Content-Type: multipart/mixed; boundary="-------------0409201712221" This is a multi-part message in MIME format. ---------------0409201712221 Content-Type: text/plain; name="04-292.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-292.keywords" nonlinear Schroedinger equation, ground states, parametric resonance, asymptotic stability ---------------0409201712221 Content-Type: application/postscript; name="resnls.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="resnls.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: resnls.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -f resnls.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.09.16:1447 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls 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1525 y Fn(j)p Fp(V)21 b Fq(\()p Fp(x)p Fq(\))p Fn(j)26 b(\024)1496 1463 y Fp(C)p 1343 1504 377 4 v 1343 1587 a Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)p Fq(\))1672 1561 y Fo(\014)1730 1525 y Fp(;)197 b(C)32 b(>)25 b Fq(0)p Fp(;)107 b(\014)30 b(>)25 b Fq(3)p Fp(;)107 b Fn(8)p Fp(x)24 b Fn(2)h Fl(R)2949 1487 y Fm(3)2994 1525 y Fp(;)630 b Fq(\(1.4\))149 1772 y(and)1524 1901 y Fn(r)p Fp(V)46 b Fn(2)24 b Fp(L)1846 1863 y Fo(p)1886 1901 y Fq(\()p Fl(R)1981 1863 y Fm(3)2026 1901 y Fq(\))p Fp(;)67 b Fn(8)p Fp(p)24 b(>)2380 1839 y Fq(3)p 2380 1880 46 4 v 2380 1963 a(2)2435 1901 y Fp(:)1189 b Fq(\(1.5\))149 2093 y(It)33 b(follo)m(ws)f(from)g(the)h(deca)m(y)h(condition)d(\(1.4\))k (on)d Fp(V)20 b Fq(\()p Fp(x)p Fq(\))34 b(that)f(the)g(op)s(erator)g Fn(H)c Fq(=)g Fn(\000)p Fq(\001)21 b(+)h Fp(V)e Fq(\()p Fp(x)p Fq(\))33 b(is)f(self-)149 2209 y(adjoin)m(t)21 b(on)f Fp(L)626 2176 y Fm(2)666 2209 y Fq(\()p Fl(R)761 2176 y Fm(3)806 2209 y Fq(\))h(with)e(domain)h Fp(H)1456 2176 y Fm(2)1495 2209 y Fq(\()p Fl(R)1590 2176 y Fm(3)1636 2209 y Fq(\),)j(see)e([12)q(,)i(Theorem)d(7.5.5].)40 b(As)20 b(a)h(result,)h(the)f(sp)s(ectrum)e(of)i Fn(H)149 2325 y Fq(consists)i(of)g(\014nite-dimensional)c(discrete)j(sp)s (ectrum)g(for)h(negativ)m(e)h(eigen)m(v)-5 b(alues,)24 b(absolutely)e(con)m(tin)m(uous)149 2442 y(sp)s(ectrum)28 b(for)g(non-negativ)m(e)h(eigen)m(v)-5 b(alues,)29 b(and)f(a)h(p)s (ossible)d(resonance)j(at)g(zero.)41 b(W)-8 b(e)30 b(assume)e(that)h (the)149 2558 y(zero-eigen)m(v)-5 b(alue)42 b(resonance)f(do)s(es)g (not)g(o)s(ccur)f(in)f(the)i(sp)s(ectrum)f(of)h Fn(H)g Fq(and)f(denote)h(the)g(pro)5 b(jection)149 2674 y(op)s(erator)33 b(on)e(the)h(absolutely)f(con)m(tin)m(uous)h(sp)s(ectrum)f(of)h Fn(H)g Fq(b)m(y)g Fn(P)2498 2688 y Fo(c)2533 2674 y Fq(.)45 b(Adding)30 b(a)i(tec)m(hnical)g(assumption)149 2790 y(on)f(the)f(discrete)g(sp)s(ectrum)g(of)g Fn(H)q Fq(,)g(w)m(e)h(form)m (ulate)g(our)e(main)h(result)f(as)i(Theorem)f(1.)149 2981 y Fg(Assumption)40 b(1.)k Ff(The)36 b(self-adjoint)i(op)-5 b(er)g(ator)39 b Fn(H)33 b Fq(=)e Fn(\000)p Fq(\001)23 b(+)f Fp(V)e Fq(\()p Fp(x)p Fq(\))37 b Ff(on)g Fp(L)2788 2948 y Fm(2)2827 2981 y Fq(\()p Fl(R)2922 2948 y Fm(3)2968 2981 y Fq(\))f Ff(has)i(a)e(unique)g(simple)149 3097 y(eigenvalue)d Fp(\025)f Ff(of)h(the)g(discr)-5 b(ete)33 b(sp)-5 b(e)g(ctrum,)34 b(such)f(that)1562 3304 y Fp(\025)25 b(<)g Fq(0)p Fp(;)202 b(\025)20 b Fq(+)g Fp(!)28 b(>)d Fq(0)p Fp(;)1226 b Fq(\(1.6\))149 3511 y Ff(with)34 b(the)f(c)-5 b(orr)g(esp)g(onding)36 b(r)-5 b(e)g(al-value)g(d)34 b(eigenve)-5 b(ctor)33 b Fp( )s Fq(\()p Fp(x)p Fq(\))26 b Fn(2)f Fp(L)2352 3478 y Fm(2)2392 3511 y Fq(\()p Fl(R)2487 3478 y Fm(3)2532 3511 y Fq(\))p Ff(.)149 3701 y Fg(Remark)35 b(1.1.)42 b Ff(Base)-5 b(d)34 b(on)f(the)g(de)-5 b(c)g(ay)34 b(of)f(the)g(p)-5 b(otential)34 b(\(1.4\),)g(it)f(fol)5 b(lows)34 b(fr)-5 b(om)33 b([11)q(])g(that)1209 3920 y Fn(j)p Fp( )s Fq(\()p Fp(x)p Fq(\))p Fn(j)27 b(\024)e Fp(C)7 b(e)1680 3882 y Fy(\000)1735 3829 y(p)p 1793 3829 97 3 v 1793 3882 a(\000)p Fo(\025)p Fy(j)p Fo(x)p Fy(j)1972 3920 y Fp(;)202 b Fn(8j)p Fp(x)p Fn(j)25 b Fp(>)g(x)2525 3934 y Fm(0)2589 3920 y Fn(\035)h Fq(1)p Fp(;)873 b Fq(\(1.7\))149 4127 y Ff(such)33 b(that)1167 4243 y Fp( )s Fq(\()p Fp(x)p Fq(\))26 b Fn(2)f Fp(L)1525 4206 y Fm(2)1525 4266 y Fo(\033)1572 4243 y Fq(\()p Fl(R)1667 4206 y Fm(3)1712 4243 y Fq(\))c Fn(\\)f Fp(L)1911 4206 y Fo(p)1950 4243 y Fq(\()p Fl(R)2045 4206 y Fm(3)2091 4243 y Fq(\))p Fp(;)41 b Fn(8)p Fp(\033)28 b Fn(\025)d Fq(0)p Fp(;)41 b Fn(8)p Fp(p)24 b Fn(\025)h Fq(2)p Fp(:)832 b Fq(\(1.8\))149 4434 y Fg(Theorem)34 b(1.)42 b Ff(L)-5 b(et)33 b(Assumption)h(1)f(hold)h(and)g(assume)f (that)1088 4688 y Fq(\000)1171 4631 y Fo(def)1190 4688 y Fq(=)1316 4626 y Fp(\015)1368 4593 y Fm(2)1363 4650 y(1)p 1316 4667 92 4 v 1339 4750 a Fq(4)1418 4688 y(Im)o Fn(h)p Fp( )1623 4650 y Fm(3)1663 4688 y Fp(;)15 b Fq(\()p Fn(H)22 b(\000)e Fp(\025)g Fn(\000)g Fp(!)j Fn(\000)d Fp(i)p Fq(0\))2374 4650 y Fy(\000)p Fm(1)2469 4688 y Fn(P)2532 4702 y Fo(c)2567 4688 y Fp( )2629 4650 y Fm(3)2669 4688 y Fn(i)26 b(6)p Fq(=)f(0)p Fp(;)753 b Fq(\(1.9\))149 4917 y Ff(Then)33 b(ther)-5 b(e)34 b(exists)f Fp(")904 4931 y Fm(0)969 4917 y Fp(>)25 b Fq(0)33 b Ff(such)g(that)h(for)f(al)5 b(l)33 b(initial)g(c)-5 b(onditions)34 b Fp(u)2564 4931 y Fm(0)2604 4917 y Fq(\()p Fp(x)p Fq(\))f Ff(satisfying)927 5159 y Fp(")994 5103 y Fo(def)1014 5159 y Fq(=)45 b(max)p Fn(fk)p Fp(u)1441 5173 y Fm(0)1481 5159 y Fn(k)1526 5179 y Fo(H)1589 5160 y Fi(1)1629 5159 y Fp(;)15 b Fn(k)p Fp(u)1766 5173 y Fm(0)1806 5159 y Fn(k)1851 5179 y Fo(L)1899 5160 y Fi(2)1899 5195 y Fh(\033)1945 5159 y Fn(g)26 b Fp(<)f(")2154 5173 y Fm(0)2194 5159 y Fp(;)201 b Fq(for)32 b(some)h Fp(\033)c(>)2978 5097 y Fq(5)p 2978 5138 46 4 v 2978 5221 a(2)3033 5159 y Fp(;)545 b Fq(\(1.10\))1970 5425 y(2)p eop %%Page: 3 3 3 2 bop 149 75 a Ff(the)38 b(initial-value)g(pr)-5 b(oblem)39 b(\(1.1\))f(is)g(glob)-5 b(al)5 b(ly)38 b(wel)5 b(l-p)-5 b(ose)g(d)40 b(and)e(satis\014es)g(the)g(de)-5 b(c)g(ay)39 b(estimates)f(for)g(al)5 b(l)149 191 y Fp(t)26 b Fn(\025)e Fq(0)p Ff(:)1321 307 y Fp(u)p Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)f Fp(e)1732 270 y Fy(\000)p Fo(i\025t)1882 307 y Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp( )s Fq(\()p Fp(x)p Fq(\))d(+)e Fp(u)2402 322 y Fo(d)2443 307 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(;)940 b Fq(\(1.11\))149 477 y Ff(wher)-5 b(e)1521 609 y Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)27 b(\024)2078 548 y Fp(C)2143 562 y Fm(0)2183 548 y Fp(")p 1875 588 555 4 v 1875 676 a Fq(\(1)21 b(+)e(4\000)p Fp(")2210 649 y Fm(4)2251 676 y Fp(t)p Fq(\))2319 649 y Fm(1)p Fo(=)p Fm(4)2439 609 y Fp(;)1139 b Fq(\(1.12\))149 824 y Ff(and)1110 1061 y Fn(k)p Fp(u)1207 1076 y Fo(d)1248 1061 y Fq(\()p Fp(t)p Fq(\))p Fn(k)1396 1084 y Fo(L)1444 1061 y Fi(2)1444 1105 y Fk(\000)p Fh(\033)1621 1061 y Fn(\024)1897 999 y Fp(C)1962 1013 y Fm(1)2001 999 y Fp(")p 1785 1040 371 4 v 1785 1127 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2045 1100 y Fm(3)p Fo(=)p Fm(2)2185 1061 y Fq(+)2470 999 y Fp(C)2535 1013 y Fm(2)2574 999 y Fp(")2616 966 y Fm(3)p 2286 1040 555 4 v 2286 1127 a Fq(\(1)h(+)f(4\000)p Fp(")2622 1100 y Fm(4)2662 1127 y Fp(t)p Fq(\))2730 1100 y Fm(3)p Fo(=)p Fm(4)3603 1061 y Fq(\(1.13\))1165 1320 y Fn(k)p Fp(u)1262 1335 y Fo(d)1303 1320 y Fq(\()p Fp(t)p Fq(\))p Fn(k)1451 1340 y Fo(L)1499 1321 y Fi(4)1621 1320 y Fn(\024)1897 1258 y Fp(C)1962 1272 y Fm(3)2001 1258 y Fp(")p 1785 1299 371 4 v 1785 1386 a Fq(\(1)h(+)f Fp(t)p Fq(\))2045 1360 y Fm(3)p Fo(=)p Fm(4)2185 1320 y Fq(+)2470 1258 y Fp(C)2535 1272 y Fm(4)2574 1258 y Fp(")2616 1225 y Fm(2)p 2286 1299 555 4 v 2286 1386 a Fq(\(1)h(+)f(4\000)p Fp(")2622 1360 y Fm(4)2662 1386 y Fp(t)p Fq(\))2730 1360 y Fm(1)p Fo(=)p Fm(2)2850 1320 y Fp(;)728 b Fq(\(1.14\))149 1572 y Ff(with)34 b(some)f(p)-5 b(ositive)34 b Fp(\017)p Ff(-indep)-5 b(endent)33 b(c)-5 b(onstants)35 b Fp(C)1945 1586 y Fm(0)1984 1572 y Ff({)p Fp(C)2095 1586 y Fm(4)2135 1572 y Ff(.)149 1763 y Fg(Remark)e(1.2.)41 b Ff(Sinc)-5 b(e)32 b Fp( )1051 1730 y Fm(3)1116 1763 y Fn(2)25 b Fp(L)1264 1730 y Fm(2)1264 1785 y Fo(\033)1310 1763 y Fq(\()p Fl(R)1405 1730 y Fm(3)1451 1763 y Fq(\))32 b Ff(due)f(to)h(R)-5 b(emark)33 b(1.1,)f(the)g(p)-5 b(ar)g(ameter)34 b Fq(\000)d Ff(is)g(wel)5 b(l-de\014ne)-5 b(d.)43 b(Mor)-5 b(e-)149 1879 y(over:)981 1995 y Fq(Im)p Fn(h)p Fp( )1187 1958 y Fm(3)1227 1995 y Fp(;)15 b Fq(\()p Fn(H)22 b(\000)e Fp(\026)g Fn(\000)g Fp(i)p Fq(0\))1769 1958 y Fy(\000)p Fm(1)1864 1995 y Fn(P)1927 2009 y Fo(c)1962 1995 y Fp( )2024 1958 y Fm(3)2064 1995 y Fn(i)26 b Fq(=)f Fp(\031)s Fn(h)p Fp( )2373 1958 y Fm(3)2413 1995 y Fp(;)15 b(E)2525 1958 y Fy(0)2549 1995 y Fq(\()p Fp(\026)p Fq(\))p Fp( )2736 1958 y Fm(3)2776 1995 y Fn(i)26 b(\025)f Fq(0)p Fp(;)149 2165 y Ff(wher)-5 b(e)40 b Fp(E)484 2132 y Fy(0)508 2165 y Fq(\()p Fp(\026)p Fq(\))p Ff(,)g Fp(\026)d Fn(2)f Fl(R)47 b Ff(is)39 b(the)g(sp)-5 b(e)g(ctr)g(al)41 b(me)-5 b(asur)g(e)40 b(induc)-5 b(e)g(d)40 b(by)e Fn(H)f Fq(=)f Fn(\000)p Fq(\001)25 b(+)f Fp(V)c Fq(\()p Fp(x)p Fq(\))p Ff(,)41 b(se)-5 b(e)39 b([5].)60 b(F)-7 b(or)40 b Fp(\026)c Fq(=)149 2282 y Fp(\025)21 b Fq(+)f Fp(!)28 b(>)d Fq(0)33 b Ff(the)g(ab)-5 b(ove)33 b(expr)-5 b(ession)34 b(is)f(generic)-5 b(al)5 b(ly)33 b(p)-5 b(ositive.)149 2472 y Fg(Remark)35 b(1.3.)42 b Ff(A)n(lthough)33 b(we)g(b)-5 b(elieve)32 b(that)i(the)f(de)-5 b(c)g(ay)34 b(estimate)f(\(1.14\))h(c)-5 b(an)34 b(b)-5 b(e)32 b(sharp)-5 b(ene)g(d)35 b(to:)1195 2740 y Fn(k)p Fp(u)1292 2755 y Fo(d)1333 2740 y Fq(\()p Fp(t)p Fq(\))p Fn(k)1481 2760 y Fo(L)1529 2741 y Fi(4)1594 2740 y Fn(\024)1832 2656 y Fq(~)1811 2679 y Fp(C)1876 2693 y Fm(3)1916 2679 y Fp(")p 1700 2719 371 4 v 1700 2806 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))1960 2780 y Fm(3)p Fo(=)p Fm(4)2100 2740 y Fq(+)2405 2656 y(~)2385 2679 y Fp(C)2450 2693 y Fm(4)2489 2679 y Fp(")2531 2646 y Fm(3)p 2201 2719 555 4 v 2201 2806 a Fq(\(1)h(+)f(4\000)p Fp(")2537 2780 y Fm(4)2577 2806 y Fp(t)p Fq(\))2645 2780 y Fm(3)p Fo(=)p Fm(4)2765 2740 y Fp(;)813 b Fq(\(1.15\))149 2997 y Ff(we)33 b(have)g(no)g(pr)-5 b(o)g(of)35 b(of)e(the)g(impr)-5 b(ove)g(d)35 b(estimate)e(\(1.15\).) 291 3188 y Fq(The)h(existence)i(of)f(an)g(asymptotically)g(stable)f (cen)m(ter)j(manifold)c(\(nonlinear)g(ground)h(state\))j(that)149 3304 y(bifurcates)28 b(from)h(the)g(linear)e(ground)h(state)i Fp( )s Fq(\()p Fp(x)p Fq(\))g(w)m(as)g(pro)m(v)m(ed)f(in)e(the)j(case)g Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))25 b(=)g Fp(\015)3169 3318 y Fm(0)3238 3304 y Fq(in)i([17)q(],)j(see)g(also)149 3420 y([19)r(,)25 b(20)q(].)39 b(As)24 b(a)i(consequence,)h(an)m(y)e (solution)e(of)i(the)g(initial-v)-5 b(alue)22 b(problem)h(\(1.1\))k(in) c(the)i(case)h Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(\015)3796 3434 y Fm(0)149 3536 y Fq(with)37 b(su\016cien)m(tly)g (small)g(initial)f(data)j Fp(u)1611 3550 y Fm(0)1650 3536 y Fq(\()p Fp(x)p Fq(\))g(in)e(\(1.10\))k(con)m(v)m(erges)f(to)f (the)g(nonlinear)d(ground)h(state.)149 3653 y(Theorem)h(1)g(sho)m(ws)f (that)h(the)g(cen)m(tral)g(manifold)e(is)g(destro)m(y)m(ed)j(b)m(y)e (the)h(time)g(p)s(erio)s(dic)d(p)s(erturbation)149 3769 y(\(1.2\))41 b(with)d Fp(\015)638 3783 y Fm(1)718 3769 y Fn(6)p Fq(=)i(0.)67 b(The)39 b(argumen)m(t)h(can)f(b)s(e)g(easily)f (adapted)i(to)g(handle)e(a)h(general)h(p)s(erio)s(dic)c(and)149 3885 y(almost)31 b(p)s(erio)s(dic)d(functions)h Fp(\015)5 b Fq(\()p Fp(t)p Fq(\).)291 4001 y(F)-8 b(or)38 b(the)g(pro)s(of)f(of)h (Theorem)f(1,)j(w)m(e)e(decomp)s(ose)g(the)g(solution)e Fp(u)p Fq(\()p Fp(t;)15 b(x)p Fq(\))39 b(on)m(to)g(pro)5 b(jections)37 b(on)g(the)149 4118 y(ground)c(state)i Fp( )s Fq(\()p Fp(x)p Fq(\))f(and)f(the)h(con)m(tin)m(uum)e(sp)s (ectrum)h(of)g(the)h(linear)e(disp)s(ersiv)m(e)f(op)s(erator)i Fn(H)q Fq(,)h(accord-)149 4234 y(ing)j(to)h(the)f(represen)m(tation)g (\(1.11\).)64 b(The)37 b(non-degeneracy)h(condition)d(\(1.9\))k(states) g(that)f(there)f(is)f(a)149 4350 y(non)m(trivial)25 b(F)-8 b(ermi)26 b(Golden)f(Rule)h(for)g(the)g(parametric)g(resonance)h(b)s (et)m(w)m(een)g Fp( )2897 4317 y Fm(3)2937 4350 y Fq(\()p Fp(x)p Fq(\))g(and)e(the)i(con)m(tin)m(uous)149 4466 y(sp)s(ectrum)j(of)g Fn(H)q Fq(,)g(whic)m(h)f(is)h(induced)e(b)m(y)i (the)h(time)f(frequency)g Fp(!)j Fq(of)e(the)f(nonlinearit)m(y)e(co)s (e\016cien)m(t)k Fp(\015)5 b Fq(\()p Fp(t)p Fq(\).)149 4582 y(W)-8 b(e)36 b(emplo)m(y)f(disp)s(ersiv)m(e)d(estimates,)37 b(con)m(traction)f(principles)31 b(in)j(a)h(hierarc)m(h)m(y)f(of)h (Banac)m(h)h(spaces)f(and)149 4699 y(comparison)d(results)e(for)i (ODE's)g(to)h(pro)m(v)m(e)g(that)f(the)g(resonance)h(leads)e(to)i(a)f (slo)m(w)g(destruction)f(of)h(an)m(y)149 4815 y(small)d(lo)s(calized)h (initial)d(v)-5 b(alue)30 b Fp(u)1308 4829 y Fm(0)1347 4815 y Fq(\()p Fp(x)p Fq(\),)i(according)e(to)h(the)g(deca)m(y)g(rates) g(\(1.12\){\(1.14\).)291 4931 y(Theorem)i(1)i(can)f(b)s(e)g(view)m(ed)f (as)h(a)h(nonlinear)d(generalization)i(of)g(the)g(linear)f(quan)m(tum)g (resonance)149 5047 y(results)21 b(in)g([2)q(,)h(7,)h(8,)f(9)q(,)g(21)q (,)g(26)q(].)38 b(Nonlinear)21 b(F)-8 b(ermi)21 b(Golden)h(rule)e(and)i (nonlinear)e(resonances)i(for)g(the)g(NLS)149 5164 y(equation)30 b(\(1.1\))h(with)e Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(\015)1267 5178 y Fm(0)1336 5164 y Fq(w)m(ere)30 b(in)m(tro)s(duced)e(in)h([18)q(,)h(22)q(].)40 b(Relaxation)30 b(of)g(excited)g(b)s(ound)d(states)1970 5425 y(3)p eop %%Page: 4 4 4 3 bop 149 75 a Fq(to)29 b(the)e(ground)g(state)i(due)e(to)h (nonlinear)d(resonances)j(w)m(as)g(recen)m(tly)g(studied)e(in)g([23)q (,)i(25)q(].)40 b(Our)26 b(analysis)149 191 y(of)45 b(global)f (existence)h(of)f(solutions)f(is)g(di\013eren)m(t)h(from)g([23)q(,)h (25)q(],)j(since)c(no)g(conserv)-5 b(ation)45 b(of)f(energy)149 307 y(exists)c(in)e(the)i(time)g(ev)m(olution)f(of)h(the)g(NLS)g (equation)f(\(1.1\))j(with)c(the)i(time-p)s(erio)s(dic)e(nonlinearit)m (y)149 423 y(co)s(e\016cien)m(t)c Fp(\015)5 b Fq(\()p Fp(t)p Fq(\).)50 b(In)33 b(addition,)f(w)m(e)h(use)g(metho)s(ds)g (based)g(on)g(con)m(traction)h(principles,)d(whic)m(h)g(pro)m(vide)149 540 y(not)42 b(only)e(a)i(more)f(systematical)h(treatmen)m(t)h(of)e (the)h(resonance)g(but)e(also)h(sharp)s(er)f(information)g(on)149 656 y(the)k(ev)m(olution)g(of)g(solution,)i(when)c(they)i(are)h (compared)e(to)i(previously)c(emplo)m(y)m(ed)j(b)s(o)s(otstrapping)149 772 y(argumen)m(ts.)39 b(Consequen)m(tly)-8 b(,)23 b(w)m(e)f(can)g (extend)f(our)g(results)g(to)h(the)g(case,)i(when)d Fp(\025)s Fq(+)s Fp(!)27 b(<)e Fq(0)d(but)f Fp(\025)s Fq(+)s Fp(n!)26 b(>)f Fq(0)149 888 y(for)j(some)g(in)m(teger)g Fp(n)p Fq(.)39 b(This)26 b(case)i(corresp)s(onds)e(to)i(a)g(higher-order)e (resonance)i(and)f(it)g(leads)g(to)i(a)e(slo)m(w)m(er)149 1005 y(deca)m(y)32 b(rate)f(of)g(the)f(solution)f Fp(u)p Fq(\()p Fp(t;)15 b(x)p Fq(\))31 b(lik)m(e)f Fp(t)1671 972 y Fy(\000)p Fm(1)p Fo(=)p Fm(\(4)p Fo(n)p Fm(\))1933 1005 y Fq(.)291 1121 y(The)f(pap)s(er)h(is)f(structured)g(as)i(follo)m (ws.)40 b(Section)30 b(2)g(giv)m(es)h(lo)s(cal)f(existence)g(results)g (for)g(the)g(problem)149 1237 y(\(1.1\).)42 b(The)25 b(pro)s(of)h(of)g(Theorem)g(1)h(is)e(giv)m(en)i(in)e(Section)h(3,)i (where)d(w)m(e)i(decomp)s(ose)g(the)f(problem)f(\(1.1\))j(in)149 1353 y(the)e(form)f(of)h(a)g(coupled)e(system)i(of)f(t)m(w)m(o)i (di\013eren)m(tial)d(equations.)39 b(The)25 b(pro)s(of)f(relies)g(on)i (analysis)e(of)h(eac)m(h)149 1469 y(of)h(the)g(t)m(w)m(o)i(equations)d (whic)m(h)g(is)f(rep)s(orted)i(in)e(Sections)i(4)g(and)f(5,)i(resp)s (ectiv)m(ely)-8 b(.)39 b(The)26 b Fp(H)3254 1436 y Fm(1)3319 1469 y Fq(estimates)g(are)149 1586 y(needed)g(to)g(prev)m(en)m(t)g(the) g(\014nite-time)e(blo)m(w)h(up)f(and)h(they)h(are)g(giv)m(en)f(in)f (Section)i(6.)39 b(Section)25 b(7)h(concludes)149 1702 y(the)33 b(pap)s(er)e(with)g(generalizations)h(and)f(op)s(en)h (problems.)45 b(App)s(endix)29 b(A)j(presen)m(ts)h(applications)d(of)i (the)149 1818 y(con)m(traction)g(principle)27 b(in)i(a)i(hierarc)m(h)m 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b(for)f(every)g Fp(u)3387 2831 y Fm(0)3452 2817 y Fn(2)d Fp(H)3620 2784 y Fm(2)3660 2817 y Fq(\()p Fl(R)3755 2784 y Fm(3)3800 2817 y Fq(\))149 2933 y Ff(ther)-5 b(e)43 b(exists)e(a)h(unique)e(solution)i Fp(u)p Fq(\()p Fp(t)p Fq(\))g Ff(of)f(the)h(initial)g(value)f(pr)-5 b(oblem)43 b(\(1.1\))f(de\014ne)-5 b(d)42 b(on)g(a)g(maximal)149 3049 y(interval)34 b Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)790 3063 y Fm(max)926 3049 y Fq(\))33 b Ff(such)g(that)1163 3223 y Fp(u)26 b Fn(2)e Fp(C)1398 3186 y Fm(1)1452 3149 y Fe(\000)1494 3223 y Fq([0)p Fp(;)15 b(T)1657 3237 y Fm(max)1794 3223 y Fq(\))p Fp(;)g(L)1931 3186 y Fm(2)1971 3149 y Fe(\001)2033 3223 y Fn(\\)20 b Fp(C)2200 3149 y Fe(\000)2242 3223 y Fq([0)p Fp(;)15 b(T)2405 3237 y Fm(max)2542 3223 y Fq(\))p Fp(;)g(H)2700 3186 y Fm(2)2740 3149 y Fe(\001)2797 3223 y Fp(;)827 b Fq(\(2.1\))149 3397 y Ff(wher)-5 b(e)1092 3513 y Fq(either)50 b Fp(T)1423 3527 y Fm(max)1584 3513 y Fq(=)25 b(+)p Fn(1)50 b Fq(or)129 b(lim)2039 3574 y Fo(t)p Fy(!)p Fo(T)2176 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5139 y Fn(T)i Fq(\()p Fp(t)f Fn(\000)f Fp(s)p Fq(\))p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)2606 5101 y Fm(2)2646 5139 y Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(ds:)723 b Fq(\(2.4\))1970 5425 y(4)p eop %%Page: 5 5 5 4 bop 149 75 a Fq(Since)43 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))44 b(is)f(uniformly)d(b)s(ounded,)45 b(the)f(map)f Fp(F)60 b Fq(=)47 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(u)p Fn(j)2422 42 y Fm(2)2463 75 y Fp(u)47 b Fq(:)g([0)p Fp(;)15 b Fn(1)p Fq(\))30 b Fn(\002)f Fp(H)3083 42 y Fm(2)3170 75 y Fn(!)47 b Fp(H)3391 42 y Fm(2)3473 75 y Fq(is)c(lo)s(cally)149 191 y(Lipsc)m(hitz,)32 b(b)m(y)g(the)g (generalization)g(of)h(Lemma)f(8.1.2)i(in)d([12)q(])h(to)h Fp(x)28 b Fn(2)g Fl(R)2708 158 y Fm(3)2753 191 y Fq(.)46 b(The)32 b(con)m(traction)h(principle)149 307 y(for)e(the)f(in)m (tegral)g(equation)g(\(2.4\))j(\014nishes)28 b(the)i(pro)s(of,)g(see)h (Theorem)f(6.1.7)i(in)e([12)q(].)p 3770 307 4 62 v 3774 249 55 4 v 3774 307 V 3828 307 4 62 v 149 497 a Fg(Theorem)35 b(3.)42 b Ff(Assume)34 b(that)g(b)-5 b(oth)35 b(c)-5 b(onditions)35 b(\(1.4\))f(and)g(\(1.5\))h(b)-5 b(e)33 b(satis\014e)-5 b(d.)45 b(Then,)34 b(for)g(every)f Fp(u)3709 511 y Fm(0)3775 497 y Fn(2)149 613 y Fp(H)232 580 y Fm(1)272 613 y Fq(\()p Fl(R)367 580 y Fm(3)412 613 y Fq(\))45 b Ff(ther)-5 b(e)45 b(exists)g(a)g(unique)e(solution)j Fp(u)p Fq(\()p Fp(t)p Fq(\))f Ff(of)g(the)f(initial-value)h(pr)-5 b(oblem)46 b(\(1.1\))g(de\014ne)-5 b(d)45 b(on)g(a)149 729 y(maximal)35 b(interval)e Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)1157 743 y Fm(max)1294 729 y Fq(\))33 b Ff(such)g(that)1125 930 y Fp(u)26 b Fn(2)e Fp(C)1360 892 y Fm(1)1414 856 y Fe(\000)1456 930 y Fq([0)p Fp(;)15 b(T)1619 944 y Fm(max)1756 930 y Fq(\))p Fp(;)g(H)1914 892 y Fy(\000)p Fm(1)2009 856 y Fe(\001)2071 930 y Fn(\\)20 b Fp(C)2238 856 y Fe(\000)2280 930 y Fq([0)p Fp(;)15 b(T)2443 944 y Fm(max)2580 930 y Fq(\))p Fp(;)g(H)2738 892 y Fm(1)2778 856 y Fe(\001)2835 930 y Fp(;)789 b Fq(\(2.5\))149 1130 y Ff(wher)-5 b(e)1092 1246 y Fq(either)50 b Fp(T)1423 1260 y Fm(max)1584 1246 y Fq(=)25 b(+)p Fn(1)50 b Fq(or)129 b(lim)2039 1307 y Fo(t)p Fy(!)p Fo(T)2176 1315 y Fi(max)2307 1246 y Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)2552 1266 y Fo(H)2615 1247 y Fi(1)2681 1246 y Fq(=)25 b Fn(1)p Fp(:)756 b Fq(\(2.6\))149 1451 y Ff(Mor)-5 b(e)g(over,)39 b Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)827 1471 y Fo(L)875 1452 y Fi(2)947 1451 y Fn(\021)32 b(k)p Fp(u)p Fq(\(0\))p Fn(k)1307 1471 y Fo(L)1355 1452 y Fi(2)1396 1451 y Ff(,)37 b Fn(8)p Fp(t)31 b Fn(2)h Fq([0)p Fp(;)15 b(T)1832 1465 y Fm(max)1968 1451 y Fq(\))p Ff(,)38 b(and)f Fp(u)p Fq(\()p Fp(t)p Fq(\))g Ff(dep)-5 b(ends)38 b(c)-5 b(ontinuously)37 b(on)g(the)g (initial)149 1567 y(data,)32 b(i.e.)41 b(if)29 b Fq(lim)755 1581 y Fo(n)p Fy(!1)959 1567 y Fp(u)1011 1534 y Fo(n)1011 1592 y Fm(0)1083 1567 y Fq(=)c Fp(u)1231 1581 y Fm(0)1300 1567 y Ff(in)30 b Fp(H)1492 1534 y Fm(1)1531 1567 y Fq(\()p Fl(R)1626 1534 y Fm(3)1672 1567 y Fq(\))g Ff(then)g(for)h(any)f(close) -5 b(d)31 b(interval)g Fp(I)h Fn(\032)25 b Fq([0)p Fp(;)15 b(T)3174 1581 y Fm(max)3311 1567 y Fq(\))30 b Ff(the)g(solution)149 1684 y Fp(u)201 1651 y Fo(n)248 1684 y Fq(\()p Fp(t)p Fq(\))41 b Ff(of)f(the)h(pr)-5 b(oblem)42 b(\(1.1\))f(with)g(initial)g (data)h Fp(u)1990 1651 y Fo(n)1990 1708 y Fm(0)2077 1684 y Ff(is)e(de\014ne)-5 b(d)41 b(on)g Fp(I)47 b Ff(for)41 b(su\016ciently)f(lar)-5 b(ge)41 b Fp(n)p Ff(,)h(and)149 1800 y Fq(lim)276 1814 y Fo(n)p Fy(!1)479 1800 y Fp(u)531 1767 y Fo(n)578 1800 y Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(u)p Fq(\()p Fp(t)p Fq(\))33 b Ff(in)f Fp(C)7 b Fq(\()p Fp(I)g(;)15 b(H)1379 1767 y Fm(1)1419 1800 y Fq(\))p Ff(.)149 1985 y(Pr)-5 b(o)g(of.)47 b Fq(Since)24 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))25 b(is)e(uniformly)f(b)s(ounded)g(on)j([0)p Fp(;)15 b Fn(1)p Fq(\),)27 b(the)e(argumen)m(t)f(of)h(Theorem)f(4.4.6)j (and)c(Remark)149 2101 y(4.4.8)33 b(in)c([1])i(is)e(applied)g(to)i(the) f(pro)s(of)g(of)g(theorem.)p 3770 2101 V 3774 2043 55 4 v 3774 2101 V 3828 2101 4 62 v 149 2386 a Fr(3)135 b(Global)46 b(existence)g(and)f(long)g(time)h(b)t(eha)l(vior)149 2593 y Fq(W)-8 b(e)42 b(decomp)s(ose)f(here)g(the)g(solution)e Fp(u)p Fq(\()p Fp(t)p Fq(\))i(in)m(to)g(the)g(slo)m(w)f(and)g(rapidly)e (v)-5 b(arying)40 b(parts.)72 b(Using)40 b(this)149 2709 y(setting,)33 b(w)m(e)f(pro)m(v)m(e)g(our)f(main)f(Theorem)h(1.)45 b(By)32 b(Theorems)f(2)h(or)f(3,)h(the)g(solution)e(of)i(the)f (initial-v)-5 b(alue)149 2825 y(problem)38 b(\(1.1\))i(satis\014es)e Fp(u)p Fq(\()p Fp(t)p Fq(\))i Fn(2)f Fp(L)1437 2792 y Fm(2)1476 2825 y Fq(\()p Fl(R)1571 2792 y Fm(3)1617 2825 y Fq(\))g(for)f(an)m(y)h Fp(t)g Fn(2)g Fq([0)p Fp(;)15 b(T)2353 2839 y Fm(max)2490 2825 y Fq(\).)66 b(By)39 b(the)g(Sp)s(ectral)f(Theorem)g(and)149 2941 y(Assumption)25 b(1,)i(an)m(y)g Fp(L)976 2908 y Fm(2)1041 2941 y Fq(function)e(can)h(b) s(e)g(written)f(as)h(a)g(sum)g(of)g(its)f(pro)5 b(jections)26 b(on)m(to)h(the)f(eigen)m(v)m(ector)149 3058 y Fp( )34 b Fq(and)c(the)h(con)m(tin)m(uous)f(sp)s(ectrum)f(of)h Fn(H)c Fq(=)f Fn(\000)p Fq(\001)20 b(+)g Fp(V)g Fq(.)41 b(Hence)31 b(w)m(e)g(can)f(write:)1475 3258 y Fp(u)p Fq(\()p Fp(t)p Fq(\))c(=)f Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp(e)1965 3221 y Fy(\000)p Fo(i\025t)2115 3258 y Fp( )f Fq(+)c Fp(u)2341 3273 y Fo(d)2381 3258 y Fq(\()p Fp(t)p Fq(\))p Fp(;)1140 b Fq(\(3.1\))149 3458 y(where)31 b Fp(A)p Fq(\()p Fp(t)p Fq(\))26 b Fn(2)f Fl(C)55 b Fq(and)30 b Fp(u)1022 3473 y Fo(d)1062 3458 y Fq(\()p Fp(t)p Fq(\))d(=)e Fn(P)1351 3472 y Fo(c)1386 3458 y Fp(u)p Fq(\()p Fp(t)p Fq(\).)42 b(Using)30 b(the)h(orthogonalit)m(y)g(b)s(et)m(w)m(een)g Fp( )k Fq(and)30 b(the)h(range)g(of)f Fn(P)3775 3472 y Fo(c)3810 3458 y Fq(,)149 3575 y(the)h(NLS)f(equation)g(\(1.1\))i(b)s (ecomes)f(the)f(coupled)g(system)g(of)h(equations)f(in)f Fp(A)p Fq(\()p Fp(t)p Fq(\))i(and)f Fp(u)3301 3590 y Fo(d)3341 3575 y Fq(\()p Fp(t)p Fq(\):)1455 3752 y(_)1421 3775 y Fp(A)83 b Fq(=)g Fn(\000)p Fp(i\015)5 b Fq(\()p Fp(t)p Fq(\))p Fp(e)2025 3738 y Fo(i\025t)2121 3775 y Fn(h)p Fp( )s(;)15 b Fn(j)p Fp(u)p Fn(j)2360 3738 y Fm(2)2401 3775 y Fp(u)p Fn(i)1161 b Fq(\(3.2\))1413 3916 y(_)-41 b Fp(u)1449 3931 y Fo(d)1572 3916 y Fq(=)83 b Fn(\000)p Fp(i)p Fn(H)q Fp(u)1958 3931 y Fo(d)2018 3916 y Fn(\000)20 b Fp(i\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(P)2358 3930 y Fo(c)2394 3916 y Fn(j)p Fp(u)p Fn(j)2496 3879 y Fm(2)2536 3916 y Fp(u)1061 b Fq(\(3.3\))149 4117 y(where)942 4317 y Fn(j)p Fp(u)p Fn(j)1044 4279 y Fm(2)1084 4317 y Fp(u)83 b Fq(=)g Fn(j)p Fp(A)p Fn(j)1491 4279 y Fm(2)1531 4317 y Fp(Ae)1641 4279 y Fy(\000)p Fo(i\025t)1791 4317 y Fp( )1853 4279 y Fm(3)1913 4317 y Fq(+)20 b(2)p Fn(j)p Fp(A)p Fn(j)2167 4279 y Fm(2)2208 4317 y Fp( )2270 4279 y Fm(2)2310 4317 y Fp(u)2362 4332 y Fo(d)2423 4317 y Fq(+)f Fp(A)2581 4279 y Fm(2)2621 4317 y Fp(e)2663 4279 y Fy(\000)p Fm(2)p Fo(i\025t)2848 4317 y Fp( )2910 4279 y Fm(2)p 2950 4267 53 4 v 2950 4317 a Fp(u)3002 4332 y Fo(d)1219 4458 y Fq(+)83 b(2)p Fp(Ae)1528 4421 y Fy(\000)p Fo(i\025t)1679 4458 y Fp( )s Fn(j)p Fp(u)1818 4473 y Fo(d)1859 4458 y Fn(j)1884 4421 y Fm(2)1944 4458 y Fq(+)p 2035 4385 69 4 v 20 w Fp(Ae)2145 4421 y Fo(i\025t)2240 4458 y Fp( )s(u)2354 4421 y Fm(2)2354 4481 y Fo(d)2415 4458 y Fq(+)20 b Fn(j)p Fp(u)2583 4473 y Fo(d)2624 4458 y Fn(j)2649 4421 y Fm(2)2689 4458 y Fp(u)2741 4473 y Fo(d)2781 4458 y Fp(:)843 b Fq(\(3.4\))149 4659 y(In)30 b(the)h(linear)d(case,)k(when)d Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))26 b Fn(\021)f Fq(0,)31 b(w)m(e)g(ha)m(v)m(e) 1186 4859 y Fp(A)p Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(A)1547 4873 y Fm(0)1612 4859 y Fq(=)f Fn(h)p Fp( )s(;)15 b(u)1896 4873 y Fm(0)1937 4859 y Fn(i)p Fp(;)198 b(u)2247 4874 y Fo(d)2287 4859 y Fq(\()p Fp(t)p Fq(\))26 b(=)f Fn(T)e(P)2648 4873 y Fo(c)2683 4859 y Fp(u)2735 4873 y Fm(0)2774 4859 y Fp(;)850 b Fq(\(3.5\))149 5059 y(where)34 b Fn(T)23 b Fq(\()p Fp(t)p Fq(\))31 b(=)h Fp(e)768 5026 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)970 5059 y Fq(is)i(the)g(group)f(of)h(unitary)f(op)s (erators)h(generated)h(b)m(y)f Fn(\000)p Fp(i)p Fn(H)h Fq(on)f Fp(L)3279 5026 y Fm(2)3318 5059 y Fq(\()p Fl(R)3413 5026 y Fm(3)3458 5059 y Fq(\).)52 b(Hence,)149 5176 y(the)36 b(solution)f Fp(u)p Fq(\()p Fp(t)p Fq(\))h(is)e(a)j(sup)s(erp)s (osition)32 b(of)k(the)f(b)s(ound)f(state)j Fp(A)2449 5190 y Fm(0)2489 5176 y Fp(e)2531 5143 y Fy(\000)p Fo(i\025t)2681 5176 y Fp( )s Fq(\()p Fp(x)p Fq(\))g(and)e(the)g(disp)s(ersiv)m(e)f (part)1970 5425 y(5)p eop %%Page: 6 6 6 5 bop 149 75 a Fp(u)201 90 y Fo(d)242 75 y Fq(\()p Fp(t)p Fq(\))35 b(=)f Fp(e)527 42 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)696 75 y Fn(P)759 89 y Fo(c)794 75 y Fp(u)846 89 y Fm(0)886 75 y Fq(.)57 b(Under)35 b(the)h(deca)m(y)h(condition)d (\(1.4\),)40 b(the)c(disp)s(ersiv)m(e)d(part)j(satis\014es)f(the)h (deca)m(y)149 191 y(estimates)31 b(for)g(all)e Fp(t)c Fn(\025)g Fq(0:)1075 390 y Fn(k)p Fp(e)1162 352 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1332 390 y Fn(P)1395 404 y Fo(c)1430 390 y Fp(f)10 b Fn(k)1530 413 y Fo(L)1578 391 y Fi(2)1578 434 y Fk(\000)p Fh(\033)1754 390 y Fn(\024)2045 328 y Fp(D)2120 342 y Fm(1)p 1918 369 371 4 v 1918 456 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2178 430 y Fm(3)p Fo(=)p Fm(2)2298 390 y Fn(k)p Fp(f)10 b Fn(k)2443 409 y Fo(L)2491 391 y Fi(2)2491 426 y Fh(\033)2537 390 y Fp(;)106 b(\033)29 b(>)2855 328 y Fq(5)p 2855 369 46 4 v 2855 452 a(2)3649 390 y(\(3.6\))1130 636 y Fn(k)p Fp(e)1217 598 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1387 636 y Fn(P)1450 650 y Fo(c)1485 636 y Fp(f)10 b Fn(k)1585 655 y Fo(L)1633 636 y Fi(4)91 b Fn(\024)2045 574 y Fp(D)2120 588 y Fm(2)p 1918 615 371 4 v 1918 702 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2178 675 y Fm(3)p Fo(=)p Fm(4)2298 636 y Fn(k)p Fp(f)10 b Fn(k)2443 661 y Fo(L)2491 642 y Fi(4)p Fh(=)p Fi(3)2592 636 y Fp(;)1032 b Fq(\(3.7\))1100 881 y Fn(k)p Fp(e)1187 844 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1356 881 y Fn(P)1419 895 y Fo(c)1454 881 y Fp(f)10 b Fn(k)1554 895 y Fo(L)1602 876 y Fk(1)1754 881 y Fn(\024)2045 820 y Fp(D)2120 834 y Fm(3)p 1918 860 V 1918 948 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2178 921 y Fm(3)p Fo(=)p Fm(2)2298 881 y Fn(k)p Fp(f)10 b Fn(k)2443 901 y Fo(L)2491 882 y Fi(1)2530 881 y Fp(;)1094 b Fq(\(3.8\))149 1101 y(where)30 b Fp(D)487 1115 y Fm(1)527 1101 y Fq(,)g Fp(D)657 1115 y Fm(2)696 1101 y Fq(,)h(and)e Fp(D)1003 1115 y Fm(3)1072 1101 y Fq(are)i(some)f(p)s(ositiv)m(e)f(constan)m(ts.)42 b(Estimates)30 b(\(3.6\))h(and)f(\(3.8\))h(w)m(ere)g(pro)m(v)m(ed)f(in)149 1218 y([5)q(])j(and)g([3)q(],)h(resp)s(ectiv)m(ely)-8 b(.)49 b(Estimate)33 b(\(3.7\))i(follo)m(ws)d(from)h(\(3.8\))i(b)m(y)e (the)g(Riesz-Thorin)e(in)m(terp)s(olation)149 1334 y(argumen)m(t,)c (since)e Fp(e)837 1301 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1031 1334 y Fq(is)f(unitary)g(on)h Fp(L)1616 1301 y Fm(2)1655 1334 y Fq(\()p Fl(R)1750 1301 y Fm(3)1795 1334 y Fq(\).)40 b(The)24 b(estimates)i(\(3.6\){\(3.8\))j(imply)23 b(that)i Fp(u)3408 1349 y Fo(d)3449 1334 y Fq(\()p Fp(t)p Fq(\))h(deca)m(ys)149 1450 y(in)e(the)h(same)g(w)m(a)m(y)g(as)g(the)g(solution)e(of)i(the)g (free)f(Sc)m(hr\177)-45 b(odinger)23 b(equation)i(in)e(w)m(eigh)m(ted)i Fp(L)3206 1417 y Fm(2)3270 1450 y Fq(and)f Fp(L)3503 1417 y Fo(p)3567 1450 y Fq(norms.)149 1566 y(W)-8 b(e)22 b(note)f(that)g(sligh)m(tly)e(stronger)h(assumptions)f(on)h Fp(V)h Fq(\()p Fp(x)p Fq(\))g(w)m(ould)e(imply)f(that)j Fp(u)2907 1581 y Fo(d)2948 1566 y Fq(\()p Fp(t)p Fq(\))f(con)m(v)m (erges)j(as)d Fp(t)26 b Fn(!)f(1)149 1682 y Fq(to)32 b(the)e(solution)f(of)i(the)f(free)h(Sc)m(hr\177)-45 b(odinger)29 b(equation)h(in)f(the)h(ab)s(o)m(v)m(e)i(spaces,)f(see)g ([27)q(].)291 1799 y(W)-8 b(e)32 b(consider)e(the)h(resolv)m(en)m(t)h (\()p Fn(H)22 b(\000)e Fp(\026)g Fn(\000)h Fp(i\016)s Fq(\))1844 1766 y Fy(\000)p Fm(1)1940 1799 y Fn(P)2003 1813 y Fo(c)2038 1799 y Fp(;)15 b(\016)30 b(>)c Fq(0)32 b(as)f(a)h(map)e(from)h Fp(L)2991 1766 y Fm(2)2991 1821 y Fo(\033)3038 1799 y Fq(\()p Fl(R)3133 1766 y Fm(3)3178 1799 y Fq(\))h(to)f Fp(L)3418 1766 y Fm(2)3418 1821 y Fy(\000)p Fo(\033)3520 1799 y Fq(\()p Fl(R)3615 1766 y Fm(3)3660 1799 y Fq(\))h(for)149 1915 y(some)f Fp(\033)d(>)563 1879 y Fm(5)p 563 1894 36 4 v 563 1946 a(2)609 1915 y Fq(.)40 b(By)30 b(Sections)g(3)h(and)e(8)i(in)d([7)q(],)j(the)f(resolv) m(en)m(t)h(con)m(v)m(erges)h(strongly)d(as)i Fp(")25 b Fn(&)g Fq(0)31 b(ev)m(en)g(when)149 2031 y Fp(\026)g Fq(is)e(in)g(the)i(sp)s(ectrum)e(of)h Fn(H)h Fq(and)f(it)g(satis\014es) g(the)g(follo)m(wing)f(deca)m(y)j(estimates)f(for)f(all)f Fp(t)c Fn(\025)g Fq(0:)621 2235 y Fn(k)p Fp(e)708 2198 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)877 2235 y Fq(\()p Fn(H)c(\000)f Fp(\026)g Fn(\000)g Fp(i)p Fq(0\))1378 2198 y Fy(\000)p Fm(1)1474 2235 y Fn(P)1537 2249 y Fo(c)1572 2235 y Fp(f)10 b Fn(k)1672 2259 y Fo(L)1720 2236 y Fi(2)1720 2280 y Fk(\000)p Fh(\033)1896 2235 y Fn(\024)2188 2174 y Fp(D)2263 2188 y Fm(4)p 2060 2214 371 4 v 2060 2301 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2320 2275 y Fm(3)p Fo(=)p Fm(2)2440 2235 y Fn(k)p Fp(f)10 b Fn(k)2585 2255 y Fo(L)2633 2236 y Fi(2)2633 2272 y Fh(\033)2679 2235 y Fp(;)106 b(\026)25 b Fn(6)p Fq(=)g(0)p Fp(;)67 b(\033)28 b(>)3309 2174 y Fq(5)p 3309 2214 46 4 v 3309 2298 a(2)3649 2235 y(\(3.9\))787 2481 y Fn(k)p Fp(e)874 2443 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1043 2481 y Fq(\()p Fn(H)21 b(\000)f Fp(i)p Fq(0\))1378 2443 y Fy(\000)p Fm(1)1474 2481 y Fn(P)1537 2495 y Fo(c)1572 2481 y Fp(f)10 b Fn(k)1672 2505 y Fo(L)1720 2482 y Fi(2)1720 2526 y Fk(\000)p Fh(\033)1896 2481 y Fn(\024)2188 2420 y Fp(D)2263 2434 y Fm(5)p 2060 2460 371 4 v 2060 2547 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2320 2521 y Fm(1)p Fo(=)p Fm(2)2440 2481 y Fn(k)p Fp(f)10 b Fn(k)2585 2501 y Fo(L)2633 2482 y Fi(2)2633 2517 y Fh(\033)2679 2481 y Fp(;)106 b(\033)29 b(>)2997 2420 y Fq(5)p 2997 2460 46 4 v 2997 2543 a(2)3052 2481 y Fp(;)526 b Fq(\(3.10\))149 2709 y(where)30 b Fp(D)487 2723 y Fm(4)527 2709 y Fq(,)p Fp(D)627 2723 y Fm(5)697 2709 y Fq(are)h(some)g(p)s(ositiv)m(e)e(constan)m(ts)j (and)d(\()p Fn(H)22 b(\000)e Fp(\026)g Fn(\000)f Fp(i)p Fq(0\))2493 2676 y Fy(\000)p Fm(1)2619 2709 y Fq(denotes)31 b(the)g(limit)d(as)i Fp(")c Fn(&)f Fq(0.)291 2825 y(In)31 b(the)i(nonlinear)e(case,)j(when)e Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))29 b Fn(6\021)g Fq(0)p Fp(;)k Fq(the)g(disp)s(ersiv)m(e)d (equation)i(\(3.3\))j(can)d(b)s(e)g(analyzed)h(with)149 2941 y(the)e(Duhamel)f(principle)d(in)i(the)h(equiv)-5 b(alen)m(t)30 b(in)m(tegral)g(form:)462 3160 y Fp(u)514 3175 y Fo(d)555 3160 y Fq(\()p Fp(t)p Fq(\))25 b(=)g Fp(e)821 3122 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)990 3160 y Fp(u)1042 3175 y Fo(d)1083 3160 y Fq(\(0\))c Fn(\000)f Fp(i)1356 3036 y Fe(Z)1448 3062 y Fo(t)1407 3242 y Fm(0)1492 3160 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(A)p Fn(j)1775 3122 y Fm(2)1816 3160 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)2039 3122 y Fy(\000)p Fo(i\025s)2197 3160 y Fp(e)2239 3122 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2550 3160 y Fn(P)2613 3174 y Fo(c)2648 3160 y Fp( )2710 3122 y Fm(3)2750 3160 y Fp(ds)20 b Fq(+)g Fp(K)3028 3174 y Fo(A)3085 3160 y Fq([)p Fp(u)3162 3175 y Fo(d)3203 3160 y Fq(]\()p Fp(t)p Fq(\))p 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3838 y Fo(d)2913 3823 y Fq(\()p Fp(s)p Fq(\))p Fp(ds)1311 4077 y Fn(\000)83 b Fq(2)p Fp(i)1556 3954 y Fe(Z)1647 3980 y Fo(t)1607 4160 y Fm(0)1692 4077 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)2080 4040 y Fy(\000)p Fo(i\025s)2238 4077 y Fp(e)2280 4040 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2592 4077 y Fn(P)2655 4091 y Fo(c)2690 4077 y Fp( )s Fn(j)p Fp(u)2829 4092 y Fo(d)2870 4077 y Fq(\()p Fp(s)p Fq(\))p Fn(j)3008 4040 y Fm(2)3048 4077 y Fp(ds)1311 4331 y Fn(\000)83 b Fp(i)1511 4208 y Fe(Z)1602 4234 y Fo(t)1562 4414 y Fm(0)1647 4331 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p 1812 4258 69 4 v Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)2035 4294 y Fo(i\025t)2131 4331 y Fp(e)2173 4294 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2484 4331 y Fn(P)2547 4345 y Fo(c)2582 4331 y Fp( )s(u)2696 4294 y Fm(2)2696 4354 y Fo(d)2737 4331 y Fq(\()p Fp(s)p Fq(\))p Fp(ds)1311 4585 y Fn(\000)83 b Fp(i)1511 4462 y Fe(Z)1602 4488 y Fo(t)1562 4668 y Fm(0)1647 4585 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(e)1854 4548 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2166 4585 y Fn(P)2229 4599 y Fo(c)2264 4585 y Fn(j)p Fp(u)2341 4600 y Fo(d)2381 4585 y Fn(j)2406 4548 y Fm(2)2446 4585 y Fp(u)2498 4600 y Fo(d)2539 4585 y Fq(\()p Fp(s)p Fq(\))p Fp(ds:)836 b Fq(\(3.12\))149 4799 y(W)-8 b(e)28 b(sho)m(w)f(in)e(Section)i(4)g(that)g Fp(K)1283 4813 y Fo(A)1340 4799 y Fq([)p Fp(u)1417 4814 y Fo(d)1458 4799 y Fq(])g(is)e(lo)s(cally)h(con)m(tractiv)m(e)i(in)e (certain)g(Banac)m(h)i(spaces)f(and)f(that)h(the)149 4915 y(solution)i Fp(u)545 4930 y Fo(d)586 4915 y Fq(\()p Fp(t)p Fq(\))i(can)f(b)s(e)g(appro)m(ximated)g(b)m(y:)791 5139 y Fp(u)843 5154 y Fo(d)884 5139 y Fq(\()p Fp(t)p Fq(\))25 b Fn(\031)g Fp(e)1150 5101 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1319 5139 y Fp(u)1371 5154 y Fo(d)1412 5139 y Fq(\(0\))c Fn(\000)f Fp(i)1685 5015 y Fe(Z)1776 5041 y Fo(t)1736 5221 y Fm(0)1821 5139 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(e)2028 5101 y Fy(\000)p Fo(i\025s)2186 5139 y Fn(j)p Fp(A)p Fn(j)2304 5101 y Fm(2)2344 5139 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)2567 5101 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2879 5139 y Fn(P)2942 5153 y Fo(c)2977 5139 y Fp( )3039 5101 y Fm(3)3079 5139 y Fp(ds;)409 b Fq(\(3.13\))1970 5425 y(6)p eop %%Page: 7 7 7 6 bop 149 75 a Fq(see)37 b(Lemma)f(4.2.)58 b(When)35 b(the)h(represen)m(tation)g(\(3.13\))i(is)d(substituted)f(in)m(to)i (the)g(amplitude)e(equation)149 191 y(\(3.2\))41 b(via)c(the)i (expression)e(\(3.4\),)42 b(w)m(e)d(obtain)f(a)g(closed)h(in)m (tegro-di\013eren)m(tial)e(equation)h(in)f Fp(A)p Fq(\()p Fp(t)p Fq(\).)65 b(W)-8 b(e)149 307 y(sho)m(w)30 b(in)e(Section)h(5)h (that)h(the)e(main)g(con)m(tribution)f(in)g(the)i(ev)m(olution)f(of)h Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)g Fq(comes)h(from)e(the)h (terms)149 423 y(that)37 b(are)f(linear)e(in)h Fp(u)931 438 y Fo(d)971 423 y Fq(\()p Fp(t)p Fq(\).)58 b(Their)34 b(leading)h(e\013ect)i(can)f(b)s(e)g(computed)f(with)g(the)h(F)-8 b(ermi)35 b(Golden)h(rule,)149 540 y(in)m(tro)s(duced)27 b(in)g([21)r(].)40 b(Comparison)27 b(results)g(for)h(ODE's)h(together)h (with)d(linearization)g(and)g(con)m(traction)149 656 y(principles)i(allo)m(w)j(us)f(to)i(con)m(trol)g(the)g(non-dominan)m(t) e(terms)h(and)g(obtain)f(the)i(deca)m(y)g(estimate)g(\(1.12\),)149 772 y(whic)m(h)d(implies)d(the)k(deca)m(y)g(estimates)g (\(1.13\)-\(1.14\))k(b)m(y)c(using)e(the)h(represen)m(tation)h (\(3.13\).)291 888 y(W)-8 b(e)38 b(note)g(that)g(the)g(pro)s(of)f(of)h (Theorem)f(1)h(relies)e(hea)m(vily)h(on)g(the)h(tigh)m(t)f(con)m(trol)h (of)g(the)g(error)f(in)149 1005 y(the)e(represen)m(tation)g(\(3.13\).) 56 b(In)34 b(this)f(analysis,)i(the)f(smallness)f(of)i(the)g(initial)d (condition,)i(de\014ned)g(b)m(y)149 1121 y(the)k(small)d(parameter)j Fp(")g Fq(in)d(\(1.10\),)42 b(is)36 b(crucial.)60 b(The)37 b(nonlinearit)m(y)e(of)i(op)s(erator)g Fp(K)3201 1135 y Fo(A)3259 1121 y Fq([)p Fp(u)3336 1136 y Fo(d)3376 1121 y Fq(])h(raises)e(t)m(w)m(o)149 1237 y(imp)s(ortan)m(t)30 b(issues)f(that)i(w)m(e)g(need)f(to)h(address)f(b)s(efore)g(w)m(e)g (can)h(treat)h(it)d(p)s(erturbativ)m(ely)-8 b(.)291 1353 y(The)27 b(\014rst)h(issue)f(is)g(related)i(to)g(the)f(presence)h(of)f (t)m(w)m(o)i(time)e(scales)g(in)f(the)i(in)m(tegral)f(equation)g (\(3.11\).)149 1469 y(The)h(\014rst)f(term)g(on)h(the)g(righ)m(t)f (hand)g(side)f(is)h(initially)d Fn(O)s Fq(\()p Fp(")p Fq(\))30 b(and)e(deca)m(ys)i(fast,)f(according)g(to)g(the)g(deca)m(y) 149 1586 y(estimates)24 b(\(3.6\)-\(3.8\).)43 b(The)22 b(second)i(term)f(is)f(initially)e Fn(O)s Fq(\()p Fp(")2251 1553 y Fm(3)2291 1586 y Fq(\))k(but)e(deca)m(ys)i(slo)m(wly)-8 b(,)25 b(as)e(sho)m(wn)g(in)f(Section)149 1702 y(4.)41 b(Because)31 b Fp(K)684 1716 y Fo(A)741 1702 y Fq([)p Fp(u)818 1717 y Fo(d)859 1702 y Fq(])e(is)f(a)i(nonlinear)d(op)s (erator)i(in)f Fp(u)1998 1717 y Fo(d)2039 1702 y Fq(\()p Fp(t)p Fq(\),)i(it)f(couples)f(the)i(t)m(w)m(o)g(time)f(scale)h(and)e (leads)h(to)149 1818 y(a)j(slo)m(wly)f(v)-5 b(arying)31 b(error)g(of)h(order)f Fn(O)s Fq(\()p Fp(")1546 1785 y Fm(2)1586 1818 y Fq(\).)45 b(W)-8 b(e)33 b(k)m(eep)f(the)g(time)f (scale)h(separated)g(and)g(obtain)f(a)h(slo)m(wly)149 1934 y(v)-5 b(arying)22 b(error)h(of)g(order)f Fn(O)s Fq(\()p Fp(")1159 1901 y Fm(4)1199 1934 y Fq(\))i(b)m(y)e(using)g(the)h (con)m(traction)h(principle)19 b(in)j(a)h(hierarc)m(h)m(y)g(of)g(Banac) m(h)h(spaces,)149 2051 y(describ)s(ed)29 b(in)g(App)s(endix)e(A.)291 2167 y(The)36 b(second)h(issue)f(is)h(raised)f(b)m(y)h(the)g(nonlinear) e(dep)s(endence)h(of)i Fp(K)2762 2181 y Fo(A)2819 2167 y Fq([)p Fp(u)2896 2182 y Fo(d)2937 2167 y Fq(])f(on)g Fp(A)p Fq(\()p Fp(t)p Fq(\).)62 b(In)36 b(order)h(to)149 2283 y(treat)32 b Fp(K)449 2297 y Fo(A)507 2283 y Fq([)p Fp(u)584 2298 y Fo(d)624 2283 y Fq(])g(p)s(erturbativ)m(ely)-8 b(,)29 b(w)m(e)j(need)f(to)g(obtain)g(some)g(information)e(on)i(the)g 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b(b)s(oth)149 4936 y(op)s(en)g(and)g(closed)g(nonempt)m(y)g(subset)g(of)h(the)f (connected)h(space)g([0)p Fp(;)15 b(T)2607 4950 y Fm(max)2744 4936 y Fq(\))31 b(w)m(e)g(ha)m(v)m(e)1727 5144 y Fp(\034)k Fq(=)25 b([0)p Fp(;)15 b(T)2061 5158 y Fm(max)2198 5144 y Fq(\))p Fp(:)1345 b Fq(\(3.16\))1970 5425 y(7)p eop %%Page: 8 8 8 7 bop 149 75 a Fq(Finally)-8 b(,)29 b(w)m(e)i(pro)m(v)m(e)h(in)d (Section)h(6)h(that)f(there)h(exists)f(a)h(p)s(ositiv)m(e)e(constan)m (t)j Fn(E)8 b Fq(,)31 b(suc)m(h)f(that)1154 282 y Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)1399 302 y Fo(H)1462 283 y Fi(1)1527 282 y Fn(\024)25 b(E)e Fq(log)r(\(1)e(+)f(4\000)p Fp(")2148 245 y Fm(4)2188 282 y Fp(t)p Fq(\))p Fp(;)197 b Fq(for)30 b Fp(t)25 b Fn(2)g Fp(\034)5 b(;)772 b Fq(\(3.17\))149 490 y(see)39 b(Lemma)g(6.1.)66 b(The)38 b(blo)m(w)f(up)h(alternativ)m (e)g(\(2.6\))j(and)c(the)i(estimate)g(\(3.17\))i(com)m(bined)c(with)g (the)149 606 y(closure)42 b(relation)g(\(3.16\))i(imply)c(that)j Fp(T)1614 620 y Fm(max)1795 606 y 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Fe(\))3378 5056 y Fp(:)1970 5425 y Fq(8)p eop %%Page: 9 9 9 8 bop 149 75 a Fq(W)-8 b(e)32 b(will)c(b)s(e)h(using)g(the)i(con)m(v) m(olution)f(estimates:)256 182 y Fe(Z)347 208 y Fo(t)306 388 y Fm(0)376 305 y Fq(\(1)21 b(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))790 268 y Fy(\000)p Fo(b)879 305 y Fq(\(1)h(+)f Fp(\017s)p Fq(\))1186 268 y Fy(\000)p Fo(a)1283 305 y Fp(ds)25 b Fn(\024)1494 177 y Fe(\032)1604 251 y Fp(D)s Fq(\(1)20 b(+)g Fp(\017t)p Fq(\))1978 218 y Fy(\000)12 b Fm(min)o(\()p Fo(a;b)p Fm(\))2440 251 y Fq(if)36 b Fp(b)25 b(>)g Fq(1)31 b(or)f Fp(a)c(>)f Fq(1)1604 364 y Fp(D)s(\017)1719 331 y Fo(b)p Fy(\000)p Fm(1)1843 364 y Fq(\(1)c(+)f Fp(\017t)p Fq(\))2140 331 y Fm(1)p Fy(\000)p Fo(a)p Fy(\000)p Fo(b)2440 364 y Fq(if)36 b(0)26 b Fp(<)f(b)g(<)g Fq(1)31 b(and)e(0)d Fp(<)f(a)g(<)g Fq(1)3649 305 y(\(4.3\))149 535 y(and)277 625 y Fe(Z)368 651 y Fo(t)328 831 y Fm(0)398 749 y Fq(\()p Fp(t)20 b Fn(\000)g Fp(s)p Fq(\))655 711 y Fy(\000)p Fo(b)744 749 y Fq(\(1)h(+)f Fp(\017s)p Fq(\))1051 711 y Fy(\000)p Fo(a)1148 749 y Fp(ds)25 b Fn(\024)1359 620 y Fe(\032)1468 694 y Fp(D)s(\017)1583 661 y Fo(b)p Fy(\000)p Fm(1)1708 694 y Fq(\(1)c(+)f Fp(\017t)p Fq(\))2005 661 y Fy(\000)11 b Fm(min)o(\()p Fo(a;b)p Fm(\))2418 694 y Fq(if)37 b(0)25 b Fp(<)g(b)h(<)f Fq(1)30 b(and)g Fp(a)25 b(>)g Fq(1)1468 807 y Fp(D)s(\017)1583 774 y Fo(b)p Fy(\000)p Fm(1)1708 807 y Fq(\(1)c(+)f Fp(\017t)p Fq(\))2005 774 y Fm(1)p Fy(\000)p Fo(a)p Fy(\000)p Fo(b)2418 807 y Fq(if)37 b(0)25 b Fp(<)g(b)h(<)f Fq(1)30 b(and)g(0)c Fp(<)f(a)g(<)g Fq(1)3649 749 y(\(4.4\))149 984 y(whic)m(h)31 b(are)h(v)-5 b(alid)29 b(for)j(0)27 b Fp(<)g(\017)g Fn(\024)g Fq(1)32 b(and)f Fp(t)c(>)f Fq(0.)45 b(Using)31 b(the)g(deca)m(y)i (estimates)f(\(3.6\){\(3.7\),)k(the)c(unitarit)m(y)149 1100 y(of)i Fp(e)298 1067 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)467 1100 y Fq(,)f Fp(t)d Fn(2)f Fl(R)42 b Fq(on)32 b Fp(L)967 1067 y Fm(2)1007 1100 y Fq(\()p Fl(R)1102 1067 y Fm(3)1147 1100 y Fq(\))h(and)g(the)g(con)m(v)m(olution)g(estimates)h (\(4.3\){\(4.4\),)k(it)32 b(is)g(straigh)m(tforw)m(ard)h(to)149 1216 y(sho)m(w)e(that)g(the)f(forcing)g(terms)g(in)f(the)i(in)m(tegral) f(equation)g(\(3.11\))j(satisfy:)2218 1399 y Fp(e)2260 1361 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)2429 1399 y Fp(u)2481 1414 y Fo(d)2522 1399 y Fq(\(0\))84 b Fn(2)e Fp(X)710 b Fq(\(4.5\))1037 1593 y Fn(\000)p Fp(i)1154 1469 y Fe(Z)1246 1496 y Fo(t)1205 1675 y Fm(0)1290 1593 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(A)p Fn(j)1573 1556 y Fm(2)1614 1593 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)1837 1556 y Fy(\000)p Fo(i\025s)1995 1593 y Fp(e)2037 1556 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2348 1593 y Fn(P)2411 1607 y Fo(c)2446 1593 y Fp( )2508 1556 y Fm(3)2548 1593 y Fp(ds)83 b Fn(2)f Fp(Y)5 b(:)702 b Fq(\(4.6\))149 1813 y(Moreo)m(v)m(er)47 b Fp(X)52 b Fq(is)43 b(con)m(tin)m(uously)g(em)m(b)s(edded)g(in)g Fp(Y)20 b Fq(.)83 b(W)-8 b(e)45 b(assume)f(without)f(loss)h(of)h (generalit)m(y)f(that)149 1930 y Fp(")191 1944 y Fm(0)257 1930 y Fn(\024)24 b Fq(\(4\000\))524 1897 y Fy(\000)p Fm(1)p Fo(=)p Fm(4)690 1930 y Fq(.)41 b(Due)31 b(to)g(the)f(b)s(ound)f (\(1.10\),)k(w)m(e)d(ha)m(v)m(e)1359 2112 y(4\000)p Fp(")1503 2075 y Fm(4)1569 2112 y Fn(\024)24 b Fq(1)92 b(and)e Fn(k)21 b(\001)f(k)2194 2126 y Fo(X)2287 2112 y Fn(\024)25 b(k)c(\001)f(k)2539 2126 y Fo(Y)2601 2112 y Fp(:)149 2295 y Fq(In)27 b(order)h(to)g(solv)m(e)g(the)g(in)m(tegral)g(equation) g(\(3.11\))i(b)m(y)d(using)g(a)h(lo)s(cal)f(con)m(traction)i (principle,)c(w)m(e)j(obtain)149 2411 y(the)j(Lipsc)m(hitz)e(prop)s (erties)g(of)h(the)h(nonlinear)d(op)s(erator)j Fp(K)2225 2425 y Fo(A)2282 2411 y Fq(.)149 2582 y Fg(Lemma)37 b(4.1.)44 b Ff(The)36 b(op)-5 b(er)g(ator)39 b Fp(K)1357 2596 y Fo(A)1450 2582 y Ff(is)d(a)g(lo)-5 b(c)g(al)5 b(ly)38 b(Lipschitz)f(op)-5 b(er)g(ator)39 b(on)d(b)-5 b(oth)37 b Fp(X)44 b Ff(and)36 b Fp(Y)20 b Ff(.)52 b(Mor)-5 b(e)36 b(pr)-5 b(e-)149 2698 y(cisely,)25 b(ther)-5 b(e)25 b(exist)e Fp(T)13 b Ff(-indep)-5 b(endent)24 b(p)-5 b(ositive)25 b(c)-5 b(onstants)25 b Fn(D)2210 2712 y Fm(1)2249 2698 y Ff(,)p Fn(D)2347 2712 y Fm(2)2387 2698 y Ff(,)g(which)f(dep)-5 b(end)25 b(only)f(on)g Fq(\000)p Ff(,)h Fq(sup)3534 2720 y Fo(t)p Fy(2)p Fd(R)3674 2698 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)149 2814 y Ff(and)31 b(the)e(disp)-5 b(ersion)31 b(c)-5 b(onstants)31 b Fp(D)1366 2828 y Fm(1)1406 2814 y Ff({)p Fp(D)1527 2828 y Fm(3)1596 2814 y Ff(in)e (\(3.6\){\(3.8\),)k(such)c(that)h(the)g(fol)5 b(lowing)30 b(estimates)g(ar)-5 b(e)30 b(met:)149 2930 y Fn(8R)277 2944 y Fm(1)342 2930 y Fp(>)25 b Fq(0)p Fp(;)41 b Fn(8)p Fp(u;)15 b(v)28 b Fn(2)d Fp(X)33 b Fq(:)51 b Fn(k)p Fp(u)p Fn(k)1176 2944 y Fo(X)1244 2930 y Fp(;)15 b Fn(k)p Fp(v)s Fn(k)1421 2944 y Fo(X)1515 2930 y Fn(\024)25 b(R)1688 2944 y Fm(1)1728 2930 y Fp(")p Ff(,)1075 3113 y Fn(k)p Fp(K)1197 3127 y Fo(A)1254 3113 y Fq([)p Fp(u)p Fq(])c Fn(\000)f Fp(K)1545 3127 y Fo(A)1602 3113 y Fq([)p Fp(v)s Fq(])p Fn(k)1744 3127 y Fo(X)1838 3113 y Fn(\024)25 b(D)2004 3127 y Fm(1)2044 3113 y Fq(\()p Fn(C)g Fq(+)20 b Fn(R)2320 3127 y Fm(1)2360 3113 y Fq(\))2395 3075 y Fm(2)2435 3113 y Fp(")2477 3075 y Fm(2)2516 3113 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2817 3127 y Fo(X)2885 3113 y Fp(;)739 b Fq(\(4.7\))149 3296 y Fn(8R)277 3310 y Fm(2)342 3296 y Fp(>)25 b Fq(0)p Fp(;)41 b Fn(8)p Fp(u;)15 b(v)28 b Fn(2)d Fp(Y)45 b Fq(:)51 b Fn(k)p Fp(u)p Fn(k)1166 3310 y Fo(Y)1228 3296 y Fp(;)15 b Fn(k)p Fp(v)s Fn(k)1405 3310 y Fo(Y)1492 3296 y Fn(\024)25 b(R)1665 3310 y Fm(2)1705 3296 y Fp(")p Ff(,)1101 3478 y Fn(k)p Fp(K)1223 3492 y Fo(A)1281 3478 y Fq([)p Fp(u)p Fq(])20 b Fn(\000)g Fp(K)1571 3492 y Fo(A)1629 3478 y Fq([)p Fp(v)s Fq(])p Fn(k)1771 3492 y Fo(Y)1858 3478 y Fn(\024)25 b(D)2024 3492 y Fm(2)2063 3478 y Fq(\()p Fn(C)h Fq(+)20 b Fn(R)2340 3492 y Fm(2)2379 3478 y Fq(\))2414 3441 y Fm(2)2454 3478 y Fp(")p Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2797 3492 y Fo(Y)2859 3478 y Fp(;)765 b Fq(\(4.8\))149 3661 y Ff(wher)-5 b(e)34 b Fn(C)k Ff(is)32 b(de\014ne)-5 b(d)34 b(in)f(\(3.14\).)149 3831 y(Pr)-5 b(o)g(of.)47 b Fq(Since)32 b Fp(K)751 3845 y Fo(A)808 3831 y Fq([0])f(=)e(0,)34 b(the)f(in)m(v)-5 b(ariance)32 b(of)h Fp(X)40 b Fq(and)32 b Fp(Y)53 b Fq(under)31 b Fp(K)2572 3845 y Fo(A)2662 3831 y Fq(will)f(follo)m(w)i(from)h(the)g(Lipsc)m(hitz)149 3947 y(estimates)k(\(4.7\){\(4.8\).)63 b(Let)37 b Fp(u;)52 b(v)38 b Fn(2)d Fp(X)7 b Fq(.)59 b(F)-8 b(or)37 b(an)m(y)g(\014xed)f (small)f Fp(")g(>)g Fq(0,)k(there)d(exists)g(large)h Fn(R)3610 3961 y Fm(1)3684 3947 y Fp(>)e Fq(0)149 4064 y(suc)m(h)30 b(that)h Fn(k)p Fp(u)p Fn(k)693 4078 y Fo(X)761 4064 y Fp(;)15 b Fn(k)p Fp(v)s Fn(k)938 4078 y Fo(X)1032 4064 y Fn(\024)25 b(R)1205 4078 y Fm(1)1245 4064 y Fp(")p Fq(.)41 b(Using)29 b(the)h(de\014nition)e(\(3.12\),)k(w)m(e)f(deriv)m (e)e(the)i(follo)m(wing)d(estimate)j(in)149 4180 y Fp(L)211 4147 y Fm(2)211 4202 y Fy(\000)p Fo(\033)313 4180 y Fq(\()p Fl(R)408 4147 y Fm(3)454 4180 y Fq(\):)557 4363 y Fn(k)p Fp(K)679 4377 y Fo(A)737 4363 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1131 4377 y Fo(A)1188 4363 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1433 4386 y Fo(L)1481 4363 y Fi(2)1481 4407 y Fk(\000)p Fh(\033)3649 4363 y Fq(\(4.9\))403 4586 y Fn(\024)83 b Fq(3)49 b(sup)617 4663 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)836 4586 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1066 4462 y Fe(Z)1158 4488 y Fo(t)1117 4668 y Fm(0)1203 4586 y Fn(j)p Fp(A)p Fn(j)1321 4548 y Fm(2)1361 4586 y Fq(\()p Fp(s)p Fq(\))p Fn(k)p Fp(e)1561 4548 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1873 4586 y Fn(P)1936 4600 y Fo(c)1971 4586 y Fp( )2033 4548 y Fm(2)2073 4586 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)2636 4609 y Fo(L)2684 4586 y Fi(2)2684 4630 y Fk(\000)p Fh(\033)2778 4586 y Fp(ds)403 4855 y Fq(+)83 b(3)49 b(sup)617 4932 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)836 4855 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1066 4731 y Fe(Z)1158 4757 y Fo(t)1117 4937 y Fm(0)1203 4855 y Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)p Fp(e)1552 4817 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1864 4855 y Fn(P)1927 4869 y Fo(c)1962 4855 y Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)22 b Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))31 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3226 4878 y Fo(L)3274 4855 y Fi(2)3274 4899 y Fk(\000)p Fh(\033)3369 4855 y Fp(ds)403 5124 y Fq(+)116 b(sup)557 5201 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)775 5124 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1005 5000 y Fe(Z)1097 5026 y Fo(t)1057 5206 y Fm(0)1142 5124 y Fn(k)p Fp(e)1229 5086 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1541 5124 y Fn(P)1604 5138 y Fo(c)1639 5124 y Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1889 5086 y Fm(2)1950 5124 y Fq(+)20 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)h Fq(+)f Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2738 5086 y Fm(2)2779 5124 y Fq(\))31 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))20 b Fn(\000)g Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3407 5147 y Fo(L)3455 5125 y Fi(2)3455 5168 y Fk(\000)p Fh(\033)3550 5124 y Fp(ds:)1970 5425 y Fq(9)p eop %%Page: 10 10 10 9 bop 149 75 a Fq(By)31 b(the)g(disp)s(ersiv)m(e)d(estimate)j (\(3.6\))h(and)e(H\177)-45 b(older)29 b(inequalities)f(w)m(e)j(ha)m(v)m (e)h(for)e(an)m(y)g(0)c Fn(\024)f Fp(s)g Fn(\024)g Fp(t)g Fn(\024)g Fp(T)13 b Fq(:)308 321 y Fn(k)p Fp(e)395 283 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))707 321 y Fn(P)770 335 y Fo(c)805 321 y Fp( )867 283 y Fm(2)907 321 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)1470 344 y Fo(L)1518 321 y Fi(2)1518 365 y Fk(\000)p Fh(\033)1695 321 y Fn(\024)2063 259 y Fp(D)2138 273 y Fm(1)p 1859 300 524 4 v 1859 387 a Fq(\(1)21 b(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))2273 361 y Fm(3)p Fo(=)p Fm(2)2393 321 y Fn(k)p Fq(\(1)h(+)f Fn(j)p Fp(x)p Fn(j)2732 283 y Fm(2)2772 321 y Fq(\))2807 283 y Fo(\033)r(=)p Fm(2)2925 321 y Fp( )2987 283 y Fm(2)3027 321 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))g Fn(\000)g Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)32 b(k)3590 340 y Fo(L)3638 321 y Fi(2)1695 580 y Fn(\024)1859 519 y Fp(D)1934 533 y Fm(1)1974 519 y Fn(k)p Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)2313 486 y Fm(2)2353 519 y Fq(\))2388 486 y Fo(\033)2435 519 y Fp( )2497 486 y Fm(2)2537 519 y Fn(k)2582 533 y Fo(L)2630 514 y Fk(1)p 1859 559 841 4 v 2017 646 a Fq(\(1)h(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))2431 620 y Fm(3)p Fo(=)p Fm(2)2710 580 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)3237 604 y Fo(L)3285 581 y Fi(2)3285 625 y Fk(\000)p Fh(\033)1695 840 y Fn(\024)1890 778 y Fp(D)1965 792 y Fm(1)2005 778 y Fn(k)p Fq(\(1)i(+)d Fn(j)p Fp(x)p Fn(j)2344 745 y Fm(2)2384 778 y Fq(\))2419 745 y Fo(\033)2467 778 y Fp( )2529 745 y Fm(2)2568 778 y Fn(k)2613 792 y Fo(L)2661 773 y Fk(1)p 1859 819 904 4 v 1859 906 a Fq(\(1)i(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))2273 879 y Fm(3)p Fo(=)p Fm(2)2383 906 y Fq(\(1)h(+)f Fp(s)p Fq(\))2653 879 y Fm(3)p Fo(=)p Fm(2)2773 840 y Fn(k)p Fp(u)g Fn(\000)g Fp(v)s Fn(k)3073 854 y Fo(X)3141 840 y Fp(;)149 1091 y Fq(and,)30 b(similarly:)912 1299 y Fn(k)p Fp(e)999 1261 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1311 1299 y Fn(P)1374 1313 y Fo(c)1409 1299 y Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)22 b Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))32 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)e Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)32 b(k)2674 1322 y Fo(L)2722 1300 y Fi(2)2722 1343 y Fk(\000)p Fh(\033)759 1507 y Fn(\024)1127 1446 y Fp(D)1202 1460 y Fm(1)p 922 1486 524 4 v 922 1573 a Fq(\(1)21 b(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))1336 1547 y Fm(3)p Fo(=)p Fm(2)1456 1507 y Fn(k)p Fq(\(1)h(+)f Fn(j)p Fp(x)p Fn(j)1795 1470 y Fm(2)1835 1507 y Fq(\))1870 1470 y Fo(\033)r(=)p Fm(2)1988 1507 y Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))i Fn(\000)e Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3222 1527 y Fo(L)3270 1508 y Fi(2)759 1774 y Fn(\024)922 1712 y Fp(D)997 1726 y Fm(1)1037 1712 y Fn(k)p Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)1376 1679 y Fm(2)1416 1712 y Fq(\))1451 1679 y Fo(\033)r(=)p Fm(2)1569 1712 y Fp( )s Fn(k)1676 1726 y Fo(L)1724 1707 y Fk(1)p 922 1753 872 4 v 1096 1840 a Fq(\(1)h(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))1510 1813 y Fm(3)p Fo(=)p Fm(2)1804 1774 y Fn(k)p Fp(u)h Fq(+)f Fp(v)s Fn(k)2105 1793 y Fo(L)2153 1774 y Fi(4)9 b Fn(k)p Fp(u)21 b Fn(\000)f Fp(v)s Fn(k)2493 1793 y Fo(L)2541 1774 y Fi(4)759 2040 y Fn(\024)922 1979 y Fq(2)p Fp(D)1042 1993 y Fm(1)1082 1979 y Fn(R)1159 1993 y Fm(1)1199 1979 y Fp(")p Fn(k)p Fq(\(1)i(+)e Fn(j)p Fp(x)p Fn(j)1581 1946 y Fm(2)1621 1979 y Fq(\))1656 1946 y Fo(\033)r(=)p Fm(2)1773 1979 y Fp( )s Fn(k)1880 1993 y Fo(L)1928 1974 y Fk(1)p 922 2019 1077 4 v 1008 2106 a Fq(\(1)h(+)f Fp(t)g Fn(\000)g Fp(s)p Fq(\))1422 2080 y Fm(3)p Fo(=)p Fm(2)1532 2106 y Fq(\(1)h(+)f Fp(s)p Fq(\))1802 2080 y Fm(3)p Fo(=)p Fm(2)2008 2040 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2309 2054 y Fo(X)2377 2040 y Fp(:)149 2311 y Fq(F)-8 b(or)28 b(the)f(third)f(term)h(in)f(the)h(estimate)h(\(4.9\),)h(w)m(e)f(a)m(v)m (oid)f(putting)f(a)i(w)m(eigh)m(t)f(lik)m(e)g(\(1)14 b(+)g Fn(j)p Fp(x)p Fn(j)3246 2278 y Fm(2)3285 2311 y Fq(\))3320 2278 y Fo(\033)r(=)p Fm(2)3438 2311 y Fq(,)28 b Fp(\033)g(>)d Fq(0)j(on)149 2428 y Fp(u)i Fq(or)f Fp(v)j Fq(and)d(switc)m(h)f(to)i Fp(L)1044 2395 y Fo(p)1113 2428 y Fq(t)m(yp)s(e)f(estimates.)41 b(This)27 b(tec)m(hnical)i(mo)s (di\014cation)f(is)g(needed)h(due)f(to)i(the)g(fact)149 2544 y(that)h(if)e Fp(u)p Fq(\()p Fp(t)p Fq(\))d Fn(2)f Fp(H)779 2511 y Fm(1)819 2544 y Fq(\()p Fl(R)914 2511 y Fm(3)959 2544 y Fq(\))30 b(is)g(the)g(solution)f(of)h(the)h(NLS)e (equation)h(\(1.1\),)j(then)d Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p 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3059 y Fo(L)1210 3040 y Fi(2)9 b Fn(k)p Fp(e)1336 3002 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1648 3040 y Fn(P)1711 3054 y Fo(c)1746 3040 y Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1996 3002 y Fm(2)2056 3040 y Fq(+)20 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2845 3002 y Fm(2)2886 3040 y Fq(\))p Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3504 3054 y Fo(L)3552 3035 y Fk(1)446 3219 y Fn(\024)736 3158 y Fq(3)p Fp(D)856 3172 y Fm(3)p 610 3198 413 4 v 610 3285 a Fq(2\()p Fp(t)21 b Fn(\000)f Fp(s)p Fq(\))913 3259 y Fm(3)p Fo(=)p Fm(2)1033 3219 y Fn(kj)p Fp(u)p Fn(j)1180 3182 y Fm(2)1240 3219 y Fq(+)g Fn(j)p Fp(v)s Fn(j)1428 3182 y Fm(2)1469 3219 y Fn(k)1514 3239 y Fo(L)1562 3220 y Fi(1)1601 3219 y Fn(k)p Fp(u)h Fn(\000)e Fp(v)s Fn(k)1901 3239 y Fo(L)1949 3220 y Fi(2)446 3479 y Fn(\024)804 3417 y Fq(3)p Fp(D)924 3431 y Fm(3)965 3417 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Fi(4)p Fh(=)p Fi(3)2668 1210 y Fp(ds)301 1474 y Fn(\024)83 b Fq(3)p Fp(D)575 1488 y Fm(2)615 1474 y Fn(R)692 1436 y Fm(2)692 1496 y(1)731 1474 y Fp(")773 1436 y Fm(2)828 1350 y Fe(Z)919 1376 y Fo(t)879 1556 y(t)904 1565 y Fi(1)949 1474 y Fq(\()p Fp(t)20 b Fn(\000)g Fp(s)p Fq(\))1206 1436 y Fy(\000)p Fm(3)p Fo(=)p Fm(4)1371 1474 y Fq(\(1)h(+)f Fp(s)p Fq(\))1641 1436 y Fy(\000)p Fm(9)p Fo(=)p Fm(4)1806 1474 y Fp(ds)p Fn(k)p Fp(u)g Fn(\000)g Fp(v)s Fn(k)2196 1488 y Fo(X)301 1686 y Fn(\024)83 b Fq(3)p Fp(D)575 1700 y Fm(2)615 1686 y Fn(R)692 1648 y Fm(2)692 1708 y(1)731 1686 y Fp(")773 1648 y Fm(2)813 1686 y Fq(\(1)21 b(+)f Fp(t)p Fq(\))1073 1648 y Fy(\000)p Fm(9)p Fo(=)p Fm(4)1238 1686 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)1539 1700 y Fo(X)1607 1686 y Fp(:)149 1893 y Fq(Com)m(bining)29 b(all)g(the)h(estimates,)i(w)m(e)e (ha)m(v)m(e:)719 2101 y Fn(k)p Fp(K)841 2115 y Fo(A)899 2101 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1293 2115 y Fo(A)1350 2101 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1595 2125 y Fo(L)1643 2102 y Fi(2)1643 2145 y Fk(\000)p Fh(\033)1764 2101 y Fn(\024)25 b Fq(\(1)c(+)f Fp(t)p Fq(\))2120 2063 y Fy(\000)p Fm(3)p Fo(=)p Fm(2)2306 2078 y Fq(~)2285 2101 y Fp(D)2360 2115 y Fm(1)2399 2101 y Fq(\()p Fn(C)26 b Fq(+)20 b Fn(R)2676 2115 y Fm(1)2716 2101 y Fq(\))2751 2063 y Fm(2)2790 2101 y Fp(")2832 2063 y Fm(2)2872 2101 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)3173 2115 y Fo(X)3241 2101 y Fp(;)337 b Fq(\(4.11\))149 2336 y(where)26 b(the)g(p)s(ositiv)m(e)e(constan)m(t)1275 2313 y(~)1253 2336 y Fp(D)1328 2350 y Fm(1)1394 2336 y Fq(dep)s(ends)g(only)h(on)g(sup)2194 2358 y Fo(t)p Fy(2)p Fd(R)2334 2336 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)27 b Fq(and)e(the)h(disp)s(ersion)c(constan)m(t)27 b Fp(D)3750 2350 y Fm(1)3790 2336 y Fq({)149 2452 y Fp(D)224 2466 y Fm(3)294 2452 y Fq(in)h(the)i(deca)m(y)h(estimates)g (\(3.6\){\(3.8\).)44 b(F)-8 b(or)31 b(the)f Fp(L)2072 2419 y Fm(4)2141 2452 y Fq(estimates)g(in)e(space)j Fp(X)7 b Fq(,)30 b(w)m(e)g(pro)s(ceed)g(similarly)149 2568 y(to)i(the)e (previous)f(computations:)518 2776 y Fn(k)p Fp(K)640 2790 y Fo(A)698 2776 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1092 2790 y Fo(A)1149 2776 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1394 2795 y Fo(L)1442 2777 y Fi(4)3603 2776 y Fq(\(4.12\))364 2970 y Fn(\024)83 b Fq(3)49 b(sup)578 3047 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)797 2970 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(D)1087 2984 y Fm(2)1143 2846 y Fe(Z)1234 2873 y Fo(t)1193 3052 y Fm(0)1279 2970 y Fn(j)p Fp(A)p Fn(j)1397 2932 y Fm(2)1437 2970 y Fq(\()p Fp(s)p Fq(\)\()p Fp(t)20 b Fn(\000)g Fp(s)p Fq(\))1807 2932 y Fy(\000)p Fm(3)p Fo(=)p Fm(4)1972 2970 y Fn(k)p Fp( )2079 2932 y Fm(2)2120 2970 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))g Fn(\000)g Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)2682 2996 y Fo(L)2730 2976 y Fi(4)p Fh(=)p Fi(3)2832 2970 y Fp(ds)364 3239 y Fq(+)83 b(3)49 b(sup)578 3316 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)797 3239 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(D)1087 3253 y Fm(2)1143 3115 y Fe(Z)1234 3142 y Fo(t)1193 3321 y Fm(0)1279 3239 y Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\()p Fp(t)21 b Fn(\000)f Fp(s)p Fq(\))1768 3201 y Fy(\000)p Fm(3)p Fo(=)p Fm(4)1933 3239 y Fn(k)p Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))32 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3243 3265 y Fo(L)3291 3245 y Fi(4)p Fh(=)p Fi(3)3392 3239 y Fp(ds)364 3508 y Fq(+)116 b(sup)518 3585 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)736 3508 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(D)1026 3522 y Fm(2)1082 3384 y Fe(Z)1173 3411 y Fo(t)1133 3590 y Fm(0)1203 3508 y Fq(\()p Fp(t)20 b Fn(\000)g Fp(s)p Fq(\))1460 3470 y Fy(\000)p Fm(3)p Fo(=)p Fm(4)1625 3508 y Fn(k)p Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1920 3470 y Fm(2)1981 3508 y Fq(+)g Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2770 3470 y Fm(2)2810 3508 y Fq(\))31 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3439 3534 y Fo(L)3487 3514 y Fi(4)p Fh(=)p Fi(3)3588 3508 y Fp(ds:)149 3768 y Fq(By)g(H\177)-45 b(older)30 b(inequalities:)500 3976 y Fn(k)p Fp( )607 3938 y Fm(2)648 3976 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)e Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)32 b(k)1211 4002 y Fo(L)1259 3982 y Fi(4)p Fh(=)p Fi(3)1443 3976 y Fq(=)83 b Fn(k)p Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)1936 3938 y Fm(2)1976 3976 y Fq(\))2011 3938 y Fo(\033)r(=)p Fm(2)2129 3976 y Fp( )2191 3938 y Fm(2)2231 3976 y Fq(\(1)h(+)f Fn(j)p Fp(x)p Fn(j)2525 3938 y Fm(2)2565 3976 y Fq(\))2600 3938 y Fy(\000)p Fo(\033)r(=)p Fm(2)2772 3976 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3335 4002 y Fo(L)3383 3982 y Fi(4)p Fh(=)p Fi(3)1443 4117 y Fn(\024)83 b(k)p Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)1936 4079 y Fm(2)1976 4117 y Fq(\))2011 4079 y Fo(\033)r(=)p Fm(2)2129 4117 y Fp( )2191 4079 y Fm(2)2231 4117 y Fn(k)2276 4137 y Fo(L)2324 4118 y Fi(4)2363 4117 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)2890 4141 y Fo(L)2938 4118 y Fi(2)2938 4162 y Fk(\000)p Fh(\033)1443 4346 y Fn(\024)1607 4285 y(k)p Fq(\(1)h(+)f Fn(j)p Fp(x)p Fn(j)1946 4252 y Fm(2)1986 4285 y Fq(\))2021 4252 y Fo(\033)r(=)p Fm(2)2139 4285 y Fp( )2201 4252 y Fm(2)2241 4285 y Fn(k)2286 4304 y Fo(L)2334 4285 y Fi(4)p 1607 4325 767 4 v 1800 4412 a Fq(\(1)h(+)f Fp(s)p Fq(\))2070 4386 y Fm(3)p Fo(=)p Fm(2)2383 4346 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2684 4360 y Fo(X)2752 4346 y Fp(;)149 4713 y Fn(k)p Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)j Fq(+)d Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))i Fn(\000)e Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(jk)1398 4739 y Fo(L)1446 4719 y Fi(4)p Fh(=)p Fi(3)1631 4713 y Fn(\024)83 b(k)p Fq(\(1)22 b(+)d Fn(j)p Fp(x)p Fn(j)2124 4676 y Fm(2)2164 4713 y Fq(\))2199 4676 y Fo(\033)r(=)p Fm(2)2317 4713 y Fp( )s Fn(k)2424 4727 y Fo(L)2472 4708 y Fk(1)2542 4713 y Fn(kj)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)j Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(jk)3170 4733 y Fo(L)3218 4714 y Fi(4)3258 4713 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)3785 4737 y Fo(L)3833 4714 y Fi(2)3833 4758 y Fk(\000)p Fh(\033)1631 4922 y Fn(\024)1883 4860 y Fq(2)p Fn(R)2005 4874 y Fm(1)2045 4860 y Fp(")p 1795 4901 380 4 v 1795 4988 a Fq(\(1)h(+)f Fp(s)p Fq(\))2065 4961 y Fm(9)p Fo(=)p Fm(4)2185 4922 y Fn(k)p Fq(\(1)h(+)f Fn(j)p Fp(x)p Fn(j)2524 4884 y Fm(2)2564 4922 y Fq(\))2599 4884 y Fo(\033)r(=)p Fm(2)2717 4922 y Fp( )s Fn(k)2824 4936 y Fo(L)2872 4917 y Fk(1)2942 4922 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)3243 4936 y Fo(X)3311 4922 y Fp(;)1947 5425 y Fq(11)p eop %%Page: 12 12 12 11 bop 149 75 a Fq(and)185 294 y Fn(k)p Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)480 257 y Fm(2)542 294 y Fq(+)20 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)h Fq(+)f Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)1330 257 y Fm(2)1371 294 y Fq(\))p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(jk)1938 320 y Fo(L)1986 300 y Fi(4)p Fh(=)p Fi(3)2171 294 y Fn(\024)2335 233 y Fq(3)p 2335 274 46 4 v 2335 357 a(2)2390 294 y Fn(kj)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)2650 257 y Fm(2)2711 294 y Fq(+)g Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)3012 257 y Fm(2)3053 294 y Fn(k)3098 314 y Fo(L)3146 295 y Fi(2)3185 294 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)3712 314 y Fo(L)3760 295 y Fi(4)2171 535 y Fn(\024)2335 467 y Fq(3)15 b(max)2580 394 y Fe(\000)2622 467 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(k)2877 434 y Fm(2)2877 500 y Fo(L)2925 481 y Fi(4)2965 467 y Fp(;)g Fn(k)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)3255 434 y Fm(2)3255 500 y Fo(L)3303 481 y Fi(4)3343 394 y Fe(\001)p 2335 514 1050 4 v 2670 601 a Fq(\(1)21 b(+)f Fp(s)p Fq(\))2940 575 y Fm(3)p Fo(=)p Fm(4)3395 535 y Fn(k)p Fp(u)h Fn(\000)e Fp(v)s Fn(k)3695 549 y Fo(X)2171 795 y Fn(\024)2403 733 y Fq(3)p Fn(R)2525 700 y Fm(2)2525 758 y(1)2565 733 y Fp(")2607 700 y Fm(2)p 2335 774 380 4 v 2335 861 a Fq(\(1)i(+)f Fp(s)p Fq(\))2605 835 y Fm(9)p Fo(=)p Fm(4)2725 795 y Fn(k)p Fp(u)g Fn(\000)g Fp(v)s Fn(k)3025 809 y Fo(X)3094 795 y Fp(:)149 1051 y Fq(By)28 b(plugging)f(the)h(ab)s(o)m(v)m(e)g (estimates)h(in)d(\(4.12\))k(and)d(using)g(the)h(con)m(v)m(olution)f (inequalit)m(y)g(\(4.4\))i(together)149 1168 y(with)h Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)c(\024)f(C)5 b Fp(")31 b Fq(for)f(0)25 b Fn(\024)g Fp(s)g Fn(\024)g Fp(T)13 b Fq(,)31 b(w)m(e)f(obtain:)759 1375 y Fn(k)p Fp(K)881 1389 y Fo(A)939 1375 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1333 1389 y Fo(A)1390 1375 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1635 1395 y Fo(L)1683 1376 y Fi(4)1749 1375 y Fn(\024)25 b Fq(\(1)c(+)f Fp(t)p Fq(\))2105 1338 y Fy(\000)p Fm(3)p Fo(=)p Fm(4)2291 1352 y Fq(~)2270 1375 y Fp(D)2345 1389 y Fm(2)2385 1375 y Fq(\()p Fn(C)25 b Fq(+)20 b Fn(R)2661 1389 y Fm(1)2701 1375 y Fq(\))2736 1338 y Fm(2)2775 1375 y Fp(")2817 1338 y Fm(2)2857 1375 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)3158 1389 y Fo(X)3603 1375 y Fq(\(4.13\))149 1583 y(where)38 b(the)h(p)s(ositiv)m(e)e(constan)m(t)1325 1560 y(~)1303 1583 y Fp(D)1378 1597 y Fm(2)1456 1583 y Fq(dep)s(ends)g(only)g(on)h(sup)2294 1605 y Fo(t)p Fy(2)p Fd(R)2434 1583 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)39 b Fq(and)f(the)g(disp)s(ersion)d(constan)m(ts)149 1699 y Fp(D)224 1713 y Fm(1)264 1699 y Fq({)p Fp(D)384 1713 y Fm(3)461 1699 y Fq(in)h(the)h(deca)m(y)i(estimates)e (\(3.6\){\(3.8\).)65 b(F)-8 b(or)38 b(the)g Fp(L)2304 1666 y Fm(2)2380 1699 y Fq(estimate)g(w)m(e)f(pro)s(ceed)g(as)g(in)f (\(4.9\))j(and)149 1815 y(\(4.12\))33 b(but,)d(for)g(the)h(\014rst)e(t) m(w)m(o)j(terms,)f(w)m(e)f(use)g(the)h(unitarit)m(y)e(of)i Fp(e)2510 1782 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)2679 1815 y Fq(,)f Fp(t)25 b Fn(2)g Fl(R)39 b Fq(on)30 b Fp(L)3162 1782 y Fm(2)3202 1815 y Fq(\()p Fl(R)3297 1782 y Fm(3)3342 1815 y Fq(\):)928 2023 y Fn(k)p Fp(K)1050 2037 y Fo(A)1108 2023 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1502 2037 y Fo(A)1560 2023 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1805 2043 y Fo(L)1853 2024 y Fi(2)3603 2023 y Fq(\(4.14\))775 2217 y Fn(\024)82 b Fq(3)49 b(sup)988 2294 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)1207 2217 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1437 2093 y Fe(Z)1529 2120 y Fo(t)1489 2299 y Fm(0)1574 2217 y Fn(j)p Fp(A)p Fn(j)1692 2180 y Fm(2)1732 2217 y Fq(\()p Fp(s)p Fq(\))p Fn(k)p Fp( )1952 2180 y Fm(2)1993 2217 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)2556 2237 y Fo(L)2604 2218 y Fi(2)2643 2217 y Fp(ds)775 2486 y Fq(+)82 b(3)49 b(sup)988 2563 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)1207 2486 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1437 2362 y Fe(Z)1529 2389 y Fo(t)1489 2569 y Fm(0)1574 2486 y Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(jk)p Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)24 b Fq(+)19 b Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))32 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3116 2506 y Fo(L)3164 2487 y Fi(2)3203 2486 y Fp(ds)775 2755 y Fq(+)928 2623 y Fe(\015)928 2678 y(\015)928 2732 y(\015)928 2787 y(\015)994 2631 y(Z)1085 2658 y Fo(t)1045 2838 y Fm(0)1130 2755 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(e)1337 2718 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1649 2755 y Fn(P)1712 2769 y Fo(c)1747 2755 y Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1997 2718 y Fm(2)2037 2755 y Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2524 2718 y Fm(2)2564 2755 y Fp(v)s Fq(\()p Fp(s)p Fq(\)\))p Fp(ds)2849 2623 y Fe(\015)2849 2678 y(\015)2849 2732 y(\015)2849 2787 y(\015)2901 2846 y Fo(L)2949 2827 y Fi(2)2988 2755 y Fp(:)149 3008 y Fq(By)31 b(the)g(H\177)-45 b(older)29 b(inequalities:)742 3216 y Fn(k)p Fp( )849 3178 y Fm(2)890 3216 y Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))20 b Fn(\000)g Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)32 b(k)1453 3236 y Fo(L)1501 3217 y Fi(2)1623 3216 y Fn(\024)82 b(k)p Fq(\(1)22 b(+)e Fn(j)p Fp(x)p Fn(j)2116 3178 y Fm(2)2156 3216 y Fq(\))2191 3178 y Fo(\033)r(=)p Fm(2)2309 3216 y Fp( )2371 3178 y Fm(2)2410 3216 y Fn(k)2455 3230 y Fo(L)2503 3211 y Fk(1)2573 3216 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)3100 3240 y Fo(L)3148 3217 y Fi(2)3148 3261 y Fk(\000)p Fh(\033)1623 3445 y Fn(\024)1786 3383 y(k)p Fq(\(1)i(+)e Fn(j)p Fp(x)p Fn(j)2126 3350 y Fm(2)2166 3383 y Fq(\))2201 3350 y Fo(\033)r(=)p Fm(2)2318 3383 y Fp( )2380 3350 y Fm(2)2420 3383 y Fn(k)2465 3397 y Fo(L)2513 3379 y Fk(1)p 1786 3424 797 4 v 1995 3511 a Fq(\(1)h(+)f Fp(s)p Fq(\))2265 3485 y Fm(3)p Fo(=)p Fm(2)2593 3445 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2894 3459 y Fo(X)149 3697 y Fq(and)332 3904 y Fn(k)p Fp( )s Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))32 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)1642 3924 y Fo(L)1690 3905 y Fi(2)1812 3904 y Fn(\024)83 b(k)p Fp( )s Fn(k)2118 3918 y Fo(L)2166 3899 y Fk(1)2237 3904 y Fn(k)30 b(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)21 b Fq(+)f Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)32 b(k)2926 3924 y Fo(L)2974 3905 y Fi(4)3013 3904 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(jk)3565 3924 y Fo(L)3613 3905 y Fi(4)1812 4090 y Fn(\024)1976 4028 y Fq(2)p Fn(R)2098 4042 y Fm(1)2138 4028 y Fp(")p Fn(k)p Fp( )s Fn(k)2332 4042 y Fo(L)2380 4023 y Fk(1)p 1976 4069 475 4 v 2023 4156 a Fq(\(1)h(+)f Fp(s)p Fq(\))2293 4130 y Fm(3)p Fo(=)p Fm(2)2461 4090 y Fn(k)p Fp(u)h Fn(\000)e Fp(v)s Fn(k)2761 4104 y Fo(X)2830 4090 y Fp(;)149 4347 y Fq(the)31 b(\014rst)f(t)m(w)m(o)h(terms)g(in)e(\(4.14\))j(are)f(b)s (ounded.)39 b(F)-8 b(or)31 b(the)f(last)h(term,)f(w)m(e)h(denote)1287 4554 y Fp(w)r Fq(\()p Fp(s)p Fq(\))26 b(=)f Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1804 4517 y Fm(2)1844 4554 y Fp(u)p Fq(\()p Fp(s)p Fq(\))c Fn(\000)f(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2331 4517 y Fm(2)2371 4554 y Fp(v)s Fq(\()p Fp(s)p Fq(\))149 4762 y(and)30 b(use)g(the)h(Stric)m (hartz)f(estimates,)h(see)g(form)m(ula)f(\(2.3.5\))j(in)c([1)q(],)761 4901 y Fe(\015)761 4955 y(\015)761 5010 y(\015)761 5064 y(\015)827 4909 y(Z)918 4935 y Fo(t)878 5115 y Fm(0)963 5032 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(e)1170 4995 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1482 5032 y Fn(P)1545 5046 y Fo(c)1580 5032 y Fp(w)r Fq(\()p Fp(s)p Fq(\))p Fp(ds)1850 4901 y Fe(\015)1850 4955 y(\015)1850 5010 y(\015)1850 5064 y(\015)1901 4927 y Fm(2)1901 5123 y Fo(L)1949 5104 y Fi(2)2013 5032 y Fn(\024)25 b Fp(C)7 b Fn(k)p Fp(w)r Fq(\()p Fp(s;)15 b(x)p Fq(\))p Fn(k)2543 5058 y Fo(L)2591 5038 y Fi(8)p Fh(=)p Fi(5)2689 5058 y Fm(\([0)p Fo(;t)p Fm(])p Fo(;L)2904 5038 y Fi(4)p Fh(=)p Fi(3)3001 5058 y Fm(\))3032 5032 y Fp(;)1947 5425 y Fq(12)p eop %%Page: 13 13 13 12 bop 149 75 a Fq(where)30 b Fp(C)37 b Fq(is)30 b(a)g(constan)m(t,) i(suc)m(h)e(that)875 300 y Fn(k)p Fp(w)r Fq(\()p Fp(s)p Fq(\))p Fn(k)1145 325 y Fo(L)1193 306 y Fi(4)p Fh(=)p Fi(3)1379 300 y Fn(\024)1543 238 y Fq(3)p 1543 279 46 4 v 1543 362 a(2)1598 300 y Fn(k)p Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1893 262 y Fm(2)1955 300 y Fq(+)20 b Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2256 262 y Fm(2)2296 300 y Fq(\)\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))i Fn(\000)e(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(\))p Fn(k)2959 325 y Fo(L)3007 306 y Fi(4)p Fh(=)p Fi(3)1379 515 y Fn(\024)1543 454 y Fq(3)p 1543 495 V 1543 578 a(2)1598 515 y Fn(k)31 b(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1889 478 y Fm(2)1950 515 y Fq(+)20 b Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2251 478 y Fm(2)2321 515 y Fn(k)2366 535 y Fo(L)2414 516 y Fi(2)2454 515 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)2981 535 y Fo(L)3029 516 y Fi(4)1379 748 y Fn(\024)1611 687 y Fq(3)p Fn(R)1733 654 y Fm(2)1733 711 y(1)1773 687 y Fp(")1815 654 y Fm(2)p 1543 727 380 4 v 1543 815 a Fq(\(1)h(+)f Fp(s)p Fq(\))1813 788 y Fm(9)p Fo(=)p Fm(4)1933 748 y Fn(k)p Fp(u)h Fn(\000)e Fp(v)s Fn(k)2233 762 y Fo(X)149 988 y Fq(and)230 1082 y Fe(\015)230 1136 y(\015)230 1191 y(\015)230 1245 y(\015)296 1090 y(Z)387 1116 y Fo(t)346 1296 y Fm(0)432 1213 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(e)639 1176 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))950 1213 y Fn(P)1013 1227 y Fo(c)1048 1213 y Fp(w)r Fq(\()p Fp(s)p Fq(\))p Fp(ds)1318 1082 y Fe(\015)1318 1136 y(\015)1318 1191 y(\015)1318 1245 y(\015)1370 1304 y Fo(L)1418 1285 y Fi(2)1482 1213 y Fn(\024)25 b Fq(3)p Fp(D)1698 1227 y Fm(1)1738 1213 y Fn(R)1815 1176 y Fm(2)1815 1236 y(1)1854 1213 y Fp(")1896 1176 y Fm(2)1984 1213 y Fq(sup)1951 1291 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)2169 1213 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)15 b(k)p Fp(u)22 b Fn(\000)e Fp(v)s Fn(k)2701 1227 y Fo(X)2769 1213 y Fn(k)p Fq(\(1)i(+)e Fp(s)p Fq(\))3085 1176 y Fy(\000)p Fm(9)p Fo(=)p Fm(4)3250 1213 y Fn(k)3295 1239 y Fo(L)3343 1219 y Fi(8)p Fh(=)p Fi(5)3440 1239 y Fm([0)p Fo(;t)p Fm(])3564 1213 y Fp(:)149 1461 y Fq(Therefore,)620 1573 y Fe(\015)620 1627 y(\015)620 1682 y(\015)620 1737 y(\015)685 1581 y(Z)776 1607 y Fo(t)736 1787 y Fm(0)821 1705 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fp(e)1028 1667 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1340 1705 y Fn(P)1403 1719 y Fo(c)1438 1705 y Fp(w)r Fq(\()p Fp(s)p Fq(\))p Fp(ds)1708 1573 y Fe(\015)1708 1627 y(\015)1708 1682 y(\015)1708 1737 y(\015)1759 1795 y Fo(L)1807 1776 y Fi(2)1871 1705 y Fn(\024)25 b Fq(3)p Fp(D)2087 1719 y Fm(1)2176 1705 y Fq(sup)2142 1782 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)2361 1705 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(jR)2653 1667 y Fm(2)2653 1727 y(1)2693 1705 y Fp(")2735 1667 y Fm(2)2805 1705 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(v)s Fn(k)3106 1719 y Fo(X)3174 1705 y Fp(:)149 1958 y Fq(Using)33 b(the)g(ab)s(o)m(v)m(e)i(estimates)e (in)f(the)i(b)s(ound)d(\(4.14\))k(and)e(taking)g(in)m(to)g(accoun)m(t)i (that)f Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)c(\024)g(C)5 b Fp(")34 b Fq(for)149 2074 y(0)26 b Fn(\024)f Fp(s)g Fn(\024)g Fp(T)13 b Fq(,)30 b(w)m(e)h(obtain:)959 2269 y Fn(k)p Fp(K)1081 2283 y Fo(A)1139 2269 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1533 2283 y Fo(A)1590 2269 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1835 2289 y Fo(L)1883 2270 y Fi(2)1949 2269 y Fn(\024)2066 2246 y Fq(~)2045 2269 y Fp(D)2120 2283 y Fm(3)2159 2269 y Fq(\()p Fn(C)26 b Fq(+)20 b Fn(R)2436 2283 y Fm(1)2476 2269 y Fq(\))2511 2232 y Fm(2)2550 2269 y Fp(")2592 2232 y Fm(2)2632 2269 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2933 2283 y Fo(X)3001 2269 y Fp(;)577 b Fq(\(4.15\))149 2465 y(where)38 b(the)h(p)s(ositiv)m(e)e(constan)m(t)1325 2442 y(~)1303 2465 y Fp(D)1378 2479 y Fm(3)1456 2465 y Fq(dep)s(ends)g(only)g(on)h(sup)2294 2486 y Fo(t)p Fy(2)p Fd(R)2434 2465 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)39 b Fq(and)f(the)g(disp)s(ersion)d(constan)m(ts)149 2581 y Fp(D)224 2595 y Fm(1)264 2581 y Fq({)p Fp(D)384 2595 y Fm(3)450 2581 y Fq(in)25 b(the)h(deca)m(y)h(estimates)g (\(3.6\){\(3.8\).)43 b(Com)m(bining)24 b(the)i(estimates)h(\(4.11\),)i (\(4.13\),)h(and)25 b(\(4.15\),)149 2697 y(w)m(e)31 b(obtain)f(the)h (estimate)g(\(4.7\),)h(where)e Fn(D)1662 2711 y Fm(1)1727 2697 y Fn(\025)25 b Fq(max)p Fn(f)2059 2674 y Fq(~)2037 2697 y Fp(D)2112 2711 y Fm(1)2152 2697 y Fp(;)2214 2674 y Fq(~)2192 2697 y Fp(D)2267 2711 y Fm(2)2307 2697 y Fp(;)2369 2674 y Fq(~)2347 2697 y Fp(D)2422 2711 y Fm(3)2462 2697 y Fn(g)p Fq(.)291 2813 y(Let)e Fp(u;)37 b(v)29 b Fn(2)c Fp(Y)20 b Fq(.)38 b(F)-8 b(or)23 b(an)m(y)g(\014xed)e(small)g Fp(")26 b(>)f Fq(0,)g(there)d(exists)g(large)h Fn(R)2633 2827 y Fm(2)2698 2813 y Fp(>)h Fq(0,)h(suc)m(h)d(that)h Fn(k)p Fp(u)p Fn(k)3416 2827 y Fo(Y)3478 2813 y Fp(;)38 b Fn(k)p Fp(v)s Fn(k)3678 2827 y Fo(Y)3765 2813 y Fn(\024)149 2930 y(R)226 2944 y Fm(2)266 2930 y Fp(")p Fq(.)82 b(The)44 b(pro)s(of)f(of)i(the)f(estimate)h(\(4.8\))h(follo)m(ws)d(closely)h (that)h(of)f(the)h(estimate)g(\(4.7\))g(with)e(the)149 3046 y(simpli\014cation)35 b(that)j(the)g(space)h Fp(Y)57 b Fq(requires)37 b(smaller)f(deca)m(y)j(in)d(time.)63 b(On)37 b(the)h(other)g(hand,)g(it)g(also)149 3162 y(requires)33 b(that)h Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)h Fq(deca)m(ys)f(on)g([0)p Fp(;)15 b(T)e Fq(],)36 b(whic)m(h)c(is)h(not)h (necessary)g(in)e(the)i(pro)s(of)f(of)h(the)g(b)s(ound)d(\(4.7\).)149 3278 y(Similarly)24 b(to)k(the)f(estimate)h(\(4.9\),)i(w)m(e)e(use)f (the)h(disp)s(ersiv)m(e)c(estimate)k(\(3.6\),)i(the)e(de\014nition)d (\(3.12\),)30 b(and)149 3394 y(the)h(deca)m(y)h(estimate)f(\(3.14\))h (and)e(obtain)g(the)g(follo)m(wing)f(estimate)i(in)f Fp(L)2718 3361 y Fm(2)2718 3417 y Fy(\000)p Fo(\033)2819 3394 y Fq(\()p Fl(R)2914 3361 y Fm(3)2960 3394 y Fq(\):)557 3590 y Fn(k)p Fp(K)679 3604 y Fo(A)737 3590 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1131 3604 y Fo(A)1188 3590 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1433 3613 y Fo(L)1481 3591 y Fi(2)1481 3634 y Fk(\000)p Fh(\033)3603 3590 y Fq(\(4.16\))403 3813 y Fn(\024)83 b Fq(3)49 b(sup)617 3890 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)836 3813 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1066 3689 y Fe(Z)1158 3715 y Fo(t)1117 3895 y Fm(0)1407 3751 y Fn(C)1460 3718 y Fm(2)1500 3751 y Fp(")1542 3718 y Fm(2)p 1213 3792 564 4 v 1213 3879 a Fq(\(1)21 b(+)f(4\000)p Fp(")1549 3853 y Fm(4)1589 3879 y Fp(s)p Fq(\))1667 3853 y Fm(1)p Fo(=)p Fm(2)2001 3751 y Fp(D)2076 3765 y Fm(1)p 1797 3792 524 4 v 1797 3879 a Fq(\(1)h(+)e Fp(t)i Fn(\000)e Fp(s)p Fq(\))2210 3853 y Fm(3)p Fo(=)p Fm(2)2330 3813 y Fn(k)p Fp( )2437 3775 y Fm(2)2478 3813 y Fq(\()p Fp(u)p Fq(\()p Fp(s)p Fq(\))i Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\)\))31 b Fn(k)3061 3832 y Fo(L)3109 3814 y Fi(2)3109 3849 y Fh(\033)3155 3813 y Fp(ds)403 4082 y Fq(+)83 b(3)49 b(sup)617 4159 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)836 4082 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1066 3958 y Fe(Z)1158 3984 y Fo(t)1117 4164 y Fm(0)1447 4020 y Fn(C)g Fp(")p 1213 4061 564 4 v 1213 4148 a Fq(\(1)21 b(+)f(4\000)p Fp(")1549 4122 y Fm(4)1589 4148 y Fp(s)p Fq(\))1667 4122 y Fm(1)p Fo(=)p Fm(4)2001 4020 y Fp(D)2076 4034 y Fm(1)p 1797 4061 524 4 v 1797 4148 a Fq(\(1)h(+)e Fp(t)i Fn(\000)e Fp(s)p Fq(\))2210 4122 y Fm(3)p Fo(=)p Fm(2)2330 4082 y Fn(k)p Fp( )s Fq(\()p Fp(u)2524 4044 y Fm(2)2565 4082 y Fq(\()p Fp(s)p Fq(\))i Fn(\000)f Fp(v)2837 4044 y Fm(2)2876 4082 y Fq(\()p Fp(s)p Fq(\)\))p Fn(k)3069 4101 y Fo(L)3117 4083 y Fi(2)3117 4118 y Fh(\033)3164 4082 y Fp(ds)403 4351 y Fq(+)116 b(sup)557 4428 y Fm(0)p Fy(\024)p Fo(s)p Fy(\024)p Fo(t)775 4351 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)1005 4227 y Fe(Z)1097 4253 y Fo(t)1057 4433 y Fm(0)1142 4351 y Fn(k)p Fp(e)1229 4313 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1541 4351 y Fn(P)1604 4365 y Fo(c)1639 4351 y Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1889 4313 y Fm(2)1950 4351 y Fq(+)20 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)h Fq(+)f Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2738 4313 y Fm(2)2779 4351 y Fq(\))31 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))20 b Fn(\000)g Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3407 4374 y Fo(L)3455 4352 y Fi(2)3455 4395 y Fk(\000)p Fh(\033)3550 4351 y Fp(ds:)149 4599 y Fq(The)d(\014rst)f(t)m(w)m(o)i(in)m(tegrals)e (are)i(b)s(ounded)c(with)i(the)h(H\177)-45 b(older)27 b(estimates)i(and)e(the)h(prop)s(erties)e(of)i(functions)149 4715 y(in)i Fp(Y)19 b Fq(:)779 4910 y Fn(k)p Fp( )886 4873 y Fm(2)926 4910 y Fq(\()p Fp(u)p Fq(\()p Fp(s)p Fq(\))i Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\)\))p Fn(k)1478 4930 y Fo(L)1526 4911 y Fi(2)1526 4947 y Fh(\033)1656 4910 y Fn(\024)83 b(k)p Fq(\(1)22 b(+)e Fn(j)p Fp(x)p Fn(j)2150 4873 y Fm(2)2189 4910 y Fq(\))2224 4873 y Fo(\033)2272 4910 y Fp( )2334 4873 y Fm(2)2374 4910 y Fn(k)2419 4924 y Fo(L)2467 4905 y Fk(1)2536 4910 y Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(k)3063 4934 y Fo(L)3111 4911 y Fi(2)3111 4955 y Fk(\000)p Fh(\033)1656 5132 y Fn(\024)1820 5071 y(k)p Fq(\(1)i(+)e Fn(j)p Fp(x)p Fn(j)2160 5038 y Fm(2)2199 5071 y Fq(\))2234 5038 y Fo(\033)2282 5071 y Fp( )2344 5038 y Fm(2)2384 5071 y Fn(k)2429 5085 y Fo(L)2477 5066 y Fk(1)p 1820 5111 727 4 v 1901 5199 a Fq(\(1)h(+)f(4\000)p Fp(")2237 5172 y Fm(4)2277 5199 y Fp(s)p Fq(\))2355 5172 y Fm(3)p Fo(=)p Fm(4)2556 5132 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)2857 5146 y Fo(Y)1947 5425 y Fq(13)p eop %%Page: 14 14 14 13 bop 149 75 a Fq(and)328 282 y Fn(k)p Fp( )s Fq(\()p Fp(u)522 245 y Fm(2)562 282 y Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)834 245 y Fm(2)873 282 y Fq(\()p Fp(s)p Fq(\)\))p Fn(k)1066 302 y Fo(L)1114 283 y Fi(2)1114 319 y Fh(\033)1244 282 y Fn(\024)83 b(k)p Fq(\(1)22 b(+)d Fn(j)p Fp(x)p Fn(j)1737 245 y Fm(2)1777 282 y Fq(\))1812 245 y Fo(\033)r(=)p Fm(2)1930 282 y Fp( )s Fn(k)2037 296 y Fo(L)2085 277 y Fk(1)2155 282 y Fn(k)31 b(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)21 b Fq(+)f Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)2844 302 y Fo(L)2892 283 y Fi(4)2932 282 y Fn(k)f(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3570 302 y Fo(L)3618 283 y Fi(4)1244 496 y Fn(\024)1408 435 y Fq(2)p Fn(R)1530 449 y Fm(2)1570 435 y Fp(")p Fn(k)p Fq(\(1)22 b(+)e Fn(j)p Fp(x)p Fn(j)1952 402 y Fm(2)1992 435 y Fq(\))2027 402 y Fo(\033)r(=)p Fm(2)2144 435 y Fp( )s Fn(k)2251 449 y Fo(L)2299 430 y Fk(1)p 1408 475 962 4 v 1697 558 a Fq(1)h(+)f(4\000)p Fp(")1998 532 y Fm(4)2038 558 y Fp(s)2380 496 y Fn(k)p Fp(u)g Fn(\000)g Fp(v)s Fn(k)2680 510 y Fo(Y)2742 496 y Fp(:)149 741 y Fq(The)30 b(last)h(in)m(tegral)f(term)g(is)f(b)s(ounded)g(with)g(the)h (estimates)h(in)e Fp(L)2451 708 y Fm(4)2491 741 y Fq(\()p Fl(R)2586 708 y Fm(3)2631 741 y Fq(\):)584 949 y Fn(k)p Fp(e)671 911 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))983 949 y Fn(P)1046 963 y Fo(c)1081 949 y Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1331 911 y Fm(2)1392 949 y Fq(+)20 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)d Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2180 911 y Fm(2)2221 949 y Fq(\))31 b Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))21 b Fn(\000)e Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)32 b(k)2850 973 y Fo(L)2898 950 y Fi(2)2898 994 y Fk(\000)p Fh(\033)431 1121 y Fn(\024)82 b(k)p Fq(\(1)22 b(+)e Fn(j)p Fp(x)p Fn(j)924 1084 y Fm(2)964 1121 y Fq(\))999 1084 y Fy(\000)p Fo(\033)1101 1121 y Fn(k)1146 1141 y Fo(L)1194 1122 y Fi(4)1233 1121 y Fn(k)p Fp(e)1320 1084 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))1632 1121 y Fn(P)1695 1135 y Fo(c)1730 1121 y Fq(\()p Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1980 1084 y Fm(2)2040 1121 y Fq(+)g Fn(j)p Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)i Fq(+)e Fn(j)p Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)2829 1084 y Fm(2)2870 1121 y Fq(\))p Fn(k)p Fp(u)p Fq(\()p Fp(s)p Fq(\))h Fn(\000)f Fp(v)s Fq(\()p Fp(s)p Fq(\))p Fn(j)31 b(k)3488 1147 y Fo(L)3536 1127 y Fi(4)p Fh(=)p Fi(3)431 1304 y Fn(\024)720 1242 y Fq(3)p Fp(D)840 1256 y Fm(2)p 594 1283 413 4 v 594 1370 a Fq(2\()p Fp(t)21 b Fn(\000)f Fp(s)p Fq(\))897 1343 y Fm(3)p Fo(=)p Fm(4)1017 1304 y Fn(kj)p Fp(u)p Fn(j)1164 1266 y Fm(2)1225 1304 y Fq(+)f Fn(j)p Fp(v)s Fn(j)1412 1266 y Fm(2)1453 1304 y Fn(k)1498 1323 y Fo(L)1546 1304 y Fi(2)9 b Fn(k)p Fp(u)21 b Fn(\000)f Fp(v)s Fn(k)1886 1323 y Fo(L)1934 1304 y Fi(4)431 1563 y Fn(\024)881 1501 y Fq(3)p Fp(D)1001 1515 y Fm(2)1041 1501 y Fn(R)1118 1468 y Fm(2)1118 1526 y(2)1157 1501 y Fp(")1199 1468 y Fm(2)p 594 1542 932 4 v 594 1629 a Fq(\()p Fp(t)h Fn(\000)f Fp(s)p Fq(\))852 1603 y Fm(3)p Fo(=)p Fm(4)961 1629 y Fq(\(1)h(+)f(4\000)p Fp(")1297 1603 y Fm(4)1337 1629 y Fp(s)p Fq(\))1415 1603 y Fm(3)p Fo(=)p Fm(2)1535 1563 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)1836 1577 y Fo(Y)1898 1563 y Fp(:)149 1820 y Fq(Plugging)26 b(the)h(ab)s(o)m(v)m(e)h (estimates)f(in)m(to)f(\(4.16\))j(and)d(using)g(the)h(con)m(v)m (olution)f(inequalities)e(\(4.3\){\(4.4\))31 b(w)m(e)149 1936 y(obtain:)650 2052 y Fn(k)p Fp(K)772 2066 y Fo(A)830 2052 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1224 2066 y Fo(A)1281 2052 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1526 2076 y Fo(L)1574 2053 y Fi(2)1574 2097 y Fk(\000)p Fh(\033)1695 2052 y Fn(\024)1812 2029 y Fq(~)1791 2052 y Fp(D)1866 2066 y Fm(4)1905 2052 y Fq(\()p Fn(C)26 b Fq(+)20 b Fn(R)2182 2066 y Fm(2)2222 2052 y Fq(\))2257 2015 y Fm(2)2296 2052 y Fp(")p Fq(\(1)i(+)e(4\000)p Fp(")2675 2015 y Fm(4)2715 2052 y Fp(t)p Fq(\))2783 2015 y Fy(\000)p Fm(3)p Fo(=)p Fm(4)2948 2052 y Fn(k)p Fp(u)h Fn(\000)e Fp(v)s Fn(k)3248 2066 y Fo(Y)3310 2052 y Fp(;)268 b Fq(\(4.17\))149 2250 y(where)28 b(the)g(p)s(ositiv)m(e)f(constan)m(t) 1283 2227 y(~)1262 2250 y Fp(D)1337 2264 y Fm(4)1404 2250 y Fq(dep)s(ends)f(only)h(on)h(\000)p Fp(;)43 b Fq(sup)2336 2271 y Fo(t)p Fy(2)p Fd(R)2476 2250 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)29 b Fq(and)e(the)h(disp)s(ersion)c (constan)m(ts)149 2366 y Fp(D)224 2380 y Fm(1)264 2366 y Fq({)p Fp(D)384 2380 y Fm(3)455 2366 y Fq(in)29 b(the)h(deca)m(y)i (estimates)f(\(3.6\){\(3.8\).)291 2482 y(The)k(estimates)h(in)f Fp(X)7 b Fq(,)37 b(giv)m(en)f(in)f(\(4.12\){\(4.13\),)42 b(extend)36 b(to)h(similar)c(estimates)j(in)f Fp(Y)20 b Fq(.)56 b(Although)149 2598 y(the)38 b(functions)d Fp(u;)15 b(v)41 b Fq(deca)m(y)d(slo)m(wly)e(in)g(time)h(but)f(the)h (slo)m(w)g(deca)m(y)h(is)e(comp)s(ensated)h(b)m(y)g(the)h(deca)m(y)g (in)149 2715 y Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fq(,)32 b(giv)m(en)e(b)m(y)h(\(3.14\).)43 b(As)30 b(a)h(result,)e(w)m (e)i(ha)m(v)m(e:)769 2922 y Fn(k)p Fp(K)891 2936 y Fo(A)949 2922 y Fq([)p Fp(u)p Fq(]\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(K)1343 2936 y Fo(A)1401 2922 y Fq([)p Fp(v)s Fq(]\()p Fp(t)p Fq(\))p Fn(k)1646 2942 y Fo(L)1694 2923 y Fi(4)1759 2922 y Fn(\024)1876 2899 y Fq(~)1855 2922 y Fp(D)1930 2936 y Fm(5)1970 2922 y Fq(\()p Fn(C)26 b Fq(+)20 b Fn(R)2247 2936 y Fm(2)2286 2922 y Fq(\))2321 2885 y Fm(2)2361 2922 y Fp(")p Fq(\(1)h(+)f Fp(t)p Fq(\))2663 2885 y Fy(\000)p Fm(1)p Fo(=)p Fm(2)2828 2922 y Fn(k)p Fp(u)h Fn(\000)f Fp(v)s Fn(k)3129 2936 y Fo(Y)3190 2922 y Fp(;)388 b Fq(\(4.18\))149 3130 y(where)30 b(the)h(p)s(ositiv)m(e)f(constan)m(t)1294 3107 y(~)1273 3130 y Fp(D)1348 3144 y Fm(5)1418 3130 y Fq(dep)s(ends)e(on)j(\000,)f(sup)2145 3151 y Fo(t)p Fy(2)p Fd(R)2285 3130 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fq(,)32 b(and)e(the)g(disp)s(ersion)e(constan)m(t)j Fp(D)3750 3144 y Fm(1)3790 3130 y Fq({)149 3246 y Fp(D)224 3260 y Fm(3)304 3246 y Fq(in)39 b(the)h(deca)m(y)h(estimates)f (\(3.6\){\(3.8\).)74 b(Com)m(bining)38 b(\(4.17\))k(and)d(\(4.18\),)45 b(w)m(e)40 b(\014nd)f(the)h(estimate)149 3362 y(\(4.8\),)33 b(where)d Fn(D)725 3376 y Fm(2)790 3362 y Fn(\025)25 b Fq(max)p Fn(f)1121 3339 y Fq(~)1100 3362 y Fp(D)1175 3376 y Fm(4)1215 3362 y Fp(;)1276 3339 y Fq(~)1255 3362 y Fp(D)1330 3376 y Fm(5)1370 3362 y Fn(g)p Fq(.)p 3770 3362 4 62 v 3774 3304 55 4 v 3774 3362 V 3828 3362 4 62 v 291 3553 a(In)j(App)s(endix)f(A)j(w)m(e)g(extend)f(the)h (classical)f(principle)d(of)j(sup)s(erp)s(osition)d(from)j(forced)h (linear)e(equa-)149 3669 y(tions)41 b(to)h(the)g(\014xed)f(p)s(oin)m (ts)f(of)h(nonlinear,)i(lo)s(cally)d(con)m(tractiv)m(e,)46 b(nonhomogeneous)c(op)s(erators)f(in)f(a)149 3786 y(hierarc)m(h)m(y)29 b(of)g(Banac)m(h)h(spaces.)40 b(In)28 b(the)h(particular)f(case)h(of)g (the)g(in)m(tegral)g(equation)f(\(3.11\),)k(this)c(princi-)149 3902 y(ple)i(implies)e(that)j(the)f(solution)f Fp(u)1354 3917 y Fo(d)1395 3902 y Fq(\()p Fp(t)p Fq(\))h(can)h(b)s(e)f(written)f (as)i(a)g(sum)e(of)i(a)g(function)e(in)g(X)h(and)g(a)h(function)149 4018 y(in)h(Y.)h(The)f(former)g(is)f(the)i(resp)s(onse)f(to)h(the)g (\014rst)f(forcing)g(term)g(and,)h(although)f(it)g(is)g(relativ)m(ely)f (large)149 4134 y(at)d(the)g(b)s(eginning)c(as)k Fp(O)s Fq(\()p Fp(")p Fq(\),)h(it)e(deca)m(ys)h(fast)f(in)f(time.)40 b(The)26 b(latter)i(can)f(b)s(e)g(in)m(terpreted)g(as)g(the)g(resp)s (onse)149 4250 y(to)36 b(the)f(second)g(forcing)f(term)h(and,)g (although)f(it)h(is)f(relativ)m(ely)g(small)f(at)i(the)g(b)s(eginning)e (as)i Fp(O)s Fq(\()p Fp(")3639 4217 y Fm(3)3679 4250 y Fq(\),)h(it)149 4367 y(deca)m(ys)k(v)m(ery)f(slo)m(wly)e(in)g(time)h (and)g(ev)m(en)m(tually)h(dominates.)64 b(Because)40 b(the)e(t)m(w)m(o)i(forcing)e(terms)g(ha)m(v)m(e)149 4483 y(di\013eren)m(t)26 b(initial)e(sizes)i(and)g(ev)m(olv)m(e)i(on)e (di\013eren)m(t)g(time)g(scales,)h(separation)g(of)f(eac)m(h)i(term)e (is)g(essen)m(tial)g(in)149 4599 y(the)31 b(pro)s(of)e(of)i(our)e(main) g(result.)40 b(W)-8 b(e)31 b(note)g(that)g(mo)s(dulation)d(equations)i (w)m(ere)h(used)e(in)g([22)q(,)h(23)q(,)h(25])g(to)149 4715 y(separate)e(the)f(time)f(scales)h(b)s(et)m(w)m(een)g(these)g(t)m (w)m(o)h(terms.)40 b(W)-8 b(e)28 b(completely)g(a)m(v)m(oid)g(using)e (the)i(mo)s(dulation)149 4832 y(equations)j(b)m(y)f(relying)f(on)h(the) g(con)m(traction)i(principle)27 b(of)j(App)s(endix)e(A.)149 5023 y Fg(Lemma)42 b(4.2.)k Ff(Ther)-5 b(e)40 b(exists)g(a)g Fp(T)13 b Ff(-indep)-5 b(endent)40 b(p)-5 b(ositive)40 b(c)-5 b(onstant)42 b Fn(D)s Ff(,)e(which)h(dep)-5 b(ends)41 b(only)f(on)g Fq(\000)p Ff(,)149 5139 y Fq(sup)286 5161 y Fo(t)p Fy(2)p Fd(R)426 5139 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Ff(,)45 b(and)d(the)g(disp)-5 b(ersion)43 b(c)-5 b(onstants)44 b Fp(D)1970 5153 y Fm(1)2009 5139 y Ff({)p Fp(D)2130 5153 y Fm(3)2212 5139 y Ff(in)d(the)h(de)-5 b(c)g(ay)43 b(estimates)f(\(3.6\){\(3.8\),)k(such)1947 5425 y Fq(14)p eop %%Page: 15 15 15 14 bop 149 75 a Ff(that)34 b(the)f(solution)h Fp(u)882 90 y Fo(d)923 75 y Fq(\()p Fp(t)p Fq(\))f Ff(of)g(the)g(inte)-5 b(gr)g(al)34 b(e)-5 b(quation)33 b(\(3.11\))h(b)-5 b(elongs)34 b(to)f Fp(Y)52 b Ff(for)1607 278 y Fp(")25 b(<)g(")1812 292 y Fm(0)1877 278 y Fn(\024)1983 216 y Fq(1)p 1983 257 46 4 v 1983 340 a(2)2039 278 y Fn(D)2112 240 y Fy(\000)p Fm(1)2206 278 y Fn(C)2259 240 y Fy(\000)p Fm(3)2353 278 y Fp(:)1225 b Fq(\(4.19\))149 469 y Ff(Mor)-5 b(e)g(over,)34 b(the)f(solution)h Fp(u)1124 484 y Fo(d)1164 469 y Fq(\()p Fp(t)p Fq(\))g Ff(c)-5 b(an)33 b(b)-5 b(e)32 b(written)i(as:)429 637 y Fp(u)481 652 y Fo(d)522 637 y Fq(\()p Fp(t)p Fq(\))25 b(=)g Fp(f)10 b Fq(\()p Fp(t)p Fq(\))20 b(+)g Fp(v)1059 652 y Fo(d)1100 637 y Fq(\()p Fp(t)p Fq(\))h(+)f Fp(E)1382 652 y Fo(d)1422 637 y Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(e)1689 599 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1858 637 y Fp(u)1910 652 y Fo(d)1950 637 y Fq(\(0\))d(+)e Fp(f)10 b Fq(\()p Fp(t)p Fq(\))20 b(+)g Fp(K)2524 651 y Fo(A)2581 637 y Fq([)p Fp(f)30 b Fq(+)19 b Fp(v)2815 652 y Fo(d)2856 637 y Fq(]\()p Fp(t)p Fq(\))i(+)f Fp(F)3154 652 y Fo(d)3195 637 y Fq(\()p Fp(t)p Fq(\))p Fp(;)280 b Fq(\(4.20\))149 805 y Ff(wher)-5 b(e)1053 952 y Fp(f)10 b Fq(\()p Fp(t)p Fq(\))25 b(=)g Fn(\000)p Fp(i)1449 828 y Fe(Z)1540 854 y Fo(t)1500 1034 y Fm(0)1585 952 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(A)p Fn(j)1868 914 y Fm(2)1908 952 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)2131 914 y Fy(\000)p Fo(i\025s)2289 952 y Fp(e)2331 914 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2642 952 y Fn(P)2705 966 y Fo(c)2740 952 y Fp( )2802 914 y Fm(3)2842 952 y Fp(ds)671 b Fq(\(4.21\))149 1140 y Ff(and)891 1256 y Fn(k)p Fp(v)980 1271 y Fo(d)1021 1256 y Fn(k)1066 1270 y Fo(X)1159 1256 y Fn(\024)25 b(D)s Fp(";)201 b Fn(k)p Fp(E)1708 1271 y Fo(d)1749 1256 y Fn(k)1794 1270 y Fo(Y)1881 1256 y Fn(\024)25 b(D)s(C)2103 1219 y Fm(5)2142 1256 y Fp(")2184 1219 y Fm(3)2224 1256 y Fp(;)201 b Fn(k)p Fp(F)2553 1271 y Fo(d)2594 1256 y Fn(k)2639 1270 y Fo(Y)2726 1256 y Fn(\024)25 b(D)s(C)2948 1219 y Fm(7)2987 1256 y Fp(")3029 1219 y Fm(4)3069 1256 y Fp(:)509 b Fq(\(4.22\))149 1414 y Ff(Pr)-5 b(o)g(of.)47 b Fq(W)-8 b(e)32 b(apply)d(Prop)s(osition) f(A.2)j(from)f(App)s(endix)e(A)i(to)i(the)e(in)m(tegral)g(equations)g (\(3.11\))j(and)1373 1583 y Fp(v)1417 1598 y Fo(d)1458 1583 y Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(e)1725 1545 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1894 1583 y Fp(u)1946 1598 y Fo(d)1986 1583 y Fq(\(0\))d(+)e Fp(K)2291 1597 y Fo(A)2348 1583 y Fq([)p Fp(v)2417 1598 y Fo(d)2458 1583 y Fq(]\()p Fp(t)p Fq(\))p Fp(:)992 b Fq(\(4.23\))149 1751 y(Hyp)s(othesis)30 b(\(A.1\),)i(\(A.2\))g(and)e(\(A.11\))i(hold)d(b)s(ecause)i(of)f(Lemma) h(4.1.)42 b(T)-8 b(o)31 b(v)m(erify)f(\(A.12\){\(A.16\),)35 b(w)m(e)149 1867 y(\014x)30 b Fp(L)340 1881 y Fm(1)405 1867 y Fq(=)25 b Fp(L)563 1881 y Fm(2)627 1867 y Fq(=)g(1)p Fp(=)p Fq(2.)42 b(Due)31 b(to)g(inclusions)c(\(4.5\){\(4.6\),)35 b(there)c(exist)f(constan)m(ts)h Fn(D)3037 1881 y Fm(3)3107 1867 y Fq(and)f Fn(D)3354 1881 y Fm(4)3393 1867 y Fq(,)h(suc)m(h)f (that)580 2036 y Fn(k)p Fp(e)667 1998 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)837 2036 y Fp(u)889 2051 y Fo(d)929 2036 y Fq(\(0\))p Fn(k)1089 2050 y Fo(X)1241 2036 y Fn(\024)83 b(D)1465 2050 y Fm(3)1505 2036 y Fp(")318 2177 y Fn(k)p Fp(e)405 2139 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)575 2177 y Fp(u)627 2192 y Fo(d)667 2177 y Fq(\(0\))21 b(+)f Fp(f)10 b Fq(\()p Fp(t)p Fq(\))p Fn(k)1097 2191 y Fo(Y)1241 2177 y Fn(\024)83 b(k)p Fp(e)1482 2139 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1652 2177 y Fp(u)1704 2192 y Fo(d)1744 2177 y Fq(\(0\))p Fn(k)1904 2191 y Fo(Y)1987 2177 y Fq(+)20 b Fn(k)p Fp(f)10 b Fq(\()p Fp(t)p Fq(\))p Fn(k)2326 2191 y Fo(Y)2412 2177 y Fn(\024)25 b(D)2578 2191 y Fm(4)2618 2177 y Fp(")p Fq(\(1)c(+)f Fn(C)2905 2139 y Fm(3)2960 2177 y Fq(max)p Fn(f)p Fp(")p Fq(\000)3273 2139 y Fy(\000)p Fm(1)p Fo(=)p Fm(4)3439 2177 y Fp(;)15 b(")3521 2139 y Fm(2)3561 2177 y Fn(g)p Fq(\))p Fp(:)149 2345 y Fq(Let)29 b Fp(R)379 2359 y Fm(1)444 2345 y Fq(=)c(2)p Fn(D)655 2359 y Fm(3)695 2345 y Fp(")k Fq(and)f Fp(R)1010 2359 y Fm(2)1074 2345 y Fq(=)d(3)p Fn(D)1285 2359 y Fm(4)1325 2345 y Fp(")p Fq(.)41 b(Then,)28 b(the)g(equations)h(\(A.13\),)i (\(A.15\))f(and)e(\(A.16\))i(hold)d(with)g(the)149 2461 y(ab)s(o)m(v)m(e)h(c)m(hoices)g(of)e Fp(L)867 2475 y Fm(1)p Fo(;)p Fm(2)988 2461 y Fq(and)g Fp(R)1230 2475 y Fm(1)p Fo(;)p Fm(2)1325 2461 y Fq(,)h(pro)m(vided)e(that)j Fn(D)f(\025)e Fq(max)q Fn(f)p Fq(\000)2406 2428 y Fy(\000)p Fm(1)p Fo(=)p Fm(4)2571 2461 y Fp(;)15 b Fq(2)2656 2428 y Fy(\000)p Fm(1)p Fo(=)p Fm(2)2822 2461 y Fn(g)p Fq(.)39 b(T)-8 b(o)28 b(simplify)23 b(the)k(expres-)149 2577 y(sion)g(of)h(the)h(Lipsc)m(hitz)d(constan)m(ts)j(in)e(the)h(b)s(ounds) e(\(4.7\){\(4.8\))32 b(w)m(e)c(assume)g(that)g Fn(D)g(\025)d Fq(max)p Fn(f)p Fq(2)p Fn(D)3554 2591 y Fm(3)3595 2577 y Fp(;)15 b Fq(3)p Fn(D)3750 2591 y Fm(4)3790 2577 y Fn(g)149 2694 y Fq(and)29 b Fn(C)h Fp(>)25 b Fn(D)s Fq(,)k(according)g (to)h(the)f(inequalit)m(y)e(\(3.15\).)43 b(The)28 b(Lipsc)m(hitz)g (constan)m(ts)i(for)f Fp(K)3205 2708 y Fo(A)3262 2694 y Fq([)p Fp(u)p Fq(])g(in)f(the)h(balls)149 2810 y(of)i(radius)d Fp(R)595 2824 y Fm(1)665 2810 y Fq(for)i Fp(X)38 b Fq(and)30 b Fp(R)1163 2824 y Fm(2)1233 2810 y Fq(for)g Fp(Y)50 b Fq(are)31 b(dominated)e(b)m(y:)1133 2978 y Fp(L)1195 2992 y Fo(X)1262 2978 y Fq(\()p Fp(R)1366 2992 y Fm(1)1406 2978 y Fq(\))c Fn(\024)g Fq(4)p Fn(D)1677 2992 y Fm(3)1717 2978 y Fn(C)1770 2941 y Fm(2)1810 2978 y Fp(")1852 2941 y Fm(2)1892 2978 y Fp(;)197 b(L)2176 2992 y Fo(Y)2237 2978 y Fq(\()p Fp(R)2341 2992 y Fm(2)2380 2978 y Fq(\))26 b Fn(\024)f Fq(4)p Fn(D)2652 2992 y Fm(4)2692 2978 y Fn(C)2745 2941 y Fm(2)2785 2978 y Fp(":)751 b Fq(\(4.24\))149 3146 y(Conditions)25 b(\(A.12\))j(and)e(\(A.14\))j(with)c Fp(L)1607 3160 y Fm(1)1672 3146 y Fq(=)g Fp(L)1830 3160 y Fm(2)1894 3146 y Fq(=)g(1)p Fp(=)p Fq(2)j(hold)d(pro)m(vided)g(that)j (the)e(b)s(ound)f(\(4.19\))k(holds.)149 3263 y(With)h(the)h(ab)s(o)m(v) m(e)g(constrain)m(ts,)g(Prop)s(osition)d(A.2)k(is)d(no)m(w)h(applied)f (to)i(our)f(case,)h(when)1120 3431 y Fn(k)p Fp(v)1209 3446 y Fo(d)1250 3431 y Fn(k)1295 3445 y Fo(X)1388 3431 y Fn(\024)25 b Fq(2)p Fn(D)1599 3445 y Fm(3)1639 3431 y Fp(";)198 b Fn(k)p Fp(f)10 b Fq(\()p Fp(t)p Fq(\))p Fn(k)2152 3445 y Fo(Y)2238 3431 y Fn(\024)25 b(D)2404 3445 y Fm(4)2444 3431 y Fq(\000)2501 3393 y Fy(\000)p Fm(1)p Fo(=)p Fm(4)2665 3431 y Fn(C)2718 3393 y Fm(3)2758 3431 y Fp(")2800 3393 y Fm(2)2840 3431 y Fp(:)738 b Fq(\(4.25\))149 3599 y(F)-8 b(or)32 b(the)f(Lipsc)m(hitz)e(constan)m(ts,)k(w)m(e)e(can) g(use)f(\(4.24\))j(whic)m(h)d(are)h(b)s(oth)f(less)g(than)g(1)p Fp(=)p Fq(2.)44 b(As)30 b(a)h(result,)f(the)149 3715 y(estimates)h(\(4.22\))i(follo)m(w)d(directly)f(from)h(\(A.19\))i(and)e (\(A.21\))i(pro)m(vided)d(that)1125 3884 y Fn(D)f(\025)d Fq(max)p Fn(f)p Fq(2)p Fn(D)1648 3898 y Fm(3)1688 3884 y Fp(;)15 b Fq(8)p Fn(D)1843 3898 y Fm(1)1884 3884 y Fn(D)1954 3898 y Fm(4)1993 3884 y Fq(\000)2050 3846 y Fy(\000)p Fm(1)p Fo(=)p Fm(4)2215 3884 y Fp(;)g Fq(32)p Fn(D)2418 3846 y Fm(2)2415 3906 y(2)2458 3884 y Fn(D)2528 3898 y Fm(4)2568 3884 y Fq(\000)2625 3846 y Fy(\000)p Fm(1)p Fo(=)p Fm(4)2790 3884 y Fn(g)p Fp(;)743 b Fq(\(4.26\))149 4052 y(where)30 b(the)h(constan)m(ts)g Fn(D)1041 4066 y Fm(1)p Fo(;)p Fm(2)1166 4052 y Fq(are)g(de\014ned)e(in)g(the)i(b)s (ounds)d(\(4.7\){\(4.8\).)p 3770 4052 4 62 v 3774 3994 55 4 v 3774 4052 V 3828 4052 4 62 v 149 4236 a Fg(Corollary)j(4.1.)38 b Ff(The)27 b(solution)i Fp(u)1381 4251 y Fo(d)1422 4236 y Fq(\()p Fp(t)p Fq(\))f Ff(of)g(the)g(inte)-5 b(gr)g(al)29 b(e)-5 b(quation)29 b(\(3.11\))g(satis\014es)f(the)h(de)-5 b(c)g(ay)29 b(estimates)149 4352 y(for)34 b Fq(0)25 b Fn(\024)g Fp(t)g Fn(\024)g Fp(T)13 b Ff(:)975 4568 y Fn(k)p Fp(u)1072 4583 y Fo(d)1113 4568 y Fq(\()p Fp(t)p Fq(\))21 b Fn(\000)f Fp(f)10 b Fq(\()p Fp(t)p Fq(\))p Fn(k)1531 4591 y Fo(L)1579 4569 y Fi(2)1579 4612 y Fk(\000)p Fh(\033)1756 4568 y Fn(\024)2047 4506 y(D)s Fp(")p 1919 4547 371 4 v 1919 4634 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2179 4608 y Fm(3)p Fo(=)p Fm(2)2320 4568 y Fq(+)2574 4506 y Fn(D)s(C)2700 4473 y Fm(7)2739 4506 y Fp(")2781 4473 y Fm(4)p 2421 4547 555 4 v 2421 4634 a Fq(\(1)h(+)e(4\000)p Fp(")2756 4608 y Fm(4)2797 4634 y Fp(t)p Fq(\))2865 4608 y Fm(3)p Fo(=)p Fm(4)3603 4568 y Fq(\(4.27\))1244 4827 y Fn(k)p Fp(u)1341 4842 y Fo(d)1382 4827 y Fq(\()p Fp(t)p Fq(\))p Fn(k)1530 4851 y Fo(L)1578 4828 y Fi(2)1578 4872 y Fk(\000)p Fh(\033)1756 4827 y Fn(\024)2047 4766 y(D)s Fp(")p 1919 4806 371 4 v 1919 4893 a Fq(\(1)i(+)f Fp(t)p Fq(\))2179 4867 y Fm(3)p Fo(=)p Fm(2)2320 4827 y Fq(+)2574 4766 y Fn(D)s(C)2700 4733 y Fm(5)2739 4766 y Fp(")2781 4733 y Fm(3)p 2421 4806 555 4 v 2421 4893 a Fq(\(1)h(+)e(4\000)p Fp(")2756 4867 y Fm(4)2797 4893 y Fp(t)p Fq(\))2865 4867 y Fm(3)p Fo(=)p Fm(4)3603 4827 y Fq(\(4.28\))1299 5087 y Fn(k)p Fp(u)1396 5102 y Fo(d)1437 5087 y Fq(\()p Fp(t)p Fq(\))p Fn(k)1585 5106 y Fo(L)1633 5087 y Fi(4)93 b Fn(\024)2047 5025 y(D)s Fp(")p 1919 5066 371 4 v 1919 5153 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))2179 5126 y Fm(3)p Fo(=)p Fm(4)2320 5087 y Fq(+)2574 5025 y Fn(D)s(C)2700 4992 y Fm(3)2739 5025 y Fp(")2781 4992 y Fm(2)p 2421 5066 555 4 v 2421 5153 a Fq(\(1)h(+)e(4\000)p Fp(")2756 5127 y Fm(4)2797 5153 y Fp(t)p Fq(\))2865 5126 y Fm(1)p Fo(=)p Fm(2)2985 5087 y Fp(:)593 b Fq(\(4.29\))1947 5425 y(15)p eop %%Page: 16 16 16 15 bop 149 75 a Ff(Pr)-5 b(o)g(of.)47 b Fq(The)28 b(b)s(ounds)d(\(4.28\))30 b(and)d(\(4.29\))j(follo)m(w)d(from)h (\(4.20\),)i(\(4.22\),)h(and)c(\(4.25\).)42 b(The)28 b(b)s(ound)d(\(4.27\))149 191 y(is)40 b(obtained)g(from)f(\(4.20\))k (and)d(\(4.23\))i(b)m(y)f(direct)e(calculation)h(of)h Fn(k)p Fp(K)2701 205 y Fo(A)2758 191 y Fq([)p Fp(v)2827 206 y Fo(d)2895 191 y Fq(+)26 b Fp(f)10 b Fq(])p Fn(k)3117 215 y Fo(L)3165 192 y Fi(2)3165 236 y Fk(\000)p Fh(\033)3299 191 y Fq(that)41 b(uses)f(the)149 327 y(estimates)31 b(\(4.25\))i(and)d(the)g(argumen)m(t)h(for)f(the)h Fp(L)1910 294 y Fm(2)1910 349 y Fy(\000)p Fo(\033)2042 327 y Fq(norms)e(in)g (Lemma)i(4.1.)p 3770 327 4 62 v 3774 268 55 4 v 3774 327 V 3828 327 4 62 v 149 609 a Fr(5)135 b(Reduction)46 b(to)f(the)g(amplitude)h(equation)149 815 y Fq(W)-8 b(e)35 b(analyze)e(here)g(the)g(amplitude)d(equation)j(\(3.2\),)j(whic)m(h)31 b(sho)m(ws)i(explicitly)d(the)k(deca)m(y)g(of)f Fp(A)p Fq(\()p Fp(t)p Fq(\))g(due)149 931 y(to)46 b(the)f(coupling)e(with)g (the)i(disp)s(ersiv)m(e)d(part)j Fp(u)1912 946 y Fo(d)1952 931 y Fq(\()p Fp(t)p Fq(\).)84 b(In)44 b(what)h(follo)m(ws)f Fn(D)s Fp(;)g Fn(D)3059 945 y Fm(1)3099 931 y Fp(;)h Fn(D)3239 945 y Fm(2)3278 931 y Fp(;)g Fq(etc.)85 b(denote)149 1047 y Fp(T)13 b Fq(-indep)s(enden)m(t)39 b(p)s(ositiv)m(e)g(constan)m (ts)j(that)f(dep)s(end)e(only)g(on)h(\000,)j(sup)2670 1069 y Fo(t)p Fy(2)p Fd(R)2810 1047 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fq(,)44 b(and)c(the)h(disp)s(ersion)149 1164 y(constan)m(ts)36 b Fp(D)631 1178 y Fm(1)670 1164 y Fq({)p Fp(D)790 1178 y Fm(4)864 1164 y Fq(in)d(the)h(deca)m(y)h (estimates)g(\(3.6\){\(3.9\).)55 b(In)33 b(general,)i(these)g(constan)m (ts)g(are)f(sligh)m(tly)149 1280 y(bigger)e(than)g(the)g(constan)m(t)i Fn(D)s Fq(,)e(de\014ned)f(in)f(Lemma)j(4.2.)46 b(Main)32 b(results)f(of)h(this)f(section)h(are)h(giv)m(en)f(b)m(y)149 1396 y(Lemmas)f(5.1,)h(5.2,)f(and)f(5.3.)291 1512 y(W)-8 b(e)26 b(eliminate)d(the)j(fast)f(oscillations)e(in)h(the)h(phase)g(of) g Fp(A)p Fq(\()p Fp(t)p Fq(\))g(in)f(the)h(amplitude)e(equation)i (\(3.2\))i(with)149 1628 y(the)k(cubic)e(term)i(\(3.4\):)1056 1799 y Fp(d)p Fn(j)p Fp(A)p Fn(j)1221 1766 y Fm(2)p 1056 1840 206 4 v 1119 1923 a Fp(dt)1355 1860 y Fq(=)82 b(2)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(A)p Fn(j)1826 1823 y Fm(2)1867 1860 y Fq(\()p Fp(t)p Fq(\))26 b(Im)2120 1760 y Fe(h)p 2163 1787 69 4 v 100 x Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp(e)2376 1823 y Fo(i\025t)2472 1860 y Fn(h)p Fp( )2569 1823 y Fm(3)2609 1860 y Fp(;)15 b(u)2701 1875 y Fo(d)2742 1860 y Fq(\()p Fp(t)p Fq(\))p Fn(i)2880 1760 y Fe(i)1355 2061 y Fq(+)82 b(2)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))27 b(Im)1858 1960 y Fe(h)p 1901 1988 V 101 x Fp(A)1969 2007 y Fm(2)2009 2061 y Fq(\()p Fp(t)p Fq(\))p Fp(e)2154 2023 y Fm(2)p Fo(i\025t)2285 2061 y Fn(h)p Fp( )2382 2023 y Fm(2)2422 2061 y Fp(;)15 b(u)2514 2023 y Fm(2)2514 2084 y Fo(d)2555 2061 y Fq(\()p Fp(t)p Fq(\))p Fn(i)2693 1960 y Fe(i)1355 2257 y Fq(+)82 b(2)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))27 b(Im)1858 2156 y Fe(h)p 1901 2184 V 101 x Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp(e)2114 2220 y Fo(i\025t)2210 2257 y Fn(h)p Fp( )s(;)15 b Fn(j)p Fp(u)2424 2272 y Fo(d)2466 2257 y Fn(j)2491 2220 y Fm(2)2531 2257 y Fp(u)2583 2272 y Fo(d)2623 2257 y Fq(\()p Fp(t)p Fq(\))p Fn(i)2761 2156 y Fe(i)2820 2257 y Fp(:)804 b Fq(\(5.1\))149 2457 y(The)38 b(leading-order)e(con)m(tribution)g(of)i Fp(u)1590 2472 y Fo(d)1631 2457 y Fq(\()p Fp(t)p Fq(\))g(app)s(ears)f (in)f(the)i(\014rst)f(term)h(on)g(the)g(righ)m(t)f(hand)g(side)f(of)149 2573 y(\(5.1\).)43 b(Let)384 2797 y Fp(g)s Fq(\()p Fp(t)p Fq(\))26 b(=)698 2736 y Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p 665 2776 223 4 v 665 2860 a Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)923 2797 y Fq(Im)1046 2696 y Fe(h)p 1089 2724 69 4 v 101 x Fp(A)1157 2743 y Fm(2)1197 2797 y Fq(\()p Fp(t)p Fq(\))p Fp(e)1342 2760 y Fm(2)p Fo(i\025t)1473 2797 y Fn(h)p Fp( )1570 2760 y Fm(2)1610 2797 y Fp(;)15 b(u)1702 2760 y Fm(2)1702 2820 y Fo(d)1743 2797 y Fq(\()p Fp(t)p Fq(\))p Fn(i)1881 2696 y Fe(i)1945 2797 y Fq(+)2079 2736 y Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p 2046 2776 223 4 v 2046 2860 a Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)2303 2797 y Fq(Im)2427 2696 y Fe(h)p 2470 2724 69 4 v 101 x Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp(e)2683 2760 y Fo(i\025t)2779 2797 y Fn(h)p Fp( )s(;)15 b Fn(j)p Fp(u)2993 2812 y Fo(d)3035 2797 y Fn(j)3060 2760 y Fm(2)3100 2797 y Fp(u)3152 2812 y Fo(d)3192 2797 y Fq(\()p Fp(t)p Fq(\))p Fn(i)3330 2696 y Fe(i)3389 2797 y Fp(;)235 b Fq(\(5.2\))149 3022 y(and)30 b(rewrite)g(the)g(amplitude)f(equation)h(\(5.1\))i(in)d (the)i(form:)1040 3185 y Fp(d)p Fn(j)p Fp(A)p Fn(j)p 1040 3226 167 4 v 1083 3309 a Fp(dt)1242 3247 y Fq(=)25 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(A)p Fn(j)p Fq(\()p Fp(t)p Fq(\))27 b(Im)1864 3146 y Fe(h)p 1907 3174 69 4 v 101 x Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp(e)2120 3209 y Fo(i\025t)2216 3247 y Fn(h)p Fp( )2313 3209 y Fm(3)2353 3247 y Fp(;)15 b(u)2445 3262 y Fo(d)2486 3247 y Fq(\()p Fp(t)p Fq(\))p Fn(i)2624 3146 y Fe(i)2688 3247 y Fq(+)20 b Fp(g)s Fq(\()p Fp(t)p Fq(\))p Fp(:)696 b Fq(\(5.3\))149 3449 y(By)31 b(the)g(H\177)-45 b(older)29 b(estimates,)j(w)m(e)e(ha)m(v)m(e)i(for)e Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(]:)975 3629 y Fn(jh)p Fp( )1097 3592 y Fm(2)1138 3629 y Fp(;)i(u)1230 3592 y Fm(2)1230 3652 y Fo(d)1271 3629 y Fq(\()p Fp(t)p Fq(\))p Fn(ij)84 b Fq(=)f Fn(jh)p Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)p Fq(\))2061 3592 y Fm(2)p Fo(\033)2144 3629 y Fp( )2206 3592 y Fm(2)2246 3629 y Fp(;)15 b Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)p Fq(\))2615 3592 y Fy(\000)p Fm(2)p Fo(\033)2753 3629 y Fp(u)2805 3592 y Fm(2)2805 3652 y Fo(d)2846 3629 y Fq(\()p Fp(t)p Fq(\))p Fn(ij)1518 3770 y(\024)83 b(k)p Fq(\(1)21 b(+)f Fn(j)p Fp(x)p Fn(j)p Fq(\))2046 3733 y Fo(\033)2094 3770 y Fp( )s Fn(k)2201 3733 y Fm(2)2201 3793 y Fo(L)2249 3774 y Fk(1)2319 3770 y Fn(k)p Fp(u)2416 3785 y Fo(d)2457 3770 y Fq(\()p Fp(t)p Fq(\))p Fn(k)2605 3733 y Fm(2)2605 3802 y Fo(L)2653 3779 y Fi(2)2653 3823 y Fk(\000)p Fh(\033)1518 4001 y Fn(\024)1736 3939 y(D)1806 3953 y Fm(1)1845 3939 y Fp(")1887 3906 y Fm(2)p 1682 3980 300 4 v 1682 4063 a Fq(\(1)h(+)e Fp(t)p Fq(\))1941 4037 y Fm(3)2011 4001 y Fq(+)2230 3939 y Fn(D)2300 3953 y Fm(1)2339 3939 y Fn(C)2392 3906 y Fm(10)2467 3939 y Fp(")2509 3906 y Fm(6)p 2112 3980 555 4 v 2112 4067 a Fq(\(1)i(+)f(4\000)p Fp(")2448 4040 y Fm(4)2488 4067 y Fp(t)p Fq(\))2556 4040 y Fm(3)p Fo(=)p Fm(2)2676 4001 y Fp(;)149 4230 y Fq(where,)31 b(for)f(the)g(last)h(inequalit)m(y)-8 b(,)29 b(w)m(e)i(ha)m(v)m(e)g(used)f(the)g(estimate)i(\(4.28\),)g(as)f (w)m(ell)e(as)i(the)g(inequalit)m(y:)1313 4409 y Fn(8)p Fp(a;)15 b(b)25 b(>)g Fq(0)h(:)116 b(\()p Fp(a)21 b Fq(+)f Fp(b)p Fq(\))2093 4372 y Fm(2)2158 4409 y Fn(\024)25 b Fq(2\()p Fp(a)2382 4372 y Fm(2)2442 4409 y Fq(+)20 b Fp(b)2572 4372 y Fm(2)2612 4409 y Fq(\))p Fp(:)977 b Fq(\(5.4\))149 4589 y(Similarly)-8 b(,)816 4769 y Fn(jh)p Fp( )s(;)15 b Fn(j)p Fp(u)1055 4784 y Fo(d)1097 4769 y Fn(j)1122 4732 y Fm(2)1162 4769 y Fp(u)1214 4784 y Fo(d)1254 4769 y Fq(\()p Fp(t)p Fq(\))p Fn(ij)84 b Fq(=)f Fn(jh)p Fq(\(1)22 b(+)e Fn(j)p Fp(x)p Fn(j)p Fq(\))2045 4732 y Fo(\033)2092 4769 y Fp( )s(;)15 b Fq(\(1)22 b(+)e Fn(j)p Fp(x)p Fn(j)p Fq(\))2524 4732 y Fy(\000)p Fo(\033)2627 4769 y Fp(u)2679 4784 y Fo(d)2719 4769 y Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(u)2899 4784 y Fo(d)2940 4769 y Fn(j)2965 4732 y Fm(2)3005 4769 y Fq(\()p Fp(t)p Fq(\))p Fn(ij)1501 4910 y(\024)83 b(k)p Fq(\(1)22 b(+)d Fn(j)p Fp(x)p Fn(j)p Fq(\))2029 4873 y Fo(\033)2077 4910 y Fp( )s Fn(k)2184 4924 y Fo(L)2232 4905 y Fk(1)2302 4910 y Fn(k)p Fp(u)2399 4925 y Fo(d)2440 4910 y Fq(\()p Fp(t)p Fq(\))p Fn(k)2588 4934 y Fo(L)2636 4911 y Fi(2)2636 4955 y Fk(\000)p Fh(\033)2731 4910 y Fn(k)p Fp(u)2828 4925 y Fo(d)2869 4910 y Fq(\()p Fp(t)p Fq(\))p Fn(k)3017 4873 y Fm(2)3017 4938 y Fo(L)3065 4919 y Fi(4)1501 5132 y Fn(\024)1772 5071 y(D)1842 5085 y Fm(2)1882 5071 y Fp(")1924 5038 y Fm(2)p 1665 5111 406 4 v 1665 5199 a Fq(\(1)i(+)f Fp(t)p Fq(\))1925 5172 y Fm(12)p Fo(=)p Fm(5)2101 5132 y Fq(+)2319 5071 y Fn(D)2389 5085 y Fm(2)2429 5071 y Fn(C)2482 5038 y Fm(12)2556 5071 y Fp(")2598 5038 y Fm(7)p 2201 5111 555 4 v 2201 5199 a Fq(\(1)h(+)f(4\000)p Fp(")2537 5172 y Fm(4)2578 5199 y Fp(t)p Fq(\))2646 5172 y Fm(7)p Fo(=)p Fm(4)2766 5132 y Fp(;)1947 5425 y Fq(16)p eop %%Page: 17 17 17 16 bop 149 75 a Fq(where,)31 b(for)f(the)g(last)h(inequalit)m(y)-8 b(,)29 b(w)m(e)i(ha)m(v)m(e)g(used)f(the)g(estimates)h (\(4.28\){\(4.29\),)36 b(as)31 b(w)m(ell)e(as:)643 320 y Fn(8)p Fp(a;)15 b(b;)g(c;)g(d)26 b(>)f Fq(0)h(:)116 b(\()p Fp(a)21 b Fq(+)f Fp(b)p Fq(\)\()p Fp(c)h Fq(+)f Fp(d)p Fq(\))1858 283 y Fm(2)1923 320 y Fn(\024)25 b Fq(2)2064 283 y Fm(4)p Fo(=)p Fm(3)2185 259 y Fp(a)2233 226 y Fm(7)p Fo(=)p Fm(3)2363 259 y Fq(+)20 b Fp(b)2493 226 y Fm(7)p Fo(=)p Fm(3)p 2185 299 419 4 v 2326 382 a Fq(7)p Fp(=)p Fq(3)2633 320 y(+)g(2)2769 283 y Fm(5)p Fo(=)p Fm(2)2889 259 y Fp(c)2928 226 y Fm(7)p Fo(=)p Fm(2)3059 259 y Fq(+)g Fp(d)3197 226 y Fm(7)p Fo(=)p Fm(2)p 2889 299 418 4 v 3030 382 a Fq(7)p Fp(=)p Fq(2)3317 320 y Fp(:)307 b Fq(\(5.5\))149 552 y(The)24 b(inequalit)m(y)e(\(5.5\)) k(can)e(b)s(e)f(obtained)g(from)h(Y)-8 b(oung)24 b(and)f(H\177)-45 b(older)24 b(inequalities.)36 b(T)-8 b(aking)23 b(in)m(to)h(accoun)m(t) 149 668 y(the)31 b(deca)m(y)h(of)e Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)i Fq(in)d(time,)h(w)m(e)h(obtain)f(the)g(estimate)h (for)f Fp(t)c Fn(2)e Fq([0)p Fp(;)15 b(T)e Fq(]:)1264 906 y Fn(j)p Fp(g)s Fq(\()p Fp(t)p Fq(\))p Fn(j)27 b(\024)1703 845 y(D)1773 859 y Fm(3)1812 845 y Fp(")1854 812 y Fm(2)p 1596 885 406 4 v 1596 973 a Fq(\(1)21 b(+)e Fp(t)p Fq(\))1855 946 y Fm(12)p Fo(=)p Fm(5)2031 906 y Fq(+)2249 845 y Fn(D)2319 859 y Fm(3)2359 845 y Fn(C)2412 812 y Fm(12)2487 845 y Fp(")2529 812 y Fm(7)p 2132 885 555 4 v 2132 973 a Fq(\(1)i(+)f(4\000)p Fp(")2468 946 y Fm(4)2508 973 y Fp(t)p Fq(\))2576 946 y Fm(7)p Fo(=)p Fm(4)2696 906 y Fp(:)928 b Fq(\(5.6\))149 1142 y(Let)33 b Fp(u)366 1157 y Fo(d)406 1142 y Fq(\()p Fp(t)p Fq(\))28 b(=)f Fp(f)10 b Fq(\()p Fp(t)p Fq(\))21 b(+)f(\()p Fp(u)992 1157 y Fo(d)1033 1142 y Fq(\()p Fp(t)p Fq(\))i Fn(\000)e Fp(f)10 b Fq(\()p Fp(t)p Fq(\)\),)33 b(where)e Fp(f)10 b Fq(\()p Fp(t)p Fq(\))31 b(is)f(the)i(forcing)f(term)h(from)f(the)h(b) s(ound)d(state,)34 b(giv)m(en)149 1258 y(in)c(\(4.21\).)42 b(W)-8 b(e)32 b(rewrite)e(the)g(amplitude)e(equation)j(\(5.3\))h(in)d (the)i(form:)1057 1427 y Fp(d)p Fn(j)p Fp(A)p Fn(j)p 1057 1468 167 4 v 1099 1551 a Fp(dt)1258 1489 y Fq(=)25 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(A)p Fn(j)p Fq(\()p Fp(t)p Fq(\))27 b(Im)1881 1388 y Fe(h)p 1924 1416 69 4 v 101 x Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fp(e)2137 1451 y Fo(i\025t)2232 1489 y Fn(h)p Fp( )2329 1451 y Fm(3)2370 1489 y Fp(;)15 b(f)10 b Fq(\()p Fp(t)p Fq(\))p Fn(i)2603 1388 y Fe(i)2666 1489 y Fq(+)20 b Fp(h)p Fq(\()p Fp(t)p Fq(\))p Fp(;)712 b Fq(\(5.7\))149 1698 y(where,)36 b(due)f(to)g(the)g(estimates)h(\(4.27\))h(and)e(\(5.6\),)j(the)d (correction)g(term)g Fp(h)p Fq(\()p Fp(t)p Fq(\))h(satis\014es)f(the)g (estimate)149 1814 y(for)c Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(]:)297 2052 y Fn(j)p Fp(h)p Fq(\()p Fp(t)p Fq(\))p Fn(j)27 b(\024)955 1991 y(D)1025 2005 y Fm(4)1064 1991 y Fn(C)1117 1958 y Fm(2)1157 1991 y Fp(")1199 1958 y Fm(3)p 635 2031 925 4 v 635 2119 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))895 2092 y Fm(3)p Fo(=)p Fm(2)1005 2119 y Fq(\(1)h(+)f(4\000)p Fp(")1341 2092 y Fm(4)1381 2119 y Fp(t)p Fq(\))1449 2092 y Fm(1)p Fo(=)p Fm(2)1589 2052 y Fq(+)1825 1991 y Fn(D)1895 2005 y Fm(4)1935 1991 y Fn(C)1988 1958 y Fm(9)2027 1991 y Fp(")2069 1958 y Fm(6)p 1690 2031 555 4 v 1690 2119 a Fq(\(1)h(+)f(4\000)p Fp(")2026 2092 y Fm(4)2066 2119 y Fp(t)p Fq(\))2134 2092 y Fm(5)p Fo(=)p Fm(4)2274 2052 y Fq(+)2482 1991 y Fn(D)2552 2005 y Fm(4)2592 1991 y Fp(")2634 1958 y Fm(2)p 2375 2031 406 4 v 2375 2119 a Fq(\(1)h(+)f Fp(t)p Fq(\))2635 2092 y Fm(12)p Fo(=)p Fm(5)2811 2052 y Fq(+)3029 1991 y Fn(D)3099 2005 y Fm(4)3139 1991 y Fn(C)3192 1958 y Fm(12)3267 1991 y Fp(")3309 1958 y Fm(7)p 2912 2031 555 4 v 2912 2119 a Fq(\(1)h(+)f(4\000)p Fp(")3248 2092 y Fm(4)3288 2119 y Fp(t)p Fq(\))3356 2092 y Fm(7)p Fo(=)p Fm(4)3476 2052 y Fp(:)148 b Fq(\(5.8\))149 2288 y(The)36 b(e\013ect)h(of)f Fp(f)10 b Fq(\()p Fp(t)p Fq(\))35 b(in)g(the)g(amplitude)f(equation)i(\(5.7\))h(b)s(ecomes)f (eviden)m(t)g(if)e(w)m(e)i(in)m(tegrate)h(b)m(y)f(parts)149 2404 y(once.)46 b(F)-8 b(or)33 b(rigorous)e(computations,)h(w)m(e)h (use)e(w)m(eigh)m(ted)h Fp(L)2241 2371 y Fm(2)2313 2404 y Fq(spaces)g(with)e(w)m(eigh)m(ts)j Fp(w)3192 2418 y Fy(\006)3278 2404 y Fq(=)28 b(\(1)22 b(+)f Fn(j)p Fp(x)p Fn(j)p Fq(\))3708 2371 y Fy(\006)p Fo(\033)3810 2404 y Fq(.)149 2520 y(Using)30 b(notations:)1234 2712 y Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))26 b(=)f Fp(\015)1558 2726 y Fm(0)1618 2712 y Fq(+)20 b Fp(\015)1756 2726 y Fm(1)1811 2712 y Fq(cos\()p Fp(!)s(t)p Fq(\))26 b(=)2288 2598 y Fm(1)2240 2625 y Fe(X)2217 2821 y Fo(j)t Fm(=)p Fy(\000)p Fm(1)2409 2712 y Fp(\015)2461 2674 y Fo(j)2498 2712 y Fp(e)2540 2674 y Fy(\000)p Fo(i!)2663 2684 y Fh(j)2696 2674 y Fo(t)2725 2712 y Fp(;)149 2970 y Fq(where)32 b Fp(\015)466 2937 y Fm(0)533 2970 y Fq(=)27 b Fp(\015)678 2984 y Fm(0)718 2970 y Fq(,)32 b Fp(\015)827 2937 y Fy(\006)p Fm(1)949 2970 y Fq(=)c Fp(\015)1095 2984 y Fm(1)1134 2970 y Fp(=)p Fq(2,)34 b Fp(!)1340 2984 y Fm(0)1406 2970 y Fq(=)28 b(0,)k(and)g Fp(!)1843 2984 y Fy(\006)p Fm(1)1964 2970 y Fq(=)c Fn(\006j)p Fp(!)s Fn(j)p Fq(,)k(w)m(e)g(compute)h(the)f(inner) e(pro)s(duct)g(in)h(\(5.7\))149 3086 y(as)g(follo)m(ws:)503 3303 y Fn(h)p Fp( )600 3266 y Fm(3)640 3303 y Fp(;)15 b(f)10 b Fq(\()p Fp(t)p Fq(\))p Fn(i)26 b Fq(=)f Fn(\000)p Fp(i)p Fn(h)p Fp( )1194 3266 y Fm(3)1234 3303 y Fp(;)1274 3179 y Fe(Z)1365 3206 y Fo(t)1325 3386 y Fm(0)1410 3303 y Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(A)p Fn(j)1693 3266 y Fm(2)1734 3303 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(e)1957 3266 y Fy(\000)p Fo(i\025s)2115 3303 y Fp(e)2157 3266 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2468 3303 y Fn(P)2531 3317 y Fo(c)2566 3303 y Fp( )2628 3266 y Fm(3)2668 3303 y Fp(ds)p Fn(i)349 3584 y Fq(=)83 b Fn(\000)p Fp(i)691 3471 y Fm(1)643 3498 y Fe(X)620 3693 y Fo(j)t Fm(=)p Fy(\000)p Fm(1)813 3584 y Fp(\015)865 3547 y Fo(j)901 3584 y Fn(h)p Fp(w)1001 3598 y Fm(+)1061 3584 y Fp( )1123 3547 y Fm(3)1163 3584 y Fp(;)1203 3461 y Fe(Z)1294 3487 y Fo(t)1253 3667 y Fm(0)1339 3584 y Fn(j)p Fp(A)p Fn(j)1457 3547 y Fm(2)1497 3584 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fp(w)1743 3598 y Fy(\000)1802 3584 y Fp(e)1844 3547 y Fy(\000)p Fo(i)p Fm(\()p Fo(\025)p Fm(+)p Fo(!)2090 3557 y Fh(j)2124 3547 y Fm(\))p Fo(s)2188 3584 y Fp(e)2230 3547 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2541 3584 y Fn(P)2604 3598 y Fo(c)2639 3584 y Fp( )2701 3547 y Fm(3)2741 3584 y Fp(ds)p Fn(i)349 3914 y Fq(=)g Fn(\000j)p Fp(A)p Fn(j)692 3877 y Fm(2)732 3914 y Fp(A)p Fq(\()p Fp(s)p Fq(\))999 3800 y Fm(1)951 3828 y Fe(X)928 4023 y Fo(j)t Fm(=)p Fy(\000)p Fm(1)1121 3914 y Fp(\015)1173 3877 y Fo(j)1209 3914 y Fn(h)p Fp(w)1309 3928 y Fm(+)1369 3914 y Fp( )1431 3877 y Fm(3)1471 3914 y Fp(;)15 b(w)1576 3928 y Fy(\000)1635 3914 y Fp(e)1677 3877 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1846 3914 y Fp(e)1888 3877 y Fo(i)p Fm(\()p Fy(H\000)p Fo(\025)p Fy(\000)p Fo(!)2194 3887 y Fh(j)2228 3877 y Fm(\))p Fo(s)2292 3914 y Fq(\()p Fn(H)21 b(\000)f Fp(\025)g Fn(\000)g Fp(!)2737 3928 y Fo(j)2793 3914 y Fn(\000)g Fp(i)p Fq(0\))2995 3877 y Fy(\000)p Fm(1)3091 3914 y Fn(P)3154 3928 y Fo(c)3189 3914 y Fp( )3251 3877 y Fm(3)3291 3914 y Fn(i)3341 3837 y Fe(\014)3341 3891 y(\014)3372 3877 y Fo(s)p Fm(=)p Fo(t)3372 3937 y(s)p Fm(=0)349 4244 y Fq(+)574 4130 y Fm(1)526 4158 y Fe(X)503 4353 y Fo(j)t Fm(=)p Fy(\000)p Fm(1)695 4244 y Fp(\015)747 4206 y Fo(j)799 4120 y Fe(Z)890 4146 y Fo(t)850 4326 y Fm(0)935 4244 y Fp(@)983 4258 y Fo(s)1020 4244 y Fq(\()p Fn(j)p Fp(A)p Fn(j)1173 4206 y Fm(2)1213 4244 y Fp(A)p Fq(\()p Fp(s)p Fq(\)\))p Fp(e)1471 4206 y Fy(\000)p Fo(i)p Fm(\()p Fo(\025)p Fm(+)p Fo(!)1717 4216 y Fh(j)1751 4206 y Fm(\))p Fo(s)1816 4244 y Fn(h)p Fp(w)1916 4258 y Fm(+)1975 4244 y Fp( )2037 4206 y Fm(3)2077 4244 y Fp(;)15 b(w)2182 4258 y Fy(\000)2242 4244 y Fp(e)2284 4206 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2595 4244 y Fq(\()p Fn(H)21 b(\000)f Fp(\025)g Fn(\000)g Fp(!)3040 4258 y Fo(j)3097 4244 y Fn(\000)f Fp(i)p Fq(0\))3298 4206 y Fy(\000)p Fm(1)3394 4244 y Fn(P)3457 4258 y Fo(c)3492 4244 y Fp( )3554 4206 y Fm(3)3594 4244 y Fn(i)p Fp(ds)349 4574 y Fq(=)83 b Fn(\000j)p Fp(A)p Fn(j)692 4536 y Fm(2)732 4574 y Fp(A)p Fq(\()p Fp(t)p Fq(\))990 4460 y Fm(1)942 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Fy(\000)1633 5090 y Fp(e)1675 5053 y Fy(\000)p Fo(i)p Fy(H)p Fo(t)1844 5090 y Fq(\()p Fn(H)21 b(\000)f Fp(\025)g Fn(\000)g Fp(!)2289 5104 y Fo(j)2345 5090 y Fn(\000)g Fp(i)p Fq(0\))2547 5053 y Fy(\000)p Fm(1)2643 5090 y Fn(P)2706 5104 y Fo(c)2741 5090 y Fp( )2803 5053 y Fm(3)2843 5090 y Fn(i)1947 5425 y Fq(17)p eop %%Page: 18 18 18 17 bop 414 149 a Fq(+)639 35 y Fm(1)591 63 y Fe(X)568 258 y Fo(j)t Fm(=)p Fy(\000)p Fm(1)761 149 y Fp(\015)813 111 y Fo(j)864 25 y Fe(Z)955 51 y Fo(t)915 231 y Fm(0)1000 149 y Fp(@)1048 163 y Fo(s)1085 149 y Fq(\()p Fn(j)p Fp(A)p Fn(j)1238 111 y Fm(2)1279 149 y Fp(A)p Fq(\()p Fp(s)p Fq(\)\))p Fp(e)1537 111 y Fy(\000)p Fo(i)p Fm(\()p Fo(\025)p Fm(+)p Fo(!)1783 121 y Fh(j)1817 111 y Fm(\))p Fo(s)1881 149 y Fn(h)p Fp(w)1981 163 y Fm(+)2040 149 y Fp( )2102 111 y Fm(3)2142 149 y Fp(;)15 b(w)2247 163 y Fy(\000)2307 149 y Fp(e)2349 111 y Fy(\000)p Fo(i)p Fy(H)p Fm(\()p Fo(t)p Fy(\000)p Fo(s)p Fm(\))2660 149 y Fq(\()p Fn(H)21 b(\000)f Fp(\025)h Fn(\000)f Fp(!)3106 163 y Fo(j)3162 149 y Fn(\000)g Fp(i)p Fq(0\))3364 111 y Fy(\000)p Fm(1)3459 149 y Fn(P)3522 163 y Fo(c)3557 149 y Fp( )3619 111 y Fm(3)3659 149 y Fn(i)p Fp(ds:)149 445 y Fq(In)30 b(order)g(to)h(estimate)977 421 y(~)976 445 y Fp(h)q Fq(\()p Fp(t)p Fq(\),)g(w)m(e)g(note)g(that:)1342 634 y Fn(j)p Fp(@)1415 648 y Fo(s)1452 634 y Fq(\()p Fn(j)p Fp(A)p Fn(j)1605 596 y Fm(2)1646 634 y Fp(A)p Fq(\()p Fp(s)p Fq(\)\))p Fn(j)26 b(\024)f Fq(3)p Fn(j)p Fp(A)p Fn(j)2172 596 y Fm(2)2213 634 y Fq(\()p Fp(s)p Fq(\))p Fn(j)p Fp(@)2399 648 y Fo(s)2436 634 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)149 823 y Fq(and,)30 b(b)m(y)h(virtue)e(of) i(the)f(amplitude)f(equation)h(\(3.2\),)759 1011 y Fn(j)p Fp(@)832 1025 y Fo(s)869 1011 y Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)c Fq(=)f Fn(j)p Fp(\015)5 b Fq(\()p Fp(s)p Fq(\))p Fn(h)p Fp( )s(;)15 b Fn(P)1587 1025 y Fo(c)1624 1011 y Fn(j)p Fp(u)p Fn(j)1726 974 y Fm(2)1766 1011 y Fp(u)p Fq(\()p Fp(s)p Fq(\))p Fn(ij)26 b(\024)f Fq(sup)14 b Fp(\015)5 b Fn(k)p Fp(w)2427 1025 y Fm(+)2487 1011 y Fp( )s Fn(k)2594 1025 y Fo(L)2642 1006 y Fk(1)2712 1011 y Fn(k)p Fp(u)p Fn(k)2854 1035 y Fo(L)2902 1012 y Fi(2)2902 1056 y Fk(\000)p Fh(\033)2997 1011 y Fn(k)p Fp(u)p Fn(k)3139 974 y Fm(2)3139 1039 y Fo(L)3187 1020 y Fi(4)3603 1011 y Fq(\(5.10\))149 1212 y(Using)31 b(the)g(decomp)s (osition)f(\(3.1\))j(and)e(the)g(estimates)h(\(3.14\))i(and)c (\(4.28\){\(4.29\))36 b(for)31 b Fp(A)p Fq(\()p Fp(s)p Fq(\))h(and)f Fp(u)3682 1227 y Fo(d)3722 1212 y Fq(\()p Fp(s)p Fq(\))149 1328 y(on)g Fp(s)25 b Fn(2)f Fq([0)p Fp(;)15 b(T)e Fq(],)32 b(w)m(e)f(obtain:)1359 1496 y Fn(j)p Fp(@)1432 1510 y Fo(s)1470 1496 y Fq(\()p Fn(j)p Fp(A)p Fn(j)1623 1459 y Fm(2)1663 1496 y Fp(A)p Fq(\()p Fp(s)p Fq(\)\))p Fn(j)26 b(\024)2171 1435 y(D)2241 1449 y Fm(5)2281 1435 y Fn(C)2334 1402 y Fm(5)2374 1435 y Fp(")2416 1402 y Fm(5)p 2036 1475 555 4 v 2036 1562 a Fq(\(1)21 b(+)f(4\000)p Fp(")2372 1536 y Fm(4)2412 1562 y Fp(t)p Fq(\))2480 1536 y Fm(5)p Fo(=)p Fm(4)2601 1496 y Fp(:)977 b Fq(\(5.11\))149 1706 y(Using)30 b(the)h(estimates)g (\(3.9\))h(and)e(\(5.11\),)i(w)m(e)f(obtain)f(for)g Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(]:)1278 1947 y Fn(j)1304 1923 y Fq(~)1303 1947 y Fp(h)q Fq(\()p Fp(t)p Fq(\))p Fn(j)26 b(\024)1705 1885 y(D)1775 1899 y Fm(6)1815 1885 y Fp(")1857 1852 y Fm(3)p 1616 1926 371 4 v 1616 2013 a Fq(\(1)21 b(+)f Fp(t)p Fq(\))1876 1987 y Fm(3)p Fo(=)p Fm(2)2016 1947 y Fq(+)2252 1885 y Fn(D)2322 1899 y Fm(6)2362 1885 y Fn(C)2415 1852 y Fm(5)2454 1885 y Fp(")2496 1852 y Fm(5)p 2117 1926 555 4 v 2117 2013 a Fq(\(1)h(+)f(4\000)p Fp(")2453 1987 y Fm(4)2493 2013 y Fp(t)p Fq(\))2561 1987 y Fm(5)p Fo(=)p Fm(4)2681 1947 y Fp(:)897 b Fq(\(5.12\))149 2185 y(After)31 b(straigh)m(tforw)m(ard)f(calculations,)g(the)g (amplitude)f(equation)h(\(5.7\))i(simpli\014es)27 b(to)k(the)g(form:) 1355 2356 y Fp(d)p Fn(j)p Fp(A)p Fn(j)p 1355 2397 167 4 v 1398 2480 a Fp(dt)1556 2418 y Fq(=)25 b(\()p Fn(\000)p Fq(\000)20 b(+)g Fp(\032)p Fq(\()p Fp(t)p Fq(\)\))p Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p 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Fc(\000)c Fv(i)p Fw(0\))2143 3030 y Fu(\000)p Ft(1)2231 3065 y Fc(P)2289 3077 y Fs(c)2323 3065 y Fv( )2380 3030 y Ft(3)2417 3065 y Fc(i)h(\000)f(h)p Fv( )2640 3030 y Ft(3)2677 3065 y Fv(;)c Fw(\()p Fc(H)20 b(\000)e Fv(\025)h Fw(+)f Fv(!)s Fw(\))3156 3030 y Fu(\000)p Ft(1)3245 3065 y Fc(P)3303 3077 y Fs(c)3336 3065 y Fv( )3393 3030 y Ft(3)3431 3065 y Fc(i)3463 2997 y Fb(\003)674 3247 y Fc(\000)82 b Fw(cos\()p Fv(!)s(t)p Fw(\))1092 3191 y Fv(\015)1135 3203 y Ft(0)1172 3191 y Fv(\015)1215 3203 y Ft(1)p 1092 3228 161 4 v 1151 3304 a Fw(2)1262 3247 y Fc(h)p Fv( )1351 3213 y Ft(3)1389 3247 y Fv(;)14 b Fw(Im\()p Fc(H)20 b(\000)e Fv(\025)g Fc(\000)g Fv(!)k Fc(\000)c Fv(i)p Fw(0\))2139 3213 y Fu(\000)p Ft(1)2227 3247 y Fc(P)2285 3259 y Fs(c)2319 3247 y Fv( )2376 3213 y Ft(3)2413 3247 y Fc(i)674 3429 y Fw(+)82 b(sin\()p Fv(!)s(t)p Fw(\))1082 3373 y Fv(\015)1125 3385 y Ft(0)1163 3373 y Fv(\015)1206 3385 y Ft(1)p 1083 3410 V 1142 3486 a Fw(2)1267 3362 y Fb(\002)1301 3429 y Fc(h)p Fv( )1390 3395 y Ft(3)1428 3429 y Fv(;)14 b Fw(Re\()p Fc(H)20 b(\000)e Fv(\025)g Fc(\000)g Fv(!)k Fc(\000)c Fv(i)p Fw(0\))2177 3395 y Fu(\000)p Ft(1)2265 3429 y Fc(P)2323 3441 y Fs(c)2357 3429 y Fv( )2414 3395 y Ft(3)2451 3429 y Fc(i)h(\000)f(h)p Fv( )2674 3395 y Ft(3)2711 3429 y Fv(;)c Fw(\()p Fc(H)20 b(\000)e Fv(\025)h Fw(+)f Fv(!)s Fw(\))3190 3395 y Fu(\000)p Ft(1)3279 3429 y Fc(P)3337 3441 y Fs(c)3370 3429 y Fv( )3427 3395 y Ft(3)3465 3429 y Fc(i)3497 3362 y Fb(\003)3623 3429 y Fw(\(5.14\))180 3648 y Fq(and)30 b(the)g(function)870 3624 y(^)869 3648 y Fp(h)q Fq(\()p Fp(t)p Fq(\))g(is)g(giv)m(en)g(b)m (y:)1162 3832 y(^)1161 3856 y Fp(h)q Fq(\()p Fp(t)p Fq(\))c(=)f Fp(h)p Fq(\()p Fp(t)p Fq(\))c(+)f Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(A)p Fn(j)p Fq(\()p Fp(t)p Fq(\))27 b(Im)2233 3756 y Fe(h)p 2275 3783 69 4 v 2275 3856 a Fp(A)q Fq(\()p Fp(t)p Fq(\))p Fp(e)2489 3819 y Fo(i\025t)2585 3832 y Fq(~)2584 3856 y Fp(h)q Fq(\()p Fp(t)p Fq(\))2740 3756 y Fe(i)2798 3856 y Fp(:)780 b Fq(\(5.15\))149 4083 y(Due)31 b(to)g(the)g(estimates)g(\(5.8\))h(and)e(\(5.12\),)j(the)d (function)2203 4059 y(^)2202 4083 y Fp(h)q Fq(\()p Fp(t)p Fq(\))g(is)g(b)s(ounded)e(for)i Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(])32 b(b)m(y:)809 4324 y Fn(j)835 4300 y Fq(^)834 4324 y Fp(h)q Fq(\()p Fp(t)p Fq(\))p Fn(j)26 b(\024)f Fp(R)q Fq(\()p Fp(t)p Fq(\))83 b(=)1877 4263 y Fn(D)1947 4277 y Fm(7)1986 4263 y Fn(C)2039 4230 y Fm(2)2079 4263 y Fp(")2121 4230 y Fm(3)p 1557 4303 925 4 v 1557 4390 a Fq(\(1)21 b(+)e Fp(t)p Fq(\))1816 4364 y Fm(3)p Fo(=)p Fm(2)1927 4390 y Fq(\(1)i(+)f(4\000)p Fp(")2263 4364 y Fm(4)2303 4390 y Fp(t)p Fq(\))2371 4364 y Fm(1)p Fo(=)p Fm(2)2511 4324 y Fq(+)2747 4263 y Fn(D)2817 4277 y Fm(7)2857 4263 y Fn(C)2910 4230 y Fm(9)2949 4263 y Fp(")2991 4230 y Fm(6)p 2612 4303 555 4 v 2612 4390 a Fq(\(1)h(+)f(4\000)p Fp(")2948 4364 y Fm(4)2988 4390 y Fp(t)p Fq(\))3056 4364 y Fm(5)p Fo(=)p Fm(4)1393 4583 y Fq(+)1664 4522 y Fn(D)1734 4536 y Fm(7)1773 4522 y Fp(")1815 4489 y Fm(2)p 1557 4563 406 4 v 1557 4650 a Fq(\(1)h(+)e Fp(t)p Fq(\))1816 4623 y Fm(12)p Fo(=)p Fm(5)1992 4583 y Fq(+)2210 4522 y Fn(D)2280 4536 y Fm(7)2320 4522 y Fn(C)2373 4489 y Fm(12)2448 4522 y Fp(")2490 4489 y Fm(7)p 2093 4563 555 4 v 2093 4650 a Fq(\(1)i(+)f(4\000)p Fp(")2429 4623 y Fm(4)2469 4650 y Fp(t)p Fq(\))2537 4623 y Fm(7)p Fo(=)p Fm(4)2657 4583 y Fp(:)921 b Fq(\(5.16\))149 4821 y(W)-8 b(e)32 b(note)f(that)g(an)f(equiv)-5 b(alen)m(t)30 b(w)m(a)m(y)h(of)g(writing)d(\000)20 b Fn(\000)g Fp(\032)p Fq(\()p Fp(t)p Fq(\))31 b(is)847 5090 y(\000)20 b Fn(\000)g Fp(\032)p Fq(\()p Fp(t)p Fq(\))26 b(=)1358 4977 y Fm(1)1310 5004 y Fe(X)1287 5199 y Fo(j)t Fm(=)p Fy(\000)p Fm(1)1479 5090 y Fn(j)p Fp(\015)1551 5104 y Fo(j)1588 5090 y Fn(j)1613 5053 y Fm(2)1653 5090 y Fn(h)p Fp(w)1753 5104 y Fm(+)1812 5090 y Fp( )1874 5053 y Fm(3)1914 5090 y Fp(;)15 b(w)2019 5104 y Fy(\000)2079 5090 y Fq(\()p Fn(H)21 b(\000)f Fp(\025)g Fn(\000)g Fp(!)2524 5104 y Fo(j)2580 5090 y Fn(\000)g Fp(i)p Fq(0\))2782 5053 y Fy(\000)p Fm(1)2878 5090 y Fn(P)2941 5104 y Fo(c)2976 5090 y Fp( )3038 5053 y Fm(3)3077 5090 y Fn(i)p Fp(;)1947 5425 y Fq(18)p eop %%Page: 19 19 19 18 bop 149 75 a Fq(suc)m(h)30 b(that)1328 191 y Fn(j)p Fq(\000)20 b Fn(\000)g Fp(\032)p Fq(\()p Fp(t)p Fq(\))p Fn(j)26 b(\024)f Fp(D)1893 205 y Fm(4)1948 191 y Fq(sup)1956 270 y Fo(t)p Fy(2)p Fd(R)2100 191 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)2305 153 y Fm(2)2345 191 y Fn(k)p Fp( )2452 153 y Fm(3)2493 191 y Fn(k)2538 211 y Fo(L)2586 192 y Fi(2)2586 227 y Fh(\033)2632 191 y Fp(;)946 b Fq(\(5.17\))149 405 y(where)38 b Fp(D)495 419 y Fm(4)572 405 y Fq(is)f(the)h(constan)m(t)h(in)d(the)i(estimate)g (\(3.9\).)65 b(In)37 b(the)h(rest)g(of)g(this)e(section,)k(w)m(e)e (analyze)g(the)149 521 y(deca)m(y)g(estimate)f(on)f(the)g(amplitude)e Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)j Fq(from)f(the)g(amplitude)e (equation)i(\(5.13\).)60 b(By)36 b(taking)g(the)149 638 y(upp)s(er)29 b(b)s(ound)f(in)h(\(5.13\),)k(w)m(e)e(consider)e(the)i 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b(treat)k(the)g(correction)f(term)h Fp(R)q Fq(\()p Fp(t)p Fq(\))f(p)s(erturbativ)m(ely)e(and)i(linearize)e (around)i(the)g(solution)149 2528 y(of)31 b(the)g(dominan)m(t)e (equation:)1250 2693 y Fp(dB)1366 2707 y Fm(0)p 1250 2733 156 4 v 1288 2817 a Fp(dt)1441 2754 y Fq(=)c(\()p Fn(\000)p Fq(\000)20 b(+)g Fp(\032)p Fq(\()p Fp(t)p Fq(\)\))p Fp(B)2070 2717 y Fm(5)2065 2777 y(0)2110 2754 y Fp(;)197 b(B)2401 2768 y Fm(0)2440 2754 y Fq(\(0\))27 b(=)e Fp(":)858 b Fq(\(5.21\))149 2963 y(Using)30 b(separation)g(of)h(v)-5 b(ariables,)29 b(w)m(e)i(ha)m(v)m(e)g(the)g(solution:)1276 3165 y Fp(B)1345 3179 y Fm(0)1384 3165 y Fq(\()p Fp(t)p Fq(\))26 b(=)1914 3104 y Fp(")p 1619 3144 634 4 v 1619 3231 a Fq(\(1)21 b(+)f(4)p Fp(")1898 3205 y Fm(4)1938 3231 y Fq(\012\()p Fp(t)p Fq(\)\))2142 3205 y Fm(1)p Fo(=)p Fm(4)2262 3165 y Fp(;)198 b(t)25 b Fn(\025)g Fq(0)p Fp(;)894 b Fq(\(5.22\))149 3401 y(where)1515 3547 y(\012\()p Fp(t)p Fq(\))25 b(=)1805 3424 y Fe(Z)1896 3450 y Fo(t)1856 3630 y Fm(0)1941 3547 y Fq(\(\000)20 b Fn(\000)g Fp(\032)p Fq(\()p Fp(s)p Fq(\)\))c Fp(ds:)1133 b Fq(\(5.23\))149 3752 y(Ev)m(en)34 b(though)f(\012\()p Fp(t)p Fq(\))h(ma)m(y)g(tak)m(e)h (negativ)m(e)g(v)-5 b(alues,)34 b Fp(B)2026 3766 y Fm(0)2065 3752 y Fq(\()p Fp(t)p Fq(\))g(is)f(de\014ned)f(for)i(all)e Fp(t)e Fn(\025)h Fq(0)p Fp(;)j Fq(if)e Fp(")i Fq(is)f(su\016cien)m(tly) 149 3868 y(small.)47 b(Indeed,)32 b(since)h Fp(\032)p Fq(\()p Fp(t)p Fq(\))g(is)f(con)m(tin)m(uous)g(and)g(p)s(erio)s(dic)e (on)j Fl(R)41 b Fq(with)32 b(mean)g(zero)i(w)m(e)f(ha)m(v)m(e)h(that)f (\012\()p Fp(t)p Fq(\))149 3985 y(is)d(con)m(tin)m(uous)g(on)g Fl(R)39 b Fq(and:)1655 4123 y(lim)1635 4179 y Fo(t)p Fy(!1)1826 4061 y Fq(\012\()p Fp(t)p Fq(\))p 1826 4102 170 4 v 1895 4185 a Fp(t)2031 4123 y Fq(=)25 b(\000)g Fp(>)g Fq(0)1253 b(\(5.24\))149 4310 y(Therefore,)29 b(there)g(exists)g Fp(N)35 b(>)25 b Fq(0)k(suc)m(h)f(that)i(\012\()p Fp(t)p Fq(\))25 b Fn(\025)g Fq(0)k(for)g Fp(t)c(>)g(N)10 b Fq(.)40 b(But)29 b(\012\()p Fp(t)p 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Fn(\000)p Fq(1)p Fp(=)p Fq(\012)2155 5190 y Fo(min)2289 5176 y Fq(\))2324 5138 y Fm(1)p Fo(=)p Fm(4)2434 5176 y Fp(;)1947 5425 y Fq(19)p eop %%Page: 20 20 20 19 bop 149 75 a Fq(and)33 b Fp(B)398 89 y Fm(0)437 75 y Fq(\()p Fp(t)p Fq(\))h(is)e(w)m(ell)g(de\014ned)g(for)h(all)f Fp(t)e Fn(\025)g Fq(0.)49 b(W)-8 b(e)35 b(note)e(that)h(\012\()p Fp(t)p Fq(\))f(can)h(b)s(e)e(estimated)i(on)f Fp(t)d Fn(2)f Fq([0)p Fp(;)15 b(N)10 b Fq(])35 b(b)m(y)149 191 y(using)c(\(5.17\))j(and)d(\(5.23\))j(in)d(terms)h(of)g(sup)1700 213 y Fo(t)p Fy(2)p Fd(R)1839 191 y Fn(j)p Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)33 b Fq(and)f(the)g(disp)s(ersiv)m(e)d (constan)m(t)k Fp(D)3276 205 y Fm(4)3348 191 y Fq(in)d(the)i(deca)m(y) 149 307 y(estimate)f(\(3.9\).)43 b(Hence)31 b(the)f(ab)s(o)m(v)m(e)h (restriction)f(on)g Fp(")2053 321 y Fm(0)2122 307 y Fq(can)h(b)s(e)e (accommo)s(dated)j(in)d(the)h(b)s(ound)e(\(5.19\).)149 423 y(Pro)s(ceeding)i(in)f(a)i(similar)d(manner,)h(w)m(e)i(sho)m(w)g (that)1539 664 y Fp(B)1608 678 y Fm(0)1647 664 y Fq(\()p Fp(t)p Fq(\))26 b Fn(\024)2083 603 y(D)2153 617 y Fm(1)2193 603 y Fp(")p 1882 643 555 4 v 1882 731 a Fq(\(1)21 b(+)f(4\000)p Fp(")2218 704 y Fm(4)2258 731 y Fp(t)p Fq(\))2326 704 y Fm(1)p Fo(=)p Fm(4)3603 664 y Fq(\(5.26\))149 916 y(where)30 b Fn(D)482 930 y Fm(1)552 916 y Fq(satis\014es)g(the)h(b)s(ounds:)1324 1190 y(sup)1272 1267 y Fm(0)p Fy(\024)p Fo(t<)p Fy(1)1528 1062 y Fe(\022)1645 1129 y Fq(1)20 b(+)g(4\000)p Fp(")1945 1096 y Fm(4)1986 1129 y Fp(t)p 1605 1169 454 4 v 1605 1252 a Fq(1)h(+)f(4)p Fp(")1849 1226 y Fm(4)1889 1252 y Fq(\012\()p Fp(t)p Fq(\))2068 1062 y Fe(\023)2135 1079 y Fm(1)p Fo(=)p Fm(4)2270 1190 y Fn(\024)25 b(D)2436 1204 y Fm(1)2501 1190 y Fp(<)g Fn(1)p Fp(:)890 b Fq(\(5.27\))149 1450 y(More)32 b(precisely)-8 b(,)29 b(w)m(e)i(in)m(tro)s(duce)e(the)i (auxiliary)d(function)1322 1705 y Fp(M)10 b Fq(\()p Fp(t)p Fq(\))26 b(=)1694 1643 y(1)21 b(+)f(4\000)p Fp(")1995 1610 y Fm(4)2035 1643 y Fp(t)p 1655 1684 V 1655 1767 a Fq(1)g(+)g(4)p Fp(")1898 1741 y Fm(4)1939 1767 y Fq(\012\()p Fp(t)p Fq(\))2143 1705 y(:)25 b([0)p Fp(;)15 b Fn(1)p Fq(\))27 b Fn(!)e Fl(R)s Fp(;)946 b Fq(\(5.28\))149 1958 y(suc)m(h)30 b(that)h Fp(M)10 b Fq(\(0\))27 b(=)e(1)31 b(and)1179 2226 y(lim)1158 2282 y Fo(t)p Fy(!1)1340 2226 y Fp(M)10 b Fq(\()p Fp(t)p Fq(\))26 b(=)45 b(lim)1663 2282 y Fo(t)p Fy(!1)1912 2123 y Fm(1)p 1912 2138 36 4 v 1917 2190 a Fo(t)1977 2159 y Fq(+)20 b(4\000)p Fp(")2212 2126 y Fm(4)p 1855 2205 445 4 v 1865 2274 a(1)p 1865 2289 36 4 v 1870 2342 a Fo(t)1930 2310 y Fq(+)g(4)p Fp(")2108 2284 y Fm(4)2158 2265 y(\012\()p Fo(t)p Fm(\))p 2158 2289 132 4 v 2211 2342 a Fo(t)2335 2226 y Fq(=)2441 2164 y(4\000)p Fp(")2585 2131 y Fm(4)p 2441 2205 185 4 v 2441 2288 a Fq(4\000)p Fp(")2585 2262 y Fm(4)2660 2226 y Fq(=)25 b(1)p Fp(:)777 b Fq(\(5.29\))149 2509 y(Hence)38 b(there)e(exist)g Fp(N)45 b(>)35 b Fq(0)i(suc)m(h)f(that)h Fp(M)10 b Fq(\()p Fp(t)p Fq(\))36 b Fn(\024)f Fq(3)p Fp(=)p Fq(2)p Fp(;)i Fq(for)g Fp(t)e(>)f(N)5 b(:)37 b 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y(\000)p Fn(D)2334 3516 y Fy(\000)p Fm(1)2331 3580 y(4)2428 3426 y Fe(\023)2510 3554 y Fp(:)1068 b Fq(\(5.37\))149 3809 y(The)30 b(ball)f(of)i(radius)d Fn(R)p Fp(")1009 3776 y Fm(2)1079 3809 y Fq(is)i(in)m(v)-5 b(arian)m(t)29 b(under)g Fp(K)7 b Fq([)p Fp(B)1988 3823 y Fm(1)2027 3809 y Fq(],)31 b(if)f(the)g(inhomogeneous)g(term)g(in)f Fp(K)37 b Fq(satis\014es:)1103 3931 y Fe(\015)1103 3985 y(\015)1103 4040 y(\015)1103 4094 y(\015)1168 3939 y(Z)1259 3965 y Fo(t)1219 4145 y Fm(0)1304 4063 y Fp(U)10 b Fq(\()p Fp(t;)15 b(s)p Fq(\))p Fp(R)q Fq(\()p Fp(s)p Fq(\))p Fp(ds)1835 3931 y Fe(\015)1835 3985 y(\015)1835 4040 y(\015)1835 4094 y(\015)1886 4153 y Fo(Z)1968 4063 y Fn(\024)25 b Fq(\(1)c Fn(\000)f Fp(L)p Fq(\))p Fn(R)p Fp(")2472 4025 y Fm(2)2537 4063 y Fn(\024)2643 4001 y Fq(1)p 2643 4042 V 2643 4125 a(2)2698 4063 y Fn(R)p Fp(")2817 4025 y Fm(2)2857 4063 y Fp(;)721 b Fq(\(5.38\))149 4314 y(whic)m(h)30 b(is)f(pro)m(v)m(ed)i(from)f(the)g(estimates)h(\(5.16\))i (and)d(\(5.35\).)43 b(As)30 b(a)h(result,)e(w)m(e)i(obtain:)1357 4436 y Fe(\015)1357 4491 y(\015)1357 4545 y(\015)1357 4600 y(\015)1423 4444 y(Z)1514 4470 y Fo(t)1473 4650 y Fm(0)1559 4568 y Fp(U)10 b Fq(\()p Fp(t;)15 b(s)p Fq(\))p Fp(R)q Fq(\()p Fp(s)p Fq(\))p Fp(ds)2090 4436 y Fe(\015)2090 4491 y(\015)2090 4545 y(\015)2090 4600 y(\015)2141 4658 y Fo(Z)2223 4568 y Fn(\024)25 b(D)2389 4582 y Fm(5)2428 4568 y Fn(C)2481 4530 y Fm(9)2521 4568 y Fp(")2563 4530 y Fm(2)2603 4568 y Fp(:)975 b Fq(\(5.39\))149 4827 y(If)34 b(w)m(e)g(c)m(ho)s(ose)g(that)g Fn(R)d Fq(=)g(2)p Fn(D)1197 4841 y Fm(5)1237 4827 y Fn(C)1290 4794 y Fm(9)1363 4827 y Fq(and)i Fn(D)g(\025)d Fq(max)q Fn(f)p Fq(2)p Fn(D)2077 4841 y Fm(5)2117 4827 y Fp(;)15 b Fq(4\000)2259 4794 y Fy(\000)p Fm(1)2354 4827 y Fn(D)2424 4841 y Fm(4)2464 4827 y Fn(D)2534 4841 y Fm(5)2573 4827 y Fn(g)p Fq(,)35 b(the)f(b)s(ounds)d(\(5.37\))36 b(and)d(\(5.38\))149 4943 y(are)j(satis\014ed)d(pro)m(vided)h(that)h(the)g(b)s(ound)e (\(5.19\))k(holds.)52 b(By)35 b(the)g(con)m(traction)h(principle,)c (applied)h(to)149 5059 y(the)k(amplitude)e(equation)i(\(5.31\),)k(the)c (solution)e Fp(B)2004 5073 y Fm(1)2043 5059 y Fq(\()p Fp(t)p Fq(\))j(exists)e(and)g(is)g(unique)f(in)g(the)i(ball)e(of)i (radius)149 5176 y Fn(R)p Fp(")268 5143 y Fm(2)345 5176 y Fn(\024)g(D)s(C)579 5143 y Fm(9)618 5176 y Fp(")660 5143 y Fm(2)700 5176 y Fq(.)62 b(As)37 b(a)h(result,)g(the)g(initial-v) -5 b(alue)34 b(problem)i(\(5.30\))k(is)c(w)m(ell)h(p)s(osed)f(on)h Fp(t)g Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(])38 b(and)1947 5425 y(21)p eop %%Page: 22 22 22 21 bop 149 75 a Fq(the)44 b(solution)e Fp(B)745 89 y Fm(1)784 75 y Fq(\()p Fp(t)p Fq(\))i(b)s(elongs)e(to)i(the)f(ball)f (of)h(radius)e(smaller)h(than)h Fn(D)s(C)2833 42 y Fm(9)2872 75 y Fp(")2914 42 y Fm(2)2997 75 y Fq(in)f(the)i(Banac)m(h)g(space)149 191 y Fp(Z)32 b Fq(=)25 b Fp(C)7 b Fq(\([0)p Fp(;)15 b(T)e Fq(])p Fp(;)i Fl(R)t Fq(\))37 b(with)29 b(the)i(norm)e(\(5.33\),) k(suc)m(h)d(that)1084 451 y Fn(j)p Fp(B)1178 465 y Fm(1)1218 451 y Fq(\()p Fp(t)p Fq(\))p Fn(j)c(\024)1631 389 y(D)s(C)1757 356 y Fm(9)1797 389 y Fp(")1839 356 y Fm(2)p 1478 430 555 4 v 1478 517 a Fq(\(1)21 b(+)f(4\000)p Fp(")1814 491 y Fm(4)1854 517 y Fp(t)p Fq(\))1922 490 y Fm(1)p Fo(=)p Fm(4)2042 451 y Fp(;)197 b Fq(for)30 b(all)g Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(])p Fp(:)703 b Fq(\(5.40\))149 707 y(Com)m(bining)29 b(the)h(b)s(ounds)e(\(5.26\))33 b(and)d(\(5.40\),)j(w)m(e)d(deriv)m(e)g(the)h(estimate)g(\(5.20\).)p 3770 707 4 62 v 3774 649 55 4 v 3774 707 V 3828 707 4 62 v 149 898 a Fg(Lemma)i(5.2.)43 b Ff(L)-5 b(et)33 b(the)g(b)-5 b(ound)33 b(\(5.19\))i(b)-5 b(e)32 b(satis\014e)-5 b(d.)44 b(Then,)33 b(the)g(amplitude)i Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)e Ff(is)g(b)-5 b(ounde)g(d)35 b(by)d(the)149 1015 y(solution)i Fp(B)5 b Fq(\()p Fp(t)p Fq(\))33 b Ff(as)g(fol)5 b(lows:)1302 1222 y Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)27 b(\024)e Fp(B)5 b Fq(\()p Fp(t)p Fq(\))p Fp(;)201 b Fq(for)32 b(all)25 b Fp(t)g Fn(2)f Fq([0)p Fp(;)15 b(T)e Fq(])p Fp(:)149 1430 y Ff(Pr)-5 b(o)g(of.)47 b Fq(By)32 b(Lemma)f(5.1,)i(there)f(exists)f(a)g(unique)f (solution)f Fp(B)5 b Fq(\()p Fp(t)p Fq(\))32 b(of)f(the)h(comparison)e (equation)h(\(5.18\))149 1546 y(on)g Fp(t)25 b Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(].)41 b(By)31 b(relations)f(\(5.13\))i(and)e (\(5.16\),)j(w)m(e)e(ha)m(v)m(e)1352 1736 y Fp(d)p Fn(j)p Fp(A)p Fn(j)p 1352 1777 167 4 v 1395 1860 a Fp(dt)1553 1798 y Fn(\024)1659 1736 y Fp(dB)p 1659 1777 121 4 v 1679 1860 a(dt)1790 1798 y(;)197 b Fq(for)30 b(all)24 b Fp(t)h Fn(2)g Fq([0)p Fp(;)15 b(T)e Fq(])p Fp(;)961 b Fq(\(5.41\))149 2023 y(suc)m(h)30 b(that)1625 2139 y Fn(j)p Fp(A)p Fq(\(0\))p Fn(j)d(\024)e Fp(")h Fq(=)f Fp(B)5 b Fq(\(0\))p Fp(:)1244 b Fq(\(5.42\))149 2309 y(Then,)30 b(the)h(lemma)f(follo)m(ws)f(from)h(the)h(comparison)e (principle)e(for)j(ODE's,)h(see)g([16)q(,)g(pp.)39 b(168].)p 3770 2309 4 62 v 3774 2251 55 4 v 3774 2309 V 3828 2309 4 62 v 149 2500 a Fg(Lemma)33 b(5.3.)42 b Ff(The)33 b(set)g Fp(\034)42 b Ff(is)33 b(op)-5 b(en)34 b(in)e Fq([0)p Fp(;)15 b(T)1720 2514 y Fm(max)1857 2500 y Fq(\))p Fp(:)149 2691 y Ff(Pr)-5 b(o)g(of.)47 b Fq(Com)m(bining)28 b(Lemmas)j(5.1)g(and) f(5.2,)i(w)m(e)f(ha)m(v)m(e:)1212 2946 y Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)26 b(\024)1604 2884 y(D)s Fp(")p Fq(\(1)21 b(+)f Fn(C)1964 2851 y Fm(9)2003 2884 y Fp(")p Fq(\))p 1565 2925 555 4 v 1565 3012 a(\(1)h(+)f(4\000)p Fp(")1901 2986 y Fm(4)1941 3012 y Fp(t)p Fq(\))2009 2986 y Fm(1)p Fo(=)p Fm(4)2129 2946 y Fp(;)198 b Fn(8)p Fp(t)24 b Fn(2)h Fq([0)p Fp(;)15 b(T)e Fq(])p Fp(;)831 b Fq(\(5.43\))149 3210 y(pro)m(vided)35 b(that)i(the)g(b)s(ound)d(\(5.19\))k(holds.)57 b(Due)37 b(to)g(inequalit)m(y)e(\(3.15\),)40 b(w)m(e)d(ha)m(v)m(e:)53 b Fn(D)s Fq(\(1)25 b(+)f Fn(C)3499 3177 y Fm(9)3539 3210 y Fp(")p Fq(\))35 b Fp(<)g Fn(C)5 b Fq(,)149 3327 y(suc)m(h)30 b(that)1212 3458 y Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)c Fp(<)1794 3397 y Fn(C)5 b Fp(")p 1565 3438 V 1565 3525 a Fq(\(1)21 b(+)f(4\000)p Fp(")1901 3498 y Fm(4)1941 3525 y Fp(t)p Fq(\))2009 3498 y Fm(1)p Fo(=)p Fm(4)2129 3458 y Fp(;)198 b Fn(8)p Fp(t)24 b Fn(2)h Fq([0)p Fp(;)15 b(T)e Fq(])p Fp(:)831 b Fq(\(5.44\))149 3678 y(Since)43 b Fp(A)p Fq(\()p Fp(t)p Fq(\))i(is)e(con)m(tin)m(uous,)48 b(there)c(exists)f Fp(\016)52 b(>)c Fq(0)c(suc)m(h)g(that)h(the)f(ab)s (o)m(v)m(e)h(inequalit)m(y)d(holds)h(for)h(all)149 3794 y Fp(t)32 b Fn(2)f Fq([0)p Fp(;)15 b(T)36 b Fq(+)23 b Fp(\016)s Fq(\).)52 b(Consequen)m(tly)-8 b(,)35 b(\()p Fp(T)h Fn(\000)23 b Fp(\016)n(;)15 b(T)36 b Fq(+)22 b Fp(\016)s Fq(\))33 b Fn(\032)e Fp(\034)10 b Fq(,)35 b(where)f(the)g (set)h Fp(\034)44 b Fq(is)33 b(de\014ned)g(in)g(\(3.14\).)54 b(Since)149 3910 y Fp(T)38 b Fn(2)25 b Fp(\034)41 b Fq(is)29 b(arbitrary)-8 b(,)30 b(w)m(e)h(infer)e(that)i Fp(\034)40 b Fq(is)29 b(op)s(en)h(in)f([0)p Fp(;)15 b(T)2104 3924 y Fm(max)2241 3910 y Fq(\).)p 3770 3910 4 62 v 3774 3852 55 4 v 3774 3910 V 3828 3910 4 62 v 149 4197 a Fr(6)135 b Fa(H)457 4153 y Fw(1)548 4197 y Fr(estimates)149 4403 y Fq(W)-8 b(e)32 b(pro)m(v)m(e)f(here)f(the)h(upp)s(er)d(b)s(ound)h (\(3.17\),)k(whic)m(h)c(excludes)g(the)i(\014nite)e(time)h(blo)m(w-up)f (at)i Fp(T)38 b(<)25 b Fn(1)p Fq(.)149 4594 y Fg(Lemma)33 b(6.1.)42 b Ff(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fn(E)f Fp(>)25 b Fq(0)33 b Ff(such)g(that)1150 4801 y Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)1395 4821 y Fo(H)1458 4802 y Fi(1)1524 4801 y Fn(\024)25 b(E)e Fq(log)r(\(1)e(+)f(4\000)p Fp(")2145 4764 y Fm(4)2185 4801 y Fp(t)p Fq(\))p Fp(;)201 b Fq(for)32 b Fp(t)26 b Fn(2)e Fp(\034)5 b(:)815 b Fq(\(6.1\))1947 5425 y(22)p eop %%Page: 23 23 23 22 bop 149 75 a Ff(Pr)-5 b(o)g(of.)47 b Fq(Assume)40 b(that)h Fp(u)1040 89 y Fm(0)1122 75 y Fn(2)h Fp(H)1308 42 y Fm(2)1347 75 y Fq(\()p Fl(R)1442 42 y Fm(3)1488 75 y Fq(\).)71 b(By)41 b(Theorem)f(2,)k(the)c(inner)f(pro)s(duct)g(of)i Fp(u)3206 89 y Fo(t)3276 75 y Fq(with)e(the)i(NLS)149 191 y(equation)31 b(\(1.1\))h(is)d(de\014ned,)h(suc)m(h)g(that)1244 384 y(Re)1366 311 y Fe(\002)1404 384 y Fn(h)p Fp(u)1491 398 y Fo(t)1521 384 y Fp(;)15 b Fq(\()p Fn(\000)p Fq(\001)21 b(+)f Fp(V)g Fq(\))p Fp(u)g Fq(+)g Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(j)p Fp(u)p Fn(j)2383 347 y Fm(2)2424 384 y Fp(u)p Fn(i)2511 311 y Fe(\003)2575 384 y Fq(=)24 b(0)p Fp(:)909 b Fq(\(6.2\))149 578 y(Since)36 b Fp(u)p Fq(\()p Fp(t)p Fq(\))f Fn(2)f Fp(H)761 545 y Fm(2)801 578 y Fq(\()p Fl(R)896 545 y Fm(3)941 578 y Fq(\),)k Fp(u)1091 592 y Fo(t)1121 578 y Fq(\()p Fp(t)p Fq(\))d Fn(2)g Fp(L)1417 545 y Fm(2)1456 578 y Fq(\()p Fl(R)1551 545 y Fm(3)1596 578 y Fq(\),)k(and)c Fn(j)p Fp(u)p Fn(j)1979 545 y Fm(2)2019 578 y Fp(u)p Fq(\()p Fp(t)p Fq(\))g Fn(2)g Fp(H)2388 545 y Fm(2)2427 578 y Fq(\()p Fl(R)2522 545 y Fm(3)2568 578 y Fq(\))h(b)m(y)g(Theorem)g(2,)i(an)e(elemen)m(tary)149 694 y(computation)31 b(transforms)e(\(6.2\))j(to)g(the)e(form)g(of)h (the)f(energy)h(conserv)-5 b(ation)30 b(equation:)1023 873 y Fp(d)p 1006 913 81 4 v 1006 997 a(dt)1112 806 y Fe(\024)1160 934 y Fn(h)p Fp(u;)15 b Fq(\()p Fn(\000)p Fq(\001)20 b(+)g Fp(V)h Fq(\))p Fp(u)p Fn(i)g Fq(+)1897 873 y(1)p 1897 913 46 4 v 1897 997 a(2)1953 934 y Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fn(k)p Fp(u)p Fn(k)2250 897 y Fm(4)2250 962 y Fo(L)2298 943 y Fi(4)2338 806 y Fe(\025)2411 934 y Fq(=)25 b Fn(\000k)p Fp(u)p Fn(k)2720 897 y Fm(4)2720 962 y Fo(L)2768 943 y Fi(4)2821 934 y Fq(_)-38 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))p Fp(:)661 b Fq(\(6.3\))149 1174 y(Using)30 b(the)h(decomp)s(osition)e(\(3.1\),)j(w)m(e)f(obtain:)772 1353 y Fp(d)p 756 1394 81 4 v 756 1477 a(dt)861 1286 y Fe(\024)909 1414 y Fp(\025)p Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)1183 1377 y Fm(2)1244 1414 y Fq(+)20 b Fn(h)p Fp(u)1422 1429 y Fo(d)1463 1414 y Fp(;)15 b Fq(\()p Fn(\000)p Fq(\001)20 b(+)g Fp(V)g Fq(\))p Fp(u)1956 1429 y Fo(d)1997 1414 y Fn(i)h Fq(+)e Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))2308 1353 y(1)p 2308 1394 46 4 v 2308 1477 a(2)2364 1414 y Fn(k)p Fp(u)p Fn(k)2506 1377 y Fm(4)2506 1442 y Fo(L)2554 1424 y Fi(4)2594 1286 y Fe(\025)2667 1414 y Fq(=)39 b(_)-39 b Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))2928 1353 y(1)p 2928 1394 V 2928 1477 a(2)2984 1414 y Fn(k)p Fp(u)p Fn(k)3126 1377 y Fm(4)3126 1442 y Fo(L)3174 1424 y Fi(4)3214 1414 y Fp(:)410 b Fq(\(6.4\))149 1649 y(As)31 b(a)g(result,)517 1843 y Fn(kr)p Fp(u)690 1858 y Fo(d)731 1843 y Fq(\()p Fp(t)p Fq(\))p Fn(k)879 1805 y Fm(2)879 1871 y Fo(L)927 1852 y Fi(2)1050 1843 y Fn(\024)83 b(k)p Fp(V)20 b Fn(k)1367 1857 y Fy(1)1442 1843 y Fq(\()p Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)1722 1805 y Fm(2)1722 1871 y Fo(L)1770 1852 y Fi(2)1831 1843 y Fq(+)g Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)2143 1805 y Fm(2)2184 1843 y Fq(\))g(+)g Fn(k)p Fp(u)p Fq(\(0\))p Fn(k)2587 1805 y Fm(2)2587 1871 y Fo(H)2650 1852 y Fi(1)2711 1843 y Fq(+)g Fn(k)p Fp(V)h Fn(k)2966 1857 y Fy(1)3041 1843 y Fn(k)p Fp(u)p Fq(\(0\))p Fn(k)3298 1805 y Fm(2)3298 1871 y Fo(L)3346 1852 y Fi(2)1050 2024 y Fn(\000)83 b Fp(\025)p Fq(\()p Fn(j)p Fp(A)p Fq(\()p Fp(t)p Fq(\))p Fn(j)1513 1987 y Fm(2)1574 2024 y Fq(+)20 b Fn(j)p Fp(A)p Fq(\(0\))p Fn(j)1898 1987 y Fm(2)1939 2024 y Fq(\))h(+)2095 1963 y(1)p 2095 2003 V 2095 2087 a(2)2151 2024 y(\()p Fn(j)p Fp(\015)2258 2038 y Fm(0)2298 2024 y Fn(j)f Fq(+)g Fn(j)p Fp(\015)2506 2038 y Fm(1)2546 2024 y Fn(j)p Fq(\)\()p Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)2886 1987 y Fm(4)2886 2052 y Fo(L)2934 2033 y Fi(4)2996 2024 y Fq(+)f Fn(k)p Fp(u)p Fq(\(0\))p Fn(k)3343 1987 y Fm(4)3343 2052 y Fo(L)3391 2033 y Fi(4)3432 2024 y Fq(\))1050 2258 y(+)83 b(4)p Fp(!)s Fn(j)p Fp(\015)1381 2272 y Fm(1)1421 2258 y Fn(j)1461 2134 y Fe(Z)1552 2161 y Fo(t)1512 2341 y Fm(0)1597 2184 y Fe(\000)1638 2258 y Fn(j)p Fp(A)p Fq(\()p Fp(s)p Fq(\))p Fn(j)1869 2221 y Fm(4)1910 2258 y Fn(k)p Fp( )s Fn(k)2062 2221 y Fm(4)2062 2286 y Fo(L)2110 2267 y Fi(4)2170 2258 y Fq(+)20 b Fn(k)p Fp(u)2358 2273 y Fo(d)2399 2258 y Fq(\()p Fp(s)p Fq(\))p Fn(k)2557 2221 y Fm(4)2557 2286 y Fo(L)2605 2267 y Fi(4)2645 2184 y Fe(\001)2701 2258 y Fp(ds:)833 b Fq(\(6.5\))149 2494 y(Since)30 b Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)632 2508 y Fm(2)698 2494 y Fq(=)25 b Fn(k)p Fp(u)p Fq(\(0\))p Fn(k)1051 2508 y Fm(2)1123 2494 y Fq(b)m(y)30 b(Theorem)g(2,)h(and)e Fp(t)c Fn(2)g Fp(\034)40 b Fq(b)m(y)30 b(Lemma)h(5.3,)g(w)m(e)g(use)f (the)g(estimates)h(\(3.14\))149 2611 y(and)g(\(4.29\))i(and)d(deriv)m (e)h(the)g Fp(H)1276 2578 y Fm(1)1346 2611 y Fq(b)s(ound)e(\(6.1\))k (in)d(the)h(case)h Fp(u)2354 2625 y Fm(0)2419 2611 y Fn(2)26 b Fp(H)2589 2578 y Fm(2)2629 2611 y Fq(\()p Fl(R)2724 2578 y Fm(3)2769 2611 y Fq(\).)43 b(By)31 b(the)g(standard)f(densit)m (y)149 2727 y(argumen)m(t)h(for)f Fp(H)778 2694 y Fm(2)843 2727 y Fp(,)-15 b Fn(!)25 b Fp(H)1052 2694 y Fm(1)1121 2727 y Fq(and)30 b(the)g(con)m(tin)m(uous)g(dep)s(endence)g(of)g Fp(u)2552 2741 y Fm(0)2622 2727 y Fq(in)e(Theorem)i(3,)h(the)g(b)s (ound)d(\(6.1\))149 2843 y(still)h(holds)g(for)h Fp(u)756 2857 y Fm(0)820 2843 y Fn(2)25 b Fp(H)989 2810 y Fm(1)1029 2843 y Fq(\()p Fl(R)1124 2810 y Fm(3)1169 2843 y Fq(\).)p 3770 2843 4 62 v 3774 2785 55 4 v 3774 2843 V 3828 2843 4 62 v 149 3127 a Fr(7)135 b(Generalizations)47 b(and)e(op)t(en)f (problems)149 3333 y Fq(The)21 b(main)g(Theorem)g(1)h(can)f(b)s(e)g (extended)g(with)g(ob)m(vious)f(mo)s(di\014cations)g(to)i(the)g(case)g (of)g(almost)f(p)s(erio)s(dic)149 3450 y(time-dep)s(enden)m(t)30 b(nonlinearit)m(y)e(co)s(e\016cien)m(t:)1085 3704 y Fp(\015)5 b Fq(\()p Fp(t)p Fq(\))25 b(=)1393 3591 y Fy(1)1362 3618 y Fe(X)1361 3812 y Fo(n)p Fm(=1)1510 3704 y Fp(\015)1557 3718 y Fo(n)1619 3704 y Fq(cos\()p Fp(!)1832 3718 y Fo(n)1879 3704 y Fp(t)20 b Fq(+)g Fp(\022)2066 3718 y Fo(n)2113 3704 y Fq(\))p Fp(;)2402 3591 y Fy(1)2371 3618 y Fe(X)2370 3812 y Fo(n)p Fm(=1)2519 3704 y Fn(j)p Fp(\015)2591 3718 y Fo(n)2638 3704 y Fn(j)25 b Fp(<)g Fn(1)p Fp(;)149 3975 y Fq(where)34 b Fp(\015)463 3989 y Fo(n)542 3975 y Fn(2)e Fl(C)18 b Fp(;)55 b(!)832 3989 y Fo(n)879 3975 y Fp(;)15 b(\022)962 3989 y Fo(n)1041 3975 y Fn(2)32 b Fl(R)r Fq(,)42 b Fn(8)p Fp(n)31 b Fn(2)g Fl(N)47 b Fq(and)34 b(there)h(exists)f Fp(\016)i(>)31 b Fq(0)k(suc)m(h)f(that)h Fn(j)p Fp(\025)23 b Fq(+)g Fp(!)3182 3989 y Fo(n)3229 3975 y Fn(j)32 b(\025)f Fp(\016)39 b Fq(whenev)m(er)149 4091 y Fp(\015)196 4105 y Fo(n)269 4091 y Fn(6)p Fq(=)25 b(0.)38 b(In)23 b(particular,)g(the)g (main)f(Theorem)h(extends)g(to)h(the)f(case)i(of)e(general)g(p)s(erio)s (dic)d(p)s(erturbations,)149 4207 y(when)32 b Fp(!)446 4221 y Fo(n)521 4207 y Fq(=)c Fp(n!)s Fq(,)k Fp(n)c Fn(2)g Fl(N)7 b Fq(,)39 b(pro)m(vided)30 b(that)j(the)g(resonance)f(b)s(et)m (w)m(een)h(the)g(b)s(ound)d(state)k(and)d(the)i(lo)m(w)m(est)149 4324 y(radiation)39 b(mo)s(de)g(is)f(excluded,)j(i.e.)68 b Fp(\025)26 b Fq(+)g Fp(n!)43 b Fn(6)p Fq(=)d(0)g(whenev)m(er)g Fp(\015)2482 4338 y Fo(n)2569 4324 y Fn(6)p Fq(=)g(0.)69 b(W)-8 b(e)41 b(need)e(to)h(exclude)f(suc)m(h)149 4440 y(resonances)32 b(b)s(ecause)f(the)g(estimate)h(\(3.10\),)i(as)d(opp)s (osed)f(to)i(the)f(estimate)h(\(3.9\),)h(is)d(not)h(su\016cien)m(t)g (for)149 4556 y(our)26 b(argumen)m(t)h(to)g(w)m(ork.)40 b(Our)25 b(preliminary)e(calculations)i(sho)m(w)h(that,)i(for)f Fp(\025)12 b Fq(+)g Fp(!)28 b(<)d Fq(0)h(and)g Fp(\025)12 b Fq(+)g Fp(n!)28 b(>)d Fq(0,)149 4672 y(the)31 b(deca)m(y)h(of)e(the)h (b)s(ound)d(state)k(o)s(ccurs)e(with)f(the)h(rate:)1577 4866 y Fn(k)p Fp(u)p Fq(\()p Fp(t)p Fq(\))p Fn(k)1822 4890 y Fo(L)1870 4867 y Fi(2)1870 4910 y Fk(\000)p Fh(\033)1991 4866 y Fn(\030)25 b Fp(t)2120 4828 y Fy(\000)p Fm(1)p Fo(=)p Fm(\(4)p Fo(n)p Fm(\))2382 4866 y Fp(:)291 5059 y Fq(There)39 b(are)h(three)g(op)s(en)f(problems)f(b)s(ey)m(ond)i(the)g (scop)s(e)g(of)g(the)g(curren)m(t)f(man)m(uscript.)68 b(The)39 b(\014rst)149 5176 y(one)34 b(is)d(to)j(extend)f(the)g(main)f (theorem)h(to)h(the)f(case)g(of)g(resonance)h(with)d(the)i(lo)m(w)m (est)h(radiation)e(mo)s(de,)1947 5425 y(23)p eop %%Page: 24 24 24 23 bop 149 75 a Fq(as)42 b(w)m(e)h(ha)m(v)m(e)g(men)m(tioned)e(ab)s (o)m(v)m(e.)76 b(The)41 b(second)h(op)s(en)f(problem)g(is)f(to)j (consider)e(the)h(NLS)f(equation)149 191 y(\(1.1\))f(in)35 b(the)j(space)g(of)f(lo)m(w)m(er)h(dimensions.)58 b(Since)37 b(the)g(b)s(ound)e(\(3.7\))40 b(on)d(the)g(disp)s(ersiv)m(e)e(part)i (in)f Fp(L)3796 158 y Fo(p)149 307 y Fq(dep)s(ends)h(on)i(the)g(space)h (dimension,)f(there)g(are)g(obstacles)g(in)f(\014nding)e(in)m(v)-5 b(arian)m(t)39 b(subspaces)f(for)g(the)149 423 y(in)m(tegral)i(op)s (erator)g Fp(K)943 437 y Fo(A)1000 423 y Fq([)p Fp(u)1077 438 y Fo(d)1118 423 y Fq(])f(in)g(the)h(space)g(of)g(one)g(and)f(t)m(w) m(o)h(dimensions.)66 b(The)39 b(obstacles)i(could)d(b)s(e)149 540 y(remo)m(v)m(ed)31 b(b)m(y)f(increasing)f(the)h(p)s(o)m(w)m(er)g (of)g(the)g(nonlinearit)m(y)e(term)i(in)f(the)h(NLS)f(equation)h (\(1.1\))i(to)f(mak)m(e)149 656 y(it)i(sup)s(ercritical)e([24)q(].)50 b(Ho)m(w)m(ev)m(er,)36 b(applied)31 b(problems)h(in)m(v)m(olving)g (disp)s(ersion-managed)e(optical)j(pulses)149 772 y([15)r(])28 b(and)f(nonlinearit)m(y-managed)f(Bose-Einstein)i(condensates)g([13)q (,)g(14)q(])g(require)f(solution)f(of)i(the)g(NLS)149 888 y(equation)k(in)f(one)h(dimension)d(with)i(cubic)g(nonlinearit)m(y) -8 b(.)43 b(The)31 b(third)f(op)s(en)i(problem)e(is)h(to)h(in)m(v)m (estigate)149 1005 y(the)27 b(nonlinear)d(F)-8 b(ermi)26 b(Golden)g(rule)f(in)g(the)i(case)g(of)f(large-norm)g(b)s(ound)f (states.)40 b(W)-8 b(e)28 b(use)e(the)g(smallness)149 1121 y(of)f(the)f(initial)e(data)j(to)g(con)m(trol)g(the)f(radiativ)m (e)g(part)g(of)h(the)f(solution.)38 b(The)23 b(blo)m(w-up)g(e\013ect)j (of)f(attractiv)m(e)149 1237 y(nonlinearit)m(y)e(for)i(large-norm)g (solutions)e(renders)h(the)h(treatmen)m(t)h(of)f(suc)m(h)g(a)g(case)h (m)m(uc)m(h)f(more)g(di\016cult.)149 1523 y Fr(A)134 b(App)t(endix:)112 b(Con)l(traction)73 b(principles)f(and)f(\014xed)g (p)t(oin)l(ts)h(esti-)385 1673 y(mates)47 b(in)d(a)i(hierarc)l(h)l(y)f (of)g(Banac)l(h)g(spaces)149 1879 y Fq(Let)36 b(\()p Fp(Y)5 b(;)15 b Fn(k)24 b(\001)f(k)612 1893 y Fo(Y)673 1879 y Fq(\))35 b(b)s(e)f(a)h(Banac)m(h)h(space)g(and)e Fp(K)39 b Fq(:)33 b Fp(Y)52 b Fn(!)32 b Fp(Y)55 b Fq(a)35 b(lo)s(cally)e(Lipsc)m(hitz)h(op)s(erator,)i(i.e.)54 b(for)34 b(an)m(y)149 1995 y Fp(u;)46 b(v)29 b Fn(2)c Fp(Y)50 b Fq(suc)m(h)30 b(that)h Fn(k)p Fp(u)p Fn(k)1078 2009 y Fo(Y)1140 1995 y Fp(;)45 b Fn(k)p Fp(v)s Fn(k)1347 2009 y Fo(Y)1435 1995 y Fn(\024)25 b Fp(R)31 b Fq(there)f(exist)g Fp(L)2140 2009 y Fo(Y)2201 1995 y Fq(\()p Fp(R)q Fq(\))c Fp(>)f Fq(0)30 b(with)f(the)i(prop)s(ert)m(y)1291 2203 y Fn(k)p Fp(K)7 b Fq([)p Fp(u)p Fq(])21 b Fn(\000)f Fp(K)7 b Fq([)p Fp(v)s Fq(])p Fn(k)1860 2217 y Fo(Y)1947 2203 y Fn(\024)25 b Fp(L)2105 2217 y Fo(Y)2166 2203 y Fq(\()p Fp(R)q Fq(\))p Fn(k)p Fp(u)c Fn(\000)f Fp(v)s Fn(k)2607 2217 y Fo(Y)2668 2203 y Fp(:)933 b Fq(\(A.1\))149 2410 y(Assume)30 b(that)1819 2527 y Fp(K)7 b Fq([0])26 b(=)f(0)1461 b(\(A.2\))149 2697 y(and)30 b(consider)g(an)g(abstract)h(equation:)1330 2904 y Fp(u)25 b Fq(=)g Fp(f)k Fq(+)20 b Fp(K)7 b Fq([)p Fp(u)p Fq(])26 b(=)f Fp(K)2053 2918 y Fm(1)2092 2904 y Fq([)p Fp(u)p Fq(])p Fp(;)198 b(f)35 b Fn(2)25 b Fp(Y)990 b Fq(\(A.3\))149 3112 y(The)30 b(classical)g(con)m(traction)h (principle)c(for)j(a)h(lo)s(cally)e(Lipsc)m(hitz)g(op)s(erator)i(is:) 149 3303 y Fg(Prop)s(osition)37 b(A.1.)42 b Ff(Assume)32 b(that)i(ther)-5 b(e)33 b(exist)g(the)g(c)-5 b(onstants)35 b Fp(L)25 b(<)g Fq(1)33 b Ff(and)g Fp(R)26 b(>)f Fq(0)33 b Ff(such)g(that:)1551 3511 y Fp(L)1613 3525 y Fo(Y)1673 3511 y Fq(\()p Fp(R)q Fq(\))84 b Fn(\024)e Fp(L)1514 b Fq(\(A.4\))1608 3652 y Fn(k)p Fp(f)10 b Fn(k)1753 3666 y Fo(Y)1897 3652 y Fn(\024)82 b Fq(\(1)21 b Fn(\000)f Fp(L)p Fq(\))p Fp(R)q(:)1192 b Fq(\(A.5\))149 3859 y Ff(Then)34 b(e)-5 b(quation)33 b(\(A.3\))h(has)f(an)h(unique)e (solution)i Fp(u)f Ff(in)g(the)g(b)-5 b(al)5 b(l)34 b(of)f(r)-5 b(adius)34 b Fp(R)g Ff(of)f(the)g(Banach)h(sp)-5 b(ac)g(e)35 b Fp(Y)5 b(:)149 3975 y Ff(Mor)-5 b(e)g(over,)34 b(for)f(any)h Fp(v)28 b Fn(2)d Fp(Y)5 b(;)48 b Fn(k)p Fp(v)s Fn(k)1323 3989 y Fo(Y)1410 3975 y Fn(\024)25 b Fp(R)33 b Ff(and)h Fp(n)25 b Fq(=)g(0)p Fp(;)15 b Fq(1)p Fp(;)g(:)g(:)g(:)35 b Ff(we)e(have:)791 4240 y Fn(k)p Fp(u)20 b Fn(\000)g Fp(K)1083 4203 y Fo(n)1076 4263 y Fm(1)1130 4240 y Fq([)p Fp(v)s Fq(])p Fn(k)1272 4254 y Fo(Y)1359 4240 y Fn(\024)1465 4179 y(k)p Fp(K)1594 4141 y Fo(n)p Fm(+1)1587 4205 y(1)1731 4179 y Fq([)p Fp(v)s Fq(])h Fn(\000)f Fp(K)2024 4146 y Fo(n)2017 4203 y Fm(1)2071 4179 y Fq([)p Fp(v)s Fq(])p Fn(k)2213 4193 y Fo(Y)p 1465 4219 810 4 v 1761 4303 a Fq(1)h Fn(\000)e Fp(L)2310 4240 y Fn(\024)2471 4179 y Fp(L)2533 4146 y Fo(n)p 2416 4219 219 4 v 2416 4303 a Fq(1)i Fn(\000)f Fp(L)2645 4240 y Fn(k)p Fp(K)2767 4254 y Fm(1)2807 4240 y Fq([)p Fp(v)s Fq(])h Fn(\000)f Fp(v)s Fn(k)3108 4254 y Fo(Y)3169 4240 y Fp(:)432 b Fq(\(A.6\))149 4473 y Ff(In)33 b(p)-5 b(articular,)35 b(for)e Fp(v)c Fq(=)c Fp(f)41 b Ff(we)33 b(get:)1072 4728 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(K)1365 4690 y Fo(n)1358 4750 y Fm(1)1411 4728 y Fq([)p Fp(f)10 b Fq(])p Fn(k)1561 4742 y Fo(Y)1648 4728 y Fn(\024)1808 4666 y Fp(L)1870 4633 y Fo(n)p 1754 4707 V 1754 4790 a Fq(1)20 b Fn(\000)g Fp(L)1982 4728 y Fn(k)p Fp(K)7 b Fq([)p Fp(f)j Fq(])p Fn(k)2261 4742 y Fo(Y)2347 4728 y Fn(\024)2463 4666 y Fp(L)2525 4633 y Fo(n)p Fm(+1)p 2453 4707 V 2453 4790 a Fq(1)21 b Fn(\000)f Fp(L)2682 4728 y Fn(k)p Fp(f)10 b Fn(k)2827 4742 y Fo(Y)2888 4728 y Fp(:)713 b Fq(\(A.7\))149 4960 y Ff(Pr)-5 b(o)g(of.)47 b Fq(Apply)30 b(the)i(con)m(traction)h(principle)c(to)j Fp(K)1910 4974 y Fm(1)1982 4960 y Fq(in)e(the)i(ball)f(of)h(radius)d(R) j(of)g(the)g(Banac)m(h)h(space)f Fp(Y)20 b Fq(,)149 5077 y(see)31 b(for)g(example)f([28)q(].)p 3770 5077 4 62 v 3774 5019 55 4 v 3774 5077 V 3828 5077 4 62 v 1947 5425 a(24)p eop %%Page: 25 25 25 24 bop 291 75 a Fq(Consider)28 b(another)j(abstract)g(equation:)1098 274 y Fp(u)25 b Fq(=)g Fp(g)f Fq(+)c Fp(f)29 b Fq(+)20 b Fp(K)7 b Fq([)p Fp(u)p Fq(])26 b(=)f Fp(K)1979 288 y Fm(2)2018 274 y Fq([)p Fp(u)p Fq(])p Fp(;)198 b(g)29 b Fn(2)c Fp(X)r(;)46 b(f)34 b Fn(2)25 b Fp(Y)759 b Fq(\(A.8\))149 473 y(where)35 b(\()p Fp(X)r(;)15 b Fn(k)24 b(\001)g(k)732 487 y Fo(X)799 473 y Fq(\))35 b(is)f(a)h(Banac)m(h)h(space)f(con)m(tin) m(uously)f(em)m(b)s(edded)f(in)h Fp(Y)5 b(;)34 b(X)40 b(,)-15 b Fn(!)33 b Fp(Y)20 b Fq(.)53 b(The)34 b(em)m(b)s(edding)149 589 y(implies)28 b(that)i(b)s(oth)g Fp(g)j Fq(and)d Fp(g)23 b Fq(+)c Fp(f)40 b Fq(are)30 b(in)f Fp(Y)5 b(:)30 b Fq(Assuming)e(that) j(\(A.5\))g(holds)e(with)g Fp(f)39 b Fq(replaced)30 b(b)m(y)g Fp(f)e Fq(+)20 b Fp(g)s(;)149 705 y Fq(the)31 b(previous)e(prop)s (osition)f(applies)g(and)i(the)h(solution)e Fp(u)h Fq(of)h(\(A.8\))g (satis\014es:)1277 956 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(K)1570 919 y Fo(n)1563 979 y Fm(2)1616 956 y Fq([)p Fp(f)30 b Fq(+)20 b Fp(g)s Fq(])p Fn(k)1923 970 y Fo(Y)2010 956 y Fn(\024)2126 895 y Fp(L)2188 862 y Fo(n)p Fm(+1)p 2116 936 219 4 v 2116 1019 a Fq(1)h Fn(\000)f Fp(L)2344 956 y Fn(k)p Fp(f)30 b Fq(+)20 b Fp(g)s Fn(k)2646 970 y Fo(Y)149 1181 y Fq(for)32 b Fp(n)27 b Fq(=)g(0)p Fp(;)15 b Fq(1)p Fp(;)g(:)g(:)g(:)s Fq(.)45 b(Ho)m(w)m(ev)m(er)34 b(the)e(ab)s(o)m(v)m (e)h(estimate)f(do)s(es)g(not)g(use)f(information)f(that)i(the)g (forcing)f(term)149 1297 y Fp(g)42 b Fq(b)s(elongs)36 b(to)j(a)f(\\b)s(etter")h(space.)63 b(Nev)m(ertheless,)41 b(assume)c(that)i(w)m(e)f(can)g(\014nd)e(a)i(solution)f Fp(v)k Fn(2)c Fp(X)45 b Fn(\032)149 1413 y Fp(Y)5 b(;)46 b Fn(k)p Fp(v)s Fn(k)415 1427 y Fo(Y)502 1413 y Fn(\024)25 b Fp(R)31 b Fq(of)g(the)f(equation:)1739 1529 y Fp(v)e Fq(=)d Fp(g)f Fq(+)c Fp(K)7 b Fq([)p Fp(v)s Fq(])1380 b(\(A.9\))149 1695 y(Then)1082 1811 y Fn(k)p Fp(f)30 b Fq(+)20 b Fp(v)s Fn(k)1385 1825 y Fo(Y)1472 1811 y Fq(=)25 b Fn(k)p Fp(f)30 b Fq(+)20 b Fp(g)k Fq(+)c Fp(K)7 b Fq([)p Fp(v)s Fq(])p Fn(k)2163 1825 y Fo(Y)2250 1811 y Fq(=)25 b Fn(k)p Fp(K)2468 1825 y Fm(2)2508 1811 y Fq([)p Fp(v)s Fq(])p Fn(k)2650 1825 y Fo(Y)2737 1811 y Fn(\024)g Fp(R)149 1977 y Fq(since)32 b Fp(v)j Fq(is)c(in)g(the)h (ball)f(of)h(radius)e Fp(R)j Fq(in)d Fp(Y)52 b Fq(whic)m(h)31 b(is)g(left)h(in)m(v)-5 b(arian)m(t)31 b(b)m(y)h Fp(K)2789 1991 y Fm(2)2829 1977 y Fp(:)g Fq(Hence)h(w)m(e)g(can)f(use)g Fp(f)e Fq(+)21 b Fp(v)149 2094 y Fq(in)30 b(\(A.6\))h(and)f(get:)973 2331 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(K)1266 2294 y Fo(n)1259 2354 y Fm(2)1312 2331 y Fq([)p Fp(f)30 b Fq(+)20 b Fp(v)s Fq(])p Fn(k)1620 2345 y Fo(Y)1765 2331 y Fn(\024)1983 2270 y Fp(L)2045 2237 y Fo(n)p 1928 2310 V 1928 2393 a Fq(1)h Fn(\000)f Fp(L)2157 2331 y Fn(k)p Fp(K)2279 2345 y Fm(2)2319 2331 y Fq([)p Fp(f)30 b Fq(+)20 b Fp(v)s Fq(])g Fn(\000)g Fp(f)30 b Fn(\000)20 b Fp(v)s Fn(k)2951 2345 y Fo(Y)1765 2558 y Fq(=)1983 2496 y Fp(L)2045 2463 y Fo(n)p 1928 2537 V 1928 2620 a Fq(1)h Fn(\000)f Fp(L)2157 2558 y Fn(k)p Fp(K)7 b Fq([)p Fp(f)30 b Fq(+)20 b Fp(v)s Fq(])g(+)g Fp(g)k Fn(\000)c Fp(v)s Fn(k)2910 2572 y Fo(Y)1765 2785 y Fq(=)1983 2723 y Fp(L)2045 2690 y Fo(n)p 1928 2764 V 1928 2847 a Fq(1)h Fn(\000)f Fp(L)2157 2785 y Fn(k)p Fp(K)7 b Fq([)p Fp(f)30 b Fq(+)20 b Fp(v)s Fq(])g Fn(\000)g Fp(K)7 b Fq([)p Fp(v)s Fq(])p Fn(k)2886 2799 y Fo(Y)1765 3025 y Fn(\024)1938 2964 y Fp(L)2000 2931 y Fo(n)p Fm(+1)p 1928 3004 V 1928 3087 a Fq(1)21 b Fn(\000)f Fp(L)2157 3025 y Fn(k)p Fp(f)10 b Fn(k)2302 3039 y Fo(Y)3581 3025 y Fq(\(A.10\))149 3249 y(for)31 b Fp(n)24 b Fq(=)h(0)p Fp(;)15 b Fq(1)p Fp(;)g(:)g(:)g(:)291 3366 y Fq(No)m(w)37 b(the)f(existence)h(of)g Fp(v)j Fq(with)35 b(the)i(ab)s(o)m(v)m(e)g (prop)s(erties)e(could)h(b)s(e)f(inferred)g(from)h(assuming)f(that)149 3482 y(the)f(\014xed)e(p)s(oin)m(t)g(problem)g(\(A.9\))j(satis\014es)d (the)i(h)m(yp)s(otheses)e(of)i(Prop)s(osition)d(A.1)j(but)e(in)g(the)h (Banac)m(h)149 3598 y(space)e Fp(X)7 b Fq(.)41 b(W)-8 b(e)32 b(ha)m(v)m(e)g(th)m(us)e(pro)m(v)m(ed)g(the)h(follo)m(wing)e (result:)149 3782 y Fg(Prop)s(osition)38 b(A.2.)k Ff(In)33 b(addition)j(to)e(\(A.1\))f(and)h(\(A.2\),)g(assume)g(that)h Fp(K)40 b Ff(is)33 b(lo)-5 b(c)g(al)5 b(ly)35 b(Lipschitz)g(in)e Fp(X)r(;)149 3898 y Ff(i.e.)42 b(for)33 b(any)g Fp(u;)48 b(v)29 b Fn(2)c Fp(X)40 b Ff(such)32 b(that)i Fn(k)p Fp(u)p Fn(k)1569 3912 y Fo(X)1637 3898 y Fp(;)48 b Fn(k)p Fp(v)s Fn(k)1847 3912 y Fo(X)1941 3898 y Fn(\024)25 b Fp(Q)33 b Ff(ther)-5 b(e)33 b(exist)g Fp(L)2642 3912 y Fo(X)2709 3898 y Fq(\()p Fp(Q)p Fq(\))26 b Fp(>)f Fq(0)33 b Ff(with)g(the)g(pr)-5 b(op)g(erty)1284 4097 y Fn(k)p Fp(K)7 b Fq([)p Fp(u)p Fq(])21 b Fn(\000)f Fp(K)7 b Fq([)p Fp(v)s Fq(])p Fn(k)1853 4111 y Fo(X)1946 4097 y Fn(\024)25 b Fp(L)2104 4111 y Fo(X)2171 4097 y Fq(\()p Fp(Q)p Fq(\))p Fn(k)p Fp(u)d Fn(\000)e Fp(v)s Fn(k)2615 4111 y Fo(Y)2676 4097 y Fp(:)880 b Fq(\(A.11\))149 4296 y Ff(F)-7 b(uthermor)i(e,)33 b(assume)d(that)h(ther)-5 b(e)31 b(exist)f(c)-5 b(onstants)31 b Fp(L)2051 4310 y Fo(i)2105 4296 y Fp(<)25 b Fq(1)p Fp(;)45 b(i)26 b Fq(=)f(1)p Fp(;)15 b Fq(2)31 b Ff(and)f Fp(R)2872 4310 y Fo(i)2926 4296 y Fp(>)25 b Fq(0)p Fp(;)45 b(i)26 b Fq(=)f(1)p Fp(;)15 b Fq(2)31 b Ff(such)e(that:)1489 4495 y Fp(L)1551 4509 y Fo(X)1618 4495 y Fq(\()p Fp(R)1722 4509 y Fm(1)1762 4495 y Fq(\))83 b Fn(\024)g Fp(L)2096 4509 y Fm(1)3581 4495 y Fq(\(A.12\))1592 4636 y Fn(k)p Fp(g)s Fn(k)1728 4650 y Fo(X)1880 4636 y Fn(\024)g Fq(\(1)21 b Fn(\000)f Fp(L)2288 4650 y Fm(1)2327 4636 y Fq(\))p Fp(R)2431 4650 y Fm(1)2471 4636 y Fp(;)1085 b Fq(\(A.13\))149 4835 y Ff(and)1523 5034 y Fp(L)1585 5048 y Fo(Y)1646 5034 y Fq(\()p Fp(R)1750 5048 y Fm(2)1789 5034 y Fq(\))84 b Fn(\024)f Fp(L)2124 5048 y Fm(2)3581 5034 y Fq(\(A.14\))1461 5176 y Fn(k)p Fp(g)24 b Fq(+)c Fp(f)10 b Fn(k)1764 5190 y Fo(Y)1908 5176 y Fn(\024)83 b Fq(\(1)21 b Fn(\000)f Fp(L)2316 5190 y Fm(2)2355 5176 y Fq(\))p Fp(R)2459 5190 y Fm(2)2499 5176 y Fp(:)1057 b Fq(\(A.15\))1947 5425 y(25)p eop %%Page: 26 26 26 25 bop 149 75 a Ff(and)34 b(also)g(assume)f(that)785 279 y Fp(B)854 293 y Fm(1)919 279 y Fq(=)25 b Fn(f)p Fp(v)k Fn(2)c Fp(X)65 b Fq(:)58 b Fn(k)p Fp(v)s Fn(k)1579 293 y Fo(X)1673 279 y Fn(\024)25 b Fp(R)1838 293 y Fm(1)1877 279 y Fn(g)h(\032)f Fp(B)2113 293 y Fm(2)2177 279 y Fq(=)g Fn(f)p Fp(u)h Fn(2)f Fp(Y)78 b Fq(:)58 b Fn(k)p Fp(u)p Fn(k)2838 293 y Fo(Y)2925 279 y Fn(\024)25 b Fp(R)3090 293 y Fm(2)3129 279 y Fn(g)p Fp(:)382 b Fq(\(A.16\))149 483 y Ff(Then)39 b(ther)-5 b(e)39 b(exist)f(a)g(unique)g(solution)h Fp(u)c Fn(2)g Fp(B)1825 497 y Fm(2)1902 483 y Ff(of)j(\(A.8\))h(and)g (a)f(unique)f(solution)j Fp(v)e Fn(2)d Fp(B)3430 497 y Fm(1)3507 483 y Ff(of)j(\(A.9\))149 600 y(with)c(the)f(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)1210 843 y Fn(k)p Fp(v)24 b Fn(\000)c Fp(g)s Fn(k)1505 857 y Fo(X)1599 843 y Fn(\024)1705 782 y(k)p Fp(K)7 b Fq([)p Fp(g)s Fq(])p Fn(k)1975 796 y Fo(X)p 1705 822 340 4 v 1746 906 a Fq(1)20 b Fn(\000)g Fp(L)1964 920 y Fm(1)2137 843 y Fn(\024)2379 782 y Fp(L)2441 796 y Fm(1)p 2301 822 258 4 v 2301 906 a Fq(1)g Fn(\000)g Fp(L)2519 920 y Fm(1)2569 843 y Fn(k)p Fp(g)s Fn(k)2705 857 y Fo(X)3581 843 y Fq(\(A.17\))1345 1095 y Fn(k)p Fp(u)h Fn(\000)f Fp(K)1638 1058 y Fo(n)1631 1118 y Fm(2)1685 1095 y Fq([)p Fp(f)29 b Fq(+)20 b Fp(v)s Fq(])p Fn(k)1992 1109 y Fo(Y)2137 1095 y Fn(\024)2330 1034 y Fp(L)2392 995 y Fo(n)p Fm(+1)2392 1060 y(2)p 2301 1074 V 2301 1158 a Fq(1)g Fn(\000)g Fp(L)2519 1172 y Fm(2)2569 1095 y Fn(k)p Fp(f)10 b Fn(k)2714 1109 y Fo(Y)3581 1095 y Fq(\(A.18\))149 1331 y Ff(for)34 b Fp(n)24 b Fq(=)h(0)p Fp(;)15 b Fq(1)p Fp(;)g Fq(2)g Fp(:)g(:)g(:)36 b Ff(In)d(p)-5 b(articular,)35 b(for)e Fp(n)25 b Fq(=)g(0)p Fp(;)15 b Fq(1)34 b Ff(we)e(have)1300 1568 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(v)j Fn(\000)d Fp(f)10 b Fn(k)1767 1582 y Fo(Y)1911 1568 y Fn(\024)2153 1507 y Fp(L)2215 1521 y Fm(2)p 2075 1547 V 2075 1631 a Fq(1)20 b Fn(\000)g Fp(L)2293 1645 y Fm(2)2342 1568 y Fn(k)p Fp(f)10 b Fn(k)2487 1582 y Fo(Y)3581 1568 y Fq(\(A.19\))1301 1801 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(g)j Fn(\000)d Fp(f)10 b Fn(k)1767 1815 y Fo(Y)1911 1801 y Fn(\024)2153 1740 y Fp(L)2215 1754 y Fm(1)p 2075 1780 V 2075 1864 a Fq(1)20 b Fn(\000)g Fp(L)2293 1878 y Fm(1)2342 1801 y Fn(k)p Fp(g)s Fn(k)2478 1815 y Fo(X)2567 1801 y Fq(+)2747 1740 y Fp(L)2809 1754 y Fm(2)p 2668 1780 V 2668 1864 a Fq(1)h Fn(\000)f Fp(L)2887 1878 y Fm(2)2936 1801 y Fn(k)p Fp(f)10 b Fn(k)3081 1815 y Fo(Y)3581 1801 y Fq(\(A.20\))843 2048 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(g)j Fn(\000)d Fp(f)30 b Fn(\000)20 b Fp(K)7 b Fq([)p Fp(v)23 b Fq(+)d Fp(f)10 b Fq(])p Fn(k)1767 2062 y Fo(Y)1911 2048 y Fn(\024)2153 1986 y Fp(L)2215 1953 y Fm(2)2215 2011 y(2)p 2075 2027 V 2075 2110 a Fq(1)20 b Fn(\000)g Fp(L)2293 2124 y Fm(2)2342 2048 y Fn(k)p Fp(f)10 b Fn(k)2487 2062 y Fo(Y)3581 2048 y Fq(\(A.21\))291 2283 y(Lemma)29 b(4.2)h(is)e(a)i(sp)s(eci\014c)e(application)f(of)j (Prop)s(osition)d(A.2.)41 b(Moreo)m(v)m(er,)32 b(an)d(inductiv)m(e)f (pro)s(cedure)149 2399 y(can)j(generalize)g(the)f(b)s(ound)e(\(A.20\)) 33 b(as)d(follo)m(ws:)414 2656 y Fn(k)p Fp(u)21 b Fn(\000)f Fp(f)668 2670 y Fm(1)727 2656 y Fn(\000)g Fp(f)863 2670 y Fm(2)922 2656 y Fn(\000)g Fp(:)15 b(:)g(:)21 b Fn(\000)f Fp(f)1275 2670 y Fo(m)1341 2656 y Fn(k)1386 2670 y Fo(m)1478 2656 y Fn(\024)1662 2594 y Fp(L)1724 2561 y Fm(2)1724 2619 y(1)p 1584 2635 V 1584 2718 a Fq(1)h Fn(\000)f Fp(L)1803 2732 y Fm(1)1852 2656 y Fn(k)p Fp(f)1942 2670 y Fm(1)1981 2656 y Fn(k)2026 2670 y Fm(1)2086 2656 y Fq(+)2266 2594 y Fp(L)2328 2561 y Fm(2)2328 2619 y(2)p 2187 2635 V 2187 2718 a Fq(1)h Fn(\000)f Fp(L)2406 2732 y Fm(2)2455 2656 y Fn(k)p Fp(f)2545 2670 y Fm(2)2584 2656 y Fn(k)2629 2670 y Fm(2)2690 2656 y Fq(+)f Fp(:)c(:)g(:)22 b Fq(+)3086 2594 y Fp(L)3148 2561 y Fm(2)3148 2617 y Fo(m)p 3008 2635 286 4 v 3008 2718 a Fq(1)e Fn(\000)g Fp(L)3226 2732 y Fo(m)3303 2656 y Fn(k)p Fp(f)3393 2670 y Fo(m)3459 2656 y Fn(k)3504 2670 y Fo(m)149 2891 y Fq(where)30 b Fp(u)h Fq(is)e(the)i(solution)e(of)893 3096 y Fp(u)d Fq(=)f Fp(f)1112 3110 y Fm(1)1171 3096 y Fq(+)20 b Fp(f)1307 3110 y Fm(2)1366 3096 y Fq(+)g Fp(:)15 b(:)g(:)h(f)1623 3110 y Fo(m)1709 3096 y Fq(+)k Fp(K)7 b Fq([)p Fp(u)p Fq(])p Fp(;)198 b(f)2254 3110 y Fo(i)2306 3096 y Fn(2)25 b Fp(X)2467 3110 y Fo(i)2496 3096 y Fp(;)46 b(i)25 b Fq(=)g(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(m)149 3300 y Fq(with)35 b(\()p Fp(X)472 3314 y Fo(i)501 3300 y Fp(;)15 b Fn(k)25 b(\001)f(k)705 3314 y Fo(i)734 3300 y Fq(\))p Fp(;)51 b(i)36 b Fq(=)e(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(m)36 b Fq(Banac)m(h)h(spaces)f(that)h(satisfy)e Fp(X)2607 3314 y Fm(1)2682 3300 y Fp(,)-15 b Fn(!)35 b Fp(X)2893 3314 y Fm(2)2967 3300 y Fp(,)-15 b Fn(!)35 b Fp(:)15 b(:)g(:)36 b(,)-15 b Fn(!)34 b Fp(X)3454 3314 y Fo(m)3557 3300 y Fq(and)i(on)149 3416 y(whic)m(h)30 b Fp(K)37 b Fq(is)29 b(con)m(tractiv)m(e)j(with)d(corresp)s(onding)g (Lipsc)m(hitz)g(constan)m(ts)i Fp(L)2726 3430 y Fo(i)2755 3416 y Fq(,)f Fp(i)c Fq(=)f(1)p Fp(;)15 b Fq(2)p Fp(;)g(:)g(:)g(:)j(m)p Fq(.)149 3702 y Fr(References)195 3909 y Fq([1])46 b(T.)29 b(Cazena)m(v)m(e,)j(Semilinear)27 b(Sc)m(hr\177)-45 b(odinger)28 b(equations,)h(Couran)m(t)g(Lecture)h(Notes)g(in)e(Mathematics,)336 4025 y(v)m(ol.)j(10,)g(New)g(Y)-8 b(ork,)31 b(Couran)m(t)g(Institute,)f (2003.)195 4215 y([2])46 b(O.)39 b(Costin,)i(R.D.)f(Costin,)g(J.L.)f (Leb)s(o)m(witz,)j(A.)d(Rokhlenk)m(o,)i(Ev)m(olution)d(of)i(a)f(mo)s (del)f(quan)m(tum)336 4331 y(system)h(under)d(time)i(p)s(erio)s(dic)e (forcing:)55 b(conditions)37 b(for)h(complete)h(ionization,)g(Comm.)f (Math.)336 4447 y(Ph)m(ys.)31 b(221)g(\(2001\))i(1-26.)195 4637 y([3])46 b(M.)37 b(Goldb)s(erg,)g(W.)f(Sc)m(hlag,)i(Disp)s(ersiv)m (e)d(estimates)i(for)e(Sc)m(hr\177)-45 b(odinger)35 b(op)s(erators)h (in)f(dimensions)336 4753 y(one)c(and)f(three,)h(preprin)m(t)d(2004.) 195 4943 y([4])46 b(S.)36 b(Inouy)m(e,)h(M.R.)f(Andrews,)g(J.)g (Stenger,)h(H.J.)f(Miesner,)h(D.M.)g(Stemp)s(er-Kurn,)e(W.)h(Ketterle,) 336 5059 y(Observ)-5 b(ation)37 b(of)h(F)-8 b(esh)m(bac)m(h)39 b(resonances)g(in)d(a)j(Bose-Einstein)e(condensate,)k(Nature)d(392)h (\(1998\))336 5176 y(151-154.)1947 5425 y(26)p eop %%Page: 27 27 27 26 bop 195 75 a Fq([5])46 b(A.)33 b(Jensen,)g(R.)f(Kato,)i(Sp)s (ectral)e(prop)s(erties)f(of)h(Sc)m(hr\177)-45 b(odinger)32 b(op)s(erators)g(and)g(time-deca)m(y)i(of)e(the)336 191 y(w)m(a)m(v)m(e)h(functions,)c(Duk)m(e)i(Math.)g(J.)f(46)i(\(1979\))g (583-611.)195 382 y([6])46 b(P)-8 b(.G.)36 b(Kevrekidis,)d(G.)i(Theo)s (c)m(haris,)g(D.J.)f(F)-8 b(ranzesk)j(akis,)36 b(B.A.)g(Malomed,)f(F)-8 b(esh)m(bac)m(h)36 b(resonance)336 498 y(managemen)m(t)c(for)e (Bose-Einstein)g(condensates,)h(Ph)m(ys.)g(Rev.)f(Lett.)i(90)f (\(2003\))i(230401.)195 689 y([7])46 b(E.)22 b(Kirr,)f(M.I.)h(W)-8 b(einstein,)22 b(P)m(arametrically)f(excited)g(Hamiltonian)f(partial)g (di\013eren)m(tial)g(equations,)336 805 y(SIAM)30 b(J.)h(Math.)g(Anal.) f(33)h(\(2001\))i(16-52.)195 996 y([8])46 b(E.)35 b(Kirr,)f(M.I.)i(W)-8 b(einstein,)35 b(Metastable)h(states)g(in)d(parametrically)h(excited)h (m)m(ultimo)s(de)d(Hamil-)336 1112 y(tonian)e(systems,)h(Comm.)f(Math.) h(Ph)m(ys.)f(236)i(\(2003\))h(335-372.)195 1303 y([9])46 b(E.)34 b(Kirr,)f(M.I.)i(W)-8 b(einstein,)34 b(Di\013usion)e(of)i(p)s (o)m(w)m(er)g(in)e(randomly)g(p)s(erturb)s(ed)g(Hamiltonian)g(partial) 336 1420 y(di\013eren)m(tial)d(equations,)h(Comm.)h(Math.)g(Ph)m(ys.)f (\(2004\),)j(to)e(b)s(e)f(published.)149 1611 y([10])47 b(F.Kh.)25 b(Ab)s(dullaev,)e(B.B.)j(Baizak)m(o)m(v,)i(S.A.)d(Darman)m (y)m(an,)h(V.V.)g(Konotop,)g(M.)f(Salerno,)g(Nonlinear)336 1727 y(excitations)31 b(in)e(arra)m(ys)i(of)f(Bose-Einstein)g (condensates,)h(Ph)m(ys.)f(Rev.)h(A)g(64)g(\(2001\))i(043606.)149 1918 y([11])47 b(A.J.)35 b(O'Connor,)f(Exp)s(onen)m(tial)f(deca)m(y)i (of)g(b)s(ound)d(state)k(w)m(a)m(v)m(e)g(functions,)e(Comm.)g(Math.)h (Ph)m(ys.)336 2034 y(32)d(\(1973\))g(319-340.)149 2225 y([12])47 b(A.)26 b(P)m(azy)-8 b(,)28 b(Semigroups)c(of)i(linear)d(op)s (erators)j(and)f(applications)e(to)k(partial)d(di\013eren)m(tial)g (equations,)336 2341 y(Applied)k(Mathematical)k(Sciences,)e(v)m(ol.)h (44,)g(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(1983.)149 2532 y([13])47 b(D.E.)31 b(P)m(elino)m(vsky)-8 b(,)31 b(P)-8 b(.G.)31 b(Kevrekidis,)d(D.J.)j(F)-8 b(ranzesk)j(akis,)31 b(Av)m(eraging)g(for)f(solitons)f(with)g(nonlin-)336 2648 y(earit)m(y)i(managemen)m(t,)h(Ph)m(ys.)e(Rev.)h(Lett.)h(91)f (\(2003\))i(240201.)149 2839 y([14])47 b(D.E.)28 b(P)m(elino)m(vsky)-8 b(,)28 b(P)-8 b(.G.)28 b(Kevrekidis,)e(D.J.)i(F)-8 b(ranzesk)j(akis,)29 b(V.)e(Zharnitsky)-8 b(,)26 b(Hamiltonian)g(a)m(v)m(erag-)336 2956 y(ing)k(of)g(solitons)g(with)f(nonlinearit)m(y)f(managemen)m(t,)k (Ph)m(ys.)e(Rev.)h(E)f(70)i(\(2004\),)h(to)e(b)s(e)f(published.)149 3147 y([15])47 b(D.E.)34 b(P)m(elino)m(vsky)-8 b(,)34 b(V.)g(Zharnitsky)-8 b(,)32 b(Av)m(eraging)i(of)f(disp)s (ersion-managed)e(pulses:)45 b(existence)33 b(and)336 3263 y(stabilit)m(y)-8 b(,)30 b(SIAM)g(J.)g(Appl.)f(Math.)j(63)f (\(2003\))i(745-776.)149 3454 y([16])47 b(L.C.)d(Piccinini,)g(G.)g (Stampacc)m(hia,)k(G.)c(Vidossic)m(h,)i(Ordinary)41 b(di\013eren)m (tial)h(equations)i(in)e Fg(R)3763 3418 y Fo(n)3810 3454 y Fq(,)336 3570 y(Applied)28 b(Mathematical)k(Sciences,)e(v)m(ol.)h (39,)g(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(1984.)149 3761 y([17])47 b(C.A.)22 b(Pillet,)f(C.E.)h(W)-8 b(a)m(yne,)24 b(In)m(v)-5 b(arian)m(t)21 b(manifolds)e(for)i(a)h(class) f(of)g(disp)s(ersiv)m(e,)g(Hamiltonian,)h(partial)336 3877 y(di\013eren)m(tial)29 b(equations,)h(J.)h(Di\013.)f(Eqs.)g(141)i (\(1997\))h(310-326.)149 4068 y([18])47 b(I.M.)h(Sigal,)i(Nonlinear)45 b(w)m(a)m(v)m(e)j(and)f(Sc)m(hr\177)-45 b(odinger)45 b(equations.)i(I:)f(Instabilit)m(y)f(of)i(p)s(erio)s(dic)d(and)336 4184 y(quasip)s(erio)s(dic)27 b(solutions,)i(Comm.)h(Math.)i(Ph)m(ys.)e (153)h(\(1993\),)j(297-320.)149 4375 y([19])47 b(A.)28 b(So\013er,)g(M.I.)g(W)-8 b(einstein,)28 b(Multic)m(hannel)d(nonlinear) g(scattering)j(for)f(nonin)m(tegrable)g(equations,)336 4491 y(Comm.)j(Math.)i(Ph)m(ys.)e(133)i(\(1990\))g(119-146.)149 4682 y([20])47 b(A.)28 b(So\013er,)g(M.I.)g(W)-8 b(einstein,)28 b(Multic)m(hannel)d(nonlinear)g(scattering)j(for)f(nonin)m(tegrable)g (equations.)336 4799 y(I)s(I:)j(The)g(case)h(of)g(anisotropic)e(p)s (oten)m(tials)h(and)g(data,)h(J.)f(Di\013.)h(Eqs.)f(98)h(\(1992\))i (376-390.)149 4990 y([21])47 b(A.)e(So\013er,)j(M.I.)d(W)-8 b(einstein,)48 b(Nonautonomous)c(Hamiltonians,)j(J.)d(Statist.)h(Ph)m (ys.)f(93)h(\(1998\))336 5106 y(359-391.)1947 5425 y(27)p eop %%Page: 28 28 28 27 bop 149 75 a Fq([22])47 b(A.)29 b(So\013er,)f(M.I.)h(W)-8 b(einstein,)28 b(Resonances,)h(radiation)e(damping)g(and)g(instabilit)m (y)e(in)i(Hamiltonian)336 191 y(nonlinear)i(w)m(a)m(v)m(e)j(equations,) e(In)m(v)m(en)m(t.)i(Math.)f(136)h(\(1999\))g(9-74.)149 382 y([23])47 b(A.)36 b(So\013er,)g(M.I.)g(W)-8 b(einstein,)36 b(Selection)f(of)g(the)h(ground)e(state)i(in)e(nonlinear)f(Sc)m(hr\177) -45 b(odinger)34 b(equa-)336 498 y(tions,)c(preprin)m(t)f(\(2001\).)149 689 y([24])47 b(C.)28 b(Sulem,)f(P)-8 b(.L.)29 b(Sulem,)e(The)h (Nonlinear)e(Sc)m(hr\177)-45 b(odinger)27 b(Equation,)h(Springer-V)-8 b(erlag,)27 b(New)i(Y)-8 b(ork,)336 805 y(1999.)149 996 y([25])47 b(T.P)-8 b(.)25 b(Tsai,)h(H.T.)f(Y)-8 b(au,)26 b(Relaxation)f(of)g(excited)g(states)h(in)d(nonlinear)g(Sc)m(hr\177)-45 b(odinger)23 b(equations,)j(In)m(t.)336 1112 y(Math.)32 b(Res.)e(Not.)i(31)f(\(2002\))i(1629-1673.)149 1303 y([26])47 b(K.)29 b(Y)-8 b(a)5 b(jima,)30 b(Scattering)f(theory)g(for)g(Sc)m (hr\177)-45 b(odinger)27 b(equations)i(with)f(p)s(oten)m(tials)g(p)s (erio)s(dic)e(in)i(time,)336 1420 y(J.)j(Math.)g(So)s(c.)f(Japan)g(29)h (\(1977\))i(729-743.)149 1611 y([27])47 b(K.)21 b(Y)-8 b(a)5 b(jima,)24 b Fp(L)849 1578 y Fo(p)888 1611 y Fq(-con)m(tin)m(uit) m(y)d(of)h(w)m(a)m(v)m(e)g(op)s(erators)g(for)e(Sc)m(hr\177)-45 b(odinger)20 b(op)s(erators)h(and)g(its)f(applications,)336 1727 y(in)40 b(Pro)s(ceedings)f(of)i(the)g(Korea-Japan)g(P)m(artial)f (Di\013eren)m(tial)g(Equations)g(Conference)g(\(T)-8 b(aejon,)336 1843 y(1996\),)33 b(Lecture)e(Notes)g(Series,)f(v)m(ol.)h (39,)g(Seoul,)f(1997.)149 2034 y([28])47 b(E.)37 b(Zeidler,)g (Nonlinear)f(functional)g(analysis)f(and)i(its)f(applications.)g(I:)h (Fixed-p)s(oin)m(t)f(theorems,)336 2150 y(Springer-V)-8 b(erlag,)30 b(New)g(Y)-8 b(ork,)32 b(1986.)1947 5425 y(28)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0409201712221--