrum)p Fp(,)43 b(J.)163 1389 y(F)-8 b(unct.)31 b(Anal)f Fd(142)h Fp(\(1996\),)i(406-445.)-24 1577 y([23])47 b(Man)m(tica)36 b(G.)f Ff(Quantum)h(intermittency)h(in)f(almost)i(p)-5 b(erio)g(dic)38 b(systems)f(derive)-5 b(d)38 b(fr)-5 b(om)37 b(their)g(sp)-5 b(e)g(c-)163 1690 y(tr)g(al)42 b(pr)-5 b(op)g(erties)p Fp(,)45 b(Ph)m(ysica)39 b(D)g Fd(103)h Fp(\(1997\),)k(576-589;)j Ff(Wave)40 b(Pr)-5 b(op)g(agation)44 b(in)d(A)n(lmost-Perio)-5 b(dic)163 1803 y(Structur)g(es)p Fp(,)32 b(Ph)m(ysica)e(D)h Fd(109)g Fp(\(1997\),)i(113-127.)-24 1990 y([24])47 b(Olsen)29 b(L.,)i Ff(A)h(multifr)-5 b(actal)35 b(F)-7 b(ormalism)p Fp(,)33 b(Adv.)d(Math.)h Fd(116)p Fp(,)h(82-195)h(\(1995\).)-24 2178 y([25])47 b(P)m(esin)28 b(Y.B.,)i Ff(Dimension)h(The)-5 b(ory)32 b(in)f(Dynamic)-5 b(als)32 b(Systems:)42 b(Contemp)-5 b(or)g(ary)34 b(Views)d(and)g(Appli-)163 2291 y(c)-5 b(ations)p Fp(,)32 b(Univ.)e(Chicago)g(Press,)g(\(1996\).)-24 2479 y([26])47 b(Piec)m(hon)33 b(F.,)g Ff(A)n(nomalous)j(Di\013usion)e (Pr)-5 b(op)g(erties)37 b(of)e(Wave)f(Packets)h(on)g(Quasip)-5 b(erio)g(dic)36 b(Chains)p Fp(,)163 2592 y(Ph)m(ys.)30 b(Rev.)h(Lett.)h Fd(76)p Fp(,)f Fo(n)1062 2559 y Fh(o)1101 2592 y Fp(23,)g(4372-4375)k(\(1996\).)-24 2779 y([27])47 b(B.)23 b(Simon,)f Ff(Schr\177)-46 b(odinger)27 b(semi-gr)-5 b(oups)p Fp(,)25 b(Bull.)c(Amer.)h(Math.)h(So)s(c.)g Fd(V)-9 b(ol.7)p Fp(,)24 b Fo(n)2850 2746 y Fm(o)2888 2779 y Fp(3,)h(p.447-526)g(\(1982\).)-24 2967 y([28])47 b(T)-8 b(a)m(ylor)26 b(S.)f(J.,)i(T)-8 b(ricot)26 b(C.,)g Ff(Packing)i(Me)-5 b(asur)g(e,)30 b(and)f(its)f(Evaluation)h(for)g(a)g (Br)-5 b(ownian)30 b(Path)p Fp(,)e(T)-8 b(rans.)163 3080 y(Amer.)31 b(Math.)g(So)s(c.)f Fd(288)p Fp(,)i(679-699)h(\(1985\).)-24 3267 y([29])47 b(Wilkinson)20 b(M.,)k(Austin)d(E.J.,)j Ff(Sp)-5 b(e)g(ctr)g(al)28 b(dimension)e(and)g(dynamics)h(for)e(Harp)-5 b(er's)27 b(e)-5 b(quation)p Fp(,)24 b(Ph)m(ys.)163 3380 y(Rev.)31 b(B)g Fd(50)g Fp(\(1994\),)i(1420-1429.)1773 5611 y(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9910191739585--