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b(They)42 b(de\014ne)g(a)h(\014ltration)e Fm(fF)1379 1642 y Fn(k)1423 1627 y Fm(g)i Fr(of)f(increasing)g Fq(\033)s Fr(-algebras)h(in)e(the)i (ob)m(vious)f(w)m(a)m(y)-8 b(.)78 b(The)42 b(follo)m(wing)-47 1740 y(prop)s(osition)28 b(is)h(the)i(analogue)g(of)f(Prop)s(osition)e (2.3)k(for)e(the)2103 1704 y Fp(3)p 2103 1719 V 2103 1771 a(4)2148 1740 y Fr(-mo)s(del.)-47 1901 y Fd(Prop)s(osition)36 b(3.1.)46 b Fg(L)-5 b(et)45 b Fq(V)64 b Fg(b)-5 b(e)44 b(as)h(in)51 b Fr(\(1.2\))46 b Fg(with)f Fq(\014)52 b(>)2063 1865 y Fp(1)p 2063 1880 V 2063 1932 a(2)2108 1901 y Fg(.)77 b(L)-5 b(et)44 b Fq(\032)j(>)f Fr(0)f Fg(and)g Fq(")i(>)f Fr(0)e Fg(b)-5 b(e)44 b(arbitr)-5 b(ary.)79 b(Then)-47 2014 y(up)40 b(to)g(pr)-5 b(ob)g(ability)42 b Fq(e)691 1981 y Fk(\000)p Fn(R)799 1957 y Fh(")838 2014 y Fg(,)f(any)g(solution) g Fr(\()p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))p Fq(;)g(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\)\))44 b Fg(of)59 b Fr(\(2.1\))42 b Fg(with)f Fm(j)p Fq(y)s 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g(that)g Fq(y)s(;)15 b(\021)32 b Fr(are)c(\014xed.)40 b(De\014ne)28 b(stopping)-47 3151 y(times)675 3323 y Fq(\034)715 3338 y Fn(`)831 3323 y Fr(:=)83 b(inf)5 b Fm(f)p Fq(t)26 b Fm(\025)f Fr(0)46 b(:)f Fm(j)p Fq(x)p Fr(\()p Fq(t)p Fr(\))p Fm(j)27 b Fr(=)d Fq(R)d Fr(+)f Fq(`R)2097 3286 y Fn(\014)2144 3323 y Fm(g)677 3461 y Fl(e)-53 b Fq(\034)715 3476 y Fn(`)831 3461 y Fr(:=)83 b(inf)5 b Fm(f)p Fq(s)25 b Fm(\025)g Fq(\034)1369 3476 y Fn(`)p Fk(\000)p Fp(1)1538 3461 y Fr(:)45 b Fm(j)p Fq(x)p Fr(\()p Fq(\034)1760 3476 y Fn(`)p Fk(\000)p Fp(1)1884 3461 y Fr(\))20 b(+)g(\()p Fq(s)h Fm(\000)e Fq(\034)2259 3476 y Fn(`)p Fk(\000)p Fp(1)2382 3461 y Fr(\))p Fq(\030)t Fr(\()p Fq(\034)2536 3476 y Fn(`)p Fk(\000)p Fp(1)2660 3461 y Fr(\))p Fm(j)26 b Fr(=)f Fq(R)c Fr(+)f Fq(`R)3131 3424 y Fn(\014)3177 3461 y Fm(g)-3269 b Fr(\(3.4\))-47 3633 y(for)30 b(those)h Fq(`)f Fr(with)f(0)d Fm(\024)f Fq(`)g Fm(\024)g Fq(R)1001 3600 y Fp(1)p Fk(\000)p Fn(\014)1138 3633 y Fr(.)41 b(Th)m(us,)29 b(b)m(y)h(T)-8 b(a)m(ylor)31 b(expansion,)234 3864 y Fq(\030)t Fr(\()p Fq(\034)353 3879 y Fn(`)p Fp(+1)476 3864 y Fr(\))21 b Fm(\000)f Fq(\030)t Fr(\()p Fq(\034)742 3879 y Fn(`)775 3864 y Fr(\))25 b(=)g Fm(\000)1017 3740 y Fl(Z)1110 3767 y Fc(e)-41 b Fn(\034)1139 3779 y Fh(`)p Fj(+1)1068 3947 y Fn(\034)1099 3959 y Fh(`)1266 3864 y Fm(r)p Fq(V)20 b Fr(\()p Fq(x)p Fr(\()p Fq(\034)1577 3879 y Fn(`)1610 3864 y Fr(\))h(+)e(\()p Fq(s)i Fm(\000)f Fq(\034)1986 3879 y Fn(`)2018 3864 y Fr(\))p Fq(\030)t Fr(\()p Fq(\034)2172 3879 y Fn(`)2206 3864 y Fr(\)\))15 b Fq(ds)598 4096 y Fr(+)p Fq(O)741 3995 y Fl(\020)794 4096 y Fm(j)p Fq(\034)859 4111 y Fn(`)p Fp(+1)1003 4096 y Fm(\000)20 b Fq(\034)1134 4111 y Fn(`)1166 4096 y Fm(j)1191 4059 y Fp(3)1246 4096 y Fm(k)p Fq(D)1369 4059 y Fp(2)1409 4096 y Fq(V)g Fm(k)1527 4115 y Fn(L)1575 4096 y Fb(1)1641 4115 y Fp(\()p Fk(S)1711 4127 y Fh(`;R)1806 4115 y Fp(\))1853 4096 y Fm(k)p Fq(D)s(V)g Fm(k)2094 4115 y Fn(L)2142 4096 y Fb(1)2208 4115 y Fp(\()p Fk(S)2278 4127 y Fh(`;R)2373 4115 y Fp(\))2404 3995 y Fl(\021)2479 4096 y Fr(+)g Fq(O)2642 3995 y Fl(\020)2696 4096 y Fm(j)r Fl(e)-53 b Fq(\034)2761 4111 y Fn(`)p Fp(+1)2904 4096 y Fm(\000)20 b Fq(\034)3035 4111 y Fn(`)p Fp(+1)3158 4096 y Fm(j)15 b(k)p Fq(D)s(V)21 b Fm(k)3440 4115 y Fn(L)3488 4096 y Fb(1)3553 4115 y Fp(\()p Fk(S)3623 4127 y Fh(`;R)3719 4115 y Fp(\))3750 3995 y Fl(\021)3804 4096 y Fq(:)-3876 b Fr(\(3.5\))-47 4317 y(Denote)33 b(the)g(in)m(tegral)f(in)f(\(3.5\))j(b)m(y)e Fq(Y)1266 4332 y Fn(`)p Fp(+1)1389 4317 y Fr(.)46 b(Since)31 b Fm(j)p Fq(\034)1764 4332 y Fn(`)p Fp(+1)1908 4317 y Fm(\000)22 b Fq(\034)2041 4332 y Fn(`)2073 4317 y Fm(j)29 b(\024)f Fq(C)2307 4281 y Fn(R)2360 4257 y Fh(\014)p 2307 4296 96 4 v 2337 4348 a Fn(\032)2444 4317 y Fr(and)k(b)m(y)g (Lemma)g(2.2,)i(with)d Fq(L)d Fr(=)g Fq(R)3719 4284 y Fn(\014)3798 4317 y Fr(and)-47 4452 y Fq(\016)h Fr(=)c Fq(C)7 b(R)260 4419 y Fk(\000)p Fn(m)p Fp(\(1\))481 4416 y Fn(L)p 481 4431 48 4 v 487 4483 a(\032)539 4452 y Fr(,)1406 4708 y Fm(j)r Fl(e)-53 b Fq(\034)1471 4723 y Fn(`)p Fp(+1)1615 4708 y Fm(\000)20 b Fq(\034)1746 4723 y Fn(`)p Fp(+1)1869 4708 y Fm(j)25 b(\024)g Fq(C)2096 4647 y(R)2166 4614 y Fp(2)p Fn(\014)s Fk(\000)p Fn(m)p Fp(\(1\))p 2096 4687 360 4 v 2233 4771 a Fq(\032)2280 4744 y Fp(3)2466 4708 y Fq(;)1903 4992 y Fr(11)p eop %%Page: 12 12 12 11 bop -47 274 a Fr(one)30 b(obtains)320 518 y Fq(\030)t Fr(\()p Fq(\034)439 533 y Fn(`)p Fp(+1)563 518 y Fr(\))20 b Fm(\000)g Fq(\030)t Fr(\()p Fq(\034)828 533 y Fn(`)861 518 y Fr(\))26 b(=)f Fq(Y)1071 533 y Fn(`)p Fp(+1)1214 518 y Fr(+)20 b Fq(O)1377 417 y Fl(\020)1441 456 y Fq(R)1511 423 y Fp(3)p Fn(\014)p 1441 497 153 4 v 1474 580 a Fq(\032)1521 554 y Fp(3)1603 518 y Fq(R)1673 480 y Fk(\000)p Fn(m)p Fp(\(1\))p Fk(\000)p Fn(m)p Fp(\(2\))2092 417 y Fl(\021)2166 518 y Fr(+)g Fq(O)2329 417 y Fl(\020)2393 456 y Fq(R)2463 423 y Fp(2)p Fn(\014)s Fk(\000)p Fp(2)p Fn(m)p Fp(\(1\))p 2393 497 395 4 v 2548 580 a Fq(\032)2595 554 y Fp(3)2798 417 y Fl(\021)2878 518 y Fr(=)25 b Fq(Y)3027 533 y Fn(`)p Fp(+1)3170 518 y Fr(+)20 b Fq(O)s Fr(\()p Fq(\032)3415 480 y Fk(\000)p Fp(3)3524 518 y Fq(R)3594 480 y Fk(\000)3659 453 y Fj(3)p 3659 465 31 3 v 3659 506 a(2)3704 518 y Fr(\))p Fq(:)-3811 b Fr(\(3.6\))-47 761 y(Clearly)-8 b(,)969 1022 y Fm(k)p Fq(Y)1067 1037 y Fn(`)1100 1022 y Fm(k)1145 1036 y Fk(1)1245 1022 y Fm(\024)25 b Fq(C)7 b Fm(j)r Fl(e)-53 b Fq(\034)1478 1037 y Fn(`)p Fp(+1)1621 1022 y Fm(\000)20 b Fq(\034)1752 1037 y Fn(`)1785 1022 y Fm(j)15 b(k)p Fq(D)s(V)21 b Fm(k)2067 1041 y Fn(L)2115 1022 y Fb(1)2180 1041 y Fp(\()p Fk(S)2250 1053 y Fh(`;R)2345 1041 y Fp(\))2402 1022 y Fm(\024)k Fq(C)2595 961 y(R)2665 928 y Fn(\014)s Fk(\000)p Fn(m)p Fp(\(1\))p 2595 1001 325 4 v 2733 1085 a Fq(\032)-47 1265 y Fr(and)1137 1432 y Fk(b)p Fn(R)1221 1408 y Fj(1)p Fb(\000)p Fh(\014)1342 1432 y Fk(c)1189 1468 y Fl(X)1196 1665 y Fn(`)p Fp(=0)1388 1554 y Fm(k)p Fq(Y)1486 1569 y Fn(`)1519 1554 y Fm(k)1564 1516 y Fp(2)1564 1576 y Fk(1)1665 1554 y Fm(\024)g Fq(C)1857 1493 y(R)1927 1460 y Fp(1+)p Fn(\014)s Fk(\000)p Fp(2)p Fn(m)p Fp(\(1\))p 1857 1533 450 4 v 2039 1616 a Fq(\032)2086 1590 y Fp(2)2343 1554 y Fr(=)2448 1493 y Fq(R)2518 1457 y Fk(\000)2583 1430 y Fj(1)p 2583 1442 31 3 v 2583 1483 a(2)2623 1457 y Fk(\000)p Fn(\014)p 2448 1533 278 4 v 2544 1616 a Fq(\032)2591 1590 y Fp(2)2735 1554 y Fq(:)-2807 b Fr(\(3.7\))-47 1842 y(Summing)28 b(up)h(\(3.6\))j(yields)1091 2030 y Fl(\014)1091 2085 y(\014)1091 2139 y(\014)1121 2135 y Fq(\030)t Fr(\()p Fq(\034)1240 2150 y Fn(`)1269 2159 y Fj(1)1308 2135 y Fr(\))20 b Fm(\000)g Fq(\030)t Fr(\()p Fq(\034)1573 2150 y Fn(`)1602 2159 y Fj(2)1641 2135 y Fr(\))g Fm(\000)1874 2020 y Fn(`)1903 2029 y Fj(1)1840 2048 y Fl(X)1787 2246 y Fn(`)p Fp(=)p Fn(`)1900 2255 y Fj(2)1934 2246 y Fp(+1)2040 2135 y Fq(Y)2093 2150 y Fn(`)2125 2030 y Fl(\014)2125 2085 y(\014)2125 2139 y(\014)2181 2135 y Fm(\024)25 b Fq(C)7 b(\032)2396 2097 y Fk(\000)p Fp(3)2505 2135 y Fq(R)2575 2097 y Fk(\000)2640 2070 y Fj(1)p 2639 2082 31 3 v 2639 2123 a(2)2680 2097 y Fk(\000)p Fn(\014)2782 2135 y Fq(:)-47 2432 y Fr(Therefore,)30 b(Lemma)h(2.1)g(and)f(\(3.7\))i(imply)c(that)781 2657 y Fo(P)p Fr([)p Fm(j)p Fq(\030)t Fr(\()p Fq(\034)1005 2672 y Fn(`)1034 2681 y Fj(1)1074 2657 y Fr(\))21 b Fm(\000)e Fq(\030)t Fr(\()p Fq(\034)1339 2672 y Fn(`)1368 2681 y Fj(2)1407 2657 y Fr(\))p Fm(j)26 b Fq(>)f(\025\032)1689 2620 y Fk(\000)p Fp(1)1783 2657 y Fq(R)1853 2620 y Fk(\000)1918 2592 y Fj(1)p 1918 2604 V 1918 2646 a(4)1958 2620 y Fk(\000)2023 2589 y Fh(\014)p 2023 2605 38 3 v 2026 2646 a Fj(2)2095 2657 y Fr(+)20 b Fq(C)7 b(\032)2305 2620 y Fk(\000)p Fp(3)2414 2657 y Fq(R)2484 2620 y Fk(\000)2549 2592 y Fj(1)p 2548 2604 31 3 v 2548 2646 a(2)2589 2620 y Fk(\000)p Fn(\014)2691 2657 y Fr(])25 b Fm(\024)g Fq(C)7 b(e)2951 2620 y Fk(\000)p Fn(c\025)3078 2596 y Fj(2)-47 2861 y Fr(for)30 b(an)m(y)h(t)m(w)m(o)g Fq(`)473 2875 y Fp(1)513 2861 y Fq(;)15 b(`)591 2875 y Fp(2)661 2861 y Fr(as)30 b(ab)s(o)m(v)m(e.)42 b(Setting)30 b Fq(\025)c Fr(=)f Fq(R)1623 2828 y Fn(")1689 2861 y Fr(one)31 b(concludes)f(that)711 3082 y Fo(P)p Fr([)142 b(sup)791 3164 y Fp(0)p Fk(\024)p Fn(t;s)p Fk(\024)p Fp(10)p Fn(R=\032)1225 3082 y Fm(j)p Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))22 b Fm(\000)e Fq(\030)t Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))p Fm(j)27 b Fq(>)e(R)2237 3044 y Fk(\000)2302 3017 y Fj(1)p 2302 3029 V 2302 3070 a(4)2342 3044 y Fk(\000)2407 3013 y Fh(\014)p 2407 3029 38 3 v 2410 3070 a Fj(2)2454 3044 y Fp(+)p Fn(")2546 3082 y Fr(])g Fm(\024)g Fq(C)2757 3096 y Fn(\032)2813 3082 y Fq(R)2883 3044 y Fp(2)2937 3082 y Fq(e)2979 3044 y Fk(\000)p Fn(cR)3118 3021 y Fj(2)p Fh(")-47 3358 y Fr(pro)m(vided)34 b Fq(R)g Fm(\025)g Fq(R)609 3372 y Fp(0)648 3358 y Fr(\()p Fq(\032;)15 b(")p Fr(\).)58 b(The)35 b(dep)s(endence)f(on)i Fq(y)s(;)15 b(\021)39 b Fr(on)c(the)h(left-hand)e(side)h(can)h(b)s(e)e (remo)m(v)m(ed)j(b)m(y)e(means)h(of)-47 3471 y(discretization.)j(It)31 b(is)e(easy)i(to)g(see)g(that)g(this)f(leads)f(to)441 3691 y 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b(constan)m(ts)f(will)c(b)s(e)j(allo)m (w)m(ed)g(to)g(dep)s(end)e(on)i Fq(R)3880 3766 y Fp(1)3920 3752 y Fr(.)-47 3865 y(F)-8 b(rom)31 b(\(2.1\))h(and)e(T)-8 b(a)m(ylor's)30 b(form)m(ula)g(one)h(obtains)766 4069 y(_)741 4092 y(\001)o(\()p Fq(t)p Fr(\))84 b(=)f Fq(\030)1197 4106 y Fp(1)1236 4092 y Fr(\()p Fq(t)p Fr(\))21 b Fm(\000)f Fq(\030)1491 4106 y Fp(0)1530 4092 y Fr(\()p Fq(t)p Fr(\))26 b(=)f Fm(\000)1841 3968 y Fl(Z)1931 3994 y Fk(1)1891 4174 y Fn(t)2006 3991 y Fl(h)2049 4092 y Fm(r)p Fq(V)20 b Fr(\()p Fq(x)2285 4106 y Fp(1)2325 4092 y Fr(\()p Fq(s)p Fr(\)\))h Fm(\000)f(r)p Fq(V)f Fr(\()p Fq(x)2820 4106 y Fp(0)2860 4092 y Fr(\()p Fq(s)p Fr(\)\))3008 3991 y Fl(i)3067 4092 y Fq(ds)1003 4334 y Fr(=)83 b Fm(\000)1323 4247 y Fl(X)1243 4448 y Fp(1)p Fk(\024j)p Fn(\013)p Fk(j\024)p Fn(N)1580 4272 y Fr(1)p 1560 4313 84 4 v 1560 4396 a Fq(\013)p Fr(!)1669 4210 y Fl(Z)1760 4236 y Fk(1)1720 4416 y Fn(t)1850 4334 y Fm(r)p Fq(D)2004 4296 y Fn(\013)2053 4334 y Fq(V)20 b Fr(\()p Fq(x)2213 4348 y Fp(0)2253 4334 y Fr(\()p Fq(s)p Fr(\)\)\001\()p Fq(s)p Fr(\))2590 4296 y Fn(\013)2655 4334 y Fq(ds)-2792 b Fr(\(3.16\))1003 4648 y Fm(\000)1237 4561 y Fl(X)1157 4763 y Fk(j)p Fn(\013)p Fk(j)p Fp(=)p Fn(N)7 b Fp(+1)1494 4586 y Fr(1)p 1475 4627 V 1475 4710 a Fq(\013)p Fr(!)1583 4524 y Fl(Z)1674 4550 y Fp(1)1634 4730 y(0)1729 4524 y Fl(Z)1820 4550 y Fk(1)1779 4730 y Fn(t)1910 4648 y Fm(r)p Fq(D)2064 4610 y Fn(\013)2113 4648 y Fq(V)20 b Fr(\()p Fq(x)2273 4663 y Fn(\022)2312 4648 y Fr(\()p Fq(s)p Fr(\)\))15 b(\001\()p Fq(s)p Fr(\))2664 4610 y Fn(\013)2730 4648 y Fq(ds)g(d\022)s(:)-3000 b Fr(\(3.17\))1903 4992 y(14)p eop %%Page: 15 15 15 14 bop -47 274 a Fr(Here)31 b Fq(N)40 b Fr(is)29 b(c)m(hosen)i(so)g (large)f(that)h(the)g(error)f(term)g(\(3.17\))j(is)c(on)i(the)f(order)g (of)452 490 y Fq(O)524 389 y Fl(\020)578 366 y(Z)669 393 y Fk(1)628 572 y Fn(t)759 490 y Fq(s)802 452 y Fk(\000)867 425 y Fj(3)p 866 437 31 3 v 866 479 a(4)907 452 y Fk(\000)p Fp(\()p Fn(N)7 b Fp(+2\))p Fn(\014)1232 490 y Fq(s)1275 452 y Fn(N)g Fp(\()1375 425 y Fj(1)p 1375 437 V 1375 479 a(2)1415 452 y Fk(\000)p Fn(")p Fp(\))1535 490 y Fm(j)p Fr(\001\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds)1879 389 y Fl(\021)2017 490 y Fr(=)82 b Fq(O)2242 389 y Fl(\020)2296 366 y(Z)2387 393 y Fk(1)2347 572 y Fn(t)2477 490 y Fq(s)2520 452 y Fk(\000)p Fp(3)2629 490 y Fm(j)p Fr(\001\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds)2973 389 y Fl(\021)3028 490 y Fq(:)2017 732 y Fr(=)82 b Fq(O)2242 631 y Fl(\020)2296 732 y Fr(\001\()p Fq(t)p Fr(\))p Fq(t)2508 694 y Fk(\000)p Fp(2)2623 732 y Fr(+)2714 608 y Fl(Z)2805 634 y Fk(1)2764 814 y Fn(t)2895 732 y Fq(s)2938 694 y Fk(\000)p Fp(2)3047 732 y Fm(j)3097 709 y Fr(_)3072 732 y(\001\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds)3391 631 y Fl(\021)-47 732 y Fr(\(3.18\))-47 945 y(Observ)m(e)25 b(that)h(in)e(order)h(to)i(obtain)e (this)f(estimate)i(w)m(e)g(ha)m(v)m(e)h(used)e(the)g(ob)m(vious)g(b)s (ound)f(on)h(the)h(deriv)-5 b(ativ)m(es)25 b(of)g Fq(V)c Fr(,)-47 1057 y(as)30 b(w)m(ell)g(as)g(\(3.15\))s(.)40 b(No)m(w)31 b(\014x)f(an)m(y)h Fq(\013)g Fr(with)e(1)c Fm(\024)g(j)p Fq(\013)p Fm(j)i(\024)e Fq(N)40 b Fr(and)30 b(denote)1294 1274 y Fq(U)1356 1288 y Fn(\013)1405 1274 y Fr(\()p Fq(t)p Fr(\))c(:=)f Fm(\000)1741 1150 y Fl(Z)1832 1176 y Fk(1)1792 1356 y Fn(t)1922 1274 y Fm(r)p Fq(D)2076 1236 y Fn(\013)2125 1274 y Fq(V)20 b Fr(\()p Fq(x)2285 1288 y Fp(0)2325 1274 y Fr(\()p Fq(s)p Fr(\)\))15 b Fq(ds:)-47 1485 y Fr(By)30 b(Lemma)h(3.3)1426 1667 y Fm(j)p Fq(U)1513 1681 y Fn(\013)1563 1667 y Fr(\()p Fq(t)p Fr(\))p Fm(j)26 b(\024)f Fq(C)c(t)1932 1630 y Fk(\000)1997 1602 y Fj(1)p 1997 1614 V 1997 1656 a(4)2037 1630 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+)2269 1602 y Fj(1)p 2269 1614 V 2269 1656 a(2)2310 1630 y Fp(\))p Fn(\014)s Fp(+)p Fn(")-47 1839 y Fr(up)29 b(to)i(probabilit)m(y)d(at)j(most)g Fq(e)1036 1806 y Fk(\000)p Fn(R)1144 1783 y Fh(")1144 1827 y Fj(1)1183 1839 y Fr(.)41 b(In)m(tegrating)31 b(b)m(y)f(parts)g (in)f(the)i Fq(\013)p Fr(-term)g(in)e(\(3.16\))k(therefore)d(yields)476 1929 y Fl(\014)476 1984 y(\014)476 2038 y(\014)476 2093 y(\014)506 1937 y(Z)597 1963 y Fk(1)557 2143 y Fn(t)687 2061 y Fm(r)p Fq(D)841 2023 y Fn(\013)890 2061 y Fq(V)20 b Fr(\()p Fq(x)1050 2075 y Fp(0)1090 2061 y Fr(\()p Fq(s)p Fr(\)\)\001\()p Fq(s)p Fr(\))1427 2023 y Fn(\013)1492 2061 y Fq(ds)1582 1929 y Fl(\014)1582 1984 y(\014)1582 2038 y(\014)1582 2093 y(\014)1638 2061 y Fm(\024)1734 1956 y Fl(\014)1734 2011 y(\014)1734 2065 y(\014)1764 1937 y(Z)1855 1963 y Fk(1)1814 2143 y Fn(t)1945 2061 y Fq(U)2007 2075 y Fn(\013)2056 2061 y Fr(\()p Fq(s)p Fr(\))2201 1999 y Fq(d)p 2179 2040 90 4 v 2179 2123 a(ds)2279 1960 y Fl(\020)2334 2061 y Fr(\001\()p Fq(s)p Fr(\))2523 2023 y Fn(\013)2572 1960 y Fl(\021)2642 2061 y Fq(ds)2732 1956 y Fl(\014)2732 2011 y(\014)2732 2065 y(\014)2782 2061 y Fr(+)2873 1956 y Fl(\014)2873 2011 y(\014)2873 2065 y(\014)2903 2061 y Fq(U)2965 2075 y Fn(\013)3015 2061 y Fr(\()p Fq(t)p Fr(\)\001\()p Fq(t)p Fr(\))3297 2023 y Fn(\013)3347 1956 y Fl(\014)3347 2011 y(\014)3347 2065 y(\014)322 2306 y Fm(\024)83 b Fq(C)562 2183 y Fl(Z)653 2209 y Fk(1)613 2389 y Fn(t)743 2306 y Fq(s)786 2269 y Fk(\000)851 2242 y Fj(1)p 851 2254 31 3 v 851 2295 a(4)891 2269 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+)1123 2242 y Fj(1)p 1123 2254 V 1123 2295 a(2)1163 2269 y Fp(\))p Fn(\014)s Fp(+)p Fn(")1340 2306 y Fq(s)1383 2269 y Fp(\()1420 2242 y Fj(1)p 1420 2254 V 1420 2295 a(2)1461 2269 y Fk(\000)p Fn(")p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fp(1\))1825 2306 y Fm(j)1875 2283 y Fr(_)1850 2306 y(\001\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds)20 b Fr(+)g Fq(C)i(t)2400 2269 y Fk(\000)2465 2242 y Fj(1)p 2465 2254 V 2465 2295 a(4)2505 2269 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+)2737 2242 y Fj(1)p 2737 2254 V 2737 2295 a(2)2777 2269 y Fp(\))p Fn(\014)s Fp(+)p Fn(")2954 2306 y Fq(t)2987 2269 y Fp(\()3024 2242 y Fj(1)p 3025 2254 V 3025 2295 a(2)3065 2269 y Fk(\000)p Fn(")p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fp(1\))3429 2306 y Fm(j)p Fr(\001\()p Fq(t)p Fr(\))p Fm(j)322 2548 y(\024)83 b Fq(C)562 2424 y Fl(Z)653 2451 y Fk(1)613 2631 y Fn(t)743 2548 y Fq(s)786 2511 y Fk(\000)851 2483 y Fj(1)p 851 2495 V 851 2537 a(4)891 2511 y Fk(\000)956 2480 y Fj(3)p Fh(\014)p 956 2495 68 3 v 975 2537 a Fj(2)1034 2511 y Fp(+)p Fn(")1140 2548 y Fm(j)1191 2525 y Fr(_)1165 2548 y(\001\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds)21 b Fr(+)f Fq(C)i(t)1716 2511 y Fk(\000)1781 2483 y Fj(1)p 1780 2495 31 3 v 1780 2537 a(4)1821 2511 y Fk(\000)1886 2480 y Fj(3)p Fh(\014)p 1886 2495 68 3 v 1904 2537 a Fj(2)1963 2511 y Fp(+)p Fn(")2070 2548 y Fm(j)p Fr(\001\()p Fq(t)p Fr(\))p Fm(j)p Fq(:)-2371 b Fr(\(3.19\))-47 2776 y(Here)37 b(w)m(e)g(ha)m(v)m(e)g(used)f(that)h Fm(j)p Fq(\013)p Fm(j)f Fr(=)f(1)i(is)f(the)g(w)m(orst)h(case.)60 b(No)m(w)37 b(\014x)f Fq(\014)41 b(>)2592 2741 y Fp(1)p 2592 2756 36 4 v 2592 2808 a(2)2674 2776 y Fr(and)35 b(c)m(ho)s(ose)j Fq(")e(>)f Fr(0)i(so)f(small)f(that)-47 2903 y Fm(\000)34 2867 y Fp(1)p 34 2882 V 34 2935 a(4)99 2903 y Fm(\000)200 2862 y Fp(3)p Fn(\014)p 200 2882 79 4 v 221 2935 a Fp(2)308 2903 y Fr(+)20 b Fq(")26 b(<)f Fm(\000)p Fr(1)20 b Fm(\000)g Fq(\016)34 b Fr(for)c(some)h Fq(\016)e(>)c Fr(0.)41 b(Com)m(bining)30 b(\(3.16\))r({\(3.19\))j(yields)1028 3128 y Fm(j)1078 3105 y Fr(_)1053 3128 y(\001\()p Fq(t)p Fr(\))p Fm(j)26 b(\024)f Fq(C)7 b(t)1484 3090 y Fk(\000)p Fp(1)p Fk(\000)p Fn(\016)1666 3128 y Fm(j)p Fr(\001\()p Fq(t)p Fr(\))p Fm(j)21 b Fr(+)f Fq(C)2094 3004 y Fl(Z)2184 3031 y Fk(1)2144 3210 y Fn(t)2274 3128 y Fq(s)2317 3090 y Fk(\000)p Fp(1)p Fk(\000)p Fn(\016)2500 3128 y Fm(j)2550 3105 y Fr(_)2525 3128 y(\001\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds:)-2916 b Fr(\(3.20\))-47 3361 y(Starting)22 b(from)h(the)g(trivial)f(b)s(ound) f Fm(j)1243 3338 y Fr(_)1218 3361 y(\001\()p Fq(t)p Fr(\))p Fm(j)26 b(\024)f Fq(C)k Fr(and)23 b(assuming)f(\(as)h(w)m(e)h(ma)m(y\)) g(that)2887 3326 y Fp(1)p 2887 3341 36 4 v 2888 3393 a Fn(\016)2955 3361 y Fr(is)e(not)i(an)f(in)m(teger)g(iteration)-47 3474 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983 46 4 v 994 1066 a(2)1050 1004 y Fm(hr)p Fq(D)1239 967 y Fp(2)1278 1004 y Fq(V)20 b Fr(\()p Fq(\022)8 b Fr(~)-50 b Fq(z)t Fr(\()p Fq(s;)15 b(p)p Fr(\))21 b(+)f(\(1)h Fm(\000)f Fq(\022)s Fr(\)\()p Fq(a)g Fr(+)g Fq(z)t Fr(\()p Fq(\034)5 b(;)15 b(p)p Fr(\))21 b Fm(\000)f Fq(y)s Fr(\()p Fq(\034)10 b Fr(\)\)\))15 b Fq(w)r Fr(\()p Fq(\034)10 b Fr(\))p Fq(;)15 b(w)r Fr(\()p Fq(\034)10 b Fr(\))p Fm(i)15 b Fq(d\022)23 b(d\034)i(ds:)-3719 b Fr(\(3.38\))-47 1254 y(Observ)m(e)30 b(that)h(the)g(triple)d(in)m(tegral)i(in)f (\(3.38\))k(is)c(no)i(larger)f(than)822 1379 y Fl(Z)913 1405 y Fk(1)872 1585 y Fn(t)1003 1379 y Fl(Z)1093 1405 y Fk(1)1053 1585 y Fn(s)1183 1503 y Fq(C)7 b(\034)1305 1465 y Fk(\000)1370 1438 y Fj(3)p 1370 1450 31 3 v 1370 1491 a(4)1410 1465 y Fk(\000)p Fp(3)p Fn(\014)1547 1503 y Fm(j)p Fq(w)r Fr(\()p Fq(\034)j Fr(\))p Fm(j)1784 1465 y Fp(2)1841 1503 y Fq(d\034)g(ds)25 b Fm(\024)g Fq(C)d Fr(sup)2239 1581 y Fn(t)p Fk(\025)p Fn(T)2388 1503 y Fm(j)p Fq(w)r Fr(\()p Fq(t)p Fr(\))p 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Fr(+)h Fq(z)t Fr(\()p Fq(t;)15 b(p)p Fr(\))21 b Fm(\000)f Fq(y)s Fr(\()p Fq(t)p Fr(\).)43 b(It)31 b(is)f(eviden)m(t)g(that)i(this)e(function)f Fq(x)i Fr(satis\014es)f(\(3.33\))s(.)42 b(It)31 b(no)m(w)g(remains)e (to)j(sho)m(w)f(that)g Fq(x)g Fr(is)-47 500 y(a)g(solution)d(of)38 b(\(2.1\))r(.)j(Di\013eren)m(tiating)31 b(\(3.38\))i(t)m(wice)d(with)f (resp)s(ect)i(to)g Fq(t)p Fr(,)f(one)h(obtains)f(that)106 738 y(\177)-51 b Fq(y)s Fr(\()p Fq(t)p Fr(\))84 b(=)22 b Fq(D)506 701 y Fp(2)545 738 y Fq(V)e Fr(\()5 b(~)-50 b Fq(z)5 b Fr(\()p Fq(t;)15 b(p)p Fr(\)\))p Fq(w)r Fr(\()p Fq(t)p Fr(\))22 b(+)1207 614 y Fl(Z)1298 641 y Fp(1)1258 821 y(0)1362 677 y Fr(1)p 1362 717 46 4 v 1362 800 a(2)1418 738 y Fm(hr)p Fq(D)1607 701 y Fp(2)1646 738 y Fq(V)e Fr(\()p Fq(\022)8 b Fr(~)-50 b Fq(z)t Fr(\()p Fq(s;)15 b(p)p Fr(\))21 b(+)f(\(1)h Fm(\000)f Fq(\022)s Fr(\)\()p Fq(a)g Fr(+)g Fq(z)t Fr(\()p Fq(\034)5 b(;)15 b(p)p Fr(\))21 b Fm(\000)f Fq(y)s Fr(\()p Fq(\034)10 b Fr(\)\)\))15 b Fq(w)r Fr(\()p Fq(\034)10 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b(3.10.)46 b Fr(F)-8 b(ollo)m(wing)30 b([9)q(],)h(let)842 1707 y Fq(E)914 1659 y Fp(\(0\))909 1731 y Fn(t)1009 1707 y Fr(\()p Fq(a;)15 b(p)p Fr(\))26 b(=)f(\()p Fq(a)20 b Fr(+)g Fq(z)t Fr(\()p Fq(t;)15 b(p)p Fr(\))p Fq(;)g(p)p Fr(\))62 b(and)e Fq(S)2215 1659 y Fp(\(0\))2210 1731 y Fn(t)2310 1707 y Fr(\()p Fq(a;)15 b(p)p Fr(\))26 b(=)f(\()p Fq(a)20 b Fr(+)g Fq(pt;)15 b(p)p Fr(\))p Fq(:)-47 1887 y Fr(The)30 b(\015o)m(w)g(of)38 b(\(2.1\))32 b(is)d(denoted)i(b)m(y)f Fq(U)1280 1901 y Fn(t)1310 1887 y Fr(.)40 b(By)31 b(the)g(theorem,)1294 2083 y(lim)1274 2139 y Fn(t)p Fk(!1)1455 2083 y Fq(U)1517 2097 y Fn(t)1567 2083 y Fm(\016)21 b(H)1711 2045 y Fp(+)1770 2083 y Fr(\()p Fq(a;)15 b(p)p Fr(\))21 b Fm(\000)f Fq(E)2158 2035 y Fp(\(0\))2153 2107 y Fn(t)2252 2083 y Fr(\()p Fq(a;)15 b(p)p Fr(\))27 b(=)e(0)-47 2321 y(for)30 b(all)f(\()p Fq(a;)15 b(p)p Fr(\))26 b Fm(2)f Fr(\006.)40 b(Moreo)m(v)m(er,)33 b Fm(H)1168 2288 y Fp(+)1257 2321 y Fr(in)m(tert)m(wines)d Fq(U)1785 2335 y Fn(t)1845 2321 y Fr(and)f Fq(S)2082 2273 y Fp(\(0\))2077 2345 y Fn(t)2177 2321 y Fr(,)h(i.e.,)1530 2518 y Fq(U)1592 2532 y Fn(t)1641 2518 y Fm(\016)21 b(H)1785 2481 y Fp(+)1869 2518 y Fr(=)k Fm(H)2043 2481 y Fp(+)2122 2518 y Fm(\016)c Fq(S)2249 2470 y Fp(\(0\))2244 2542 y Fn(t)2343 2518 y Fq(:)-47 2698 y Fr(In)27 b(this)h(resp)s(ect,)h Fm(H)653 2665 y Fp(+)740 2698 y Fr(enjo)m(ys)g(man)m(y)f(prop)s(erties) f(of)i(a)f(classical)g(w)m(a)m(v)m(e)i(op)s(erator.)41 b(Ho)m(w)m(ev)m(er,)31 b(it)d(is)f(in)g(general)i(not)-47 2811 y(a)f(canonical)g(transformation.)40 b(It)28 b(is)f(imp)s(ortan)m (t)h(to)h(realize)f(that)h(it)f(is)f(necessary)i(to)g(consider)e (iterations)h Fq(z)3761 2825 y Fn(n)3836 2811 y Fr(for)-47 2924 y(man)m(y)i Fq(n)p Fr(,)g(see)h(Lemma)g(3.8.)42 b(This)28 b(is)i(in)f(con)m(trast)i(to)h([1])f(and)f([2],)h(where)f (only)f(one)i(iteration)f(is)g(used.)-47 3207 y Fs(4)134 b(Solving)46 b(the)f(Hamilton-Jacobi)i(equation)-47 3409 y Fr(In)29 b(this)h(section)g(w)m(e)h(construct)g(solutions,)e(with)g (probabilit)m(y)f(one,)j(of)f(the)h(equation)1238 3623 y Fq(@)1286 3637 y Fn(t)1316 3623 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))21 b Fm(\000)1686 3561 y Fr(1)p 1686 3602 46 4 v 1686 3685 a(2)1742 3623 y Fm(j)p Fq(\030)t Fm(j)1836 3585 y Fp(2)1896 3623 y Fm(\000)f Fq(V)g Fr(\()p Fq(D)2170 3638 y Fn(\030)2208 3623 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))27 b(=)e(0)-2706 b(\(4.1\))-47 3839 y(on)36 b(a)g(subset)f(of)h(\(0)p Fq(;)15 b Fm(1)p Fr(\))25 b Fm(\002)f(f)p Fq(\030)39 b Fm(2)34 b Fo(R)1203 3806 y Fn(d)1308 3839 y Fr(:)59 b Fq(\030)38 b Fm(6)p Fr(=)c(0)p Fm(g)p Fr(.)58 b(As)36 b(in)f([10)q(],)i(the)g(metho)s(d)e (of)h(c)m(haracteristics)g(is)f(used.)57 b(This)-47 3952 y(requires)26 b(b)s(ounding)f(deriv)-5 b(ativ)m(es)27 b(of)h(the)g(c)m(haracteristics,)h(whic)m(h)e(are)h(precisely)e(the)i (tra)5 b(jectories)29 b(of)35 b(\(2.1\))r(,)28 b(with)-47 4065 y(resp)s(ect)35 b(to)h(the)f(initial)d(conditions.)54 b(Abusing)33 b(notation)j(sligh)m(tly)d(\(in)h(the)h(previous)f (section)h(this)f(notation)i(is)-47 4178 y(used)24 b(with)h(a)g(sligh)m (tly)f(di\013eren)m(t)h(meaning\),)h(w)m(e)g(let)g Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))p Fq(;)g(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\))29 b(denote)d(a)f (solution)f(of)33 b(\(2.1\))27 b(with)e(initial)-47 4291 y(data)32 b Fq(x)p Fr(\()p Fq(t)281 4305 y Fp(0)321 4291 y Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))28 b(=)g Fq(t)692 4305 y Fp(0)731 4291 y Fq(y)34 b Fr(and)d Fq(\030)t Fr(\()p Fq(t)1100 4305 y Fp(0)1140 4291 y Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))29 b(=)e Fq(\021)s Fr(.)45 b(F)-8 b(or)32 b(the)g(sak)m(e)h(of)e (simplicit)m(y)-8 b(,)30 b(w)m(e)j(shall)c(restrict)j(ourselv)m(es)f (to)i(the)-47 4404 y(case)i(1)c Fm(\024)g(j)p Fq(\021)s Fm(j)g(\024)g Fr(2.)51 b(Then)33 b(one)h(requires)f Fm(j)p Fq(y)25 b Fm(\000)d Fq(\021)s Fm(j)32 b Fq(<)f(c)1870 4418 y Fp(0)1943 4404 y Fr(for)j(some)g(small)e(constan)m(t)j Fq(c)2967 4418 y Fp(0)3007 4404 y Fr(,)g(sa)m(y)f Fq(c)3266 4418 y Fp(0)3337 4404 y Fr(=)3466 4368 y Fp(1)p 3449 4383 71 4 v 3449 4435 a(10)3529 4404 y Fr(.)51 b(W)-8 b(e)35 b(refer)-47 4517 y(to)30 b(suc)m(h)e Fq(y)s(;)15 b(\021)33 b Fr(as)d Fg(admissible)p Fr(.)41 b(The)29 b(extension)g(to)g(other)h(dy)m(adic)e(shells)g Fq(\032)d(<)g Fm(j)p Fq(\030)t Fm(j)h Fq(<)f Fr(2)p Fq(\032)k Fr(is)f(straigh)m (tforw)m(ard)h(and)g(is)-47 4630 y(analogous)i(to)h(the)f(previous)e (section.)42 b(The)31 b(reader)g(will)d(easily)i(v)m(erify)g(that)i (small)d Fq(\032)i Fr(can)g(b)s(e)f(comp)s(ensated)h(for)-47 4743 y(b)m(y)f(taking)g Fq(t)392 4757 y Fp(0)462 4743 y Fr(large.)1903 4992 y(22)p eop %%Page: 23 23 23 22 bop -47 274 a Fd(Lemma)32 b(4.1.)47 b Fg(L)-5 b(et)30 b Fq(")25 b(>)g Fr(0)30 b Fg(b)-5 b(e)30 b(su\016ciently)g(smal)5 b(l)31 b(dep)-5 b(ending)31 b(on)f Fq(\014)g(>)2462 238 y Fp(1)p 2462 253 36 4 v 2462 305 a(2)2507 274 y Fg(.)41 b(Then)30 b(ther)-5 b(e)31 b(exists)f(a)g(lar)-5 b(ge)31 b(time)f Fq(t)3810 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Fm(j)p Fq(\013)p Fm(j)26 b Fr(=)f(1)p Fq(:)-2968 b Fr(\(4.2\))-47 897 y Fg(Pr)-5 b(o)g(of.)47 b Fr(Fix)30 b(admissible)e Fq(y)s(;)15 b(\021)s Fr(.)43 b(F)-8 b(or)32 b(simplicit)m(y)-8 b(,)29 b(let)i Fq(x)p Fr(\()p Fq(t)p Fr(\))c(=)f Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))32 b(and)f Fq(\030)t Fr(\()p Fq(t)p Fr(\))26 b(=)g Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\).)44 b(Denote)33 b Fq(X)h Fr(=)26 b Fq(D)3851 911 y Fn(\021)3893 897 y Fq(x)-47 1010 y Fr(and)k Fq(Y)45 b Fr(=)25 b Fq(D)399 1024 y Fn(\021)441 1010 y Fq(\030)t Fr(.)41 b(Di\013eren)m(tiating)31 b(\(2.1\))h(with)d (resp)s(ect)h(to)i Fq(\021)h Fr(sho)m(ws)d(that)1375 1157 y(_)1339 1180 y Fq(X)j Fr(=)24 b Fq(Y)103 b(;)1831 1157 y Fr(_)1807 1180 y Fq(Y)20 b Fr(\()p Fq(t)p Fr(\))26 b(=)e Fm(\000)p Fq(D)2253 1142 y Fp(2)2292 1180 y Fq(V)d Fr(\()p Fq(x)p Fr(\()p Fq(t)p Fr(\)\))p Fq(X)-2713 b Fr(\(4.3\))1223 1318 y Fq(X)7 b Fr(\()p Fq(t)1373 1332 y Fp(0)1413 1318 y Fr(\))26 b(=)f(0)83 b Fq(;)h(Y)20 b Fr(\()p Fq(t)1948 1332 y Fp(0)1987 1318 y Fr(\))26 b(=)f Fq(I)-47 1487 y Fr(where)30 b Fq(I)37 b Fr(is)29 b(the)i Fq(d)21 b Fm(\002)e Fq(d)31 b Fr(iden)m(tit)m(y)-8 b(.)41 b(Our)29 b(goal)i(is)e(to)i(sho)m(w)g(that)f Fq(Y)51 b Fr(remains)29 b(in)g(a)i(small)e(neigh)m(b)s(orho)s(o)s(d)f(of)i Fq(I)7 b Fr(.)41 b(Let)1389 1696 y Fq(U)10 b Fr(\()p Fq(t)p Fr(\))26 b(=)e Fm(\000)1771 1572 y Fl(Z)1862 1598 y Fk(1)1822 1778 y Fn(t)1952 1696 y Fq(D)2030 1658 y Fp(2)2069 1696 y Fq(V)d Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(\)\))15 b Fq(ds:)-47 1944 y Fr(Then)30 b Fm(j)p Fq(U)10 b Fr(\()p Fq(t)p Fr(\))p Fm(j)27 b(\024)e Fq(C)d(t)659 1908 y Fk(\000)724 1881 y Fj(1)p 724 1893 31 3 v 724 1935 a(4)764 1908 y Fk(\000)829 1878 y Fj(3)p Fh(\014)p 829 1893 68 3 v 847 1935 a Fj(2)907 1908 y Fp(+)p Fn(")1029 1944 y Fr(for)30 b Fq(t)c Fm(\025)g Fq(t)1357 1958 y Fp(0)1427 1944 y Fr(up)k(to)h(probabilit)m(y)e(at)i(most)g Fq(e)2512 1911 y Fk(\000)p Fn(t)2592 1888 y Fh(")2592 1932 y Fj(0)2632 1944 y Fr(,)g(see)g(Lemma)g(3.3.)43 b(In)m(tegrating)32 b(out)-47 2057 y(\(4.3\))g(yields)496 2283 y Fq(Y)20 b Fr(\()p Fq(t)p Fr(\))h Fm(\000)f Fq(Y)g Fr(\()p Fq(t)925 2297 y Fp(0)965 2283 y Fr(\))83 b(=)g Fm(\000)1323 2159 y Fl(Z)1413 2186 y Fn(t)1373 2366 y(t)1398 2375 y Fj(0)1458 2283 y Fq(D)1536 2246 y Fp(2)1576 2283 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(\)\))p Fq(X)7 b Fr(\()p Fq(s)p Fr(\))15 b Fq(ds)27 b Fr(=)2307 2159 y Fl(Z)2398 2186 y Fn(t)2357 2366 y(t)2382 2375 y Fj(0)2442 2283 y Fq(U)10 b Fr(\()p Fq(s)p Fr(\))2664 2260 y(_)2627 2283 y Fq(X)e Fr(\()p Fq(s)p Fr(\))15 b Fq(ds)21 b Fm(\000)f Fq(U)10 b Fr(\()p Fq(t)p Fr(\))p Fq(X)d Fr(\()p Fq(t)p Fr(\))1083 2546 y(=)1237 2423 y Fl(Z)1328 2449 y Fn(t)1287 2629 y(t)1312 2638 y Fj(0)1372 2546 y Fq(U)j Fr(\()p Fq(s)p Fr(\))p Fq(Y)21 b Fr(\()p Fq(s)p Fr(\))15 b Fq(ds)20 b Fm(\000)g Fq(U)10 b Fr(\()p Fq(t)p Fr(\))2150 2423 y Fl(Z)2242 2449 y Fn(t)2201 2629 y(t)2226 2638 y Fj(0)2287 2546 y Fq(Y)20 b Fr(\()p Fq(s)p Fr(\))15 b Fq(ds:)-47 2766 y Fr(This)28 b(clearly)i(implies)e(that)1174 2987 y Fm(k)p Fq(Y)21 b Fr(\()p Fq(t)p Fr(\))g Fm(\000)f Fq(I)7 b Fm(k)25 b(\024)g Fq(C)1823 2863 y Fl(Z)1914 2889 y Fn(t)1874 3069 y(t)1899 3078 y Fj(0)1959 2987 y Fq(s)2002 2949 y Fk(\000)2067 2922 y Fj(1)p 2066 2934 31 3 v 2066 2975 a(4)2107 2949 y Fk(\000)2172 2919 y Fj(3)p Fh(\014)p 2172 2934 68 3 v 2190 2975 a Fj(2)2249 2949 y Fp(+)p Fn(")2341 2987 y Fm(k)p Fq(Y)20 b Fr(\()p Fq(s)p Fr(\))p Fm(k)15 b Fq(ds)-47 3207 y Fr(so)30 b(that)1097 3394 y(max)1058 3455 y Fn(t)1083 3464 y Fj(0)1118 3455 y Fk(\024)p Fn(t)p Fk(\024)p Fn(T)1319 3394 y Fm(k)p Fq(Y)21 b Fr(\()p Fq(t)p Fr(\))g Fm(\000)f Fq(I)7 b Fm(k)25 b(\024)g Fq(C)d(t)1996 3310 y Fj(3)p 1996 3322 31 3 v 1996 3363 a(4)2036 3337 y Fk(\000)2101 3306 y Fj(3)p Fh(\014)p 2101 3322 68 3 v 2120 3363 a Fj(2)2179 3337 y Fp(+)p Fn(")1986 3420 y Fp(0)2324 3394 y Fr(max)2285 3455 y Fn(t)2310 3464 y Fj(0)2346 3455 y Fk(\024)p Fn(t)p Fk(\024)p Fn(T)2547 3394 y Fm(k)p Fq(Y)e Fr(\()p Fq(t)p Fr(\))p Fm(k)p Fq(:)-2885 b Fr(\(4.4\))-47 3602 y(Hence)1627 3772 y(max)1588 3832 y Fn(t)1613 3841 y Fj(0)1649 3832 y Fk(\024)p Fn(t)p Fk(\024)p Fn(T)1850 3772 y Fm(k)p Fq(Y)20 b Fr(\()p Fq(t)p Fr(\))p Fm(k)27 b(\024)d Fr(2)p Fq(;)-47 3986 y Fr(sa)m(y)-8 b(,)28 b(pro)m(vided)d Fq(t)527 4000 y Fp(0)592 3986 y Fr(w)m(as)i(c)m(hosen)f(large)g(enough.)39 b(Reinserting)25 b(this)g(b)s(ound)f(in)m(to)i(\(4.4\))i(and)d(letting)h Fq(T)38 b Fm(!)25 b(1)h Fr(sho)m(ws)-47 4099 y(that)1385 4286 y(sup)1383 4363 y Fn(t)p Fk(\025)p Fn(t)1488 4372 y Fj(0)1539 4286 y Fm(k)p Fq(Y)20 b Fr(\()p Fq(t)p Fr(\))h Fm(\000)f Fq(I)7 b Fm(k)25 b(\024)g Fq(C)d(t)2215 4202 y Fj(3)p 2215 4214 31 3 v 2215 4255 a(4)2255 4229 y Fk(\000)2320 4198 y Fj(3)p Fh(\014)p 2320 4214 68 3 v 2339 4255 a Fj(2)2398 4229 y Fp(+)p Fn(")2205 4312 y Fp(0)2489 4286 y Fq(;)-47 4517 y Fr(as)35 b(desired.)53 b(No)m(w)36 b(let)f Fq(X)41 b Fr(=)33 b Fq(D)1070 4531 y Fn(y)1111 4517 y Fq(x)i Fr(and)g Fq(Y)53 b Fr(=)33 b Fq(D)1665 4531 y Fn(y)1707 4517 y Fq(\030)t Fr(.)54 b(These)35 b(quan)m(tities)g(satisfy)f(the)i(system)f(\(4.3\))i(with)d(initial)-47 4630 y(conditions)f Fq(X)7 b Fr(\()p Fq(t)542 4644 y Fp(0)582 4630 y Fr(\))34 b(=)f Fq(t)788 4644 y Fp(0)842 4630 y Fq(I)42 b Fr(and)35 b Fq(Y)20 b Fr(\()p Fq(t)1247 4644 y Fp(0)1287 4630 y Fr(\))33 b(=)g(0.)56 b(Virtually)32 b(the)k(same)f(argumen)m(t)h(as)f(ab)s(o)m(v)m(e)i(no)m(w)e(implies)d (that)k Fq(Y)-47 4743 y Fr(satis\014es)30 b(\(4.2\))q(,)h(as)g (claimed.)p 3879 4743 4 62 v 3883 4685 55 4 v 3883 4743 V 3937 4743 4 62 v 1903 4992 a(23)p eop %%Page: 24 24 24 23 bop -47 274 a Fr(The)34 b(follo)m(wing)g(corollary)h(is)f (basically)f(an)j(in)m(tegrated)f(v)m(ersion)g(of)g(the)h(previous)e (lemma.)54 b(It)36 b(will)c(b)s(e)j(used)f(in)-47 387 y(the)f(construction)h(of)f(solutions)f(of)41 b(\(4.1\))r(.)50 b(The)33 b(exact)i(form)e(of)h(the)f(exp)s(onen)m(t)h(in)e(\(4.5\))j (is)e(not)h(relev)-5 b(an)m(t,)35 b(only)-47 500 y(that)c(it)f(can)g(b) s(e)g(made)g(strictly)g(less)g(than)g(one.)-47 680 y Fd(Corollary)35 b(4.2.)46 b Fg(Under)33 b(the)g(same)g(hyp)-5 b(otheses)35 b(as)e(in)g(the)g(pr)-5 b(evious)33 b(lemma)h(one)f(has)h (the)f(b)-5 b(ound)926 787 y Fl(\014)926 842 y(\014)926 897 y(\014)957 892 y Fq(D)1035 854 y Fn(\013)1032 915 y(y)r(;\021)1131 892 y Fr(\()p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))22 b Fm(\000)e Fq(y)j Fm(\000)d Fr(\()p Fq(t)g Fm(\000)g Fq(t)1981 906 y Fp(0)2020 892 y Fr(\))p Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\)\))2413 787 y Fl(\014)2413 842 y(\014)2413 897 y(\014)2471 892 y Fm(\024)25 b Fq(C)d(t)2697 827 y Fj(7)p 2697 839 31 3 v 2697 881 a(4)2737 854 y Fk(\000)2802 824 y Fj(3)p Fh(\014)p 2802 839 68 3 v 2820 881 a Fj(2)2880 854 y Fp(+)p Fn(")-47 892 y Fr(\(4.5\))-47 1110 y Fg(for)33 b(any)g Fm(j)p Fq(\013)p Fm(j)26 b Fr(=)f(1)33 b Fg(and)h(al)5 b(l)33 b(times)g Fq(t)25 b Fm(\025)g Fq(t)1318 1124 y Fp(0)1357 1110 y Fg(.)-47 1289 y(Pr)-5 b(o)g(of.)47 b Fr(Fix)29 b(some)i Fq(\013)g Fr(of)g(length)e(one.)41 b(As)31 b(ab)s(o)m(v)m(e,)h(let)1288 1523 y Fq(U)10 b Fr(\()p Fq(t)p Fr(\))25 b(:=)h Fm(\000)1696 1400 y Fl(Z)1786 1426 y Fk(1)1746 1606 y Fn(t)1876 1523 y Fq(D)1954 1486 y Fp(2)1993 1523 y Fq(V)21 b Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\)\))g Fq(ds:)-47 1764 y Fr(Using)31 b(\(2.1\))h(and)e(in)m(tegrating)g(b)m(y)g(parts)g(one)h (obtains)497 2015 y Fq(D)575 1978 y Fn(\013)572 2038 y(y)r(;\021)671 2015 y Fr(\()p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))22 b Fm(\000)e Fq(y)j Fm(\000)d Fr(\()p Fq(t)g Fm(\000)g Fq(t)1521 2029 y Fp(0)1560 2015 y Fr(\))p Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\)\))28 b(=)d Fq(D)2155 1978 y Fn(\013)2152 2038 y(y)r(;\021)2266 1892 y Fl(Z)2357 1918 y Fn(t)2316 2098 y(t)2341 2107 y Fj(0)2386 1914 y Fl(h)2429 2015 y Fq(\030)t Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))22 b Fm(\000)e Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))3198 1914 y Fl(i)3257 2015 y Fq(ds)343 2279 y Fr(=)83 b Fq(D)575 2241 y Fn(\013)572 2301 y(y)r(;\021)686 2155 y Fl(Z)777 2181 y Fn(t)736 2361 y(t)761 2370 y Fj(0)806 2279 y Fr(\()p Fq(s)21 b Fm(\000)e Fq(t)1028 2293 y Fp(0)1068 2279 y Fr(\))c Fm(r)p Fq(V)20 b Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\)\))g Fq(ds)28 b Fr(=)1907 2155 y Fl(Z)1997 2181 y Fn(t)1957 2361 y(t)1982 2370 y Fj(0)2027 2279 y Fr(\()p Fq(s)20 b Fm(\000)g Fq(t)2249 2293 y Fp(0)2288 2279 y Fr(\))15 b Fq(D)2416 2241 y Fp(2)2456 2279 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\)\))p Fq(D)3018 2241 y Fn(\013)3015 2301 y(y)r(;\021)3116 2279 y Fq(x)p Fr(\()p Fq(s)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\))g Fq(ds)343 2542 y Fr(=)83 b Fm(\000)583 2418 y Fl(Z)673 2444 y Fn(t)633 2624 y(t)658 2633 y Fj(0)718 2542 y Fq(U)10 b Fr(\()p Fq(s)p Fr(\))903 2441 y Fl(h)946 2542 y Fq(D)1024 2504 y Fn(\013)1021 2564 y(y)r(;\021)1120 2542 y Fq(x)p Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))22 b(+)e(\()p Fq(s)g Fm(\000)g Fq(t)1796 2556 y Fp(0)1835 2542 y Fr(\))p Fq(D)1948 2504 y Fn(\013)1945 2564 y(y)r(;\021)2045 2542 y Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))2368 2441 y Fl(i)2427 2542 y Fq(ds)20 b Fr(+)g(\()p Fq(t)h Fm(\000)f Fq(t)2841 2556 y Fp(0)2880 2542 y Fr(\))p Fq(U)10 b Fr(\()p Fq(t)p Fr(\))15 b Fq(D)3183 2504 y Fn(\013)3180 2564 y(y)r(;\021)3280 2542 y Fq(x)p Fr(\()p Fq(t)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\))p Fq(:)-47 2786 y Fr(Since)1074 3014 y Fq(D)1152 2977 y Fn(\013)1149 3037 y(y)r(;\021)1248 3014 y Fq(x)p Fr(\()p Fq(t)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\))26 b(=)f Fq(D)1779 2977 y Fn(\013)1776 3037 y(y)r(;\021)1891 3014 y Fq(y)e Fr(+)2050 2890 y Fl(Z)2141 2917 y Fn(t)2100 3097 y(t)2125 3106 y Fj(0)2185 3014 y Fq(D)2263 2977 y Fn(\013)2260 3037 y(y)r(;\021)2359 3014 y Fq(\030)t Fr(\()p Fq(s)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))g Fq(ds;)-47 3259 y Fr(Lemma)30 b(4.1)i(implies)27 b(the)k(b)s(ound)d(for)i(all)g Fq(t)25 b Fm(\025)g Fq(t)1622 3273 y Fp(0)1558 3454 y Fm(j)p Fq(D)1661 3417 y Fn(\013)1658 3477 y(y)r(;\021)1757 3454 y Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))p Fm(j)27 b(\024)e Fq(C)7 b(t)-47 3658 y Fr(up)28 b(to)j(probabilit)m(y)c(at)j(most)g Fq(e)1032 3625 y Fk(\000)p Fn(t)1112 3601 y Fh(")1112 3645 y Fj(0)1181 3658 y Fr(and)f(pro)m(vided)f Fq(t)1763 3672 y Fp(0)1832 3658 y Fr(is)h(large.)41 b(By)29 b(Lemma)h(3.3)h(and)e(Lemma)h(4.1)g (therefore,)h(up)-47 3770 y(to)g(probabilit)m(y)d(at)j(most)g Fq(e)905 3737 y Fk(\000)p Fn(t)985 3714 y Fh(")985 3758 y Fj(0)1024 3770 y Fr(,)227 3912 y Fl(\014)227 3967 y(\014)227 4021 y(\014)258 4017 y Fq(D)336 3979 y Fn(\013)333 4039 y(y)r(;\021)432 4017 y Fr(\()p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))22 b Fm(\000)e Fq(y)j Fm(\000)d Fr(\()p Fq(t)g Fm(\000)g Fq(t)1282 4031 y Fp(0)1321 4017 y Fr(\))p Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\)\))1714 3912 y Fl(\014)1714 3967 y(\014)1714 4021 y(\014)1772 4017 y Fm(\024)25 b Fq(C)1954 3893 y Fl(Z)2045 3919 y Fn(t)2005 4099 y(t)2030 4108 y Fj(0)2090 4017 y Fq(s)2133 3979 y Fk(\000)2198 3952 y Fj(1)p 2198 3964 31 3 v 2198 4005 a(4)2238 3979 y Fk(\000)2303 3948 y Fj(3)p Fh(\014)p 2303 3964 68 3 v 2322 4005 a Fj(2)2381 3979 y Fp(+)p Fn(")2487 4017 y Fq(s)15 b(ds)20 b Fr(+)g Fq(C)7 b(t)2861 3952 y Fj(7)p 2860 3964 31 3 v 2860 4005 a(4)2901 3979 y Fk(\000)2966 3948 y Fj(3)p Fh(\014)p 2966 3964 68 3 v 2984 4005 a Fj(2)3043 3979 y Fp(+)p Fn(")3160 4017 y Fm(\024)25 b Fq(C)7 b(t)3371 3952 y Fj(7)p 3371 3964 31 3 v 3371 4005 a(4)3411 3979 y Fk(\000)3476 3948 y Fj(3)p Fh(\014)p 3476 3964 68 3 v 3494 4005 a Fj(2)3554 3979 y Fp(+)p Fn(")3645 4017 y Fq(;)-47 4262 y Fr(as)30 b(claimed.)p 3879 4262 4 62 v 3883 4204 55 4 v 3883 4262 V 3937 4262 4 62 v -47 4448 a(In)f(the)i(follo)m(wing)e (lemma)h(w)m(e)h(estimate)g(deriv)-5 b(ativ)m(es)29 b(of)i(higher)e (order.)40 b(Let)1227 4680 y Fq(\026)p Fr(\()p Fq(j)5 b Fr(\))26 b(=)f(\(1)c Fm(\000)f Fq(\014)5 b Fr(\))p Fq(j)27 b Fm(\000)1963 4619 y Fr(1)p 1963 4659 46 4 v 1963 4743 a(4)2039 4680 y Fm(\000)2140 4619 y Fq(\014)p 2140 4659 57 4 v 2145 4743 a Fr(2)2267 4680 y(for)60 b Fq(j)31 b Fm(\025)25 b Fr(2)p Fq(:)-2717 b Fr(\(4.6\))1903 4992 y(24)p eop %%Page: 25 25 25 24 bop -47 274 a Fd(Lemma)32 b(4.3.)47 b Fg(L)-5 b(et)44 b Fq(")k Fr(=)f Fq(")p Fr(\()p Fq(\014)5 b Fr(\))48 b Fq(>)f Fr(0)e Fg(b)-5 b(e)44 b(su\016ciently)h(smal)5 b(l)46 b(dep)-5 b(ending)45 b(on)g Fq(\014)5 b Fg(.)78 b(Then)45 b(ther)-5 b(e)46 b(exists)f(a)g(lar)-5 b(ge)-47 387 y(time)43 b Fq(t)203 401 y Fp(0)287 387 y Fr(=)h Fq(t)435 401 y Fp(0)474 387 y Fr(\()p Fq(")p Fr(\))g Fg(so)g(that)g(up)g(to)g(pr)-5 b(ob)g(ability)45 b(at)f(most)g Fq(e)2053 354 y Fk(\000)p Fn(t)2133 330 y Fh(")2133 375 y Fj(0)2216 387 y Fg(the)f(fol)5 b(lowing)44 b(pr)-5 b(op)g(erty)46 b(holds:)65 b(A)n(ny)43 b(solution)-47 500 y Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))p Fq(;)g(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\))36 b Fg(satis\014es)d(the)g(estimates)1151 597 y Fl(\014)1151 651 y(\014)1151 706 y(\014)1181 701 y Fq(D)1259 664 y Fn(\013)1256 724 y(y)r(;\021)1356 701 y Fr(\()p Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))p Fq(;)g(x)p Fr(\()p Fq(t)p Fr(;)g Fq(y)s(;)g(\021)s Fr(\))p Fq(=t)p Fr(\))2198 597 y Fl(\014)2198 651 y(\014)2198 706 y(\014)2258 701 y Fm(\024)25 b Fq(C)c(t)2473 664 y Fn(\026)p Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(\)+)p Fn(")-47 701 y Fr(\(4.7\))-47 903 y Fg(for)33 b(any)g(multi-index)g Fm(j)p Fq(\013)p Fm(j)26 b(\025)f Fr(2)33 b Fg(and)h Fq(t)25 b Fm(\025)g Fq(t)1423 917 y Fp(0)1462 903 y Fg(.)42 b(The)33 b(c)-5 b(onstant)34 b Fq(C)39 b Fg(dep)-5 b(ends)34 b(on)f Fq(\014)5 b(;)15 b(")p Fg(.)-47 1069 y(Pr)-5 b(o)g(of.)47 b Fr(First)29 b(consider)h(the)g(case)i Fm(j)p Fq(\013)p Fm(j)26 b Fr(=)f(2)31 b(and)e(deriv)-5 b(ativ)m(es)30 b(in)f Fq(\021)s Fr(.)41 b(Then)30 b(one)g(has)g(the)h(system)813 1242 y Fq(@)866 1209 y Fp(2)921 1242 y Fr(_)-41 b Fq(x)p 747 1282 277 4 v 747 1366 a(@)5 b(\021)845 1380 y Fn(j)882 1366 y Fq(@)g(\021)980 1381 y Fn(k)1116 1303 y Fr(=)1349 1242 y Fq(@)1402 1209 y Fp(2)1442 1242 y Fq(\030)p 1280 1282 V 1280 1366 a(@)g(\021)1378 1380 y Fn(j)1415 1366 y Fq(@)g(\021)1513 1381 y Fn(k)817 1533 y Fq(@)870 1500 y Fp(2)929 1509 y Fr(_)909 1533 y Fq(\030)p 747 1573 V 747 1657 a(@)g(\021)845 1671 y Fn(j)882 1657 y Fq(@)g(\021)980 1672 y Fn(k)1116 1594 y Fr(=)83 b Fm(\000)1427 1481 y Fn(d)1379 1508 y Fl(X)1356 1702 y Fn(p;q)r Fp(=1)1560 1533 y Fm(r)p Fq(@)1689 1500 y Fp(2)1728 1533 y Fq(V)21 b Fr(\()p Fq(x)p Fr(\))p 1560 1573 365 4 v 1598 1657 a Fq(@)5 b(x)1703 1671 y Fn(p)1743 1657 y Fq(@)g(x)1848 1671 y Fn(q)1944 1532 y Fq(@)g(x)2049 1546 y Fn(p)p 1944 1573 145 4 v 1949 1657 a Fq(@)g(\021)2047 1671 y Fn(j)2109 1532 y Fq(@)g(x)2214 1546 y Fn(q)p 2109 1573 144 4 v 2110 1657 a Fq(@)g(\021)2208 1672 y Fn(k)2282 1594 y Fm(\000)2421 1481 y Fn(d)2373 1508 y Fl(X)2376 1702 y Fn(p)p Fp(=1)2530 1533 y Fm(r)p Fq(@)g(V)20 b Fr(\()p Fq(x)p Fr(\))p 2530 1573 325 4 v 2620 1657 a Fq(@)5 b(x)2725 1671 y Fn(p)2921 1532 y Fq(@)2974 1499 y Fp(2)3013 1532 y Fq(x)3065 1546 y Fn(p)p 2875 1573 277 4 v 2875 1657 a Fq(@)g(\021)2973 1671 y Fn(j)3010 1657 y Fq(@)g(\021)3108 1672 y Fn(k)-47 1594 y Fr(\(4.8\))-47 1861 y(with)29 b(initial)e(conditions)i(at)i(time)f Fq(t)1208 1875 y Fp(0)1278 1861 y Fr(b)s(oth)g(equal)g(to)h(zero.)41 b(Let)1276 2091 y Fq(U)1338 2105 y Fn(pq)1412 2091 y Fr(\()p Fq(t)p Fr(\))83 b(=)g Fm(\000)1838 1967 y Fl(Z)1929 1993 y Fk(1)1888 2173 y Fn(t)2029 2029 y Fm(r)p Fq(@)2158 1996 y Fp(2)2197 2029 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(\)\))p 2029 2070 478 4 v 2123 2153 a Fq(@)5 b(x)2228 2167 y Fn(p)2268 2153 y Fq(@)g(x)2373 2167 y Fn(q)2532 2091 y Fq(ds)1286 2342 y(W)1372 2356 y Fn(p)1412 2342 y Fr(\()p Fq(t)p Fr(\))83 b(=)g Fm(\000)1838 2219 y Fl(Z)1929 2245 y Fk(1)1888 2425 y Fn(t)2029 2281 y Fm(r)p Fq(@)5 b(V)20 b Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(\)\))p 2029 2321 439 4 v 2175 2405 a Fq(@)5 b(x)2280 2419 y 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y(in)c(the)h(resulting)f(expression)g(leads)h(to)h(\(with)e Fq(D)1684 3068 y Fn(j)1746 3054 y Fr(=)1886 3018 y Fn(@)p 1852 3033 110 4 v 1852 3085 a(@)t(\021)1928 3095 y Fh(j)2002 3054 y Fr(etc.)i(and)f(using)f(the)i(summation)e(con)m(v)m(en)m(tion\)) 609 3317 y Fq(D)687 3280 y Fp(2)684 3341 y Fn(j)t(k)759 3317 y Fq(\030)t Fr(\()p Fq(t)p Fr(\))84 b(=)f(2)1204 3194 y Fl(Z)1295 3220 y Fn(t)1255 3400 y(t)1280 3409 y Fj(0)1340 3317 y Fq(U)1402 3331 y Fn(pq)1476 3317 y Fr(\()p Fq(s)p Fr(\))p Fq(D)1664 3331 y Fn(j)1701 3317 y Fq(\030)1741 3331 y Fn(p)1780 3317 y Fr(\()p Fq(s)p Fr(\))p Fq(D)1968 3332 y Fn(k)2011 3317 y Fq(x)2063 3331 y Fn(q)2101 3317 y Fr(\()p Fq(s)p Fr(\))15 b Fq(ds)21 b Fm(\000)f Fq(U)2493 3331 y Fn(pq)2567 3317 y Fr(\()p Fq(t)p Fr(\))p Fq(D)2745 3331 y Fn(j)2782 3317 y Fq(x)2834 3331 y Fn(p)2874 3317 y Fr(\()p Fq(t)p Fr(\))p Fq(D)3052 3332 y Fn(k)3095 3317 y Fq(x)3147 3331 y Fn(q)3185 3317 y Fr(\()p Fq(t)p Fr(\))1144 3581 y(+)1230 3457 y Fl(Z)1320 3483 y Fn(t)1280 3663 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Fk(\000)2610 4365 y Fj(3)p Fh(\014)p 2610 4381 68 3 v 2629 4422 a Fj(2)2688 4396 y Fp(+)p Fn(")2795 4310 y Fl(Z)2886 4336 y Fn(t)2845 4516 y(t)2870 4525 y Fj(0)2930 4433 y Fm(j)p Fq(D)3033 4396 y Fp(2)3030 4456 y Fn(\021)3073 4433 y Fq(\030)t Fr(\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds)935 4697 y Fm(\024)83 b Fq(C)22 b(t)1219 4632 y Fj(7)p 1219 4644 31 3 v 1219 4685 a(4)1259 4659 y Fk(\000)1324 4628 y Fj(5)p Fh(\014)p 1324 4644 68 3 v 1342 4685 a Fj(2)1402 4659 y Fp(+)p Fn(")1513 4697 y Fr(+)e Fq(C)1691 4573 y Fl(Z)1782 4599 y Fn(t)1741 4779 y(t)1766 4788 y Fj(0)1827 4697 y Fq(s)1870 4659 y Fk(\000)1935 4632 y Fj(1)p 1934 4644 31 3 v 1934 4685 a(4)1974 4659 y Fk(\000)2039 4628 y Fj(3)p Fh(\014)p 2039 4644 68 3 v 2058 4685 a Fj(2)2117 4659 y Fp(+)p Fn(")2209 4697 y Fm(j)p Fq(D)2312 4659 y Fp(2)2309 4719 y Fn(\021)2351 4697 y Fq(\030)t Fr(\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds:)-2710 b Fr(\(4.11\))1903 4992 y(25)p eop %%Page: 26 26 26 25 bop -47 274 a Fr(Gron)m(w)m(all's)30 b(inequalit)m(y)e(therefore) j(implies)858 509 y Fm(j)p Fq(D)961 472 y Fp(2)958 532 y Fn(\021)1000 509 y Fq(\030)t Fr(\()p Fq(t)p Fr(\))p Fm(j)26 b(\024)f Fq(C)d(t)1424 445 y Fj(7)p 1423 457 31 3 v 1423 498 a(4)1464 472 y Fk(\000)1529 441 y Fj(5)p Fh(\014)p 1529 457 68 3 v 1547 498 a Fj(2)1606 472 y Fp(+)p Fn(")1718 509 y Fr(+)e Fq(C)1896 386 y Fl(Z)1987 412 y Fn(t)1946 592 y(t)1971 601 y Fj(0)2031 509 y Fq(s)2084 445 y Fj(3)p 2084 457 31 3 v 2084 498 a(2)2124 472 y Fk(\000)p Fp(4)p Fn(\014)s Fp(+2)p Fn(")2400 509 y Fq(ds)25 b Fm(\024)g Fq(C)c(t)2740 445 y Fj(7)p 2740 457 V 2740 498 a(4)2781 472 y Fk(\000)2846 441 y Fj(5)p Fh(\014)p 2845 457 68 3 v 2864 498 a Fj(2)2923 472 y Fp(+)p Fn(")3015 509 y Fq(;)-47 749 y Fr(as)34 b(claimed.)50 b(The)33 b(same)h(argumen)m(t)h(also)e(applies)f(to)j(deriv)-5 b(ativ)m(es)33 b(in)g Fq(y)s Fr(,)h(since)g(\(4.2\))h(con)m(tin)m(ues)f (to)h(hold)d(for)i Fq(y)s Fr(.)-47 862 y(The)26 b(only)g(mo)s (di\014cation)g(that)h(needs)g(to)g(b)s(e)g(made)g(arises)f(from)2224 826 y Fn(@)t(x)p 2224 841 81 4 v 2225 893 a(@)t(y)2340 862 y Fr(=)f Fq(t)2469 876 y Fp(0)2508 862 y Fq(I)7 b Fr(,)28 b(cf.)f(\(4.10\))i(and)e(\(4.11\))r(.)40 b(But)27 b(this)f(can)-47 990 y(easily)32 b(b)s(e)h(seen)h(to)g(b)s(e)f (insigni\014can)m(t.)47 b(Finally)-8 b(,)33 b(the)h(b)s(ound)d(on)j Fq(D)2309 957 y Fp(2)2306 1012 y Fn(y)r(;\021)2405 990 y Fr(\()p Fq(x=t)p Fr(\))g(follo)m(ws)e(b)m(y)i(in)m(tegrating)f(the)h (b)s(ound)-47 1103 y(w)m(e)d(just)e(obtained)h(for)g Fq(\030)35 b Fr(in)29 b(time.)-47 1216 y(F)-8 b(or)39 b(larger)f Fq(\013)h Fr(one)f(uses)g(induction.)63 b(Therefore,)40 b(let)e Fq(\013)h Fr(b)s(e)f(a)h(m)m(ulti-index)c(of)k(length)f Fm(j)p Fq(\013)p Fm(j)h(\025)g Fr(3.)65 b(T)-8 b(aking)38 b Fq(D)3896 1183 y Fn(\013)3893 1238 y(\021)-47 1329 y Fr(deriv)-5 b(ativ)m(es)30 b(of)37 b(\(2.1\))32 b(leads)e(to)h(an)f (expression)g(of)g(the)h(form)893 1600 y Fq(D)971 1562 y Fn(\013)968 1622 y(\021)1040 1576 y Fr(_)1021 1600 y Fq(\030)e Fr(=)1186 1513 y Fl(X)1332 1600 y Fr(co)s(e\013)1601 1538 y Fm(r)p Fq(@)1730 1505 y Fn(m)1792 1514 y Fj(1)1827 1505 y Fp(+)p Fn(:::)n Fp(+)p Fn(m)2057 1517 y Fh(`)2092 1538 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p 1549 1579 790 4 v 1549 1662 a Fq(@)1602 1636 y Fn(m)1664 1645 y Fj(1)1704 1662 y Fq(x)1756 1676 y Fn(p)1792 1685 y Fj(1)1850 1662 y Fm(\001)h Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(@)2120 1636 y Fn(m)2182 1648 y Fh(`)2217 1662 y Fq(x)2269 1676 y Fn(p)2305 1688 y Fh(`)2408 1486 y Fn(`)2364 1513 y Fl(Y)2365 1709 y Fn(i)p Fp(=1)2515 1484 y Fn(m)2577 1494 y Fh(i)2502 1513 y Fl(Y)2495 1711 y Fn(k)r Fp(=1)2639 1600 y Fq(D)2714 1641 y Fn(\021)2751 1606 y Fj(\()p Fh(i)p Fj(\))2749 1665 y Fh(k)2846 1600 y Fq(x)2898 1614 y Fn(p)2934 1624 y Fh(i)2964 1600 y Fq(;)-3036 b Fr(\(4.12\))-47 1863 y(where)30 b(co)s(e\013)h(stands)f(for)g(suitable)f(co)s (e\016cien)m(ts)i(and)e(the)i(sum)e(extends)i(o)m(v)m(er)g(all)f(terms) g(so)h(that)1312 1997 y Fn(`)1260 2025 y Fl(X)1269 2220 y Fn(i)p Fp(=1)1428 1995 y Fn(m)1490 2005 y Fh(i)1407 2025 y Fl(X)1408 2222 y Fn(k)r Fp(=1)1553 2111 y Fm(j)p Fq(\021)1626 2063 y Fp(\()p Fn(i)p Fp(\))1623 2140 y Fn(k)1710 2111 y Fm(j)25 b Fr(=)g Fm(j)p Fq(\013)p Fm(j)62 b Fr(with)d Fm(j)p Fq(\021)2336 2063 y Fp(\()p Fn(i)p Fp(\))2333 2140 y Fn(k)2420 2111 y Fm(j)26 b(\025)f Fr(1)p Fq(:)-2684 b Fr(\(4.13\))-47 2374 y(There)30 b(is)f(exactly)i(one)g (term)f(with)f Fq(`)c Fr(=)g(1,)31 b(namely)1621 2526 y Fn(d)1573 2553 y Fl(X)1576 2748 y Fn(p)p Fp(=1)1730 2578 y Fm(r)p Fq(@)5 b(V)20 b Fr(\()p Fq(x)p Fr(\))p 1730 2619 325 4 v 1820 2702 a Fq(@)5 b(x)1925 2716 y Fn(p)2065 2640 y Fq(D)2143 2602 y Fn(\013)2140 2662 y(\021)2192 2640 y Fq(x)2244 2654 y Fn(p)2284 2640 y Fq(;)-2356 b Fr(\(4.14\))-47 2917 y(cf.)51 b(the)35 b(second)f(term)g(in)e(\(4.8\))r (.)52 b(The)33 b(other)h(terms)g(only)f(con)m(tain)h(deriv)-5 b(ativ)m(es)34 b(of)g Fq(x)g Fr(of)g(orders)f Fq(<)e Fm(j)p Fq(\013)p Fm(j)k Fr(and)e(are)-47 3030 y(therefore)k(estimated)h (b)m(y)f(means)g(of)g(the)h(induction)d(h)m(yp)s(othesis.)59 b(More)38 b(precisely)-8 b(,)38 b(consider)e(a)i(t)m(ypical)f(term)-47 3143 y(with)29 b Fq(`)c Fm(\025)g Fr(2)31 b(as)f(on)h(the)f(righ)m (t-hand)f(side)h(of)37 b(\(4.12\))c(and)d(let)1192 3380 y Fq(U)10 b Fr(\()p Fq(t)p Fr(\))26 b(=)f Fm(\000)1575 3256 y Fl(Z)1665 3282 y Fk(1)1625 3462 y Fn(t)1765 3318 y Fm(r)p Fq(@)1894 3285 y Fn(m)1956 3294 y Fj(1)1991 3285 y Fp(+)p Fn(:::)o Fp(+)p Fn(m)2222 3297 y Fh(`)2256 3318 y Fq(V)c Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(\)\))p 1765 3359 801 4 v 1770 3442 a Fq(@)1823 3416 y Fn(m)1885 3425 y Fj(1)1925 3442 y Fq(x)1977 3456 y Fn(p)2013 3465 y Fj(1)2071 3442 y Fm(\001)g Fq(:)15 b(:)g(:)21 b Fm(\001)g Fq(@)2342 3416 y Fn(m)2404 3428 y Fh(`)2439 3442 y Fq(x)2491 3456 y Fn(p)2527 3468 y Fh(`)2591 3380 y Fq(ds:)-47 3611 y Fr(By)30 b(Lemma)h(3.3)g(one)g(has)1458 3805 y Fm(j)p Fq(U)10 b Fr(\()p Fq(t)p Fr(\))p Fm(j)26 b(\024)f Fq(C)7 b(t)1910 3768 y Fk(\000)1975 3741 y Fj(1)p 1975 3753 31 3 v 1975 3794 a(4)2015 3768 y Fk(\000)p Fp(\()p Fn(M)g Fp(+)2237 3741 y Fj(1)p 2237 3753 V 2237 3794 a(2)2277 3768 y Fp(\))p Fn(\014)s Fp(+)p Fn(")-47 3989 y Fr(up)34 b(to)j(probabilit)m(y)c(at)j(most)g Fq(e)1062 3956 y Fk(\000)p Fn(t)1142 3933 y Fh(")1142 3977 y Fj(0)1181 3989 y Fr(.)57 b(Here)36 b(w)m(e)g(ha)m(v)m(e)h(set)f Fq(M)44 b Fr(=)2221 3921 y Fl(P)2317 3947 y Fn(`)2317 4016 y(i)p Fp(=1)2450 3989 y Fq(m)2530 4003 y Fn(i)2558 3989 y Fr(.)56 b(In)m(tegrating)37 b(b)m(y)e(parts)g(as)h(in)f (\(4.11\))-47 4102 y(and)29 b(in)m(v)m(oking)g(the)h(induction)d(h)m (yp)s(othesis,)i(as)h(w)m(ell)e(as)i(\(4.13\))i(yields)c(an)h(in)m (tegral)h(of)g(the)f(form)h(\(the)g(b)s(oundary)-47 4215 y(term)g(is)g(dominated)f(b)m(y)h(the)h(in)m(tegral)f(as)h(b)s(efore\)) 788 4486 y Fm(j)p Fq(U)10 b Fr(\()p Fq(t)p Fr(\))p Fm(j)84 b(\024)e Fq(C)1337 4362 y Fl(Z)1428 4389 y Fn(t)1387 4569 y(t)1412 4578 y Fj(0)1473 4486 y Fq(s)1516 4449 y Fk(\000)1581 4421 y Fj(1)p 1580 4433 V 1580 4475 a(4)1621 4449 y Fk(\000)p Fp(\()p Fn(M)7 b Fp(+)1843 4421 y Fj(1)p 1843 4433 V 1843 4475 a(2)1883 4449 y Fp(\))p Fn(\014)s Fp(+)p Fn(")2060 4486 y Fq(s)2103 4449 y Fn(M)g Fk(\000)p Fp(1)2346 4373 y Fn(`)2302 4400 y Fl(Y)2303 4595 y Fn(i)p Fp(=1)2454 4370 y Fn(m)2516 4380 y Fh(i)2440 4400 y Fl(Y)2434 4597 y Fn(k)r Fp(=1)2577 4486 y Fq(s)2620 4449 y Fn(\026)p Fp(\()p Fk(j)p Fn(\021)2746 4413 y Fj(\()p Fh(i)p Fj(\))2744 4472 y Fh(k)2822 4449 y Fk(j)p Fp(\)+)p Fn(")2957 4458 y Fj(1)3020 4486 y Fq(ds)-3157 b Fr(\(4.15\))1097 4743 y Fm(\024)82 b Fq(C)7 b(t)1355 4705 y Fk(\000)1420 4678 y Fj(1)p 1419 4690 V 1419 4731 a(4)1460 4705 y Fk(\000)p Fp(\()p Fn(M)g Fp(+)1682 4678 y Fj(1)p 1682 4690 V 1682 4731 a(2)1722 4705 y Fp(\))p Fn(\014)s Fp(+)p Fn(")p Fp(+)p Fn(M)g Fp(+\(1)p Fk(\000)p Fn(\014)s Fp(\))p Fk(j)p Fn(\013)p Fk(j\000)p Fn(M)g Fp(\()2504 4678 y Fj(1)p 2505 4690 V 2505 4731 a(4)2545 4705 y Fp(+)2610 4674 y Fh(\014)p 2610 4690 38 3 v 2613 4731 a Fj(2)2657 4705 y Fp(\)+)p Fn(M)g(")2847 4714 y Fj(1)2886 4743 y Fq(:)-2958 b Fr(\(4.16\))1903 4992 y(26)p eop %%Page: 27 27 27 26 bop -47 274 a Fr(Here)25 b Fq(")g(>)g Fr(0)g(is)e(arbitrary)g (small)g(but)g(\014xed,)i(but)f(some)h(care)g(needs)f(to)h(b)s(e)e(tak) m(en)j(with)d(the)h(c)m(hoice)h(of)f Fq(")3522 288 y Fp(1)3562 274 y Fr(.)39 b(Observ)m(e)-47 398 y(that)e(w)m(e)h(can)f (use)g(the)g(induction)e(h)m(yp)s(othesis)h(only)g(when)g Fm(j)p Fq(\021)2186 350 y Fp(\()p Fn(i)p Fp(\))2183 427 y Fn(k)2270 398 y Fm(j)g(\025)h Fr(2.)61 b(Nev)m(ertheless,)39 b(since)e Fq(\026)p Fr(\(1\))g(=)3681 362 y Fp(3)p 3681 377 36 4 v 3681 429 a(4)3751 398 y Fm(\000)3857 357 y Fp(3)p Fn(\014)p 3857 377 79 4 v 3878 429 a Fp(2)-47 525 y Fr(estimate)29 b(\(4.15\))i(remains)c(correct)j(pro)m(vided)d (one)i(c)m(ho)s(oses)g Fq(")2116 539 y Fp(1)2181 525 y Fr(=)2287 483 y Fp(3)p Fn(\014)p 2287 504 V 2309 556 a Fp(2)2392 525 y Fm(\000)2489 489 y Fp(3)p 2489 504 36 4 v 2489 556 a(4)2551 525 y Fr(+)16 b Fq(")29 b Fr(so)g(that)g Fq(\026)p Fr(\(1\))18 b(+)e Fq(")3331 539 y Fp(1)3396 525 y Fr(=)25 b Fq(")p Fr(.)41 b(It)28 b(is)g(no)m(w)-47 637 y(a)34 b(simple)f(matter)i(to)g(c)m(hec)m(k)h(that)f(the)f(exp)s (onen)m(t)g(in)f(\(4.16\))k(is)c(maximized)g(if)g Fq(M)42 b Fr(=)32 b Fm(j)p Fq(\013)p Fm(j)j Fr(\(clearly)-8 b(,)36 b(that)e(case)i(is)-47 766 y(the)30 b(analogue)h(of)g(the)f(double)f (sum)h(in)f(\(4.8\))q(.)41 b(In)30 b(particular,)f(all)g Fm(j)p Fq(\021)2350 718 y Fp(\()p Fn(i)p Fp(\))2347 796 y Fn(k)2434 766 y Fm(j)c Fr(=)g(1\).)42 b(Indeed,)837 1020 y Fm(\000)918 958 y Fr(1)p 918 999 46 4 v 918 1082 a(4)994 1020 y Fm(\000)19 b Fr(\()p Fq(M)31 b Fr(+)1339 958 y(1)p 1339 999 V 1339 1082 a(2)1394 1020 y(\))p Fq(\014)26 b Fr(+)20 b Fq(")h Fr(+)f Fq(M)30 b Fr(+)20 b(\(1)h Fm(\000)f Fq(\014)5 b Fr(\))p Fm(j)p Fq(\013)p Fm(j)22 b(\000)d Fq(M)10 b Fr(\()2606 958 y(1)p 2606 999 V 2606 1082 a(4)2682 1020 y(+)2783 958 y Fq(\014)p 2783 999 57 4 v 2788 1082 a Fr(2)2850 1020 y(\))20 b(+)g Fq(M)10 b(")3136 1034 y Fp(1)3201 1020 y Fm(\024)684 1235 y(\024)82 b(\000)918 1174 y Fr(1)p 918 1215 46 4 v 918 1298 a(4)994 1235 y Fm(\000)19 b Fq(\014)5 b Fr(\()p Fm(j)p Fq(\013)p Fm(j)22 b Fr(+)1406 1174 y(1)p 1406 1215 V 1406 1298 a(2)1462 1235 y(\))e(+)g(\()p Fm(j)p Fq(\013)p Fm(j)i Fr(+)d(1\))p Fq(")j Fr(+)e Fm(j)p Fq(\013)p Fm(j)p Fq(:)-47 1461 y Fr(By)30 b(our)g(c)m(hoice)i(of)e Fq(")673 1475 y Fp(1)743 1461 y Fr(this)f(is)h(in)f(fact)i(an)f(equalit)m(y)-8 b(.)41 b(The)30 b(conclusion)f(from)h(the)g(preceding)g(is)f(the)i(b)s (ound)562 1716 y Fm(j)p Fq(D)665 1679 y Fn(\013)662 1739 y(\021)715 1716 y Fq(\030)t Fr(\()p Fq(t)p Fr(\))p Fm(j)26 b(\024)f Fq(C)1095 1592 y Fl(Z)1186 1619 y Fn(t)1146 1799 y(t)1171 1808 y Fj(0)1231 1716 y Fq(s)1274 1679 y Fk(\000)1339 1652 y Fj(1)p 1338 1664 31 3 v 1338 1705 a(4)1379 1679 y Fk(\000)p Fn(\014)s Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+)1654 1652 y Fj(1)p 1653 1664 V 1653 1705 a(2)1694 1679 y Fp(\)+)p Fn(")1828 1716 y Fq(s)1871 1679 y Fk(j)p Fn(\013)p Fk(j\000)p Fp(1)2065 1716 y Fq(ds)20 b Fr(+)g Fq(C)2352 1592 y Fl(Z)2443 1619 y Fn(t)2403 1799 y(t)2428 1808 y Fj(0)2488 1716 y Fq(s)2531 1679 y Fk(\000)2596 1652 y Fj(1)p 2596 1664 V 2596 1705 a(4)2636 1679 y Fk(\000)2701 1648 y Fj(3)p Fh(\014)p 2701 1664 68 3 v 2719 1705 a Fj(2)2779 1679 y Fp(+)p Fn(")2870 1716 y Fm(j)p Fq(D)2973 1679 y Fn(\013)2970 1739 y(\021)3023 1716 y Fq(\030)t Fr(\()p Fq(s)p Fr(\))p Fm(j)15 b Fq(ds:)-47 1976 y Fr(As)31 b(b)s(efore,)g(Gron)m(w)m(all)g(\014nishes)e(the)j(pro) s(of.)43 b(The)30 b(c)m(hanges)j(required)c(for)i Fq(D)2632 1943 y Fn(\013)2629 1998 y(y)2713 1976 y Fr(and)f Fq(x=t)i Fr(are)g(analogous)f(to)h(those)-47 2089 y(for)e Fm(j)p Fq(\013)p Fm(j)c Fr(=)f(2)31 b(and)f(will)d(b)s(e)j(skipp)s(ed.)p 3879 2089 4 62 v 3883 2031 55 4 v 3883 2089 V 3937 2089 4 62 v -47 2276 a(W)-8 b(e)30 b(are)f(no)m(w)g(able)g(to)h(solv)m(e)f (equation)g(\(4.1\))i(b)m(y)d(means)h(of)h(the)f(metho)s(d)f(of)h(c)m (haracteristics.)41 b(This)27 b(is)h(standard,)-47 2389 y(see)j(for)f(example)g(Section)g(A.3)h(in)e([6)q(].)41 b(W)-8 b(e)32 b(will)27 b(call)j(a)h Fq(C)1993 2356 y Fk(1)2067 2389 y Fr(-)f(function)g Fq( )j Fr(on)e Fo(R)2762 2356 y Fn(d)2829 2389 y Fm(n)21 b(f)p Fr(0)p Fm(g)31 b Fg(admissible)h Fr(if)1602 2641 y Fm(jr)p Fq( )s Fr(\()p Fq(\030)t Fr(\))21 b Fm(\000)f Fq(\030)t Fm(j)26 b(\024)2191 2580 y(j)p Fq(\030)t Fm(j)p 2191 2620 95 4 v 2193 2704 a Fr(10)-47 2882 y(for)k(all)f Fq(\030)g Fm(6)p Fr(=)c(0.)41 b(The)30 b(example)g(to)h(k)m(eep)h(in)d(mind)f(is)h(of)i(course)f Fq( )s Fr(\()p Fq(\030)t Fr(\))d(=)2476 2846 y Fp(1)p 2476 2861 36 4 v 2476 2913 a(2)2521 2882 y Fm(j)p Fq(\030)t Fm(j)2615 2849 y Fp(2)2655 2882 y Fr(.)-47 3069 y Fd(Prop)s(osition)36 b(4.4.)46 b Fg(Given)38 b Fr(0)e Fq(<)f(\032)h(<)f Fr(1)k Fg(arbitr)-5 b(ary)40 b(and)f Fq(")d(>)f Fr(0)k Fg(su\016ciently)g (smal)5 b(l,)40 b(ther)-5 b(e)39 b(exists)g(some)g(lar)-5 b(ge)-47 3182 y(time)28 b Fq(t)188 3196 y Fp(0)227 3182 y Fr(\()p Fq(";)15 b(\032)p Fr(\))29 b Fg(so)f(that)h(the)g(fol)5 b(lowing)28 b(holds)i(up)e(to)g(pr)-5 b(ob)g(ability)30 b Fq(e)2204 3149 y Fk(\000)p Fn(t)2284 3126 y Fh(")2284 3170 y Fj(0)2324 3182 y Fg(:)39 b(F)-7 b(or)29 b(any)f(admissible)h Fq( )i Fg(the)d(e)-5 b(quation)36 b Fr(\(4.1\))-47 3295 y Fg(has)d(a)g(unique)f Fq(C)559 3262 y Fk(1)633 3295 y Fg(-solution)i(on)f(the)g(set)1061 3517 y Fr(\012)1127 3531 y Fn(t)1152 3540 y Fj(0)1186 3531 y Fn(;\032)1271 3517 y Fr(:=)26 b([)p Fq(t)1451 3531 y Fp(0)1490 3517 y Fq(;)15 b Fm(1)p Fr(\))21 b Fm(\002)1768 3416 y Fl(n)1829 3517 y Fq(\030)29 b Fm(2)c Fo(R)2043 3480 y Fn(d)2135 3517 y Fr(:)46 b Fq(\032)25 b Fm(\024)g(j)p Fq(\030)t Fm(j)h(\024)f Fr(2)p Fq(\032)2682 3480 y Fk(\000)p Fp(1)2777 3416 y Fl(o)-47 3744 y Fg(with)33 b(initial)g(c)-5 b(ondition)35 b Fq(S)5 b Fr(\()p Fq(t)944 3758 y Fp(0)983 3744 y Fq(;)15 b(\030)t Fr(\))26 b(=)f Fq(t)1257 3758 y Fp(0)1296 3744 y Fq( )s Fr(\()p Fq(\030)t Fr(\))p Fg(.)43 b(Mor)-5 b(e)g(over,)34 b(with)f Fq(\026)g Fg(as)g(in)39 b Fr(\(4.6\))r Fg(,)33 b(one)g(has)744 3948 y Fm(j)p Fq(D)844 3963 y Fn(\030)883 3948 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)21 b Fr(+)f Fm(j)p Fq(D)1371 3911 y Fp(2)1368 3971 y Fn(\030)1410 3948 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)84 b(\024)f Fq(C)7 b(t)1258 4105 y Fm(j)p Fq(D)1361 4067 y Fn(\013)1358 4128 y(\030)1410 4105 y Fq(S)e Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)84 b(\024)f Fq(C)1986 4119 y Fn(\013)2051 4105 y Fq(t)2084 4067 y Fp(1+)p Fn(\026)p Fp(\()p Fk(j)p Fn(\013)p Fk(j\000)p Fp(1\)+)p Fn(")2602 4105 y Fg(for)34 b(al)5 b(l)33 b Fm(j)p Fq(\013)p Fm(j)26 b(\025)f Fr(3)-3200 b(\(4.17\))-47 4309 y Fg(in)32 b Fr(\012)130 4323 y Fn(t)155 4332 y Fj(0)190 4323 y Fn(;\032)250 4309 y Fg(.)42 b(Mor)-5 b(e)g(over,)33 b(in)g Fr(\012)925 4323 y Fn(t)950 4332 y Fj(0)985 4323 y Fn(;\032)1043 4538 y Fm(j)p Fq(D)1143 4553 y Fn(\030)1182 4538 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))21 b Fm(\000)f Fq(t)1575 4552 y Fp(0)1614 4538 y Fm(r)p Fq( )s Fr(\()p Fq(\030)t Fr(\))h Fm(\000)f Fr(\()p Fq(t)g Fm(\000)g Fq(t)2190 4552 y Fp(0)2230 4538 y Fr(\))p Fq(\030)t Fm(j)25 b(\024)g Fq(C)d(t)2585 4473 y Fj(3)p 2585 4485 31 3 v 2585 4527 a(4)2625 4501 y Fk(\000)2690 4470 y Fh(\014)p 2690 4485 38 3 v 2693 4527 a Fj(2)2737 4501 y Fp(+)p Fn(")2829 4538 y Fq(:)-2901 b Fr(\(4.18\))1903 4992 y(27)p eop %%Page: 28 28 28 27 bop -47 274 a Fg(Pr)-5 b(o)g(of.)47 b Fr(Uniqueness)24 b(is)g(completely)i(standard)e(and)i(will)c(not)k(b)s(e)f(presen)m (ted,)i(see)f([6)q(])g(for)f(example.)39 b(Let)26 b Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))-47 387 y(and)43 b Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(y)s(;)g(\021)s Fr(\))45 b(b)s(e)d(the)i(solutions)e(of)51 b(\(2.1\))45 b(from)e(the)h(b)s(eginning)c(of)k(this)f(section.)80 b(W)-8 b(e)44 b(no)m(w)g(sp)s(ecialize)e(to)-47 500 y Fq(y)28 b Fr(=)d Fm(r)p Fq( )s Fr(\()p Fq(\021)s Fr(\).)39 b(Recall)22 b(that)h(this)f(means)g(solving)g(\(2.1\))i(with)e(initial) d(conditions)i Fq(x)p Fr(\()p Fq(t)2846 514 y Fp(0)2886 500 y Fr(\))k(=)g Fq(t)3075 514 y Fp(0)3115 500 y Fm(r)p Fq( )s Fr(\()p Fq(\021)s Fr(\))e(and)f Fq(\030)t Fr(\()p Fq(t)3675 514 y Fp(0)3715 500 y Fr(\))j(=)g Fq(\021)s Fr(.)-47 613 y(W)-8 b(e)29 b(denote)g(these)f(solutions)f(b)m(y)g Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fq(;)g(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(\021)s Fr(\).)43 b(F)-8 b(or)29 b(simplicit)m(y)c(w)m(e)k(set)f Fq(\032)e Fr(=)f(1,)k(but)e(tak)m(e)j Fq(\021)e Fm(2)d Fo(R)3405 580 y Fn(d)3479 613 y Fr(in)i(a)h(sligh)m (tly)-47 726 y(larger)44 b(region,)49 b(namely)882 690 y Fp(1)p 882 705 36 4 v 882 757 a(2)976 726 y Fq(<)g Fm(j)p Fq(\021)s Fm(j)i Fq(<)e Fr(4.)84 b(By)46 b(Prop)s(osition)c(3.1) k(this)e(insures)f(that)i Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))46 b(remains)e(in)g(the)-47 839 y(region)236 803 y Fp(1)p 236 818 V 236 870 a(4)306 839 y Fm(\024)25 b(j)p Fq(\030)t Fm(j)h(\024)f Fr(8)31 b(for)f(all)f(times)h(pro)m (vided)f Fq(t)1609 853 y Fp(0)1678 839 y Fr(is)h(tak)m(en)h(to)g(b)s(e) f(large)h(enough.)40 b(No)m(w)31 b(de\014ne)576 1077 y Fq(Q)p Fr(\()p Fq(t;)15 b(\021)s Fr(\))27 b(:=)e Fq(t)1020 1091 y Fp(0)1059 1077 y Fq( )s Fr(\()p Fq(\021)s Fr(\))d(+)1352 953 y Fl(Z)1443 979 y Fn(t)1402 1159 y(t)1427 1168 y Fj(0)1472 976 y Fl(h)1525 1015 y Fr(1)p 1525 1056 46 4 v 1525 1139 a(2)1581 1077 y Fm(j)p Fq(\030)t Fr(\()p Fq(s)p Fr(;)15 b Fq(\021)s Fr(\))p Fm(j)1876 1039 y Fp(2)1937 1077 y Fr(+)20 b Fq(V)g Fr(\()p Fq(x)p Fr(\()p Fq(s)p Fr(;)15 b Fq(\021)s Fr(\)\))22 b(+)e Fm(h)p Fq(x)p Fr(\()p Fq(s)p Fr(;)15 b Fq(\021)s Fr(\))p Fq(;)2886 1053 y Fr(_)2865 1077 y Fq(\030)6 b Fr(\()p Fq(s)p Fr(;)15 b Fq(\021)s Fr(\))p Fm(i)3147 976 y Fl(i)3206 1077 y Fq(ds:)-3368 b Fr(\(4.19\))-47 1300 y(Then)990 1466 y(_)959 1489 y Fq(Q)p Fr(\()p Fq(t;)15 b(\021)s Fr(\))27 b(=)1355 1428 y(1)p 1355 1468 V 1355 1551 a(2)1410 1489 y Fm(j)p Fq(\030)t Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fm(j)1695 1452 y Fp(2)1756 1489 y Fr(+)20 b Fq(V)h Fr(\()p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\)\))22 b(+)e Fm(h)p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fq(;)2686 1465 y Fr(_)2665 1489 y Fq(\030)6 b Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fm(i)-47 1702 y Fr(and)30 b(therefore,)h(with)e Fq(@)792 1716 y Fn(j)854 1702 y Fr(=)994 1666 y Fn(@)p 960 1681 110 4 v 960 1733 a(@)t(\021)1036 1743 y Fh(j)1109 1702 y Fr(and)h(suppressing)d(the)k(argumen)m(ts)g(for)f(simplicit)m(y) -8 b(,)743 1907 y Fq(@)791 1921 y Fn(j)858 1884 y Fr(_)828 1907 y Fq(Q)82 b Fr(=)h Fm(h)p Fq(\030)t(;)15 b(@)1303 1921 y Fn(j)1341 1907 y Fq(\030)t Fm(i)20 b Fr(+)g Fm(hr)p Fq(V)g Fr(\()p Fq(x)p Fr(\))p Fq(;)15 b(@)1925 1921 y Fn(j)1963 1907 y Fq(x)p Fm(i)21 b Fr(+)f Fm(h)p Fq(@)2245 1921 y Fn(j)2282 1907 y Fq(x;)15 b Fm(\000r)p Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p Fm(i)h Fr(+)f Fm(h)p Fq(x;)15 b(@)3038 1921 y Fn(j)3095 1883 y Fr(_)3076 1907 y Fq(\030)s Fm(i)982 2091 y Fr(=)1163 2029 y Fq(d)p 1146 2070 81 4 v 1146 2153 a(dt)1236 2091 y Fm(h)p Fq(x;)g(@)1411 2105 y Fn(j)1449 2091 y Fq(\030)t Fm(i)p Fq(:)-47 2290 y Fr(Observ)m(e)30 b(that)h(w)m(e)g(ha)m(v)m(e)g(used)f(\(2.1\))i(in)d (t)m(w)m(o)j(places.)40 b(This)29 b(implies)f(that)635 2463 y Fq(@)683 2477 y Fn(j)719 2463 y Fq(Q)p Fr(\()p Fq(t;)15 b(\021)s Fr(\))22 b Fm(\000)e(h)p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fq(;)g(@)1461 2477 y Fn(j)1500 2463 y Fq(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(\021)s Fr(\))p Fm(i)84 b Fr(=)f Fq(@)2056 2477 y Fn(j)2093 2463 y Fq(Q)p Fr(\()p Fq(t)2233 2477 y Fp(0)2272 2463 y Fq(;)15 b(\021)s Fr(\))22 b Fm(\000)e(h)p Fq(x)p Fr(\()p Fq(t)2663 2477 y Fp(0)2703 2463 y Fr(;)15 b Fq(\021)s Fr(\))p Fq(;)g(@)2914 2477 y Fn(j)2952 2463 y Fq(\030)t Fr(\()p Fq(t)3064 2477 y Fp(0)3103 2463 y Fr(;)g Fq(\021)s Fr(\))p Fm(i)1854 2601 y Fr(=)83 b Fq(t)2041 2615 y Fp(0)2080 2601 y Fq(@)2128 2615 y Fn(j)2165 2601 y Fq( )s Fr(\()p Fq(\021)s Fr(\))22 b Fm(\000)e(h)p Fq(t)2526 2615 y Fp(0)2565 2601 y Fm(r)p Fq( )s Fr(\()p Fq(\021)s Fr(\))p Fq(;)27 b(~)-57 b(e)2903 2615 y Fn(j)2942 2601 y Fm(i)26 b Fr(=)f(0)-47 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4992 y Fr(28)p eop %%Page: 29 29 29 28 bop -47 274 a Fr(whic)m(h)29 b(pro)m(v)m(es)i(the)f(fundamen)m (tal)g(relation)1496 478 y Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))26 b(=)f Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))p Fq(:)-2446 b Fr(\(4.21\))-47 682 y(T)-8 b(o)31 b(c)m(hec)m(k)h(that)f Fq(S)k Fr(solv)m(es)30 b(\(4.1\))j(one)d(computes,)h(using)f(\(4.20\))r (,)352 974 y Fq(@)400 988 y Fn(t)430 974 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))84 b(=)e Fq(@)963 988 y Fn(t)993 974 y Fq(Q)p Fr(\()p Fq(t;)15 b(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))22 b(+)1604 860 y Fn(d)1556 887 y Fl(X)1561 1083 y Fn(j)t Fp(=1)1703 974 y Fq(@)1751 988 y Fn(j)1788 974 y Fq(Q)p Fr(\()p Fq(t;)15 b(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))30 b(_)-55 b Fq(\021)2283 988 y Fn(j)2321 974 y Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))762 1307 y(=)82 b Fq(@)963 1321 y Fn(t)993 1307 y Fq(Q)p Fr(\()p Fq(t;)15 b(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))22 b(+)1556 1206 y Fl(D)1612 1307 y Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))p Fq(;)2132 1193 y Fn(d)2085 1220 y Fl(X)2089 1416 y Fn(j)t Fp(=1)2231 1307 y Fq(@)2279 1321 y Fn(j)2316 1307 y Fq(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))30 b(_)-55 b Fq(\021)2783 1321 y Fn(j)2822 1307 y Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))3009 1206 y Fl(E)762 1591 y Fr(=)925 1530 y(1)p 925 1570 46 4 v 925 1653 a(2)981 1591 y Fm(j)p Fq(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))p Fm(j)1453 1554 y Fp(2)1515 1591 y Fr(+)20 b Fq(V)1679 1517 y Fl(\000)1721 1591 y Fm(r)p Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\()p Fq(t)p Fr(;)g Fq(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\)\))2423 1517 y Fl(\001)2488 1591 y Fr(+)k Fm(h)p Fq(x)p Fr(\()p Fq(t)p Fr(;)c Fq(\021)s Fr(\()p Fq(t)p Fr(;)g Fq(\030)t Fr(\)\))p Fq(;)3106 1567 y Fr(_)3083 1591 y Fq(\030)7 b Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\()p Fq(t)p 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b(then)1283 2574 y Fm(j)p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)26 b(\024)93 b Fr(sup)1791 2639 y Fj(1)p 1791 2651 31 3 v 1791 2692 a(2)1832 2666 y Fn(<)p Fk(j)p Fn(\021)r Fk(j)p Fn(<)p Fp(4)2069 2574 y Fm(j)p Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fm(j)27 b(\024)e Fq(C)7 b(t:)-47 2869 y Fr(whereas)30 b(if)f Fm(j)p Fq(\013)p Fm(j)e Fr(=)d(2,)31 b(then)877 3074 y Fm(j)p Fq(D)980 3036 y Fp(2)1019 3074 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)26 b(\024)93 b Fr(sup)1424 3138 y Fj(1)p 1424 3150 V 1424 3192 a(2)1465 3166 y Fn(<)p Fk(j)p Fn(\021)r Fk(j)p Fn(<)p Fp(4)1702 3074 y Fm(j)p Fq(@)1775 3088 y Fn(\021)1817 3074 y Fq(x)p Fr(\()p Fq(t)p Fr(;)15 b Fq(\021)s Fr(\))p Fm(j)75 b Fr(sup)2100 3156 y Fp(1)p Fn(<)p Fk(j)p Fn(\030)s Fk(j)p Fn(<)p Fp(2)2370 3074 y Fm(j)p Fq(D)2470 3089 y Fn(\030)2509 3074 y Fq(\021)s Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))p Fm(j)27 b(\024)d Fq(C)7 b(t:)-47 3364 y Fr(Here)30 b(w)m(e)h(ha)m(v)m(e)g(used)f(Lemma)g(4.1)h(and)f 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Fm(j)27 b(\024)e Fq(C)d(t)2781 4579 y Fj(3)p 2781 4591 V 2781 4633 a(4)2821 4606 y Fk(\000)2886 4576 y Fh(\014)p 2886 4591 38 3 v 2889 4633 a Fj(2)2933 4606 y Fp(+)p Fn(")1903 4992 y Fr(29)p eop %%Page: 30 30 30 29 bop -47 274 a Fr(for)30 b(all)f Fq(t)c Fm(\025)g Fq(t)405 288 y Fp(0)444 274 y Fr(.)41 b(In)30 b(particular,)f(one)h (can)h(set)g Fq(\021)e Fr(=)c Fq(\021)s Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))31 b(where)f(1)c Fq(<)f Fm(j)p Fq(\030)t Fm(j)g Fq(<)g Fr(2)31 b(whic)m(h)e(leads)h(to)940 491 y Fm(j)p Fq(D)1040 506 y Fn(\030)1078 491 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))21 b Fm(\000)f Fq(t)1471 505 y Fp(0)1510 491 y Fm(r)p Fq( )s Fr(\()p Fq(\021)s Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\)\))23 b Fm(\000)d Fr(\()p Fq(t)g Fm(\000)g Fq(t)2279 505 y Fp(0)2318 491 y Fr(\))p Fq(\030)t Fm(j)26 b(\024)f Fq(C)d(t)2674 426 y Fj(3)p 2673 438 31 3 v 2673 479 a(4)2714 453 y Fk(\000)2779 422 y Fh(\014)p 2779 438 38 3 v 2782 479 a Fj(2)2826 453 y Fp(+)p Fn(")2918 491 y Fq(;)-2990 b Fr(\(4.22\))-47 738 y(see)28 b(\(4.21\))r(.)39 b(T)-8 b(o)28 b(replace)f Fq(\021)s Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))29 b(in)d Fm(r)p Fq( )s Fr(,)i(one)f(uses)g(that)h Fm(j)p Fq(\021)17 b Fm(\000)d Fq(\030)t Fr(\()p Fq(t)p Fr(;)h Fq(\021)s Fr(\))p Fm(j)27 b(\024)e Fq(C)7 b(t)2594 681 y Fk(\000)2659 654 y Fj(1)p 2658 666 31 3 v 2658 707 a(4)2698 681 y Fk(\000)2763 650 y Fh(\014)p 2763 666 38 3 v 2766 707 a Fj(2)2810 681 y Fp(+)p Fn(")2594 764 y Fp(0)2902 738 y Fr(.)39 b(This)26 b(in)g(turn)g(implies)e(that)-47 895 y Fm(j)p Fq(\030)g Fm(\000)c Fq(\021)s Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))p Fm(j)27 b(\024)e Fq(C)7 b(t)621 838 y Fk(\000)686 811 y Fj(1)p 685 823 31 3 v 685 864 a(4)725 838 y Fk(\000)790 807 y Fh(\014)p 790 823 38 3 v 793 864 a Fj(2)838 838 y Fp(+)p Fn(")621 921 y Fp(0)929 895 y Fr(.)41 b(Inserting)29 b(this)g(inequalit)m(y)g(in)m(to)h (\(4.22\))j(leads)d(to)h(the)f(desired)f(b)s(ound)f(\(4.18\))s(.)p 3879 895 4 62 v 3883 837 55 4 v 3883 895 V 3937 895 4 62 v -47 1082 a(Belo)m(w)j(it)f(will)d(b)s(e)j(necessary)h(to)g (estimate)g(a)m(v)m(erages)i(of)d(the)h(form)1459 1212 y Fl(\014)1459 1266 y(\014)1459 1321 y(\014)1489 1193 y(Z)1580 1219 y Fk(1)1540 1399 y Fn(t)1670 1316 y Fq(D)1748 1279 y Fn(\013)1797 1316 y Fq(V)21 b Fr(\()p Fq(D)1981 1331 y Fn(\030)2019 1316 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))g Fq(ds)2407 1212 y Fl(\014)2407 1266 y(\014)2407 1321 y(\014)-47 1558 y Fr(where)34 b Fm(j)p Fq(\013)p Fm(j)e(\025)g Fr(1.)53 b(This)32 b(is)i(of)g(course)g(v)m (ery)h(close)g(to)g(Lemma)f(3.3.)54 b(In)33 b(view)h(of)g(the)h (argumen)m(ts)f(in)f(Lemma)i(3.8)-47 1671 y(and)44 b(Theorem)g(3.9)h (it)f(migh)m(t)g(seem)h(natural)e(to)j(appro)m(ximate)e(the)h(path)f Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))45 b(b)m(y)f(means)g(of)h (the)f(line)-47 1784 y Fq(t)-14 1798 y Fp(0)25 1784 y Fm(r)p Fq( )s Fr(\()p Fq(\030)t Fr(\))27 b(+)f(\()p Fq(t)h Fm(\000)f Fq(t)626 1798 y Fp(0)665 1784 y Fr(\))p Fq(\030)44 b Fr(and)39 b(in)m(tegrate)i(b)m(y)f(parts.)68 b(This,)41 b(ho)m(w)m(ev)m(er,)i(do)s(es)d(not)g(w)m(ork)f(as)h(w)m(ell)f(as)h(b)s (efore.)68 b(F)-8 b(or)-47 1907 y(example,)27 b(let)g Fq(z)t Fr(\()p Fq(t;)15 b(\030)t Fr(\))26 b(=)f Fq(t)847 1921 y Fp(0)887 1907 y Fm(r)p Fq( )s Fr(\()p Fq(\030)t Fr(\))13 b(+)g(\()p Fq(t)g Fm(\000)g Fq(t)1434 1921 y Fp(0)1473 1907 y Fr(\))p Fq(\030)31 b Fr(and)26 b(estimate)2113 1834 y Fl(R)2174 1860 y Fp(2)p Fn(T)2156 1939 y(T)2279 1907 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))g Fq(dt)29 b Fr(b)m(y)d(means)h(of)g(appro)m(ximating) -47 2020 y Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))31 b(with)e Fq(z)t Fr(\()p Fq(t;)15 b(\030)t Fr(\).)42 b(T)-8 b(a)m(ylor)30 b(expansion)f(giv)m(es)i(rise)f(to)h(terms)f(of)h(whic)m (h)e(the)h(\014rst)g(is)1183 2153 y Fl(Z)1274 2179 y Fp(2)p Fn(T)1234 2359 y(T)1380 2277 y Fq(D)s(V)20 b Fr(\()p Fq(z)t Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))g(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))23 b Fm(\000)d Fq(z)t 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Fq(T)3863 3483 y Fn(\014)3910 3516 y Fr(\))-47 3629 y(rather)d(than)g(b)m(y)g (linearization)f(o)m(v)m(er)j(the)e(en)m(tire)g(in)m(terv)-5 b(al)30 b([)p Fq(T)8 b(;)15 b Fr(2)p Fq(T)e Fr(].)-47 3810 y Fd(Lemma)32 b(4.5.)47 b Fg(L)-5 b(et)34 b Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))35 b Fg(b)-5 b(e)34 b(the)g(solution)i(c)-5 b(onstructe)g(d)35 b(in)f(the)h(pr)-5 b(evious)35 b(pr)-5 b(op)g(osition.)50 b(Then,)34 b(up)g(to)h(pr)-5 b(ob)g(a-)-47 3923 y(bility)32 b(at)i(most)f Fq(e)552 3890 y Fk(\000)p Fn(t)632 3866 y Fh(")632 3910 y Fj(0)672 3923 y Fg(,)f(one)h(has)h(the)f(b)-5 b(ounds)33 b(for)h(al)5 b(l)33 b Fq(t)25 b Fm(\025)g Fq(t)1980 3937 y Fp(0)1043 4053 y Fl(\014)1043 4107 y(\014)1043 4162 y(\014)1073 4034 y(Z)1164 4060 y Fk(1)1123 4240 y Fn(t)1254 4157 y Fq(D)1332 4120 y Fn(\013)1381 4157 y Fq(V)20 b Fr(\()p Fq(D)1564 4172 y Fn(\030)1603 4157 y Fq(S)5 b Fr(\()p Fq(s;)15 b(\030)t Fr(\)\))g Fq(ds)2001 4053 y Fl(\014)2001 4107 y(\014)2001 4162 y(\014)2057 4157 y Fm(\024)25 b Fq(C)2218 4171 y Fn(\013)2283 4157 y Fq(t)2316 4120 y Fk(\000)2381 4093 y Fj(1)p 2380 4105 31 3 v 2380 4146 a(4)2421 4120 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j\000)2653 4093 y Fj(1)p 2653 4105 V 2653 4146 a(2)2693 4120 y Fp(\))p Fn(\014)s Fp(+)p Fn(")-47 4399 y Fg(and)33 b(al)5 b(l)33 b(multi-indic)-5 b(es)33 b Fm(j)p Fq(\013)p Fm(j)27 b(\025)e Fr(1)p Fg(,)32 b(wher)-5 b(e)g(as)35 b(for)e Fq(\013)26 b Fr(=)f(0)1253 4546 y Fl(\014)1253 4601 y(\014)1253 4656 y(\014)1283 4527 y(Z)1374 4554 y Fn(t)1334 4733 y(t)1359 4742 y Fj(0)1419 4651 y Fq(V)20 b Fr(\()p Fq(D)1602 4666 y Fn(\030)1640 4651 y Fq(S)5 b Fr(\()p Fq(s;)15 b(\030)t Fr(\)\))g Fq(ds)2038 4546 y Fl(\014)2038 4601 y(\014)2038 4656 y(\014)2095 4651 y Fm(\024)25 b Fq(C)d(t)2311 4613 y Fk(\000)2376 4586 y Fj(1)p 2375 4598 V 2375 4640 a(4)2416 4613 y Fp(+)2481 4583 y Fh(\014)p 2481 4598 38 3 v 2484 4640 a Fj(2)2528 4613 y Fp(+)p Fn(")2620 4651 y Fq(:)1903 4992 y Fr(30)p eop %%Page: 31 31 31 30 bop -47 274 a Fg(Pr)-5 b(o)g(of.)47 b Fr(Let)32 b Fq(T)41 b(>)28 b Fr(0)33 b(b)s(e)f(arbitrary)e(but)i(\014xed.)46 b(Let)32 b Fq(T)45 b Fr(b)s(e)32 b(suc)m(h)g(that)g Fm(j)p Fq(D)2510 289 y Fn(\030)2549 274 y Fq(S)5 b Fr(\()p Fq(T)j(;)15 b(\030)t Fr(\))p Fm(j)29 b Fr(=)f Fq(R)33 b Fr(and)e(denote)i(b)m(y)f Fq(T)3747 241 y Fk(\003)3819 274 y Fr(the)-47 387 y(time)c(for)g(whic)m (h)f Fm(j)p Fq(D)653 402 y Fn(\030)692 387 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)26 b Fr(=)f(2)p Fq(R)q Fr(.)40 b(As)29 b(in)e(the)i(pro)s(of)e(of)i(Lemma)g(3.3,)h (de\014ne)e(stopping)f(times)h Fq(\034)3398 402 y Fn(`)3431 387 y Fq(;)e Fl(e)-62 b Fq(\034)3511 402 y Fn(`)3572 387 y Fr(b)m(y)29 b(means)-47 500 y(of)452 696 y Fq(\034)492 711 y Fn(`)608 696 y Fr(:=)83 b(inf)6 b Fm(f)p Fq(t)25 b Fm(\025)g Fr(0)46 b(:)g Fm(j)p Fq(D)1359 711 y Fn(\030)1397 696 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)26 b Fr(=)f Fq(R)c Fr(+)f Fq(`R)2081 658 y Fn(\014)2128 696 y Fm(g)455 834 y Fl(e)-54 b Fq(\034)492 849 y Fn(`)608 834 y Fr(:=)83 b(inf)6 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Fn(`)2742 1142 y Fr(;)15 b Fq(\030)t Fr(\))29 b(is)f(de\014ned)f(en)m(tirely)h(in)f (terms)-47 1255 y(of)40 b(the)g(random)f(v)-5 b(ariables)38 b(inside)g Fm(j)p Fq(x)p Fm(j)j(\024)g Fq(R)27 b Fr(+)f Fq(`R)1785 1222 y Fn(\014)1832 1255 y Fr(.)69 b(This)38 b(of)i(course)g(requires)e(that)i(the)g(c)m(haracteristics)h(of)-47 1368 y(the)32 b(Hamilton-Jacobi)f(PDE)h(b)s(elong)f(to)i(the)g (outgoing)f(region)g(of)g(phase)f(space,)j(whic)m(h)d(is)g(guaran)m (teed)i(b)m(y)f(the)-47 1481 y(assumption)d(on)h(the)g(initial)e(data)j (\()p Fq( )j Fr(is)c(admissible\).)38 b(T)-8 b(o)31 b(v)m(erify)e(our)h (claim,)g(recall)g(that)1419 1677 y Fq(D)1494 1692 y Fn(\030)1532 1677 y Fq(S)5 b Fr(\()p Fq(\034)1668 1692 y Fn(`)1701 1677 y Fr(;)15 b Fq(\030)t Fr(\))26 b(=)f Fq(x)p Fr(\()p Fq(\034)2069 1692 y Fn(`)2102 1677 y Fr(;)15 b Fq(\021)s Fr(\()p Fq(\034)2265 1692 y Fn(`)2299 1677 y Fr(;)g Fq(\030)t Fr(\)\))p Fq(;)-47 1873 y Fr(see)31 b(\(4.21\))r(.)42 b(Let)31 b Fq(\021)s Fr(\()p Fq(\034)685 1888 y Fn(`)719 1873 y Fr(;)15 b Fq(\030)t Fr(\))26 b(=)g Fq(\021)1006 1887 y Fp(0)1076 1873 y Fr(so)31 b(that)h Fq(\030)d Fr(=)d Fq(\030)t Fr(\()p Fq(\034)1671 1888 y Fn(`)1704 1873 y Fr(;)15 b Fq(\021)1789 1887 y Fp(0)1829 1873 y Fr(\).)42 b(Since)30 b(b)m(y)h(de\014nition)d Fm(j)p Fq(x)p Fr(\()p Fq(\034)2852 1888 y Fn(`)2885 1873 y Fr(;)15 b Fq(\021)2970 1887 y Fp(0)3010 1873 y Fr(\))p Fm(j)27 b Fr(=)e Fq(R)c Fr(+)g Fq(`R)3483 1840 y Fn(\014)3529 1873 y Fr(,)31 b(it)g(follo)m(ws)-47 1986 y(that)36 b Fq(S)5 b Fr(\()p Fq(\034)291 2001 y Fn(`)324 1986 y Fr(;)15 b Fq(\030)t Fr(\))36 b(dra)m(ws)f(only)g(on)g(those)h(random)f(v)-5 b(ariables)34 b(inside)f Fm(j)p Fq(x)p Fm(j)h(\024)g Fq(R)24 b Fr(+)f Fq(`R)2844 1953 y Fn(\014)2891 1986 y Fr(,)37 b(as)f(claimed.)55 b(This)34 b(implies)-47 2099 y(that)1173 2295 y Fq(z)1215 2310 y Fn(`)1248 2295 y Fr(\()p Fq(t)p Fr(\))26 b(:=)f Fq(D)1573 2310 y Fn(\030)1611 2295 y Fq(S)5 b Fr(\()p Fq(\034)1747 2310 y Fn(`)1780 2295 y Fq(;)15 b(\030)t Fr(\))21 b(+)f(\()p Fq(t)h Fm(\000)e Fq(\034)2230 2310 y Fn(`)2263 2295 y Fr(\))p Fq(D)2373 2310 y Fn(\030)2412 2295 y Fq(S)2468 2309 y Fn(t)2497 2295 y Fr(\()p Fq(\034)2572 2310 y Fn(`)2605 2295 y Fq(;)c(\030)t Fr(\))-47 2491 y(is)29 b(measurable)h(with)f(resp)s(ect)h(to)h(those)g (v)-5 b(ariables)29 b(as)i(w)m(ell.)39 b(By)31 b(T)-8 b(a)m(ylor)31 b(expansion)157 2620 y Fl(Z)247 2647 y Fn(T)298 2623 y Fb(\003)207 2826 y Fn(T)354 2744 y Fq(D)432 2706 y Fn(\013)481 2744 y Fq(V)20 b Fr(\()p Fq(D)664 2759 y Fn(\030)703 2744 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))g Fq(dt)26 b Fr(=)1203 2658 y Fl(X)1254 2855 y Fn(`)1350 2620 y Fl(Z)1444 2647 y Fp(~)-38 b Fn(\034)1472 2659 y Fh(`)p Fj(+1)1400 2826 y Fn(\034)1431 2838 y Fh(`)1598 2744 y Fq(D)1676 2706 y Fn(\013)1726 2744 y Fq(V)20 b Fr(\()p Fq(z)1876 2759 y Fn(`)1909 2744 y Fr(\()p Fq(t)p Fr(\)\))15 b Fq(dt)338 3002 y Fr(+)424 2916 y Fl(X)475 3113 y Fn(`)571 3002 y Fq(O)643 2901 y Fl(\020)697 3002 y Fm(j)5 b Fr(~)-50 b Fq(\034)762 3017 y Fn(`)p Fp(+1)905 3002 y Fm(\000)20 b Fq(\034)1036 3017 y Fn(`)p Fp(+1)1159 3002 y Fm(j)15 b(k)p Fq(D)1322 2965 y Fn(\013)1372 3002 y Fq(V)20 b Fm(k)1490 3021 y Fn(L)1538 3002 y Fb(1)1604 3021 y Fp(\()p Fk(S)1674 3033 y Fh(`;R)1769 3021 y Fp(\))1821 3002 y Fr(+)g Fm(j)p Fq(\034)1977 3017 y Fn(`)p Fp(+1)2120 3002 y Fm(\000)g Fq(\034)2251 3017 y Fn(`)2284 3002 y Fm(j)2309 2965 y Fp(3)2348 3002 y Fm(kr)p Fq(D)2547 2965 y Fn(\013)2597 3002 y Fq(V)g Fm(k)2715 3021 y Fn(L)2763 3002 y Fb(1)2829 3021 y Fp(\()p Fk(S)2899 3033 y Fh(`;R)2994 3021 y Fp(\))3140 3002 y Fr(sup)3041 3079 y Fn(\034)3072 3091 y Fh(`)3102 3079 y Fk(\024)p Fn(t)p Fk(\024)t Fp(~)-39 b Fn(\034)3268 3091 y Fh(`)p Fj(+1)3391 3002 y Fm(j)p Fq(D)3491 3017 y Fn(\030)3529 3002 y Fq(S)3585 3016 y Fn(tt)3640 3002 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)3852 2901 y Fl(\021)-47 3002 y Fr(\(4.25\))-47 3282 y(where)28 b(the)g(sum)g(extends)g(o)m(v)m(er)i(all)e(shells)e Fm(S)1502 3297 y Fn(`;R)1637 3282 y Fr(that)j(lie)e(b)s(et)m(w)m(een)i Fq(R)g Fr(and)f(2)p Fq(R)q Fr(.)40 b(By)29 b(the)g(equation)f(\(using)g (formal)-47 3395 y(notation\),)420 3591 y Fq(D)495 3606 y Fn(\030)533 3591 y Fq(S)589 3605 y Fn(t)618 3591 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))84 b(=)f Fq(\030)24 b Fr(+)c Fm(r)p Fq(V)g Fr(\()p Fq(D)1457 3606 y Fn(\030)1495 3591 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(D)1856 3553 y Fp(2)1853 3614 y Fn(\030)1897 3591 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))-2192 b(\(4.26\))416 3740 y Fq(D)494 3703 y Fp(2)491 3764 y Fn(\030)533 3740 y Fq(S)589 3754 y Fn(t)618 3740 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))84 b(=)f Fq(I)27 b Fr(+)20 b Fq(D)1279 3703 y Fp(2)1276 3763 y Fn(x)1320 3740 y Fq(V)g Fr(\()p Fq(D)1503 3755 y Fn(\030)1542 3740 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\)\()p Fq(D)1938 3703 y Fp(2)1935 3764 y Fn(\030)1979 3740 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))2262 3703 y Fp(2)2322 3740 y Fr(+)20 b Fm(r)p Fq(V)g Fr(\()p Fq(D)2672 3755 y Fn(\030)2711 3740 y Fq(S)5 b Fr(\))p Fq(D)2885 3703 y Fp(3)2882 3764 y Fn(\030)2924 3740 y Fq(S)-3027 b Fr(\(4.27\))394 3890 y Fq(D)469 3905 y Fn(\030)507 3890 y Fq(S)563 3904 y Fn(tt)618 3890 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))84 b(=)f Fq(D)1121 3853 y Fp(2)1118 3913 y Fn(x)1162 3890 y Fq(V)20 b Fr(\()p Fq(D)1345 3905 y Fn(\030)1384 3890 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(D)1742 3905 y Fn(\030)1781 3890 y Fq(S)1837 3904 y Fn(t)1866 3890 y Fr(\()p Fq(t;)g(\030)t Fr(\))p Fq(D)2131 3853 y Fp(2)2128 3913 y Fn(\030)2171 3890 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))21 b(+)f Fm(r)p Fq(V)g Fr(\()p Fq(D)2790 3905 y Fn(\030)2829 3890 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(D)3190 3853 y Fp(2)3187 3913 y Fn(\030)3230 3890 y Fq(S)3286 3904 y Fn(t)3316 3890 y Fr(\()p Fq(t;)g(\030)t Fr(\))-3550 b(\(4.28\))-47 4086 y(By)32 b(\(4.26\))g(and)e(\(4.17\))j(one)d(has)h Fq(D)1203 4101 y Fn(\030)1241 4086 y Fq(S)1297 4100 y Fn(t)1351 4086 y Fr(=)25 b Fq(O)s Fr(\(1\),)32 b(and)d(b)m(y)j (\(4.27\))844 4302 y Fq(D)922 4264 y Fp(2)919 4325 y Fn(\030)961 4302 y Fq(S)1017 4316 y Fn(t)1046 4302 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))27 b(=)e Fq(O)s Fr(\()p Fq(t)1496 4264 y Fk(\000)1561 4237 y Fj(3)p 1560 4249 31 3 v 1560 4290 a(4)1601 4264 y Fk(\000)p Fp(2)p Fn(\014)1738 4302 y Fq(t)1771 4264 y Fp(2)1831 4302 y Fr(+)20 b Fq(t)1955 4264 y Fk(\000)2020 4237 y Fj(3)p 2019 4249 V 2019 4290 a(4)2060 4264 y Fk(\000)p Fn(\014)2162 4302 y Fq(t)2195 4264 y Fp(1+)p Fn(\026)p Fp(\(2\)+)p Fn(")2509 4302 y Fr(\))26 b(=)e Fq(O)s Fr(\()p Fq(t)2815 4237 y Fj(5)p 2815 4249 V 2815 4290 a(4)2856 4264 y Fk(\000)p Fp(2)p Fn(\014)2993 4302 y Fr(\))p Fq(:)-47 4498 y Fr(Inserting)29 b(these)i(estimates)g(in)m(to)f(\(4.28\))j(sho)m(ws)d(that)916 4709 y Fq(D)991 4724 y Fn(\030)1029 4709 y Fq(S)1085 4723 y Fn(tt)1140 4709 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))26 b(=)f Fq(O)s Fr(\()p Fq(t)1589 4671 y Fk(\000)1654 4644 y Fj(3)p 1654 4656 V 1654 4697 a(4)1695 4671 y Fk(\000)p Fp(2)p Fn(\014)1832 4709 y Fq(t)20 b Fr(+)g Fq(t)2009 4671 y Fk(\000)2074 4644 y Fj(3)p 2073 4656 V 2073 4697 a(4)2114 4671 y Fk(\000)p Fn(\014)2216 4709 y Fq(t)2259 4644 y Fj(5)p 2259 4656 V 2259 4697 a(4)2299 4671 y Fk(\000)p Fp(2)p Fn(\014)2436 4709 y Fr(\))26 b(=)f Fq(O)s Fr(\()p Fq(t)2743 4644 y Fj(1)p 2743 4656 V 2743 4697 a(4)2783 4671 y Fk(\000)p Fp(2)p Fn(\014)2921 4709 y Fr(\))p Fq(:)-3028 b Fr(\(4.29\))1903 4992 y(31)p eop %%Page: 32 32 32 31 bop -47 274 a Fr(T)-8 b(o)25 b(estimate)g(the)f(di\013erence)29 b(~)-50 b Fq(\034)1030 289 y Fn(`)p Fp(+1)1161 274 y Fm(\000)8 b Fq(\034)1280 289 y Fn(`)p Fp(+1)1427 274 y Fr(via)24 b(Lemma)g(2.2,)j(one)e(uses)f(\(4.26\))i(to)g(conclude)d (that)i Fq(\016)j Fr(in)23 b(that)i(lemma)-47 387 y(satis\014es)449 591 y Fq(\016)k Fm(\024)93 b Fr(sup)614 669 y Fn(T)10 b Fk(\024)p Fn(t)p Fk(\024)p Fn(T)851 650 y Fb(\003)902 591 y Fm(j)16 b Fr(_)-41 b Fq(z)969 606 y Fn(`)1003 591 y Fr(\()p Fq(t)p Fr(\))21 b Fm(\000)e Fq(D)1292 606 y Fn(\030)1330 591 y Fq(S)1386 605 y Fn(t)1416 591 y Fr(\()p Fq(t;)c(\030)t Fr(\))p Fm(j)26 b(\024)93 b Fr(sup)1750 669 y Fn(T)10 b Fk(\024)p Fn(t)p Fk(\024)p Fn(T)1987 650 y Fb(\003)2038 591 y Fm(j)p Fq(D)2138 606 y Fn(\030)2177 591 y Fq(S)2233 605 y Fn(t)2262 591 y Fr(\()p Fq(\034)2337 606 y Fn(`)2370 591 y Fq(;)15 b(\030)t Fr(\))21 b Fm(\000)f Fq(D)2676 606 y Fn(\030)2714 591 y Fq(S)2770 605 y Fn(t)2799 591 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)27 b(\024)e Fq(C)7 b(T)3281 526 y Fj(1)p 3281 538 31 3 v 3281 580 a(4)3321 554 y Fk(\000)p Fn(\014)3423 591 y Fq(:)-47 851 y Fr(Hence)1536 1056 y Fm(j)e Fr(~)-50 b Fq(\034)1601 1071 y Fn(`)p Fp(+1)1744 1056 y Fm(\000)20 b Fq(\034)1875 1071 y Fn(`)p Fp(+1)1998 1056 y Fm(j)26 b(\024)f Fq(C)7 b(T)2292 991 y Fj(1)p 2292 1003 V 2292 1044 a(4)2336 1056 y Fq(:)-2408 b Fr(\(4.30\))-47 1260 y(The)30 b(error)g(terms)g(in) f(\(4.25\))k(are)d(therefore)h(b)s(ounded)d(b)m(y)393 1393 y Fl(X)444 1590 y Fn(`)540 1479 y Fq(O)s Fr(\()p Fq(T)723 1414 y Fj(1)p 723 1426 V 723 1467 a(4)767 1479 y Fq(T)833 1441 y Fk(\000)898 1414 y Fj(3)p 898 1426 V 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1966 y Fn(\013)2019 2004 y Fq(V)20 b Fr(\()p Fq(z)2169 2019 y Fn(`)2202 2004 y Fr(\()p Fq(t)p Fr(\)\))15 b Fq(dt:)-47 2293 y Fr(One)30 b(c)m(hec)m(ks)h(that)g Fm(k)p Fq(Y)720 2308 y Fn(`)754 2293 y Fm(k)799 2307 y Fk(1)899 2293 y Fm(\024)25 b Fq(C)7 b(T)1133 2257 y Fk(\000)1198 2230 y Fj(3)p 1197 2242 V 1197 2283 a(4)1238 2257 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j\000)p Fp(1\))p Fn(\014)1599 2293 y Fr(and)30 b(hence)1301 2452 y Fl(\020)1355 2466 y(X)1406 2664 y Fn(`)1501 2552 y Fm(k)p Fq(Y)1599 2567 y Fn(`)1632 2552 y Fm(k)1677 2515 y Fp(2)1677 2575 y Fk(1)1753 2452 y Fl(\021)1817 2447 y Fj(1)p 1817 2459 V 1817 2500 a(2)1887 2552 y Fm(\024)25 b Fq(C)7 b(T)2121 2515 y Fk(\000)2186 2488 y Fj(1)p 2185 2500 V 2185 2541 a(4)2225 2515 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j\000)2457 2488 y Fj(1)p 2457 2500 V 2457 2541 a(2)2497 2515 y Fp(\))p Fn(\014)2572 2552 y Fq(:)-2644 b Fr(\(4.32\))-47 2860 y(One)34 b(easily)f(c)m(hec)m(ks)j(that)e(the)h 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Fq(t)1159 4312 y Fp(0)1199 4298 y Fq( )s Fr(\()p Fq(\030)t Fr(\))h Fm(\000)f Fr(\()p Fq(t)g Fm(\000)g Fq(t)1699 4312 y Fp(0)1738 4298 y Fr(\))p Fm(j)p Fq(\030)t Fm(j)1867 4261 y Fp(2)1908 4298 y Fq(=)p Fr(2)1998 4224 y Fl(\003)2036 4221 y(\014)2036 4275 y(\014)2092 4298 y Fm(\024)25 b Fq(C)2253 4312 y Fn(\013)2318 4298 y Fq(t)2351 4261 y Fp(1+)p Fn(\027)t Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(\)+)p Fn(")2776 4298 y Fg(for)33 b(al)5 b(l)33 b Fm(j)p Fq(\013)p Fm(j)26 b(\024)f Fr(2)-3373 b(\(4.33\))-47 4521 y Fg(Her)-5 b(e)32 b Fq(\027)6 b Fr(\(0\))26 b(=)468 4480 y Fn(\014)p 468 4500 44 4 v 472 4552 a Fp(2)541 4521 y Fm(\000)642 4485 y Fp(5)p 642 4500 36 4 v 642 4552 a(4)688 4521 y Fg(,)32 b Fq(\027)6 b Fr(\(1\))26 b(=)f Fm(\000)1117 4485 y Fp(1)p 1117 4500 V 1117 4552 a(4)1182 4521 y Fm(\000)1283 4480 y Fn(\014)p 1283 4500 44 4 v 1287 4552 a Fp(2)1336 4521 y Fg(,)32 b(and)i Fq(\027)6 b Fr(\(2\))26 b(=)1871 4485 y Fp(3)p 1871 4500 36 4 v 1871 4552 a(4)1936 4521 y Fm(\000)2037 4480 y Fp(3)p 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b(are)f(going)g(to)h(pull)c Fq(V)20 b Fr(\()p Fq(x)p Fr(\))38 b(inside)c(the)k(\014rst)e(in)m(tegral.)60 b(This)35 b(can)i(b)s(e)f(accomplished)g(b)m(y)h(means)g(of)-47 4743 y(T)-8 b(a)m(ylor)30 b(expansion)g(and)f(the)i(follo)m(wing)e (simple)f(tec)m(hnical)i(lemma.)1903 4992 y(35)p eop %%Page: 36 36 36 35 bop -47 274 a Fd(Lemma)32 b(5.1.)47 b Fg(F)-7 b(or)33 b(any)g(multi-index)g Fq(\013)g Fg(one)g(has)490 470 y Fq(e)532 432 y Fn(i)p Fp(\010)612 470 y Fr(\()p Fq(D)s Fr(\010\))826 432 y Fn(\013)901 470 y Fr(=)25 b(\()p Fm(\000)p Fq(i)p Fr(\))1169 432 y Fk(j)p Fn(\013)p Fk(j)1258 470 y Fq(D)1336 432 y Fn(\013)1385 470 y Fq(e)1427 432 y Fn(i)p Fp(\010)1527 470 y Fr(+)1859 384 y Fl(X)1618 585 y Fn(m)p Fk(\025)p Fp(1)p Fn(;)p Fk(j)p Fn(\027)1845 594 y Fj(1)1879 585 y Fk(j)p Fn(;:::)10 b(;)p Fk(j)p Fn(\027)2064 593 y Fh(m)2121 585 y Fk(j\025)p Fp(2)1661 651 y Fn(\027)1696 660 y Fj(1)1730 651 y Fp(+)p Fn(:::)o Fp(+)p Fn(\027)1934 659 y Fh(m)1992 651 y Fp(+)p Fn(\015)t Fp(=)p Fn(\013)2246 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Fr(standard)c(basis)h(v)m(ector.)42 b(Th)m(us)226 1406 y Fq(e)268 1368 y Fn(i)p Fp(\010)347 1406 y Fr(\()p Fq(D)s Fr(\010\))561 1368 y Fn(\013)p Fp(+)p Fn(e)694 1380 y Fh(`)811 1406 y Fr(=)83 b Fm(\000)p Fq(iD)1142 1421 y Fn(`)1175 1406 y Fr(\()p Fq(e)1252 1368 y Fn(i)p Fp(\010)1332 1406 y Fr(\()p Fq(D)s Fr(\010\))1546 1368 y Fn(\013)1596 1406 y Fr(\))20 b(+)g Fq(ie)1815 1368 y Fn(i)p Fp(\010)1895 1406 y Fq(D)1970 1421 y Fn(`)2003 1305 y Fl(h)2046 1406 y Fr(\()p Fq(D)s Fr(\010\))2260 1368 y Fn(\013)2310 1305 y Fl(i)811 1603 y Fr(=)83 b Fm(\000)p Fq(iD)1142 1618 y Fn(`)1175 1502 y Fl(h)1218 1603 y Fr(\()p Fm(\000)p Fq(i)p Fr(\))1390 1565 y Fk(j)p Fn(\013)p Fk(j)1480 1603 y Fq(D)1558 1565 y Fn(\013)1607 1603 y Fq(e)1649 1565 y Fn(i)p Fp(\010)1749 1603 y Fr(+)2081 1516 y Fl(X)1840 1718 y Fn(m)p Fk(\025)p Fp(1)p Fn(;)p Fk(j)p Fn(\027)2067 1727 y Fj(1)2101 1718 y Fk(j)p Fn(;:::)9 b(;)p Fk(j)p Fn(\027)2285 1726 y Fh(m)2343 1718 y Fk(j\025)p Fp(2)1883 1783 y Fn(\027)1918 1792 y Fj(1)1952 1783 y Fp(+)p Fn(:::)o Fp(+)p Fn(\027)2156 1791 y Fh(m)2214 1783 y Fp(+)p Fn(\015)t Fp(=)p Fn(\013)2468 1603 y Fr(co)s(e\013)16 b Fq(D)2743 1565 y Fn(\027)2778 1574 y Fj(1)2816 1603 y Fr(\010)k Fm(\001)g Fq(:)15 b(:)g(:)22 b Fm(\001)e Fq(D)3197 1565 y Fn(\027)3232 1573 y Fh(m)3294 1603 y Fr(\010)25 b Fq(D)3463 1565 y Fn(\015)3508 1603 y Fq(e)3550 1565 y Fn(i)p Fp(\010)3629 1502 y Fl(i)-47 1603 y Fr(\(5.6\))1147 2007 y(+)p Fq(ie)1291 1970 y Fn(i)p Fp(\010)1433 1894 y Fn(d)1386 1921 y Fl(X)1390 2116 y Fn(j)t Fp(=1)1532 2007 y Fq(\013)1590 2021 y Fn(j)1627 2007 y Fr(\()p Fq(D)s Fr(\010\))1841 1970 y Fn(\013)p Fk(\000)p Fn(e)1974 1980 y Fh(j)2011 2007 y Fq(D)2086 2022 y Fn(j)t(`)2151 2007 y Fr(\010)p Fq(:)-2289 b Fr(\(5.7\))-47 2289 y(Applying)30 b(the)j(pro)s(duct)e(rule)h(in)f(\(5.6\))k(and)d (the)g(induction)f(h)m(yp)s(othesis)g(in)g(\(5.7\))k(yields)c(an)h (expression)g(of)h(the)-47 2402 y(desired)c(form.)p 3879 2402 4 62 v 3883 2344 55 4 v 3883 2402 V 3937 2402 4 62 v -47 2587 a(F)-8 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4548 y Fn(\015)1868 4512 y Fl(\002)1905 4586 y Fq(D)1983 4548 y Fn(\027)2018 4557 y Fj(1)2057 4586 y Fq(S)25 b Fm(\001)20 b Fq(:)15 b(:)g(:)22 b Fm(\001)e Fq(D)2433 4548 y Fn(\027)2468 4556 y Fh(m)2530 4586 y Fq(S)31 b(D)2695 4548 y Fn(\013)2692 4608 y(x)2744 4586 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))g Fq(a)p Fr(\()p Fq(t;)g(\030)t Fr(\))3475 4562 y(^)3463 4586 y Fq(\036)s Fr(\()p Fq(\030)t Fr(\))3634 4512 y Fl(\003)3687 4586 y Fq(d\030)t(:)-3850 b Fr(\(5.11\))1903 4992 y(36)p eop %%Page: 37 37 37 36 bop -47 274 a Fr(Observ)m(e)22 b(that)h(here)f(w)m(e)h(ha)m(v)m (e)g(replaced)f Fq(D)1428 241 y Fn(\027)1463 251 y Fh(j)1499 274 y Fr(\010)g(with)f Fq(D)1864 241 y Fn(\027)1899 251 y Fh(j)1935 274 y Fq(S)27 b Fr(since)22 b Fm(j)p Fq(\027)2303 288 y Fn(j)2340 274 y Fm(j)j(\025)g Fr(2.)39 b(In)21 b(order)h(to)h(obtain)f(a)h Fq(\036)p Fr(-indep)s(enden)m(t)-47 387 y(de\014nition)32 b(of)j(the)g(amplitude)e Fq(a)p Fr(,)j(one)f(needs)g(to)g(c)m(hec)m(k)i(that)e(those)g(terms)g(in)f (\(5.10\))j(and)d(\(5.11\))j(in)d(whic)m(h)f(at)-47 500 y(least)28 b(one)h(deriv)-5 b(ativ)m(e)28 b(falls)e(on)1067 476 y(^)1055 500 y Fq(\036)p Fr(\()p Fq(\030)t Fr(\))j(are)f (negligible,)f(in)g(the)h(sense)h(that)f(they)h(giv)m(e)g(a)f Fq(t)p Fr(-in)m(tegrable)g(con)m(tribution)-47 613 y(to)41 b(\(5.5\))r(.)72 b(This)39 b(will)f(require)i(using)f(the)i(follo)m (wing)e(b)s(ounds)g(on)h Fq(S)46 b Fr(from)40 b(the)h(previous)e (section,)44 b(see)e(\(4.17\))-47 726 y(and)30 b(\(4.33\))r(:)41 b(There)29 b(is)h Fq(c)25 b(<)952 690 y Fp(1)p 952 705 36 4 v 952 757 a(2)1028 726 y Fr(\(in)k(fact,)j Fq(c)25 b Fr(=)g(1)c Fm(\000)f Fq(\014)5 b Fr(\))31 b(and)f Fq(")25 b(>)g Fr(0)31 b(so)g(that)g(for)f(all)f(large)h Fq(t)808 927 y Fm(j)p Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)21 b Fr(+)f Fm(j)p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)27 b Fe(.)e Fq(t;)40 b Fm(j)p Fq(D)1918 889 y Fn(\013)1968 927 y Fq(S)5 b 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Fr(\))47 b(=)f(\(1)29 b Fm(\000)f Fq(\014)5 b Fr(\))p Fq(j)35 b Fm(\000)1513 1393 y Fp(1)p 1513 1408 36 4 v 1513 1460 a(4)1587 1429 y Fm(\000)1696 1388 y Fn(\014)p 1696 1408 44 4 v 1700 1460 a Fp(2)1795 1429 y Fr(=)46 b(\(1)29 b Fm(\000)f Fq(\014)5 b Fr(\)\()p Fq(j)35 b Fm(\000)29 b Fr(1\))g(+)2636 1393 y Fp(3)p 2636 1408 36 4 v 2636 1460 a(4)2710 1429 y Fm(\000)2819 1388 y Fp(3)p Fn(\014)p 2819 1408 79 4 v 2840 1460 a Fp(2)2907 1429 y Fr(.)78 b(Since)3270 1393 y Fp(3)p 3270 1408 36 4 v 3270 1460 a(4)3344 1429 y Fm(\000)3453 1388 y Fp(3)p Fn(\014)p 3453 1408 79 4 v 3475 1460 a Fp(2)3570 1429 y Fr(+)28 b Fq(")46 b Fm(\024)g Fr(0,)-47 1542 y(the)27 b(estimate)h(ab)s(o)m(v)m(e)g(holds.)38 b(\\Large")29 b Fq(t)e Fr(means)g(that)h Fq(t)d Fm(\025)g Fq(t)2051 1556 y Fp(0)2117 1542 y Fr(with)h(some)h(random)f Fq(t)2908 1556 y Fp(0)2975 1542 y Fr(that)h(also)g(dep)s(ends)f(on)h (the)-47 1655 y(compact)i(subset)f(of)h Fo(R)752 1622 y Fn(d)815 1655 y Fm(n)17 b(f)p Fr(0)p Fm(g)29 b Fr(to)g(whic)m(h)f Fq(\030)k Fr(b)s(elongs.)39 b(Clearly)-8 b(,)27 b(\(5.13\))k(ensures)c (that)i(the)g(phase)f(\010)g(has)g(a)h(unique)-47 1768 y(critical)g(p)s(oin)m(t)h(on)g(the)h(supp)s(ort)e(of)1230 1744 y(^)1218 1768 y Fq(\036)h Fr(pro)m(vided)g Fq(t)g Fr(is)g(su\016cien)m(tly)f(large.)41 b(W)-8 b(e)32 b(denote)f(this)f (critical)f(p)s(oin)m(t)h(b)m(y)g Fq(\030)3880 1782 y Fp(0)3920 1768 y Fr(.)-47 1881 y(F)-8 b(or)31 b(the)f(follo)m(wing)f (lemma)h(w)m(e)h(need)f(to)h(assume)f(that)h(for)g(all)e(large)h Fq(t)g Fr(and)g(an)m(y)h(m)m(ulti-index)d Fq(\015)1323 2068 y Fm(j)p Fq(D)1426 2030 y Fn(\015)1471 2068 y Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)21 b Fr(+)f Fm(j)p Fq(D)1946 2030 y Fn(\015)1990 2068 y Fq(a)2038 2082 y Fn(t)2068 2068 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)26 b Fe(.)f Fq(t)2435 2030 y Fn(c)p Fk(j)p Fn(\015)t Fk(j)2549 2068 y Fq(:)-2621 b Fr(\(5.14\))-47 2255 y(This)36 b(b)s(ound)g(will)f (b)s(e)i(pro)m(v)m(en)h(b)s(elo)m(w)f(when)g(w)m(e)h(construct)g(the)g (amplitude)e Fq(a)p Fr(.)63 b(The)37 b(follo)m(wing)g(lemma)g(is)g(an) -47 2368 y(instance)j(of)g(the)h(w)m(ell-kno)m(wn)e(fact)i(that)g(with) e(high)g(probabilit)m(y)e(a)k(particle)f(lies)e(in)h(the)i(\\classical) f(region")-47 2481 y Fm(j)p Fq(x)p Fm(j)25 b(\020)g Fq(t)p Fr(.)41 b(Because)32 b(of)e(this)f(prop)s(ert)m(y)h(it)g(su\016ces)g (to)h(restrict)f Fq(L)2162 2448 y Fp(2)2162 2503 y Fn(x)2236 2481 y Fr(in)g(\(5.5\))i(to)f Fq(L)2733 2448 y Fp(2)2772 2481 y Fr(\()p Fm(j)p Fq(x)p Fm(j)26 b(\020)f Fq(t)p Fr(\).)-47 2654 y Fd(Lemma)32 b(5.2.)47 b Fg(Under)32 b(the)h(assumption)42 b Fr(\(5.14\))35 b Fg(one)e(has)-7 2900 y Fl(\015)-7 2955 y(\015)-7 3009 y(\015)43 3005 y Fq(V)21 b Fr(\()p Fq(x)p Fr(\))254 2881 y Fl(Z)305 3087 y Fi(R)353 3068 y Fh(d)408 3005 y Fq(e)450 2967 y Fn(i)p Fp(\010\()p Fn(t;x)p Fp(;)p Fn(\030)s Fp(\))748 3005 y Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))995 2981 y(^)983 3005 y Fq(\036)q Fr(\()p Fq(\030)t Fr(\))g Fq(d\030)25 b Fm(\000)1370 2881 y Fl(Z)1421 3087 y Fi(R)1469 3068 y Fh(d)1524 3005 y Fq(e)1566 2967 y Fn(i)p Fp(\010\()p Fn(t;x)p Fp(;)p Fn(\030)s Fp(\))1864 3005 y Fr([)p Fq(V)20 b Fr(\()p Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(a)p Fr(\()p Fq(t;)g(\030)t Fr(\))23 b(+)d Fq(ia)2761 3019 y Fn(t)2791 3005 y Fr(\()p Fq(t;)15 b(\030)t Fr(\)])3015 2981 y(^)3003 3005 y Fq(\036)r Fr(\()p Fq(\030)t Fr(\))g Fq(d\030)3279 2900 y Fl(\015)3279 2955 y(\015)3279 3009 y(\015)3330 3068 y Fn(L)3378 3049 y Fj(2)3413 3068 y Fp(\()p Fk(j)p Fn(x)p Fk(j6\020)p Fn(t)p Fp(\))3656 3005 y Fe(.)25 b Fq(t)3785 2967 y Fk(\000)p Fp(2)3880 3005 y Fq(:)-47 2839 y Fr(\(5.15\))-47 3246 y Fg(Her)-5 b(e)37 b Fm(j)p Fq(x)p Fm(j)e(6\020)e Fq(t)38 b Fg(me)-5 b(ans)38 b(that)h(either)f Fm(j)p Fq(x)p Fm(j)c(\024)g Fq(A)1534 3213 y Fk(\000)p Fp(1)1629 3246 y Fq(t)j Fg(or)h Fm(j)p Fq(x)p Fm(j)d(\025)e Fq(At)38 b Fg(for)g(some)g(c)-5 b(onstant)39 b Fq(A)c(>)e Fr(0)38 b Fg(that)h(only)f(dep)-5 b(ends)-47 3358 y(on)33 b Fr(supp)n(\()317 3334 y(^)305 3358 y Fq(\036)q Fr(\))p Fg(.)42 b(In)32 b(p)-5 b(articular,)35 b(it)e(su\016c)-5 b(es)33 b(to)g(pr)-5 b(ove)40 b Fr(\(5.5\))35 b Fg(in)d(the)h(r)-5 b(e)g(gion)34 b Fm(j)p Fq(x)p Fm(j)26 b(\020)f Fq(t)p Fg(.)-47 3532 y(Pr)-5 b(o)g(of.)47 b Fr(Fix)31 b(some)h Fq(\036)d Fm(2)e(S)39 b Fr(with)31 b(supp)n(\()1339 3508 y(^)1327 3532 y Fq(\036)p Fr(\))e Fm(\032)e(f)p Fr(2)p Fq(A)1701 3499 y Fk(\000)p Fp(1)1825 3532 y Fq(<)g Fm(j)p Fq(\030)t Fm(j)i Fq(<)e(A=)p Fr(2)p Fm(g)p Fr(,)34 b(and)e Fq(A)g Fr(some)g(large)g(constan)m(t.)47 b(In)31 b(view)h(of)-47 3645 y(\(4.18\))g(one)f(has)1335 3847 y Fm(j)p Fq(D)s Fr(\010\()p Fq(t;)15 b(x)p Fr(;)g Fq(\030)t Fr(\))p Fm(j)27 b Fr(=)e Fm(j)p Fq(x)20 b Fm(\000)g Fq(t\030)t Fm(j)g Fr(+)g Fq(O)s Fr(\()p Fq(t)2482 3782 y Fj(1)p 2482 3794 31 3 v 2482 3835 a(2)2527 3847 y Fr(\))-47 4034 y(for)30 b(large)g Fq(t)h Fr(and)e Fq(A)622 4001 y Fk(\000)p Fp(1)742 4034 y Fm(\024)c(j)p Fq(\030)t Fm(j)h(\024)f Fq(A)p Fr(.)40 b(Th)m(us,)797 4221 y Fm(j)p Fq(D)s Fr(\010\()p Fq(t;)15 b(x)p Fr(;)g Fq(\030)t Fr(\))p Fm(j)27 b(\020)e(j)p Fq(x)p Fm(j)20 b Fr(+)g Fq(t)61 b Fr(in)29 b(the)h(region)61 b Fm(j)p Fq(x)p Fm(j)25 b Fq(>)g(At;)41 b Fm(j)p Fq(x)p Fm(j)26 b Fq(<)f(A)2948 4184 y Fk(\000)p Fp(1)3042 4221 y Fq(t:)-47 4409 y Fr(W)-8 b(e)38 b(will)d(in)m(tegrate)j(eac)m(h)g(of) g(the)f Fq(\030)t Fr(-in)m(tegrals)g(in)f(\(5.15\))k(b)m(y)d(parts)g (separately)-8 b(.)61 b(In)37 b(order)g(to)h(do)f(so,)i(one)f(\014rst) -47 4522 y(c)m(hec)m(ks)32 b(that)1304 4709 y Fm(j)p Fq(D)1407 4671 y Fn(\013)1457 4709 y Fr([)p Fq(V)20 b Fr(\()p Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(a)p Fr(\()p Fq(t;)g(\030)t Fr(\)])p Fm(j)29 b Fe(.)c Fq(t)2369 4671 y Fn(c)p Fk(j)p Fn(\013)p Fk(j\000)2550 4644 y Fj(3)p 2549 4656 V 2549 4697 a(4)-47 4709 y Fr(\(5.16\))1903 4992 y(37)p eop %%Page: 38 38 38 37 bop -47 274 a Fr(b)m(y)31 b(\(5.14\))i(and)d(\(5.12\))r(.)41 b(Indeed,)328 475 y Fq(D)406 438 y Fn(\013)455 475 y Fr([)p Fq(V)21 b Fr(\()p Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\)\)])27 b(=)1395 389 y Fl(X)1073 591 y Fk(j)p Fn(\013)1138 568 y Fb(0)1161 591 y Fk(j)p Fp(=)p Fn(m)p Fk(\025)p Fp(1)p Fn(;)p Fk(j)p Fn(\027)1463 600 y Fj(1)1496 591 y Fk(j)p Fn(;:::)10 b(;)p Fk(j)p Fn(\027)1681 599 y Fh(m)1738 591 y Fk(j\025)p Fp(2)1128 667 y Fk(j)p Fn(\027)1183 676 y Fj(1)1216 667 y Fk(j)p Fp(+)p Fn(:::)o Fp(+)p Fk(j)p Fn(\027)1460 675 y Fh(m)1518 667 y Fk(j)p Fp(=)p Fk(j)p Fn(\013)p Fk(j)p Fp(+)p Fn(m)1863 475 y Fr(co)s(e\013)16 b Fq(D)2138 438 y Fn(\013)2183 414 y Fb(0)2135 498 y Fn(x)2210 475 y Fq(V)k Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))g Fq(D)2772 438 y Fn(\027)2807 447 y Fj(1)2847 475 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)3223 438 y Fn(\027)3258 446 y Fh(m)3321 475 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))-47 844 y(so)30 b(that)h(\(with)f(the)g(sum)g(o)m(v)m(er)h(the)g(same)g (indices\))720 1046 y Fm(j)p Fq(D)823 1008 y Fn(\013)873 1046 y Fr([)p Fq(V)20 b Fr(\()p Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\)\)])p Fm(j)28 b Fe(.)1516 960 y Fl(X)1662 1046 y Fq(t)1695 1008 y Fk(\000)1760 981 y Fj(3)p 1760 993 31 3 v 1760 1035 a(4)1800 1008 y Fk(\000)p Fn(\014)s Fk(j)p Fn(\013)1963 985 y Fb(0)1985 1008 y Fk(j)2024 1046 y Fq(t)2057 1006 y Fn(m)p Fp(+)p Fn(c)p Fp(\()2232 956 y Fc(P)2307 976 y Fh(m)2307 1027 y(j)s Fj(=1)2428 1006 y Fk(j)p Fn(\027)2483 1016 y Fh(j)2515 1006 y Fk(j\000)p Fp(2)p Fn(m)p Fp(\))2744 1046 y Fe(.)d Fq(t)2873 1008 y Fk(\000)2938 981 y Fj(3)p 2938 993 V 2938 1035 a(4)2978 1008 y Fp(+)p Fn(c)p Fk(j)p Fn(\013)p Fk(j)3152 1046 y Fq(:)-47 1247 y Fr(Com)m(bining)35 b(this)i(with)g (\(5.14\))j(via)e(the)g(pro)s(duct)e(rule)h(yields)f(\(5.16\))r(.)64 b(Moreo)m(v)m(er,)42 b(one)c(has)g Fm(j)p Fq(D)3416 1214 y Fn(\013)3465 1247 y Fr(\010\()p Fq(t;)15 b(x)p Fr(;)g Fq(\030)t Fr(\))p Fm(j)39 b Fr(=)-47 1360 y Fm(j)p Fq(D)56 1327 y Fn(\013)105 1360 y Fq(S)5 b Fr(\()p Fq(t)p Fr(;)15 b Fq(\030)t Fr(\))p Fm(j)27 b Fe(.)e Fq(t)534 1327 y Fp(1+)p Fn(c)p Fp(\()p Fk(j)p Fn(\013)p Fk(j\000)p Fp(2\))918 1360 y Fr(for)k Fm(j)p Fq(\013)p Fm(j)e(\025)e Fr(2)30 b(b)m(y)h(\(5.12\))r(.)41 b(Th)m(us)29 b(eac)m(h)i(of)f(the)g Fq(\030)t Fr(-in)m(tegrals)g(in)e(\(5.15\))33 b(is)c(no)g(larger)h (than)-47 1473 y(\()p Fq(t)20 b Fr(+)g Fm(j)p Fq(x)p Fm(j)p Fr(\))269 1440 y Fk(\000)p Fn(p)364 1473 y Fq(t)397 1440 y Fn(cp)498 1473 y Fr(where)30 b Fq(p)g Fr(is)f(the)i(n)m(um)m(b)s (er)e(of)h(partial)f(in)m(tegrations.)41 b(Hence,)32 b(\(5.15\))g(itself)d(is)h(con)m(trolled)g(b)m(y)1087 1615 y Fl(\020)1141 1592 y(Z)1192 1799 y Fi(R)1240 1780 y Fh(d)1280 1716 y Fr(\()p Fq(t)20 b Fr(+)g Fm(j)p Fq(x)p Fm(j)p Fr(\))1596 1679 y Fk(\000)p Fp(2)p Fn(p)1727 1716 y Fq(t)1760 1679 y Fp(2)p Fn(cp)1881 1716 y Fq(dx)1980 1615 y Fl(\021)2044 1611 y Fj(1)p 2044 1623 V 2044 1664 a(2)2114 1716 y Fe(.)25 b Fq(t)2243 1679 y Fn(cp)p Fp(+)2375 1651 y Fh(d)p 2374 1663 33 3 v 2375 1705 a Fj(2)2435 1716 y Fq(t)2468 1679 y Fk(\000)p Fn(p)2588 1716 y Fe(.)f Fq(t)2716 1679 y Fk(\000)p Fp(2)-47 1945 y Fr(pro)m(vided)29 b Fq(p)h Fr(is)f(tak)m(en)j(to)f(b)s(e)f(large.)p 3879 1945 4 62 v 3883 1887 55 4 v 3883 1945 V 3937 1945 4 62 v -47 2129 a(W)-8 b(e)27 b(no)m(w)g(return)e(to)i(those)g(terms)f (in)f(\(5.10\))k(and)d(\(5.11\))i(in)d(whic)m(h)g(at)i(least)g(one)g (deriv)-5 b(ativ)m(e)26 b(falls)e(on)3517 2105 y(^)3505 2129 y Fq(\036)p Fr(\()p Fq(\030)t Fr(\).)40 b(Since)-47 2242 y(the)29 b(v)m(olume)g(of)g(the)g(region)f Fm(j)p Fq(x)p Fm(j)e(\020)f Fq(t)k Fr(is)e Fq(t)1356 2209 y Fn(d)1397 2242 y Fr(,)i(the)g(follo)m(wing)f(lemma)g(sho)m(ws)h(that)g (the)g(con)m(tribution)f(to)h Fq(L)3578 2209 y Fp(2)3618 2242 y Fr(\()p Fm(j)p Fq(x)p Fm(j)d(\020)f Fq(t)p Fr(\))-47 2355 y(in)33 b(\(5.5\))k(of)e(those)h(terms)e(is)g(in)m(tegrable)h(in)e (time)i(and)f(therefore)i(negligible.)52 b(W)-8 b(e)36 b(shall)d(mak)m(e)j(the)f(restriction)-47 2468 y Fm(j)p Fq(x)p Fm(j)25 b(\020)g Fq(t)31 b Fr(henceforth)f(without)f(further)g (men)m(tion.)-47 2642 y Fd(Lemma)j(5.3.)47 b Fg(Under)37 b(the)h(assumption)46 b Fr(\(5.14\))39 b Fg(the)f(inte)-5 b(gr)g(als)39 b(in)44 b Fr(\(5.10\))c Fg(and)47 b Fr(\(5.11\))40 b Fg(in)d(which)h(at)g(le)-5 b(ast)38 b(one)-47 2771 y(derivative)33 b(fal)5 b(ls)33 b(on)707 2747 y Fr(^)695 2771 y Fq(\036)g Fg(ar)-5 b(e)33 b Fe(.)25 b Fq(t)1065 2735 y Fk(\000)1130 2708 y Fj(5)p 1130 2720 31 3 v 1130 2761 a(4)1170 2735 y Fk(\000)1235 2708 y Fh(d)p 1235 2720 33 3 v 1236 2761 a Fj(2)1281 2771 y Fg(.)-47 2944 y(Pr)-5 b(o)g(of.)47 b Fr(W)-8 b(e)30 b(claim)f(that)h(the)g(in)m (tegrals)f(in)f(\(5.10\))k(and)c(\(5.11\))k(in)c(whic)m(h)h(at)h(least) g(one)f(deriv)-5 b(ativ)m(e)29 b(falls)f(on)3752 2920 y(^)3740 2944 y Fq(\036)h Fr(are)-47 3057 y(of)h(the)h(form)863 3140 y Fl(Z)914 3346 y Fi(R)962 3327 y Fh(d)1017 3264 y Fq(e)1059 3226 y Fn(i)p Fp(\010\()p Fn(t;x)p Fp(;)p Fn(\030)s Fp(\))1357 3264 y Fq(b)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))g Fq(d\030)66 b Fr(where)75 b(sup)2111 3343 y Fn(\030)2211 3264 y Fm(j)p Fq(D)2314 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(the)h(error)f(term)g(\(5.9\))j(can)d(b)s(e)g(treated)h(b)m(y)g(the)f (same)h(metho)s(d.)-47 2012 y Fd(Lemma)h(5.4.)47 b Fg(Under)28 b(the)h(assumption)37 b Fr(\(5.14\))30 b Fg(the)f(c)-5 b(ontribution)29 b(of)48 b Fr(\(5.9\))30 b Fg(is)e(no)h(lar)-5 b(ger)29 b(than)g Fq(t)3389 1976 y Fk(\000)p Fp(2)p Fk(\000)3544 1949 y Fh(d)p 3544 1961 33 3 v 3545 2002 a Fj(2)3618 2012 y Fg(pr)-5 b(ovide)g(d)-47 2125 y Fq(N)42 b Fg(is)33 b(lar)-5 b(ge.)-47 2303 y(Pr)g(o)g(of.)47 b Fr(Fix)c(some)h Fq(\013)h Fr(with)d Fm(j)p Fq(\013)p Fm(j)49 b Fr(=)f Fq(N)39 b Fr(+)29 b(1.)82 b(Let)44 b(\011\()p Fq(t;)15 b(x)p Fr(;)g Fq(\021)s Fr(\))46 b(b)s(e)d(de\014ned)g(b)m(y)i(\(5.19\)) h(and)d(set)i Fq(G)3492 2317 y Fn(\013)3542 2303 y Fr(\()p Fq(t;)15 b(x)p Fr(;)g Fq(\021)s Fr(\))49 b(=)-47 2416 y Fq(F)11 2430 y Fn(\013)61 2416 y Fr(\()p Fq(t;)15 b(x)p Fr(;)g Fq(\021)s(=t)387 2383 y Fn(c)423 2416 y Fr(\))31 b(where)1008 2668 y Fq(F)1066 2682 y Fn(\013)1116 2668 y Fr(\()p Fq(t;)15 b(x)p Fr(;)g Fq(\030)t Fr(\))26 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Fn(\013)2493 3372 y Fr(\()p Fq(t;)g(x)p Fr(;)g Fq(\021)s Fr(\))27 b(\()p Fq(@)2886 3386 y Fn(\021)2929 3372 y Fr(\011\()p Fq(t;)15 b(x)p Fr(;)g Fq(\021)s Fr(\)\))3318 3335 y Fn(\013)3384 3372 y Fq(d\021)s(:)-3551 b Fr(\(5.23\))-47 3670 y(As)22 b(b)s(efore,)i(one)e(applies)f(the)h(metho)s(d)g(of)g (stationary)h(phase)f(to)h(those)g(terms)f(in)f(\(5.23\))j(for)e(whic)m (h)f Fm(j)p Fq(t\030)3472 3684 y Fp(0)3515 3670 y Fm(\000)t Fq(g)s Fm(j)26 b(\024)f Fr(100,)-47 3783 y(where)34 b Fq(\030)260 3797 y Fp(0)333 3783 y Fr(is)f(the)i(critical)e(p)s(oin)m (t)g(of)h(\010\()p Fq(t;)15 b(x)p Fr(;)g Fq(\030)t Fr(\).)54 b(This)32 b(requires)h(deriv)-5 b(ativ)m(e)34 b(b)s(ounds)e(on)i Fq(G)3167 3797 y Fn(\013)3251 3783 y Fr(and)g(the)g(phase)g(\011.)-47 3896 y(The)c(latter)g(ones)h(w)m(ere)g(explained)d(in)h(the)i(previous) e(pro)s(of,)h(whereas)g Fq(G)2509 3910 y Fn(\013)2589 3896 y Fr(satis\014es)686 4099 y Fm(j)p Fq(@)764 4062 y Fn(\015)759 4122 y(\021)809 4099 y Fq(G)880 4113 y Fn(\013)930 4099 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b(error)h(terms,)k(w)m(e)c (conclude)g(from)g(the)h(preceding)e(\(see)j(in)d(particular)g(\(5.5\)) r(,)-47 1860 y(\(5.10\))32 b(and)e(\(5.11\))r(\))h(that)g(w)m(e)g (should)d(c)m(ho)s(ose)k Fq(a)e Fr(suc)m(h)g(that)382 2115 y Fq(ia)461 2129 y Fn(t)574 2115 y Fr(=)808 2028 y Fl(X)728 2230 y Fp(1)p Fk(\024j)p Fn(\013)p Fk(j\024)p Fn(N)1065 2053 y Fr(1)p 1046 2094 84 4 v 1046 2177 a Fq(\013)p Fr(!)1139 1986 y Fl(\032)1207 2115 y Fr(\()p Fm(\000)p Fq(i)p Fr(\))1379 2077 y Fk(j)p Fn(\013)p Fk(j)1469 2115 y Fq(D)1547 2077 y Fn(\013)1596 2041 y Fl(\002)1634 2115 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(a)p Fr(\()p Fq(t;)g(\030)t Fr(\))2338 2041 y Fl(\003)819 2420 y Fr(+)1145 2334 y Fl(X)905 2535 y Fn(m)p Fk(\025)p Fp(1)p Fn(;)p Fk(j)p Fn(\027)1132 2544 y Fj(1)1166 2535 y Fk(j)p Fn(;:::)9 b(;)p Fk(j)p Fn(\027)1350 2543 y Fh(m)1408 2535 y Fk(j\025)p Fp(2)948 2601 y Fn(\027)983 2610 y Fj(1)1017 2601 y Fp(+)p Fn(:::)o Fp(+)p Fn(\027)1221 2609 y Fh(m)1279 2601 y 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Fm(j)p Fq(D)1586 4644 y Fn(\015)1631 4682 y Fq(a)1679 4696 y Fn(j)1715 4682 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)27 b Fe(.)e Fq(t)2083 4644 y Fk(\000)p Fn(j)2180 4617 y Fh(\016)p 2180 4629 31 3 v 2180 4670 a Fj(2)2220 4644 y Fp(+)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)2389 4682 y Fq(:)-2461 b Fr(\(5.26\))1903 4992 y(41)p eop %%Page: 42 42 42 41 bop -47 274 a Fg(The)32 b(c)-5 b(onstants)34 b(implicit)e(in)g (this)h(notation)h(ar)-5 b(e)32 b(r)-5 b(andom)35 b(but)c(uniform)i (for)g Fq(\030)i Fg(in)d(c)-5 b(omp)g(act)34 b(subsets)e(of)h Fo(R)3651 241 y Fn(d)3716 274 y Fm(n)20 b(f)p Fr(0)p Fm(g)p Fg(.)-47 387 y(Final)5 b(ly,)916 603 y Fm(j)p Fq(D)1019 566 y Fn(\015)1063 603 y Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)27 b Fe(.)d Fq(t)1478 566 y Fk(\000)1543 539 y Fh(\016)p 1543 551 31 3 v 1543 592 a Fj(2)1584 566 y Fp(+)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)1818 603 y Fg(and)66 b Fm(j)p Fq(D)2130 566 y Fn(\015)2174 603 y Fq(a)2222 617 y Fn(t)2252 603 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)26 b Fe(.)f Fq(t)2619 566 y Fk(\000)2684 539 y Fj(3)p 2684 551 V 2684 592 a(4)2724 566 y Fk(\000)p Fn(\016)r Fp(+)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)-47 603 y Fr(\(5.27\))-47 808 y Fg(for)33 b(al)5 b(l)33 b Fq(\015)5 b Fg(,)33 b(which)g(pr)-5 b(oves)42 b Fr(\(5.14\))r Fg(.)-47 995 y(Pr)-5 b(o)g(of.)47 b Fr(By)38 b(induction.)63 b(Set)39 b Fq(a)1065 1009 y Fp(0)1143 995 y Fr(=)f(1)h(and)f(supp)s(ose)f(that)i Fq(a)2124 1009 y Fp(0)2164 995 y Fq(;)15 b(a)2252 1009 y Fp(1)2292 995 y Fq(;)g(:)g(:)g(:)31 b(;)15 b(a)2556 1009 y Fn(j)t Fk(\000)p Fp(1)2722 995 y Fr(ha)m(v)m(e)40 b(b)s(een)d(constructed)i(satisfy-)-47 1108 y(ing)31 b(\(5.26\))r(.)41 b(Then)29 b(de\014ne)399 1357 y Fq(a)447 1371 y Fn(j)484 1357 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))26 b(:=)f Fm(\000)p Fq(i)939 1270 y Fl(X)723 1472 y Fn(j)t Fp(=)p Fn(k)r Fk(\000)p Fp(1+4\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\))896 1543 y Fn(k)r Fk(\024)p Fn(j)t Fk(\000)p Fp(1)1301 1357 y Fr(co)s(e\013)1513 1233 y Fl(Z)1604 1259 y Fk(1)1564 1439 y 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b(:)g(:)g(:)32 b(;)15 b Fm(j)p Fq(\027)1450 2064 y Fn(m)1517 2050 y Fm(j)26 b(\025)f Fr(2)61 b(and)f Fq(\027)2022 2064 y Fp(1)2081 2050 y Fr(+)20 b Fq(:)15 b(:)g(:)22 b Fr(+)d Fq(\027)2434 2064 y Fn(m)2521 2050 y Fr(+)h Fq(\015)30 b Fr(=)25 b Fq(\013;)-47 2254 y Fr(see)32 b(\(5.25\))r(.)44 b(The)31 b(meaning)f(of)i(the)g(condition)e Fq(j)i Fr(=)27 b Fq(k)d Fm(\000)d Fr(1)g(+)f(4\()p Fm(j)p Fq(\013)p Fm(j)j(\000)e Fq(m)p Fr(\))31 b(will)e(b)s(ecome)j(clear)f(in)f(the)i(pro)s(of.)43 b(F)-8 b(or)-47 2367 y(the)37 b(momen)m(t)g(w)m(e)g(only)f(p)s(oin)m(t) g(out)h(that)g(it)f(insures)f(that)i(the)g(sum)f(in)f(\(5.28\))k(is)d (\014nite,)h(since)f Fm(j)p Fq(\013)p Fm(j)h(\025)e Fr(2)p Fq(m)p Fr(.)60 b(T)-8 b(o)-47 2480 y(v)m(erify)30 b(\(5.26\))h(for)f Fq(j)35 b Fr(w)m(e)29 b(b)s(egin)f(with)g(an)i(in)m(tegral)f(as)g(in)g (\(5.28\))i(that)f(do)s(es)f(not)h(con)m(tain)g(a)f(term)h Fq(D)3443 2447 y Fn(\015)3487 2480 y Fq(a)3535 2495 y Fn(k)3578 2480 y Fr(,)f(in)g(other)-47 2593 y(w)m(ords)22 b 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Fn(`)2033 2833 y Fq(;)c Fl(e)-62 b Fq(\034)2113 2848 y Fn(`)2176 2833 y Fr(as)31 b(in)e(\(4.24\))r(.)41 b(Th)m(us,)29 b(with)1200 3038 y Fq(z)1242 3053 y Fn(`)1275 3038 y Fr(\()p Fq(t)p Fr(\))d(:=)f Fq(@)1573 3053 y Fn(\030)1611 3038 y Fq(S)5 b Fr(\()p Fq(\034)1747 3053 y Fn(`)1780 3038 y Fq(;)15 b(\030)t Fr(\))21 b(+)f(\()p Fq(t)h Fm(\000)e Fq(\034)2230 3053 y Fn(`)2263 3038 y Fr(\))p Fq(@)2346 3053 y Fn(\030)2385 3038 y Fq(S)2441 3052 y Fn(t)2470 3038 y Fr(\()p Fq(\034)2545 3053 y Fn(`)2578 3038 y Fq(;)c(\030)t Fr(\))-47 3242 y(one)30 b(has)193 3362 y Fl(Z)284 3388 y Fn(T)335 3365 y Fb(\003)243 3568 y Fn(T)390 3485 y Fq(D)468 3448 y Fn(\013)465 3508 y(x)517 3485 y Fq(V)21 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(D)1065 3448 y Fn(\027)1100 3457 y Fj(1)1139 3485 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)g Fq(D)1516 3448 y Fn(\027)1551 3456 y Fh(m)1613 3485 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))g Fq(dt)284 3747 y Fr(=)380 3660 y Fl(X)431 3858 y Fn(`)526 3623 y Fl(Z)619 3649 y Fc(e)-41 b Fn(\034)648 3661 y Fh(`)p Fj(+1)577 3829 y Fn(\034)608 3841 y Fh(`)775 3747 y Fq(D)853 3709 y Fn(\013)850 3769 y(x)902 3747 y Fq(V)20 b Fr(\()p Fq(z)1052 3762 y Fn(`)1086 3747 y Fr(\()p Fq(t)p Fr(\)\))15 b Fq(D)1317 3709 y Fn(\027)1352 3718 y Fj(1)1391 3747 y Fq(S)5 b Fr(\()p Fq(\034)1527 3762 y Fn(`)1560 3747 y Fq(;)15 b(\030)t Fr(\))21 b Fm(\001)g Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)1995 3709 y Fn(\027)2030 3717 y Fh(m)2093 3747 y Fq(S)5 b Fr(\()p Fq(\034)2229 3762 y Fn(`)2262 3747 y Fq(;)15 b(\030)t Fr(\))g Fq(dt)-2523 b Fr(\(5.29\))284 4005 y(+)370 3918 y Fl(X)421 4116 y Fn(`)516 4005 y Fq(O)588 3904 y Fl(\020)642 4005 y Fm(j)p Fq(\034)707 4020 y Fn(`)p Fp(+1)851 4005 y Fm(\000)22 b Fl(e)-53 b Fq(\034)982 4020 y Fn(`)p Fp(+1)1105 4005 y Fm(j)137 b Fr(sup)1170 4087 y Fn(R)p Fk(\024j)p Fn(x)p Fk(j\024)p Fp(2)p Fn(R)1516 4005 y Fm(j)p Fq(D)1619 3967 y Fn(\013)1616 4027 y(x)1669 4005 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p Fm(j)42 b Fr(sup)1933 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4548 y Fn(\027)2911 4556 y Fh(m)2973 4586 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)])p Fm(j)3271 4485 y Fl(\021)3327 4586 y Fq(:)-3399 b Fr(\(5.32\))1903 4992 y(42)p eop %%Page: 43 43 43 42 bop -47 278 a Fr(By)32 b(\(4.30\))g(one)f(has)f Fm(j)p Fq(\034)753 293 y Fn(`)p Fp(+1)896 278 y Fm(\000)23 b Fl(e)-54 b Fq(\034)1027 293 y Fn(`)p Fp(+1)1150 278 y Fm(j)26 b Fe(.)f Fq(T)1372 215 y Fj(1)p 1372 227 31 3 v 1372 268 a(4)1417 278 y Fr(.)41 b(Another)30 b(estimate)h(that)g(w) m(e)g(shall)e(use)h(is)584 414 y Fl(\014)584 468 y(\014)614 491 y Fq(D)692 454 y Fn(\027)727 463 y Fj(1)766 491 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))21 b Fm(\001)f Fq(:)15 b(:)g(:)21 b Fm(\001)g Fq(D)1330 454 y Fn(\027)1365 462 y Fh(m)1427 491 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(j)27 b Fe(.)d Fq(t)1855 451 y Fn(m)p Fp(+)p Fn(c)2015 401 y Fc(P)2089 422 y Fh(m)2089 472 y(j)s Fj(=1)2199 451 y Fp(\()p Fk(j)p Fn(\027)2281 461 y Fh(j)2313 451 y Fk(j\000)p Fp(2\))2480 491 y Fr(=)h Fq(t)2609 454 y Fp(\(1)p 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Fq(V)f Fr(.)37 b(Observ)m(e)21 b(that)g(w)m(e)h(ha)m(v)m(e)g (included)c Fq(\015)25 b Fr(in)20 b(\(5.33\))r(.)38 b(Despite)21 b(the)g(fact)g(that)h(here)e Fq(\015)31 b Fr(=)25 b(0)c(since)f(w)m(e)h (are)g(assuming)-47 1344 y(that)34 b Fq(k)h Fr(=)30 b(0,)36 b(it)d(will)f(b)s(e)h(con)m(v)m(enien)m(t)i(to)g(include)c(it)j(in)e (the)i(estimates)h(for)f(later)g(applications.)49 b(Finally)-8 b(,)34 b(recall)-47 1457 y(that)c(there)f(are)h Fq(T)597 1424 y Fp(1)p Fk(\000)p Fn(\014)759 1457 y Fr(=)25 b Fq(T)921 1424 y Fn(c)985 1457 y Fr(man)m(y)30 b(terms)f(in)f(the)h (sums)g(o)m(v)m(er)h Fq(`)g Fr(ab)s(o)m(v)m(e.)41 b(Consequen)m(tly)29 b(the)g(error)g(terms)h(ab)s(o)m(v)m(e)-47 1570 y(can)g(b)s(e)g (estimated)h(as)f(follo)m(ws:)430 1775 y(\(5.30\))85 b Fe(.)e Fq(T)965 1737 y Fn(c)1015 1775 y Fq(T)1090 1710 y Fj(1)p 1090 1722 V 1090 1763 a(4)1150 1775 y Fq(T)1216 1737 y Fk(\000)1281 1710 y Fj(3)p 1281 1722 V 1281 1763 a(4)1321 1737 y Fk(\000j)p Fn(\013)p Fk(j)p Fn(\014)1523 1775 y Fq(T)1589 1734 y Fn(m)p Fp(+)p Fn(c)1748 1684 y Fc(P)1823 1755 y Fh(j)1855 1734 y Fp(\()p Fk(j)p Fn(\027)1937 1744 y Fh(j)1969 1734 y Fk(j\000)p Fp(2\))745 1927 y Fr(=)g Fq(T)975 1863 y Fj(1)p 975 1875 V 975 1916 a(2)1015 1890 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+1\)\(1)p Fk(\000)p Fn(c)p Fp(\))1495 1927 y Fq(T)1561 1890 y Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\))p Fn(m)p Fp(+)p Fn(c)p Fk(j)p Fn(\013)p Fk(j)2023 1927 y Fq(T)2089 1890 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)2283 1927 y Fr(=)25 b Fq(T)2445 1890 y Fn(c)p Fk(\000)2541 1863 y Fj(1)p 2540 1875 V 2540 1916 a(2)2600 1927 y Fq(T)2666 1890 y Fk(\000)p Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\))3207 1927 y Fq(T)3273 1890 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)430 2065 y Fr(\(5.31\))85 b Fe(.)e Fq(T)965 2028 y Fn(c)1015 2065 y Fq(T)1081 2028 y Fp(3)p Fn(\014)1178 2065 y Fq(T)1244 2028 y Fk(\000)1309 2000 y Fj(3)p 1309 2012 V 1309 2054 a(4)1349 2028 y Fk(\000)p Fp(\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+1\))p Fn(\014)1696 2065 y Fq(T)1772 2000 y Fj(1)p 1772 2012 V 1772 2054 a(4)1812 2028 y Fk(\000)p Fp(2)p Fn(\014)1949 2065 y Fq(T)2015 2028 y Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\))p Fn(m)p Fp(+)p Fn(c)p Fk(j)p Fn(\013)p Fk(j)2478 2065 y Fq(T)2544 2028 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)745 2218 y Fr(=)g Fq(T)965 2180 y Fn(c)p Fk(\000)1061 2153 y Fj(1)p 1060 2165 V 1060 2206 a(2)1120 2218 y Fq(T)1186 2180 y Fk(\000j)p Fn(\013)p Fk(j)p Fp(\(1)p Fk(\000)p Fn(c)p Fp(\))1520 2218 y Fq(T)1586 2180 y Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\))p Fn(m)p Fp(+)p Fn(c)p Fk(j)p Fn(\013)p Fk(j)2048 2218 y Fq(T)2114 2180 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)2308 2218 y Fr(=)25 b Fq(T)2470 2180 y Fn(c)p Fk(\000)2566 2153 y Fj(1)p 2566 2165 V 2566 2206 a(2)2625 2218 y Fq(T)2691 2180 y Fk(\000)p Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\))3233 2218 y Fq(T)3299 2180 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)430 2356 y Fr(\(5.32\))85 b Fe(.)e Fq(T)965 2318 y Fn(c)1015 2356 y Fq(T)1081 2318 y Fp(2)p Fn(\014)1178 2356 y Fq(T)1244 2318 y Fk(\000)1309 2291 y Fj(3)p 1309 2303 V 1309 2344 a(4)1349 2318 y Fk(\000j)p Fn(\013)p Fk(j)p Fn(\014)1551 2356 y Fq(T)1617 2318 y Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\))p Fn(m)p Fp(+)p Fn(c)p Fk(j)p Fn(\013)p Fk(j)2079 2356 y Fq(T)2145 2318 y Fk(\000)2210 2291 y Fj(3)p 2210 2303 V 2210 2344 a(4)2250 2318 y Fk(\000)p Fp(1+2)p Fn(c)2480 2356 y Fq(T)2546 2318 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)-47 2356 y Fr(\(5.35\))745 2508 y(=)g Fq(T)965 2471 y Fn(c)p Fk(\000)1061 2444 y Fj(1)p 1060 2456 V 1060 2497 a(2)1120 2508 y Fq(T)1186 2471 y Fk(\000)p Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\))1727 2508 y Fq(T)1793 2471 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)1962 2508 y Fq(:)-47 2713 y Fr(T)-8 b(o)31 b(pass)f(to)h(\(5.35\))h(w)m(e)f (used)f(that)21 2882 y Fl(\014)21 2936 y(\014)21 2991 y(\014)51 2986 y Fq(@)99 3000 y Fn(t)129 2986 y Fr([)p Fq(D)232 2949 y Fn(\027)267 2958 y Fj(1)305 2986 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))21 b Fm(\001)g Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)869 2949 y Fn(\027)904 2957 y Fh(m)967 2986 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)])1240 2882 y Fl(\014)1240 2936 y(\014)1240 2991 y(\014)1296 2986 y Fe(.)1427 2873 y Fn(m)1392 2900 y Fl(X)1397 3095 y Fn(j)t Fp(=1)1539 2986 y Fm(j)p Fq(D)1642 2949 y Fn(\027)1677 2959 y Fh(j)1713 2986 y Fr([)p Fq(@)1786 3000 y Fn(t)1816 2986 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)])p Fm(j)2186 2873 y Fn(m)2159 2900 y Fl(Y)2156 3094 y Fn(s)p Fp(=1)2158 3151 y Fn(s)p Fk(6)p Fp(=)p Fn(j)2294 2986 y Fq(t)2327 2949 y Fp(1+)p Fn(c)p Fp(\()p Fk(j)p Fn(\027)2530 2957 y Fh(s)2563 2949 y Fk(j\000)p Fp(2\))2729 2986 y Fe(.)25 b Fq(t)2858 2949 y Fn(m)p Fp(+)p Fn(c)p Fp(\()p Fk(j)p Fn(\013)p Fk(j\000j)p Fn(\015)t Fk(j\000)p Fp(2)p Fn(m)p Fp(\))3462 2986 y Fq(t)3495 2949 y Fk(\000)3560 2922 y Fj(3)p 3559 2934 V 3559 2975 a(4)3604 2986 y Fq(t)3637 2949 y Fk(\000)p Fp(1+2)p Fn(c)3852 2986 y Fq(;)-47 3340 y Fr(see)31 b(\(5.34\))r(.)40 b(No)m(w)31 b(let)g Fq(Y)789 3355 y Fn(`)851 3340 y Fr(b)s(e)f(one)h(of)f(the)g (summands)f(in)g(\(5.29\))r(.)41 b(By)30 b(the)g(measurabilit)m(y)f (considerations)g(in)g(the)-47 3453 y(pro)s(of)h(of)i(Lemma)f(4.5)i (the)e Fm(f)p Fq(Y)1024 3468 y Fn(`)1058 3453 y Fm(g)g Fr(form)g(a)h(martingale)f(di\013erence)g(sequence)g(with)f(resp)s(ect) i(to)g(the)g(usual)d(family)-47 3566 y(of)h(increasing)f Fq(\033)s Fr(-algebras.)41 b(Hence)32 b(Lemma)e(2.1)i(implies)27 b(that)k(up)e(to)j(probabilit)m(y)27 b(at)32 b(most)e Fq(e)3286 3533 y Fk(\000)p Fn(T)3392 3510 y Fh(")408 3821 y Fm(j)448 3735 y Fl(X)500 3932 y Fn(`)595 3821 y Fq(Y)648 3836 y Fn(`)681 3821 y Fm(j)83 b(\024)g Fq(T)1009 3784 y Fn(")1045 3720 y Fl(\020)1099 3735 y(X)1151 3932 y Fn(`)1246 3821 y Fm(k)p Fq(Y)1344 3836 y Fn(`)1377 3821 y Fm(k)1422 3784 y Fp(2)1422 3844 y Fk(1)1497 3720 y Fl(\021)1561 3716 y Fj(1)p 1561 3728 V 1561 3769 a(2)1631 3821 y Fe(.)25 b Fq(T)1793 3784 y 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b(alen)m(t)30 b(to)513 500 y(\(1)21 b Fm(\000)f Fr(2)p Fq(c)h Fm(\000)f Fr(2)p Fq(\016)s Fr(\)\()p Fm(j)p Fq(\013)p Fm(j)j(\000)d Fq(m)p Fr(\))26 b Fm(\025)1527 439 y Fr(1)p 1527 480 46 4 v 1527 563 a(4)1583 500 y(\(1)21 b Fm(\000)f Fr(2)p Fq(c)h Fm(\000)f Fr(2)p Fq(\016)s Fr(\))i(+)d Fq(")61 b Fr(or)g(0)26 b Fq(<)f(")g Fm(\024)2791 439 y Fr(1)p 2791 480 V 2791 563 a(4)2847 500 y(\(1)c Fm(\000)f Fr(2)p Fq(c)h Fm(\000)f Fr(2)p Fq(\016)s Fr(\))p Fq(:)-3430 b Fr(\(5.37\))-47 730 y(The)34 b(\014nal)g(statemen)m(t)j(holds)c(as)i Fm(j)p Fq(\013)p Fm(j)24 b(\000)f Fq(m)33 b Fm(\025)g(j)p Fq(\013)p Fm(j)p Fq(=)p Fr(2)h Fm(\025)f Fr(1)p Fq(=)p Fr(2)j(and)e(1)24 b Fm(\000)f Fr(2)p Fq(c)h Fm(\000)f Fr(2)p Fq(\016)37 b Fr(=)c(2\()p Fq(\014)c Fm(\000)3122 694 y Fp(1)p 3122 709 36 4 v 3122 762 a(2)3190 730 y Fm(\000)23 b Fq(\016)s Fr(\))34 b Fq(>)f Fr(0.)55 b(W)-8 b(e)36 b(no)m(w)-47 843 y(compare)i(the)g(b)s(ound)e(on)i Fm(j)950 775 y Fl(P)1046 870 y Fn(`)1095 843 y Fq(Y)1148 858 y Fn(`)1180 843 y Fm(j)h Fr(from)e(\(5.36\))j(with)d(the)h(b)s (ound)e(on)i(the)g(error)f(terms)h(obtained)g(in)e(\(5.35\))s(,)-47 973 y(whic)m(h)c(is)g Fq(T)376 938 y Fn(c)p Fk(\000)472 910 y Fj(1)p 472 922 31 3 v 472 964 a(2)531 973 y Fq(T)597 940 y Fk(\000)p Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\))1124 973 y Fr(.)50 b(This)31 b(latter)j(quan)m(tit)m(y)g(is)e(indeed)g(smaller)g(than)h (the)h(\014rst)f(b)s(ound)e(in)h(\(5.36\))s(,)-47 1086 y(as)74 1050 y Fp(1)p 74 1065 36 4 v 74 1118 a(4)140 1086 y Fm(\000)243 1050 y Fn(c)p 241 1065 V 241 1118 a Fp(2)311 1086 y Fq(>)25 b(c)c Fm(\000)568 1050 y Fp(1)p 568 1065 V 568 1118 a(2)613 1086 y Fr(.)40 b(W)-8 b(e)32 b(ha)m(v)m(e)g(arriv)m(ed)d(at)i(the)g(conclusion)e(that)866 1246 y Fl(\014)866 1300 y(\014)866 1355 y(\014)896 1227 y(Z)987 1253 y Fn(T)1038 1230 y Fb(\003)947 1433 y Fn(T)1093 1350 y Fq(D)1171 1313 y Fn(\013)1168 1373 y(x)1221 1350 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\)\))p Fq(D)1780 1313 y Fn(\027)1815 1322 y Fj(1)1855 1350 y Fq(S)25 b Fm(\001)20 b Fq(:)15 b(:)g(:)22 b Fm(\001)e Fq(D)2231 1313 y Fn(\027)2266 1321 y Fh(m)2328 1350 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))g Fq(dt)2671 1246 y Fl(\014)2671 1300 y(\014)2671 1355 y(\014)2728 1350 y Fe(.)25 b Fq(T)2890 1313 y Fk(\000)p Fn(j)2987 1286 y Fh(\016)p 2987 1298 31 3 v 2987 1339 a Fj(2)-47 1600 y Fr(up)k(to)i(probabilit)m(y)d(at)j(most)g Fq(e)1036 1567 y Fk(\000)p Fn(T)1142 1544 y Fh(")1180 1600 y Fr(.)40 b(In)30 b(fact,)h(w)m(e)g(ha)m(v)m(e)h(sho)m(wn)e(the)g(stronger)h (statemen)m(t)1069 1832 y(sup)994 1911 y Fn(T)10 b Fk(\024)p Fn(\034)e Fk(\024)p Fn(T)1245 1892 y Fb(\003)1281 1728 y Fl(\014)1281 1782 y(\014)1281 1837 y(\014)1311 1709 y(Z)1402 1735 y Fn(\034)1361 1915 y(T)1460 1832 y Fq(D)1538 1795 y Fn(\013)1535 1855 y(x)1587 1832 y Fq(V)21 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\)\))p Fq(D)2147 1795 y Fn(\027)2182 1804 y Fj(1)2221 1832 y Fq(S)26 b Fm(\001)20 b Fq(:)15 b(:)g(:)21 b Fm(\001)g Fq(D)2598 1795 y Fn(\027)2633 1803 y Fh(m)2695 1832 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))g Fq(dt)3038 1728 y Fl(\014)3038 1782 y(\014)3038 1837 y(\014)994 2063 y Fe(.)25 b Fq(T)1166 1999 y Fj(1)p 1166 2011 V 1166 2052 a(4)1206 2026 y Fk(\000)1272 1999 y Fh(c)p 1271 2011 V 1271 2052 a Fj(2)1311 2026 y Fp(+)p Fn(")1418 2063 y Fq(T)1484 2026 y Fk(\000)p Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\))2026 2063 y Fq(T)2092 2023 y Fn(c)p Fp(\()2150 1973 y Fc(P)2224 2044 y Fh(j)2268 2023 y Fk(j)p Fn(\027)2323 2033 y Fh(j)2355 2023 y Fk(j\000j)p Fn(\013)p Fk(j)p Fp(\))2571 2063 y Fe(.)g Fq(T)2733 2026 y Fk(\000)p Fn(j)2830 1999 y Fh(\016)p 2830 2011 V 2830 2052 a Fj(2)2875 2063 y Fq(:)-2947 b Fr(\(5.38\))-47 2257 y(By)30 b(Borel-Can)m(telli)g(one)g (no)m(w)h(obtains)f(that,)h(with)e(probabilit)m(y)f(one)i(and)g(some)h (random)e(constan)m(t)j Fq(C)3583 2271 y Fn(\032)3623 2257 y Fr(,)747 2498 y(sup)647 2582 y Fn(\032<)p Fk(j)p Fn(\030)s Fk(j)p Fn(<\032)903 2563 y Fb(\000)p Fj(1)985 2394 y Fl(\014)985 2448 y(\014)985 2503 y(\014)1015 2374 y(Z)1106 2401 y Fk(1)1065 2581 y Fn(t)1196 2498 y Fq(D)1274 2461 y Fn(\013)1271 2521 y(x)1323 2498 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\)\))p Fq(D)1882 2461 y Fn(\027)1917 2470 y Fj(1)1957 2498 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)2333 2461 y Fn(\027)2368 2469 y Fh(m)2430 2498 y Fq(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\))g Fq(d\034)2802 2394 y Fl(\014)2802 2448 y(\014)2802 2503 y(\014)2860 2498 y Fm(\024)25 b Fq(C)3021 2512 y Fn(\032)3076 2498 y Fq(t)3109 2461 y Fk(\000)p Fn(j)3206 2433 y Fh(\016)p 3206 2445 V 3206 2487 a Fj(2)-47 2760 y Fr(for)30 b(all)f Fq(t)h Fr(and)g(arbitrary)f Fq(\032)c(>)g Fr(0,)31 b(as)g(desired.)39 b(The)30 b(same)h(argumen)m(t)g(also)f(sho)m(ws)g(that)656 2992 y(sup)556 3076 y Fn(\032<)p Fk(j)p Fn(\030)s Fk(j)p Fn(<\032)812 3057 y Fb(\000)p Fj(1)894 2888 y Fl(\014)894 2942 y(\014)894 2997 y(\014)924 2992 y Fq(D)1002 2955 y Fn(\015)1061 2868 y Fl(Z)1152 2895 y Fk(1)1112 3074 y Fn(t)1242 2992 y Fq(D)1320 2955 y Fn(\013)1317 3015 y(x)1369 2992 y Fq(V)21 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\)\))p Fq(D)1929 2955 y Fn(\027)1964 2964 y Fj(1)2003 2992 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)g Fq(D)2380 2955 y Fn(\027)2415 2963 y Fh(m)2477 2992 y Fq(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\))g Fq(d\034)2849 2888 y Fl(\014)2849 2942 y(\014)2849 2997 y(\014)2906 2992 y Fe(.)25 b Fq(t)3035 2955 y Fk(\000)p Fn(j)3132 2927 y Fh(\016)p 3132 2939 V 3132 2981 a Fj(2)3172 2955 y Fp(+)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)-47 3259 y Fr(for)g(an)m(y)g(m)m(ulti-index)e Fq(\015)5 b Fr(.)39 b(Indeed,)25 b(consider)g(the)g(case)h(when)e(all)g(the)i(deriv)-5 b(ativ)m(es)24 b(fall)g(on)h Fq(D)3153 3226 y Fn(\027)3188 3235 y Fj(1)3227 3259 y Fq(S)15 b Fm(\001)10 b Fq(:)15 b(:)g(:)10 b Fm(\001)g Fq(D)3561 3226 y Fn(\027)3596 3234 y Fh(m)3659 3259 y Fq(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\).)-47 3371 y(In)32 b(view)g(of)39 b(\(5.12\))c(this)d (giv)m(es)h(an)f(increase)h(b)m(y)f Fq(t)1699 3338 y Fk(j)p Fn(\015)t Fk(j)p Fn(c)1813 3371 y Fr(,)i(as)f(claimed.)46 b(All)31 b(other)i(cases)h(are)f(of)g(the)f(same)i(order)e(of)-47 3484 y(magnitude.)-47 3597 y(Next,)h(w)m(e)f(turn)e(to)i(those)h(terms) e(in)f(\(5.28\))k(that)e(con)m(tain)g Fq(a)2085 3612 y Fn(k)2128 3597 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))32 b(for)f(some)h(0)c Fq(<)f(k)j Fm(\024)d Fq(j)g Fm(\000)21 b Fr(1.)44 b(The)31 b(martingale)-47 3710 y(argumen)m(t)h(that)h(w)m (as)f(used)f(for)h Fq(k)f Fr(=)d(0)k(do)s(es)g(not)g(immediately)e (apply)h(to)i(the)f(case)h Fq(k)e Fm(6)p Fr(=)d(0)k(as)g Fq(a)3361 3725 y Fn(k)3404 3710 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))33 b(dep)s(ends)-47 3823 y(on)c(all)f(random)g(v)-5 b(ariables.)39 b(This)27 b(issue,)i(ho)m(w)m(ev)m(er,)i(can)e(b)s(e)f (circum)m(v)m(en)m(ted)i(b)m(y)f(inductiv)m(ely)e(reducing)g(oneself)i (to)-47 3936 y(the)h(case)i Fq(k)c 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Fq(D)2409 4409 y Fn(\027)2444 4417 y Fh(m)2506 4446 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))g Fq(dt)-2896 b Fr(\(5.39\))733 4706 y Fm(\000)819 4582 y Fl(Z)910 4608 y Fn(T)961 4585 y Fb(\003)869 4788 y Fn(T)1016 4706 y Fq(D)1094 4668 y Fn(\013)1091 4728 y(x)1143 4706 y Fq(V)21 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(D)1691 4668 y Fn(\027)1726 4677 y Fj(1)1765 4706 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)2141 4668 y Fn(\027)2176 4676 y Fh(m)2239 4706 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))g Fq(D)2580 4668 y Fn(\015)2625 4706 y Fr([)p Fq(a)2698 4721 y Fn(k)2741 4706 y Fr(\()p Fq(T)8 b(;)15 b(\030)t Fr(\))21 b Fm(\000)f Fq(a)3116 4721 y Fn(k)3159 4706 y Fr(\()p Fq(t;)15 b(\030)t Fr(\)])g Fq(dt:)-3538 b Fr(\(5.40\))1903 4992 y(44)p eop %%Page: 45 45 45 44 bop -47 274 a Fr(The)33 b(\014rst)g(expression)f(\(5.39\))k(is)d (con)m(trolled)g(b)m(y)h(means)g(of)g(the)g(induction)d(h)m(yp)s (othesis)h(and)h(the)h(previous)e(cal-)-47 387 y(culation)27 b(with)h Fq(k)g Fr(=)d(0.)40 b(Indeed,)29 b(using)f(\(5.26\))j(with)c Fq(k)h(<)d(j)34 b Fr(and)28 b(the)h(one)g(b)s(efore)f(last)g(estimate)h (in)e(\(5.36\))k(sho)m(ws)-47 500 y(that)1062 704 y(\(5.39\))85 b Fe(.)e Fq(T)1597 666 y Fk(\000)p Fn(k)1700 639 y Fh(\016)p 1700 651 31 3 v 1700 693 a Fj(2)1740 666 y Fp(+)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)1935 704 y Fq(T)2001 666 y Fk(\000)2066 639 y Fh(\016)p 2065 651 V 2065 693 a Fj(2)2106 666 y Fp([)p Fk(\000)p Fp(1+4\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\)])2601 704 y Fq(T)2667 666 y Fk(\000)p Fn(c)p Fk(j)p Fn(\015)t Fk(j)1377 859 y Fe(.)g Fq(T)1597 822 y Fk(\000)1662 795 y Fh(\016)p 1661 807 V 1661 848 a Fj(2)1702 822 y Fp([)p Fk(\000)p Fp(1+)p Fn(k)r Fp(+4\()p Fk(j)p Fn(\013)p Fk(j\000)p Fn(m)p Fp(\)])2301 859 y Fr(=)25 b Fq(T)2463 822 y Fk(\000)p Fn(j)2560 795 y Fh(\016)p 2560 807 V 2560 848 a Fj(2)2604 859 y Fq(;)-47 1063 y Fr(since)k Fq(j)i Fr(=)25 b Fm(\000)p Fr(1)c(+)f Fq(k)j Fr(+)d(4\()p Fm(j)p Fq(\013)p Fm(j)i(\000)e Fq(m)p Fr(\).)41 b(Clearly)-8 b(,)29 b(the)i(same)g(calculation)e(also)i(sho)m(ws)f (that)1501 1285 y Fm(j)p Fq(D)1604 1248 y Fn(\027)1647 1285 y Fr(\(5.39\))r Fm(j)c Fe(.)f Fq(T)2092 1248 y Fk(\000)p Fn(j)2189 1220 y Fh(\016)p 2189 1232 V 2189 1274 a Fj(2)2229 1248 y Fp(+)p Fk(j)p Fn(\027)t Fk(j)p Fn(c)-47 1489 y Fr(for)30 b(an)m(y)h(m)m(ulti-index)c Fq(\027)6 b Fr(.)41 b(By)30 b(de\014nition)f(\(5.28\))j(the)f(in)m(tegral)f(in)f(\(5.40\))j (is)e(the)g(sum)g(of)g(terms)h(of)f(the)h(form)-47 1753 y Fl(Z)44 1780 y Fn(T)95 1756 y Fb(\003)3 1959 y Fn(T)150 1877 y Fq(D)228 1839 y Fn(\013)225 1899 y(x)277 1877 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(D)824 1839 y Fn(\027)859 1848 y Fj(1)899 1877 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)1275 1839 y Fn(\027)1310 1847 y Fh(m)1373 1877 y Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))1636 1753 y Fl(Z)1727 1780 y Fn(t)1687 1959 y(T)1772 1877 y Fq(D)1850 1839 y Fn(\013)1895 1816 y Fb(0)1847 1899 y Fn(x)1922 1877 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\)\))p Fq(D)2481 1839 y Fn(\027)2520 1816 y Fb(0)2516 1860 y Fj(1)2556 1877 y Fq(S)25 b Fm(\001)c Fq(:)15 b(:)g(:)21 b Fm(\001)f Fq(D)2932 1837 y Fn(\027)2971 1813 y Fb(0)2967 1867 y Fh(m)3021 1853 y Fb(0)3052 1877 y Fq(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\))g Fq(D)3405 1839 y Fn(\015)3445 1816 y Fb(0)3473 1877 y Fq(a)3521 1892 y Fn(k)3560 1873 y Fb(0)3586 1877 y Fr(\()p Fq(\034)5 b(;)15 b(\030)t Fr(\))g Fq(d\034)37 b(dt)-47 1691 y Fr(\(5.41\))-47 2122 y(where)30 b(the)g(v)-5 b(arious)30 b(parameters)g(satisfy)g(the)h (relations)1131 2326 y Fq(\013)83 b Fr(=)g Fq(\027)1471 2340 y Fp(1)1530 2326 y Fr(+)20 b Fq(:)15 b(:)g(:)22 b Fr(+)e Fq(\027)1884 2340 y Fn(m)1970 2326 y Fr(+)g Fq(\015)5 b(;)107 b Fm(j)p Fq(\013)p Fm(j)26 b(\025)f Fr(1)p Fq(;)41 b Fm(j)p Fq(\027)2656 2340 y Fn(j)2693 2326 y Fm(j)25 b(\025)g Fr(2)944 2464 y Fq(\013)1002 2427 y Fk(0)1046 2464 y Fr(+)20 b Fq(\015)88 b Fr(=)83 b Fq(\027)1477 2427 y 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y Fe(.)25 b Fq(T)1562 4558 y Fj(1)p 1562 4570 V 1562 4611 a(2)1602 4585 y Fk(\000)p Fn(c)p Fp(+2)p Fn(")1840 4622 y Fq(T)1906 4585 y Fk(\000)p Fp(\(1)p Fk(\000)p Fp(2)p Fn(c)p Fp(\)\()p Fk(j)p Fn(\013)p Fk(j)p Fp(+)p Fk(j)p Fn(\013)2403 4561 y Fb(0)2426 4585 y Fk(j\000)p Fn(m)p Fk(\000)p Fn(m)2680 4561 y Fb(0)2702 4585 y Fp(\))2759 4622 y Fe(.)g Fq(T)2921 4585 y Fk(\000)p Fn(j)3018 4558 y Fh(\016)p 3018 4570 V 3018 4611 a Fj(2)3063 4622 y Fq(;)-3135 b Fr(\(5.46\))1903 4992 y(46)p eop %%Page: 47 47 47 46 bop -47 274 a Fr(whic)m(h)38 b(agrees)j(with)d(the)i(informal)e (calculation)h(in)f(\(5.45\))r(.)69 b(There)39 b(are)h(three)g(error)f (terms)h(in)m(v)m(olv)m(ed)f(in)g(the)-47 387 y(appro)m(ximation)i(of) 49 b(\(5.41\))44 b(b)m(y)1105 319 y Fl(P)1201 414 y Fn(`)1249 387 y Fq(Z)1311 402 y Fn(`)1344 387 y Fr(.)75 b(One)42 b(deriv)m(es)f(from)h(replacing)f Fq(\034)2631 402 y Fn(`)p Fp(+1)2795 387 y Fr(with)i Fl(e)-53 b Fq(\034)3054 402 y Fn(`)p Fp(+1)3177 387 y Fr(,)45 b(another)d(from)g(the)-47 500 y(linearization)35 b(of)i Fq(D)s(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\),)40 b(and)d(the)g(\014nal)f(one)h(from)g(freezing)g Fq(t)g Fr(at)h Fq(\034)2503 515 y Fn(`)2535 500 y Fr(.)62 b(The)36 b(\014rst)h(t)m(w)m(o)h(error)f(terms)g(can)g(b)s(e)-47 613 y(treated)30 b(in)e(the)i(same)g(w)m(a)m(y)h(as)e(those)h(in)e (\(5.30\))k(and)d(\(5.31\))r(,)h(resp)s(ectiv)m(ely)-8 b(,)30 b(and)f(w)m(e)h(therefore)g(skip)e(the)i(details.)-47 726 y(On)c(the)h(other)g(hand,)f(freezing)h(the)g(second)g(in)m(tegral) f(in)g(\(5.41\))j(at)e(time)g Fq(t)e Fr(=)g Fq(\034)2707 741 y Fn(l)2759 726 y Fr(in)m(tro)s(duces)h(a)h(new)f(t)m(yp)s(e)h(of)g (error)-47 839 y(that)k(w)m(e)g(no)m(w)f(estimate.)41 b(Clearly)-8 b(,)30 b(the)h(error)f(in)f(question)g(is)h(b)s(ounded)e (b)m(y)781 1075 y Fq(T)g(T)928 1037 y Fk(\000)993 1010 y Fj(3)p 992 1022 31 3 v 992 1063 a(4)1033 1037 y Fk(\000j)p Fn(\013)p Fk(j)p Fn(\014)1235 1075 y Fq(T)1301 1034 y Fn(m)p Fp(+)p Fn(c)p Fp(\()1476 985 y 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y Fl(P)271 1386 y Fn(m)271 1455 y(j)t Fp(=1)413 1428 y Fm(j)p Fq(\027)483 1442 y Fn(j)519 1428 y Fm(j)h Fr(=)f Fm(j)p Fq(\013)p Fm(j)20 b(\000)e(j)p Fq(\015)5 b Fm(j)30 b Fr(and)1192 1360 y Fl(P)1288 1386 y Fn(m)1350 1363 y Fb(0)1288 1455 y Fn(j)t Fp(=1)1429 1428 y Fm(j)p Fq(\027)1505 1395 y Fk(0)1499 1453 y Fn(j)1536 1428 y Fm(j)c Fr(=)f Fm(j)p Fq(\013)1766 1395 y Fk(0)1790 1428 y Fm(j)19 b Fr(+)f Fm(j)p Fq(\015)5 b Fm(j)p Fr(.)41 b(Comparing)29 b(this)f(with)h(the)g(desired)g(b)s(ound)e(\(5.46\))s(,) -47 1552 y(one)37 b(sees)h(that)f(\(5.47\))j(is)c(smaller)g(than)g (\(5.46\))k(b)m(y)d Fq(T)1895 1519 y Fk(\000)p Fp(2)p Fn(")2021 1552 y Fr(,)i(so)f(that)f(it)g(can)g(b)s(e)g(ignored.)60 b(This)35 b(concludes)i(the)-47 1665 y(inductiv)m(e)29 b(pro)s(of)g(of)38 b(\(5.26\))r(.)-47 1778 y(No)m(w)f(\014x)f(some)h (large)g Fq(N)47 b Fr(and)36 b(let)g Fq(a)g Fr(=)1395 1709 y Fl(P)1490 1736 y Fn(M)1490 1804 y(j)t Fp(=0)1632 1778 y Fq(a)1680 1792 y Fn(j)1753 1778 y Fr(with)g(another)h(large)g 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Fm(j)e(\000)e Fq(m)p Fr(\).)41 b(Summing)28 b(o)m(v)m(er)j Fq(j)g Fm(\025)25 b Fr(1)31 b(pro)m(v)m(es)g(\(5.27\))i(and)c(th)m(us)h(the)h(prop)s(osition.)p 3879 4734 4 62 v 3883 4676 55 4 v 3883 4734 V 3937 4734 4 62 v 1903 4992 a(47)p eop %%Page: 48 48 48 47 bop -47 274 a Fr(It)45 b(should)f(b)s(e)h(clear)h(from)f(the)h (previous)e(pro)s(of)h(that)h(the)g(usual)e(de\014nition)f(of)j(mo)s (di\014ed)d(w)m(a)m(v)m(e)48 b(op)s(erators)-47 387 y(with)37 b Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))40 b(=)f(1)g(do)s(es)g(not)g (apply)e(in)h(our)g(situation.)64 b(Indeed,)41 b(the)e(\014rst)f (signi\014can)m(t)f(term)i(in)e(the)i(station-)-47 500 y(ary)30 b(phase)g(expansion)f(of)1163 617 y Fl(Z)1213 823 y Fi(R)1261 804 y Fh(d)1317 741 y Fq(e)1359 703 y Fn(i)p Fp(\010\()p Fn(t;x)p Fp(;)p Fn(\030)s Fp(\))1656 741 y Fr([)p Fq(V)21 b Fr(\()p Fq(x)p Fr(\))g Fm(\000)e Fq(V)i Fr(\()p Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\)\)])2470 717 y(^)2458 741 y Fq(\036)r Fr(\()p Fq(\030)t Fr(\))g Fq(d\030)-47 984 y Fr(is)838 1083 y Fl(Z)889 1289 y Fi(R)937 1270 y Fh(d)992 1207 y Fq(e)1034 1169 y Fn(i)p Fp(\010\()p Fn(t;x)p Fp(;)p Fn(\030)s Fp(\))1332 1207 y Fm(h)p Fq(D)1445 1169 y Fp(2)1442 1229 y Fn(x)1486 1207 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\)\()p Fq(x)i Fm(\000)d Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\)\))p Fq(;)g(x)23 b Fm(\000)c Fq(@)5 b(S)g Fr(\()p Fq(t;)15 b(\030)t Fr(\))p Fm(i)2769 1183 y Fr(^)2757 1207 y Fq(\036)r Fr(\()p Fq(\030)t Fr(\))g Fq(d\030)t(:)-3105 b Fr(\(5.49\))-47 1487 y(By)30 b(the)h(metho)s(d)f(of)g(stationary)h(phase)f(this)f(in)m(tegral)h(is)g (of)g(the)h(order)f Fq(D)2548 1454 y Fp(2)2545 1509 y Fn(x)2589 1487 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p Fq(t)15 b(t)2865 1451 y Fk(\000)2930 1424 y Fh(d)p 2930 1436 33 3 v 2931 1477 a Fj(2)2976 1487 y Fr(.)41 b(Therefore,)1270 1691 y Fm(k)p Fr(\(5.49\))r Fm(k)1592 1711 y Fn(L)1640 1692 y Fj(2)1675 1711 y Fp(\()p Fk(j)p Fn(x)p Fk(j\020)p Fn(t)p Fp(\))1919 1691 y Fe(.)36 b Fr(sup)2015 1773 y Fk(j)p Fn(x)p Fk(j\020)p Fn(t)2189 1691 y Fm(j)p Fq(D)2292 1653 y Fp(2)2289 1713 y Fn(x)2334 1691 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p Fm(j)15 b Fq(t:)-47 1975 y Fr(In)44 b(the)h(con)m(text)i (of)f(standard)e(scattering)i(theory)f([6)q(],)k(whic)m(h)44 b(requires)g Fq(t)2673 1942 y Fk(\000)p Fp(2)p Fk(\000)p Fn(")2899 1975 y Fr(of)i(the)f(second)g(deriv)-5 b(ativ)m(es,)-47 2088 y(this)44 b(latter)h(expression)f(is)g(in)m(tegrable)g(in)g(time.) 84 b(In)45 b(our)f(case,)50 b(ho)m(w)m(ev)m(er,)h Fm(j)p Fq(D)2846 2055 y Fp(2)2843 2110 y Fn(x)2887 2088 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p Fq(t)p Fm(j)46 b Fr(is)e(t)m(ypically)g(on)h (the)-47 2214 y(order)d(of)g Fq(t)351 2178 y Fk(\000)416 2151 y Fj(3)p 416 2163 31 3 v 416 2204 a(4)456 2178 y Fk(\000)p Fn(")590 2214 y Fr(and)g(therefore)h(not)f(in)m(tegrable.)77 b(W)-8 b(e)43 b(circum)m(v)m(en)m(t)g(this)f(di\016cult)m(y)e(here)j(b) m(y)f(means)g(of)h(the)-47 2327 y(amplitude)26 b Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\).)41 b(In)28 b(fact,)i(it)d(is)h(easy)h (to)g(see)g(from)f(\(5.28\))i(and)e(also)g(natural)g(in)f(view)g(of)i (the)g(aforemen)m(tioned)-47 2439 y(problem)g(with)g Fq(D)592 2406 y Fp(2)631 2439 y Fq(V)20 b Fr(\()p Fq(x)p Fr(\))p Fq(t)31 b Fr(that)g(the)f(\014rst)g(nonzero)h(term)f(after)h Fq(a)2250 2453 y Fp(0)2315 2439 y Fr(=)25 b(1)30 b(is)987 2688 y Fq(a)1035 2702 y Fp(3)1075 2688 y Fr(\()p Fq(t;)15 b(\030)t Fr(\))26 b(=)f(const)1622 2564 y Fl(Z)1713 2590 y Fk(1)1672 2770 y Fn(t)1803 2688 y Fq(D)1881 2650 y Fp(2)1878 2710 y Fn(x)1922 2688 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\)\))g Fq(D)2496 2650 y Fp(2)2537 2688 y Fq(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\))g Fq(d\034)-47 2938 y Fr(with)34 b(a)j(suitable)d (constan)m(t)j(const)g(\(this)e(allo)m(ws)g(one)h(to)h(sligh)m(tly)d (strengthen)i(\(5.27\))r(,)h(but)e(w)m(e)i(do)f(not)g(exploit)-47 3051 y(this)28 b(fact\).)42 b(Generally)28 b(sp)s(eaking,)g(the)i (patien)m(t)f(reader)h(will)c(easily)i(\014nd)g(the)i(\014rst)e(few)h Fq(a)3084 3065 y Fn(j)3150 3051 y Fr(b)m(y)g(means)g(of)g(explicit)-47 3164 y(in)m(tegration)34 b(b)m(y)h(parts.)54 b(It)34 b(turns)g(out)h(that)g Fq(a)1592 3178 y Fp(1)1664 3164 y Fr(=)d Fq(a)1815 3178 y Fp(2)1887 3164 y Fr(=)h Fq(a)2039 3178 y Fp(4)2111 3164 y Fr(=)f Fq(a)2262 3178 y Fp(5)2334 3164 y Fr(=)g Fq(a)2485 3178 y Fp(8)2557 3164 y Fr(=)g(0,)37 b(whereas)d(the)h(\014rst)f(few)h(nonzero)-47 3287 y(terms)40 b(deca)m(y)i(lik)m(e)d Fq(t)688 3252 y Fk(\000)753 3224 y Fj(3)p Fh(\016)p 753 3236 61 3 v 768 3278 a Fj(2)827 3287 y Fq(;)15 b(t)900 3254 y Fk(\000)p Fp(3)p Fn(\016)1069 3287 y Fr(and)40 b Fq(t)1289 3252 y Fk(\000)1354 3224 y Fj(7)p Fh(\016)p 1354 3236 V 1368 3278 a Fj(2)1428 3287 y Fr(,)j(resp)s(ectiv)m(ely)-8 b(.)71 b(In)39 b(the)i(con)m(text)h (of)48 b(\(5.28\))42 b(the)f(zero)g(terms)g Fq(a)3747 3301 y Fn(j)3823 3287 y Fr(are)-47 3400 y(iden)m(ti\014ed)28 b(b)m(y)i(means)h(of)f(non-divisibilit)m(y)c(b)m(y)k(four)f(of)i Fq(j)26 b Fr(+)20 b(1)g Fm(\000)g Fq(k)s Fr(.)-47 3513 y(W)-8 b(e)31 b(ha)m(v)m(e)h(no)m(w)e(arriv)m(ed)g(at)h(our)f(main)f (result.)-47 3701 y Fd(Theorem)34 b(5.6.)46 b Fg(With)30 b(pr)-5 b(ob)g(ability)31 b(one)f(the)g(fol)5 b(lowing)31 b(holds:)42 b(F)-7 b(or)30 b(any)h Fq(\036)25 b Fm(2)g(S)7 b Fr(\()p Fo(R)2892 3668 y Fn(d)2938 3701 y Fr(\))30 b Fg(with)g Fr(supp)o(\()3432 3677 y(^)3420 3701 y Fq(\036)p Fr(\))c Fm(\032)f Fo(R)3691 3668 y Fp(2)3750 3701 y Fm(n)14 b(f)p Fr(0)p Fm(g)-47 3814 y Fg(let)1085 4037 y Fr(\()p Fq(U)c Fr(\()p Fq(t)p Fr(\))p Fq(\036)p Fr(\)\()p Fq(x)p Fr(\))27 b(:=)1654 3913 y Fl(Z)1705 4120 y Fi(R)1753 4101 y Fh(d)1808 4037 y Fq(e)1850 4000 y Fn(i)p Fp([)p Fn(x)p Fk(\001)p Fn(\030)s Fk(\000)p Fn(S)t Fp(\()p Fn(t;\030)s Fp(\)])2261 4037 y Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))2524 4013 y(^)2511 4037 y Fq(\036)q Fr(\()p Fq(\030)t Fr(\))g Fq(d\030)t(;)-47 4287 y Fg(wher)-5 b(e)46 b Fq(S)5 b Fr(\()p Fq(t;)15 b(\030)t Fr(\))46 b Fg(and)g Fq(a)p Fr(\()p Fq(t;)15 b(\030)t Fr(\))46 b Fg(ar)-5 b(e)46 b(the)f Fq(C)1388 4254 y Fk(1)1507 4287 y Fg(functions)h(c)-5 b(onstructe)g(d)47 b(in)e(Cor)-5 b(ol)5 b(lary)48 b(4.8)d(and)h(Pr)-5 b(op)g(osition)48 b(5.5,)-47 4400 y(r)-5 b(esp)g(e)g(ctively.)43 b(F)-7 b(or)34 b(every)e(such)h Fq(\036)g Fg(ther)-5 b(e)33 b(exists)g Fq(f)i Fm(2)24 b Fq(L)1871 4367 y Fp(2)1911 4400 y Fr(\()p Fo(R)2006 4367 y Fn(d)2052 4400 y Fr(\))33 b Fg(such)g(that)1462 4605 y Fr(lim)1442 4661 y Fn(t)p Fk(!1)1609 4527 y Fl(\015)1609 4582 y(\015)1659 4605 y Fq(e)1701 4567 y Fn(itH)1818 4605 y Fq(f)d Fm(\000)20 b Fq(U)10 b Fr(\()p Fq(t)p Fr(\))p Fq(\036)2213 4527 y Fl(\015)2213 4582 y(\015)2289 4605 y Fr(=)25 b(0)p Fq(:)-2502 b Fr(\(5.50\))1903 4992 y(48)p eop %%Page: 49 49 49 48 bop -47 274 a Fg(In)41 b(other)h(wor)-5 b(ds,)44 b Fq(W)717 241 y Fp(+)816 274 y Fr(:=)c Fq(s)26 b Fm(\000)g Fr(lim)1244 288 y Fn(t)p Fk(!1)1430 274 y Fq(e)1472 241 y Fk(\000)p Fn(itH)1644 274 y Fq(U)10 b Fr(\()p Fq(t)p Fr(\))41 b Fg(exists,)i Fq(W)2246 241 y Fp(+)2345 274 y Fg(is)e(an)g(isometry,)j(and)e Fq(W)3284 241 y Fp(+)3383 274 y Fg(intertwines)g Fq(H)-47 387 y Fg(and)139 351 y Fp(1)p 139 366 36 4 v 139 418 a(2)185 387 y Fm(4)p Fg(,)32 b(i.e.,)g(for)h(al)5 b(l)33 b Fq(t)25 b Fm(\025)g Fr(0)1547 613 y Fq(e)1589 576 y Fn(itH)1706 613 y Fq(W)1805 576 y Fp(+)1889 613 y Fr(=)g Fq(W)2084 576 y Fp(+)2142 613 y Fq(e)2184 576 y Fn(i)2222 548 y Fh(t)p 2218 560 31 3 v 2218 602 a Fj(2)2259 576 y Fk(4)2326 613 y Fq(:)-2398 b Fr(\(5.51\))-47 817 y Fg(Final)5 b(ly,)33 b(this)g(implies)h(that)g Fq(\033)1017 831 y Fn(a:c)1108 817 y Fr(\()p Fq(H)7 b Fr(\))26 b(=)f([0)p Fq(;)15 b Fm(1)p Fr(\))p Fg(.)-47 1005 y(Pr)-5 b(o)g(of.)47 b Fr(The)25 b(amplitude)f Fq(a)h Fr(w)m(as)h(constructed)g(in)f(suc)m(h)g(a)h(w)m(a)m(y)h(that)f (\(5.2\))i(con)m(v)m(erges,)h(whic)m(h)24 b(implies)f(that)k(\(5.50\)) -47 1134 y(holds.)67 b(As)40 b Fm(k)p Fq(\036)27 b Fm(\000)g Fq(U)10 b Fr(\()p Fq(t)p Fr(\))p Fq(\036)p Fm(k)896 1148 y Fp(2)977 1134 y Fe(.)41 b Fq(t)1122 1098 y Fk(\000)1187 1071 y Fh(\016)p 1187 1083 V 1187 1124 a Fj(2)1271 1134 y Fr(b)m(y)g(\(5.26\))r(,)i(it)c(follo)m(ws)g(that)i Fq(W)2422 1101 y Fp(+)2520 1134 y Fr(is)e(an)h(isometry)f(on)h Fq(L)3336 1101 y Fp(2)3375 1134 y Fr(\()p Fo(R)3470 1101 y Fn(d)3517 1134 y Fr(\).)69 b(F)-8 b(or)41 b(the)-47 1247 y(in)m(tert)m(wining)28 b(prop)s(ert)m(y)i(one)g(computes)243 1451 y Fq(e)285 1413 y Fn(itH)417 1451 y Fq(W)516 1413 y Fp(+)575 1451 y Fq(\036)83 b Fr(=)107 b(lim)866 1505 y Fn(s)p Fk(!1)1055 1451 y Fq(e)1097 1413 y Fk(\000)p Fn(i)p Fp(\()p Fn(s)p Fk(\000)p Fn(t)p Fp(\))p Fn(H)1426 1451 y Fq(U)10 b Fr(\()p Fq(s)p Fr(\))p Fq(\036)712 1643 y Fr(=)107 b(lim)866 1698 y Fn(s)p Fk(!1)1055 1643 y Fq(e)1097 1605 y Fk(\000)p Fn(i)p Fp(\()p Fn(s)p Fk(\000)p Fn(t)p Fp(\))p Fn(H)1426 1643 y Fq(U)10 b Fr(\()p Fq(s)20 b Fm(\000)g Fq(t)p Fr(\))15 b Fq(e)1812 1605 y Fn(i)1850 1578 y Fh(t)p 1847 1590 V 1847 1631 a Fj(2)1887 1605 y Fk(4)1954 1643 y Fq(\036)21 b Fr(+)43 b(lim)2120 1698 y Fn(s)p Fk(!1)2308 1643 y Fq(e)2350 1605 y Fk(\000)p Fn(i)p Fp(\()p Fn(s)p Fk(\000)p Fn(t)p Fp(\))p Fn(H)2680 1643 y Fr(\()p Fq(U)10 b Fr(\()p Fq(s)p Fr(\))21 b Fm(\000)f Fq(U)10 b Fr(\()p Fq(s)20 b Fm(\000)g Fq(t)p Fr(\))15 b Fq(e)3398 1605 y Fn(i)3436 1578 y Fh(t)p 3432 1590 V 3432 1631 a Fj(2)3473 1605 y Fk(4)3540 1643 y Fr(\))p Fq(\036:)-3701 b Fr(\(5.52\))-47 1900 y(The)36 b(\014rst)f(expression)h (on)g(the)h(righ)m(t-hand)e(side)g(is)h Fq(W)1917 1867 y Fp(+)1991 1900 y Fq(e)2033 1864 y Fn(i)2070 1837 y Fh(t)p 2067 1849 V 2067 1890 a Fj(2)2107 1864 y Fk(4)2174 1900 y Fq(\036)p Fr(,)j(whereas)d(the)h(second)f(is)g(\(up)f(to)j(the)e (unitary)-47 2013 y(factor)31 b Fq(e)255 1980 y Fk(\000)p Fn(i)p Fp(\()p Fn(s)p Fk(\000)p Fn(t)p Fp(\))p Fn(H)569 2013 y Fr(\))g(equal)f(to)157 2259 y(\()p Fq(U)10 b Fr(\()p Fq(s)p Fr(\))21 b Fm(\000)f Fq(U)10 b Fr(\()p Fq(s)20 b Fm(\000)g Fq(t)p Fr(\))15 b Fq(e)875 2221 y Fn(i)913 2194 y Fh(t)p 910 2206 V 910 2247 a Fj(2)950 2221 y Fk(4)1017 2259 y Fr(\))p Fq(\036)p Fr(\()p Fq(x)p Fr(\))26 b(=)1350 2135 y Fl(Z)1456 2259 y Fq(e)1498 2221 y Fn(ix)p Fk(\001)p Fn(\030)1620 2158 y Fl(h)1663 2259 y Fq(e)1705 2221 y Fk(\000)p Fn(iS)t Fp(\()p Fn(s;\030)s Fp(\))1991 2259 y Fq(a)p Fr(\()p Fq(s;)15 b(\030)t Fr(\))20 b Fm(\000)g Fq(e)2389 2221 y Fk(\000)p Fn(i)2482 2194 y Fh(t)p 2479 2206 V 2479 2247 a Fj(2)2519 2221 y Fk(j)p Fn(\030)s Fk(j)2593 2198 y Fj(2)2626 2221 y Fk(\000)p Fn(iS)t Fp(\()p Fn(s)p Fk(\000)p Fn(t;\030)s Fp(\))2992 2259 y Fq(a)p Fr(\()p Fq(s)h Fm(\000)e Fq(t;)c(\030)t Fr(\))3381 2158 y Fl(i)3452 2235 y Fr(^)3440 2259 y Fq(\036)p Fr(\()p Fq(\030)t Fr(\))g Fq(d\030)t(:)-47 2499 y Fr(Since)566 2725 y Fq(S)5 b Fr(\()p Fq(s;)15 b(\030)t Fr(\))21 b Fm(\000)f Fq(S)5 b Fr(\()p Fq(s)20 b Fm(\000)g Fq(t;)15 b(\030)t Fr(\))26 b(=)1476 2663 y Fq(t)p 1470 2704 46 4 v 1470 2787 a Fr(2)1525 2725 y Fm(j)p Fq(\030)t Fm(j)1619 2687 y Fp(2)1679 2725 y Fr(+)1770 2601 y Fl(Z)1861 2627 y Fn(s)1821 2807 y(s)p Fk(\000)p Fn(t)1953 2725 y Fq(V)20 b Fr(\()p Fq(D)s(S)5 b Fr(\()p Fq(\034)g(;)15 b(\030)t Fr(\))g Fq(d\034)37 b Fm(!)2671 2663 y Fq(t)p 2664 2704 V 2664 2787 a Fr(2)2720 2725 y Fm(j)p Fq(\030)t Fm(j)2814 2687 y Fp(2)2914 2725 y Fr(as)61 b Fq(s)25 b Fm(!)g(1)-47 3009 y Fr(and)i Fm(j)p Fq(a)p Fr(\()p Fq(s;)15 b(\030)t Fr(\))h Fm(\000)f Fq(a)p Fr(\()p Fq(s)g Fm(\000)g Fq(t;)g(\030)t Fr(\))p Fm(j)26 b Fe(.)f Fq(s)1068 2973 y Fk(\000)1133 2946 y Fj(3)p 1133 2958 31 3 v 1133 2999 a(4)1173 2973 y Fk(\000)p Fn(\016)1281 3009 y Fq(t)j Fr(b)m(y)h(\(5.27\))r(,)g(with)d (b)s(oth)h(estimates)i(holding)d(uniformly)f(for)i Fq(\030)32 b Fr(in)27 b(compact)-47 3122 y(subsets)h(of)h Fo(R)428 3089 y Fn(d)492 3122 y Fm(n)18 b(f)p Fr(0)p Fm(g)p Fr(,)31 b(one)f(concludes)e(that)i(the)f(term)g(in)f(brac)m(k)m(ets)i(go)s(es)g (to)g(zero)g(as)f Fq(s)c Fm(!)g(1)p Fr(.)40 b(Th)m(us)28 b(the)i(second)-47 3234 y(term)e(in)g(\(5.52\))j(go)s(es)e(to)g(zero)h (b)m(y)e(Planc)m(herel)g(and)g(the)h(dominated)e(con)m(v)m(ergence)k (theorem.)41 b(Finally)-8 b(,)28 b(the)g(W)-8 b(eyl)-47 3347 y(criteria)35 b(implies)f(that)j(the)f(deterministic)e(essen)m (tial)i(sp)s(ectrum)g Fq(\033)2328 3361 y Fn(ess)2430 3347 y Fr(\()p Fq(H)7 b Fr(\))37 b(of)f(the)h(Sc)m(hr\177)-45 b(odinger)35 b(op)s(erator)h(with)-47 3460 y(p)s(oten)m(tial)j(\(1.2\)) i(coincides)d(with)g(the)i(essen)m(tial)f(sp)s(ectrum)f(of)i(the)f(unp) s(erturb)s(ed)d(op)s(erator)k Fm(\000)p Fr(\001,)h Fq(\033)3578 3474 y Fn(ess)3680 3460 y Fr(\()p Fq(H)7 b Fr(\))41 b(=)-47 3573 y([0)p Fq(;)15 b Fm(1)p Fr(\).)42 b(Therefore)30 b(w)m(e)h(conclude,)e(from)h(the)h(in)m(tert)m(wining)d(prop)s(erties)h (of)h(the)h(w)m(a)m(v)m(e)h(op)s(erator)e Fq(W)3457 3540 y Fp(+)3516 3573 y Fr(,)g(that)h(with)-47 3686 y(probabilit)m(y)d(one)i Fq(\033)636 3700 y Fn(a:c:)748 3686 y Fr(\()p Fq(H)7 b Fr(\))26 b(=)f([0)p Fq(;)15 b Fm(1)p Fr(\).)p 3879 3686 4 62 v 3883 3628 55 4 v 3883 3686 V 3937 3686 4 62 v -47 3973 a Fs(References)-2 4175 y Fr([1])47 b(Alsholm,)29 b(P)-8 b(.,)31 b(Kato,)g(T.)g Fg(Sc)-5 b(attering)33 b(with)h(long)f(r)-5 b(ange)34 b(p)-5 b(otentials.)32 b Fr(P)m(artial)e(di\013eren)m(tial)f(equations)h(\(Pro)s(c.)140 4288 y(Symp)s(os.)i(Pure)h(Math.,)j(V)-8 b(ol.)34 b(XXI)s(I)s(I,)f (Univ.)f(California,)h(Berk)m(eley)-8 b(,)36 b(Calif.,)d(1971\),)k(pp.) c(393{399.)k(Amer.)140 4401 y(Math.)31 b(So)s(c.,)g(Pro)m(vidence,)f (R.I.,)h(1973.)-2 4589 y([2])47 b(Buslaev,)27 b(V.)g(S.,)h(Matv)m(eev,) i(V.)d(B.)g Fg(Wave)i(op)-5 b(er)g(ators)33 b(for)d(the)g(Schr\177)-46 b(odinger)31 b(e)-5 b(quation)30 b(with)g(slow)5 b(ly)30 b(de)-5 b(cr)g(e)g(as-)140 4702 y(ing)32 b(p)-5 b(otential.)32 b Fr(\(Russian\))e(T)-8 b(eoret.)32 b(Mat.)f(Fiz.)g(2)g(\(1970\),)i (no.)d(3,)h(367{376.)1903 4992 y(49)p eop %%Page: 50 50 50 49 bop -2 274 a Fr([3])47 b(Christ,)31 b(M.)i Fg(One-dimensional)j (Schr\177)-46 b(odinger)36 b(op)-5 b(er)g(ators)38 b(with)d(slow)5 b(ly)36 b(de)-5 b(c)g(aying)36 b(p)-5 b(otentials:)48 b(Sp)-5 b(e)g(ctr)g(a)37 b(and)140 387 y(asymptotics.)50 b Fr(Lectures,)k(W)-8 b(orkshop)48 b(on)g(Oscillatory)f(In)m(tegrals)i (and)f(Disp)s(ersiv)m(e)f(Equations,)52 b(IP)-8 b(AM,)140 500 y(Marc)m(h)31 b(2001.)-2 682 y([4])47 b(Christ,)29 b(M.,)i(Kiselev,)f(A.,)h Fg(A)n(bsolutely)i(c)-5 b(ontinuous)34 b(sp)-5 b(e)g(ctrum)34 b(for)f(one-dimensional)i(Schr\177)-46 b(odinger)34 b(op)-5 b(er)g(a-)140 795 y(tors)32 b(with)g(slow)5 b(ly)32 b(de)-5 b(c)g(aying)32 b(p)-5 b(otentials:)43 b(some)32 b(optimal)h(r)-5 b(esults.)30 b Fr(J.)e(Amer.)h(Math.)g(So)s (c.)g(11)h(\(1998\),)h(no.)e(4,)140 908 y(771{797.)-2 1090 y([5])47 b(Christ,)52 b(M.,)j(Kiselev,)e(A.,)i Fg(WKB)50 b(asymptotic)i(b)-5 b(ehavior)51 b(of)g(almost)h(al)5 b(l)50 b(gener)-5 b(alize)g(d)52 b(eigenfunctions)140 1203 y(for)37 b(one-dimensional)h(Schr\177)-46 b(odinger)39 b(op)-5 b(er)g(ators)40 b(with)d(slow)5 b(ly)38 b(de)-5 b(c)g(aying)38 b(p)-5 b(otentials.)37 b Fr(J.)e(F)-8 b(unct.)35 b(Anal.)g(179)140 1316 y(\(2001\),)e(no.)d(2,)h(426{447.)-2 1498 y([6])47 b(Derezi)s(\023)-48 b(nski,)32 b(J.,)h(G)m(\023)-43 b(erard,)33 b(C.)g Fg(Sc)-5 b(attering)35 b(the)-5 b(ory)36 b(of)f(classic)-5 b(al)36 b(and)f(quantum)g Fq(N)10 b Fg(-p)-5 b(article)35 b(systems.)e Fr(T)-8 b(exts)140 1611 y(and)29 b(Monographs)i(in)e(Ph)m(ysics.)h(Springer-V)-8 b(erlag,)29 b(Berlin,)g(1997.)-2 1793 y([7])47 b(Deift,)32 b(P)-8 b(.,)33 b(Killip,)28 b(R.,)k Fg(On)i(the)g(absolutely)h(c)-5 b(ontinuous)35 b(sp)-5 b(e)g(ctrum)35 b(of)g(one-dimensional)g (Schr\177)-46 b(odinger)35 b(op-)140 1906 y(er)-5 b(ators)34 b(with)g(squar)-5 b(e)33 b(summable)h(p)-5 b(otentials.)32 b Fr(Comm.)e(Math.)h(Ph)m(ys.)g(203)g(\(1999\),)i(no.)e(2,)g(341{347.) -2 2088 y([8])47 b(Dollard,)38 b(J.)f(D.)g Fg(Asymptotic)j(c)-5 b(onver)g(genc)g(e)40 b(and)g(the)f(Coulomb)h(inter)-5 b(action.)38 b Fr(J.)f(Mathematical)h(Ph)m(ys.)f(5)140 2201 y(\(1964\),)c(729{738.)-2 2383 y([9])47 b(Herbst,)25 b(I.)g(W.)g Fg(Classic)-5 b(al)29 b(sc)-5 b(attering)28 b(with)g(long)g(r)-5 b(ange)28 b(for)-5 b(c)g(es.)26 b Fr(Comm.)e(Math.)h(Ph)m(ys.)g(35)g(\(1974\),)j(193{214.)-47 2565 y([10])47 b(H\177)-45 b(ormander,)29 b(L.)h Fg(The)i(existenc)-5 b(e)32 b(of)g(wave)h(op)-5 b(er)g(ators)35 b(in)d(sc)-5 b(attering)33 b(the)-5 b(ory.)31 b Fr(Math.)g(Z.)e(146)i(\(1976\),)h (no.)e(1,)140 2678 y(69{91.)-47 2860 y([11])47 b(H\177)-45 b(ormander,)25 b(L.)g Fg(The)j(analysis)h(of)e(line)-5 b(ar)29 b(p)-5 b(artial)30 b(di\013er)-5 b(ential)28 b(op)-5 b(er)g(ators)31 b(I.)25 b Fr(Springer)d(V)-8 b(erlag,)27 b(Heidelb)s(erg,)140 2973 y(1990.)-47 3155 y([12])47 b(Jak)-5 b(\024)-41 b(si)m(\023)e(c,)30 b(V.,)f(Last,)g(Y.)f Fg(Sp)-5 b(e)g(ctr)g(al)32 b(structur)-5 b(e)31 b(of)g(A)n(nderson)g (typ)-5 b(e)31 b(Hamiltonians.)e Fr(In)m(v)m(en)m(t.)g(Math.)g(141)g (\(2000\),)140 3268 y(no.)h(3,)h(561{577.)-47 3450 y([13])47 b(Kirsc)m(h,)24 b(W.,)j(Krishna,)c(M.,)k(Ob)s(ermeit,)d(J.)g Fg(A)n(nderson)k(mo)-5 b(del)29 b(with)f(de)-5 b(c)g(aying)28 b(r)-5 b(andomness:)42 b(mobility)28 b(e)-5 b(dge.)140 3563 y Fr(Math.)31 b(Z.)f(235)i(\(2000\),)h(no.)d(3,)h(421{433.)-47 3745 y([14])47 b(Krishna,)23 b(M.)h Fg(A)n(nderson)k(mo)-5 b(del)28 b(with)f(de)-5 b(c)g(aying)28 b(r)-5 b(andomness)30 b(existenc)-5 b(e)26 b(of)h(extende)-5 b(d)28 b(states.)d Fr(Pro)s(c.)f(Indian)140 3858 y(Acad.)31 b(Sci.)e(Math.)j(Sci.)d(100)j (\(1990\),)h(no.)e(3,)g(285{294.)-47 4040 y([15])47 b(Molc)m(hano)m(v,) 36 b(S.,)e(No)m(vitskii,)g(M.,)h(V)-8 b(ain)m(b)s(erg,)35 b(B.)f Fg(First)i(KdV)g(inte)-5 b(gr)g(als)37 b(and)g(absolutely)g(c)-5 b(ontinuous)36 b(sp)-5 b(e)g(c-)140 4153 y(trum)33 b(for)g(1-D)f (Schr\177)-46 b(odinger)34 b(op)-5 b(er)g(ator.)34 b Fr(Comm.)c(Math.)h(Ph)m(ys.)f(216)i(\(2001\),)h(no.)e(1,)g(195{213.)-47 4335 y([16])47 b(Reed,)30 b(M.,)g(Simon,)f(B.)h Fg(Metho)-5 b(ds)34 b(of)e(mo)-5 b(dern)34 b(mathematic)-5 b(al)34 b(physics.)f(III.)f(Sc)-5 b(attering)33 b(the)-5 b(ory.)32 b Fr(Academic)140 4448 y(Press)e([Harcourt)h(Brace)g(Jo)m(v)-5 b(ano)m(vic)m(h,)32 b(Publishers],)27 b(New)k(Y)-8 b(ork-London,)31 b(1979.)-47 4630 y([17])47 b(Simon,)22 b(B.)h Fg(Schr\177)-46 b(odinger)27 b(op)-5 b(er)g(ators)28 b(in)e(the)f(twenty-\014rst)h(c)-5 b(entury.)23 b Fr(Mathematical)g(ph)m(ysics)f(2000,)j(283{288,)140 4743 y(Imp.)k(Coll.)g(Press,)i(London,)e(2000.)1903 4992 y(50)p eop %%Page: 51 51 51 50 bop -47 274 a Fr([18])47 b(Y)-8 b(afaev,)37 b(D.)e(R.)g Fg(Mathematic)-5 b(al)38 b(sc)-5 b(attering)38 b(the)-5 b(ory.)38 b(Gener)-5 b(al)37 b(the)-5 b(ory.)37 b Fr(T)-8 b(ranslated)33 b(from)h(the)h(Russian)e(b)m(y)140 387 y(J.)38 b(R.)h(Sc)m(h)m(ulen)m(b)s(erger.)f(T)-8 b(ranslations)37 b(of)i(Mathematical)h(Monographs,)h(105.)g(American)d(Mathematical)140 500 y(So)s(ciet)m(y)-8 b(,)31 b(Pro)m(vidence,)f(RI,)h(1992.)-47 712 y Fa(R)m(odnianski:)41 b(Dep)-6 b(ar)g(tment)25 b(of)i(Ma)-6 b(thema)g(tics,)27 b(Princeton)g(University,)h(Fine)g(Hall,)g (Princeton)-47 825 y(N.J.)34 b(08544,)h(U.S.A.)-47 938 y Fg(email:)47 b Fd(iro)s(d@math.princeton.edu)-47 1101 y Fa(Schla)n(g:)71 b(Dep)-6 b(ar)g(tment)47 b(of)h(Ma)-6 b(thema)g(tics,)49 b(Princeton)f(University,)j(Fine)d(Hall,)j (Princeton)-47 1214 y(N.J.)34 b(08544,)h(U.S.A.)-47 1327 y Fg(email:)47 b Fd(sc)m(hlag@math.princeton.edu)1903 4992 y Fr(51)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0105161442620--