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(e)h(do)f(not)h(indicate)e(the)i(Gevrey)g(constan)m(ts)g Fo(L)2293 4620 y Fn(1)2333 4606 y Fp(,)f Fo(L)2450 4620 y Fn(2)2489 4606 y Fp(.)117 4719 y(Let)44 b Fo(H)376 4686 y Fn(0)460 4719 y Fp(b)s(e)f(a)h(completely)g(in)m(tegrable)g (real)f(v)-5 b(alued)43 b(Gevrey)i(smo)s(oth)e(Hamiltonian)g Fm(T)3354 4686 y Fl(n)3430 4719 y Fk(\002)29 b Fo(D)3608 4686 y Fn(0)3695 4719 y Fk(3)-24 4832 y Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(\))49 b Fk(!)f Fo(H)450 4799 y Fn(0)490 4832 y Fp(\()p Fo(I)7 b Fp(\))49 b Fk(2)f Fm(R)p Fp(.)84 b(W)-8 b(e)45 b(supp)s(ose)e(that)i Fo(H)1772 4799 y Fn(0)1856 4832 y Fp(is)e(non-degenerate,)50 b(whic)m(h)43 b(means)h(that)h(the)g (map)-24 4945 y Fk(r)p Fo(H)135 4912 y Fn(0)206 4945 y Fp(:)32 b Fo(D)341 4912 y Fn(0)412 4945 y Fk(!)g Fp(\012)601 4912 y Fn(0)674 4945 y Fp(is)i(a)g(di\013eomorphism.)50 b(Denote)36 b(b)m(y)e Fo(g)2028 4912 y Fn(0)2100 4945 y Fk(2)e Fo(C)2265 4912 y Fi(1)2339 4945 y Fp(\(\012)2440 4912 y Fn(0)2479 4945 y Fp(\))j(the)f(Legendre)h(transform)f(of)g Fo(H)3716 4912 y Fn(0)-24 5058 y Fp(\(then)k Fk(r)p Fo(g)348 5025 y Fn(0)426 5058 y Fp(:)g(\012)555 5025 y Fn(0)633 5058 y Fk(!)g Fo(D)840 5025 y Fn(0)917 5058 y Fp(is)f(the)i(in)m(v)m (erse)f(map)f(to)i Fk(r)p Fo(H)1977 5025 y Fn(0)2016 5058 y Fp(\).)65 b(W)-8 b(e)39 b(supp)s(ose)e(also)h(that)g(there)h (are)f(p)s(ositiv)m(e)-24 5171 y(constan)m(ts)31 b Fo(\032)26 b(>)f Fp(1,)31 b Fo(A)716 5185 y Fn(0)780 5171 y Fo(>)25 b Fp(0,)31 b(and)f Fo(L)1216 5185 y Fn(0)1281 5171 y Fk(\024)25 b Fo(L)1439 5185 y Fn(2)1508 5171 y Fp(suc)m(h)30 b(that)h Fo(H)1993 5138 y Fn(0)2058 5171 y Fk(2)25 b(G)2203 5127 y Fl(\032)2198 5198 y(L)2246 5207 y Fj(0)2285 5171 y Fp(\()p Fo(D)2398 5138 y Fn(0)2437 5171 y Fp(\),)31 b Fo(g)2574 5138 y Fn(0)2639 5171 y Fk(2)25 b(G)2784 5127 y Fl(\032)2779 5198 y(L)2827 5207 y Fj(0)2866 5171 y Fp(\(\012)2967 5138 y Fn(0)3007 5171 y Fp(\),)31 b(and)1396 5374 y Fk(k)p Fo(H)1524 5337 y Fn(0)1564 5374 y Fk(k)1609 5388 y Fl(L)1657 5397 y Fj(0)1711 5374 y Fo(;)46 b Fk(k)p Fo(g)1873 5337 y Fn(0)1914 5374 y Fk(k)1959 5388 y Fl(L)2007 5397 y Fj(0)2101 5374 y Fk(\024)56 b Fo(A)2296 5388 y Fn(0)3569 5374 y Fp(\(1.2\))-24 5565 y(in)30 b(the)i(corresp)s(onding)e (norms,)h(de\014ned)g(as)h(in)e(\(1.1\).)47 b(In)31 b(particular,)g (\012)2547 5532 y Fn(0)2617 5565 y Fp(is)g(a)h(b)s(ounded)e(domain.)43 b(Giv)m(en)-24 5678 y(a)32 b(sub)s(domain)c Fo(D)34 b Fp(of)e Fo(D)808 5645 y Fn(0)878 5678 y Fp(w)m(e)g(set)g(\012)26 b(:=)h Fk(r)p Fo(H)1530 5645 y Fn(0)1569 5678 y Fp(\()p Fo(D)s Fp(\))h Fk(\032)e Fp(\012)1908 5645 y Fn(0)1947 5678 y Fp(.)44 b(Fix)31 b Fo(\034)37 b(>)26 b(n)21 b Fk(\000)f Fp(1)32 b(and)f Fo(\024)c(>)f Fp(0.)44 b(W)-8 b(e)33 b(denote)f(b)m(y)f(\012)3711 5692 y Fl(\024)p eop %%Page: 2 2 2 1 bop -24 573 a Fp(the)34 b(set)h(of)g(all)e(frequencies)g Fo(!)i Fk(2)c Fp(\012)j(ha)m(ving)g(distance)g Fk(\025)e Fo(\024)i Fp(to)h(the)g(b)s(oundary)d(of)j(\012)f(and)f(also)i (satisfying)-24 686 y(the)30 b(Diophan)m(tine)g(condition)1119 904 y Fk(jh)p Fo(!)s(;)15 b(k)s Fk(ij)57 b(\025)1628 842 y Fo(\024)p 1582 883 144 4 v 1582 966 a Fk(j)p Fo(k)s Fk(j)1682 940 y Fl(\034)1751 904 y Fo(;)92 b Fp(for)30 b(all)f(0)d Fk(6)p Fp(=)f Fo(k)j Fk(2)d Fm(Z)2525 866 y Fl(n)2587 904 y Fo(;)957 b Fp(\(1.3\))-24 1157 y(where)30 b Fk(j)p Fo(k)s Fk(j)c Fp(=)f Fk(j)p Fo(k)533 1171 y Fn(1)573 1157 y Fk(j)20 b Fp(+)g Fk(\001)15 b(\001)g(\001)21 b Fp(+)f Fk(j)p Fo(k)998 1171 y Fl(n)1046 1157 y Fk(j)p Fp(.)117 1270 y(W)-8 b(e)29 b(are)f(going)f(to)h(\014nd)e(a)i(Gevrey)h (family)c(of)j(KAM)g(in)m(v)-5 b(arian)m(t)27 b(tori)g(with)f (frequencies)g(in)h(\012)3340 1284 y Fl(\024)3412 1270 y Fp(for)g(small)-24 1382 y(p)s(erturbations)32 b(of)i Fo(H)739 1349 y Fn(0)812 1382 y Fp(in)f Fk(G)981 1338 y Fl(\032)976 1410 y(L)1024 1419 y Fj(1)1058 1410 y Fl(;L)1126 1419 y Fj(2)1165 1382 y Fp(.)51 b(In)33 b(what)h(follo)m(ws)f(w)m(e)i (\014x)e(the)h(constan)m(ts)h Fo(A)2795 1396 y Fn(0)2869 1382 y Fp(and)e Fo(L)3111 1396 y Fn(0)3150 1382 y Fp(,)i(and)f(allo)m (w)f(the)-24 1495 y(constan)m(ts)23 b Fo(L)432 1509 y Fn(2)497 1495 y Fk(\025)i Fo(L)655 1509 y Fn(1)719 1495 y Fk(\025)g Fp(1)e(to)g(b)s(e)e(arbitrary)g(large.)39 b(This)20 b(o)s(ccurs)i(in)f(the)h(case)h(of)g(the)f(elliptic)e (equilibrium)e(for)-24 1608 y(example)24 b(\()p Fo(L)423 1622 y Fn(2)488 1608 y Fk(\035)h Fp(1\).)40 b(Giv)m(en)25 b Fo(!)j Fk(2)c Fp(\012,)i(w)m(e)f(denote)h(b)m(y)e Fk(L)1893 1622 y Fl(!)1969 1608 y Fp(=)h Fk(h)p Fo(!)s(;)15 b(@)2248 1622 y Fl(')2299 1608 y Fk(i)25 b Fp(=)2455 1544 y Ff(P)2543 1570 y Fl(n)2543 1631 y(j)t Fn(=1)2685 1608 y Fo(!)2742 1622 y Fl(j)2778 1608 y Fo(@)5 b(=@)g(')2988 1622 y Fl(j)3050 1608 y Fp(the)25 b(corresp)s(onding)-24 1721 y(v)m(ector\014eld)31 b(on)f Fm(T)615 1688 y Fl(n)661 1721 y Fp(.)41 b(Fix)30 b(0)c Fo(<)f(&)32 b Fk(\024)25 b Fp(1.)-24 1909 y Fm(Theorem)34 b(1.1)46 b Fe(L)-5 b(et)43 b Fo(H)849 1876 y Fn(0)931 1909 y Fe(b)-5 b(e)43 b(a)h(r)-5 b(e)g(al)44 b(value)-5 b(d)44 b(non-de)-5 b(gener)g(ate)44 b Fk(G)2308 1876 y Fl(\032)2349 1909 y Fe(-smo)-5 b(oth)45 b(Hamiltonian)g(,)g Fo(\032)f(>)g Fp(1)p Fe(,)i(de-)-24 2022 y(p)-5 b(ending)37 b(only)f(on)h Fo(I)h Fk(2)31 b Fo(D)896 1989 y Fn(0)971 2022 y Fe(and)37 b(satisfying)f(\(1.2\).)53 b(L)-5 b(et)36 b Fo(D)i Fe(b)-5 b(e)36 b(a)g(sub)-5 b(domain)38 b(of)e Fo(D)2943 1989 y Fn(0)3018 2022 y Fe(with)p 3219 1950 78 4 v 37 w Fo(D)e Fk(\032)d Fo(D)3508 1989 y Fn(0)3547 2022 y Fe(,)36 b(and)-24 2135 y Fp(\012)d(=)g Fk(r)p Fo(H)338 2102 y Fn(0)377 2135 y Fp(\()p Fo(D)s Fp(\))p Fe(.)56 b(Fix)37 b Fo(L)837 2149 y Fn(2)909 2135 y Fk(\025)d Fo(L)1076 2149 y Fn(1)1148 2135 y Fk(\025)f Fp(1)38 b Fe(and)g Fo(\024)33 b Fk(\024)g Fo(L)1767 2096 y Fi(\000)p Fn(1)p Fi(\000)p Fl(&)1767 2159 y Fn(2)1984 2135 y Fe(such)38 b(that)g Fo(L)2448 2149 y Fn(2)2520 2135 y Fk(\025)33 b Fo(L)2686 2149 y Fn(0)2763 2135 y Fe(and)38 b Fp(\012)3010 2149 y Fl(\024)3088 2135 y Fk(6)p Fp(=)33 b Fk(;)p Fe(.)55 b(Then)37 b(ther)-5 b(e)-24 2248 y(exists)31 b Fo(N)k Fp(=)25 b Fo(N)10 b Fp(\()p Fo(n;)15 b(\032;)g(\034)10 b Fp(\))26 b Fo(>)f Fp(0)31 b Fe(and)g Fo(\017)25 b(>)g Fp(0)31 b Fe(indep)-5 b(endent)32 b(of)f Fo(\024)p Fe(,)g Fo(L)2198 2262 y Fn(1)2237 2248 y Fe(,)g Fo(L)2358 2262 y Fn(2)2397 2248 y Fe(,)f(and)i(of)e(the)h(domain)h Fo(D)c Fk(\032)d Fo(D)3483 2215 y Fn(0)3523 2248 y Fe(,)30 b(such)-24 2360 y(that)k(for)f(any)g Fo(H)f Fk(2)25 b(G)734 2316 y Fl(\032)729 2388 y(L)777 2397 y Fj(1)812 2388 y Fl(;L)880 2397 y Fj(2)918 2360 y Fp(\()p Fm(T)1026 2327 y Fl(n)1093 2360 y Fk(\002)p 1184 2288 V 20 w Fo(D)s Fp(\))33 b Fe(with)h(norm)1185 2584 y Fo(\017)1222 2598 y Fl(H)1315 2584 y Fp(:=)25 b Fo(\024)1488 2546 y Fi(\000)p Fn(2)1582 2584 y Fk(k)p Fo(H)j Fk(\000)20 b Fo(H)1905 2546 y Fn(0)1944 2584 y Fk(k)1989 2598 y Fl(L)2037 2607 y Fj(1)2072 2598 y Fl(;L)2140 2607 y Fj(2)2204 2584 y Fk(\024)25 b Fo(\017L)2399 2546 y Fi(\000)p Fl(N)2399 2609 y Fn(1)2521 2584 y Fo(;)-24 2788 y Fe(ther)-5 b(e)45 b(exists)g(a)g(map)p 777 2716 66 4 v 45 w Fp(\010)h(:=)h(\()p 1067 2716 72 4 v Fo(U)10 b(;)p 1179 2716 74 4 v 15 w(V)21 b Fp(\))47 b(:)g Fm(T)1480 2755 y Fl(n)1555 2788 y Fk(\002)29 b Fp(\012)46 b Fk(!)g Fo(D)h Fe(of)e(an)g(anisotr)-5 b(opic)47 b(Gevr)-5 b(ey)45 b(class)g Fk(G)3374 2755 y Fl(\032;\032)3466 2732 y Fd(0)3493 2788 y Fe(,)i Fo(\032)3615 2755 y Fi(0)3685 2788 y Fp(=)-24 2901 y Fo(\032)p Fp(\()p Fo(\034)30 b Fp(+)20 b(1\))h(+)f(1)p Fe(,)33 b(such)g(that)62 3089 y Fp(\(i\))45 b Fe(F)-7 b(or)36 b(e)-5 b(ach)36 b Fo(!)c Fk(2)d Fp(\012)827 3103 y Fl(\024)872 3089 y Fe(,)35 b Fp(\003)998 3103 y Fl(!)1078 3089 y Fp(:=)29 b Fk(f)p Fp(\()p 1283 3017 66 4 v(\010)q(\()p Fo(\022)s(;)15 b(!)s Fp(\)\))30 b(:)f Fo(\022)j Fk(2)d Fm(T)1923 3056 y Fl(n)1970 3089 y Fk(g)35 b Fe(is)g(an)g(emb)-5 b(e)g(dde)g(d)37 b(L)-5 b(agr)g(angian)37 b(invariant)f(torus)203 3202 y(of)d Fo(H)39 b Fe(and)34 b Fo(X)677 3216 y Fl(H)765 3202 y Fk(\016)p 830 3130 V 20 w Fp(\010\()p Fk(\001)p Fo(;)15 b(!)s Fp(\))27 b(=)e Fo(D)p 1292 3130 V 3 w Fp(\010)o(\()p Fk(\001)p Fo(;)15 b(!)s Fp(\))21 b Fk(\001)g(L)1682 3216 y Fl(!)1732 3202 y Fe(.)36 3389 y Fp(\(ii\))45 b Fe(Ther)-5 b(e)34 b(ar)-5 b(e)33 b(c)-5 b(onstants)35 b Fo(A;)15 b(C)32 b(>)25 b Fp(0)p Fe(,)33 b(indep)-5 b(endent)34 b(of)f Fo(\024)p Fe(,)g Fo(L)2212 3403 y Fn(1)2251 3389 y Fe(,)f Fo(L)2373 3403 y Fn(2)2413 3389 y Fe(,)g(and)h(of)g Fo(D)s Fe(,)g(such)f(that)937 3517 y Ff(\014)937 3567 y(\014)937 3617 y(\014)965 3614 y Fo(@)1018 3581 y Fl(\013)1013 3641 y(\022)1068 3614 y Fo(@)1121 3581 y Fl(\014)1116 3637 y(!)1168 3614 y Fp(\()p 1203 3542 72 4 v Fo(U)11 b Fp(\()p Fo(\022)s Fp(;)k Fo(!)s Fp(\))20 b Fk(\000)g Fo(\022)s Fp(\))1684 3517 y Ff(\014)1684 3567 y(\014)1684 3617 y(\014)1731 3614 y Fp(+)g Fo(\024)1874 3581 y Fi(\000)p Fn(1)1984 3517 y Ff(\014)1984 3567 y(\014)1984 3617 y(\014)2012 3614 y Fo(@)2065 3581 y Fl(\013)2060 3641 y(\022)2114 3614 y Fo(@)2167 3581 y Fl(\014)2162 3637 y(!)2215 3614 y Fp(\()p 2250 3542 74 4 v Fo(V)g Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))21 b Fk(\000)f(r)p Fo(g)2773 3581 y Fn(0)2813 3614 y Fp(\()p Fo(!)s Fp(\)\))2978 3517 y Ff(\014)2978 3567 y(\014)2978 3617 y(\014)937 3897 y Fk(\024)58 b Fo(A)15 b(C)1221 3850 y Fi(j)p Fl(\013)p Fi(j)1214 3922 y Fn(1)1324 3803 y Ff(\020)1374 3897 y Fo(C)1439 3911 y Fn(2)1478 3897 y Fo(\024)1530 3860 y Fi(\000)p Fn(1)1625 3803 y Ff(\021)1675 3826 y Fi(j)p Fl(\014)s Fi(j)1791 3897 y Fo(\013)p Fp(!)1889 3860 y Fl(\032)1930 3897 y Fo(\014)5 b Fp(!)2026 3860 y Fl(\032)p Fn(\()p Fl(\034)j Fn(+1\)+1)2357 3897 y Fo(L)2419 3850 y Fl(N)q(=)p Fn(2)2419 3922 y(1)2551 3835 y Fk(p)p 2626 3835 105 4 v 2626 3897 a Fo(\017)2663 3911 y Fl(H)2746 3897 y Fo(;)203 4130 y Fe(uniformly)34 b(in)e Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))26 b Fk(2)f Fm(T)1130 4097 y Fl(n)1197 4130 y Fk(\002)19 b Fp(\012)33 b Fe(and)g(for)g(any)h Fo(\013;)15 b(\014)31 b Fk(2)25 b Fm(N)2230 4097 y Fl(n)2277 4130 y Fe(,)32 b(wher)-5 b(e)34 b Fo(C)2659 4144 y Fn(1)2723 4130 y Fp(=)25 b Fo(C)7 b(L)2953 4144 y Fn(1)3025 4130 y Fe(and)33 b Fo(C)3266 4144 y Fn(2)3331 4130 y Fp(=)25 b Fo(C)7 b(L)3561 4092 y Fl(\034)h Fn(+1)3561 4155 y(1)3694 4130 y Fe(.)-24 4333 y Fp(Note)36 b(that)p 400 4261 66 4 v 34 w(\010)e(b)s(elongs)g(to)h Fk(G)1005 4289 y Fl(\032;\032)1097 4266 y Fd(0)1008 4353 y Fn(\026)1000 4370 y Fl(L)1048 4379 y Fj(1)1082 4370 y Fl(;)1110 4353 y Fn(\026)1102 4370 y Fl(L)1150 4379 y Fj(2)1189 4333 y Fp(\()p Fm(T)1297 4300 y Fl(n)1366 4333 y Fk(\002)p 1460 4261 78 4 v 23 w Fo(D)s Fp(\))f(with)f(Gevrey)i(constan)m(ts)2554 4310 y(\026)2543 4333 y Fo(L)2605 4347 y Fn(1)2676 4333 y Fp(=)c Fo(C)7 b(L)2912 4347 y Fn(1)2985 4333 y Fp(and)3177 4310 y(\026)3166 4333 y Fo(L)3228 4347 y Fn(2)3299 4333 y Fp(=)32 b Fo(C)3467 4347 y Fn(2)3506 4333 y Fo(\024)3558 4300 y Fi(\000)p Fn(1)3685 4333 y Fk(\025)-24 4472 y Fo(C)7 b(L)110 4434 y Fl(\034)h Fn(+1)110 4497 y(1)243 4472 y Fo(L)305 4434 y Fn(1+)p Fl(&)305 4497 y Fn(2)430 4472 y Fp(.)40 b(As)30 b(a)g(consequence,)h(w)m(e)f(obtain)f(a)h (symplectic)f(normal)g(form)g(of)h Fo(H)37 b Fp(near)29 b(the)h(union)e(of)i(the)-24 4585 y(in)m(v)-5 b(arian)m(t)30 b(tori.)41 b(W)-8 b(e)32 b(sa)m(y)f(that)h(a)f(real)f(v)-5 b(alued)30 b(function)f(\010)c Fk(2)h Fo(C)2223 4552 y Fi(1)2297 4585 y Fp(\()p Fm(R)2410 4552 y Fl(n)2478 4585 y Fk(\002)20 b Fo(D)s Fp(\))31 b(is)f(a)h(generating)g(function)e (of)-24 4698 y(an)f(exact)j(symplectic)c(map)i Fo(\037)c Fp(:)g Fm(T)1190 4665 y Fl(n)1254 4698 y Fk(\002)17 b Fo(D)28 b Fk(!)d Fm(T)1634 4665 y Fl(n)1697 4698 y Fk(\002)17 b Fo(D)31 b Fp(if)d(\010\()p Fo(x;)15 b(I)7 b Fp(\))18 b Fk(\000)e(h)p Fo(x;)f(I)7 b Fk(i)30 b Fp(is)e(2)p Fo(\031)s Fp(-p)s(erio)s(dic)f(with)g(resp)s(ect)i(to)-24 4811 y Fo(x)p Fp(,)h Fk(j)p Fp(Id)20 b Fk(\000)g Fp(\010)369 4825 y Fl(I)409 4811 y Fk(j)25 b Fo(<)g Fp(1,)31 b(and)128 5015 y Fk(f)p Fp(\()p Fo(';)15 b(I)7 b Fp(;)15 b Fo(\037)p Fp(\()p Fo(';)g(I)7 b Fp(\)\))29 b(:)41 b(\()p Fo(';)15 b(I)7 b Fp(\))27 b Fk(2)e Fm(T)1199 4978 y Fl(n)1266 5015 y Fk(\002)20 b 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(oin)m(ts)g(of)g(a)h(p)s(ositiv)m(e)f(Leb)s(esgue)g(densit)m(y)-24 5456 y(in)j(\012)153 5470 y Fl(\024)197 5456 y Fp(.)54 b(In)35 b(other)g(w)m(ords,)h Fo(!)f Fk(2)1125 5429 y Ff(e)1115 5456 y Fp(\012)1181 5470 y Fl(\024)1260 5456 y Fp(if)f(for)h(an)m(y)g(neigh)m(b)s(orho)s(o)s(d)e Fo(U)44 b Fp(of)36 b Fo(!)h Fp(in)d(\012)g(the)i(Leb)s(esgue)e(measure)h(of)-24 5569 y Fo(U)30 b Fk(\\)20 b Fp(\012)215 5583 y Fl(\024)290 5569 y Fp(is)29 b(p)s(ositiv)m(e.)40 b(Ob)m(viously)-8 b(,)1207 5542 y Ff(e)1198 5569 y Fp(\012)1264 5583 y Fl(\024)1339 5569 y Fp(and)29 b(\012)1581 5583 y Fl(\024)1656 5569 y Fp(ha)m(v)m(e)j(the)e(same)h(Leb)s(esgue)f(measure.)1843 5927 y(2)p eop %%Page: 3 3 3 2 bop -24 573 a Fm(Corollary)35 b(1.2)46 b Fe(Supp)-5 b(ose)40 b(that)g(the)e(hyp)-5 b(othesis)41 b(of)e(The)-5 b(or)g(em)40 b(1.1)g(hold)f(and)h Fp(0)c Fo(<)g(\024)g Fk(\024)3182 570 y Ff(e)3179 573 y Fo(\024)q Fe(.)59 b(Then)38 b(ther)-5 b(e)-24 686 y(exists)44 b Fo(N)55 b Fp(=)45 b Fo(N)10 b Fp(\()p Fo(n;)15 b(\032;)g(\034)10 b Fp(\))47 b Fo(>)e Fp(0)f Fe(and)g Fo(\017)i(>)f Fp(0)f Fe(indep)-5 b(endent)45 b(of)f Fo(\024)p Fe(,)j Fo(L)2414 700 y Fn(1)2453 686 y Fe(,)f Fo(L)2589 700 y Fn(2)2629 686 y Fe(,)g(and)f Fo(D)s Fe(,)h(such)d(that)i(for)g(any)-24 815 y Fo(H)f Fk(2)36 b(G)252 770 y Fl(\032)247 842 y(L)295 851 y Fj(1)330 842 y Fl(;L)398 851 y Fj(2)436 815 y Fp(\()p Fm(T)544 782 y Fl(n)616 815 y Fk(\002)p 711 742 78 4 v 24 w Fo(D)s Fp(\))k Fe(with)f Fo(")1109 829 y Fl(H)1214 815 y Fk(\024)d Fo(\017L)1420 767 y Fi(\000)p Fl(N)7 b Fi(\000)p Fn(2\()p Fl(\034)h Fn(+2\))1420 839 y(1)1855 815 y Fe(ther)-5 b(e)40 b(is)f(a)h Fk(G)2337 782 y Fl(\032)2373 758 y Fd(0)2400 815 y Fe(-di\013e)-5 b(omorphism)41 b Fo(!)f Fp(:)d Fo(D)j Fk(!)c Fp(\012)p Fe(,)k(and)-24 947 y(an)32 b(exact)g(symple)-5 b(ctic)33 b(tr)-5 b(ansformation)35 b Fo(\037)25 b Fk(2)g(G)1621 914 y Fl(\032;\032)1713 891 y Fd(0)1740 947 y Fp(\()p Fm(T)1848 914 y Fl(n)1913 947 y Fk(\002)17 b Fo(D)s(;)e Fm(T)2192 914 y Fl(n)2257 947 y Fk(\002)i Fo(D)s Fp(\))32 b Fe(de\014ne)-5 b(d)33 b(by)e(a)h(gener)-5 b(ating)32 b(function)-24 1060 y Fp(\010\()p Fo(x;)15 b(I)7 b Fp(\))26 b(=)f Fk(h)p Fo(x;)15 b(I)7 b Fk(i)15 b Fp(+)g Fo(\036)p Fp(\()p Fo(x;)g(I)7 b Fp(\))p Fe(,)32 b Fo(\036)25 b Fk(2)g(G)1230 1027 y Fl(\032;\032)1322 1004 y Fd(0)1349 1060 y Fp(\()p Fm(T)1457 1027 y Fl(n)1519 1060 y Fk(\002)15 b Fo(D)s Fp(\))p Fe(,)30 b(such)g(that)h(the)g(tr)-5 b(ansforme)g(d)33 b(Hamiltonian)3357 1037 y Fp(~)3334 1060 y Fo(H)7 b Fp(\()p Fo(';)15 b(I)7 b Fp(\))26 b(:=)-24 1173 y Fo(H)7 b Fp(\()p Fo(\037)p Fp(\()p Fo(';)15 b(I)7 b Fp(\)\))35 b Fe(b)-5 b(elongs)33 b(to)h Fk(G)917 1140 y Fl(\032;\032)1009 1117 y Fd(0)1036 1173 y Fp(\()p Fm(T)1144 1140 y Fl(n)1211 1173 y Fk(\002)20 b Fo(D)s Fp(\))33 b Fe(and)h(for)f(e)-5 b(ach)34 b Fo(I)f Fk(2)25 b Fo(!)2194 1140 y Fi(\000)p Fn(1)2288 1173 y Fp(\()2333 1147 y Ff(e)2323 1173 y Fp(\012)2389 1187 y Fl(\024)2434 1173 y Fp(\))p Fe(,)33 b Fm(T)2603 1140 y Fl(n)2670 1173 y Fk(\002)20 b(f)p Fo(I)7 b Fk(g)34 b Fe(is)f(an)g(invariant)h(torus)-24 1286 y(of)107 1263 y Fp(~)83 1286 y Fo(H)7 b Fe(.)41 b(The)33 b(functions)g Fo(K)7 b Fp(\()p Fo(I)g Fp(\))26 b(:=)1188 1263 y(~)1164 1286 y Fo(H)7 b Fp(\(0)p Fo(;)15 b(I)7 b Fp(\))34 b Fe(and)g Fo(R)q Fp(\()p Fo(';)15 b(I)7 b Fp(\))27 b(:=)2118 1263 y(~)2094 1286 y Fo(H)7 b Fp(\()p Fo(';)15 b(I)7 b Fp(\))22 b Fk(\000)d Fo(K)7 b Fp(\()p Fo(I)g Fp(\))34 b Fe(satisfy)304 1490 y Fk(8)15 b Fo(\013)25 b Fk(2)g Fm(N)621 1452 y Fl(n)700 1490 y Fo(;)48 b Fk(8)15 b Fp(\()p Fo(';)g(I)7 b Fp(\))27 b Fk(2)e Fm(T)1241 1452 y Fl(n)1308 1490 y Fk(\002)20 b Fo(!)1459 1452 y Fi(\000)p Fn(1)1553 1490 y Fp(\()1598 1463 y Ff(e)1588 1490 y Fp(\012)1654 1504 y Fl(\024)1699 1490 y Fp(\))32 b Fo(;)109 b(@)1953 1452 y Fl(\013)2003 1490 y Fk(r)p Fo(K)7 b Fp(\()p Fo(I)g Fp(\))40 b(=)g Fo(@)2484 1452 y Fl(\013)2534 1490 y Fo(!)s Fp(\()p Fo(I)7 b Fp(\))15 b Fo(;)49 b(@)2853 1452 y Fl(\013)2848 1512 y(I)2903 1490 y Fo(R)q Fp(\()p Fo(';)15 b(I)7 b Fp(\))41 b(=)g(0)15 b Fo(:)-24 1694 y Fe(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)33 b(exist)g Fo(A;)15 b(C)32 b(>)25 b Fp(0)33 b Fe(indep)-5 b(endent)34 b Fo(\024)p Fe(,)f Fo(L)1900 1708 y Fn(1)1939 1694 y Fe(,)g Fo(L)2062 1708 y Fn(2)2101 1694 y Fe(,)f(and)i(of)f(the)g(domain)h Fo(D)s Fe(,)e(such)h(that)553 1816 y Ff(\014)553 1866 y(\014)553 1916 y(\014)580 1913 y Fo(@)633 1880 y Fl(\013)628 1936 y(')683 1913 y Fo(@)736 1869 y Fl(\014)731 1941 y(I)784 1913 y Fo(\036)p Fp(\()p Fo(';)15 b(I)7 b Fp(\))1054 1816 y Ff(\014)1054 1866 y(\014)1054 1916 y(\014)1119 1913 y Fp(+)1225 1816 y Ff(\014)1225 1866 y(\014)1225 1916 y(\014)1252 1913 y Fo(@)1305 1869 y Fl(\014)1300 1941 y(I)1353 1913 y Fp(\()p Fo(!)s Fp(\()p Fo(I)g Fp(\))21 b Fk(\000)f(r)p Fo(H)1836 1880 y Fn(0)1875 1913 y Fp(\()p Fo(I)7 b Fp(\)\))2027 1816 y Ff(\014)2027 1866 y(\014)2027 1916 y(\014)2091 1913 y Fp(+)2197 1816 y Ff(\014)2197 1866 y(\014)2197 1916 y(\014)2225 1913 y Fo(@)2278 1880 y Fl(\013)2273 1936 y(')2328 1913 y Fo(@)2381 1869 y Fl(\014)2376 1941 y(I)2428 1913 y Fp(\()2487 1890 y(~)2463 1913 y Fo(H)h Fp(\()p Fo(';)15 b(I)7 b Fp(\))21 b Fk(\000)f Fo(H)2958 1880 y Fn(0)2998 1913 y Fp(\()p Fo(I)7 b Fp(\)\))3150 1816 y Ff(\014)3150 1866 y(\014)3150 1916 y(\014)553 2178 y Fk(\024)57 b Fo(A\024C)873 2130 y Fi(j)p Fl(\013)p Fi(j)866 2203 y Fn(1)977 2109 y Ff(\000)1015 2178 y Fo(C)1080 2192 y Fn(2)1120 2178 y Fo(\024)1172 2145 y Fi(\000)p Fn(1)1266 2109 y Ff(\001)1304 2129 y Fi(j)p Fl(\014)s Fi(j)1421 2178 y Fo(\013)p Fp(!)1516 2145 y Fl(\032)1557 2178 y Fo(\014)5 b Fp(!)1650 2145 y Fl(\032)1686 2122 y Fd(0)1728 2178 y Fo(L)1790 2130 y Fl(N)q(=)p Fn(2)1790 2203 y(1)1922 2120 y Fk(p)p 1997 2120 105 4 v 1997 2178 a Fo(\017)2034 2192 y Fl(H)2117 2178 y Fo(;)-24 2390 y Fe(uniformly)47 b(with)f(r)-5 b(esp)g(e)g(ct)48 b(to)e Fp(\()p Fo(';)15 b(I)7 b Fp(\))51 b Fk(2)d Fm(T)1495 2357 y Fl(n)1572 2390 y Fk(\002)29 b Fo(D)48 b Fe(and)f(for)f(any)h Fo(\013;)15 b(\014)55 b Fk(2)48 b Fm(N)2726 2357 y Fl(n)2773 2390 y Fe(,)h(wher)-5 b(e)47 b Fo(C)3185 2404 y Fn(1)3273 2390 y Fp(=)i Fo(C)7 b(L)3527 2404 y Fn(1)3611 2390 y Fe(and)-24 2503 y Fo(C)41 2517 y Fn(2)106 2503 y Fp(=)24 b Fo(C)7 b(L)335 2464 y Fl(\034)h Fn(+1)335 2527 y(1)468 2503 y Fe(.)-24 2690 y Fp(Denote)40 b(b)m(y)f(\012)g Fk(3)g Fo(!)j Fk(7!)d Fo(I)7 b Fp(\()p Fo(!)s Fp(\))40 b Fk(2)f Fo(D)i Fp(the)e(in)m(v)m(erse)g(map)f(to)i(the)f (di\013eomorphism)d Fo(I)46 b Fk(7!)39 b Fo(!)s Fp(\()p Fo(I)7 b Fp(\).)67 b(Then)38 b(for)-24 2803 y(eac)m(h)31 b Fo(!)d Fk(2)360 2777 y Ff(e)351 2803 y Fp(\012)417 2817 y Fl(\024)461 2803 y Fp(,)i(the)g(restriction)e(of)i(the)f (Hamiltonian)f(\015o)m(w)i(of)2201 2780 y(~)2178 2803 y Fo(H)36 b Fp(to)30 b(the)g(in)m(v)-5 b(arian)m(t)29 b(torus)g Fm(T)3242 2770 y Fl(n)3307 2803 y Fk(\002)18 b(f)p Fo(I)7 b Fp(\()p Fo(!)s Fp(\))p Fk(g)31 b Fp(is)-24 2924 y(giv)m(en)h(b)m(y)f(\()p Fo(t;)15 b(';)g(I)7 b Fp(\))30 b Fk(7!)d Fp(\()p Fo(')22 b Fp(+)f Fo(t)p Fk(r)p Fo(K)7 b Fp(\()p Fo(I)g Fp(\))p Fo(;)15 b(I)7 b 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3150 y Fp(and)30 b Fo(m)25 b Fk(2)g Fm(N)181 3388 y Fk(j)p Fo(@)259 3351 y Fl(\013)254 3411 y(')309 3388 y Fo(@)362 3344 y Fl(\014)357 3415 y(I)409 3388 y Fo(R)q Fp(\()p Fo(';)15 b(I)7 b Fp(\))p Fk(j)57 b(\024)f Fo(\024A)15 b(C)1111 3340 y Fi(j)p Fl(\013)p Fi(j)1104 3413 y Fn(1)1215 3294 y Ff(\020)1264 3388 y Fo(C)1329 3402 y Fn(2)1369 3388 y Fo(\024)1421 3351 y Fi(\000)p Fn(1)1515 3294 y Ff(\021)1565 3317 y Fi(j)p Fl(\014)s Fi(j)p Fn(+)p Fl(m)1799 3388 y Fo(\013)p Fp(!)1897 3351 y Fl(\032)1938 3388 y Fo(\014)5 b Fp(!)2034 3351 y Fl(\032)2070 3327 y Fd(0)2113 3388 y Fo(m)p Fp(!)2233 3351 y Fl(\032)2269 3327 y Fd(0)2291 3351 y Fi(\000)p Fn(1)2386 3388 y Fo(;)106 b Fp(\()p Fo(';)15 b(I)7 b Fp(\))27 b Fk(2)e Fm(T)2919 3351 y Fl(n)2986 3388 y Fk(\002)19 b Fo(D)f(;)46 b(I)d(=)-56 b Fk(2)25 b Fo(E)3465 3402 y Fl(\024)3525 3388 y Fo(;)-24 3603 y Fp(where)33 b(the)i(p)s(ositiv)m (e)d(constan)m(ts)k Fo(A;)15 b(C)1322 3617 y Fn(1)1362 3603 y Fo(;)g(C)1467 3617 y Fn(2)1540 3603 y Fp(are)35 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Fo(<)f(\016)k Fk(\024)c Fp(1,)31 b(and)f(for)g(an)m (y)h(0)26 b Fo(<)f(a)g Fk(\024)g Fo(a)2616 5692 y Fn(0)2681 5678 y Fk(\034)g Fp(1.)1843 5927 y(3)p eop %%Page: 4 4 4 3 bop 117 573 a Fp(As)23 b(in)f([10)q(])i(the)f(symplectic)f(normal)g (form)h(in)f(Corollary)g(1.2)i(can)g(b)s(e)e(used)g(to)i(obtain)f (Gevrey)h(quan)m(tum)-24 686 y(in)m(tegrabilit)m(y)36 b(o)m(v)m(er)k(the)e(corresp)s(onding)e(family)g(of)i(in)m(v)-5 b(arin)m(t)37 b(tori)h(and)f(to)i(construct)f(quasimo)s(des)f(with)-24 799 y(exp)s(onen)m(tially)31 b(small)h(discrepancy)g(in)g(the)h (semi-classical)f(limit)f(for)h(Sc)m(hr\177)-45 b(odinger)32 b(t)m(yp)s(e)h(op)s(erators)h(with)-24 912 y(Gevrey)40 b(co)s(e\016cien)m(ts.)70 b(Similar)37 b(results)i(could)g(b)s(e)g (obtained)g(for)h(more)g(general)g(classes)g(of)g(non)f(quasi-)-24 1024 y(analytic)30 b(Hamiltonians)e(as)j(w)m(ell.)117 1137 y(The)38 b(idea)f(of)i(the)f(pro)s(of)g(of)g(Theorem)g(1.1)h(is)e 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b Fp(1)31 b(dep)s(ends)-24 4690 y(only)e(on)i Fo(L)364 4704 y Fn(0)403 4690 y Fp(,)f Fo(\032)p Fp(,)h Fo(\034)40 b Fp(and)30 b Fo(n)p Fp(.)40 b(Then)30 b(using)f(\(1.1\),)j (\(1.2\),)g(and)e(Prop)s(osition)f(A.3)i(w)m(e)g(obtain)716 4877 y Fk(j)p Fo(@)794 4839 y Fl(\013)789 4899 y(\022)844 4877 y Fo(@)897 4832 y Fl(\014)892 4904 y(I)945 4877 y Fo(@)998 4839 y Fl(\015)993 4899 y(!)1044 4877 y Fo(P)13 b Fp(\()p Fo(\022)s(;)i(I)7 b Fp(;)15 b Fo(!)s Fp(\))p Fk(j)26 b(\024)1565 4782 y Ff(\020)1614 4877 y Fo(A)1682 4891 y Fn(0)1722 4877 y Fo(R)1792 4839 y Fn(2)1851 4877 y Fp(+)20 b Fo(\024)1994 4839 y Fn(2)2034 4877 y Fo(\017)2071 4891 y Fl(H)2138 4782 y Ff(\021)2203 4877 y Fo(L)2265 4829 y Fi(j)p Fl(\013)p Fi(j)2265 4901 y Fn(1)2353 4877 y Fo(L)2415 4829 y Fi(j)p Fl(\014)s Fi(j)p Fn(+)p Fi(j)p Fl(\015)t Fi(j)2415 4901 y Fn(2)2636 4877 y Fp(\()p Fo(\013)p Fp(!)p Fo(\014)5 b Fp(!)p Fo(\015)g Fp(!\))2947 4839 y Fl(\032)2990 4877 y Fo(;)-24 5063 y Fp(for)34 b(an)m(y)g Fo(\013;)15 b(\014)41 b 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Fn(1)1841 5379 y Fo(L)1903 5332 y Fi(\000j)p Fl(\014)s Fi(j\000j)p Fl(\015)t Fi(j)1903 5404 y Fn(2)2179 5379 y Fp(\()p Fo(\013)p Fp(!)p Fo(\014)5 b Fp(!)p Fo(\015)g Fp(!\))2490 5342 y Fi(\000)p Fl(\032)2587 5285 y Ff(\021)2662 5379 y Fk(\024)25 b Fo(A)2826 5393 y Fn(0)2866 5379 y Fo(R)2936 5342 y Fn(2)2995 5379 y Fp(+)20 b Fo(\024)3138 5342 y Fn(2)3178 5379 y Fo(\017)3215 5393 y Fl(H)3282 5379 y Fo(;)262 b Fp(\(2.2\))-24 5565 y(where)38 b(the)i(sup)d(is)h (tak)m(en)i(o)m(v)m(er)h(all)d(m)m(ulti-indices)d Fo(\013;)15 b(\014)5 b(;)15 b(\015)46 b Fp(and)38 b(for)h(all)f(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(!)s Fp(\))40 b Fk(2)f Fm(T)3100 5532 y Fl(n)3173 5565 y Fk(\002)26 b Fo(B)k Fk(\002)c Fp(\012.)66 b(Fix)-24 5678 y(0)26 b Fo(<)f(&)32 b Fk(\024)25 b Fp(1.)41 b(W)-8 b(e)32 b(can)e(no)m(w)h(form)m(ulate)f (our)g(main)f(result)h(in)f(this)g(section.)1843 5927 y(4)p eop %%Page: 5 5 5 4 bop -24 573 a Fm(Theorem)34 b(2.1)46 b Fe(Supp)-5 b(ose)41 b(that)g Fo(H)46 b Fe(is)39 b(given)g(by)g(\(2.1\))i(wher)-5 b(e)40 b Fo(P)51 b Fk(2)37 b(G)2504 529 y Fl(\032)2499 600 y(L)2547 609 y Fj(1)2582 600 y Fl(;L)2650 609 y Fj(2)2684 600 y Fl(;L)2752 609 y Fj(2)2790 573 y Fp(\()p Fm(T)2898 540 y Fl(n)2971 573 y Fk(\002)p 3066 501 74 4 v 24 w Fo(B)3140 587 y Fl(R)3198 573 y Fp(\(0\))26 b Fk(\002)p 3435 501 66 4 v 25 w Fp(\012)o(\))p Fe(.)63 b(Fix)-24 702 y Fo(\024)26 b(>)e Fp(0)31 b Fe(and)h Fo(r)27 b(>)e Fp(0)31 b Fe(such)g(that)g Fo(\024;)15 b(r)29 b(<)c(L)1348 664 y Fi(\000)p Fn(1)p Fi(\000)p Fl(&)1348 727 y Fn(2)1558 702 y Fe(and)31 b Fo(r)d Fk(\024)d Fo(R)q Fe(.)41 b(Then)31 b(ther)-5 b(e)31 b(is)f Fo(N)36 b Fp(=)24 b Fo(N)10 b Fp(\()p Fo(n;)15 b(\032;)g(\034)10 b Fp(\))27 b Fo(>)e Fp(0)31 b Fe(and)g Fo(\017)25 b(>)g Fp(0)-24 815 y Fe(indep)-5 b(endent)34 b(of)f Fo(\024)p Fe(,)g Fo(L)762 829 y Fn(1)801 815 y Fe(,)f Fo(L)923 829 y Fn(2)963 815 y Fe(,)g Fo(r)s Fe(,)g Fo(R)q Fe(,)g(and)i(of)f Fp(\012)24 b Fk(\032)h Fp(\012)1793 829 y Fn(0)1833 815 y Fe(,)32 b(such)h(that)h(if)1549 1020 y Fk(k)p Fo(P)13 b Fk(k)27 b(\024)57 b Fo(\017\024r)s(L)2060 981 y Fi(\000)p Fl(N)2060 1044 y Fn(1)3569 1020 y Fp(\(2.3\))-24 1224 y Fe(then)25 b(ther)-5 b(e)25 b(exist)f(maps)i Fo(\036)f Fk(2)g(G)1044 1191 y Fl(\032)1080 1167 y Fd(0)1107 1224 y Fp(\(\012)p Fo(;)15 b Fp(\012\))25 b Fe(and)g Fp(\010)g(=)g(\()p Fo(U;)15 b(V)21 b Fp(\))26 b Fk(2)f(G)2146 1191 y Fl(\032;\032)2238 1167 y Fd(0)2264 1224 y Fp(\()p Fm(T)2372 1191 y Fl(n)2422 1224 y Fk(\002)r Fp(\012)p Fo(;)15 b Fm(T)2674 1191 y Fl(n)2722 1224 y Fk(\002)r Fo(B)2864 1238 y Fl(R)2922 1224 y Fp(\(0\)\))p Fe(,)27 b Fo(\032)3174 1191 y Fi(0)3223 1224 y Fp(=)e Fo(\032)p Fp(\()p Fo(\034)12 b Fp(+)r(1\))r(+)r(1)p Fe(,)-24 1337 y(satisfying)55 1524 y(\(i\))46 b(F)-7 b(or)41 b(e)-5 b(ach)40 b Fo(!)g Fk(2)d Fp(\012)852 1538 y Fl(\024)897 1524 y Fe(,)k(the)e(map)i Fp(\010)1396 1538 y Fl(!)1484 1524 y Fp(:=)c(\010\()p Fk(\001)p Fo(;)15 b(!)s Fp(\))38 b(:)g Fm(T)2052 1491 y Fl(n)2136 1524 y Fk(!)g Fm(T)2338 1491 y Fl(n)2410 1524 y Fk(\002)24 b Fo(B)2574 1538 y Fl(R)2632 1524 y Fp(\(0\))41 b Fe(is)e(an)h Fk(G)3088 1491 y Fl(\032)3168 1524 y Fe(emb)-5 b(e)g(dding)40 b(and)203 1637 y Fp(\003)266 1651 y Fl(!)342 1637 y Fp(:=)25 b(\010)529 1651 y Fl(!)579 1637 y Fp(\()p Fm(T)687 1604 y Fl(n)734 1637 y Fp(\))31 b Fe(is)g(an)g(emb)-5 b(e)g(dde)g(d)33 b(L)-5 b(agr)g(angian)33 b(torus)f(invariant)g(with)g(r)-5 b(esp)g(e)g(ct)32 b(to)f(the)h(Hamiltonian)203 1750 y(\015ow)i(of)f Fo(H)579 1769 y Fl(\036)p Fn(\()p Fl(!)r Fn(\))726 1750 y Fp(\()p Fo(';)15 b(I)7 b Fp(\))27 b(:=)e Fo(H)7 b Fp(\()p Fo(';)15 b(I)7 b Fp(;)15 b Fo(\036)p Fp(\()p Fo(!)s Fp(\)\))p Fe(.)44 b(Mor)-5 b(e)g(over,)34 b Fo(X)2188 1764 y Fl(H)2246 1778 y Fg(\036)p Fj(\()p Fg(!)r Fj(\))2400 1750 y Fk(\016)21 b Fp(\010)2532 1764 y Fl(!)2607 1750 y Fp(=)k Fo(D)s Fp(\010)2847 1764 y Fl(!)2917 1750 y Fk(\001)c(L)3026 1764 y Fl(!)3108 1750 y Fe(on)33 b Fm(T)3311 1717 y Fl(n)3358 1750 y Fe(.)28 1938 y(\(ii\))45 b(Ther)-5 b(e)34 b(exist)e Fo(A;)15 b(C)33 b(>)25 b Fp(0)p Fe(,)32 b(indep)-5 b(endent)35 b(of)d Fo(\024)p Fe(,)h Fo(L)1869 1952 y Fn(1)1909 1938 y Fe(,)f Fo(L)2031 1952 y Fn(2)2070 1938 y Fe(,)h Fo(r)s Fe(,)f(and)h(of)g Fp(\012)25 b Fk(\032)g Fp(\012)2771 1952 y Fn(0)2810 1938 y Fe(,)32 b(such)h(that)744 2065 y Ff(\014)744 2115 y(\014)744 2165 y(\014)771 2163 y Fo(@)824 2130 y Fl(\013)819 2189 y(\022)874 2163 y Fo(@)927 2130 y Fl(\014)922 2185 y(!)975 2163 y Fp(\()p Fo(U)10 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))20 b Fk(\000)g Fo(\022)s Fp(\))1490 2065 y Ff(\014)1490 2115 y(\014)1490 2165 y(\014)1553 2163 y Fp(+)35 b Fo(r)1703 2130 y Fi(\000)p Fn(1)1812 2065 y Ff(\014)1812 2115 y(\014)1812 2165 y(\014)1839 2163 y Fo(@)1892 2130 y Fl(\013)1887 2189 y(\022)1942 2163 y Fo(@)1995 2130 y Fl(\014)1990 2185 y(!)2043 2163 y Fo(V)20 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))2332 2065 y Ff(\014)2332 2115 y(\014)2332 2165 y(\014)2380 2163 y Fp(+)20 b Fo(\024)2523 2130 y Fi(\000)p Fn(1)2633 2065 y Ff(\014)2633 2115 y(\014)2633 2165 y(\014)2660 2163 y Fo(@)2713 2130 y Fl(\014)2708 2185 y(!)2761 2163 y Fp(\()p Fo(\036)p Fp(\()p Fo(!)s Fp(\))h Fk(\000)f Fo(!)s Fp(\))3187 2065 y Ff(\014)3187 2115 y(\014)3187 2165 y(\014)744 2465 y Fk(\024)57 b Fo(A)15 b(C)1027 2417 y Fi(j)p Fl(\013)p Fi(j)1020 2490 y Fn(1)1146 2396 y Ff(\000)1184 2465 y Fo(C)1249 2479 y Fn(2)1288 2465 y Fo(\024)1340 2432 y Fi(\000)p Fn(1)1435 2396 y Ff(\001)1473 2415 y Fi(j)p Fl(\014)s Fi(j)1590 2465 y Fo(\013)p Fp(!)1685 2432 y Fl(\032)1726 2465 y Fo(\014)5 b Fp(!)1819 2432 y Fl(\032)1855 2409 y Fd(0)1907 2403 y Fk(k)p Fo(P)13 b Fk(k)p Fo(L)2130 2370 y Fl(N)2130 2427 y Fn(1)p 1907 2444 292 4 v 2004 2527 a Fo(\024r)203 2708 y Fe(uniformly)34 b(in)e Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))26 b Fk(2)f Fm(T)1130 2675 y Fl(n)1197 2708 y Fk(\002)19 b Fp(\012)33 b Fe(and)g(for)g(any)h Fo(\013)f Fe(and)g Fo(\014)5 b Fe(,)33 b(wher)-5 b(e)34 b Fo(C)2588 2722 y Fn(1)2652 2708 y Fp(=)25 b Fo(C)7 b(L)2882 2722 y Fn(1)2953 2708 y Fe(and)34 b Fo(C)3195 2722 y Fn(2)3260 2708 y Fp(=)25 b Fo(C)7 b(L)3490 2670 y Fl(\034)h Fn(+1)3490 2733 y(1)3622 2708 y Fe(.)-24 2896 y(R)-5 b(emark)p Fp(.)41 b(Note)30 b(that)e(the)g(constan)m(t)i Fo(e)p Fp(\()p Fo(!)s Fp(\))f(in)d (\(2.1\))k(pla)m(ys)e(no)g(role)f(in)g(Theorem)h(2.1)h(and)e(from)h(no) m(w)g(on)g(w)m(e)-24 3009 y(supp)s(ose)h Fo(e)p Fp(\()p Fo(!)s Fp(\))d Fk(\021)f Fp(0.)117 3121 y(Theorem)31 b(2.1)g(will)e(b)s(e)h(pro)m(v)m(ed)h(in)e(the)i(next)g(section.)43 b(Theorem)30 b(1.1)i(and)e(Corollary)g(1.2)i(follo)m(w)e(from)-24 3234 y(Theorem)g(2.1)h(and)f(they)h(will)c(b)s(e)j(pro)m(v)m(ed)h(in)e (Sect.)41 b(4.)-24 3521 y Fr(3)135 b Fq(Pr)-6 b(o)g(of)42 b(of)f(The)-6 b(or)g(em)41 b(2.1.)-24 3724 y Fp(W)-8 b(e)38 b(divide)c(the)j(pro)s(of)f(of)g(Theorem)h(2.1)g(in)f(sev)m (eral)h(steps.)59 b(First,)38 b(using)d(Theorem)h(3.7,)k(w)m(e)d (extend)f Fo(P)-24 3837 y Fp(to)d(a)g(Gevrey)g(function)861 3810 y Ff(e)841 3837 y Fo(P)45 b Fp(of)33 b(the)g(class)f Fk(G)1483 3792 y Fl(\032)1481 3878 y Ff(e)1478 3888 y Fl(L)1526 3897 y Fj(1)1561 3888 y Fl(;)1584 3878 y Ff(e)1581 3888 y Fl(L)1629 3897 y Fj(2)1667 3837 y Fp(\()p Fm(T)1775 3804 y Fl(n)1844 3837 y Fk(\002)21 b Fm(R)2014 3804 y Fn(2)p Fl(n)2096 3837 y Fp(\))33 b(suc)m(h)f(that)h Fk(k)2636 3810 y Ff(e)2615 3837 y Fo(P)14 b Fk(k)29 b(\024)f Fo(A)p Fk(k)p Fo(P)13 b Fk(k)p Fp(,)3159 3810 y Ff(e)3149 3837 y Fo(L)3211 3851 y Fn(1)3279 3837 y Fp(=)28 b Fo(C)7 b(L)3512 3851 y Fn(1)3551 3837 y Fp(,)33 b(and)-14 3969 y Ff(e)-24 3996 y Fo(L)38 4010 y Fn(2)118 3996 y Fp(=)41 b Fo(C)7 b(L)364 4010 y Fn(2)403 3996 y Fp(,)42 b(where)e(the)g (constan)m(ts)h Fo(A;)15 b(C)48 b(>)41 b Fp(0)f(are)g(indep)s(enden)m (t)e(of)i Fo(P)13 b Fp(,)43 b Fo(R)d Fp(and)f(\012.)69 b(T)-8 b(o)41 b(simplify)36 b(the)-24 4109 y(notations)26 b(w)m(e)f(drop)g Fk(\030)p Fp(.)39 b(Multiplying)22 b Fo(P)38 b Fp(with)24 b(a)i(suitable)e(cut-o\013)j(function)d(w)m(e)i (assume)f(that)h(the)g(supp)s(ort)-24 4222 y(of)k Fo(P)44 b Fp(with)29 b(resp)s(ect)h(to)h(\()p Fo(I)7 b(;)15 b(!)s Fp(\))32 b(is)d(con)m(tained)i(in)e Fo(B)1737 4236 y Fn(1)1776 4222 y Fp(\(0\))21 b Fk(\002)f Fo(B)2087 4229 y Fn(\026)2072 4245 y Fl(R)2130 4222 y Fp(\(0\),)2321 4199 y(\026)2302 4222 y Fo(R)26 b Fk(\035)f Fp(1.)-24 4465 y Fc(3.1)112 b Fb(Appr)-5 b(oximation)34 b(L)-5 b(emma)34 b(for)h(Gevr)-5 b(ey)35 b(functions)-24 4637 y Fp(W)-8 b(e)36 b(\014x)f(0)f Fo(<)f(&)40 b Fk(\024)33 b Fp(1)j(and)e(c)m(ho)s(ose)j(three)e(strictly)f(decreasing)h (sequences)h(of)f(p)s(ositiv)m(e)f(n)m(um)m(b)s(ers)g Fk(f)p Fo(u)3521 4651 y Fl(j)3558 4637 y Fk(g)3603 4604 y Fi(1)3603 4661 y Fl(j)t Fn(=0)3730 4637 y Fp(,)-24 4750 y Fk(f)p Fo(v)65 4764 y Fl(j)102 4750 y Fk(g)147 4717 y Fi(1)147 4774 y Fl(j)t Fn(=0)304 4750 y Fp(and)c Fk(f)p Fo(w)591 4764 y Fl(j)628 4750 y Fk(g)673 4717 y Fi(1)673 4774 y Fl(j)t Fn(=0)831 4750 y Fp(tending)f(to)i(0)g(and)f (suc)m(h)g(that)768 4967 y Fk(8)p Fo(j)h Fk(2)24 b Fm(N)56 b Fp(:)116 b Fo(v)1295 4981 y Fl(j)1332 4967 y Fo(L)1394 4981 y Fn(2)1433 4967 y Fo(;)31 b(w)1554 4981 y Fl(j)1590 4967 y Fo(L)1652 4981 y Fn(2)1717 4967 y Fk(\024)25 b Fo(u)1865 4981 y Fl(j)1901 4967 y Fo(L)1963 4981 y Fn(1)2028 4967 y Fk(\024)g Fp(1)15 b Fo(;)107 b(v)2360 4981 y Fn(0)2399 4967 y Fo(;)31 b(w)2520 4981 y Fn(0)2585 4967 y Fk(\024)25 b Fo(L)2743 4929 y Fi(\000)p Fn(1)p Fi(\000)p Fl(&)2743 4992 y Fn(2)2938 4967 y Fo(:)606 b Fp(\(3.1\))-24 5172 y(Consider)40 b(the)i(complex)f(sets)h Fk(U)1157 5139 y Fl(m)1148 5196 y(j)1223 5172 y Fp(,)j Fo(m)f Fp(=)g(1)p Fo(;)15 b Fp(2,)46 b(in)40 b Fm(C)1926 5139 y Fl(n)1972 5172 y Fo(=)p Fp(2)p Fo(\031)s Fm(Z)2181 5139 y Fl(n)2257 5172 y Fk(\002)27 b Fm(C)2431 5139 y Fl(n)2505 5172 y Fk(\002)h Fm(C)2680 5139 y Fl(n)2768 5172 y Fp(consisting)40 b(of)i(all)f(\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)s Fp(\))-24 5284 y(with)36 b(real)g(parts)h(Re)16 b Fo(\022)38 b Fk(2)f Fm(T)988 5251 y Fl(n)1034 5284 y Fp(,)i(Re)16 b Fo(I)43 b Fk(2)36 b Fo(B)1470 5298 y Fn(2)1510 5284 y Fp(\(0\))i(and)f(Re)15 b Fo(!)39 b Fk(2)d Fo(B)2246 5291 y Fn(\026)2231 5308 y Fl(R)q Fn(+1)2379 5284 y Fp(\(0\),)k(and)d (suc)m(h)g(that)g Fk(j)p Fp(Im)15 b Fo(\022)3350 5299 y Fl(k)3393 5284 y Fk(j)37 b(\024)f Fo(mu)3694 5298 y Fl(j)3730 5284 y Fp(,)-24 5397 y Fk(j)p Fp(Im)15 b Fo(I)165 5412 y Fl(k)207 5397 y Fk(j)27 b(\024)e Fo(mv)479 5411 y Fl(j)516 5397 y Fp(,)31 b Fk(j)p Fp(Im)15 b Fo(!)778 5412 y Fl(k)820 5397 y Fk(j)27 b(\024)e Fo(mw)1113 5411 y Fl(j)1150 5397 y Fp(,)31 b(for)f(eac)m(h)i(1)27 b Fk(\024)f Fo(k)j Fk(\024)d Fo(n)p Fp(.)41 b(Set)31 b Fk(U)2227 5411 y Fl(j)2290 5397 y Fp(=)25 b Fk(U)2452 5364 y Fn(1)2443 5422 y Fl(j)2522 5397 y Fp(and)31 b(denote)g(b)m(y)g Fo(A)p Fp(\()p Fk(U)3280 5411 y Fl(j)3317 5397 y Fp(\))g(the)g(set)g (of)-24 5510 y(all)e(real-analytic)h(b)s(ounded)e(functions)h(in)g Fk(U)1551 5524 y Fl(j)1618 5510 y Fp(equipp)s(ed)f(with)h(the)i (sup-norm)d Fk(j)21 b(\001)f(j)2891 5524 y Fi(U)2935 5534 y Fg(j)2972 5510 y Fp(.)1843 5927 y(5)p eop %%Page: 6 6 6 5 bop -24 573 a Fm(Prop)s(osition)36 b(3.1)g(\(Appro)m(ximation)e (Lemma\))43 b Fe(L)-5 b(et)41 b Fo(P)54 b Fk(2)40 b(G)2334 529 y Fl(\032)2329 600 y(L)2377 609 y Fj(1)2412 600 y Fl(;L)2480 609 y Fj(2)2518 573 y Fp(\()p Fm(T)2626 540 y Fl(n)2700 573 y Fk(\002)26 b Fm(R)2875 540 y Fn(2)p Fl(n)2957 573 y Fp(\))p Fe(.)68 b(Supp)-5 b(ose)43 b(that)f(the)-24 686 y(supp)-5 b(ort)44 b(of)e Fo(P)54 b Fe(with)43 b(r)-5 b(esp)g(e)g(ct)43 b(to)f Fp(\()p Fo(I)7 b(;)15 b(!)s Fp(\))43 b Fe(is)f(in)f Fo(B)1721 700 y Fn(1)1761 686 y Fp(\(0\))28 b Fk(\002)e Fo(B)2085 693 y Fn(\026)2070 709 y Fl(R)2128 686 y Fp(\(0\))p Fe(,)45 b(and)d(assume)h(\(3.1\).)70 b(Then)42 b(ther)-5 b(e)42 b(is)g(a)-24 799 y(se)-5 b(quenc)g(e)32 b Fo(P)403 813 y Fl(j)465 799 y Fk(2)25 b Fo(A)p Fp(\()p Fk(U)711 813 y Fl(j)748 799 y Fp(\))p Fe(,)33 b Fo(j)e Fk(\025)25 b Fp(0)p Fe(,)33 b(such)f(that)521 1026 y Fk(j)p Fo(P)604 1040 y Fl(j)t Fn(+1)752 1026 y Fk(\000)19 b Fo(P)900 1040 y Fl(j)937 1026 y Fk(j)962 1040 y Fi(U)1006 1050 y Fg(j)s Fj(+1)1203 1026 y Fk(\024)83 b Fo(C)1422 1040 y Fn(0)1477 1026 y Fo(L)1539 993 y Fl(n)1539 1049 y Fn(1)1616 1026 y Fp(exp)1770 932 y Ff(\020)1819 1026 y Fk(\000)1900 991 y Fn(3)p 1900 1006 36 4 v 1900 1058 a(4)1945 1026 y Fp(\()p Fo(\032)21 b Fk(\000)f Fp(1\)\(2)p Fo(L)2361 1040 y Fn(1)2402 1026 y Fo(u)2454 1040 y Fl(j)2490 1026 y Fp(\))2525 981 y Fi(\000)2630 954 y Fj(1)p 2590 966 111 4 v 2590 1007 a Fg(\032)p Fd(\000)p Fj(1)2715 932 y Ff(\021)2780 1026 y Fk(k)p Fo(P)13 b Fk(k)p Fo(;)889 1246 y Fk(j)p Fo(P)972 1260 y Fn(0)1012 1246 y Fk(j)1037 1260 y Fi(U)1081 1269 y Fj(0)1203 1246 y Fk(\024)83 b Fo(C)1422 1260 y Fn(0)1477 1152 y Ff(\020)1526 1246 y Fp(1)21 b(+)f Fo(L)1745 1213 y Fl(n)1745 1269 y Fn(1)1807 1246 y Fp(exp)1961 1152 y Ff(\020)2010 1246 y Fk(\000)2091 1210 y Fn(3)p 2091 1225 36 4 v 2091 1277 a(4)2136 1246 y Fp(\()p Fo(\032)h Fk(\000)f Fp(1\)\(2)p Fo(L)2552 1260 y Fn(1)2593 1246 y Fo(u)2645 1260 y Fn(0)2684 1246 y Fp(\))2719 1201 y Fi(\000)2824 1173 y Fj(1)p 2784 1185 111 4 v 2784 1227 a Fg(\032)p Fd(\000)p Fj(1)2909 1152 y Ff(\021)o(\021)3023 1246 y Fk(k)p Fo(P)13 b Fk(k)p Fo(:)-24 1481 y Fe(wher)-5 b(e)34 b Fo(C)298 1495 y Fn(0)362 1481 y Fp(=)478 1454 y Ff(e)458 1481 y Fo(C)523 1495 y Fn(0)563 1481 y Fp(\()p Fo(n;)15 b(\032;)g(&)7 b Fp(\)\()911 1458 y(\026)890 1481 y Fo(R)960 1448 y Fl(n)1028 1481 y Fp(+)20 b(1\))p Fe(.)43 b(Mor)-5 b(e)g(over,)315 1722 y Fp(sup)14 b Fk(j)p Fo(@)545 1685 y Fl(\013)540 1745 y(\022)595 1722 y Fo(@)648 1678 y Fl(\014)643 1750 y(I)695 1722 y Fo(@)748 1685 y Fl(\015)743 1745 y(!)794 1722 y Fp(\()p Fo(P)34 b Fk(\000)20 b Fo(P)1070 1736 y Fl(j)1106 1722 y Fp(\)\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)s Fp(\))p Fk(j)60 b(\024)d Fo(C)1722 1736 y Fn(0)1777 1722 y Fo(L)1839 1685 y Fl(n)1839 1745 y Fn(1)1901 1722 y Fo(L)1963 1736 y Fn(2)2032 1722 y Fp(exp)2186 1603 y Ff(\022)2247 1722 y Fk(\000)2328 1661 y Fp(3)p 2328 1701 46 4 v 2328 1785 a(4)2383 1722 y(\()p Fo(\032)21 b Fk(\000)f Fp(1\)\(2)p Fo(L)2799 1736 y Fn(1)2840 1722 y Fo(u)2892 1736 y Fl(j)2928 1722 y Fp(\))2963 1677 y Fi(\000)3068 1650 y Fj(1)p 3029 1662 111 4 v 3029 1703 a Fg(\032)p Fd(\000)p Fj(1)3153 1603 y Ff(\023)3229 1722 y Fk(k)p Fo(P)13 b Fk(k)p Fo(;)-24 1964 y Fe(in)32 b Fm(T)160 1931 y Fl(n)227 1964 y Fk(\002)20 b Fo(B)387 1978 y Fn(1)427 1964 y Fp(\(0\))h Fk(\002)f Fo(B)738 1971 y Fn(\026)723 1987 y Fl(R)781 1964 y Fp(\(0\))33 b Fe(for)g Fk(j)p Fo(\013)p Fk(j)22 b Fp(+)e Fk(j)p Fo(\014)5 b Fk(j)20 b Fp(+)g Fk(j)p Fo(\015)5 b Fk(j)26 b(\024)f Fp(1)p Fe(.)-24 2173 y(R)-5 b(emark)p Fp(.)50 b(Instead)33 b(of)g(3)p Fo(=)p Fp(4)h(w)m(e)g(can)f(tak)m(e)h(ab)s(o)m(v)m(e)h(an)m (y)e(p)s(ositiv)m(e)f(n)m(um)m(b)s(er)g(less)g(than)h(1)g(in)f(order)g (to)i(absorb)-24 2286 y(certain)40 b(p)s(olynomials)d(of)j(\()p Fo(L)1014 2300 y Fn(1)1054 2286 y Fo(u)1106 2300 y Fl(j)1143 2286 y Fp(\))1178 2253 y Fi(\000)p Fn(1)1272 2286 y Fp(.)70 b(Similar)37 b(estimates)k(can)g(b)s(e)e(obtained)g(without)h(the)g (inequalities)-24 2399 y Fo(v)20 2413 y Fn(0)59 2399 y Fo(;)15 b(w)164 2413 y Fn(0)230 2399 y Fk(\024)25 b Fo(L)388 2360 y Fi(\000)p Fn(1)p Fi(\000)p Fl(&)388 2423 y Fn(2)568 2399 y Fp(.)39 b(In)25 b(this)f(case)1121 2372 y Ff(e)1101 2399 y Fo(C)1166 2413 y Fn(0)1231 2399 y Fp(=)1347 2372 y Ff(e)1327 2399 y Fo(C)1392 2413 y Fn(0)1431 2399 y Fp(\()p Fo(n;)15 b(\032)p Fp(\))26 b(but)f(the)h(righ) m(t)f(hand)g(side)f(of)i(the)g(estimates)g(ab)s(o)m(v)m(e)h(should)-24 2512 y(b)s(e)g(m)m(ultiplied)e(b)m(y)i Fo(L)712 2479 y Fn(2)p Fl(n)712 2535 y Fn(2)795 2512 y Fp(.)39 b(Using)28 b(the)g(`standard')f(pro)s(of)h(of)g(the)g(Appro)m(ximation)e(Lemma)i ([12)r(])g(one)g(obtains)-24 2648 y(for)i(an)m(y)h Fo(\016)e(>)c Fp(0)31 b(an)f(appro)m(ximation)f(mo)s(dulo)g Fo(C)7 b Fp(\()p Fo(\032;)15 b(\016)s Fp(\))g(exp)2029 2554 y Ff(\020)2078 2648 y Fk(\000)p Fo(c)p Fp(\()p Fo(\032)p Fp(\)\()p Fo(L)2402 2662 y Fn(1)2443 2648 y Fo(u)2495 2662 y Fl(j)2531 2648 y Fp(\))2566 2602 y Fi(\000)2632 2575 y Fj(1)p 2631 2587 32 4 v 2631 2628 a Fg(\032)2673 2602 y Fn(+)p Fl(\016)2766 2554 y Ff(\021)2831 2648 y Fk(k)p Fo(P)e Fk(k)p Fp(,)31 b Fo(C)q(;)15 b(c)26 b(>)f Fp(0.)-24 2842 y Fe(Pr)-5 b(o)g(of.)42 b Fp(W)-8 b(e)31 b(divide)d(the)j(pro)s(of)f(in)m(to)g(t)m(w)m(o)i(parts.)-24 3026 y Fe(1.)41 b(A)n(lmost)29 b(analytic)h(extension)g(of)f(P)p Fp(.)d(There)f(is)g(a)i(constan)m(t)h Fo(C)7 b Fp(\()p Fo(\032)p Fp(\))25 b Fk(\025)g Fp(1,)j(dep)s(ending)23 b(only)j(on)g Fo(\032)p Fp(,)h(suc)m(h)f(that)1046 3250 y Fo(t)f Fk(2)g Fp(\(0)p Fo(;)15 b Fp(2])g Fo(;)48 b(m)25 b Fk(2)g Fm(N)15 b Fo(;)106 b Fp(1)26 b Fk(\024)f Fo(m)g Fk(\024)g Fo(t)2300 3204 y Fi(\000)2404 3177 y Fj(1)p 2364 3189 111 4 v 2364 3230 a Fg(\032)p Fd(\000)p Fj(1)2509 3250 y Fp(+)19 b(1)c Fo(;)885 b Fp(\(3.2\))-24 3451 y(implies)1146 3564 y Fo(t)1179 3526 y Fl(m)1261 3564 y Fo(m)p Fp(!)1378 3526 y Fl(\032)p Fi(\000)p Fn(1)1564 3564 y Fk(\024)55 b Fo(C)7 b Fp(\()p Fo(\032)p Fp(\))p Fo(m)1959 3526 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(=)p Fn(2)2215 3564 y Fo(e)2257 3526 y Fi(\000)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(m)2560 3564 y Fo(:)984 b Fp(\(3.3\))-24 3729 y(Indeed,)30 b(b)m(y)g(Stirling's)d(form)m(ula,)j(w)m(e)h(get)607 3930 y Fo(t)640 3892 y Fl(m)722 3930 y Fo(m)p Fp(!)839 3892 y Fl(\032)p Fi(\000)p Fn(1)1025 3930 y Fk(\024)55 b Fo(C)1216 3944 y Fn(1)1255 3930 y Fp(\()p Fo(\032)p Fp(\))p Fo(m)1452 3892 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(=)p Fn(2)1709 3930 y Fo(e)1751 3892 y Fi(\000)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(m)2069 3930 y Fp(exp)p Fk(f)p Fo(m)p Fp([\()p Fo(\032)21 b Fk(\000)f Fp(1\))15 b(ln)g Fo(m)20 b Fp(+)g(ln)13 b Fo(t)p Fp(])p Fk(g)791 4154 y Fp(=)55 b Fo(C)982 4168 y Fn(1)1021 4154 y Fp(\()p Fo(\032)p Fp(\))p Fo(m)1218 4116 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(=)p Fn(2)1475 4154 y Fo(e)1517 4116 y Fi(\000)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(m)1835 4154 y Fp(exp)1989 4060 y Ff(n)2044 4154 y Fp(\()p Fo(\032)21 b Fk(\000)f Fp(1\))p Fo(m)15 b Fp(ln)2504 4060 y Ff(h)2544 4154 y Fo(m)g(t)2721 4081 y Fj(1)p 2681 4093 V 2681 4134 a Fg(\032)p Fd(\000)p Fj(1)2806 4060 y Ff(io)2915 4154 y Fo(:)-24 4328 y Fp(Moreo)m(v)m(er,) 807 4459 y Fo(m)g Fp(ln)993 4365 y Ff(h)1032 4459 y Fo(m)g(t)1210 4387 y Fj(1)p 1170 4399 V 1170 4440 a Fg(\032)p Fd(\000)p Fj(1)1294 4365 y Ff(i)1359 4459 y Fk(\024)25 b Fo(m)15 b Fp(ln)1641 4365 y Ff(h)1680 4459 y Fp(1)21 b(+)e Fo(t)1919 4387 y Fj(1)p 1879 4399 V 1879 4440 a Fg(\032)p Fd(\000)p Fj(1)2004 4365 y Ff(i)2068 4459 y Fk(\024)25 b Fo(m)15 b(t)2342 4387 y Fj(1)p 2302 4399 V 2302 4440 a Fg(\032)p Fd(\000)p Fj(1)2451 4459 y Fk(\024)25 b Fp(1)c(+)f(2)2799 4387 y Fj(1)p 2759 4399 V 2759 4440 a Fg(\032)p Fd(\000)p Fj(1)2899 4459 y Fo(;)-24 4640 y Fp(whic)m(h)29 b(pro)m(v)m(es)i (\(3.3\).)117 4753 y(W)-8 b(e)32 b(de\014ne)d(an)h(almost)h(analytic)f (extensions)g Fo(F)1798 4767 y Fl(j)1865 4753 y Fp(of)h Fo(P)43 b Fp(in)29 b Fk(U)2242 4720 y Fn(2)2233 4777 y Fl(j)2312 4753 y Fp(as)h(follo)m(ws)402 5035 y Fo(F)460 5049 y Fl(j)497 5035 y Fp(\()p Fo(\022)22 b Fp(+)e Fo(i)726 5008 y Ff(e)719 5035 y Fo(\022)s(;)15 b(I)28 b Fp(+)20 b Fo(i)1006 5009 y Ff(e)995 5035 y Fo(I)7 b(;)15 b(!)23 b Fp(+)d Fo(i)1291 5032 y Ff(e)1284 5035 y Fo(!)s Fp(\))56 b(=)1695 4954 y Ff(X)1562 5143 y Fn(\()p Fl(\013;\014)s(;\015)t Fn(\))p Fi(2M)1915 5153 y Fg(j)1978 5035 y Fo(@)2031 4998 y Fl(\013)2026 5058 y(\022)2081 5035 y Fo(@)2134 4991 y Fl(\014)2129 5063 y(I)2181 5035 y Fo(@)2234 4998 y Fl(\015)2229 5058 y(!)2280 5035 y Fo(P)13 b Fp(\()p Fo(\022)s(;)i(I)7 b(;)15 b(!)s Fp(\))2664 4974 y(\()p Fo(i)2737 4946 y Ff(e)2730 4974 y Fo(\022)t Fp(\))2812 4941 y Fl(\013)2861 4974 y Fp(\()p Fo(i)2938 4947 y Ff(e)2927 4974 y Fo(I)8 b Fp(\))3010 4941 y Fl(\014)3058 4974 y Fp(\()p Fo(i)3131 4971 y Ff(e)3124 4974 y Fo(!)s Fp(\))3219 4941 y Fl(\015)p 2664 5014 600 4 v 2843 5098 a Fo(\013)p Fp(!)p Fo(\014)d Fp(!)p Fo(\015)g Fp(!)3304 5035 y Fo(:)240 b Fp(\(3.4\))-24 5335 y(The)41 b(index)f(set)j Fk(M)693 5349 y Fl(j)771 5335 y Fp(consists)e(of)h(all)e(m)m(ulti-indices)f Fo(\013)45 b Fp(=)f(\()p Fo(\013)2230 5349 y Fn(1)2269 5335 y Fo(;)15 b(:)g(:)g(:)i(;)e(\013)2529 5349 y Fl(n)2577 5335 y Fp(\),)45 b Fo(\014)k Fp(=)44 b(\()p Fo(\014)2983 5349 y Fn(1)3023 5335 y Fo(;)15 b(:)g(:)g(:)i(;)e(\014)3276 5349 y Fl(n)3323 5335 y Fp(\))42 b(and)f Fo(\015)50 b Fp(=)-24 5448 y(\()p Fo(\015)58 5462 y Fn(1)98 5448 y Fo(;)15 b(:)g(:)g(:)h(;)f(\015)346 5462 y Fl(n)394 5448 y Fp(\))30 b(suc)m(h)g(that)h Fo(\013)919 5463 y Fl(k)988 5448 y Fk(\024)24 b Fo(N)1156 5462 y Fn(1)1196 5448 y Fp(,)31 b Fo(\014)1303 5463 y Fl(k)1371 5448 y Fk(\024)25 b Fo(N)1540 5462 y Fn(2)1610 5448 y Fp(and)k Fo(\015)1833 5463 y Fl(k)1901 5448 y Fk(\024)c Fo(N)2070 5462 y Fn(3)2110 5448 y Fp(,)30 b Fo(k)f Fp(=)c(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)p Fp(,)30 b(where)195 5672 y Fo(N)268 5686 y Fn(1)333 5672 y Fp(=)429 5578 y Ff(h)468 5672 y Fp(\(2)p Fo(L)610 5686 y Fn(1)650 5672 y Fo(u)702 5686 y Fl(j)739 5672 y Fp(\))774 5627 y Fi(\000)879 5599 y Fj(1)p 839 5611 111 4 v 839 5653 a Fg(\032)p Fd(\000)p Fj(1)963 5578 y Ff(i)1023 5672 y Fp(+)19 b(1)31 b Fo(;)46 b(N)1333 5686 y Fn(2)1398 5672 y Fp(=)1494 5578 y Ff(h)1533 5672 y Fp(\(2)p Fo(L)1675 5686 y Fn(2)1715 5672 y Fo(v)1759 5686 y Fl(j)1796 5672 y Fp(\))1831 5627 y Fi(\000)1936 5599 y Fj(1)p 1896 5611 V 1896 5653 a Fg(\032)p Fd(\000)p Fj(1)2020 5578 y Ff(i)2080 5672 y Fp(+)19 b(1)31 b Fo(;)46 b(N)2390 5686 y Fn(3)2455 5672 y Fp(=)2551 5578 y Ff(h)2590 5672 y Fp(\(2)p Fo(L)2732 5686 y Fn(2)2772 5672 y Fo(w)2837 5686 y Fl(j)2873 5672 y Fp(\))2908 5627 y Fi(\000)3014 5599 y Fj(1)p 2974 5611 V 2974 5653 a Fg(\032)p Fd(\000)p Fj(1)3098 5578 y Ff(i)3157 5672 y Fp(+)20 b(1)31 b Fo(;)220 b Fp(\(3.5\))1843 5927 y(6)p eop %%Page: 7 7 7 6 bop -24 573 a Fp(and)30 b([)p Fo(t)p Fp(])g(stands)g(for)g(the)h (in)m(teger)g(part)f(of)g Fo(t)p Fp(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)527 749 y Fk(j)p Fo(F)610 763 y Fl(j)648 749 y Fk(j)673 773 y Fi(U)724 750 y Fj(2)717 796 y Fg(j)818 749 y Fk(\024)55 b(k)p Fo(P)13 b Fk(k)1254 668 y Ff(X)1121 857 y Fn(\()p Fl(\013;\014)s(;\015)t Fn(\))p Fi(2M)1474 867 y Fg(j)1523 749 y Fp(\(2)p Fo(L)1665 763 y Fn(1)1705 749 y Fo(u)1757 763 y Fl(j)1793 749 y Fp(\))1828 712 y Fi(j)p Fl(\013)p Fi(j)1917 749 y Fp(\(2)p Fo(L)2059 763 y Fn(2)2099 749 y Fo(v)2143 763 y Fl(j)2180 749 y Fp(\))2215 712 y Fi(j)p Fl(\014)s Fi(j)2302 749 y Fp(\(2)p Fo(L)2444 763 y Fn(2)2484 749 y Fo(w)2549 763 y Fl(j)2586 749 y Fp(\))2621 712 y Fi(j)p Fl(\015)t Fi(j)2704 749 y Fp(\()p Fo(\013)p Fp(!)p Fo(\014)5 b Fp(!)p Fo(\015)g Fp(!\))3015 712 y Fl(\032)p Fi(\000)p Fn(1)3179 749 y Fo(:)-24 1020 y Fp(F)-8 b(or)31 b Fo(\013)197 1035 y Fl(k)240 1020 y Fo(;)15 b(\014)331 1035 y Fl(k)374 1020 y Fo(;)g(\015)461 1035 y Fl(k)530 1020 y Fk(6)p Fp(=)25 b(0)30 b(w)m(e)h(estimate)g(eac)m(h)h(term)478 1196 y(\(2)p Fo(L)620 1210 y Fn(1)660 1196 y Fo(u)712 1210 y Fl(j)749 1196 y Fp(\))784 1159 y Fl(\013)829 1171 y Fg(k)872 1196 y Fo(\013)930 1211 y Fl(k)973 1196 y Fp(!)1010 1159 y Fl(\032)p Fi(\000)p Fn(1)1170 1196 y Fo(;)46 b Fp(\(2)p Fo(L)1383 1210 y Fn(2)1423 1196 y Fo(v)1467 1210 y Fl(j)1504 1196 y Fp(\))1539 1159 y Fl(\014)1579 1171 y Fg(k)1621 1196 y Fo(\014)1672 1211 y Fl(k)1715 1196 y Fp(!)1752 1159 y Fl(\032)p Fi(\000)p Fn(1)1913 1196 y Fo(;)f Fp(\(2)p Fo(L)2125 1210 y Fn(2)2166 1196 y Fo(w)2231 1210 y Fl(j)2267 1196 y Fp(\))2302 1159 y Fl(\015)2338 1171 y Fg(k)2381 1196 y Fo(\015)2428 1211 y Fl(k)2471 1196 y Fp(!)2508 1159 y Fl(\032)p Fi(\000)p Fn(1)2669 1196 y Fo(;)g(k)29 b Fp(=)c(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)g(;)-24 1373 y Fp(b)m(y)32 b Fo(C)7 b Fp(\()p Fo(\032)p Fp(\))p Fo(m)373 1340 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(=)p Fn(2)629 1373 y Fo(e)671 1340 y Fi(\000)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(m)974 1373 y Fp(,)34 b(where)e Fo(m)c Fk(\025)h Fp(1)k(stands)f(for)h Fo(\013)2070 1388 y Fl(k)2112 1373 y Fo(;)15 b(\014)2203 1388 y Fl(k)2247 1373 y Fp(,)33 b(and)f Fo(\015)2531 1388 y Fl(k)2606 1373 y Fp(resp)s(ectiv)m(ely)-8 b(.)48 b(T)-8 b(o)33 b(this)e(end,)i(w)m(e)-24 1486 y(put)24 b Fo(t)h Fp(=)g(2)p Fo(L)398 1500 y Fn(1)438 1486 y Fo(u)490 1500 y Fl(j)527 1486 y Fo(;)30 b Fp(2)p Fo(L)689 1500 y Fn(2)729 1486 y Fo(v)773 1500 y Fl(j)809 1486 y Fo(;)h Fp(2)p Fo(L)972 1500 y Fn(2)1012 1486 y Fo(w)1077 1500 y Fl(j)1113 1486 y Fp(,)c(resp)s(ectiv)m(ely)-8 b(,)26 b(and)e(w)m(e)i(get)g Fo(t)f Fk(2)g Fp(\(0)p Fo(;)15 b Fp(2])27 b(in)c(view)h(of)i(\(3.1\).)40 b(No)m(w)26 b(\(3.2\))h(holds)-24 1598 y(b)s(ecause)j(of)h(\(3.5\).)42 b(Then)30 b(using)f(\(3.3\))j(w)m(e)f(obtain)621 1857 y Fk(j)p Fo(F)704 1871 y Fl(j)741 1857 y Fk(j)766 1880 y Fi(U)817 1858 y Fj(2)810 1904 y Fg(j)911 1857 y Fk(\024)55 b(k)p Fo(P)13 b Fk(k)1213 1713 y Ff( )1280 1857 y Fp(1)21 b(+)f Fo(C)7 b Fp(\()p Fo(\032)p Fp(\))1682 1751 y Fi(1)1657 1776 y Ff(X)1641 1957 y Fl(m)p Fn(=1)1809 1857 y Fo(m)1889 1819 y Fn(\(1)p Fi(\000)p Fl(\032)p Fn(\))p Fl(=)p Fn(2)2144 1857 y Fo(e)2186 1819 y Fi(\000)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fl(m)2489 1713 y Ff(!)2555 1736 y Fn(3)p Fl(n)2693 1857 y Fp(=)55 b Fo(C)2884 1871 y Fn(1)2924 1857 y Fk(k)p Fo(P)13 b Fk(k)p Fo(:)-24 2125 y Fp(Set)30 b Fo(z)174 2140 y Fl(k)243 2125 y Fp(=)25 b Fo(\022)382 2140 y Fl(k)444 2125 y Fp(+)20 b Fo(i)573 2097 y Ff(e)566 2125 y Fo(\022)609 2140 y Fl(k)652 2125 y Fp(.)40 b(Then)30 b(applying)1337 2101 y(\026)1326 2125 y Fo(@)1374 2139 y Fl(z)1407 2151 y Fg(k)1475 2125 y Fp(=)25 b(\()p Fo(@)5 b(=@)g(\022)1800 2140 y Fl(k)1863 2125 y Fp(+)20 b Fo(i@)5 b(=@)2143 2097 y Ff(e)2136 2125 y Fo(\022)2179 2140 y Fl(k)2223 2125 y Fp(\))p Fo(=)p Fp(2)32 b(to)f Fo(F)2549 2139 y Fl(j)2616 2125 y Fp(w)m(e)g(obtain)173 2372 y(2)230 2348 y(\026)218 2372 y Fo(@)266 2386 y Fl(z)299 2398 y Fg(k)342 2372 y Fo(F)400 2386 y Fl(j)437 2372 y Fp(\()p Fo(\022)23 b Fp(+)d Fo(i)667 2344 y Ff(e)660 2372 y Fo(\022)r(;)15 b(I)28 b Fp(+)20 b Fo(i)946 2345 y Ff(e)935 2372 y Fo(I)7 b(;)15 b(!)24 b Fp(+)c Fo(i)1232 2369 y Ff(e)1225 2372 y Fo(!)s Fp(\))56 b(=)1624 2291 y Ff(X)1512 2467 y Fj(\()p Fg(\013;\014)s(;\015)s Fj(\))p Fd(2M)1824 2483 y Fg(j)1580 2536 y(\013)1620 2554 y(k)1658 2536 y Fj(=)p Fg(N)1753 2551 y Fj(1)1896 2372 y Fo(@)1949 2334 y Fl(\013)1944 2394 y(\022)1999 2372 y Fo(@)2052 2328 y Fl(\014)2047 2399 y(I)2100 2372 y Fo(@)2153 2334 y Fl(\015)2148 2394 y(!)2198 2372 y Fo(@)2246 2387 y Fl(\022)2279 2399 y Fg(k)2322 2372 y Fo(P)13 b Fp(\()p Fo(\022)s(;)i(I)7 b(;)15 b(!)s Fp(\))2706 2310 y(\()p Fo(i)2779 2283 y Ff(e)2772 2310 y Fo(\022)s Fp(\))2853 2277 y Fl(\013)2903 2310 y Fp(\()p Fo(i)2980 2284 y Ff(e)2969 2310 y Fo(I)8 b Fp(\))3052 2277 y Fl(\014)3099 2310 y Fp(\()p Fo(i)3172 2307 y Ff(e)3165 2310 y Fo(!)t Fp(\))3261 2277 y Fl(\015)p 2706 2351 600 4 v 2885 2434 a Fo(\013)p Fp(!)p Fo(\014)d Fp(!)p Fo(\015)g Fp(!)3346 2372 y Fo(:)198 b Fp(\(3.6\))-24 2697 y(W)-8 b(e)31 b(estimate)h(eac)m(h)f(term)g(in)e(the)h(sum)g(b)m (y)737 2873 y Fo(L)799 2887 y Fn(1)838 2873 y Fp(\(2)p Fo(L)980 2887 y Fn(1)1020 2873 y Fo(u)1072 2887 y Fl(j)1109 2873 y Fp(\))1144 2836 y Fi(j)p Fl(\013)p Fi(j)1233 2873 y Fp(\(2)p Fo(L)1375 2887 y 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3264 y Fp(\))2369 3218 y Fi(\000)2474 3191 y Fj(1)p 2434 3203 V 2434 3244 a Fg(\032)p Fd(\000)p Fj(1)2579 3264 y Fp(+)20 b(1)p Fo(;)-24 3440 y Fp(w)m(e)31 b(obtain)f(from)f(\(3.3\))k(\(with)c Fo(t)c Fp(=)g(2)p Fo(L)1328 3454 y Fn(1)1368 3440 y Fo(u)1420 3454 y Fl(j)1487 3440 y Fp(and)k Fo(m)d Fp(=)f Fo(N)1938 3454 y Fn(1)1977 3440 y Fp(\))383 3643 y(\(2)p Fo(L)525 3657 y Fn(1)565 3643 y Fo(u)617 3657 y Fl(j)654 3643 y Fp(\))689 3606 y Fl(\013)734 3618 y Fg(k)792 3643 y Fo(\013)850 3658 y Fl(k)893 3643 y Fp(!)930 3606 y Fl(\032)p Fi(\000)p Fn(1)1060 3643 y Fp(\()p Fo(\013)1153 3658 y Fl(k)1216 3643 y Fp(+)20 b(1\))1387 3606 y Fl(\032)1454 3643 y Fk(\024)25 b Fo(C)1622 3606 y Fi(0)1644 3643 y Fp(\()p Fo(L)1741 3657 y Fn(1)1781 3643 y Fo(u)1833 3657 y Fl(j)1870 3643 y Fp(\))1905 3598 y Fi(\000)2009 3566 y Fg(\032)p 1970 3583 V 1970 3624 a(\032)p Fd(\000)p Fj(1)2090 3598 y Fi(\000)2155 3571 y Fj(1)p 2155 3583 31 4 v 2155 3624 a(2)2215 3643 y Fp(exp)2369 3549 y Ff(\020)2418 3643 y Fk(\000)p 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3979 y Fi(\000)2845 3952 y Fj(1)p 2805 3964 111 4 v 2805 4005 a Fg(\032)p Fd(\000)p Fj(1)2929 3930 y Ff(\021)3009 4024 y Fk(k)p Fo(P)13 b Fk(k)1016 4244 y(j)1053 4220 y Fp(\026)1041 4244 y Fo(@)1089 4258 y Fl(z)1122 4270 y Fg(k)1165 4244 y Fo(F)1223 4258 y Fl(j)1260 4244 y Fk(j)1285 4268 y Fi(U)1336 4245 y Fj(2)1329 4291 y Fg(j)1430 4244 y Fk(\024)56 b Fo(C)21 b(L)1705 4258 y Fn(1)1775 4244 y Fp(exp)1929 4150 y Ff(\020)1978 4244 y Fk(\000)2059 4208 y Fn(3)p 2059 4223 36 4 v 2059 4275 a(4)2119 4244 y Fp(\()p Fo(\032)g Fk(\000)f Fp(1\)\(2)p Fo(L)2535 4258 y Fn(1)2576 4244 y Fo(u)2628 4258 y Fl(j)2664 4244 y Fp(\))2699 4199 y Fi(\000)2805 4171 y Fj(1)p 2765 4183 111 4 v 2765 4225 a Fg(\032)p Fd(\000)p Fj(1)2889 4150 y Ff(\021)2969 4244 y Fk(k)p Fo(P)13 b Fk(k)i Fo(;)-24 4441 y Fp(where)30 b Fo(C)h Fp(=)25 b Fo(C)7 b Fp(\()p Fo(\032;)15 b(n)p Fp(\))26 b Fo(>)f Fp(0.)41 b(In)30 b(the)g(same)h(w)m(a)m(y)-8 b(,)32 b(di\013eren)m(tiating)d(\(3.6\),)j(w)m(e)f(get)h(with)d Fo(\014)c Fp(+)20 b Fo(\015)31 b Fp(=)25 b(1)854 4638 y Fk(j)p Fo(@)932 4594 y Fl(\014)927 4666 y(\022)960 4674 y Fg(p)1001 4638 y Fo(@)1054 4594 y Fl(\015)1049 4681 y Ff(e)1049 4691 y Fl(\022)1082 4699 y Fg(p)1134 4614 y Fp(\026)1122 4638 y Fo(@)1170 4652 y Fl(z)1203 4664 y Fg(k)1246 4638 y Fo(F)1304 4652 y Fl(j)1341 4638 y Fk(j)1366 4662 y Fi(U)1417 4639 y Fj(2)1410 4685 y Fg(j)1486 4638 y Fo(;)45 b Fk(j)p Fo(@)1634 4594 y Fl(\014)1629 4666 y(I)1660 4674 y Fg(p)1701 4638 y Fo(@)1754 4594 y Fl(\015)1752 4680 y Ff(e)1749 4690 y Fl(I)1780 4698 y Fg(p)1831 4614 y Fp(\026)1820 4638 y Fo(@)1868 4652 y Fl(z)1901 4664 y Fg(k)1943 4638 y Fo(F)2001 4652 y Fl(j)2038 4638 y Fk(j)2063 4662 y Fi(U)2114 4639 y Fj(2)2107 4685 y Fg(j)2183 4638 y Fo(;)h Fk(j)p Fo(@)2332 4605 y Fl(\014)2327 4661 y(!)2371 4669 y Fg(p)2412 4638 y Fo(@)2465 4594 y Fl(\015)2460 4680 y Ff(e)-46 b Fl(!)2504 4688 y Fg(p)2556 4614 y Fp(\026)2544 4638 y Fo(@)2592 4652 y Fl(z)2625 4664 y Fg(k)2667 4638 y Fo(F)2725 4652 y Fl(j)2762 4638 y Fk(j)2787 4662 y Fi(U)2838 4639 y Fj(2)2831 4685 y Fg(j)1096 4858 y Fk(\024)25 b Fo(C)7 b(L)1326 4872 y Fn(1)1364 4858 y Fo(L)1426 4872 y Fn(2)1481 4858 y Fp(exp)1635 4764 y Ff(\020)1684 4858 y Fk(\000)1765 4822 y Fn(3)p 1765 4837 36 4 v 1765 4889 a(4)1825 4858 y Fp(\()p Fo(\032)21 b Fk(\000)f Fp(1\)\(2)p Fo(L)2241 4872 y Fn(1)2282 4858 y Fo(u)2334 4872 y Fl(j)2370 4858 y Fp(\))2405 4812 y Fi(\000)2510 4785 y Fj(1)p 2471 4797 111 4 v 2471 4838 a Fg(\032)p Fd(\000)p Fj(1)2595 4764 y Ff(\021)2675 4858 y Fk(k)p Fo(P)13 b Fk(k)i Fo(;)-24 5050 y Fp(where)27 b Fo(C)32 b Fp(=)25 b Fo(C)7 b Fp(\()p Fo(\032;)15 b(n)p Fp(\))26 b Fo(>)e Fp(0)29 b(\(w)m(e)g(recall)e(that)i Fo(L)1572 5064 y Fn(2)1636 5050 y Fk(\025)c Fo(L)1794 5064 y Fn(1)1859 5050 y Fk(\025)g Fp(1\).)40 b(Using)28 b(\(3.1\),)i(w)m(e)f(obtain)e(the)h(same)h(estimates)-24 5163 y(for)127 5139 y(\026)116 5163 y Fo(@)164 5177 y Fl(I)195 5189 y Fg(k)237 5163 y Fo(F)295 5177 y Fl(j)332 5163 y Fp(,)400 5139 y(\026)389 5163 y Fo(@)437 5177 y Fl(!)481 5189 y Fg(k)523 5163 y Fo(F)581 5177 y Fl(j)618 5163 y Fp(,)j(and)f(for)g(their)f(deriv)-5 b(ativ)m(es)31 b(of)g(order)g(one)h(in)e Fk(U)2349 5130 y Fn(2)2340 5187 y Fl(j)2388 5163 y Fp(.)43 b(Indeed,)31 b(putting)f Fo(z)3144 5178 y Fl(k)3214 5163 y Fp(=)d Fo(I)3352 5178 y Fl(k)3415 5163 y Fp(+)21 b Fo(i)3549 5136 y Ff(e)3538 5163 y Fo(I)3578 5178 y Fl(k)3652 5163 y Fp(w)m(e)-24 5276 y(obtain)887 5452 y Fk(j)924 5428 y Fp(\026)912 5452 y Fo(@)960 5466 y Fl(z)993 5478 y Fg(k)1036 5452 y Fo(F)1094 5466 y Fl(j)1131 5452 y Fk(j)1156 5476 y Fi(U)1207 5453 y Fj(2)1200 5499 y Fg(j)1271 5452 y Fk(\024)k Fo(C)1439 5419 y Fi(0)1462 5452 y Fo(L)1524 5466 y Fn(2)1563 5452 y Fp(\()p Fo(L)1660 5466 y Fn(2)1700 5452 y Fo(v)1744 5466 y Fl(j)1780 5452 y Fp(\))1815 5407 y Fi(\000)1920 5376 y Fg(\032)p 1880 5392 V 1880 5433 a(\032)p Fd(\000)p Fj(1)2001 5407 y Fi(\000)2066 5380 y Fj(1)p 2065 5392 31 4 v 2065 5433 a(2)2125 5452 y Fp(exp)2279 5358 y Ff(\020)2329 5452 y Fk(\000)p Fp(\()p Fo(\032)20 b Fk(\000)g Fp(1\)\(2)p Fo(L)2815 5466 y Fn(2)2856 5452 y Fo(v)2900 5466 y Fl(j)2936 5452 y Fp(\))2971 5407 y Fi(\000)3077 5380 y Fj(1)p 3037 5392 111 4 v 3037 5433 a Fg(\032)p Fd(\000)p Fj(1)3161 5358 y Ff(\021)3241 5452 y Fk(k)p Fo(P)13 b Fk(k)329 5672 y(\024)24 b Fo(C)e Fp(exp)665 5578 y Ff(\020)715 5672 y Fk(\000)796 5636 y Fn(3)p 796 5651 36 4 v 796 5703 a(4)841 5672 y Fp(\()p Fo(\032)e Fk(\000)g Fp(1\)\(2)p Fo(L)1256 5686 y Fn(2)1297 5672 y Fo(v)1341 5686 y Fl(j)1378 5672 y Fp(\))1413 5627 y Fi(\000)1518 5599 y Fj(1)p 1478 5611 111 4 v 1478 5653 a Fg(\032)p Fd(\000)p Fj(1)1602 5578 y Ff(\021)1667 5672 y Fk(k)p Fo(P)13 b Fk(k)26 b(\024)f Fo(C)d Fp(exp)2191 5578 y Ff(\020)2240 5672 y Fk(\000)2321 5636 y Fn(3)p 2321 5651 36 4 v 2321 5703 a(4)2366 5672 y Fp(\()p Fo(\032)f Fk(\000)f Fp(1\)\(2)p Fo(L)2782 5686 y Fn(1)2823 5672 y Fo(u)2875 5686 y Fl(j)2911 5672 y Fp(\))2946 5627 y Fi(\000)3051 5599 y Fj(1)p 3011 5611 111 4 v 3011 5653 a Fg(\032)p Fd(\000)p Fj(1)3136 5578 y Ff(\021)3200 5672 y Fk(k)p Fo(P)13 b Fk(k)i Fo(;)1843 5927 y Fp(7)p eop %%Page: 8 8 8 7 bop -24 573 a Fp(where)38 b Fo(C)45 b Fp(=)38 b Fo(C)7 b Fp(\()p Fo(\032;)15 b(n;)g(&)7 b Fp(\))40 b Fo(>)e Fp(0,)j(since)d Fo(L)1383 587 y Fn(2)1461 573 y Fk(\024)g Fp(\()p Fo(L)1667 587 y Fn(2)1707 573 y Fo(v)1751 587 y Fl(j)1788 573 y Fp(\))1823 540 y Fi(\000)p Fn(1)p Fl(=&)2022 573 y Fp(b)m(y)g(\(3.1\).)67 b(W)-8 b(e)40 b(generalize)e(these)h (estimates)g(as)-24 686 y(follo)m(ws.)g(Set)30 b Fo(z)g Fp(=)25 b(\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)s Fp(\))26 b Fk(2)f Fm(C)1124 653 y Fl(n)1170 686 y Fo(=)p Fp(2)p Fo(\031)s Fm(Z)1379 653 y Fl(n)1446 686 y Fk(\002)18 b Fm(C)1611 653 y Fn(2)p Fl(n)1693 686 y Fp(,)30 b(denote)g(b)m(y)g Fo(x)2218 701 y Fl(k)2290 686 y Fp(and)f Fo(y)2511 701 y Fl(k)2553 686 y Fp(,)h(resp)s(ectiv)m(ely)-8 b(,)30 b(the)g(real)f(and)g(the)-24 799 y(imaginary)34 b(part)h(of)g Fo(z)760 814 y Fl(k)803 799 y Fp(,)h(1)e Fk(\024)f Fo(k)j Fk(\024)d Fp(3)p Fo(n)p Fp(,)j(and)f(put)1760 775 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y Fp(8)p eop %%Page: 9 9 9 8 bop -24 573 a Fp(By)43 b(con)m(tin)m(uit)m(y)g(last)f(form)m(ula)g (remains)g(true)g(for)h(Re)15 b Fo(z)1975 587 y Fn(1)2061 573 y Fp(=)45 b Fk(\006)p Fo(\031)s Fp(.)78 b(Set)43 b Fo(F)2633 587 y Fl(j;)p Fn(0)2721 573 y Fp(\()p Fo(z)t Fp(\))k(:=)e Fo(F)3083 587 y Fl(j)3120 573 y Fp(\()p Fo(z)t Fp(\))f(and)e Fk(U)3535 540 y Fn(2)3526 597 y Fl(j;)p Fn(1)3659 573 y Fp(:=)-24 686 y Fk(U)42 653 y Fn(2)33 710 y Fl(j)111 686 y Fk(\\)29 b(fj)p Fp(Im)15 b Fo(z)437 700 y Fn(1)477 686 y Fk(j)49 b(\024)g Fo(u)723 700 y Fl(j)759 686 y Fk(g)p Fp(.)84 b(W)-8 b(e)45 b(claim)f(that)h(for) f(an)m(y)h(m)m(ulti-index)d Fo(\013)49 b Fp(=)f(\(0)p Fo(;)15 b(\013)2793 700 y Fn(2)2834 686 y Fo(;)g(:)g(:)g(:)i(;)e(\013) 3094 700 y Fn(3)p Fl(n)3177 686 y Fp(\))49 b Fk(2)f Fm(N)3452 653 y Fn(3)p Fl(n)3579 686 y Fp(with)-24 799 y(0)26 b Fk(\024)f Fo(\013)201 813 y Fl(m)293 799 y Fk(\024)f Fp(1,)31 b(1)26 b Fk(\024)f Fo(m)g Fk(\024)g Fp(3)p Fo(n)p Fp(,)31 b(an)m(y)f(index)f Fo(k)s Fp(,)i(and)f Fo(\014)5 b(;)15 b(\015)31 b Fk(2)25 b Fm(N)30 b Fp(suc)m(h)g(that)h(0)26 b Fk(\024)f Fo(\014)g Fp(+)20 b Fo(\015)30 b Fk(\024)25 b Fp(1,)31 b(w)m(e)g(ha)m(v)m(e)350 941 y Ff(\014)350 991 y(\014)350 1041 y(\014)378 1038 y Fo(@)431 1001 y Fl(\014)426 1061 y(z)459 1073 y Fg(k)512 1014 y Fp(\026)501 1038 y Fo(@)554 1001 y Fl(\015)549 1061 y(z)582 1073 y Fg(k)636 1014 y Fp(\026)624 1038 y Fo(@)677 1001 y Fl(\013)672 1061 y(z)727 1038 y Fp(\()p Fo(F)820 1052 y Fl(j;)p Fn(1)929 1038 y Fk(\000)19 b Fo(F)1077 1052 y Fl(j;)p Fn(0)1165 1038 y Fp(\))1200 941 y Ff(\014)1200 991 y(\014)1200 1041 y(\014)1228 1094 y Fi(U)1279 1072 y Fj(2)1272 1118 y Fg(j;)p Fj(1)1411 1038 y Fk(\024)55 b Fo(C)22 b(L)1686 1001 y Fl(n)1686 1061 y Fn(1)1748 1038 y Fo(L)1810 994 y Fl(\014)s Fn(+)p Fl(\015)1810 1063 y Fn(2)1982 1038 y Fp(exp)2136 919 y Ff(\022)2197 1038 y Fk(\000)2278 977 y Fp(3)p 2278 1017 46 4 v 2278 1101 a(4)2333 1038 y(\()p Fo(\032)f Fk(\000)f Fp(1\)\(2)p Fo(L)2749 1052 y Fn(1)2790 1038 y Fo(u)2842 1052 y Fl(j)2878 1038 y Fp(\))2913 993 y Fi(\000)3018 966 y Fj(1)p 2979 978 111 4 v 2979 1019 a Fg(\032)p Fd(\000)p Fj(1)3103 919 y Ff(\023)3194 1038 y Fk(k)p Fo(P)13 b Fk(k)p Fo(;)189 b Fp(\(3.9\))-24 1295 y(where)38 b Fo(C)46 b Fp(=)39 b Fo(C)7 b Fp(\()p Fo(n;)15 b(\032;)g(&)7 b Fp(\))40 b Fo(>)f Fp(0.)66 b(F)-8 b(or)40 b Fo(k)j Fk(6)p Fp(=)c(1)g(it)f (directly)g(follo)m(ws)f(from)i(\(3.7\))h(di\013eren)m(tiating)e(under) f(the)-24 1408 y(in)m(tegral.)j(T)-8 b(o)31 b(pro)m(v)m(e)g(it)f(for)g Fo(k)f Fp(=)c(1,)31 b(w)m(e)g(use)f(the)g(same)h(argumen)m(t)g(for)939 1584 y(1)p 895 1625 132 4 v 895 1708 a(2)p Fo(\031)s(i)1052 1531 y Ff(Z)1098 1719 y Fl(D)1156 1728 y Fj(1)1237 1622 y Fp(\026)1225 1646 y Fo(@)1273 1660 y Fl(\021)1308 1669 y Fj(1)1348 1646 y Fo(F)1406 1660 y Fl(j)1443 1646 y Fp(\()p Fo(\021)1523 1660 y Fn(1)1563 1646 y Fo(;)15 b(z)1645 1660 y Fn(2)1685 1646 y Fo(;)g(:)g(:)g(:)h(;)f(z)1928 1660 y Fn(3)p Fl(n)2011 1646 y Fp(\))p Fo(K)2123 1660 y Fn(1)2163 1646 y Fp(\()p Fo(\021)2243 1660 y Fn(1)2283 1646 y Fo(;)g(z)2365 1660 y Fn(1)2405 1646 y Fp(\))g Fo(d\021)2547 1660 y Fn(1)2608 1646 y Fk(^)20 b Fo(d)7 b Fp(\026)-52 b Fo(\021)2781 1660 y Fn(1)2821 1646 y Fo(:)-24 1891 y Fp(On)29 b(the)i(other)g(hand,)1138 1993 y(1)p 1095 2033 V 1095 2117 a(2)p Fo(\031)s(i)1252 1939 y Ff(Z)1298 2128 y Fl(D)1356 2137 y Fj(1)1447 1969 y Fp(\026)1435 1993 y Fo(@)1483 2007 y Fl(\021)1518 2016 y Fj(1)1557 1993 y Fo(F)1615 2007 y Fl(j)1652 1993 y Fp(\()p Fo(\021)1732 2007 y Fn(1)1772 1993 y Fo(;)15 b(z)1854 2007 y Fn(2)1894 1993 y Fo(;)g(:)g(:)g(:)i(;)e(z)2138 2007 y Fn(3)p Fl(n)2221 1993 y Fp(\))p 1435 2033 822 4 v 1707 2117 a Fo(\021)1752 2131 y Fn(1)1812 2117 y Fk(\000)20 b Fo(z)1945 2131 y Fn(1)2281 2054 y Fo(d\021)2373 2068 y Fn(1)2434 2054 y Fk(^)f Fo(d)7 b Fp(\026)-52 b Fo(\021)2606 2068 y Fn(1)518 2337 y Fp(=)25 b Fk(\000)5 b Fp(\026)-50 b Fo(z)727 2351 y Fn(1)778 2313 y Fp(\026)766 2337 y Fo(@)814 2351 y Fl(z)847 2360 y Fj(1)886 2337 y Fo(F)944 2351 y Fl(j)981 2337 y Fp(\()p Fo(z)t Fp(\))21 b(+)1262 2275 y(1)p 1219 2316 132 4 v 1219 2399 a(2)p Fo(\031)s(i)1376 2222 y Ff(Z)1422 2410 y Fl(D)1480 2419 y Fj(1)1571 2251 y Fp(\026)1559 2275 y Fo(@)1607 2289 y Fl(\021)1642 2298 y Fj(1)1682 2275 y Fo(F)1740 2289 y Fl(j)1777 2275 y Fp(\()p Fo(\021)1857 2289 y Fn(1)1897 2275 y Fo(;)15 b(z)1979 2289 y Fn(2)2019 2275 y Fo(;)g(:)g(:)g(:)h(;)f (z)2262 2289 y Fn(3)p Fl(n)2345 2275 y Fp(\))21 b Fk(\000)2503 2251 y Fp(\026)2492 2275 y Fo(@)2540 2289 y Fl(z)2573 2298 y Fj(1)2611 2275 y Fo(F)2669 2289 y Fl(j)2706 2275 y Fp(\()p Fo(z)t Fp(\))p 1559 2316 1265 4 v 2053 2399 a Fo(\021)2098 2413 y Fn(1)2157 2399 y Fk(\000)f Fo(z)2290 2413 y Fn(1)2849 2337 y Fo(d\021)2941 2351 y Fn(1)3001 2337 y Fk(^)f Fo(d)7 b Fp(\026)-52 b Fo(\021)3173 2351 y Fn(1)-24 2550 y Fp(for)26 b Fo(z)153 2564 y Fn(1)218 2550 y Fk(2)f Fo(D)379 2564 y Fn(1)419 2550 y Fp(,)j(whic)m(h)d(follo)m (ws)h(from)g(\(3.8\))j(applied)24 b(to)k Fo(f)10 b Fp(\()p Fo(z)2002 2564 y Fn(1)2041 2550 y Fp(\))26 b(=)k(\026)-50 b Fo(z)2240 2564 y Fn(1)2279 2550 y Fp(.)40 b(Di\013eren)m(tiating)26 b(the)h(last)g(equalit)m(y)f(and)-24 2663 y(using)j(\(3.7\))j(w)m(e)f (get)g(the)g(estimate.)41 b(Moreo)m(v)m(er,)33 b(if)c Fk(j)p Fo(\013)p Fk(j)e(\025)e Fp(1,)31 b(then)f(\(3.7\))i(and)e (\(3.9\))i(imply)521 2810 y Ff(\014)521 2860 y(\014)521 2910 y(\014)549 2908 y Fo(@)602 2870 y Fl(\014)597 2930 y(z)630 2942 y Fg(k)684 2884 y Fp(\026)672 2908 y Fo(@)725 2870 y Fl(\015)720 2930 y(z)753 2942 y Fg(k)807 2884 y Fp(\026)796 2908 y Fo(@)849 2870 y Fl(\013)844 2930 y(z)898 2908 y Fo(F)956 2922 y Fl(j;)p Fn(1)1044 2810 y Ff(\014)1044 2860 y(\014)1044 2910 y(\014)1072 2964 y Fi(U)1123 2941 y Fj(2)1116 2987 y Fg(j;)p Fj(1)1254 2908 y Fk(\024)56 b Fo(C)21 b(L)1529 2870 y Fl(n)1529 2930 y Fn(1)1591 2908 y Fo(L)1653 2863 y Fl(\014)s Fn(+)p Fl(\015)1653 2932 y Fn(2)1811 2908 y Fp(exp)1965 2789 y Ff(\022)2026 2908 y Fk(\000)2107 2846 y Fp(3)p 2107 2887 46 4 v 2107 2970 a(4)2162 2908 y(\()p Fo(\032)f Fk(\000)g Fp(1\)\(2)p Fo(L)2577 2922 y Fn(1)2618 2908 y Fo(u)2670 2922 y Fl(j)2707 2908 y Fp(\))2742 2862 y Fi(\000)2847 2835 y Fj(1)p 2807 2847 111 4 v 2807 2888 a Fg(\032)p Fd(\000)p Fj(1)2931 2789 y Ff(\023)3023 2908 y Fk(k)p Fo(P)13 b Fk(k)p Fo(:)314 b Fp(\(3.10\))-24 3172 y(W)-8 b(e)39 b(de\014ne)e(b)m(y)h(recurrence)g Fo(F)1050 3186 y Fl(j;m)1165 3172 y Fp(\()p Fo(z)t Fp(\),)k(2)c Fk(\024)g Fo(m)g Fk(\024)g Fo(n)p Fp(,)h(and)f(w)m(e)g(pro)m(v)m(e)h (that)g(it)f(satis\014es)f(\(3.9\))j(in)d Fk(U)3516 3139 y Fn(2)3507 3197 y Fl(j;m)3659 3172 y Fp(:=)-24 3285 y Fk(U)42 3252 y Fn(2)33 3309 y Fl(j;m)p Fi(\000)p Fn(1)258 3285 y Fk(\\)20 b(fj)p Fp(Im)15 b Fo(z)575 3299 y Fl(m)642 3285 y Fk(j)26 b(\024)g Fo(u)842 3299 y Fl(j)878 3285 y Fk(g)31 b Fp(for)g Fo(\013)26 b Fp(=)g(\(0)p Fo(;)15 b(:)g(:)g(:)i(;)e Fp(0)p Fo(;)g(\013)1700 3299 y Fl(m)p Fn(+1)1858 3285 y Fo(;)g(:)g(:)g(:)i(;)e(\013)2118 3299 y Fn(3)p Fl(n)2201 3285 y Fp(\).)42 b(Moreo)m(v)m(er,)33 b Fo(F)2786 3299 y Fl(j;m)2901 3285 y Fp(\()p Fo(z)t Fp(\))e(satis\014es)g(\(3.10\))i(for)-24 3398 y Fk(j)p Fo(\013)p Fk(j)26 b(\025)f Fp(1.)117 3511 y(F)-8 b(or)31 b Fo(n)25 b(<)g(m)g Fk(\024)g Fp(3)p Fo(n)30 b Fp(w)m(e)h(de\014ne)368 3753 y Fo(F)426 3767 y Fl(j;m)541 3753 y Fp(\()p Fo(z)t Fp(\))26 b(=)833 3691 y(1)p 789 3732 132 4 v 789 3815 a(2)p Fo(\031)s(i)946 3638 y Ff(Z)993 3826 y Fl(@)t(D)1092 3834 y Fg(m)1195 3691 y Fo(F)1253 3705 y Fl(j;m)p Fi(\000)p Fn(1)1458 3691 y Fp(\()p Fo(z)1535 3705 y Fn(1)1575 3691 y Fo(;)15 b(:)g(:)g(:)h(;)f(z)1818 3705 y Fl(m)p Fi(\000)p Fn(1)1976 3691 y Fo(;)g(\021)2061 3705 y Fl(m)2128 3691 y Fo(;)g(z)2210 3705 y Fl(m)p Fn(+1)2367 3691 y Fo(;)g(:)g(:)g(:)i(;)e (z)2611 3705 y Fn(3)p Fl(n)2694 3691 y Fp(\))p 1195 3732 1535 4 v 1796 3815 a Fo(\021)1841 3829 y Fl(m)1928 3815 y Fk(\000)20 b Fo(z)2061 3829 y Fl(m)2754 3753 y Fo(d\021)2846 3767 y Fl(m)2944 3753 y Fo(;)106 b(z)30 b Fk(2)24 b(U)3298 3715 y Fn(2)3289 3775 y Fl(j)3338 3753 y Fo(;)-24 4010 y Fp(and)31 b(set)i Fk(U)364 3977 y Fn(2)355 4034 y Fl(j;m)498 4010 y Fp(:=)28 b Fk(U)688 3977 y Fn(2)679 4034 y Fl(j;m)p Fi(\000)p Fn(1)905 4010 y Fk(\\)21 b(fj)p Fp(Im)15 b Fo(z)1223 4024 y Fl(m)1290 4010 y Fk(j)29 b(\024)f Fo(p)1489 4024 y Fl(m)1555 4010 y Fk(g)p Fp(,)33 b(where)f Fo(p)1969 4024 y Fl(m)2063 4010 y Fp(=)c Fo(v)2206 4024 y Fl(j)2275 4010 y Fp(for)k Fo(n)20 b Fp(+)i(1)28 b Fk(\024)g Fo(m)g Fk(\024)g Fp(2)p Fo(n)k Fp(and)g Fo(p)3320 4024 y Fl(m)3414 4010 y Fp(=)c Fo(w)3578 4024 y Fl(j)3647 4010 y Fp(for)-24 4123 y(2)p Fo(n)23 b Fp(+)g(1)34 b Fk(\024)f Fo(m)g Fk(\024)h Fp(3)p Fo(n)p Fp(.)55 b(By)35 b(recurrence)g(with)f(resp)s(ect)h(to)h Fo(m)p Fp(,)h(w)m(e)e(obtain)g(\(3.9\))i(for)e Fo(F)3002 4137 y Fl(j;m)3152 4123 y Fp(in)f Fk(U)3329 4090 y Fn(2)3320 4147 y Fl(j;m)3470 4123 y Fp(for)h(an)m(y)-24 4236 y Fo(n)27 b(<)g(m)g Fk(\024)g Fp(3)p Fo(n)p Fp(,)32 b Fo(\013)c Fp(=)f(\(0)p Fo(;)15 b(:)g(:)g(:)i(;)e Fp(0)p Fo(;)g(\013)1127 4250 y Fl(m)p Fn(+1)1285 4236 y Fo(;)g(:)g(:)g(:)i(;)e(\013)1545 4250 y Fn(3)p Fl(n)1628 4236 y Fp(\),)32 b(0)c Fk(\024)f Fo(\013)1949 4250 y Fl(q)2014 4236 y Fk(\024)g Fp(1)32 b(\()p Fo(\013)d Fp(=)e(0)32 b(if)e Fo(m)d Fp(=)g(3)p Fo(n)p Fp(\),)33 b(for)e(an)m(y)h(index)e Fo(k)35 b Fp(and)-24 4349 y Fo(\014)5 b(;)15 b(\015)40 b Fp(suc)m(h)33 b(that)i(0)d Fk(\024)f Fo(\014)d Fp(+)23 b Fo(\015)36 b Fk(\024)c Fp(1.)52 b(Moreo)m(v)m(er,)37 b Fo(F)1715 4363 y Fl(j;m)1830 4349 y Fp(\()p Fo(z)t Fp(\))e(satis\014es)f(\(3.10\))i(for)e Fk(j)p Fo(\013)p Fk(j)f(\025)e Fp(1)k(and)e Fo(m)e(<)h Fp(3)p Fo(n)p Fp(.)52 b(F)-8 b(or)-24 4462 y(2)p Fo(n)25 b(<)g(m)g Fk(\024)g Fp(3)p Fo(n)j Fp(the)h(constan)m(t)g Fo(C)35 b Fp(should)26 b(b)s(e)h(replaced)h(b)m(y)g Fo(C)2105 4429 y Fi(0)2128 4462 y Fp(\(1)16 b(+)2330 4439 y(\026)2311 4462 y Fo(R)p Fp(\))2415 4429 y Fl(m)p Fi(\000)p Fn(2)p Fl(n)2615 4462 y Fp(,)29 b(where)f Fo(C)3002 4429 y Fi(0)3050 4462 y Fp(=)d Fo(C)3218 4429 y Fi(0)3241 4462 y Fp(\()p Fo(\032;)15 b(n;)g(&)7 b Fp(\).)41 b(The)-24 4575 y(factor)31 b(1)21 b(+)413 4552 y(\026)393 4575 y Fo(R)31 b Fp(comes)g(from)f(the)g (measure)h(of)f Fo(D)1665 4589 y Fl(m)1732 4575 y Fp(,)h(2)p Fo(n)25 b(<)g(m)g Fk(\024)g Fp(3)p Fo(n)p Fp(.)117 4688 y(Set)31 b Fo(P)332 4702 y Fl(j)394 4688 y Fp(=)25 b Fo(F)548 4702 y Fl(j;)p Fn(3)p Fl(n)678 4688 y Fp(.)41 b(Then)29 b(for)i(an)m(y)f(index)f Fo(k)34 b Fp(and)c Fo(`)25 b Fp(=)g(0)p Fo(;)15 b Fp(1,)31 b(w)m(e)g(obtain)603 4936 y Fk(j)p Fo(@)681 4899 y Fl(`)676 4959 y(x)716 4971 y Fg(k)758 4936 y Fp(\()p Fo(P)851 4950 y Fl(j)908 4936 y Fk(\000)20 b Fo(F)1057 4950 y Fl(j)1094 4936 y Fp(\))p Fk(j)1154 4950 y Fi(U)1198 4960 y Fg(j)1291 4936 y Fk(\024)55 b Fo(C)22 b(L)1566 4899 y Fl(n)1566 4959 y Fn(1)1628 4936 y Fo(L)1690 4899 y Fl(`)1690 4959 y Fn(2)1745 4936 y Fp(exp)1899 4817 y Ff(\022)1960 4936 y Fk(\000)2041 4875 y Fp(3)p 2041 4915 46 4 v 2041 4998 a(4)2096 4936 y(\()p Fo(\032)e Fk(\000)g Fp(1\)\(2)p Fo(L)2511 4950 y Fn(1)2552 4936 y Fo(u)2604 4950 y Fl(j)2641 4936 y Fp(\))2676 4891 y Fi(\000)2781 4863 y Fj(1)p 2741 4875 111 4 v 2741 4917 a Fg(\032)p Fd(\000)p Fj(1)2865 4817 y Ff(\023)2942 4936 y Fk(k)p Fo(P)13 b Fk(k)p Fo(:)-24 5176 y Fp(In)30 b(particular,)732 5304 y Fk(j)p Fo(P)815 5318 y Fl(j)t Fn(+1)962 5304 y Fk(\000)20 b Fo(P)1111 5318 y Fl(j)1148 5304 y Fk(j)1173 5318 y Fi(U)1217 5328 y Fg(j)s Fj(+1)1356 5304 y Fk(\024)25 b(j)p Fo(P)1535 5318 y Fl(j)t Fn(+1)1683 5304 y Fk(\000)20 b Fo(F)1832 5318 y Fl(j)t Fn(+1)1959 5304 y Fk(j)g Fp(+)g Fk(j)p Fo(P)2178 5318 y Fl(j)2236 5304 y Fk(\000)f Fo(F)2384 5318 y Fl(j)2421 5304 y Fk(j)i Fp(+)f Fk(j)p Fo(F)2641 5318 y Fl(j)t Fn(+1)2788 5304 y Fk(\000)g Fo(F)2937 5318 y Fl(j)2974 5304 y Fk(j)1327 5524 y(\024)25 b Fo(C)c(L)1571 5491 y Fl(n)1571 5547 y Fn(1)1649 5524 y Fp(exp)1803 5430 y Ff(\020)1852 5524 y Fk(\000)1933 5488 y Fn(3)p 1933 5503 36 4 v 1933 5555 a(4)1978 5524 y Fp(\()p Fo(\032)g Fk(\000)f Fp(1\)\(2)p Fo(L)2394 5538 y Fn(1)2435 5524 y Fo(u)2487 5538 y Fl(j)2523 5524 y Fp(\))2558 5479 y Fi(\000)2663 5451 y Fj(1)p 2623 5463 111 4 v 2623 5505 a Fg(\032)p Fd(\000)p Fj(1)2748 5430 y Ff(\021)2812 5524 y Fk(k)p Fo(P)13 b Fk(k)p Fo(:)1843 5927 y Fp(9)p eop %%Page: 10 10 10 9 bop -24 573 a Fp(Moreo)m(v)m(er,)33 b(for)d(an)m(y)g(index)f Fo(k)34 b Fp(and)c Fo(`)25 b Fp(=)g(0)p Fo(;)15 b Fp(1,)545 801 y Fk(j)p Fo(@)623 763 y Fl(`)618 823 y(x)658 835 y Fg(k)700 801 y Fp(\()p Fo(P)793 815 y Fl(j)831 801 y Fp(\()p Fo(x)p Fp(\))20 b Fk(\000)g Fo(P)13 b Fp(\()p Fo(x)p Fp(\)\))p Fk(j)57 b(\024)e Fo(C)22 b(L)1649 763 y Fl(n)1649 823 y Fn(1)1711 801 y Fo(L)1773 763 y Fl(`)1773 823 y Fn(2)1828 801 y Fp(exp)1982 682 y Ff(\022)2043 801 y Fk(\000)2124 739 y Fp(3)p 2124 780 46 4 v 2124 863 a(4)2179 801 y(\()p Fo(\032)e Fk(\000)g Fp(1\)\(2)p Fo(L)2594 815 y Fn(1)2635 801 y Fo(u)2687 815 y Fl(j)2724 801 y Fp(\))2759 755 y Fi(\000)2864 728 y Fj(1)p 2824 740 111 4 v 2824 781 a Fg(\032)p Fd(\000)p Fj(1)2948 682 y Ff(\023)3025 801 y Fk(k)p Fo(P)13 b Fk(k)-24 1033 y Fp(in)29 b Fk(U)139 1047 y Fl(j)196 1033 y Fk(\\)19 b(f)p Fp(Im)d Fo(z)29 b Fp(=)c(0)p Fk(g)p Fp(,)32 b(since)d Fo(F)1040 1047 y Fl(j)1077 1033 y Fp(\()p Fo(x)p Fp(\))d(=)f Fo(P)13 b Fp(\()p Fo(x)p Fp(\))31 b(for)f Fo(x)g Fp(real.)41 b(Finally)-8 b(,)367 1270 y Fk(j)p Fo(P)450 1284 y Fn(0)490 1270 y Fk(j)515 1284 y Fi(U)559 1293 y Fj(0)623 1270 y Fk(\024)25 b(j)p Fo(F)802 1284 y Fn(0)843 1270 y Fk(j)20 b Fp(+)g Fk(j)p Fo(P)1062 1284 y Fn(0)1122 1270 y Fk(\000)g Fo(F)1271 1284 y Fn(0)1311 1270 y Fk(j)25 b(\024)56 b Fo(C)1574 1151 y Ff(\022)1635 1270 y Fp(1)21 b(+)f Fo(L)1854 1233 y Fl(n)1854 1293 y Fn(1)1916 1270 y Fp(exp)2070 1151 y Ff(\022)2131 1270 y Fk(\000)2212 1209 y Fp(3)p 2212 1249 46 4 v 2212 1332 a(4)2267 1270 y(\()p Fo(\032)h Fk(\000)f Fp(1\)\(2)p Fo(L)2683 1284 y Fn(1)2724 1270 y Fo(u)2776 1284 y Fn(0)2815 1270 y Fp(\))2850 1225 y Fi(\000)2955 1197 y Fj(1)p 2915 1209 111 4 v 2915 1251 a Fg(\032)p Fd(\000)p Fj(1)3040 1151 y Ff(\023\023)3177 1270 y Fk(k)p Fo(P)13 b Fk(k)p Fo(:)-24 1498 y Fp(This)28 b(completes)j(the)g(pro)s(of)e(of)i(the)f(prop)s(osition.)1925 b Fa(2)-24 1858 y Fc(3.2)112 b Fb(The)34 b(KAM)i(step)-24 2029 y Fp(In)m(tro)s(duce)30 b(the)g(complex)g(domains)726 2221 y Fo(D)801 2235 y Fl(s;r)917 2221 y Fp(=)25 b Fk(f)p Fo(\022)j Fk(2)d Fm(C)1291 2184 y Fl(n)1337 2221 y Fo(=)p Fp(2)p Fo(\031)s Fm(Z)1546 2184 y Fl(n)1619 2221 y Fp(:)56 b Fk(j)p Fp(Im)15 b Fo(\022)s Fk(j)25 b Fo(<)g(s)p Fk(g)20 b(\002)g(f)p Fo(I)33 b Fk(2)25 b Fm(C)2520 2184 y Fl(n)2591 2221 y Fp(:)56 b Fk(j)p Fo(I)7 b Fk(j)26 b Fo(<)f(r)s Fk(g)p Fo(;)1237 2414 y(O)1306 2429 y Fl(h)1377 2414 y Fp(=)g Fk(f)p Fo(!)j Fk(2)d Fm(C)1765 2376 y Fl(n)1837 2414 y Fp(:)56 b Fk(j)p Fo(!)23 b Fk(\000)d Fp(\012)2180 2428 y Fl(\024)2224 2414 y Fk(j)26 b Fo(<)f(h)p Fk(g)p Fo(:)-24 2575 y Fp(The)g(sup-norm)g(of)h(functions)e(in)h Fk(V)32 b Fp(:=)25 b Fo(D)1431 2589 y Fl(s;r)1533 2575 y Fk(\002)11 b Fo(O)1684 2590 y Fl(h)1755 2575 y Fp(will)23 b(b)s(e)i(denoted)h(b)m(y)g Fk(j)11 b(\001)g(j)2599 2590 y Fl(s;r)n(;h)2747 2575 y Fp(.)39 b(Fix)25 b(0)h Fo(<)f(\035)j(<)d Fp(1)p Fo(=)p Fp(6)j(and)d(set)-18 2685 y Ff(e)-24 2688 y Fo(\035)k Fp(=)24 b(1)p Fo(=)p Fp(2)31 b Fk(\000)14 b Fp(3)p Fo(\035)30 b Fp(\(w)m(e)e(shall)e(c)m(ho)s(ose)i(later)f Fo(\035)h Fp(=)d(1)p Fo(=)p Fp(54)k(and)1956 2685 y Ff(e)1950 2688 y Fo(\035)f Fp(=)d(4)p Fo(=)p Fp(9)k(\).)40 b(Fix)26 b(0)g Fo(<)f(s;)15 b(r)28 b(<)d Fp(1)p Fo(;)43 b Fp(0)26 b Fo(<)f(\021)j(<)d Fp(1)p Fo(=)p Fp(8)p Fo(;)44 b Fp(0)26 b Fo(<)-24 2801 y(\033)47 b(<)d(s=)p Fp(5)p Fo(;)58 b(K)51 b Fk(\025)43 b Fp(1)p Fo(:)g Fp(Consider)d(the)i(real)f(v)-5 b(alued)41 b(Hamiltonian)f Fo(H)7 b Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(!)s Fp(\))44 b(=)g Fo(N)10 b Fp(\()p Fo(I)d Fp(;)15 b Fo(!)s Fp(\))29 b(+)e Fo(H)3387 2815 y Fn(1)3426 2801 y Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(!)s Fp(\),)-24 2913 y Fo(N)10 b Fp(\()p Fo(I)d Fp(;)15 b Fo(!)s Fp(\))31 b(=)e Fo(e)p Fp(\()p Fo(!)s Fp(\))23 b(+)f Fk(h)p Fo(!)s(;)15 b(I)7 b Fk(i)p Fp(.)50 b(W)-8 b(e)34 b(shall)d(denote)j(b)m(y)f(`Const.')49 b(a)34 b(p)s(ositiv)m(e)e(constan)m(t)i(dep)s(ending)d(only)h(on)h Fo(n)-24 3026 y Fp(and)d Fo(\034)40 b Fp(and)30 b(b)m(y)g(`const.')42 b(if)29 b(it)h(is)f Fk(\024)c Fp(1.)41 b(W)-8 b(e)32 b(recall)e(from)g(P\177)-45 b(osc)m(hel)30 b([9)q(])g(the)h(follo)m (wing)-24 3227 y Fm(Prop)s(osition)36 b(3.2)46 b Fe(L)-5 b(et)33 b Fo(H)40 b Fe(b)-5 b(e)32 b(r)-5 b(e)g(al)34 b(analytic)g(in)e Fk(V)7 b Fe(.)42 b(Supp)-5 b(ose)35 b(that)e Fk(j)p Fo(H)28 b Fk(\000)20 b Fo(N)10 b Fk(j)2725 3242 y Fl(s;r)n(;h)2897 3227 y Fk(\024)25 b Fo(")33 b Fe(with)42 3405 y Fp(\(a\))46 b Fo(")26 b Fk(\024)f Fp(const)p Fo(:)16 b(\024\021)s(r)s(\033)814 3372 y Fl(\034)8 b Fn(+1)948 3405 y Fo(;)36 3588 y Fp(\(b\))46 b Fo(")26 b Fk(\024)f Fp(const)p Fo(:)16 b(\035)s(hr)m(;)47 3791 y Fp(\(c\))46 b Fo(h)26 b Fk(\024)492 3729 y Fo(\024)p 387 3770 263 4 v 387 3853 a Fp(2)p Fo(K)516 3827 y Fl(\034)8 b Fn(+1)-24 3989 y Fe(Then)33 b(ther)-5 b(e)33 b(exists)g(a)g(r)-5 b(e)g(al)34 b(analytic)g(tr)-5 b(ansformation)564 4182 y Fk(F)35 b Fp(=)25 b(\(\010)p Fo(;)15 b(\036)p Fp(\))33 b Fo(;)108 b Fp(\010)25 b(:)h Fo(D)1373 4196 y Fl(s)p Fi(\000)p Fn(5)p Fl(\033)n(;\021)r(r)1650 4182 y Fk(\002)20 b Fo(O)1810 4197 y Fl(h)1880 4182 y Fk(\000)-15 b(!)25 b Fo(D)2127 4196 y Fl(s;r)2238 4182 y Fk(\002)20 b Fo(O)2398 4197 y Fl(h)2458 4182 y Fo(;)48 b(\036)25 b Fp(:)h Fo(O)2729 4210 y Ff(e)-45 b Fl(\035)s(h)2841 4182 y Fk(\000)-15 b(!)25 b Fo(O)3082 4197 y Fl(h)3142 4182 y Fo(;)-24 4374 y Fe(of)37 b(the)h(form)g Fp(\010\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(!)s Fp(\))34 b(=)f(\()p Fo(U)10 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))p Fo(;)g(V)21 b Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(!)s Fp(\)\))p Fe(,)39 b(with)f Fo(V)57 b Fe(a\016ne)38 b(line)-5 b(ar)38 b(with)g(r)-5 b(esp)g(e)g(ct)39 b(to)f Fo(I)7 b Fe(,)38 b(wher)-5 b(e)38 b(the)-24 4487 y(tr)-5 b(ansformation)33 b Fp(\010\()p Fk(\001)p Fp(;)15 b Fo(!)s Fp(\))29 b Fe(is)h(c)-5 b(anonic)g(al)31 b(for)e(e)-5 b(ach)30 b Fo(!)s Fe(,)g(and)g(such)f (that)h Fo(H)20 b Fk(\016)13 b(F)34 b Fp(=)25 b Fo(N)2814 4501 y Fn(+)2886 4487 y Fp(+)13 b Fo(P)3028 4501 y Fn(+)3115 4487 y Fe(with)30 b Fo(N)3382 4501 y Fn(+)3441 4487 y Fp(\()p Fo(I)7 b Fp(;)15 b Fo(!)s Fp(\))27 b(=)-24 4600 y Fo(e)18 4614 y Fn(+)77 4600 y Fp(\()p Fo(!)s Fp(\))21 b(+)f Fk(h)p Fo(!)s(;)15 b(I)7 b Fk(i)p Fe(,)33 b(and)781 4858 y Fk(j)p Fo(P)864 4872 y Fn(+)923 4858 y Fk(j)948 4886 y Fl(s)p Fi(\000)p Fn(5)p Fl(\033)n(;\021)r(r)n(;)o Ff(e)-45 b Fl(\035)t(h)1328 4858 y Fk(\024)25 b Fp(Const)p Fo(:)1712 4714 y Ff( )1889 4796 y Fo(")1931 4763 y Fn(2)p 1788 4837 285 4 v 1788 4920 a Fo(\024r)s(\033)1939 4894 y Fl(\034)8 b Fn(+1)2102 4858 y Fp(+)20 b(\()p Fo(\021)2276 4820 y Fn(2)2337 4858 y Fp(+)g Fo(K)2512 4820 y Fl(n)2558 4858 y Fo(e)2600 4820 y Fi(\000)p Fl(K)5 b(\033)2767 4858 y Fp(\))p Fo(")2844 4714 y Ff(!)2925 4858 y Fo(:)573 b Fp(\(3.11\))-24 5109 y Fe(Mor)-5 b(e)g(over,)847 5236 y Fk(j)p Fo(W)13 b Fp(\(\010)20 b Fk(\000)g Fo(id)p Fp(\))p Fk(j)p Fo(;)49 b Fk(j)p Fo(W)13 b Fp(\()p Fo(D)s Fp(\010)20 b Fk(\000)g Fp(Id)o(\))p Fo(W)2026 5198 y Fi(\000)p Fn(1)2120 5236 y Fk(j)26 b(\024)e Fp(Const)p Fo(:)2671 5174 y(")p 2550 5215 V 2550 5298 a(\024r)s(\033)2701 5272 y Fl(\034)8 b Fn(+1)2859 5236 y Fo(;)1192 5455 y Fk(j)p Fo(\036)21 b Fk(\000)f Fo(id)p Fk(j)p Fo(;)48 b(\035)s(h)p Fk(j)p Fo(D)s(\036)21 b Fk(\000)f Fp(Id)p Fk(j)25 b(\024)g Fp(Const)p Fo(:)2446 5394 y(")p 2445 5434 44 4 v 2445 5518 a(r)2514 5455 y(;)-24 5650 y Fe(uniformly)34 b(on)f Fo(D)596 5664 y Fl(s)p Fi(\000)p Fn(5)p Fl(\033)n(;\021)r(r)873 5650 y Fk(\002)20 b Fo(O)1033 5665 y Fl(h)1111 5650 y Fe(and)33 b Fo(O)1355 5678 y Ff(e)-45 b Fl(\035)s(h)1442 5650 y Fe(,)32 b(r)-5 b(esp)g(e)g(ctively,)34 b(wher)-5 b(e)34 b Fo(W)k Fp(=)25 b(diag)2680 5581 y Ff(\000)2718 5650 y Fo(\033)2773 5617 y Fi(\000)p Fn(1)2868 5650 y Fp(Id)14 b Fo(;)h(r)3050 5617 y Fi(\000)p Fn(1)3144 5650 y Fp(Id)3243 5581 y Ff(\001)3281 5650 y Fe(.)1820 5927 y Fp(10)p eop %%Page: 11 11 11 10 bop -24 573 a Fm(Remark)34 b(3.3)46 b Fe(Set)p 707 501 99 4 v 32 w Fo(W)37 b Fp(=)25 b(diag)1124 504 y Ff(\000)1163 573 y Fo(\033)1218 540 y Fi(\000)p Fn(1)1312 573 y Fp(Id)15 b Fo(;)g(r)1495 540 y Fi(\000)p Fn(1)1589 573 y Fp(Id)f Fo(;)h(h)1779 540 y Fi(\000)p Fn(1)1874 573 y Fp(Id)1973 504 y Ff(\001)2042 573 y Fe(and)32 b(supp)-5 b(ose)33 b(that)g Fo(h)25 b Fk(\024)g Fo(\024\033)3013 540 y Fl(\034)8 b Fn(+1)3147 573 y Fe(.)41 b(Sinc)-5 b(e)32 b Fp(1)17 b Fk(\000)3613 570 y Ff(e)3607 573 y Fo(\035)29 b Fk(\025)-24 686 y Fp(1)p Fo(=)p Fp(3)p Fe(,)34 b(using)e(the)h(Cauchy)g(estimate)h(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)f Fo(!)s Fe(,)g(we)f(obtain)693 922 y Fk(j)p 718 850 V Fo(W)12 b Fp(\()p Fk(F)30 b(\000)20 b Fo(id)p Fp(\))p Fk(j)p Fo(;)49 b Fk(j)p 1274 850 V Fo(W)13 b Fp(\()p Fo(D)s Fk(F)30 b(\000)20 b Fp(Id)o(\))p 1790 850 V Fo(W)1889 869 y Fi(\000)p Fn(1)1983 922 y Fk(j)26 b(\024)2140 860 y Fo(C)7 b(")p 2140 901 114 4 v 2149 984 a(r)s(h)2279 922 y(;)48 b(C)31 b Fp(=)25 b Fo(C)7 b Fp(\()p Fo(n;)15 b(\034)10 b Fp(\))26 b Fo(>)f Fp(0)15 b Fo(;)-24 1137 y Fe(uniformly)34 b(on)f Fo(D)596 1151 y Fl(s)p Fi(\000)p Fn(5)p Fl(\033)n(;\021)r(r)873 1137 y Fk(\002)20 b Fo(O)1032 1165 y Ff(e)-45 b Fl(\035)s(h)1119 1137 y Fe(,)32 b(wher)-5 b(e)34 b Fo(D)s Fk(F)42 b Fe(stands)34 b(for)f(the)g(Jac)-5 b(obian)34 b(of)e Fk(F)9 b Fe(.)-24 1339 y Fp(The)43 b(pro)s(of)g(of)g(Prop)s(osition)f(3.2)i(is)f(giv)m(en)g(in)g([9].)81 b(The)42 b(only)h(di\013erence)g(b)s(et)m(w)m(een)h(the)g(statemen)m(t) h(of)-24 1452 y(Prop)s(osition)23 b(1.3)k(and)d(that)i(of)g(the)f(KAM)g (step)h(in)e([9])i(app)s(ears)e(in)g(the)i(transformation)e(of)i(the)f (frequencies)-24 1565 y(\()17 1562 y Ff(e)11 1565 y Fo(\035)k Fp(=)c(1)p Fo(=)p Fp(4)h(in)e([9]\).)40 b(T)-8 b(o)25 b(pro)m(v)m(e)h(the)f(prop)s(osition)e(with)1841 1562 y Ff(e)1835 1565 y Fo(\035)29 b Fp(as)c(ab)s(o)m(v)m(e)h(w)m(e)f(use)g (the)g(follo)m(wing)f(analog)h(of)g(Lemma)-24 1678 y(A.3)31 b([9)q(].)-24 1857 y Fm(Lemma)i(3.4)46 b Fe(Supp)-5 b(ose)40 b Fo(f)45 b Fp(:)36 b Fo(O)1098 1872 y Fl(h)1179 1857 y Fk(!)g Fm(C)1382 1824 y Fl(n)1467 1857 y Fe(is)i(r)-5 b(e)g(al)40 b(analytic)g(with)f(b)-5 b(ounde)g(d)40 b Fk(j)p Fo(f)10 b Fk(j)2759 1872 y Fl(h)2804 1857 y Fe(.)59 b(L)-5 b(et)39 b Fp(0)d Fo(<)g(\035)j(<)d Fp(1)p Fo(=)p Fp(6)j Fe(and)-18 1967 y Ff(e)-24 1970 y Fo(\035)29 b Fp(=)24 b(1)p Fo(=)p Fp(2)37 b Fk(\000)20 b Fp(3)p Fo(\035)s Fe(.)43 b(If)32 b Fk(j)p Fo(f)e Fk(\000)20 b Fo(id)p Fk(j)969 1985 y Fl(h)1040 1970 y Fk(\024)25 b Fo(\035)s(h)p Fe(,)33 b(then)g Fo(f)42 b Fe(has)33 b(a)g(r)-5 b(e)g(al)34 b(analytic)g(inverse)f Fo(f)h Fp(:)26 b Fo(O)2859 1999 y Ff(e)-45 b Fl(\035)s(h)2971 1970 y Fk(!)25 b Fo(O)3156 1999 y Fn(2\()p Fl(\035)r Fn(+)o Ff(e)-45 b Fl(\035)t Fn(\))p Fl(h)3460 1970 y Fe(and)1069 2173 y Fk(j)p Fo(\036)20 b Fk(\000)g Fo(id)p Fk(j)1361 2201 y Ff(e)-45 b Fl(\035)t(h)1481 2173 y Fo(;)48 b Fp(3)p Fo(\035)s(h)p Fk(j)p Fo(D)s(\036)22 b Fk(\000)d Fo(id)p Fk(j)2074 2201 y Ff(e)-45 b Fl(\035)t(h)2187 2173 y Fk(\024)25 b(j)p Fo(f)30 b Fk(\000)20 b Fo(id)p Fk(j)2577 2188 y Fl(h)2638 2173 y Fo(:)-24 2367 y Fp(A)36 b(sk)m(etc)m(h)i(of)e(pro)s(of)f(of)h(the)h(Lemma)f(is)f(giv)m(en)h(in) f(the)h(App)s(endix.)56 b(W)-8 b(e)37 b(are)f(going)h(to)f(prepare)g (the)g(next)-24 2480 y(iteration.)k(W)-8 b(e)32 b(c)m(ho)s(ose)f(a)g (`w)m(eigh)m(ted)g(error')f(0)c Fo(<)f(E)30 b(<)25 b Fp(1,)31 b(\014x)f(0)c Fo(<)2316 2477 y Ff(b)2310 2480 y Fo(")f Fk(\024)g Fp(1,)31 b(and)f(set)1054 2674 y Fo(\021)f Fp(=)c Fo(E)1296 2636 y Fn(1)p Fl(=)p Fn(2)1406 2674 y Fo(;)46 b(")25 b Fp(=)1646 2671 y Ff(b)1640 2674 y Fo(")q(\024E)5 b(r)s(\033)1906 2636 y Fl(\034)j Fn(+1)2039 2674 y Fo(;)46 b Fp(0)26 b Fo(<)f(E)30 b(<)25 b Fp(1)p Fo(=)p Fp(64)p Fo(:)-24 2868 y Fp(W)-8 b(e)31 b(de\014ne)f Fo(K)37 b Fp(and)30 b Fo(h)h Fp(b)m(y)1323 2995 y Fo(K)1407 2957 y Fl(n)1454 2995 y Fo(e)1496 2957 y Fi(\000)p Fl(K)5 b(\033)1688 2995 y Fp(=)25 b Fo(E)5 b(;)46 b(h)25 b Fp(=)2215 2933 y Fo(\024)p 2110 2974 263 4 v 2110 3057 a Fp(2)p Fo(K)2239 3031 y Fl(\034)8 b Fn(+1)2383 2995 y Fo(:)-24 3187 y Fp(Setting)32 b Fo(x)d Fp(=)g Fo(K)7 b(\033)35 b Fp(w)m(e)e(get)h(the)f(equation)f Fo(x)1517 3154 y Fl(n)1564 3187 y Fo(e)1606 3154 y Fi(\000)p Fl(x)1734 3187 y Fp(=)c Fo(E)5 b(\033)1960 3154 y Fl(n)2008 3187 y Fp(,)33 b(whic)m(h)f(has)g(an)g(unique)f(solution)g(with)g(resp)s (ect)-24 3300 y(to)g Fo(x)25 b Fk(2)g Fp([1)p Fo(;)15 b Fp(+)p Fk(1)p Fp(\),)32 b(since)d(0)d Fo(<)f(E)31 b(<)25 b Fp(1)p Fo(=)p Fp(64)i Fo(<)e Fp(1)p Fo(=e)p Fp(.)42 b(Then)29 b Fo(K)j Fp(=)25 b Fo(x\033)2248 3267 y Fi(\000)p Fn(1)2368 3300 y Fo(>)g Fp(1.)41 b(W)-8 b(e)31 b(set)1229 3494 y Fo(r)1270 3508 y Fn(+)1354 3494 y Fp(=)25 b Fo(\021)s(r)m(;)47 b(s)1651 3508 y Fn(+)1735 3494 y Fp(=)25 b Fo(s)20 b Fk(\000)f Fp(5)p Fo(\033)n(;)47 b(\033)2203 3508 y Fn(+)2287 3494 y Fp(=)25 b Fo(\016)s(\033)n(;)-24 3689 y Fp(where)33 b(0)f Fo(<)f(\016)k(<)c Fp(1.)52 b(Later)35 b(w)m(e)f(shall)e(c)m(ho)s (ose)j Fo(\016)g Fp(=)c Fo(\016)s Fp(\()p Fo(\032)p Fp(\))36 b(as)e(a)g(function)f(of)h Fo(\032)g Fp(only)-8 b(.)51 b(No)m(w)35 b(the)f(KAM)g(step)-24 3801 y(giv)m(es)d(the)f(estimate)350 4006 y Fk(j)p Fo(P)433 4020 y Fn(+)493 4006 y Fk(j)518 4035 y Fl(s)551 4044 y Fj(+)601 4035 y Fl(;r)653 4044 y Fj(+)704 4035 y Fl(;)o Ff(e)-45 b Fl(\035)r(h)834 4006 y Fo(<)25 b Fp(Const)p Fo(:)1209 4003 y Ff(b)1203 4006 y Fo("\024r)s(\033)1396 3969 y Fl(\034)8 b Fn(+1)1545 3912 y Ff(\020)1595 4006 y Fo(E)1667 3969 y Fn(2)1727 4006 y Fp(+)20 b(\()p Fo(\021)1901 3969 y Fn(2)1961 4006 y Fp(+)g Fo(K)2136 3969 y Fl(n)2183 4006 y Fo(e)2225 3969 y Fi(\000)p Fl(K)5 b(\033)2391 4006 y Fp(\))p Fo(E)2498 3912 y Ff(\021)2573 4006 y Fp(=)25 b(Const)p Fo(:)2948 4003 y Ff(b)2943 4006 y Fo("\024r)s(\033)3136 3969 y Fl(\034)8 b Fn(+1)3269 4006 y Fo(E)3341 3969 y Fn(2)1211 4235 y Fp(=)25 b(Const)p Fo(:)15 b(\016)s Fp(\()p Fo(\032)p Fp(\))1740 4197 y Fi(\000)p Fl(\034)8 b Fi(\000)p Fn(1)1935 4232 y Ff(b)1930 4235 y Fo("\024r)2065 4249 y Fn(+)2124 4235 y Fo(\033)2179 4197 y Fl(\034)g Fn(+1)2176 4257 y(+)2313 4235 y Fo(E)2385 4197 y Fn(3)p Fl(=)p Fn(2)2495 4235 y Fo(:)-24 4397 y Fp(Hence)31 b(there)g(is)e(a)i(constan)m(t)g Fo(c)1051 4411 y Fn(1)1117 4397 y Fo(>)24 b Fp(1)31 b(dep)s(ending)d (only)i(on)g Fo(n)p Fp(,)g Fo(\032)g Fp(and)g Fo(\034)40 b Fp(suc)m(h)30 b(that)1173 4624 y Fk(j)p Fo(P)1256 4638 y Fn(+)1316 4624 y Fk(j)1341 4653 y Fl(s)1374 4662 y Fj(+)1424 4653 y Fl(;r)1476 4662 y Fj(+)1526 4653 y Fl(;)o Ff(e)-45 b Fl(\035)s(h)1657 4624 y Fk(\024)1763 4562 y Fp(1)p 1763 4603 46 4 v 1763 4686 a(2)1824 4621 y Ff(b)1818 4624 y Fo("c)1899 4576 y Fn(1)p Fl(=)p Fn(2)1899 4649 y(1)2010 4624 y Fo(\024r)2103 4638 y Fn(+)2162 4624 y Fo(\033)2217 4586 y Fl(\034)8 b Fn(+1)2214 4646 y(+)2351 4624 y Fo(E)2423 4586 y Fn(3)p Fl(=)p Fn(2)2533 4624 y Fo(:)-24 4874 y Fp(W)-8 b(e)33 b(\014x)e(the)h(w)m(eigh)m(ted)g (error)f(for)h(the)g(iteration)f(b)m(y)h Fo(E)1890 4888 y Fn(+)1976 4874 y Fp(=)27 b Fo(c)2113 4826 y Fn(1)p Fl(=)p Fn(2)2113 4898 y(1)2224 4874 y Fo(E)2296 4841 y Fn(3)p Fl(=)p Fn(2)2406 4874 y Fp(,)32 b(set)h Fo(")2649 4888 y Fn(+)2736 4874 y Fp(=)2840 4871 y Ff(b)2834 4874 y Fo("\024r)2969 4888 y Fn(+)3029 4874 y Fo(\033)3084 4835 y Fl(\034)8 b Fn(+1)3081 4896 y(+)3217 4874 y Fo(E)3284 4888 y Fn(+)3343 4874 y Fp(,)33 b(and)e(then)-24 4986 y(de\014ne)h Fo(\021)286 5000 y Fn(+)345 4986 y Fo(;)47 b(x)469 5000 y Fn(+)528 4986 y Fo(;)h(K)678 5000 y Fn(+)737 4986 y Fp(,)34 b(and)d Fo(h)1026 5000 y Fn(+)1118 4986 y Fp(as)i(ab)s(o)m(v)m(e.)48 b(Notice)33 b(that,)h Fo(c)2082 5000 y Fn(1)2122 4986 y Fo(E)2189 5000 y Fn(+)2277 4986 y Fp(=)28 b(\()p Fo(c)2450 5000 y Fn(1)2490 4986 y Fo(E)5 b Fp(\))2597 4954 y Fn(3)p Fl(=)p Fn(2)2708 4986 y Fp(.)47 b(W)-8 b(e)33 b(require)e(also)i Fo(c)3470 5000 y Fn(1)3510 4986 y Fo(E)h(<)28 b Fp(1)-24 5099 y(whic)m(h)h(leads)h(to)h(an)f(exp)s (onen)m(tially)f(con)m(v)m(erging)i(sc)m(heme.)41 b(Supp)s(ose)29 b(that)1646 5294 y Fo(h)1698 5308 y Fn(+)1813 5294 y Fk(\024)1945 5291 y Ff(e)1940 5294 y Fo(\035)18 b(h:)1439 b Fp(\(3.12\))-24 5488 y(Then)29 b(w)m(e)i(obtain)1444 5616 y Fk(j)p Fo(P)1527 5630 y Fn(+)1587 5616 y Fk(j)1612 5631 y Fl(s)1645 5640 y Fj(+)1696 5631 y Fl(;r)1748 5640 y Fj(+)1798 5631 y Fl(;h)1859 5640 y Fj(+)1969 5616 y Fk(\024)2105 5555 y Fp(1)p 2105 5595 V 2105 5678 a(2)2160 5616 y Fo(")2202 5630 y Fn(+)2262 5616 y Fo(:)1236 b Fp(\(3.13\))1820 5927 y(11)p eop %%Page: 12 12 12 11 bop -24 573 a Fc(3.3)112 b Fb(Setting)35 b(the)g(p)-5 b(ar)g(ameters)34 b(and)g(iter)-5 b(ation)-24 744 y Fp(As)27 b(in)f([9)q(])i(w)m(e)g(are)g(going)f(to)h(iterate)g(the)g(KAM)g(step)f (in\014nitely)e(man)m(y)i(times)g(c)m(ho)s(osing)g(appropriately)f(the) -24 857 y(parameters)j(0)d Fo(<)f(s;)15 b(r)m(;)g(\033)n(;)g(h;)g(\021) 31 b(<)25 b Fp(1)k(and)g(so)g(on.)40 b(Our)28 b(goal)h(is)f(to)i(get)g (a)g(con)m(v)m(ergen)m(t)h(sc)m(heme)f(in)d(the)j(Gevrey)-24 970 y(spaces)43 b Fk(G)326 937 y Fl(\032;\032)p Fn(\()p Fl(\034)8 b Fn(+1\)+1)696 970 y Fp(.)77 b(W)-8 b(e)44 b(are)e(going)h(to)g(de\014ne)f(suitable)f(strictly)g(decreasing)h (sequences)h(of)f(p)s(ositiv)m(e)-24 1083 y(n)m(um)m(b)s(ers)29 b Fk(f)p Fo(s)431 1097 y Fl(j)467 1083 y Fk(g)512 1050 y Fi(1)512 1107 y Fl(j)t Fn(=0)640 1083 y Fp(,)h Fk(f)p Fo(r)781 1097 y Fl(j)818 1083 y Fk(g)863 1050 y Fi(1)863 1107 y Fl(j)t Fn(=0)1021 1083 y Fp(and)f Fk(f)p Fo(h)1294 1097 y Fl(j)1332 1083 y Fk(g)1377 1050 y Fi(1)1377 1107 y Fl(j)t Fn(=0)1504 1083 y Fp(,)h(tending)g(to)h(0,)g(and)f(denote)1078 1287 y Fo(D)1153 1301 y Fl(j)1215 1287 y Fp(=)25 b Fo(D)1386 1301 y Fl(s)1419 1311 y Fg(j)1452 1301 y Fl(;r)1504 1311 y Fg(j)1555 1287 y Fo(;)46 b(O)1695 1301 y Fl(j)1757 1287 y Fp(=)25 b Fo(O)1922 1302 y Fl(h)1963 1312 y Fg(j)2015 1287 y Fo(;)45 b Fk(V)2141 1301 y Fl(j)2203 1287 y Fp(=)25 b Fo(D)2374 1301 y Fl(j)2431 1287 y Fk(\002)20 b Fo(O)2591 1301 y Fl(j)2628 1287 y Fo(:)-24 1492 y Fp(Fix)30 b Fo(\016)f Fk(2)c Fp(\(0)p Fo(;)15 b Fp(1\))32 b(\()p Fo(\016)j Fp(will)27 b(dep)s(end)i(only)g(on)i Fo(\032)p Fp(\))f(and)g(set)1082 1696 y Fo(s)1125 1710 y Fl(j)1186 1696 y Fp(=)25 b Fo(s)1325 1710 y Fn(0)1364 1696 y Fo(\016)1407 1658 y Fl(j)1460 1696 y Fo(;)45 b(\033)1582 1710 y Fl(j)1644 1696 y Fp(=)25 b Fo(\033)1792 1710 y Fn(0)1832 1696 y Fo(\016)1875 1658 y Fl(j)1927 1696 y Fo(;)46 b(s)2041 1710 y Fn(0)2080 1696 y Fp(\(1)21 b Fk(\000)f Fo(\016)s Fp(\))26 b(=)f(5)p Fo(\033)2569 1710 y Fn(0)2624 1696 y Fo(:)-24 1900 y Fp(Ob)m(viously)-8 b(,)29 b Fo(s)464 1914 y Fl(j)t Fn(+1)615 1900 y Fp(=)c Fo(s)754 1914 y Fl(j)811 1900 y Fk(\000)20 b Fp(5)p Fo(\033)999 1914 y Fl(j)1066 1900 y Fp(and)30 b Fo(\033)1295 1914 y Fl(j)1356 1900 y Fp(=)25 b(5)1497 1867 y Fi(\000)p Fn(1)1592 1900 y Fp(\(1)c Fk(\000)f Fo(\016)s Fp(\))p Fo(s)1905 1914 y Fl(j)1973 1900 y Fp(for)30 b Fo(j)h Fk(\025)25 b Fp(0.)41 b(W)-8 b(e)31 b(set)996 2104 y Fo(u)1048 2118 y Fl(j)1110 2104 y Fp(=)25 b(4)p Fo(s)1294 2118 y Fl(j)1356 2104 y Fp(=)g(4)p Fo(s)1540 2118 y Fn(0)1579 2104 y Fo(\016)1622 2067 y Fl(j)1690 2104 y Fo(;)45 b(v)1804 2118 y Fl(j)1866 2104 y Fp(=)25 b(4)p Fo(r)2048 2118 y Fn(0)2088 2104 y Fo(\016)2131 2067 y Fl(j)2199 2104 y Fo(;)46 b(w)2335 2118 y Fl(j)2396 2104 y Fp(=)25 b(4)p Fo(h)2589 2118 y Fn(0)2630 2104 y Fo(\016)2673 2067 y Fl(j)2710 2104 y Fo(;)-24 2309 y Fp(and)41 b(denote)h(b)m(y)g Fk(U)663 2323 y Fl(j)741 2309 y Fp(the)g(corresp)s(onding)e(complex)h(sets)h(de\014ned)f(in)f (Sect.)75 b(3.1.)h(W)-8 b(e)43 b(assume)f(for)f(the)-24 2422 y(momen)m(t)31 b(that)g(these)g(sequences)f(v)m(erify)g(\(3.1\).) 43 b(Then)29 b(applying)f(Prop)s(osition)h(3.1)i(w)m(e)g(obtain)507 2647 y Fk(j)p Fo(P)590 2661 y Fn(0)630 2647 y Fk(j)655 2661 y Fi(U)699 2670 y Fj(0)821 2647 y Fk(\024)83 b Fo(C)1040 2661 y Fn(0)1094 2647 y Fo(L)1156 2614 y Fl(n)1156 2670 y Fn(1)1218 2647 y Fk(k)p Fo(P)13 b Fk(k)216 2821 y(j)p Fo(P)299 2835 y Fl(j)356 2821 y Fk(\000)20 b Fo(P)505 2835 y Fl(j)t Fi(\000)p Fn(1)632 2821 y Fk(j)657 2835 y Fi(U)701 2845 y Fg(j)821 2821 y Fk(\024)113 b Fo(C)1070 2835 y Fn(0)1140 2821 y Fo(L)1202 2784 y Fl(n)1202 2844 y Fn(1)1264 2821 y Fk(k)p Fo(P)13 b Fk(k)i Fo(e)1482 2784 y Fi(\000)1549 2774 y Ff(e)1537 2784 y Fl(B)1590 2793 y Fj(0)1638 2784 y Fl(s)1671 2794 y Fg(j)1703 2742 y Fd(\000)1801 2723 y Fj(1)p 1761 2732 111 4 v 1761 2767 a Fg(\032)p Fd(\000)p Fj(1)1914 2821 y Fp(=)56 b Fo(C)2106 2835 y Fn(0)2175 2821 y Fo(L)2237 2784 y Fl(n)2237 2844 y Fn(1)2299 2821 y Fk(k)p Fo(P)13 b Fk(k)i Fo(e)2517 2784 y Fi(\000)p Fl(B)2625 2793 y Fj(0)2673 2784 y Fl(\033)2713 2794 y Fg(j)2746 2742 y Fd(\000)2844 2723 y Fj(1)p 2804 2732 V 2804 2767 a Fg(\032)p Fd(\000)p Fj(1)2963 2821 y Fo(;)46 b(j)30 b Fk(\025)25 b Fp(1)15 b Fo(;)3523 2727 y Fp(\(3.14\))-24 3091 y(where)30 b(the)g(p)s(ositiv)m(e)g(constan)m (ts)1156 3065 y Ff(e)1135 3091 y Fo(B)1204 3105 y Fn(0)1243 3091 y Fo(L)1355 2997 y Fj(1)p 1315 3009 V 1315 3050 a Fg(\032)p Fd(\000)p Fj(1)1305 3116 y Fn(1)1469 3091 y Fp(and)g Fo(B)1715 3105 y Fn(0)1755 3091 y Fo(L)1866 2997 y Fj(1)p 1827 3009 V 1827 3050 a Fg(\032)p Fd(\000)p Fj(1)1817 3116 y Fn(1)1981 3091 y Fp(dep)s(end)f(only)h(on)g Fo(\032)g Fp(and)g Fo(\016)s Fp(.)42 b(Giv)m(en)30 b Fo(N)41 b Fp(and)29 b Fo(a)d(>)f Fp(0)-24 3204 y(w)m(e)31 b(set)1328 3314 y Ff(b)1323 3317 y Fo(")25 b Fp(:=)g Fk(k)p Fo(P)13 b Fk(k)p Fo(L)1734 3279 y Fl(N)7 b Fi(\000)p Fn(2)1734 3342 y(1)1893 3317 y Fp(\()p Fo(a\024r)s Fp(\))2107 3280 y Fi(\000)p Fn(1)2227 3317 y Fk(\024)25 b Fp(1)15 b Fo(;)1115 b Fp(\(3.15\))-24 3484 y(and)30 b(w)m(e)g(in)m(tro)s(duce) 1192 3594 y Ff(e)1186 3597 y Fo(")1228 3611 y Fl(j)1290 3597 y Fp(=)1392 3594 y Ff(b)1386 3597 y Fo("\024r)1521 3611 y Fn(0)1561 3597 y Fo(\033)1616 3559 y Fl(\034)8 b Fn(+1)1613 3622 y(0)1765 3597 y Fp(exp)1919 3503 y Ff(\020)1968 3597 y Fk(\000)p Fo(B)2108 3611 y Fn(0)2163 3597 y Fo(\033)2215 3611 y Fl(j)2251 3551 y Fi(\000)2356 3524 y Fj(1)p 2316 3536 V 2316 3577 a Fg(\032)p Fd(\000)p Fj(1)2440 3503 y Ff(\021)2520 3597 y Fo(:)-24 3779 y Fp(W)-8 b(e)46 b(will)41 b(c)m(ho)s(ose)46 b(later)e Fo(N)59 b Fp(=)48 b Fo(N)10 b Fp(\()p Fo(n;)15 b(\034)5 b(;)15 b(\032)p Fp(\))46 b(and)d Fo(a)49 b(>)g Fp(0)44 b(indep)s(enden)m(t)f(of)h Fo(\024)p Fp(,)49 b Fo(L)2869 3793 y Fn(1)2908 3779 y Fp(,)f Fo(L)3043 3793 y Fn(2)3082 3779 y Fp(,)h(and)43 b Fo(r)s Fp(,)48 b(so)d(that)-24 3892 y Fk(j)p Fo(P)59 3906 y Fn(0)99 3892 y Fk(j)124 3906 y Fi(U)168 3915 y Fj(0)232 3892 y Fk(\024)334 3889 y Ff(e)328 3892 y Fo(")370 3906 y Fn(0)440 3892 y Fp(and)30 b Fk(j)p Fo(P)700 3906 y Fl(j)757 3892 y Fk(\000)20 b Fo(P)906 3906 y Fl(j)t Fi(\000)p Fn(1)1033 3892 y Fk(j)1058 3906 y Fi(U)1102 3916 y Fg(j)1165 3892 y Fk(\024)1266 3889 y Ff(e)1261 3892 y Fo(")1303 3906 y Fl(j)1370 3892 y Fp(for)30 b Fo(j)h Fk(\025)25 b Fp(1.)41 b(No)m(w)31 b(w)m(e)g(put)695 4149 y Fo(E)762 4163 y Fl(j)824 4149 y Fp(:=)26 b Fo(c)985 4111 y Fi(\000)p Fn(1)985 4174 y(1)1094 4149 y Fp(exp)1248 4055 y Ff(\020)1298 4149 y Fk(\000)p Fo(B)5 b(\033)1495 4163 y Fl(j)1531 4104 y Fi(\000)1635 4077 y Fj(1)p 1596 4089 V 1596 4130 a Fg(\032)p Fd(\000)p Fj(1)1720 4055 y Ff(\021)1876 4149 y Fp(with)89 b Fo(B)30 b Fp(:=)2373 4088 y Fo(B)2442 4102 y Fn(0)p 2373 4128 109 4 v 2405 4212 a Fp(2)2507 4055 y Ff(\020)2556 4149 y Fo(\016)2599 4104 y Fi(\000)2705 4077 y Fj(1)p 2665 4089 111 4 v 2665 4130 a Fg(\032)p Fd(\000)p Fj(1)2809 4149 y Fk(\000)20 b Fp(1)2945 4055 y Ff(\021)3011 4149 y Fo(;)-24 4380 y Fp(where)30 b Fo(c)278 4394 y Fn(1)343 4380 y Fo(>)25 b Fp(1)31 b(is)e(the)h(constan)m(t)i(in) d(the)i(KAM)f(step.)41 b(W)-8 b(e)32 b(\014nd)c Fo(\016)i Fk(2)24 b Fp(\(0)p Fo(;)15 b Fp(1\))33 b(from)d(the)g(equalities)1298 4600 y Fk(8)p Fo(j)h Fk(2)24 b Fm(N)31 b Fo(;)106 b(E)1813 4614 y Fl(j)t Fn(+1)1965 4600 y Fp(=)25 b Fo(c)2100 4552 y Fn(1)p Fl(=)p Fn(2)2100 4624 y(1)2210 4600 y Fo(E)2282 4552 y Fn(3)p Fl(=)p Fn(2)2277 4625 y Fl(j)2408 4600 y Fo(:)-24 4877 y Fp(This)33 b(is)g(equiv)-5 b(alen)m(t)34 b(to)h Fo(\033)886 4891 y Fl(j)t Fn(+1)1045 4877 y Fp(=)d(\(2)p Fo(=)p Fp(3\))1353 4844 y Fl(\032)p Fi(\000)p Fn(1)1486 4877 y Fo(\033)1538 4891 y Fl(j)1574 4877 y Fp(,)k(and)e(w)m(e)h(get)h Fo(\016)g Fp(=)2289 4758 y Ff(\022)2360 4815 y Fp(2)p 2360 4856 46 4 v 2360 4939 a(3)2415 4758 y Ff(\023)2476 4781 y Fl(\032)p Fi(\000)p Fn(1)2607 4877 y Fp(,)g(whic)m(h)d(implies)f Fo(B)k Fp(=)c Fo(B)3522 4891 y Fn(0)3562 4877 y Fo(=)p Fp(4)h(=)-24 5067 y Fo(A)44 5081 y Fn(0)83 5067 y Fo(L)145 5081 y Fn(1)185 5021 y Fi(\000)290 4994 y Fj(1)p 250 5006 111 4 v 250 5047 a Fg(\032)p Fd(\000)p Fj(1)374 5067 y Fp(,)g(where)e Fo(A)764 5081 y Fn(0)832 5067 y Fp(=)d Fo(A)999 5081 y Fn(0)1038 5067 y Fp(\()p Fo(\032)p Fp(\))h Fo(>)f Fp(0)33 b(dep)s(ends)d(only)h(on)h Fo(\032)p Fp(.)46 b(No)m(w)33 b(w)m(e)f(set)h Fo(\021)2694 5081 y Fl(j)2759 5067 y Fp(=)27 b Fo(E)2929 5019 y Fn(1)p Fl(=)p Fn(2)2924 5093 y Fl(j)3040 5067 y Fp(,)32 b Fo(r)3138 5081 y Fl(j)t Fn(+1)3293 5067 y Fp(=)c Fo(\021)3437 5081 y Fl(j)3474 5067 y Fo(r)3515 5081 y Fl(j)3551 5067 y Fp(,)33 b(and)-24 5180 y(put)1521 5293 y Fo(")1563 5307 y Fl(j)1625 5293 y Fp(=)1727 5290 y Ff(b)1721 5293 y Fo("\024r)1856 5307 y Fl(j)1893 5293 y Fo(\033)1948 5254 y Fl(\034)8 b Fn(+1)1945 5319 y Fl(j)2082 5293 y Fo(E)2149 5307 y Fl(j)2185 5293 y Fo(:)-24 5460 y Fp(The)32 b(c)m(hoice)h(of)g (the)g(`w)m(eigh)m(ted)g(error')f Fo(E)1425 5474 y Fl(j)1495 5460 y Fp(ab)s(o)m(v)m(e)h(is)f(motiv)-5 b(ated)33 b(b)m(y)f(the)h (inequalit)m(y)2994 5457 y Ff(e)2988 5460 y Fo(")3030 5474 y Fl(j)3096 5460 y Fk(\024)28 b Fo(")3237 5474 y Fl(j)t Fn(+1)3364 5460 y Fo(=)p Fp(2,)35 b Fo(j)f Fk(\025)29 b Fp(0,)-24 5572 y(whic)m(h)g(will)f(b)s(e)h(pro)m(v)m(ed)i(in)e (\(3.20\).)43 b(This)29 b(inequalit)m(y)f(will)g(allo)m(w)i(us)g(to)h (put)e Fo(P)2719 5586 y Fl(j)2776 5572 y Fk(\000)20 b Fo(P)2925 5586 y Fl(j)t Fi(\000)p Fn(1)3083 5572 y Fp(in)29 b(the)h(error)g(term)1820 5927 y(12)p eop %%Page: 13 13 13 12 bop -24 573 a Fp(of)33 b(the)h(iteration)f(of)g(order)g Fo(j)5 b Fp(.)50 b(Next)35 b(w)m(e)e(determine)g Fo(K)1945 587 y Fl(j)2015 573 y Fp(from)g(the)g(equation)g Fo(K)2850 540 y Fl(n)2843 597 y(j)2897 573 y Fo(e)2939 540 y Fi(\000)p Fl(K)3054 550 y Fg(j)3086 540 y Fl(\033)3126 550 y Fg(j)3193 573 y Fp(=)d Fo(E)3361 587 y Fl(j)3398 573 y Fp(.)50 b(Setting)-24 686 y Fo(x)28 700 y Fl(j)90 686 y Fp(=)25 b Fo(K)263 700 y Fl(j)299 686 y Fo(\033)351 700 y Fl(j)418 686 y Fp(w)m(e)31 b(obtain)1007 825 y Fo(x)1059 788 y Fl(n)1059 848 y(j)1106 825 y Fo(e)1148 788 y Fi(\000)p Fl(x)1243 798 y Fg(j)1305 825 y Fp(=)25 b Fo(E)1468 839 y Fl(j)1504 825 y Fo(\033)1559 788 y Fl(n)1556 848 y(j)1632 825 y Fp(=)g Fo(c)1767 787 y Fi(\000)p Fn(1)1767 850 y(1)1876 825 y Fo(\033)1931 788 y Fl(n)1928 848 y(j)2009 825 y Fp(exp)2163 731 y Ff(\020)2213 825 y Fk(\000)p Fo(B)5 b(\033)2410 839 y Fl(j)2445 780 y Fi(\000)2550 753 y Fj(1)p 2510 765 111 4 v 2510 806 a Fg(\032)p Fd(\000)p Fj(1)2634 731 y Ff(\021)2699 825 y Fo(:)-24 994 y Fp(Consider)28 b(the)j(equation)1061 1146 y Fo(x)1113 1160 y Fl(j)1170 1146 y Fk(\000)19 b Fo(n)c Fp(ln)f Fo(x)1473 1160 y Fl(j)1535 1146 y Fp(=)25 b Fo(B)5 b(\033)1760 1079 y Fi(\000)1864 1052 y Fj(1)p 1824 1064 V 1824 1105 a Fg(\032)p Fd(\000)p Fj(1)1757 1172 y Fl(j)1969 1146 y Fk(\000)20 b Fo(n)15 b Fp(ln)n(\()p Fo(\033)2292 1160 y Fl(j)2329 1146 y Fp(\))21 b(+)f(ln)13 b Fo(c)2605 1160 y Fn(1)2645 1146 y Fo(:)853 b Fp(\(3.16\))-24 1304 y(W)-8 b(e)31 b(set)767 1417 y Fo(\033)819 1431 y Fn(0)899 1417 y Fp(=)40 b Fo(\033)19 b(L)1143 1379 y Fi(\000)p Fn(1)1143 1442 y(1)1252 1417 y Fp(\(ln)o(\()p Fo(L)1460 1431 y Fn(1)1520 1417 y Fp(+)h Fo(e)p Fp(\)\))1724 1375 y Fi(\000)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))1994 1417 y Fo(;)107 b Fp(0)25 b Fo(<)g(\033)k Fk(\024)2473 1414 y Ff(e)2469 1417 y Fo(\033)s Fp(\()p Fo(n;)15 b(\032)p Fp(\))26 b Fk(\034)f Fp(1)15 b Fo(:)560 b Fp(\(3.17\))-24 1575 y(Ob)m(viously)-8 b(,)29 b Fo(\033)473 1589 y Fn(0)512 1575 y Fo(L)574 1589 y Fn(1)639 1575 y Fo(<)c(\033)j Fk(\024)916 1572 y Ff(e)911 1575 y Fo(\033)s Fp(\()p Fo(n;)15 b(\032)p Fp(\))26 b Fk(\034)f Fp(1,)31 b(and)f(for)g(an)m(y)h Fo(j)g Fk(2)25 b Fm(N)30 b Fp(w)m(e)h(obtain)448 1806 y Fo(B)5 b(\033)577 1739 y Fi(\000)681 1712 y Fj(1)p 641 1724 V 641 1765 a Fg(\032)p Fd(\000)p Fj(1)574 1832 y Fl(j)786 1806 y Fk(\000)20 b Fo(n)15 b Fp(ln)n(\()p Fo(\033)1109 1820 y Fl(j)1146 1806 y Fp(\))21 b(+)f(ln)14 b Fo(c)1423 1820 y Fn(1)1488 1806 y Fk(\025)25 b Fo(B)5 b(\033)1713 1739 y Fi(\000)1817 1712 y Fj(1)p 1777 1724 V 1777 1765 a Fg(\032)p Fd(\000)p Fj(1)1710 1831 y Fn(0)1927 1806 y Fp(=)25 b Fo(A)2091 1820 y Fn(0)2130 1806 y Fp(\()p Fo(L)2227 1820 y Fn(1)2267 1806 y Fo(\033)2319 1820 y Fn(0)2358 1806 y Fp(\))2393 1761 y Fi(\000)2498 1734 y Fj(1)p 2459 1746 V 2459 1787 a Fg(\032)p Fd(\000)p Fj(1)2608 1806 y Fo(>)g(A)2772 1820 y Fn(0)2812 1806 y Fo(\033)2867 1761 y Fi(\000)2972 1734 y Fj(1)p 2932 1746 V 2932 1787 a Fg(\032)p Fd(\000)p Fj(1)3081 1806 y Fk(\035)h Fp(1)15 b Fo(:)-24 1993 y Fp(Hence,)33 b(c)m(ho)s(osing)f (0)d Fo(<)e(\033)k Fk(\024)1003 1990 y Ff(e)998 1993 y Fo(\033)s Fp(\()p Fo(n;)15 b(\032)p Fp(\))29 b Fk(\034)f Fp(1,)33 b(w)m(e)f(obtain)g(for)f(eac)m(h)j Fo(j)f Fk(2)28 b Fm(N)j Fp(an)h(unique)e(solution)h Fo(x)3378 2007 y Fl(j)3442 1993 y Fp(=)d Fo(x)3593 2007 y Fl(j)3629 1993 y Fp(\()p Fo(\033)s Fp(\))-24 2106 y(of)i(\(3.16\))j(suc)m(h)d(that)967 2263 y Fo(x)1019 2277 y Fl(j)1080 2263 y Fk(\025)25 b Fo(x)1228 2277 y Fl(j)1285 2263 y Fk(\000)20 b Fo(n)15 b Fp(ln)f Fo(x)1589 2277 y Fl(j)1650 2263 y Fk(\025)25 b Fo(B)5 b(\033)1875 2196 y Fi(\000)1980 2169 y Fj(1)p 1940 2181 V 1940 2222 a Fg(\032)p Fd(\000)p Fj(1)1872 2289 y Fl(j)2089 2263 y Fk(\025)25 b Fo(A)2253 2277 y Fn(0)2293 2263 y Fo(\033)2348 2218 y Fi(\000)2453 2190 y Fj(1)p 2413 2202 V 2413 2244 a Fg(\032)p Fd(\000)p Fj(1)2563 2263 y Fk(\035)g Fp(1)15 b Fo(:)-24 2421 y Fp(Then)29 b Fo(x)265 2435 y Fl(j)322 2421 y Fk(\000)20 b Fo(n)15 b Fp(ln)e Fo(x)625 2435 y Fl(j)687 2421 y Fp(=)25 b Fo(x)835 2435 y Fl(j)872 2421 y Fp(\(1)c(+)f Fo(o)p Fp(\(1\)\))31 b(as)g Fo(\033)e Fk(&)c Fp(0.)41 b(On)29 b(the)i(other)f(hand,)g(using)f(again)h(\(3.17\))j(w)m(e)e(get)95 2656 y Fo(x)147 2670 y Fl(j)203 2656 y Fk(\000)20 b Fo(n)15 b Fp(ln)f Fo(x)507 2670 y Fl(j)626 2656 y Fk(\024)83 b Fo(B)5 b(\033)909 2589 y Fi(\000)1014 2562 y Fj(1)p 974 2574 V 974 2615 a Fg(\032)p Fd(\000)p Fj(1)906 2682 y Fl(j)1113 2562 y Ff(h)1152 2656 y Fp(1)21 b Fk(\000)f Fo(nA)1432 2618 y Fi(\000)p Fn(1)1432 2681 y(0)1526 2656 y Fp(\()p Fo(L)1623 2670 y Fn(1)1663 2656 y Fo(\033)1715 2670 y Fl(j)1751 2656 y Fp(\))1836 2583 y Fj(1)p 1796 2595 V 1796 2636 a Fg(\032)p Fd(\000)p Fj(1)1936 2656 y Fp(ln)o(\()p Fo(L)2109 2670 y Fn(1)2148 2656 y Fo(\033)2200 2670 y Fl(j)2237 2656 y Fp(\))g(+)g Fo(nA)2506 2618 y Fi(\000)p Fn(1)2506 2681 y(0)2600 2656 y Fp(\()p Fo(L)2697 2670 y Fn(1)2737 2656 y Fo(\033)2789 2670 y Fn(0)2828 2656 y Fp(\))2914 2583 y Fj(1)p 2874 2595 V 2874 2636 a Fg(\032)p Fd(\000)p Fj(1)2998 2656 y Fp(\(ln)14 b Fo(L)3186 2670 y Fn(1)3246 2656 y Fp(+)20 b(ln)o(\()p Fo(c)3487 2670 y Fn(1)3527 2656 y Fp(\)\))3597 2562 y Ff(i)626 2897 y Fp(=)83 b Fo(B)5 b(\033)909 2830 y Fi(\000)1014 2803 y Fj(1)p 974 2815 V 974 2856 a Fg(\032)p Fd(\000)p Fj(1)906 2923 y Fl(j)1098 2897 y Fp(\(1)21 b(+)f Fo(o)p Fp(\(1\)\))15 b Fo(;)-24 3086 y Fp(uniformly)27 b(with)i(resp)s(ect)i (to)g Fo(j)g Fk(2)25 b Fm(N)p Fp(.)40 b(Hence,)1012 3312 y Fo(B)5 b(\033)1141 3245 y Fi(\000)1245 3218 y Fj(1)p 1205 3230 V 1205 3271 a Fg(\032)p Fd(\000)p Fj(1)1138 3338 y Fl(j)1355 3312 y Fk(\024)25 b Fo(x)1503 3326 y Fl(j)1565 3312 y Fk(\024)g Fo(B)5 b(\033)1790 3245 y Fi(\000)1894 3218 y Fj(1)p 1854 3230 V 1854 3271 a Fg(\032)p Fd(\000)p Fj(1)1787 3338 y Fl(j)1979 3312 y Fp(\(1)21 b(+)f Fo(o)p Fp(\(1\)\))15 b Fo(;)47 b(\033)28 b Fk(&)e Fp(0)p Fo(;)804 b Fp(\(3.18\))-24 3514 y(uniformly)27 b(with)i(resp)s(ect)i(to)g Fo(j)g Fk(2)25 b Fm(N)p Fp(.)40 b(W)-8 b(e)32 b(set)f Fo(h)1672 3528 y Fl(j)1734 3514 y Fp(=)25 b Fo(\024)15 b Fp(2)1942 3477 y Fi(\000)p Fn(1)2037 3514 y Fo(K)2121 3476 y Fi(\000)p Fl(\034)8 b Fi(\000)p Fn(1)2114 3540 y Fl(j)2339 3514 y Fp(and)30 b(\014x)g Fo(\035)e Fp(=)d(1)p Fo(=)p Fp(54.)117 3627 y(W)-8 b(e)36 b(are)f(going)g(to)g(c)m(hec)m(k)h(the)f(h)m(yp)s(othesis)e(\(a\))j (and)e(\(b\))h(in)e(Prop)s(osition)g(3.2)j(for)e(an)m(y)h Fo(j)j Fk(\025)33 b Fp(0)i(\()g(\(c\))g(is)-24 3740 y(ful\014lled)c(b)m (y)k(de\014nition\).)53 b(T)-8 b(o)35 b(pro)m(v)m(e)h(\()p Fo(a)p Fp(\))f(w)m(e)h(use)f(\(3.15\))i(and)d(that)i Fo(\021)2466 3707 y Fn(2)2463 3764 y Fl(j)2538 3740 y Fp(=)d Fo(E)2709 3754 y Fl(j)2778 3740 y Fp(=)g Fo(o)p Fp(\(1\))j(as)f Fo(\033)h Fk(&)d Fp(0.)55 b(Using)-24 3853 y(\(3.17\))33 b(and)c(\(3.18\))k(w)m(e)e(obtain)540 4031 y Fl(")573 4041 y Fg(j)p 503 4058 138 4 v 503 4110 a Fl(r)535 4120 y Fg(j)568 4110 y Fl(h)609 4120 y Fg(j)691 4079 y Fp(=)40 b(2)853 4076 y Ff(b)847 4079 y Fo(")q(E)957 4093 y Fl(j)994 4079 y Fo(x)1046 4093 y Fl(j)1082 4046 y(\034)8 b Fn(+1)1256 4079 y Fk(\024)40 b Fp(2)p Fo(c)1451 4041 y Fi(\000)p Fn(1)1451 4104 y(1)1561 4079 y Fp(exp)1715 3985 y Ff(\020)1765 4079 y Fk(\000)p Fo(B)5 b(\033)1962 4093 y Fl(j)1998 4033 y Fi(\000)2102 4006 y Fj(1)p 2063 4018 111 4 v 2063 4059 a Fg(\032)p Fd(\000)p Fj(1)2187 3985 y Ff(\021)15 b(\020)2301 4079 y Fo(B)5 b(\033)2427 4093 y Fl(j)2463 4033 y Fi(\000)2568 4006 y Fj(1)p 2528 4018 V 2528 4059 a Fg(\032)p Fd(\000)p Fj(1)2652 3985 y Ff(\021)2702 4000 y Fl(\034)j Fn(+1)2850 4079 y Fp(\(1)21 b(+)f Fo(o)p Fp(\(1\)\))1032 4316 y Fk(\024)25 b Fo(c)p Fp(\()p Fo(\032;)15 b(\034)10 b Fp(\))1389 4222 y Ff(\020)1441 4316 y Fk(\000)1522 4279 y Fl(A)1575 4288 y Fj(0)p 1521 4295 88 4 v 1547 4347 a Fn(2)1619 4316 y Fp(\()p Fo(L)1716 4330 y Fn(1)1755 4316 y Fo(\033)1807 4330 y Fl(j)1844 4316 y Fp(\))1879 4271 y Fi(\000)1984 4243 y Fj(1)p 1944 4255 111 4 v 1944 4297 a Fg(\032)p Fd(\000)p Fj(1)2068 4222 y Ff(\021)2143 4316 y Fk(\024)25 b Fo(c)p Fp(\()p Fo(\032;)15 b(\034)10 b Fp(\))2500 4197 y Ff(\022)2563 4316 y Fk(\000)2644 4279 y Fl(A)2697 4288 y Fj(0)p 2644 4295 88 4 v 2670 4347 a Fn(2)2741 4316 y Fp(\()p Fo(\033)s(\016)2874 4283 y Fl(j)2912 4316 y Fp(\))2947 4265 y Fi(\000)3052 4238 y Fj(1)p 3012 4250 111 4 v 3012 4291 a Fg(\032)p Fd(\000)p Fj(1)3136 4197 y Ff(\023)3213 4316 y Fo(:)-24 4551 y Fp(This)33 b(implies)g Fo(")544 4565 y Fl(j)581 4551 y Fp(\()p Fo(r)657 4565 y Fl(j)694 4551 y Fo(h)746 4565 y Fl(j)783 4551 y Fp(\))818 4518 y Fi(\000)p Fn(1)946 4551 y Fk(\034)h Fp(const)p Fo(:)16 b(\035)38 b Fp(for)e(0)e Fo(<)f(\033)k Fk(\024)1932 4548 y Ff(e)1927 4551 y Fo(\033)s Fp(\()p Fo(n;)15 b(\032;)g(\034)10 b Fp(\))35 b Fk(\034)f Fp(1)h(whic)m(h)g(pro)m(v)m(es)h(\(b\).)56 b(In)34 b(the)i(same)-24 4664 y(w)m(a)m(y)31 b(w)m(e)g(obtain)1100 4756 y Fi(1)1082 4781 y Ff(Y)1074 4963 y Fl(j)t Fn(=0)1211 4718 y Ff( )1277 4862 y Fp(1)21 b(+)1452 4800 y Fo(C)7 b(")1566 4814 y Fl(j)p 1444 4841 167 4 v 1444 4924 a Fo(r)1485 4938 y Fl(j)1521 4924 y Fo(h)1573 4938 y Fl(j)1620 4718 y Ff(!)1726 4862 y Fk(\024)40 b Fp(exp)1991 4693 y Ff(0)1991 4842 y(@)2090 4756 y Fi(1)2065 4781 y Ff(X)2064 4963 y Fl(j)t Fn(=0)2220 4800 y Fo(C)7 b(")2334 4814 y Fl(j)p 2212 4841 V 2212 4924 a Fo(r)2253 4938 y Fl(j)2289 4924 y Fo(h)2341 4938 y Fl(j)2388 4693 y Ff(1)2388 4842 y(A)2501 4862 y Fk(\024)40 b Fp(2)866 b(\(3.19\))-24 5110 y(for)29 b(0)d Fo(<)f(\033)j Fk(\024)462 5107 y Ff(e)457 5110 y Fo(\033)t Fp(\()p Fo(n;)15 b(\032;)g(\034)10 b Fp(\))26 b Fk(\034)f Fp(1,)30 b(where)g Fo(C)h Fp(=)25 b Fo(C)7 b Fp(\()p Fo(n;)15 b(\034)10 b Fp(\))26 b Fo(>)f Fp(0)30 b(is)e(the)i(constan)m(t)h(in)d(Remark)i(3.3.)41 b(W)-8 b(e)31 b(are)f(going)-24 5246 y(to)h(c)m(hec)m(k)h(\(3.12\))h(with)c Fo(\035)f Fp(=)e(1)p Fo(=)p Fp(54.)42 b(Using)30 b(\(3.18\))i(w)m(e)f (obtain)f Fo(x)2210 5260 y Fl(j)2247 5246 y Fo(=x)2344 5260 y Fl(j)t Fn(+1)2496 5246 y Fp(=)25 b(\()p Fo(\033)2679 5260 y Fl(j)t Fn(+1)2806 5246 y Fo(=\033)2903 5260 y Fl(j)2940 5246 y Fp(\))3025 5173 y Fj(1)p 2985 5185 111 4 v 2985 5226 a Fg(\032)p Fd(\000)p Fj(1)3125 5246 y Fp(\(1)c(+)f Fo(o)p Fp(\(1\)\).)42 b(This)-24 5359 y(implies)316 5561 y Fo(h)368 5575 y Fl(j)t Fn(+1)p 316 5601 180 4 v 361 5685 a Fo(h)413 5699 y Fl(j)530 5622 y Fp(=)626 5478 y Ff( )747 5561 y Fo(x)799 5575 y Fl(j)p 702 5601 179 4 v 702 5685 a Fo(x)754 5699 y Fl(j)t Fn(+1)891 5478 y Ff(!)956 5501 y Fl(\034)8 b Fn(+1)1105 5478 y Ff( )1180 5561 y Fo(\033)1232 5575 y Fl(j)t Fn(+1)p 1180 5601 V 1225 5685 a Fo(\033)1277 5699 y Fl(j)1369 5478 y Ff(!)1435 5501 y Fl(\034)g Fn(+1)1593 5622 y Fp(=)25 b Fo(\016)1732 5577 y Fn(\()p Fl(\034)8 b Fn(+1\))1966 5545 y Fg(\032)p 1927 5562 111 4 v 1927 5603 a(\032)p Fd(\000)p Fj(1)2067 5622 y Fp(\(1)21 b(+)f Fo(o)p Fp(\(1\)\)\))27 b(=)2611 5503 y Ff(\022)2682 5561 y Fp(2)p 2682 5601 46 4 v 2682 5685 a(3)2737 5503 y Ff(\023)2798 5526 y Fl(\032)p Fn(\()p Fl(\034)8 b Fn(+1\))3038 5622 y Fp(\(1)21 b(+)f Fo(o)p Fp(\(1\)\))1820 5927 y(13)p eop %%Page: 14 14 14 13 bop -24 573 a Fp(for)30 b Fo(\033)e Fk(&)e Fp(0,)31 b(uniformly)c(with)i(resp)s(ect)h(to)i Fo(j)e Fk(2)25 b Fm(N)p Fp(.)41 b(Since)29 b Fo(\032)c(>)g Fp(1)31 b(and)f Fo(\034)g Fp(+)20 b(1)26 b Fo(>)f(n)g Fk(\025)g Fp(2,)31 b(w)m(e)f(obtain)1409 751 y Fo(h)1461 765 y Fl(j)t 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Fo(")1793 1695 y Fn(1)1871 1681 y Fp(=)1986 1678 y Ff(b)1981 1681 y Fo("\024r)2116 1695 y Fn(1)2156 1681 y Fo(\033)2211 1642 y Fl(\034)8 b Fn(+1)2208 1705 y(1)2344 1681 y Fo(E)2411 1695 y Fn(1)2490 1681 y Fp(=)2605 1678 y Ff(b)2599 1681 y Fo(")q(\024r)2735 1695 y Fn(0)2774 1681 y Fo(\033)2829 1642 y Fl(\034)g Fn(+1)2826 1705 y(0)2963 1681 y Fo(\016)3006 1648 y Fl(\034)g Fn(+1)3140 1681 y Fo(c)3179 1633 y Fn(1)p Fl(=)p Fn(2)3179 1705 y(1)3289 1681 y Fo(E)3361 1648 y Fn(2)3356 1704 y(0)3401 1681 y Fp(,)41 b(and)d(w)m(e)-24 1794 y(obtain)820 1945 y Ff(e)814 1948 y Fo(")856 1962 y Fn(0)896 1948 y Fo(=")983 1962 y Fn(1)1049 1948 y Fp(=)25 b Fo(c)1184 1900 y Fi(\000)p Fn(1)p Fl(=)p Fn(2)1184 1972 y(1)1364 1829 y Ff(\022)1435 1886 y Fp(3)p 1435 1927 V 1435 2010 a(2)1490 1829 y Ff(\023)1552 1852 y Fn(\()p Fl(\034)8 b Fn(+1\)\()p Fl(\032)p Fi(\000)p Fn(1\))1936 1948 y Fp(exp)2090 1854 y Ff(\020)2140 1948 y Fk(\000)p Fp(2)p Fo(B)d(\033)2382 1962 y Fn(0)2421 1902 y Fi(\000)2525 1875 y Fj(1)p 2486 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Fl(j)t Fn(+1)2065 5662 y Fo(c)2104 5614 y Fn(1)p Fl(=)p Fn(2)2104 5686 y(1)2214 5662 y Fo(E)2286 5614 y Fn(3)p Fl(=)p Fn(2)2281 5688 y Fl(j)2422 5662 y Fp(=)2528 5600 y Fo(")2570 5614 y Fl(j)t Fn(+1)p 2528 5641 170 4 v 2589 5724 a Fp(2)2707 5662 y Fo(:)1820 5927 y Fp(15)p eop %%Page: 16 16 16 15 bop -24 573 a Fp(Moreo)m(v)m(er,)31 b Fk(F)464 587 y Fl(j)529 573 y Fp(satis\014es)c(\(3.23\))k(in)26 b(view)i(of)g(Remark)g(3.3)h(and)f(as)g(in)f([9)q(])h(w)m(e)h(obtain)e (\(3.24\).)42 b(W)-8 b(e)30 b(are)e(going)-24 686 y(to)j(sho)m(w)f (that)1277 799 y Fk(F)1351 761 y Fl(j)t Fn(+1)1503 799 y Fp(:)c Fo(D)1629 813 y Fl(j)t Fn(+1)1776 799 y Fk(\002)20 b Fo(O)1936 813 y Fl(j)t Fn(+1)2118 799 y Fk(\000)-15 b(!)55 b(U)2377 813 y Fl(j)2429 799 y Fo(:)1069 b Fp(\(3.25\))-24 966 y(This)28 b(inequalit)m(y)h(com)m(bined)h(with)f(\(3.20\))j(and)e (\(3.21\))j(implies)813 1189 y Fk(j)p Fp(\()p Fo(P)931 1203 y Fl(j)989 1189 y Fk(\000)20 b Fo(P)1138 1203 y Fl(j)t Fi(\000)p Fn(1)1265 1189 y Fp(\))g Fk(\016)h(F)1460 1151 y Fl(j)t Fn(+1)1587 1189 y Fk(j)1612 1203 y Fl(j)t Fn(+1)1764 1189 y Fk(\024)k(j)p Fo(P)1943 1203 y Fl(j)2001 1189 y Fk(\000)20 b Fo(P)2150 1203 y Fl(j)t Fi(\000)p Fn(1)2277 1189 y Fk(j)2302 1203 y Fi(U)2346 1213 y Fg(j)2408 1189 y Fk(\024)2509 1186 y Ff(e)2504 1189 y Fo(")2546 1203 y Fl(j)2608 1189 y Fk(\024)2714 1127 y Fo(")2756 1141 y Fl(j)t Fn(+1)p 2714 1167 170 4 v 2776 1251 a Fp(2)2893 1189 y Fo(;)-24 1427 y Fp(and)30 b(w)m(e)g(obtain)g Fo(H)645 1441 y Fl(j)702 1427 y Fk(\016)20 b(F)841 1394 y Fl(j)t Fn(+1)994 1427 y Fp(=)25 b Fo(N)1163 1441 y Fl(j)t Fn(+1)1309 1427 y Fp(+)20 b Fo(R)1469 1441 y Fl(j)t Fn(+1)1596 1427 y Fp(,)31 b(where)f Fk(j)p Fo(R)2009 1441 y Fl(j)t Fn(+1)2135 1427 y Fk(j)2160 1441 y Fl(j)t Fn(+1)2313 1427 y Fk(\024)25 b Fo(")2451 1441 y Fl(j)t Fn(+1)2578 1427 y Fp(.)117 1540 y(T)-8 b(o)31 b(pro)m(v)m(e)g(\(3.25\))i(w)m(e)e(note)g(that)658 1758 y Fk(j)p 683 1686 99 4 v Fo(W)782 1773 y Fl(k)p 824 1686 V 824 1758 a Fo(W)923 1705 y Fi(\000)p Fn(1)923 1780 y Fl(k)r Fn(+1)1056 1758 y Fk(j)25 b Fp(=)g(sup)14 b Fk(f)q Fo(s)1443 1773 y Fl(k)r Fn(+1)1575 1758 y Fo(=s)1663 1773 y Fl(k)1706 1758 y Fo(;)h(r)1787 1773 y Fl(k)r Fn(+1)1920 1758 y Fo(=r)2006 1773 y Fl(k)2049 1758 y Fo(;)g(h)2141 1773 y Fl(k)r Fn(+1)2275 1758 y Fo(=h)2372 1773 y Fl(k)2415 1758 y Fk(g)26 b Fp(=)f Fo(s)2625 1773 y Fl(k)r Fn(+1)2757 1758 y Fo(=s)2845 1773 y Fl(k)2913 1758 y Fp(=)g Fo(\016)n(;)-24 1985 y Fp(since)31 b Fo(r)241 2000 y Fl(k)r Fn(+1)374 1985 y Fo(=r)460 2000 y Fl(k)530 1985 y Fp(=)c Fo(E)700 1937 y Fn(1)p Fl(=)p Fn(2)695 2013 y Fl(k)837 1985 y Fk(\034)g Fo(\016)s Fp(,)33 b(and)e Fo(h)1286 2000 y Fl(k)r Fn(+1)1419 1985 y Fo(=h)1516 2000 y Fl(k)1587 1985 y Fk(\030)26 b Fo(\016)1727 1952 y Fn(\()p Fl(\034)8 b Fn(+1\))p Fl(\032)p Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))2126 1928 y Fd(\000)p Fj(1)2244 1985 y Fo(<)26 b(\016)36 b Fp(for)31 b(an)m(y)h Fo(k)e Fk(2)d Fm(N)p Fp(.)44 b(Then)30 b(using)g(\(3.19\))-24 2098 y(and)g(\(3.23\),)j(w)m(e)d(estimate)i(the) e(Jacobian)g(of)h Fk(F)1653 2065 y Fl(j)t Fn(+1)1810 2098 y Fp(in)e Fo(D)1991 2112 y Fl(j)t Fn(+1)2138 2098 y Fk(\002)20 b Fo(O)2298 2112 y Fl(j)t Fn(+1)2455 2098 y Fp(as)31 b(follo)m(ws)f(\(see)h([9)q(]\))227 2229 y Ff(\014)227 2279 y(\014)227 2329 y(\014)p 254 2254 V 254 2327 a Fo(W)353 2341 y Fn(0)392 2327 y Fo(D)s Fk(F)544 2294 y Fl(j)t Fn(+1)p 671 2254 V 671 2327 a Fo(W)770 2274 y Fi(\000)p Fn(1)770 2349 y Fl(j)864 2229 y Ff(\014)864 2279 y(\014)864 2329 y(\014)892 2383 y Fl(j)t Fn(+1)1116 2327 y Fp(=)1285 2229 y Ff(\014)1285 2279 y(\014)1285 2329 y(\014)p 1313 2254 V -2 x Fo(W)1411 2341 y Fn(0)1451 2327 y Fo(D)s Fp(\()p Fk(F)1629 2341 y Fn(0)1689 2327 y Fk(\016)21 b(\001)15 b(\001)g(\001)21 b(\016)g(F)2012 2341 y Fl(j)2048 2327 y Fp(\))p 2083 2254 V Fo(W)2182 2274 y Fi(\000)p Fn(1)2182 2349 y Fl(j)2277 2229 y Ff(\014)2277 2279 y(\014)2277 2329 y(\014)2304 2383 y Fl(j)t Fn(+1)1116 2564 y Fk(\024)1270 2500 y Ff(Q)1349 2520 y Fl(j)t Fi(\000)p Fn(1)1349 2592 y Fl(k)r Fn(=0)1497 2445 y Ff(\022)1558 2466 y(\014)1558 2516 y(\014)1558 2566 y(\014)p 1585 2492 V 1585 2564 a Fo(W)1684 2579 y Fl(k)1726 2564 y Fo(D)s Fk(F)1869 2579 y Fl(k)p 1912 2492 V 1912 2564 a Fo(W)2011 2511 y Fi(\000)p Fn(1)2011 2586 y Fl(k)2105 2466 y Ff(\014)2105 2516 y(\014)2105 2566 y(\014)2133 2620 y Fl(k)r Fn(+1)2296 2466 y Ff(\014)2296 2516 y(\014)2296 2566 y(\014)p 2324 2492 V -2 x Fo(W)2422 2579 y Fl(k)p 2465 2492 V 2465 2564 a Fo(W)2563 2511 y Fi(\000)p Fn(1)2563 2586 y Fl(k)r Fn(+1)2696 2466 y Ff(\014)2696 2516 y(\014)2696 2566 y(\014)2724 2445 y(\023)2800 2466 y(\014)2800 2516 y(\014)2800 2566 y(\014)p 2828 2492 V -2 x Fo(W)2926 2578 y Fl(j)2963 2564 y Fo(D)s Fk(F)3106 2578 y Fl(j)p 3142 2492 V 3142 2564 a Fo(W)3241 2511 y Fi(\000)p Fn(1)3241 2586 y Fl(j)3335 2466 y Ff(\014)3335 2516 y(\014)3335 2566 y(\014)3363 2620 y Fl(j)t Fn(+1)1116 2781 y Fk(\024)83 b Fo(\016)1313 2748 y Fl(j)1366 2717 y Ff(Q)1444 2743 y Fi(1)1444 2804 y Fl(j)t Fn(=0)1601 2687 y Ff(\020)1651 2781 y Fp(1)20 b(+)1826 2734 y Fl(C)5 b(")1914 2744 y Fg(j)p 1817 2760 138 4 v 1817 2813 a Fl(r)1849 2823 y Fg(j)1882 2813 y Fl(h)1923 2823 y Fg(j)1965 2687 y Ff(\021)2070 2781 y Fk(\024)55 b Fp(2)15 b Fo(\016)2299 2748 y Fl(j)2352 2781 y Fo(:)-24 3001 y Fp(Set)30 b Fo(z)g Fp(=)25 b(\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)s Fp(\))26 b(=)f Fo(x)19 b Fp(+)h Fo(iy)28 b Fk(2)d Fo(D)1152 3015 y Fl(j)t Fn(+1)1299 3001 y Fk(\002)20 b Fo(O)1459 3015 y Fl(j)t Fn(+1)1586 3001 y Fp(,)30 b(where)g Fo(x)g Fp(and)f Fo(y)k Fp(are)e(the)f(real)g (and)g(the)g(imaginary)f(part)h(of)-24 3114 y Fo(z)t Fp(.)41 b(Then)905 3225 y Fk(F)979 3192 y Fl(j)t Fn(+1)1106 3225 y Fp(\()p Fo(x)20 b Fp(+)g Fo(iy)s Fp(\))41 b(=)f Fk(F)1644 3192 y Fl(j)t Fn(+1)1771 3225 y Fp(\()p Fo(x)p Fp(\))21 b(+)p 2005 3153 99 4 v 20 w Fo(W)2103 3173 y Fi(\000)p Fn(1)2103 3248 y(0)2198 3225 y Fo(T)2251 3239 y Fl(j)t Fn(+1)2377 3225 y Fp(\()p Fo(x;)15 b(y)s Fp(\))p 2587 3153 V Fo(W)2687 3239 y Fl(j)2739 3225 y Fo(y)j(;)1009 3472 y(T)1062 3486 y Fl(j)t Fn(+1)1189 3472 y Fp(\()p Fo(x;)d(y)s Fp(\))26 b(=)f Fo(i)1583 3357 y Ff(Z)1666 3383 y Fn(1)1629 3545 y(0)p 1736 3399 V 1736 3472 a Fo(W)1834 3486 y Fn(0)1874 3472 y Fo(D)s Fk(F)2026 3434 y Fl(j)t Fn(+1)2153 3472 y Fp(\()p Fo(x)20 b Fp(+)g Fo(ity)s Fp(\))p 2498 3399 V Fo(W)2597 3419 y Fi(\000)p Fn(1)2597 3494 y Fl(j)2706 3472 y Fo(dt)c(:)-24 3690 y Fp(Moreo)m(v)m(er,)33 b Fk(j)p Fo(T)479 3704 y Fl(j)t Fn(+1)606 3690 y Fp(\()p Fo(x;)15 b(y)s Fp(\))p Fk(j)26 b(\024)f Fp(2)15 b Fo(\016)1066 3657 y Fl(j)1134 3690 y Fp(and)30 b(using)f(that)i Fk(j)p 1771 3618 V Fo(W)1870 3704 y Fl(j)1921 3690 y Fo(y)s Fk(j)26 b(\024)f(j)p 2141 3618 V Fo(W)2239 3704 y Fl(j)t Fn(+1)2381 3690 y Fo(y)s Fk(j)h(\024)2576 3615 y(p)p 2651 3615 46 4 v 2651 3690 a Fp(3)31 b(w)m(e)g(get)890 3894 y Fk(j)p Fo(T)968 3908 y Fl(j)t Fn(+1)1095 3894 y Fp(\()p Fo(x;)15 b(y)s Fp(\))p 1305 3822 99 4 v Fo(W)1405 3908 y Fl(j)1456 3894 y Fo(y)s Fk(j)26 b(\024)f Fp(4)15 b Fo(\016)1754 3856 y Fl(j)1807 3894 y Fo(;)106 b(x)20 b Fp(+)g Fo(iy)29 b Fk(2)24 b Fo(D)2366 3908 y Fl(j)t Fn(+1)2514 3894 y Fk(\002)c Fo(O)2674 3908 y Fl(j)t Fn(+1)2816 3894 y Fo(:)-24 4098 y Fp(This)27 b(implies)e Fk(F)562 4065 y Fl(j)t Fn(+1)689 4098 y Fp(\()p Fo(x)17 b Fp(+)f Fo(iy)s Fp(\))26 b Fk(2)e(U)1162 4112 y Fl(j)1199 4098 y Fp(,)29 b(since)f Fk(F)1548 4065 y Fl(j)t Fn(+1)1675 4098 y Fp(\()p Fo(x)p Fp(\))h(is)e(real,)i(and)f(w)m(e)h(complete)f (the)h(pro)s(of)f(of)g(Prop)s(osition)-24 4211 y(3.5.)3541 b Fa(2)117 4442 y Fp(W)-8 b(e)32 b(are)e(going)h(to)g(pro)m(v)m(e)g (suitable)e(Gevrey)i(estimates)g(for)f Fk(F)2290 4409 y Fl(j)2327 4442 y Fp(.)41 b(W)-8 b(e)31 b(set)434 4620 y Ff(e)414 4646 y Fo(D)489 4660 y Fl(j)551 4646 y Fp(=)25 b Fk(f)p Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(\))26 b Fk(2)e Fo(D)1081 4660 y Fl(j)1144 4646 y Fp(:)40 b Fk(j)p Fp(Im)15 b Fo(\022)s Fk(j)25 b Fo(<)g(s)1593 4660 y Fl(j)1629 4646 y Fo(=)p Fp(2)p Fk(g)32 b Fo(;)1947 4620 y Ff(e)1927 4646 y Fo(O)1996 4660 y Fl(j)2058 4646 y Fp(=)25 b Fk(f)p Fo(!)k Fk(2)24 b Fm(C)2446 4609 y Fl(n)2518 4646 y Fp(:)56 b Fk(j)p Fo(!)23 b Fk(\000)d Fp(\012)2861 4660 y Fl(\024)2906 4646 y Fk(j)25 b Fo(<)g(h)3104 4660 y Fl(j)3141 4646 y Fo(=)p Fp(2)p Fk(g)15 b Fo(;)-24 4851 y Fp(and)33 b(w)m(e)h(denote)g Fk(S)651 4818 y Fl(j)718 4851 y Fp(=)c Fk(F)893 4818 y Fl(j)t Fn(+1)1043 4851 y Fk(\000)22 b(F)1210 4818 y Fl(j)1247 4851 y Fp(.)50 b(F)-8 b(or)34 b(an)m(y)g(m)m(ulti-indices)d Fo(\013)j Fp(and)f Fo(\014)39 b Fp(and)33 b Fo(m)d Fk(2)g Fm(N)k Fp(with)e Fk(j)p Fo(\014)5 b Fk(j)31 b(\024)f Fo(m)p Fp(,)35 b(w)m(e)-24 4964 y(denote)228 5168 y Fo(R)298 5130 y Fl(m)297 5190 y(!)343 5171 y Fd(0)385 5074 y Ff(\020)435 5168 y Fo(@)488 5130 y Fl(\013)483 5190 y(\022)537 5168 y Fo(@)590 5130 y Fl(\014)585 5190 y(!)638 5168 y Fk(S)700 5130 y Fl(j)736 5074 y Ff(\021)801 5168 y Fp(\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)s Fp(\))26 b(:=)f Fo(@)1304 5130 y Fl(\013)1299 5190 y(\022)1354 5168 y Fo(@)1407 5130 y Fl(\014)1402 5190 y(!)1454 5168 y Fk(S)1516 5130 y Fl(j)1553 5168 y Fp(\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)s Fp(\))21 b Fk(\000)2055 5087 y Ff(X)1968 5275 y Fi(j)p Fl(\014)s Fn(+)p Fl(\015)t Fi(j\024)p Fl(m)2262 5168 y Fp(\()p Fo(!)j Fk(\000)c Fo(!)2529 5130 y Fi(0)2552 5168 y Fp(\))2587 5130 y Fl(\015)2631 5168 y Fo(@)2684 5130 y Fl(\013)2679 5190 y(\022)2734 5168 y Fo(@)2787 5130 y Fl(\014)s Fn(+)p Fl(\015)2782 5190 y(!)2930 5168 y Fk(S)2992 5130 y Fl(j)3028 5168 y Fp(\()p Fo(\022)s(;)15 b(I)7 b(;)15 b(!)3296 5130 y Fi(0)3320 5168 y Fp(\))p Fo(=\015)5 b Fp(!)p Fo(:)-24 5470 y Fp(Recall)30 b(that)h Fo(\032)494 5437 y Fi(0)542 5470 y Fp(=)25 b Fo(\032)p Fp(\()p Fo(\034)31 b Fp(+)20 b(1\))h(+)f(1.)1820 5927 y(16)p eop %%Page: 17 17 17 16 bop -24 573 a Fm(Lemma)33 b(3.6)46 b Fe(Under)32 b(the)h(assumptions)i(of)e(Pr)-5 b(op)g(osition)35 b(3.5)f(we)e(have) 354 788 y Fk(j)p 379 716 99 4 v Fo(W)477 802 y Fn(0)517 788 y Fo(@)570 755 y Fl(\013)565 814 y(\022)620 788 y Fo(@)673 755 y Fl(\014)668 810 y(!)720 788 y Fk(S)782 755 y Fl(j)818 788 y Fp(\()p Fo(\022)s(;)15 b Fp(0)p Fo(;)g(!)s Fp(\))p Fk(j)100 b(\024)1419 785 y Ff(b)1413 788 y Fo(")15 b(A)g(C)1625 755 y Fi(j)p Fl(\013)p Fn(+)p Fl(\014)s Fi(j)1827 788 y Fo(L)1889 740 y Fi(j)p Fl(\013)p Fi(j)p Fn(+)p Fi(j)p Fl(\014)s Fi(j)p Fn(\()p Fl(\034)8 b Fn(+1\)+1)1889 813 y(1)2389 788 y Fo(\024)2441 755 y Fi(\000j)p Fl(\014)s Fi(j)2598 788 y Fo(\013)p Fp(!)2693 755 y Fl(\032)2733 788 y Fo(\014)d Fp(!)2826 755 y Fl(\032)2862 732 y Fd(0)2889 788 y Fo(E)2961 740 y Fn(1)p Fl(=)p Fn(2)2956 814 y Fl(j)3072 788 y Fo(;)1398 1006 y Fp(\()p Fo(\022)s(;)15 b Fp(0)p Fo(;)g(!)s Fp(\))26 b Fk(2)1832 979 y Ff(e)1811 1006 y Fo(D)1886 1020 y Fl(j)t Fn(+1)2033 1006 y Fk(\002)2144 979 y Ff(e)2124 1006 y Fo(O)2193 1020 y Fl(j)t Fn(+1)2335 1006 y Fo(;)141 1270 y Fk(j)p 166 1198 V Fo(W)265 1284 y Fn(0)304 1270 y Fo(R)374 1233 y Fl(m)373 1293 y(!)419 1274 y Fd(0)446 1270 y Fp(\()p Fo(@)534 1233 y Fl(\013)529 1293 y(\022)584 1270 y Fo(@)637 1233 y Fl(\014)632 1293 y(!)685 1270 y Fk(S)747 1233 y Fl(j)783 1270 y Fp(\()p Fo(\022)s(;)15 b Fp(0)p Fo(;)g(!)s Fp(\)\))p Fk(j)100 b(\024)1419 1267 y Ff(b)1413 1270 y Fo(")15 b(A)g(C)1625 1237 y Fl(m)p Fn(+)p Fi(j)p Fl(\013)p Fi(j)p Fn(+1)1921 1270 y Fo(L)1983 1223 y Fi(j)p Fl(\013)p Fi(j)p Fn(+\()p Fl(m)p Fn(+1\)\()p Fl(\034)8 b Fn(+1\)+1)1983 1295 y(1)2608 1270 y Fo(\024)2660 1237 y Fi(\000)p Fl(m)p Fi(\000)p Fn(1)1244 1491 y Fk(\002)1408 1446 y Fi(j)p Fl(!)r Fi(\000)p Fl(!)1575 1423 y Fd(0)1597 1446 y Fi(j)1617 1423 y Fg(m)p Fd(\000j)p Fg(\014)s Fd(j)p Fj(+1)p 1408 1469 468 4 v 1459 1523 a Fn(\()p Fl(m)p Fi(\000j)p Fl(\014)s Fi(j)p Fn(+1\)!)1900 1491 y Fo(\013)p Fp(!)1998 1458 y Fl(\032)2055 1491 y Fp(\()p Fo(m)20 b Fp(+)g(1\)!)2401 1458 y Fl(\032)2437 1434 y Fd(0)2465 1491 y Fo(E)2537 1443 y Fn(1)p Fl(=)p Fn(2)2532 1516 y Fl(j)2662 1491 y Fo(;)109 b(\022)27 b Fk(2)e Fm(T)3025 1458 y Fl(n)3072 1491 y Fo(;)48 b(!)s(;)15 b(!)3305 1458 y Fi(0)3353 1491 y Fk(2)25 b Fp(\012)3505 1505 y Fl(\024)3565 1491 y Fo(;)-24 1702 y Fe(for)33 b(any)g Fo(m)25 b Fk(2)g Fm(N)p Fo(;)48 b(\013;)15 b(\014)31 b Fk(2)25 b Fm(N)989 1669 y Fl(n)1036 1702 y Fo(;)48 b Fk(j)p Fo(\014)5 b Fk(j)26 b(\024)f Fo(m)p Fe(,)32 b(wher)-5 b(e)34 b(the)f(c)-5 b(onstants)34 b Fo(A;)15 b(C)40 b Fe(dep)-5 b(end)34 b(only)f(on)h Fo(\034)10 b Fe(,)32 b Fo(\032)p Fe(,)g Fo(n)h Fe(and)g Fo(&)7 b Fe(.)-24 1886 y(Pr)-5 b(o)g(of)p Fp(.)41 b(Using)26 b(\(3.24\))k(and)c (the)h(Cauc)m(h)m(y)h(estimate,)h(w)m(e)e(ev)-5 b(aluate)28 b Fo(@)2336 1853 y Fl(\013)2331 1912 y(\022)2386 1886 y Fo(@)2439 1853 y Fl(\014)2434 1908 y(!)2486 1886 y Fk(S)2548 1853 y Fl(j)2612 1886 y Fp(for)e(an)m(y)i Fo(j)j Fk(\025)25 b Fp(0)i(and)g Fk(j)p Fo(\013)14 b Fp(+)g Fo(\014)5 b Fk(j)25 b(\025)g Fp(1)-24 1998 y(in)103 1972 y Ff(e)82 1998 y Fo(D)157 2012 y Fl(j)t Fn(+1)304 1998 y Fk(\002)415 1972 y Ff(e)395 1998 y Fo(O)464 2012 y Fl(j)t Fn(+1)591 1998 y Fp(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)566 2274 y Fo(M)654 2289 y Fl(j;\013;\014)854 2274 y Fp(:=)976 2176 y Ff(\014)976 2226 y(\014)976 2276 y(\014)1003 2274 y Fo(W)1089 2288 y Fn(0)1129 2274 y Fo(@)1182 2236 y Fl(\013)1177 2296 y(\022)1231 2274 y Fo(@)1284 2236 y Fl(\014)1279 2296 y(!)1332 2274 y Fk(S)1394 2236 y Fl(j)1430 2176 y Ff(\014)1430 2226 y(\014)1430 2276 y(\014)1498 2274 y Fk(\024)40 b Fo(c)1658 2212 y Fp(2)1703 2179 y Fi(j)p Fl(\013)p Fn(+)p Fl(\014)s Fi(j)1906 2212 y Fo(\013)p Fp(!)15 b Fo(\014)5 b Fp(!)15 b Fo(")2142 2226 y Fl(j)p 1658 2253 522 4 v 1661 2361 a Fo(r)1702 2375 y Fl(j)1740 2361 y Fo(h)1792 2375 y Fl(j)1828 2361 y Fo(s)1871 2313 y Fi(j)p Fl(\013)p Fi(j)1871 2386 y Fl(j)t Fn(+1)1998 2361 y Fo(h)2050 2313 y Fi(j)p Fl(\014)s Fi(j)2050 2386 y Fl(j)t Fn(+1)2231 2274 y Fp(=)40 b Fo(c\024)2439 2271 y Ff(b)2433 2274 y Fo(")2486 2202 y Fp(2)2531 2169 y Fi(j)p Fl(\013)p Fn(+)p Fl(\014)s Fi(j)2733 2202 y Fo(\013)p Fp(!)15 b Fo(\014)5 b Fp(!)15 b Fo(E)2994 2216 y Fl(j)3032 2202 y Fo(\033)3087 2169 y Fl(\034)3084 2227 y(j)p 2486 2253 645 4 v 2590 2361 a Fo(h)2642 2375 y 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720 y Fl(m)97 780 y(j;\013;\014)370 753 y Fk(\024)707 673 y Ff(X)554 861 y Fi(j)p Fl(\015)t Fi(j\025)p Fl(m)p Fi(\000j)p Fl(\014)s Fi(j)p Fn(+1)1009 753 y Fk(j)p Fo(!)1094 716 y Fi(0)1137 753 y Fk(\000)20 b Fo(!)s Fk(j)1313 716 y Fi(j)p Fl(\015)t Fi(j)1412 753 y Fo(M)1500 768 y Fl(j;\013;\014)s Fn(+)p Fl(\015)1756 753 y Fp(\()p Fo(\022)s(;)15 b Fp(0)p Fo(;)g(!)2022 716 y Fi(0)2046 753 y Fp(\))p Fo(=\015)5 b Fp(!)370 1060 y Fk(\024)83 b Fo(c)15 b(\013)p Fp(!\()p Fo(m)22 b Fp(+)e(1\)!)1020 998 y Fk(j)p Fo(!)1105 965 y Fi(0)1148 998 y Fk(\000)g Fo(!)s Fk(j)1324 965 y Fl(m)p Fi(\000j)p Fl(\014)s Fi(j)p Fn(+1)p 1020 1039 599 4 v 1044 1122 a Fp(\()p Fo(m)g Fk(\000)g(j)p Fo(\014)5 b Fk(j)21 b Fp(+)f(1\)!)1716 998 y(4)1761 965 y Fi(j)p Fl(\013)p Fi(j)p Fn(+)p Fl(m)p Fn(+1)2057 998 y Fo(")2099 1012 y Fl(j)p 1653 1039 545 4 v 1653 1146 a Fo(r)1694 1160 y Fl(j)1731 1146 y Fo(h)1783 1160 y Fl(j)1820 1146 y Fo(s)1863 1099 y Fi(j)p Fl(\013)p Fi(j)1863 1172 y Fl(j)t Fn(+1)1989 1146 y Fo(h)2041 1108 y 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Fl(m)659 5620 y(!)705 5601 y Fd(0)732 5593 y Fo(@)785 5560 y Fl(\013)780 5620 y(\022)835 5593 y Fo(f)10 b Fp(\))924 5624 y Fl(\014)987 5593 y Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))p Fk(j)98 b(\024)g Fo(A)15 b(C)1650 5545 y Fi(j)p Fl(\013)p Fi(j)1643 5618 y Fn(1)1739 5593 y Fo(C)1811 5555 y Fl(m)p Fn(+1)1804 5618 y(2)1992 5532 y Fk(j)p Fo(!)23 b Fk(\000)d Fo(!)2248 5499 y Fi(0)2271 5532 y Fk(j)2296 5499 y Fl(m)p Fi(\000j)p Fl(\014)s Fi(j)p Fn(+1)p 1992 5572 V 2016 5656 a Fp(\()p Fo(m)g Fk(\000)g(j)p Fo(\014)5 b Fk(j)21 b Fp(+)f(1\)!)2615 5593 y Fo(\013)p Fp(!)2713 5556 y Fl(\032)2770 5593 y Fp(\()p Fo(m)g Fp(+)g(1\)!)3113 5556 y Fl(\032)3149 5532 y Fd(0)3192 5593 y Fo(:)3523 5478 y Fp(\(3.28\))1820 5927 y(18)p eop %%Page: 19 19 19 18 bop -24 573 a Fm(Theorem)34 b(3.7)46 b Fe(Ther)-5 b(e)30 b(exist)f(p)-5 b(ositive)30 b(c)-5 b(onstants)31 b Fo(A)1853 587 y Fn(0)1922 573 y Fe(and)f Fo(C)2160 587 y Fn(0)2228 573 y Fe(and)g(for)g(any)g(c)-5 b(omp)g(act)31 b(set)e Fo(K)36 b Fe(and)30 b(any)g(jet)-24 686 y Fo(f)40 b Fp(=)31 b(\()p Fo(f)253 653 y Fl(\014)300 686 y Fp(\))p Fe(,)37 b Fo(\014)f Fk(2)31 b Fm(N)661 653 y Fl(n)708 686 y Fe(,)36 b(satisfying)h(\(3.28\))g(ther)-5 b(e)37 b(exists)1958 658 y Ff(e)1939 686 y Fo(f)j Fk(2)31 b(G)2175 653 y Fl(\032;\032)2267 629 y Fd(0)2294 686 y Fp(\()p Fm(T)2402 653 y Fl(n)2472 686 y Fk(\002)22 b Fm(R)2643 653 y Fl(n)2690 686 y Fp(\))36 b Fe(such)g(that)i Fo(@)3214 653 y Fl(\013)3209 712 y(\022)3263 686 y Fo(@)3316 653 y Fl(\014)3311 708 y(!)3383 658 y Ff(e)3364 686 y Fo(f)i Fp(=)31 b Fo(@)3604 653 y Fl(\013)3599 712 y(\022)3654 686 y Fo(f)3709 653 y Fl(\014)-24 799 y Fe(on)i Fo(K)39 b Fe(for)33 b(e)-5 b(ach)34 b Fo(\013)p Fe(,)f Fo(\014)5 b Fe(,)32 b(and)601 1003 y Fk(j)p Fo(@)679 965 y Fl(\013)674 1025 y(\022)729 1003 y Fo(@)782 965 y Fl(\014)777 1025 y(!)849 976 y Ff(e)830 1003 y Fo(f)9 b Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))p Fk(j)41 b(\024)f Fo(A)15 b(A)1428 1017 y Fn(0)1498 1003 y Fp(max\()p Fo(C)1767 1017 y Fn(1)1807 1003 y Fo(;)g Fp(1\))g(\()p Fo(C)2042 1017 y Fn(0)2083 1003 y Fo(C)2148 1017 y Fn(1)2188 1003 y Fp(\))2223 965 y Fi(j)p Fl(\013)p Fi(j)p Fn(+1)2417 1003 y Fp(\()p Fo(C)2517 1017 y Fn(0)2557 1003 y Fo(C)2622 1017 y Fn(2)2661 1003 y Fp(\))2696 965 y Fi(j)p Fl(\014)s Fi(j)2798 1003 y Fo(\013)p Fp(!)2893 965 y Fl(\032)2934 1003 y Fo(\014)5 b Fp(!)3027 965 y Fl(\032)3063 942 y Fd(0)3105 1003 y Fo(:)-24 1207 y Fe(R)-5 b(emark)p Fp(.)50 b(W)-8 b(e)35 b(p)s(oin)m(t)d(out)h(that)h(the)f(p)s(ositiv)m(e)g(constan)m(ts)h Fo(A)2090 1221 y Fn(0)2163 1207 y Fp(and)e Fo(C)2407 1221 y Fn(0)2480 1207 y Fp(do)h(not)g(dep)s(end)e(on)j(the)f(the)g(jet) h Fo(f)10 b Fp(,)-24 1320 y(on)30 b(compact)i(set)f Fo(K)37 b Fp(nor)30 b(on)g(the)g(constan)m(ts)i Fo(A;)f(C)1758 1334 y Fn(1)1827 1320 y Fp(and)f Fo(C)2069 1334 y Fn(2)2109 1320 y Fp(.)-24 1433 y Fe(Pr)-5 b(o)g(of.)129 b Fp(Consider)29 b(the)h(F)-8 b(ourier)30 b(co)s(e\016cien)m(ts)834 1665 y Fo(f)889 1621 y Fl(\014)879 1693 y(k)935 1665 y Fp(\()p Fo(!)s Fp(\))56 b(=)g(\(2)p Fo(\031)s Fp(\))1418 1628 y Fi(\000)p Fl(n)1551 1550 y Ff(Z)1597 1739 y Fh(T)1654 1720 y Fg(n)1731 1665 y Fo(e)1773 1628 y Fi(\000)p Fl(i)p Fi(h)p Fl(k)r(;')p Fi(i)2030 1665 y Fo(f)2085 1628 y Fl(\014)2132 1665 y Fp(\()p Fo(';)15 b(!)s Fp(\))p Fo(d')32 b(;)46 b(k)28 b Fk(2)d Fm(Z)2795 1628 y Fl(n)2872 1665 y Fo(;)-24 1936 y Fp(and)31 b(denote)i(b)m(y)f Fk(A)g Fp(the)g(set)h(of)f(all)f(Whitney)h(jets)g Fo(g)1807 1951 y Fl(k)1878 1936 y Fp(=)1977 1842 y Ff(\020)2027 1936 y Fo(g)2073 1892 y Fl(\014)2070 1964 y(k)2121 1842 y Ff(\021)2170 1936 y Fp(,)h Fo(\014)h Fk(2)27 b Fm(N)2483 1903 y Fl(n)2530 1936 y Fp(,)33 b(where)f Fo(g)2899 1892 y Fl(\014)2896 1964 y(k)2975 1936 y Fp(=)27 b Fo(e)3115 1903 y Fl(r)r Fi(j)p Fl(k)r Fi(j)3228 1880 y Fj(1)p Fg(=\032)3345 1936 y Fo(f)3400 1892 y Fl(\014)3390 1964 y(k)3446 1936 y Fp(\()p Fo(!)s Fp(\))33 b(and)-24 2094 y Fo(k)d Fk(2)d Fm(Z)205 2061 y Fl(n)252 2094 y Fp(,)32 b Fk(j)p Fo(k)s Fk(j)c Fp(=)f Fk(j)p Fo(k)607 2108 y Fn(1)647 2094 y Fk(j)21 b Fp(+)g Fk(\001)15 b(\001)g(\001)22 b Fp(+)f Fk(j)p Fo(k)1076 2108 y Fl(n)1124 2094 y Fk(j)p Fp(.)44 b(Cho)s(osing)30 b Fo(r)g Fp(=)d Fo(c)1824 2108 y Fn(0)1864 2094 y Fo(C)1936 2046 y Fi(\000)p Fn(1)p Fl(=\032)1929 2119 y Fn(1)2133 2094 y Fp(with)j(0)e Fo(<)f(c)2551 2108 y Fn(0)2618 2094 y Fp(=)g Fo(c)2755 2108 y Fn(0)2795 2094 y Fp(\()p Fo(n;)15 b(\032)p Fp(\))27 b Fk(\034)h Fp(1,)k(w)m(e)g(are)g(going)-24 2217 y(to)f(sho)m(w)f(that)h(\()p Fo(g)590 2173 y Fl(\014)587 2246 y(k)638 2217 y Fp(\))26 b Fk(2)f(A)30 b Fp(satisfy)1015 2350 y Ff(\014)1015 2400 y(\014)1015 2450 y(\014)1043 2448 y Fo(g)1089 2404 y Fl(\014)1086 2476 y(k)1137 2448 y Fp(\()p Fo(!)s Fp(\))1267 2350 y Ff(\014)1267 2400 y(\014)1267 2450 y(\014)1393 2448 y Fk(\024)98 b Fo(A)1630 2462 y Fn(2)1669 2448 y Fo(C)1741 2400 y Fi(j)p Fl(\014)s Fi(j)1734 2473 y Fn(2)1842 2448 y Fo(\014)5 b Fp(!)1935 2415 y Fl(\032)2006 2448 y Fo(;)46 b(!)28 b Fk(2)d Fo(K)738 2707 y Fk(j)15 b Fp(\()p Fo(R)883 2674 y Fl(m)882 2734 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y Fi(0)2656 5519 y Fk(j)2681 5486 y Fl(m)p Fi(\000j)p Fl(\014)s Fi(j)p Fn(+1)p 2377 5559 599 4 v 2401 5643 a Fp(\()p Fo(m)g Fk(\000)g(j)p Fo(\014)5 b Fk(j)21 b Fp(+)f(1\)!)3000 5580 y(\()p Fo(m)h Fp(+)f(1\)!)3344 5543 y Fl(\032)3380 5519 y Fd(0)1820 5927 y Fp(19)p eop %%Page: 20 20 20 19 bop -24 573 a Fp(for)37 b Fo(C)194 535 y Fi(\000)p Fn(1)187 598 y(2)288 573 y Fo(!)s(;)15 b(C)460 535 y Fi(\000)p Fn(1)453 598 y(2)554 573 y Fo(!)614 540 y Fi(0)674 573 y Fk(2)37 b Fo(K)7 b Fp(.)62 b(Hence,)41 b(w)m(e)d(can)g(supp)s (ose)e(that)i(\(12\))h(and)e(\(13\),)k([1)q(],)f(hold)c(with)g Fo(")i Fp(=)f Fo(A)g Fp(=)g(1)-24 686 y(\()p Fo(K)i Fp(is)30 b(scaled)i(to)g(another)g(compact)h(still)d(denoted)i(b)m(y)f Fo(K)7 b Fp(\).)45 b(Then)31 b(it)g(is)g(easy)h(to)h(see)f(that)g(the)g (constan)m(ts)-24 799 y Fo(A)g Fp(and)g Fo(")h Fp(in)e(\(19\),)k([1)q (],)e(are)g(indep)s(enden)m(t)d(of)i Fo(g)k Fp(and)c Fo(K)7 b Fp(.)47 b(Moreo)m(v)m(er,)35 b(the)e(di\013eren)m(t)f(constan) m(ts)h(in)e(Lemma)-24 912 y(3.2)j(and)f(3.3,)j([1],)f(do)e(not)h(dep)s (end)e(on)h Fo(g)k Fp(and)c Fo(K)7 b Fp(,)34 b(and)f(w)m(e)h(obtain)e (that)i(the)g(constan)m(ts)h(in)d(\(26\),)k([1],)f(are)-24 1024 y(indep)s(enden)m(t)26 b(of)j Fo(g)j Fp(and)c Fo(K)7 b Fp(.)40 b(Scaling)27 b(bac)m(k)i(b)m(y)g Fo(C)1701 1038 y Fn(2)1769 1024 y Fp(w)m(e)g(obtain)f(the)h(desired)e(estimates)i (with)e(constan)m(ts)j Fo(A)3716 1038 y Fn(0)-24 1137 y Fp(and)g Fo(C)218 1151 y Fn(0)287 1137 y Fp(indep)s(enden)m(t)f(of)h Fo(g)s Fp(,)h Fo(K)7 b Fp(,)30 b Fo(A)1215 1151 y Fn(2)1285 1137 y Fp(and)g Fo(C)1527 1151 y Fn(2)1567 1137 y Fp(.)2065 b Fa(2)117 1368 y Fp(Applying)27 b(Theorem)h(3.8)i(to)g(the)f(family)e (of)i(jets)h Fk(A)p Fp(,)f(w)m(e)g(obtain)f(a)i(family)d(of)i (functions)3235 1365 y Ff(e)3233 1368 y Fo(g)3276 1383 y Fl(k)3344 1368 y Fk(2)c(G)3489 1335 y Fl(\032)3525 1312 y Fd(0)3581 1368 y Fp(suc)m(h)-24 1481 y(that)31 b Fo(@)226 1448 y Fl(\014)221 1504 y(!)276 1478 y Ff(e)273 1481 y Fo(g)316 1496 y Fl(k)385 1481 y 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3706 y Fn(0)1073 3692 y Fk(\024)25 b Fo(\024\033)1276 3654 y Fl(\034)8 b Fn(+1)1273 3717 y(0)1435 3692 y Fo(<)25 b(\024)p Fp(,)31 b(and)f(w)m(e)h(obtain)630 3799 y Ff(\014)630 3848 y(\014)630 3898 y(\014)658 3896 y Fo(@)711 3863 y Fl(\013)706 3922 y(\022)760 3896 y Fo(@)813 3863 y Fl(\014)808 3919 y(!)861 3896 y Fp(\()p Fo(U)10 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))21 b Fk(\000)e Fo(\022)s Fp(\))1376 3799 y Ff(\014)1376 3848 y(\014)1376 3898 y(\014)1439 3896 y Fp(+)35 b Fo(r)1589 3863 y Fi(\000)p Fn(1)1698 3799 y Ff(\014)1698 3848 y(\014)1698 3898 y(\014)1726 3896 y Fo(@)1779 3863 y Fl(\013)1774 3922 y(\022)1829 3896 y Fo(@)1882 3863 y Fl(\014)1877 3919 y(!)1929 3896 y Fo(V)20 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))2218 3799 y Ff(\014)2218 3848 y(\014)2218 3898 y(\014)2266 3896 y Fp(+)20 b Fo(\024)2409 3863 y Fi(\000)p Fn(1)2519 3799 y Ff(\014)2519 3848 y(\014)2519 3898 y(\014)2547 3896 y Fo(@)2600 3863 y Fl(\014)2595 3919 y(!)2647 3896 y Fp(\()p Fo(\036)p Fp(\()p Fo(!)s Fp(\))i Fk(\000)d Fo(!)s Fp(\))3073 3799 y Ff(\014)3073 3848 y(\014)3073 3898 y(\014)630 4151 y Fk(\024)55 b Fo(A)834 4089 y Fk(k)p Fo(P)13 b Fk(k)p Fo(L)1057 4056 y Fl(N)1057 4112 y Fn(1)p 834 4130 292 4 v 932 4213 a Fo(\024r)1150 4151 y Fp(\()p Fo(C)7 b(L)1319 4165 y Fn(1)1359 4151 y Fp(\))1394 4113 y Fi(j)p Fl(\013)p Fi(j)1498 4151 y Fp(\()p Fo(C)g(L)1667 4113 y Fl(\034)h Fn(+1)1667 4176 y(1)1800 4151 y Fo(\024)1852 4113 y Fi(\000)p Fn(1)1947 4151 y Fp(\))1982 4113 y Fi(j)p Fl(\014)s Fi(j)2068 4151 y Fo(\013)p Fp(!)2166 4113 y Fl(\032)2207 4151 y Fo(\014)d Fp(!)2303 4113 y Fl(\032)2339 4090 y Fd(0)-24 4358 y Fp(for)28 b(some)i(p)s(ositiv)m(e)d(constan)m (ts)j Fo(A)f Fp(and)f Fo(C)35 b Fp(as)29 b(ab)s(o)m(v)m(e.)42 b(Cho)s(osing)27 b Fo(\017)e(<)g Fp(1)p Fo(=)-5 b(A)30 b Fp(in)e(\(2.3\))i(w)m(e)f(obtain)g Fk(j)p Fo(V)20 b Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))p Fk(j)26 b(\024)-24 4471 y Fo(A)p Fk(k)p Fo(P)13 b Fk(k)p Fo(L)267 4438 y Fl(N)267 4494 y Fn(1)335 4471 y Fo(\024)387 4438 y Fi(\000)p Fn(1)513 4471 y Fk(\024)31 b Fo(A\017r)k(<)c(r)j Fk(\024)d Fo(R)q Fp(.)51 b(In)34 b(the)g(same)h(w)m(a)m(y)g(w)m(e)f(get)i Fo(\036)p Fp(\()p Fo(!)s Fp(\))c Fk(2)f Fp(\012)i(for)h Fo(!)h Fk(2)c Fp(\012)3012 4485 y Fl(\024)3056 4471 y Fp(.)52 b(This)32 b(pro)m(v)m(es)j(the)-24 4600 y(estimates)h(in)d (Theorem)i(2.1.)56 b(As)35 b(in)f(Sect.)55 b(5.d,)37 b([9],)g(w)m(e)f(obtain)e(that)2540 4502 y Ff(\014)2540 4552 y(\014)2540 4602 y(\014)2567 4600 y Fo(X)2642 4614 y Fl(H)2700 4624 y Fg(j)2758 4600 y Fk(\016)20 b(F)2897 4567 y Fl(j)2954 4600 y Fk(\000)g Fo(D)s Fp(\010)3189 4567 y Fl(j)3245 4600 y Fk(\001)h Fo(X)3366 4614 y Fl(N)3433 4502 y Ff(\014)3433 4552 y(\014)3433 4602 y(\014)3494 4600 y Fk(\024)3629 4552 y Fl(c")3693 4562 y Fg(j)p 3608 4579 138 4 v 3608 4631 a Fl(r)3640 4641 y Fg(j)3672 4631 y Fl(h)3713 4641 y Fg(j)-24 4728 y Fp(on)32 b Fm(T)177 4695 y Fl(n)246 4728 y Fk(\002)21 b(f)p Fp(0)p Fk(g)i(\002)e Fp(\012)654 4742 y Fl(\024)731 4728 y Fp(for)33 b(all)e Fo(j)j Fk(\025)29 b Fp(0,)34 b(where)e Fo(X)1616 4742 y Fl(H)1674 4752 y Fg(j)1743 4728 y Fp(and)g Fo(X)1997 4742 y Fl(N)2097 4728 y Fp(stand)g(for)g(the)h(Hamiltonian)e(v)m(ector) j(\014elds)d(of)-24 4841 y Fo(H)52 4855 y Fl(j)88 4841 y Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(!)s Fp(\))34 b(and)e Fo(N)40 b Fp(=)29 b Fk(h)p Fo(!)s(;)15 b(I)7 b Fk(i)p Fp(,)35 b(resp)s(ectiv)m(ely)-8 b(.)48 b(On)33 b(the)g(other)g(hand,)g Fk(r)p Fo(H)2587 4855 y Fl(j)2656 4841 y Fp(con)m(v)m(erges)i(uniformly)30 b(to)k Fk(r)p Fo(H)-24 4954 y Fp(as)c Fo(j)h Fk(!)26 b(1)k Fp(in)f(view)g(of)i(Prop)s (osition)d(3.1,)k(hence,)1364 5135 y Fo(X)1439 5154 y Fl(H)5 b Fn(\()p Fi(\001)p Fn(;)p Fl(\036)p Fn(\()p Fl(!)r Fn(\)\))1764 5135 y Fk(\016)20 b Fp(\010)25 b(=)g Fo(D)s Fp(\010)20 b Fk(\001)g Fo(X)2300 5149 y Fl(N)-24 5317 y Fp(on)29 b Fm(T)174 5284 y Fl(n)238 5317 y Fk(\002)18 b(f)p Fp(0)p Fk(g)h(\002)e Fp(\012)635 5331 y Fl(\024)680 5317 y Fp(.)40 b(Then)28 b Fk(f)p Fp(\010\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))26 b(:)40 b Fo(\022)28 b Fk(2)d Fm(T)1629 5284 y Fl(n)1675 5317 y Fk(g)30 b Fp(is)e(an)h(em)m(b)s (edded)f(in)m(v)-5 b(arian)m(t)28 b(torus)h(of)h(the)f(Hamiltonian)-24 5430 y Fo(H)7 b Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(\036)p Fp(\()p Fo(!)s Fp(\)\))29 b(with)e(frequency)h Fo(!)g Fk(2)d Fp(\012)1367 5444 y Fl(\024)1411 5430 y Fp(.)40 b(It)29 b(is)e(Lagrangian)h(b)m(y)g(construction)g(\(see)h (also)f([2)q(],)h(Sect.)41 b(I.3.2\).)-24 5543 y(This)28 b(completes)j(the)g(pro)s(of)e(of)i(Theorem)f(1.1.)2032 b Fa(2)1820 5927 y Fp(20)p eop %%Page: 21 21 21 20 bop -24 573 a Fc(3.6)112 b Fb(R)-5 b(e)g(al)35 b(analytic)f(hamiltonians.)-24 863 y Fp(Consider)28 b(a)j(real)f (analytic)g(Hamiltonian)f Fo(P)38 b Fk(2)25 b(G)1718 830 y Fn(1)1713 888 y Fl(L)1761 897 y Fj(1)1796 888 y Fl(;L)1864 897 y Fj(2)1898 888 y Fl(;L)1966 897 y Fj(2)2004 863 y Fp(\()p Fm(T)2112 830 y Fl(n)2179 863 y Fk(\002)20 b Fo(B)25 b Fk(\002)20 b Fp(\012\),)31 b Fo(B)e Fp(=)c Fo(B)2875 877 y Fl(R)2933 863 y Fp(\(0\),)32 b(with)d(norm)511 1094 y Fk(k)p Fo(P)13 b Fk(k)672 1108 y Fl(L)720 1117 y Fj(1)756 1108 y Fl(;L)824 1117 y Fj(2)887 1094 y Fp(=)25 b(sup)1135 1000 y Ff(\020)1185 1094 y Fk(j)p Fo(@)1263 1057 y Fl(\013)1258 1117 y(\022)1313 1094 y Fo(@)1366 1050 y Fl(\014)1361 1121 y(I)1413 1094 y Fo(@)1466 1057 y Fl(\015)1461 1117 y(!)1512 1094 y Fo(P)13 b Fp(\()p Fo(\022)s(;)i(I)7 b Fp(;)15 b Fo(!)s Fp(\))p Fk(j)p Fo(L)1973 1046 y Fi(\000j)p Fl(\013)p Fi(j)1973 1119 y Fn(1)2117 1094 y Fo(L)2179 1046 y Fi(\000j)p Fl(\014)s Fi(j\000j)p Fl(\015)t Fi(j)2179 1119 y Fn(2)2455 1094 y Fp(\()p Fo(\013)p Fp(!)p Fo(\014)5 b Fp(!)p Fo(\015)g Fp(!\))2766 1057 y Fi(\000)p Fn(1)2862 1000 y Ff(\021)2937 1094 y Fo(<)25 b Fp(+)p Fk(1)p Fo(:)-24 1317 y Fp(Then)40 b Fo(P)54 b Fp(can)41 b(b)s(e)f(extended)h(as)h(an)e(analytic)h(Hamiltonian)e(in) h Fo(D)2385 1331 y Fl(s;r)2503 1317 y Fk(\002)27 b Fo(O)2670 1332 y Fl(h)2715 1317 y Fp(,)44 b(where)c Fo(s)j Fp(=)f(\(2)p Fo(L)3398 1331 y Fn(1)3438 1317 y Fp(\))3473 1284 y Fi(\000)p Fn(1)3609 1317 y Fp(and)-24 1430 y Fo(r)i Fp(=)e Fo(h)g Fp(=)g(\(2)p Fo(L)523 1444 y Fn(2)563 1430 y Fp(\))598 1397 y Fi(\000)p Fn(1)692 1430 y Fp(,)h(and)d(with)f(sup-norm)g (satisfying)g Fk(k)p Fo(P)13 b Fk(k)2155 1444 y Fn(2)p Fl(L)2238 1453 y Fj(1)2274 1444 y Fl(;)p Fn(2)p Fl(L)2377 1453 y Fj(2)2457 1430 y Fk(\024)42 b(k)p Fo(P)13 b Fk(k)2731 1445 y Fl(s;r)n(;h)2920 1430 y Fk(\024)42 b Fo(C)7 b Fk(k)p Fo(P)13 b Fk(k)3266 1444 y Fl(L)3314 1453 y Fj(1)3348 1444 y Fl(;L)3416 1453 y Fj(2)3455 1430 y Fp(,)43 b(where)-24 1543 y Fo(C)33 b Fp(=)25 b Fo(C)7 b Fp(\()p Fo(n)p Fp(\))26 b Fo(>)g Fp(0.)43 b(Fix)30 b Fo(\034)816 1510 y Fi(0)866 1543 y Fo(>)c(\034)36 b(>)26 b(n)20 b Fk(\000)g Fp(1.)43 b(W)-8 b(e)32 b(can)g(sligh)m(tly)d(impro)m(v)m(e)i(Theorem)f(3.1)i ([10)q(].)43 b(Giv)m(en)31 b Fo(s)26 b(>)g Fp(0)31 b(w)m(e)-24 1656 y(denote)g(b)m(y)f Fk(U)452 1670 y Fl(s)519 1656 y Fp(the)h(set)g(of)f(all)f Fo(\022)f Fk(2)d Fm(C)1280 1623 y Fl(n)1326 1656 y Fo(=)p Fp(2)p Fo(\031)s Fm(Z)1535 1623 y Fl(n)1613 1656 y Fp(suc)m(h)30 b(that)h Fk(j)p Fp(Im)15 b Fo(\022)s Fk(j)25 b Fo(<)g(s)p Fp(.)-24 1840 y Fm(Theorem)34 b(3.9)46 b Fe(Supp)-5 b(ose)41 b(that)g Fo(H)47 b Fe(is)39 b(given)g(by)h(\(2.1\))h(wher)-5 b(e)40 b Fo(P)52 b Fk(2)37 b(G)2507 1807 y Fn(1)2502 1865 y Fl(L)2550 1874 y Fj(1)2585 1865 y Fl(;L)2653 1874 y Fj(2)2687 1865 y Fl(;L)2755 1874 y Fj(2)2793 1840 y Fp(\()p Fm(T)2901 1807 y Fl(n)2974 1840 y Fk(\002)25 b Fo(B)3139 1854 y Fl(R)3197 1840 y Fp(\(0\))h Fk(\002)f Fp(\012\))p Fe(.)63 b(Fix)-24 1968 y Fo(\024)35 b(>)f Fp(0)k Fe(and)h Fo(r)e(>)d Fp(0)k Fe(such)g(that)h Fo(\024;)15 b(r)37 b(<)d(L)1439 1929 y Fi(\000)p Fn(1)1439 1992 y(2)1571 1968 y Fe(and)39 b Fo(r)e Fk(\024)d Fo(R)q Fe(.)57 b(Then)38 b(ther)-5 b(e)38 b(is)g Fp(0)d Fo(<)f(s)2895 1982 y Fn(0)2969 1968 y Fk(\024)g Fo(s)j Fe(and)i Fo(\017)34 b(>)g Fp(0)k Fe(b)-5 b(oth)-24 2080 y(indep)g(endent)43 b(of)e Fo(\024)p Fe(,)i Fo(L)789 2094 y Fn(2)829 2080 y Fe(,)g Fo(r)s Fe(,)f Fo(R)q Fe(,)h(and)f(of)g Fp(\012)e Fk(\032)g Fp(\012)1739 2094 y Fn(0)1778 2080 y Fe(,)k(such)d(that)h(if)f Fk(k)p Fo(P)13 b Fk(k)42 b(\024)81 b Fo(\017\024r)44 b Fe(then)e(ther)-5 b(e)42 b(exist)f(maps)-24 2193 y Fo(\036)25 b Fk(2)g(G)200 2160 y Fl(\034)239 2137 y Fd(0)262 2160 y Fn(+2)356 2193 y Fp(\(\012)p Fo(;)15 b Fp(\012\))33 b Fe(and)g Fp(\010)25 b(=)g(\()p Fo(U;)15 b(V)21 b Fp(\))26 b Fk(2)e(G)1410 2160 y Fn(1)p Fl(;\034)1504 2137 y Fd(0)1527 2160 y Fn(+2)1621 2193 y Fp(\()p Fk(U)1713 2212 y Fl(s)1746 2221 y Fj(0)1781 2212 y Fl(=)p Fn(2)1875 2193 y Fk(\002)19 b Fp(\012)p Fo(;)31 b Fk(U)2144 2207 y Fl(s)2177 2216 y Fj(0)2235 2193 y Fk(\002)19 b Fo(B)2394 2207 y Fl(R)2451 2193 y Fp(\(0\)\))p Fe(,)34 b(satisfying)f(\(i\),)f(The)-5 b(or)g(em)35 b(2.1,)-24 2306 y(with)e Fo(\032)26 b Fp(=)f(1)p Fe(.)42 b(Mor)-5 b(e)g(over,)33 b(ther)-5 b(e)34 b(exist)f Fo(A;)15 b(C)32 b(>)25 b Fp(0)p Fe(,)33 b(indep)-5 b(endent)34 b(of)f Fo(\024)p Fe(,)g Fo(L)2516 2320 y Fn(2)2555 2306 y Fe(,)f Fo(r)s Fe(,)g(and)i Fp(\012)p Fe(,)e(such)h(that)200 2451 y Ff(\014)200 2501 y(\014)200 2551 y(\014)228 2549 y Fo(@)281 2511 y Fl(\014)276 2571 y(!)329 2549 y Fp(\()p Fo(U)10 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))20 b Fk(\000)g Fo(\022)s Fp(\))844 2451 y Ff(\014)844 2501 y(\014)844 2551 y(\014)907 2549 y Fp(+)35 b Fo(r)1057 2511 y Fi(\000)p Fn(1)1166 2451 y Ff(\014)1166 2501 y(\014)1166 2551 y(\014)1193 2549 y Fo(@)1246 2511 y Fl(\014)1241 2571 y(!)1294 2549 y Fo(V)20 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))1583 2451 y Ff(\014)1583 2501 y(\014)1583 2551 y(\014)1631 2549 y Fp(+)20 b Fo(\024)1774 2511 y Fi(\000)p Fn(1)1884 2451 y Ff(\014)1884 2501 y(\014)1884 2551 y(\014)1912 2549 y Fo(@)1965 2511 y Fl(\014)1960 2571 y(!)2012 2549 y Fp(\()p Fo(\036)p Fp(\()p Fo(!)s Fp(\))h Fk(\000)f Fo(!)s Fp(\))2438 2451 y Ff(\014)2438 2501 y(\014)2438 2551 y(\014)2491 2549 y Fk(\024)58 b Fo(A)15 b(C)2775 2511 y Fi(j)p Fl(\014)s Fi(j)2876 2549 y Fo(\024)2928 2511 y Fi(\000j)p Fl(\014)s Fi(j)3085 2549 y Fo(\014)5 b Fp(!)3178 2511 y Fl(\034)3217 2488 y Fd(0)3240 2511 y Fn(+2)3359 2487 y Fk(k)p Fo(P)13 b Fk(k)p 3359 2528 162 4 v 3392 2611 a Fo(\024r)-24 2774 y Fe(uniformly)34 b(in)e Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))25 b Fk(2)g(U)886 2793 y Fl(s)919 2802 y Fj(0)953 2793 y Fl(=)p Fn(2)1048 2774 y Fk(\002)20 b Fp(\012)32 b Fe(and)i(for)f(any)g Fo(\014)e Fk(2)25 b Fm(N)1983 2741 y Fl(n)2030 2774 y Fe(.)-24 2958 y(Pr)-5 b(o)g(of)p Fp(.)83 b(Fix)44 b Fo(\032)k Fp(=)g(\()p Fo(\034)770 2925 y Fi(0)823 2958 y Fk(\000)29 b Fo(\034)10 b Fp(\)\()p Fo(\034)40 b Fp(+)30 b(1\))1304 2925 y Fi(\000)p Fn(1)1428 2958 y Fp(+)f(1)45 b(and)e(set)i(as)f(ab)s(o)m(v)m(e)i Fo(\033)2413 2972 y Fl(j)2497 2958 y Fp(=)i Fo(\033)2668 2972 y Fn(0)2708 2958 y Fo(\016)2751 2925 y Fl(j)2788 2958 y Fp(,)g(and)43 b Fo(s)3094 2972 y Fl(j)t Fn(+1)3269 2958 y Fp(=)48 b Fo(s)3431 2972 y Fl(j)3496 2958 y Fk(\000)29 b Fp(5)p Fo(\033)3693 2972 y Fl(j)3730 2958 y Fp(,)-24 3071 y Fo(j)38 b Fk(\025)33 b Fp(0,)k(where)d Fo(s)572 3085 y Fn(0)611 3071 y Fp(\(1)24 b Fk(\000)f Fo(\016)s Fp(\))34 b(=)f(20)p Fo(\033)1167 3085 y Fn(0)1242 3071 y Fp(\(then)i Fo(s)1532 3085 y Fl(j)1601 3071 y Fk(!)e Fp(3)p Fo(s)1813 3085 y Fn(0)1852 3071 y Fo(=)p Fp(4\))j(and)f Fo(s)2238 3085 y Fn(0)2310 3071 y Fk(\024)d Fo(s)h Fp(=)f(\(2)p Fo(L)2734 3085 y Fn(1)2775 3071 y Fp(\))2810 3038 y Fi(\000)p Fn(1)2904 3071 y Fp(.)55 b(As)35 b(ab)s(o)m(v)m(e)h(w)m(e)f(de\014ne) -24 3232 y Fo(E)43 3246 y Fl(j)113 3232 y Fp(:=)f Fo(c)282 3194 y Fi(\000)p Fn(1)282 3257 y(1)392 3232 y Fp(exp\()p Fk(\000)p Fo(\033)692 3165 y Fi(\000)797 3138 y Fj(1)p 757 3150 111 4 v 757 3191 a Fg(\032)p Fd(\000)p Fj(1)689 3258 y Fl(j)881 3232 y Fp(\))i(and)f(set)h Fo(")1323 3246 y Fl(j)1394 3232 y Fp(=)1504 3229 y Ff(b)1498 3232 y Fo(")q(\024r)1634 3246 y Fl(j)1671 3232 y Fo(\033)1726 3194 y Fl(\034)8 b Fn(+1)1723 3258 y Fl(j)1859 3232 y Fo(E)1926 3246 y Fl(j)1963 3232 y Fp(,)37 b(where)2298 3229 y Ff(b)2293 3232 y Fo(")d Fp(=)f Fk(k)p Fo(P)13 b Fk(k)2634 3247 y Fl(s;r)n(;h)2782 3232 y Fp(\()p Fo(a\024r)s Fp(\))2996 3199 y Fi(\000)p Fn(1)3091 3232 y Fp(.)56 b(Cho)s(ose)35 b Fo(r)3534 3246 y Fn(0)3607 3232 y Fp(=)f Fo(r)-24 3345 y Fp(and)28 b Fo(h)203 3359 y Fn(0)269 3345 y Fp(=)c Fo(h)30 b Fp(and)f(de\014ne)f Fo(r)924 3359 y Fl(j)961 3345 y Fp(,)h Fo(x)1067 3359 y Fl(j)1104 3345 y Fp(,)g Fo(K)1235 3359 y Fl(j)1301 3345 y Fp(and)g Fo(h)1529 3359 y Fl(j)1595 3345 y Fp(as)g(ab)s(o)m(v)m(e.)42 b(W)-8 b(e)30 b(consider)e Fo(H)k Fp(=)25 b Fk(h)p Fo(!)s(;)15 b(I)7 b Fk(i)19 b Fp(+)f Fo(P)42 b Fp(in)28 b Fo(D)3315 3359 y Fl(j)3369 3345 y Fk(\002)18 b Fo(O)3527 3359 y Fl(j)3564 3345 y Fp(.)40 b(W)-8 b(e)-24 3458 y(\014x)29 b Fo(\033)156 3472 y Fn(0)220 3458 y Fp(=)c Fo(\033)368 3472 y Fn(0)408 3458 y Fp(\()p Fo(n;)15 b(\032;)g(\034)5 b(;)15 b(s)p Fp(\))26 b Fk(\024)f Fo(s)p Fp(\(1)19 b Fk(\000)e Fo(\016)s Fp(\))p Fo(=)p Fp(20,)33 b(so)c(that)h(a\),)h(b\),) e(c\))i(in)d(Prop)s(osition)f(3.1)j(and)f(\(3.12\))j(hold)c(for)h(an)m (y)-24 3616 y Fo(j)i Fk(\025)25 b Fp(0.)41 b(Next)30 b(w)m(e)g(c)m(ho)s(ose)h Fo(a)26 b Fp(=)e Fo(a)p Fp(\()p Fo(n;)15 b(\032;)g(\034)5 b(;)15 b(L)1474 3630 y Fn(1)1515 3616 y Fp(\))26 b Fk(\024)f Fo(c)1711 3577 y Fi(\000)p Fn(1)1711 3640 y(1)1805 3616 y Fo(\033)1860 3577 y Fi(\000)p Fl(\034)8 b Fi(\000)p Fn(1)1857 3640 y(0)2064 3616 y Fp(exp\()p Fk(\000)p Fo(\033)2364 3549 y Fi(\000)2469 3521 y Fj(1)p 2429 3533 V 2429 3575 a Fg(\032)p Fd(\000)p Fj(1)2361 3640 y Fn(0)2553 3616 y Fp(\).)41 b(Then)29 b Fk(j)p Fo(P)13 b Fk(j)3012 3630 y Fn(0)3077 3616 y Fk(\024)25 b Fo(")3215 3630 y Fn(0)3255 3616 y Fp(,)30 b(and)f(w)m(e)h(can)-24 3728 y(apply)f(Prop)s(osition)f(3.5.)42 b(Moreo)m(v)m(er,)33 b(as)d(in)g(Lemma)g(3.6)h(w)m(e)g(obtain)f(with)f Fo(\032)2681 3696 y Fi(0)2730 3728 y Fp(=)c Fo(\032)p Fp(\()p Fo(\034)30 b Fp(+)20 b(1\))h(+)f(1)26 b(=)f Fo(\034)3478 3696 y Fi(0)3521 3728 y Fp(+)20 b(2)445 3949 y Fk(j)p 470 3877 99 4 v Fo(W)569 3963 y Fn(0)608 3949 y Fo(@)661 3916 y Fl(\014)656 3972 y(!)708 3949 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b(the)f(pro)s(of)g(of) g(the)h(theorem)g(as)f(ab)s(o)m(v)m(e.)1927 b Fa(2)-24 5163 y Fr(4)135 b Fq(Pr)-6 b(o)g(of)42 b(of)f(The)-6 b(or)g(em)41 b(1.1)g(and)g(Cor)-6 b(ol)6 b(lary)41 b(1.2.)-24 5366 y Fe(Pr)-5 b(o)g(of)31 b(of)g(The)-5 b(or)g(em)32 b(1.1)p Fp(.)40 b(Set)28 b Fo(r)g Fp(=)d Fo(R)h Fp(=)f Fo(\024)1444 5307 y Fk(p)p 1520 5307 105 4 v 59 x Fo(\017)1557 5380 y Fl(H)1624 5366 y Fp(.)40 b(Then)26 b(\(2.2\))k(implies)24 b Fk(k)p Fo(P)13 b Fk(k)26 b(\024)f Fp(\()p Fo(A)15 b Fp(+)g(1\))p Fo(\024r)3106 5307 y Fk(p)p 3182 5307 V 59 x Fo(\017)3219 5380 y Fl(H)3314 5366 y Fp(and)27 b(w)m(e)h(can)-24 5479 y(apply)h(Theorem)h(2.1.)42 b(Consider)28 b(the)j(map)p 1530 5407 66 4 v 30 w(\010)25 b(:)g Fm(T)1744 5446 y Fl(n)1811 5479 y Fk(\002)20 b Fp(\012)25 b Fk(!)g Fm(T)2182 5446 y Fl(n)2249 5479 y Fk(\002)20 b Fo(D)33 b Fp(giv)m(en)d(b)m(y)p 1043 5606 V 1043 5678 a(\010\()p Fo(\022)s(;)15 b(!)s Fp(\))25 b(=)g(\()p Fo(U)10 b Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\))p Fo(;)g Fk(r)p Fo(g)1931 5641 y Fn(0)1972 5678 y Fp(\()p Fo(\036)p Fp(\()p Fo(!)s Fp(\)\))21 b(+)f Fo(V)h Fp(\()p Fo(\022)s Fp(;)15 b Fo(!)s Fp(\)\))p Fo(;)1820 5927 y Fp(21)p eop %%Page: 22 22 22 21 bop -24 573 a Fp(where)35 b Fo(U)46 b Fp(and)35 b Fo(V)56 b Fp(are)37 b(obtained)e(in)g(Theorem)g(2.1.)59 b(W)-8 b(e)37 b(ha)m(v)m(e)g Fo(H)7 b Fp(\()p Fo(\022)s(;)15 b(I)7 b Fp(;)15 b Fo(\036)p Fp(\()p Fo(!)s Fp(\)\))36 b(=)e Fo(H)7 b Fp(\()p Fo(\022)s(;)15 b Fk(r)p Fo(g)3234 540 y Fn(0)3274 573 y Fp(\()p Fo(\036)p Fp(\()p Fo(!)s Fp(\)\))25 b(+)f Fo(I)7 b Fp(\),)-24 686 y Fo(I)37 b Fk(2)30 b Fo(B)213 700 y Fl(R)271 686 y Fp(\(0\),)36 b(in)c(the)h(notations)h(of)g(Sect.)50 b(2.1.)h(Then)33 b(Theorem)g(2.1,)i(\(i\),)g(implies)30 b(that)k(\003)3190 700 y Fl(!)3271 686 y Fp(=)c Fk(f)p 3417 614 66 4 v Fp(\010)q(\()p Fo(\022)s(;)15 b(!)s Fp(\))30 b(:)-24 799 y Fo(\022)37 b Fk(2)e Fm(T)225 766 y Fl(n)271 799 y Fk(g)p Fp(,)k Fo(!)f Fk(2)c Fp(\012)636 813 y Fl(\024)681 799 y Fp(,)k(is)d(an)h(em)m (b)s(edded)f(Lagrangian)h(in)m(v)-5 b(arian)m(t)36 b(torus)g(of)g(the)g (Hamiltonian)f Fo(H)43 b Fp(for)36 b(eac)m(h)-24 912 y Fo(!)46 b Fk(2)c Fp(\012)248 926 y Fl(\024)334 912 y Fp(with)d(frequency)i Fo(!)s Fp(.)72 b(The)41 b(corresp)s(onding)e (estimates)i(for)p 2484 839 V 41 w(\010)g(follo)m(w)f(directly)g(from)g (those)i(in)-24 1024 y(Theorem)30 b(2.1.)3157 b Fa(2)-24 1256 y Fe(Pr)-5 b(o)g(of)32 b(of)f(Cor)-5 b(ol)5 b(lary)33 b(1.2)p Fp(.)41 b(The)28 b(pro)s(of)f(is)h(close)g(to)h(that)g(of)g (Theorem)f(2.1,)i([10)q(])e(and)g(w)m(e)h(will)d(b)s(e)h(concerned)-24 1368 y(mainly)j(with)h(the)i(corresp)s(onding)d(Gevrey)j(estimates.)48 b(Let)33 b Fo(")2197 1382 y Fl(H)2293 1368 y Fk(\024)28 b Fo(\017L)2491 1330 y Fi(\000)p Fl(N)7 b Fi(\000)p Fn(2)2491 1393 y(1)2703 1368 y Fp(.)47 b(Then)31 b(in)g(the)i(notations)f(of)-24 1499 y(\(ii\),)27 b(Theorem)f(1.1,)i(w)m(e)f(get)h Fo(AC)1109 1513 y Fn(1)1149 1499 y Fo(L)1211 1451 y Fl(N)q(=)p Fn(2)1211 1524 y(1)1342 1441 y Fk(p)p 1418 1441 110 4 v 58 x Fo(")1460 1513 y Fl(H)1553 1499 y Fk(\024)d Fo(AC)7 b(\017)p Fp(.)39 b(Cho)s(osing)25 b Fo(\017)h Fp(small)f(enough)h(and)g(using)f(Prop)s (osition)-24 1612 y(A.2)j(as)g(w)m(ell)f(as)h(\(ii\),)f(Theorem)h(1.1,) h(w)m(e)f(obtain)f(a)h(solution)e Fo(\022)i Fp(=)d Fo(\022)s Fp(\()p Fo(';)15 b(!)s Fp(\))28 b(the)g(equation)p 3084 1540 72 4 v 27 w Fo(U)10 b Fp(\()p Fo(\022)s(;)15 b(!)s Fp(\))26 b(=)f Fo(')j Fp(suc)m(h)-24 1725 y(that)38 b Fo(\022)s Fp(\()p Fo(';)15 b(!)s Fp(\))25 b Fk(\000)f Fo(')38 b Fp(satis\014es)e(the)i(same)f(Gevrey)h(estimates)g(as)p 2257 1653 V 37 w Fo(U)10 b Fp(.)61 b(Set)37 b Fo(F)13 b Fp(\()p Fo(';)i(!)s Fp(\))38 b(=)p 3023 1653 74 4 v 36 w Fo(V)21 b Fp(\()p Fo(\022)s Fp(\()p Fo(';)15 b(!)s Fp(\))p Fo(;)g(!)s Fp(\).)62 b(W)-8 b(e)-24 1838 y(ha)m(v)m(e)31 b(\003)248 1852 y Fl(!)324 1838 y Fp(=)25 b Fk(f)p Fp(\()p Fo(';)15 b(F)e Fp(\()p Fo(';)i(!)s Fp(\)\))29 b(:)41 b Fo(')25 b Fk(2)g Fm(T)1272 1805 y Fl(n)1319 1838 y Fk(g)31 b Fp(for)f(eac)m(h)h Fo(!)e Fk(2)24 b Fp(\012)1975 1852 y Fl(\024)2020 1838 y Fp(.)40 b(Moreo)m(v)m(er,)33 b(b)m(y)e(Prop)s(osition)d(A.4)j(w)m(e)g(obtain)448 1951 y Ff(\014)448 2001 y(\014)448 2051 y(\014)476 2048 y Fo(@)529 2011 y Fl(\013)524 2071 y(')579 2048 y Fo(@)632 2011 y Fl(\014)627 2071 y(!)679 2048 y Fp(\()p Fo(F)13 b Fp(\()p Fo(';)i(!)s Fp(\))22 b Fk(\000)e(r)p Fo(g)1249 2011 y Fn(0)1289 2048 y Fp(\()p Fo(!)s Fp(\)\))1454 1951 y Ff(\014)1454 2001 y(\014)1454 2051 y(\014)1523 2048 y Fk(\024)55 b Fo(\024AC)1841 2001 y Fi(j)p Fl(\013)p Fi(j)1834 2073 y Fn(1)1930 2048 y Fp(\()p Fo(C)2030 2062 y Fn(2)2070 2048 y Fo(\024)2122 2011 y Fi(\000)p Fn(1)2217 2048 y Fp(\))2252 2011 y Fi(j)p Fl(\014)s Fi(j)2338 2048 y Fo(\013)p Fp(!)2436 2011 y Fl(\032)2477 2048 y Fo(\014)5 b Fp(!)2573 2011 y Fl(\032)p Fn(\()p Fl(\034)j Fn(+1\)+1)2904 2048 y Fo(L)2966 2001 y Fl(N)q(=)p Fn(2)2966 2073 y(1)3098 1986 y Fk(p)p 3173 1986 110 4 v 3173 2048 a Fo(")3215 2062 y Fl(H)-24 2272 y Fp(uniformly)28 b(in)h(\()p Fo(\022)s(;)15 b(!)s Fp(\))26 b Fk(2)f Fm(T)898 2239 y Fl(n)965 2272 y Fk(\002)20 b Fp(\012.)42 b(Hereafter,)32 b Fo(A)p Fp(,)f Fo(C)1803 2286 y Fn(1)1868 2272 y Fp(=)26 b Fo(C)7 b(L)2099 2286 y Fn(1)2137 2272 y Fp(,)31 b(and)f Fo(C)2435 2286 y Fn(2)2501 2272 y Fp(=)25 b Fo(C)7 b(L)2731 2233 y Fl(\034)h Fn(+1)2731 2296 y(1)2894 2272 y Fp(are)32 b(p)s(ositiv)m(e)d(constan)m (ts)-24 2385 y(as)36 b(in)f(Theorem)g(1.1.)59 b(Denote)37 b(b)m(y)f Fo(p)e Fp(:)h Fm(R)1460 2352 y Fl(n)1542 2385 y Fk(!)f Fm(T)1740 2352 y Fl(n)1823 2385 y Fp(the)i(natural)f(pro)5 b(jection.)57 b(As)36 b(in)e(Lemma)i(2.2)h(,)h([10)q(],)-24 2497 y(w)m(e)32 b(shall)e(\014nd)f Fo( )i Fk(2)c(G)744 2464 y Fl(\032;\032)836 2441 y Fd(0)863 2497 y Fp(\()p Fm(R)976 2464 y Fl(n)1044 2497 y Fk(\002)21 b Fp(\012\))31 b(and)g Fo(R)d Fk(2)f(G)1690 2464 y Fl(\032)1726 2441 y Fd(0)1753 2497 y Fp(\(\012\))32 b(suc)m(h)f(that)h Fo(Q)p Fp(\()p Fo(x;)15 b(!)s Fp(\))28 b(:=)f Fo( )s Fp(\()p Fo(x;)15 b(!)s Fp(\))22 b Fk(\000)f(h)p Fo(x;)15 b(R)q Fp(\()p Fo(!)s Fp(\))p Fk(i)32 b Fp(is)f(2)p Fo(\031)-24 2610 y Fp(p)s(erio)s(dic)d(with)h(resp)s(ect)h(to)h Fo(x)f Fp(and)62 2790 y(\(i\))45 b Fk(8)p Fp(\()p Fo(x;)15 b(!)s Fp(\))26 b Fk(2)f Fm(R)666 2757 y Fl(n)733 2790 y Fk(\002)20 b Fp(\012)890 2804 y Fl(\024)935 2790 y Fp(,)30 b Fk(r)1066 2804 y Fl(x)1110 2790 y Fo( )s Fp(\()p Fo(x;)15 b(!)s Fp(\))26 b(=)f Fo(F)13 b Fp(\()p Fo(p)p Fp(\()p Fo(x)p Fp(\))p Fo(;)i(!)s Fp(\),)36 2990 y(\(ii\))203 2893 y Ff(\014)203 2943 y(\014)203 2992 y(\014)231 2990 y Fo(@)284 2957 y Fl(\013)279 3013 y(x)334 2990 y Fo(@)387 2957 y Fl(\014)382 3013 y(!)434 2990 y Fo(Q)p Fp(\()p Fo(x;)g(!)s Fp(\))728 2893 y Ff(\014)728 2943 y(\014)728 2992 y(\014)792 2990 y Fp(+)898 2893 y Ff(\014)898 2943 y(\014)898 2992 y(\014)926 2990 y Fo(@)979 2946 y Fl(\014)974 3018 y(I)1026 2990 y Fp(\()p Fo(R)q Fp(\()p Fo(!)s Fp(\))21 b Fk(\000)f(r)p Fo(g)1495 2957 y Fn(0)1535 2990 y Fp(\()p Fo(!)s Fp(\)\))1700 2893 y Ff(\014)1700 2943 y(\014)1700 2992 y(\014)1784 2990 y Fk(\024)55 b Fo(\024AC)2102 2942 y Fi(j)p Fl(\013)p Fi(j)2095 3015 y Fn(1)2191 2990 y Fp(\()p Fo(C)2291 3004 y Fn(2)2331 2990 y Fo(\024)2383 2957 y Fi(\000)p Fn(1)2477 2990 y Fp(\))2512 2957 y Fi(j)p Fl(\014)s Fi(j)2599 2990 y Fo(\013)p Fp(!)2697 2957 y Fl(\032)2738 2990 y Fo(\014)5 b Fp(!)2834 2957 y Fl(\032)p Fn(\()p Fl(\034)j Fn(+1\)+1)3164 2990 y Fo(L)3226 2942 y Fl(N)q(=)p Fn(2)3226 3015 y(1)3358 2932 y Fk(p)p 3434 2932 V 58 x Fo(")3476 3004 y Fl(H)3544 2990 y Fo(;)-24 3186 y Fp(for)30 b(\()p Fo(x;)15 b(!)s Fp(\))26 b Fk(2)f Fm(R)527 3153 y Fl(n)595 3186 y Fk(\002)20 b Fp(\012.)40 b(T)-8 b(o)31 b(obtain)f Fo( )j Fp(w)m(e)e(consider)e (the)i(function)623 3402 y Ff(e)604 3430 y Fo( )t Fp(\()p Fo(x;)15 b(!)s Fp(\))56 b(=)1072 3315 y Ff(Z)1118 3504 y Fl(\015)1154 3512 y Fg(x)1228 3430 y Fo(\033)i Fp(=)1465 3315 y Ff(Z)1548 3341 y Fn(1)1511 3504 y(0)1602 3430 y Fk(h)p Fo(F)13 b Fp(\()p Fo(p)p Fp(\()p Fo(tx)p Fp(\))p Fo(;)i(!)s Fp(\))p Fo(;)32 b(x)p Fk(i)15 b Fo(dt;)107 b Fp(\()p 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b(Remark)h(A.1)h(w)m(e)f(sho)m(w) g(that)-24 1550 y(the)c(in)m(v)m(erse)h(map)f Fo(D)e Fk(3)d Fo(I)32 b Fk(7!)25 b Fo(!)s Fp(\()p Fo(I)7 b Fp(\))26 b Fk(2)f Fp(\012)30 b(is)f(in)g Fk(G)1654 1517 y Fl(\032)1690 1494 y Fd(0)1748 1550 y Fp(and)767 1673 y Ff(\014)767 1723 y(\014)767 1773 y(\014)794 1770 y Fo(@)847 1733 y Fl(\013)912 1676 y Ff(\020)962 1770 y Fo(!)s Fp(\()p Fo(I)7 b Fp(\))21 b Fk(\000)f(r)p Fo(H)1410 1733 y Fn(0)1449 1770 y Fp(\()p Fo(I)7 b Fp(\))1566 1676 y Ff(\021)1616 1673 y(\014)1616 1723 y(\014)1616 1773 y(\014)1699 1770 y Fk(\024)56 b Fo(\024A)15 b Fp(\()p Fo(C)2061 1784 y Fn(2)2101 1770 y Fo(\024)2153 1733 y Fi(\000)p Fn(1)2248 1770 y Fp(\))2283 1733 y Fi(j)p Fl(\013)p Fi(j)2387 1770 y Fo(\013)p Fp(!)2482 1733 y Fl(\032)2518 1710 y Fd(0)2545 1770 y Fo(L)2607 1723 y Fl(N)q(=)p Fn(2)2607 1795 y(1)2739 1708 y Fk(p)p 2815 1708 110 4 v 62 x Fo(")2857 1784 y Fl(H)2939 1770 y Fo(;)-24 1990 y Fp(uniformly)38 b(with)j(resp)s(ect)g (to)i(\()p Fo(';)15 b(!)s Fp(\))45 b Fk(2)e Fm(T)1516 1957 y Fl(n)1591 1990 y Fk(\002)27 b Fp(\012)41 b(and)g(for)g(an)m(y)h Fo(\013;)15 b(\014)50 b Fk(2)44 b Fm(N)2703 1957 y Fl(n)2749 1990 y Fp(.)75 b(No)m(w)42 b(w)m(e)g(set)g(\010\()p Fo(x;)15 b(I)7 b Fp(\))45 b(=)-24 2103 y Fo( )s Fp(\()p Fo(x;)15 b(!)s Fp(\()p Fo(I)7 b Fp(\)\).)42 b(Then)27 b(\010\()p Fo(x;)15 b(I)7 b Fp(\))16 b Fk(\000)f(h)p Fo(x;)g(I)7 b Fk(i)29 b Fp(is)e(2)p Fo(\031)s Fp(-p)s(erio)s(dic)f(with)h(resp)s (ect)h(to)h Fo(x)p Fp(,)f(and)g(using)e(Prop)s(osition)g(A.3)j(w)m(e) -24 2216 y(get)i(the)g(estimates)482 2334 y Ff(\014)482 2384 y(\014)482 2433 y(\014)510 2431 y Fo(@)563 2394 y Fl(\013)558 2454 y(x)613 2431 y Fo(@)666 2387 y Fl(\014)661 2458 y(I)713 2431 y Fp(\(\010\()p Fo(x;)15 b(I)7 b Fp(\))21 b Fk(\000)f(h)p Fo(x;)15 b(I)7 b Fk(i)p Fp(\))1379 2334 y Ff(\014)1379 2384 y(\014)1379 2433 y(\014)1464 2431 y Fk(\024)55 b Fo(\024AC)1782 2383 y Fi(j)p Fl(\013)p Fi(j)1775 2456 y Fn(1)1871 2431 y Fp(\()p Fo(C)1971 2445 y Fn(2)2011 2431 y Fo(\024)2063 2394 y 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y Fp(=)1403 4976 y Ff(X)1381 5165 y Fi(j)p Fl(p)p Fi(j\025)p Fn(2)1608 4976 y Ff(X)1561 5160 y Fl(\013)p Fi(2)p Fh(N)1716 5141 y Fg(m)1816 4996 y Fo(')1875 4963 y Fl(p;\013)p 1816 5036 165 4 v 1821 5120 a Fo(p)p Fp(!)p Fo(\013)p Fp(!)2005 4888 y Ff(0)2005 5038 y(@)2078 5057 y Fo(y)23 b Fp(+)2258 4976 y Ff(X)2236 5165 y Fi(j)p Fl(q)r Fi(j\025)p Fn(2)2461 4976 y Ff(X)2415 5161 y Fl(\014)s Fi(2)p Fh(N)2568 5142 y Fg(m)2667 4996 y Fo(u)2719 4963 y Fl(q)r(;\014)p 2667 5036 153 4 v 2668 5120 a Fo(q)s Fp(!)p Fo(\014)5 b Fp(!)2830 5057 y Fo(y)2878 5020 y Fl(q)2916 5057 y Fo(!)2976 5020 y Fl(\014)3023 4888 y Ff(1)3023 5038 y(A)3095 4911 y Fl(p)3150 5057 y Fo(!)3210 5020 y Fl(\013)3260 5057 y Fo(:)-24 5352 y Fp(Denote)32 b(b)m(y)e Fo(v)s Fp(\()p Fo(y)s(;)15 b(!)s Fp(\))31 b(the)g(p)s(o)m(w)m(er)f(series)1261 5592 y Fo(v)s Fp(\()p Fo(y)s(;)15 b(!)s Fp(\))27 b(=)1671 5511 y Ff(X)1649 5699 y Fi(j)p Fl(p)p Fi(j\025)p Fn(2)1876 5511 y Ff(X)1829 5695 y Fl(\013)p Fi(2)p Fh(N)1984 5676 y Fg(m)2085 5530 y Fo(v)2132 5497 y Fl(p;\013)p 2083 5571 155 4 v 2083 5654 a Fo(p)p Fp(!)p Fo(\013)p Fp(!)2248 5592 y Fo(y)2296 5554 y Fl(p)2335 5592 y Fo(!)2395 5554 y Fl(\013)2445 5592 y Fo(;)1820 5927 y Fp(25)p eop %%Page: 26 26 26 25 bop -24 573 a Fp(and)30 b(supp)s(ose)f(that)i(it)e(is)h(a)h (formal)e(solution)g(of)711 822 y Fo(v)s Fp(\()p Fo(y)s(;)15 b(!)s Fp(\))26 b(=)f Fo("A)1246 741 y Ff(X)1224 929 y Fi(j)p Fl(p)p Fi(j\025)p Fn(2)1451 741 y Ff(X)1404 925 y Fl(\013)p Fi(2)p Fh(N)1559 906 y Fg(m)1659 752 y Fo(M)1747 771 y Fi(j)p Fl(p)p Fi(j)p 1659 801 168 4 v 1707 884 a Fo(p)p Fp(!)1846 752 y Fo(N)1919 771 y Fi(j)p Fl(\013)p Fi(j)p 1846 801 162 4 v 1885 884 a Fo(\013)p Fp(!)2033 822 y Fo(h)2085 784 y Fi(j)p Fl(p)p Fi(j)2164 822 y Fo(h)2216 784 y Fi(j)p Fl(\013)p Fi(j)2320 822 y Fp(\()p Fo(y)e Fp(+)d Fo(v)s Fp(\()p Fo(y)s(;)15 b(!)s Fp(\)\))2814 784 y Fl(p)2871 822 y Fo(!)2931 784 y Fl(\013)2995 822 y Fo(:)526 b Fp(\(A.7\))-24 1117 y(W)-8 b(e)32 b(claim)e(that)h(suc)m (h)f(a)i(solution)d(exists)h(and)g(it)h(is)f(unique,)f Fo(v)2179 1084 y Fl(p;\013)2310 1117 y Fo(>)c Fp(0,)32 b(and)e Fo(u)c Fk(\034)g Fo(y)d Fp(+)d Fo(v)s Fp(.)42 b(Indeed,)30 b(for)h(an)m(y)-24 1230 y Fk(j)p Fo(p)p Fk(j)25 b(\025)g Fp(2,)30 b Fo(\013)c Fk(2)f Fm(N)545 1197 y Fl(m)611 1230 y Fp(,)k(and)g(1)c Fk(\024)g Fo(j)31 b Fk(\024)25 b Fo(n)p Fp(,)k(there)g(is)f(a)h(p)s(olynomial)d Fo(Q)2215 1244 y Fl(p;\013;j)2397 1230 y Fk(6\021)f Fp(0)k(of)g(the)g (v)-5 b(ariables)27 b(\()p Fo(t)3267 1245 y Fl(q)r(;\014)3368 1230 y Fo(;)15 b(s)3451 1244 y Fl(r)n(;\015)3545 1230 y Fp(\))29 b(and)-24 1343 y(with)21 b(non-negativ)m(e)j(co)s(e\016cien) m(ts,)h(where)e Fo(q)s(;)15 b(r)28 b Fk(2)d Fm(N)1759 1310 y Fl(n)1805 1343 y Fp(,)g Fo(\014)5 b(;)15 b(\015)31 b Fk(2)25 b Fm(N)2197 1310 y Fl(m)2263 1343 y Fp(,)g(2)g Fk(\024)g(j)p Fo(q)s Fk(j)h(\024)f(j)p Fo(p)p Fk(j)p Fp(,)f Fo(\014)31 b Fk(\024)25 b Fo(\013)p Fp(,)g(2)h Fk(\024)f(j)p Fo(r)s Fk(j)g(\024)g(j)p Fo(p)p Fk(j)5 b(\000)g Fp(1,)-24 1455 y Fo(\015)37 b Fk(\024)31 b Fo(\013)p Fp(,)36 b(and)d(suc)m(h)h(that)h Fo(u)923 1411 y Fl(p;\013)923 1481 y(j)1028 1455 y Fp(,)g(resp)s(ectiv)m(ely)-8 b(,)35 b Fo(v)1653 1411 y Fl(p;\013)1650 1481 y(j)1758 1455 y Fp(,)h(is)d(the)h(v)-5 b(alue)34 b(of)g Fo(Q)2492 1469 y Fl(p;\013;j)2683 1455 y Fp(for)g Fo(t)2859 1470 y Fl(q)r(;\014)2991 1455 y Fp(=)d Fo(')3152 1422 y Fl(q)r(;\014)3253 1455 y Fp(,)36 b Fo(s)3357 1469 y Fl(r)n(;\015)3482 1455 y Fp(=)31 b Fo(u)3636 1411 y Fl(r)n(;\015)3636 1481 y(j)3730 1455 y Fp(,)-24 1585 y(resp)s(ectiv)m(ely)-8 b(,)44 b(for)d Fo(t)686 1600 y Fl(q)r(;\014)831 1585 y Fp(=)i Fo("AM)1143 1604 y Fi(j)p Fl(p)p Fi(j)1223 1585 y Fo(N)1296 1604 y Fi(j)p Fl(\013)p Fi(j)1385 1585 y Fo(h)1437 1552 y Fi(j)p Fl(p)p Fi(j)p Fn(+)p Fi(j)p Fl(\013)p Fi(j)1655 1585 y Fp(,)i Fo(s)1768 1599 y Fl(r)n(;\015)1905 1585 y Fp(=)f Fo(v)2067 1541 y Fl(r)n(;\015)2064 1611 y(j)2161 1585 y Fp(.)74 b(F)-8 b(or)42 b Fk(j)p Fo(p)p Fk(j)i Fp(=)g(2)e(the)g(p)s(olynomial)c Fo(Q)3496 1599 y Fl(p;\013;j)3694 1585 y Fp(is)-24 1698 y(indep)s(enden)m(t)25 b(of)i Fo(s)631 1712 y Fl(r)n(;\015)724 1698 y Fp(,)h Fk(j)p Fo(r)s Fk(j)d(\025)g Fp(2.)40 b(Since)26 b(the)h(co)s(e\016cien)m(ts)h(of)f Fo(Q)2118 1712 y Fl(p;\013;j)2302 1698 y Fp(are)g(non-negativ)m(e,)i(w) m(e)e(get)h Fk(j)p Fo(u)3363 1654 y Fl(r)n(;\013)3363 1724 y(j)3463 1698 y Fk(j)d(\024)g Fo(v)3656 1654 y Fl(r)n(;\013)3653 1724 y(j)-24 1811 y Fp(for)30 b Fk(j)p Fo(r)s Fk(j)25 b Fp(=)g(2)31 b(and)f(eac)m(h)h Fo(\013)p Fp(.)41 b(By)31 b(recurrence)f(w)m(e)h(obtain)f Fk(j)p Fo(u)1988 1767 y Fl(r)n(;\013)1988 1837 y(j)2087 1811 y Fk(j)c(\024)f Fo(v)2281 1767 y Fl(r)n(;\013)2278 1837 y(j)2410 1811 y Fp(for)30 b(eac)m(h)i Fk(j)p Fo(r)s Fk(j)25 b(\025)g Fp(2)31 b(and)f Fo(\013)c Fk(2)e Fm(N)3473 1778 y Fl(m)3540 1811 y Fp(.)117 1924 y(No)m(w)31 b(w)m(e)g(set)g Fo(y)d Fp(=)d(\()p Fo(s;)15 b(:)g(:)g(:)h(;)f(s)p Fp(\),)31 b Fo(s)25 b(>)g Fp(0,)31 b(and)f(consider)902 2189 y Fo(\036)p Fp(\()p Fo(s;)15 b(!)s Fp(\))26 b(=)1318 2083 y Fi(1)1293 2108 y Ff(X)1291 2289 y Fl(r)r Fn(=2)1477 2108 y Ff(X)1430 2292 y Fl(\013)p Fi(2)p Fh(N)1585 2273 y Fg(m)1685 2128 y Fo(\036)1739 2095 y Fl(r)n(;\013)p 1685 2168 154 4 v 1685 2251 a Fo(r)s Fp(!)p Fo(\013)p Fp(!)1848 2189 y Fo(s)1891 2152 y Fl(r)1928 2189 y Fo(!)1988 2152 y Fl(\013)2038 2189 y Fo(;)45 b(\036)2162 2152 y Fl(r)n(;\013)2287 2189 y Fp(=)2404 2108 y Ff(X)2383 2296 y Fi(j)p Fl(p)p Fi(j)p Fn(=)p Fl(r)2561 2189 y Fo(v)2608 2152 y Fl(p;\013)2724 2128 y Fo(r)s Fp(!)p 2723 2168 71 4 v 2723 2251 a Fo(p)p Fp(!)2804 2189 y Fo(:)-24 2489 y Fp(Note)25 b(that)g(for)e(suc)m(h)h Fo(y)j Fp(the)d(righ)m(t)f(hand)g (side)g(of)h(the)g(equation)g(\(A.7\))h(is)e(in)m(v)-5 b(arian)m(t)24 b(under)e(an)m(y)i(p)s(erm)m(utation)-24 2602 y(of)39 b(the)g(comp)s(onen)m(ts)h Fo(v)807 2616 y Fl(j)843 2602 y Fp(\()p Fo(y)s(;)15 b(!)s Fp(\),)43 b(1)d Fk(\024)f Fo(j)46 b Fk(\024)39 b Fo(n)p Fp(,)i(and)e(using)e(the) j(uniqueness)d(of)i(the)g(solution,)h(w)m(e)g(obtain)-24 2715 y Fo(\036)30 2729 y Fn(1)69 2715 y Fp(\()p Fo(s;)15 b(!)s Fp(\))26 b(=)f Fk(\001)15 b(\001)g(\001)27 b Fp(=)e Fo(\036)686 2729 y Fl(n)733 2715 y Fp(\()p Fo(s;)15 b(!)s Fp(\))25 b(=)1079 2691 y(~)1067 2715 y Fo(\036)q Fp(\()p Fo(s;)15 b(!)s Fp(\).)41 b(Then)29 b Fo(t)c Fp(=)g Fo(s)20 b Fp(+)1958 2691 y(~)1946 2715 y Fo(\036)p Fp(\()p Fo(s;)15 b(!)s Fp(\))31 b(is)e(a)i(formal)f(solution)f(of)917 2980 y Fo(t)d Fp(=)e Fo(s)c Fp(+)g Fo("A)1378 2874 y Fi(1)1353 2899 y Ff(X)1351 3080 y Fl(r)r Fn(=2)1537 2899 y Ff(X)1490 3083 y Fl(\013)p Fi(2)p Fh(N)1645 3064 y Fg(m)1757 2899 y Ff(X)1735 3087 y Fi(j)p Fl(p)p Fi(j)p Fn(=)p Fl(r)1924 2918 y Fo(r)s Fp(!)p 1923 2959 V 1923 3042 a Fo(p)p Fp(!)2029 2918 y Fo(M)2117 2932 y Fl(r)p 2029 2959 127 4 v 2058 3042 a Fo(r)s Fp(!)2175 2911 y Fo(N)2248 2929 y Fi(j)p Fl(\013)p Fi(j)p 2175 2959 162 4 v 2214 3042 a Fo(\013)p Fp(!)2347 2980 y(\()p Fo(ht)p Fp(\))2502 2942 y Fl(r)2541 2980 y Fp(\()p Fo(h!)s Fp(\))2723 2942 y Fl(\013)2789 2980 y Fo(;)-24 3299 y Fp(where)239 3234 y Ff(P)326 3322 y Fi(j)p Fl(p)p Fi(j)p Fn(=)p Fl(r)520 3263 y(r)r Fn(!)p 519 3278 56 4 v 519 3330 a Fl(p)p Fn(!)609 3299 y Fk(\024)25 b Fo(n)760 3266 y Fl(r)798 3299 y Fp(.)41 b(Let)30 b Fo(t)c Fp(=)e Fo( )s Fp(\()p Fo(s;)15 b(!)s Fp(\))27 b(:=)e Fo(s)20 b Fp(+)1757 3234 y Ff(P)1844 3261 y Fi(1)1844 3322 y Fl(r)r Fn(=2)1988 3234 y Ff(P)2075 3322 y Fl(\013)p Fi(2)p Fh(N)2230 3303 y Fg(m)2334 3258 y Fl( )2382 3235 y Fg(r)n(;\013)p 2334 3278 139 4 v 2344 3330 a Fl(r)r Fn(!)p Fl(\013)p Fn(!)2483 3299 y Fo(s)2526 3266 y Fl(r)2564 3299 y Fo(!)2624 3266 y Fl(\013)2703 3299 y Fp(b)s(e)30 b(the)g(formal)g(solution)f(of)1025 3585 y Fo(t)c Fp(=)g Fo(s)20 b Fp(+)g Fo("A)1485 3479 y Fi(1)1460 3504 y Ff(X)1458 3685 y Fl(r)r Fn(=2)1660 3504 y Ff(X)1613 3688 y Fl(\013)p Fi(2)p Fh(N)1768 3669 y Fg(m)1867 3524 y Fo(M)1955 3538 y Fl(r)p 1867 3564 127 4 v 1896 3647 a Fo(r)s Fp(!)2013 3516 y Fo(N)2086 3534 y Fi(j)p Fl(\013)p Fi(j)p 2013 3564 162 4 v 2052 3647 a Fo(\013)p Fp(!)2185 3585 y(\()p Fo(hnt)p Fp(\))2395 3548 y Fl(r)2434 3585 y Fp(\()p Fo(h!)s Fp(\))2616 3548 y Fl(\013)2681 3585 y Fo(:)840 b Fp(\(A.8\))-24 3885 y(W)-8 b(e)33 b(obtain)e(as)h(ab)s(o)m(v)m(e)h Fo(s)20 b Fp(+)957 3861 y(~)945 3885 y Fo(\036)27 b Fk(\034)h Fo( )s Fp(,)k(whic)m(h)f(implies)e Fk(j)p Fo(u)1913 3841 y Fl(p;\013)1913 3911 y(j)2018 3885 y Fk(j)e(\024)g Fo( )2230 3852 y Fi(j)p Fl(p)p Fi(j)p Fl(;\013)2406 3885 y Fp(for)k(1)d Fk(\024)f Fo(j)33 b Fk(\024)27 b Fo(n)32 b Fp(and)f(for)g(an)m(y)h Fk(j)p Fo(p)p Fk(j)c(\025)f Fp(2,)-24 3998 y Fo(\013)f Fk(2)e Fm(N)227 3965 y Fl(m)294 3998 y Fp(.)117 4111 y(W)-8 b(e)32 b(are)e(going)h(to)g(estimate)g Fo( )1205 4078 y Fl(r)n(;\013)1304 4111 y Fp(.)41 b(Set)31 b Fo(C)7 b Fp(\()p Fo(!)s Fp(\))25 b(=)g Fo(A)1948 4047 y Ff(P)2036 4134 y Fl(\013)p Fi(2)p Fh(N)2191 4115 y Fg(m)2295 4056 y Fl(N)2351 4070 y Fd(j)p Fg(\013)p Fd(j)p 2295 4090 139 4 v 2331 4142 a Fl(\013)p Fn(!)2443 4111 y Fp(\()p Fo(h!)s Fp(\))2625 4078 y Fl(\013)2675 4111 y Fp(.)41 b(Then)29 b(\(A.8\))j(b)s(ecomes)1077 4384 y Fo(s)25 b Fp(=)g Fo(k)s Fp(\()p Fo(t;)15 b(!)s Fp(\))26 b(:=)f Fo(t)20 b Fk(\000)g Fo("C)7 b Fp(\()p Fo(!)s Fp(\))2072 4279 y Fi(1)2047 4304 y Ff(X)2045 4485 y Fl(r)r Fn(=2)2209 4323 y Fo(M)2297 4337 y Fl(r)p 2209 4363 127 4 v 2238 4447 a Fo(r)s Fp(!)2345 4384 y(\()p Fo(hnt)p Fp(\)\))2590 4347 y Fl(r)2629 4384 y Fo(:)-24 4661 y Fp(Recall)33 b(that)i(0)c Fo(<)g(")g Fk(\024)g Fp(1.)51 b(W)-8 b(e)35 b(supp)s(ose)d(also)i(that)g Fo(A)e Fk(\025)e Fp(1)k(and)g Fo(h)d Fk(\025)g Fp(1.)51 b(As)34 b(in)e([3)q(],)j(using)d(the)i (Lagrange)-24 4774 y(expansion)29 b(theorem,)i(w)m(e)g(obtain)f(for)g Fo(r)e Fk(\025)d Fp(2)307 5038 y Fo( )369 5005 y Fl(r)n(;\013)551 5038 y Fp(=)750 4894 y Ff(")799 5038 y Fo(@)852 5001 y Fl(\013)847 5061 y(!)917 4894 y Ff(\()983 4919 y(\022)1071 4977 y Fo(d)p 1054 5017 81 4 v 1054 5101 a(dt)1145 4919 y Ff(\023)1206 4940 y Fl(r)r Fi(\000)p Fn(1)1349 4919 y Ff(\022)1531 4977 y Fo(t)p 1420 5017 254 4 v 1420 5101 a(k)s Fp(\()p Fo(t;)15 b(!)s Fp(\))1684 4919 y Ff(\023)1745 4942 y Fl(r)1783 4894 y Ff(\))1865 5038 y Fk(j)1890 5052 y Fl(t)p Fn(=0)2010 4894 y Ff(#)2074 5038 y Fk(j)2099 5052 y Fl(!)r Fn(=0)551 5377 y Fk(\024)83 b Fo(")762 5208 y Ff(2)762 5358 y(4)818 5377 y Fo(@)871 5340 y Fl(\013)866 5400 y(!)936 5205 y Ff(8)936 5280 y(<)936 5429 y(:)1009 5258 y(\022)1097 5316 y Fo(d)p 1080 5356 81 4 v 1080 5440 a(dt)1171 5258 y Ff(\023)1232 5279 y Fl(r)r Fi(\000)p Fn(1)1375 5208 y Ff(0)1375 5358 y(@)1474 5272 y Fi(1)1450 5296 y Ff(X)1448 5478 y Fl(q)r Fn(=0)1587 5208 y Ff(0)1587 5358 y(@)1659 5377 y Fo(C)7 b Fp(\()p Fo(!)s Fp(\))p Fo(hn)1984 5272 y Fl(r)r Fi(\000)p Fn(1)1986 5296 y Ff(X)1984 5478 y Fl(p)p Fn(=1)2174 5316 y Fo(M)2262 5330 y Fl(p)p Fn(+1)p 2134 5356 299 4 v 2134 5440 a Fp(\()p Fo(p)21 b Fp(+)f(1\)!)2443 5377 y(\()p Fo(hnt)p Fp(\))2653 5340 y Fl(p)2693 5208 y Ff(1)2693 5358 y(A)2765 5231 y Fl(q)2803 5208 y Ff(1)2803 5358 y(A)2876 5229 y Fl(r)2914 5205 y Ff(9)2914 5280 y(=)2914 5429 y(;)3003 5377 y Fk(j)3028 5391 y Fl(t)p Fn(=0)3148 5208 y Ff(3)3148 5358 y(5)3219 5377 y Fk(j)3244 5391 y Fl(!)r Fn(=0)3400 5377 y Fo(;)1820 5927 y Fp(26)p eop %%Page: 27 27 27 26 bop -24 573 a Fp(and)30 b(w)m(e)g(get)483 877 y Fo( )545 844 y Fl(r)n(;\013)727 877 y Fk(\024)83 b Fo("M)1011 891 y Fl(r)1065 877 y Fp(\(4)p Fo(H)7 b(hn)p Fp(\))1370 840 y Fl(r)r Fi(\000)p Fn(1)1529 683 y Ff(2)1529 830 y(6)1529 883 y(4)1585 877 y Fo(@)1638 840 y Fl(\013)1633 900 y(!)1703 708 y Ff(0)1703 858 y(@)1775 877 y Fp(1)21 b(+)f Fo(Ahn)2134 796 y Ff(X)2122 981 y Fl(\014)s Fi(\024)p Fl(\013)2306 808 y Fo(N)2379 826 y Fi(j)p Fl(\014)s Fi(j)p 2306 856 160 4 v 2319 940 a Fk(j)p Fo(\014)5 b Fk(j)p Fp(!)2475 877 y(\()p Fo(h!)s Fp(\))2657 840 y Fl(\014)2705 708 y Ff(1)2705 858 y(A)2778 731 y Fl(r)r Fi(\000)p Fn(1)2906 683 y Ff(3)2906 830 y(7)2906 883 y(5)2976 877 y Fk(j)3001 891 y Fl(!)r Fn(=0)727 1236 y Fk(\024)83 b Fo("M)1011 1250 y Fl(r)1050 1236 y Fp(\(8)p Fo(AH)7 b(h)1333 1199 y Fn(2)1373 1236 y Fo(n)1428 1199 y Fn(2)1467 1236 y Fp(\))1502 1199 y Fl(r)r Fi(\000)p Fn(1)1646 1043 y Ff(2)1646 1189 y(6)1646 1242 y(4)1701 1236 y Fo(@)1754 1199 y Fl(\013)1749 1259 y(!)1819 1068 y Ff(0)1819 1217 y(@)1892 1156 y(X)1930 1340 y Fl(\014)2042 1092 y Ff( )2108 1117 y(\022)2179 1167 y Fo(N)2252 1185 y Fi(j)p Fl(\013)p Fi(j)p 2179 1215 162 4 v 2193 1299 a Fk(j)p Fo(\013)p Fk(j)p Fp(!)2351 1117 y Ff(\023)2412 1131 y Fn(1)p Fl(=)p Fi(j)p Fl(\013)p Fi(j)2586 1236 y Fo(h!)2698 1092 y Ff(!)2764 1108 y Fl(\014)2811 1068 y Ff(1)2811 1217 y(A)2884 1088 y Fl(r)r Fi(\000)p Fn(1)3012 1043 y Ff(3)3012 1189 y(7)3012 1242 y(5)3083 1236 y Fk(j)3108 1250 y Fl(!)r Fn(=0)727 1501 y Fk(\024)83 b Fo("M)1011 1515 y Fl(r)1050 1501 y Fo(N)1123 1520 y Fi(j)p Fl(\013)p Fi(j)1211 1501 y Fp(\(8)p Fo(AH)7 b(h)1494 1464 y Fn(2)1535 1501 y Fo(n)1590 1464 y Fn(2)1629 1501 y Fp(\))1664 1464 y Fl(r)r Fi(\000)p Fn(1)1793 1501 y Fo(h)1845 1464 y Fi(j)p Fl(\013)p Fi(j)1934 1501 y Fp(2)1979 1464 y Fl(m)p Fn(\()p Fl(r)r Fi(\000)p Fn(2\)+)p Fi(j)p Fl(\013)p Fi(j)2364 1501 y Fo(:)-24 1703 y Fp(W)-8 b(e)31 b(ha)m(v)m(e)h(used)e(the)g(inequalities)e(\(A.6\))k(for)e Fo(M)1651 1717 y Fl(j)1718 1703 y Fp(and)g Fo(N)1968 1717 y Fl(j)2004 1703 y Fp(,)h(as)g(w)m(ell)e(as)h(the)h(iden)m(tit)m (y)1299 1825 y Ff(0)1299 1974 y(@)1418 1913 y(X)1372 2097 y Fl(\014)s Fi(2)p Fh(N)1525 2079 y Fg(m)1599 1994 y Fo(!)1659 1956 y Fl(\014)1706 1825 y Ff(1)1706 1974 y(A)1779 1848 y Fl(q)1842 1994 y Fp(=)1985 1913 y Ff(X)1938 2097 y Fl(\013)p Fi(2)p Fh(N)2093 2078 y Fg(m)2183 1994 y Fo(C)2248 2008 y Fl(\013)2297 1994 y Fo(!)2357 1956 y Fl(\013)2407 1994 y Fo(;)-24 2273 y Fp(where)f Fo(q)e Fp(=)d Fo(r)e Fk(\000)c Fp(1)26 b Fk(\025)f Fp(1)31 b(and)710 2553 y Fo(C)775 2567 y Fl(\013)850 2553 y Fp(=)946 2384 y Ff(0)946 2533 y(@)1060 2486 y Fo(q)23 b Fp(+)d Fo(\013)1273 2500 y Fn(1)1332 2486 y Fk(\000)g Fp(1)1215 2633 y Fo(\013)1273 2647 y Fn(1)1510 2384 y Ff(1)1510 2533 y(A)1598 2553 y Fk(\001)15 b(\001)g(\001)1719 2384 y Ff(0)1719 2533 y(@)1834 2486 y Fo(q)23 b Fp(+)c Fo(\013)2046 2500 y Fl(m)2133 2486 y Fk(\000)h Fp(1)1989 2633 y Fo(\013)2047 2647 y Fl(m)2311 2384 y Ff(1)2311 2533 y(A)2439 2553 y Fk(\024)56 b Fp(2)2611 2515 y Fl(m)p Fn(\()p Fl(q)r Fi(\000)p Fn(1\)+)p Fi(j)p Fl(\013)p Fi(j)2996 2553 y Fo(:)-24 2832 y Fp(Th)m(us)29 b(w)m(e)i(obtain)948 2945 y Fk(8)15 b Fp(\()p Fo(p;)g(\013;)g(j)5 b Fp(\))p Fo(;)108 b Fk(j)p Fo(u)1520 2901 y Fl(p;\013)1520 2971 y(j)1625 2945 y Fk(j)26 b(\024)f Fo( )1834 2908 y Fi(j)p Fl(p)p Fi(j)p Fl(;\013)2003 2945 y Fk(\024)g Fo("A)15 b(M)2312 2964 y Fi(j)p Fl(p)p Fi(j)2392 2945 y Fo(N)2465 2964 y Fi(j)p Fl(\013)p Fi(j)2554 2945 y Fo(h)2606 2908 y Fl(r)r Fn(+)p Fi(j)p Fl(\013)p Fi(j)-24 3107 y Fp(where)28 b Fo(A;)15 b(h)26 b(>)f Fp(0)j(dep)s(end)f(only)g(on)h Fo(\032)p Fp(,)h Fo(\032)1375 3074 y Fi(0)1399 3107 y Fp(,)f Fo(n)p Fp(,)h(and)f Fo(m)p Fp(.)39 b(Changing)27 b(bac)m(k)i(the)g(v)-5 b(ariables)27 b(and)g(rescaling)h(bac)m(k)-24 3220 y(\012,)i(w)m(e)h(obtain)f(the)g(desired)f(estimates)i(for)f Fo(g)s Fp(.)117 3333 y(Finally)k(w)m(e)i(recall)f(the)h(Gevrey)g (estimates)h(for)e(the)h(comp)s(osition)f(of)g(t)m(w)m(o)j(functions)c (in)g(anisotropic)-24 3446 y(Gevrey)d(classes.)-24 3625 y Fm(Prop)s(osition)36 b(A.)f(3)46 b Fe(L)-5 b(et)39 b Fo(g)h Fk(2)c(G)1184 3581 y Fl(\026)1179 3652 y(C)1229 3661 y Fj(1)1268 3625 y Fp(\(\012)p Fo(;)15 b(Y)21 b Fp(\))39 b Fe(and)g Fo(f)46 b Fk(2)36 b(G)1986 3581 y Fl(\032;\026)1981 3652 y(B)s(;C)2107 3661 y Fj(2)2147 3625 y Fp(\()p Fo(X)c Fk(\002)24 b Fo(Y)c Fp(\))p Fe(,)41 b(wher)-5 b(e)40 b Fo(\026)c Fk(\025)g Fo(\032)h Fk(\025)f Fp(1)p Fe(,)41 b Fo(B)5 b Fe(,)39 b Fo(C)3533 3639 y Fn(1)3611 3625 y Fe(and)-24 3738 y Fo(C)41 3752 y Fn(2)123 3738 y Fe(ar)-5 b(e)43 b(p)-5 b(ositive)44 b(c)-5 b(onstants)45 b(and)e Fo(X)7 b Fe(,)46 b Fo(Y)62 b Fe(and)44 b Fp(\012)e Fe(ar)-5 b(e)44 b(op)-5 b(en)44 b(sets)f(in)f Fm(R)2564 3705 y Fl(n)2612 3738 y Fe(.)71 b(Supp)-5 b(ose)45 b(that)f Fk(k)p 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4450 y Fo(\013)p Fp(!)p 2352 4491 383 4 v 2352 4574 a Fo(\013)2410 4548 y Fn(1)2450 4574 y Fp(!)15 b Fk(\001)g(\001)g(\001)i Fo(\013)2670 4548 y Fl(p)2710 4574 y Fp(!)2776 4512 y(\()p Fo(@)2864 4474 y Fl(\013)2909 4451 y Fj(1)2859 4534 y Fl(!)2948 4512 y Fo(g)s Fp(\)\()p Fo(!)s Fp(\))e Fk(\001)g(\001)g(\001) j Fp(\()p Fo(@)3385 4474 y Fl(\013)3430 4451 y Fg(p)3380 4534 y Fl(!)3471 4512 y Fo(g)s Fp(\)\()p Fo(!)s Fp(\))p Fo(:)-24 4894 y Fp(Since)29 b Fk(j)p Fo(@)291 4861 y Fl(\013)286 4916 y(!)341 4894 y Fo(g)s Fp(\()p Fo(!)s Fp(\))p Fk(j)e(\024)e Fo(A)733 4908 y Fn(1)773 4894 y Fo(C)845 4846 y Fi(j)p Fl(\013)p Fi(j)838 4919 y Fn(1)933 4894 y Fo(\013)p Fp(!)1028 4861 y Fl(\026)1075 4894 y Fp(,)30 b(and)g Fo(j)5 b Fp(!)26 b Fk(\024)f Fp(2)1541 4861 y Fl(j)1578 4894 y Fp(\()p Fo(j)h Fk(\000)20 b Fp(1\)!)32 b(for)e Fo(j)h Fk(\025)25 b Fp(1,)31 b(w)m(e)g(estimate)g(ab)s(o)m(v)m (e)g Fk(j)q Fo(@)3142 4861 y Fl(\015)3137 4916 y(x)3186 4894 y Fo(@)3239 4861 y Fl(\013)3234 4916 y(!)3289 4894 y 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Fp(1)3013 5701 y Fo(p)g Fk(\000)g Fp(1)3308 5526 y Ff(\023)3403 5645 y Fp(\(see)35 b([11)q(],)1820 5927 y(27)p eop %%Page: 28 28 28 27 bop -24 573 a Fp(Sect.)50 b(1.2\).)i(Then)32 b(the)i(n)m(um)m(b)s (er)e(of)i(partitions)e Fo(\013)1761 540 y Fn(1)1823 573 y Fp(+)22 b Fk(\001)15 b(\001)g(\001)23 b Fp(+)f Fo(\013)2195 540 y Fl(p)2265 573 y Fp(=)31 b Fo(\013)p Fp(,)j Fo(\013)2542 540 y Fl(j)2610 573 y Fk(2)c Fm(N)2783 540 y Fl(n)2830 573 y Fp(,)k(of)g Fo(\013)c Fp(=)g(\()p Fo(\013)3278 587 y Fn(1)3318 573 y Fo(;)15 b(:)g(:)g(:)i(;)e(\013)3578 587 y Fl(n)3626 573 y Fp(\))33 b(is)-24 686 y(giv)m(en)d(b)m(y)915 717 y Ff(\022)992 780 y Fo(\013)1050 794 y Fn(1)1110 780 y Fp(+)19 b Fo(p)h Fk(\000)g Fp(1)1096 893 y Fo(p)g Fk(\000)g Fp(1)1418 717 y Ff(\023)1494 836 y Fk(\001)15 b(\001)g(\001)1615 717 y Ff(\022)1692 780 y Fo(\013)1750 794 y Fl(n)1817 780 y Fp(+)20 b Fo(p)g Fk(\000)g Fp(1)1800 893 y Fo(p)g Fk(\000)g Fp(1)2125 717 y Ff(\023)2242 836 y Fk(\024)55 b Fp(2)2413 799 y Fi(j)p Fl(\013)p Fi(j)2502 836 y Fp(2)2547 799 y Fl(n)p Fn(\()p Fl(p)p Fi(\000)p Fn(1\))2791 836 y Fo(;)-24 1041 y Fp(whic)m(h)29 b(implies)-6 1259 y Fk(j)p Fo(@)72 1222 y Fl(\015)67 1282 y(x)117 1259 y Fo(@)170 1222 y Fl(\013)165 1282 y(!)219 1259 y Fo(F)13 b Fp(\()p Fo(x)p Fp(\))p Fk(j)27 b(\024)e Fo(A)628 1273 y Fn(2)667 1259 y Fp(2)712 1222 y Fi(\000)p Fl(n)815 1259 y Fo(B)889 1222 y Fi(j)p Fl(\015)t Fi(j)987 1190 y Ff(\000)1025 1259 y Fp(2)1070 1222 y Fl(n)p Fn(+)p Fl(\026)1215 1259 y Fo(n)1270 1222 y Fl(\026)1316 1259 y Fo(C)1381 1273 y Fn(1)1435 1259 y Fp(max)q(\(1)p Fo(;)15 b(A)1793 1273 y Fn(1)1833 1259 y Fo(C)1898 1273 y Fn(2)1938 1259 y Fp(\))1973 1190 y Ff(\001)2011 1208 y Fi(j)p Fl(\013)p Fi(j)2130 1259 y Fo(\015)5 b Fp(!)2222 1222 y Fl(\032)2278 1259 y Fo(\013)p Fp(!)2373 1222 y Fl(\026)2474 1179 y Ff(X)2435 1363 y Fl(\014)s Fi(2)p Fh(N)2588 1344 y Fg(n)2647 1259 y Fp(2)2692 1222 y Fi(\000)p Fl(n)p Fi(j)p Fl(\014)s Fi(j)2901 1259 y Fo(<)25 b(A)3065 1273 y Fn(2)3105 1259 y Fo(B)3179 1222 y Fi(j)p Fl(\015)t Fi(j)3262 1259 y Fo(C)3334 1222 y Fi(j)p Fl(\013)p Fi(j)3422 1259 y Fo(\015)5 b Fp(!)3514 1222 y Fl(\032)3570 1259 y Fo(\013)p Fp(!)3665 1222 y Fl(\026)3712 1259 y Fo(:)-24 1547 y Fp(In)30 b(the)g(same)h(w)m (a)m(y)g(w)m(e)g(obtain)-24 1735 y Fm(Prop)s(osition)36 b(A.)f(4)46 b Fe(L)-5 b(et)39 b Fo(g)h Fk(2)c(G)1184 1690 y Fl(\032;\026)1179 1762 y(B)1232 1771 y Fj(1)1267 1762 y Fl(;C)1337 1771 y Fj(1)1375 1735 y Fp(\()p Fo(X)d Fk(\002)24 b Fp(\012)p Fo(;)15 b(Y)20 b Fp(\))39 b Fe(and)h Fo(f)46 b Fk(2)36 b(G)2296 1690 y Fl(\032;\026)2291 1762 y(B)2344 1771 y Fj(2)2379 1762 y Fl(;C)2449 1771 y Fj(2)2487 1735 y Fp(\()p Fo(Y)45 b Fk(\002)25 b Fp(\012\))p Fe(,)40 b(wher)-5 b(e)40 b Fo(\026)c Fk(\025)g Fo(\032)g Fk(\025)g Fp(1)p Fe(,)41 b Fo(B)3719 1749 y Fl(j)-24 1848 y Fe(and)46 b Fo(C)230 1862 y Fl(j)266 1848 y Fe(,)i Fo(j)53 b Fp(=)47 b(1)p Fo(;)15 b Fp(2)p Fe(,)49 b(ar)-5 b(e)46 b(p)-5 b(ositive)46 b(c)-5 b(onstants,)50 b(and)c Fo(X)7 b Fe(,)48 b Fo(Y)20 b Fe(,)48 b(and)e Fp(\012)e Fe(ar)-5 b(e)46 b(op)-5 b(en)46 b(sets)f(in)g Fm(R)3288 1815 y Fl(n)3335 1848 y Fe(.)79 b(Supp)-5 b(ose)-24 1961 y(that)40 b Fk(k)p Fo(g)s Fk(k)304 1975 y Fl(B)357 1984 y Fj(1)393 1975 y Fl(;C)463 1984 y Fj(1)539 1961 y Fp(=)d Fo(A)715 1975 y Fn(1)793 1961 y Fe(and)j Fk(k)p Fo(f)10 b Fk(k)1121 1975 y Fl(B)1174 1984 y Fj(2)1209 1975 y Fl(;C)1279 1984 y Fj(2)1355 1961 y Fp(=)36 b Fo(A)1530 1975 y Fn(2)1609 1961 y Fe(in)j(the)g(c)-5 b(orr)g(esp)g(onding)42 b(Gevr)-5 b(ey)40 b(norms)g(with)g Fo(A)3335 1975 y Fn(1)3375 1961 y Fo(;)30 b(A)3498 1975 y Fn(2)3575 1961 y Fo(>)36 b Fp(0)p Fe(.)-24 2073 y(Then)k(the)g(c)-5 b(omp)g(osition)43 b Fp(\()p Fo(x;)15 b(!)s Fp(\))39 b Fk(7!)f Fo(F)13 b Fp(\()p Fo(x;)i(!)s Fp(\))39 b(:=)f Fo(f)10 b Fp(\()p Fo(g)s Fp(\()p Fo(x;)15 b(!)s Fp(\))p Fo(;)g(!)s Fp(\))42 b Fe(b)-5 b(elongs)40 b(to)g Fk(G)2769 2029 y Fl(\032;\026)2764 2101 y(B)s(;C)2900 2073 y Fp(\()p Fo(X)33 b Fk(\002)25 b Fp(\012\))p Fe(,)41 b(wher)-5 b(e)41 b Fo(B)i Fp(=)-24 2195 y(2)21 2162 y Fl(n)p Fn(+)p Fl(\032)159 2195 y Fp(\(2)p Fo(n)p Fp(\))329 2162 y Fl(\032)370 2195 y Fo(B)439 2209 y Fn(1)494 2195 y Fp(max\(1)p Fo(;)15 b(A)851 2209 y Fn(1)892 2195 y Fo(B)961 2209 y Fn(2)1000 2195 y Fp(\))p Fe(,)33 b Fo(C)f Fp(=)25 b Fo(C)1354 2209 y Fn(2)1413 2195 y Fp(+)20 b(2)1549 2162 y Fl(n)p Fn(+)p Fl(\032)1688 2195 y Fp(\(2)p Fo(n)p Fp(\))1858 2162 y Fl(\032)1899 2195 y Fo(C)1964 2209 y Fn(1)2018 2195 y Fp(max\(1)p Fo(;)15 b(A)2375 2209 y Fn(1)2416 2195 y Fo(B)2485 2209 y Fn(2)2525 2195 y Fp(\))p Fe(,)32 b(and)i Fk(k)p Fo(F)13 b Fk(k)2958 2209 y Fl(B)s(;C)3120 2195 y Fk(\024)24 b Fo(A)3283 2209 y Fn(2)3323 2195 y Fe(.)-24 2383 y Fp(T)-8 b(o)38 b(pro)m(v)m(e)g(the)g(Gevrey)h(estimates)f(w)m(e)g(set)g Fo(z)k Fp(=)37 b(\()p Fo(x;)15 b(!)s Fp(\))38 b(and)f(using)f(Leibnitz) h(form)m(ula)f(and)h(the)h(F)-8 b(aa)39 b(de)-24 2496 y(Bruno)30 b(form)m(ula)f(w)m(e)i(write)520 2749 y Fo(@)573 2716 y Fl(\015)568 2772 y(x)618 2749 y Fo(@)671 2716 y Fl(\013)666 2772 y(!)721 2749 y Fo(F)13 b Fp(\()p Fo(x;)i(!)s Fp(\))84 b(=)1275 2668 y Ff(X)1251 2853 y Fl(\013)1296 2834 y Fd(0)1319 2853 y Fi(\024)p Fl(\013)1434 2630 y Ff(\022)1522 2693 y Fo(\013)1511 2806 y(\013)1569 2773 y Fi(0)1608 2630 y Ff(\023)1884 2668 y(X)1684 2856 y Fn(1)p Fi(\024j)p Fl(\014)s Fi(j)p Fn(=)p Fl(p)p Fi(\024j)p Fl(\015)t Fn(+)p Fl(\013)2163 2838 y Fd(0)2184 2856 y Fi(j)2259 2679 y Fp(\()p Fo(@)2347 2646 y Fl(\014)2342 2702 y(y)2395 2679 y Fo(@)2448 2646 y Fl(\013)p Fi(\000)p Fl(\013)2593 2623 y Fd(0)2443 2702 y Fl(\021)2620 2679 y Fo(f)10 b Fp(\)\()p Fo(g)s Fp(\()p Fo(z)t Fp(\))p Fo(;)15 b(\021)s Fp(\))p Fk(j)3055 2693 y Fl(\021)r Fn(=)p Fl(!)p 2259 2728 943 4 v 2689 2811 a Fo(\014)5 b Fp(!)1098 3006 y Fk(\002)1486 2925 y Ff(X)1261 3117 y Fg(\016)1291 3096 y Fj(1)1326 3117 y(+)p Fd(\001\001\001)o Fj(+)p Fg(\016)1506 3096 y(p)1542 3117 y Fj(=)p Fg(\016)r Fj(=\()p Fg(\015)s(;\013)1784 3096 y Fd(0)1806 3117 y Fj(\))1317 3185 y Fd(j)p Fg(\016)1366 3171 y Fj(1)1400 3185 y Fd(j\025)p Fj(1)p Fg(;:::;)p Fd(j)p Fg(\016)1641 3170 y(p)1677 3185 y Fd(j\025)p Fj(1)2038 2945 y Fo(\016)s Fp(!)p 1895 2985 354 4 v 1895 3068 a Fo(\016)1938 3042 y Fn(1)1979 3068 y Fp(!)15 b Fk(\001)g(\001)g(\001)h Fo(\016)2183 3042 y Fl(p)2224 3068 y Fp(!)2289 3006 y(\()p Fo(@)2377 2968 y Fl(\016)2410 2945 y Fj(1)2372 3028 y Fl(z)2450 3006 y Fo(g)s Fp(\)\()p Fo(z)t Fp(\))f Fk(\001)g(\001)g(\001) k Fp(\()p Fo(@)2874 2968 y Fl(\013)2919 2945 y Fg(p)2869 3028 y Fl(z)2960 3006 y Fo(g)s Fp(\)\()p Fo(z)t Fp(\))r Fo(;)-24 3383 y Fp(where)30 b Fo(\016)282 3350 y Fl(j)344 3383 y Fp(=)25 b(\()p Fo(\015)527 3350 y Fl(j)564 3383 y Fo(;)15 b(\013)662 3350 y Fl(j)700 3383 y Fp(\).)41 b(On)29 b(the)i(other)f(hand,)490 3608 y(\()p Fo(\015)577 3575 y Fn(1)617 3608 y Fp(!)15 b Fk(\001)g(\001)g(\001)h Fo(\015)830 3575 y Fl(p)870 3608 y Fp(!\))930 3575 y Fl(\032)p Fi(\000)p Fn(1)1061 3608 y Fp(\()p Fo(\013)1154 3575 y Fn(1)1195 3608 y Fp(!)f Fk(\001)g(\001)g(\001)h Fo(\013)1414 3575 y Fl(p)1454 3608 y Fp(!\))1514 3575 y Fl(\026)p Fi(\000)p Fn(1)1652 3608 y Fo(\014)5 b Fp(!)1748 3575 y Fl(\032)p Fi(\000)p Fn(1)490 3755 y Fk(\024)25 b Fp(2)631 3722 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fi(j)p Fl(\015)t Fn(+)p Fl(\013)971 3698 y Fd(0)995 3722 y Fi(j)1034 3755 y Fp(\(\()p Fk(j)p Fo(\015)1181 3722 y Fn(1)1242 3755 y Fp(+)19 b Fo(\013)1390 3722 y Fn(1)1430 3755 y Fk(j)i(\000)f Fp(1\)!)15 b Fk(\001)g(\001)g(\001)i Fp(\()p Fk(j)p Fo(\015)1921 3722 y Fl(p)1982 3755 y Fp(+)j Fo(\013)2131 3722 y Fl(p)2171 3755 y Fk(j)g(\000)g Fp(1\)!)p Fo(p)p Fp(!\))2518 3722 y Fl(\032)p Fi(\000)p Fn(1)2650 3755 y Fp(\()p Fo(\013)2743 3722 y Fn(1)2783 3755 y Fp(!)15 b Fk(\001)g(\001)g(\001)i Fo(\013)3003 3722 y Fl(p)3043 3755 y Fp(!\))3103 3722 y Fl(\026)p Fi(\000)p Fl(\032)490 3901 y Fk(\024)25 b Fp(2)631 3868 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fi(j)p Fl(\015)t Fn(+)p Fl(\013)971 3845 y Fd(0)995 3868 y Fi(j)1019 3901 y Fp(\(2)p Fo(n)p Fp(\))1189 3868 y Fn(\()p Fl(\032)p Fi(\000)p Fn(1\))p Fi(j)p Fl(\015)t Fn(+)p Fl(\013)1529 3845 y Fd(0)1553 3868 y Fi(j)1577 3901 y Fo(\015)1629 3868 y Fl(\032)1669 3901 y Fo(\013)1727 3868 y Fi(0)1751 3901 y Fp(!)1776 3868 y Fl(\026)-24 4108 y Fp(and)30 b(as)g(ab)s(o)m(v)m(e)i(w)m(e)f (complete)g(the)f(pro)s(of)g(of)g(the)h(prop)s(osition.)-24 4339 y Fe(Pr)-5 b(o)g(of)40 b(of)f(L)-5 b(emma)40 b(3.4)p Fp(.)61 b(First,)38 b(using)d(the)j(Cauc)m(h)m(y)f(form)m(ula,)h(w)m(e) f(obtain)g(as)g(in)e([9)q(],)k(Lemma)e(A.3,)j(that)-24 4452 y Fo(f)57 b Fp(:)47 b Fo(O)219 4471 y Fn(2\()5 b(~)-40 b Fl(\035)t Fn(+)p Fl(\035)r Fn(\))p Fl(h)539 4452 y Fk(!)47 b Fm(C)c Fp(is)g(one-to-one)j(\(injectiv)m(e\).)81 b(Note)45 b(that)f(2\()6 b(~)-51 b Fo(\035)34 b Fp(+)28 b Fo(\035)s Fp(\))49 b(=)e(1)30 b Fk(\000)e Fp(4)p Fo(\035)s Fp(.)82 b(T)-8 b(o)44 b(sho)m(w)g(that)-24 4565 y Fo(O)50 4580 y Fn(~)-40 b Fl(\035)s(h)156 4565 y Fk(\032)25 b Fo(f)10 b Fp(\()p Fo(O)411 4583 y Fn(\(1)p Fi(\000)p Fn(4)p Fl(\035)r Fn(\))p Fl(h)677 4565 y Fp(\),)28 b(w)m(e)g(tak)m(e)g Fo(w)g Fk(2)d Fo(O)1342 4580 y Fn(~)-40 b Fl(\035)s(h)1450 4565 y Fp(and)27 b Fo(!)h Fk(2)d Fp(\012)1861 4579 y Fl(\024)1932 4565 y Fp(suc)m(h)i(that)h Fk(j)p Fo(w)16 b Fk(\000)e Fo(!)s Fk(j)25 b Fo(<)31 b Fp(~)-51 b Fo(\035)t(h)p Fp(.)40 b(Set)27 b Fo(f)34 b Fp(=)25 b(id)j Fk(\000)14 b Fo(F)f Fp(.)39 b(Then)-24 4678 y Fk(j)p Fo(F)13 b Fk(j)97 4693 y Fl(h)168 4678 y Fk(\024)25 b Fo(\035)s(h)31 b Fp(and)f Fk(j)p Fo(D)s(F)13 b Fk(j)775 4696 y Fn(\(1)p Fi(\000)p Fn(4)p Fl(\035)r Fn(\))p Fl(h)1067 4678 y Fk(\024)25 b Fp(1)p Fo(=)p Fp(4)32 b(b)m(y)e(the)h(Cauc)m(h)m(y)g(estimates.)42 b(Put)30 b Fo(u)2605 4692 y Fn(0)2670 4678 y Fp(=)25 b Fo(!)34 b Fp(and)29 b Fo(u)3085 4693 y Fl(k)r Fn(+1)3244 4678 y Fp(=)c Fo(F)13 b Fp(\()p Fo(u)3498 4693 y Fl(k)3541 4678 y Fp(\))21 b(+)f Fo(w)-24 4791 y Fp(for)30 b Fo(k)e Fk(\025)d Fp(0.)42 b(By)30 b(recurrence)g(w)m(e)h(pro)m(v)m(e)465 4995 y Fk(j)p Fo(u)542 5010 y Fl(k)r Fn(+1)695 4995 y Fk(\000)20 b Fo(u)838 5010 y Fl(k)881 4995 y Fk(j)25 b(\024)g Fp(2)1072 4957 y Fi(\000)p Fn(1)1167 4995 y Fp(4)1212 4957 y Fi(\000)p Fl(k)1310 4995 y Fp(\(1)c Fk(\000)f Fp(4)p Fo(\035)s Fp(\))p Fo(h)32 b(;)106 b Fk(j)p Fo(u)1926 5010 y Fl(k)1989 4995 y Fk(\000)20 b Fo(!)s Fk(j)25 b(\024)g Fp(\(2)p Fo(=)p Fp(3\)\(1)f Fk(\000)19 b Fp(4)2730 4957 y Fi(\000)p Fl(k)2828 4995 y Fp(\)\(1)j Fk(\000)e Fp(4)p Fo(\035)s Fp(\))p Fo(h:)-24 5199 y Fp(T)-8 b(aking)29 b(the)g(limit)e(w)m(e)j(\014nd)d Fo(u)f Fk(2)f Fo(B)1195 5213 y Fl(r)1232 5199 y Fp(\()p Fo(!)s Fp(\),)31 b Fo(r)c Fp(=)e(\(1)19 b Fk(\000)f Fp(4)p Fo(\035)s Fp(\))p Fo(h)p Fp(,)31 b(suc)m(h)e(that)g Fo(u)d Fp(=)f Fo(F)13 b Fp(\()p Fo(u)p Fp(\))18 b(+)g Fo(w)r Fp(.)41 b(The)28 b(corresp)s(onding)-24 5312 y(estimates)j(of)f Fo(\036)h Fp(follo)m(w)f(from)f(the)i(argumen)m(ts)g(in)e([9],)i(Lemma)g(A.3.) 1271 b Fa(2)1820 5927 y Fp(28)p eop %%Page: 29 29 29 28 bop -24 573 a Fr(References)67 776 y Fp([1])46 b(J.)21 b(Bruna,)i(An)d(extension)h(theorem)g(of)g(Whitney)f(t)m(yp)s (e)i(for)e(non)h(quasi-analytic)e(classes)i(of)h(functions,)208 889 y Fe(J.)32 b(L)-5 b(ondon)35 b(Math.)e(So)-5 b(c.)p Fp(,)31 b(V)-8 b(ol.)31 b Fm(22)p Fp(,)g(1980,)h(2,)f(pp.)f(495-505.)67 1076 y([2])46 b(M.)39 b(Herman,)h(In)m(\023)-43 b(egalit)m(\023)g(es)40 b(a)f(priori)d(p)s(our)h(des)h(tores)h(lagrangiens)e(in)m(v)-5 b(arian)m(ts)38 b(par)g(des)g(di\013)m(\023)-43 b(eomor-)208 1189 y(phismes)29 b(symplectiques,)g Fe(Publ.)j(Math.)h(I.H.E.S.)p Fp(,)c(V)-8 b(ol.)31 b Fm(70)p Fp(,)g(1989,)h(pp.)e(47-101.)67 1377 y([3])46 b(H.)30 b(Komatsu,)g(The)g(implicit)c(function)j(theorem) h(for)f(ultradi\013eren)m(tiable)f(mappings,)g Fe(Pr)-5 b(o)g(c.)33 b(Jap)-5 b(an)208 1490 y(A)g(c)g(ad.)p Fp(,)32 b Fm(55)p Fp(,)f(Ser.)f(A,)h(1979,)h(pp.)e(69-72.)67 1677 y([4])46 b(S.)29 b(Kuksin,)e(Analysis)h(of)h(Hamiltonian)e(PDE's,) j(Oxford)e(Lecture)i(Series)e(in)f(Mathematics)j(and)f(its)208 1790 y(Applications,)g Fm(19)p Fp(,)i(2000.)67 1978 y([5])46 b(V.)35 b(Lazutkin,)g(KAM)f(theory)h(and)e(semiclassical)g(appro)m (ximations)g(to)i(eigenfunctions,)f(Springer-)208 2091 y(V)-8 b(erlag,)32 b(Berlin,)d(1993.)67 2278 y([6])46 b(P)-8 b(.)43 b(Lo)s(c)m(hak,)j(Canonical)41 b(p)s(erturbation)g (theory:an)h(approac)m(h)h(based)f(on)g(join)m(t)g(appro)m(ximations,) 208 2391 y Fe(Usp)-5 b(ekhi)42 b(Mat.)f(Nauk)p Fp(,)h(V)-8 b(ol.)40 b Fm(47)p Fp(,)j(6,)g(1992,)h(pp.)39 b(59-140)k(\(in)38 b(Russian\);)44 b(translation)39 b(in:)58 b Fe(R)n(ussian)208 2504 y(Math.)33 b(Surveys)p Fp(,)e(V)-8 b(ol.)31 b Fm(47)p Fp(,)g(6,)g(1992,)h(pp.)e(57-133.)67 2692 y([7])46 b(J.-P)-8 b(.)36 b(Marco,)i(D.)e(Sauzin,)f(Stabilit)m(y)f(and)g(instabilit)m(y)f (for)i(Gevrey)h(quasi-cin)m(v)m(ex)e(near-in)m(tegrable)208 2805 y(Hamiltonian)29 b(systems,)i(Publ.)d(Math.)j(IHES,)f Fm(96)p Fp(,)h(2003)i(\(to)e(app)s(ear\).)67 2992 y([8])46 b(J.)25 b(P\177)-45 b(osc)m(hel,)26 b(In)m(tegrabilit)m(y)e(of)h (Hamiltonian)f(systems)h(on)f(Can)m(tor)i(tori,)g Fe(Comm.)i(Pur)-5 b(e)28 b(Appl.)g(Math.)p Fp(,)208 3105 y(V)-8 b(ol.)31 b Fm(35)p Fp(,)g(1982,)i(pp.)c(653-695.)67 3293 y([9])46 b(J.)35 b(P\177)-45 b(osc)m(hel,)35 b(A)g(Lecture)g(on)f(the)h (Classical)e(KAM)h(Theorem,)i(Pro)s(ceedings)d(of)i(Simp)s(osia)d(in)h (Pure)208 3406 y(Mathematics,)f(V)-8 b(ol.)31 b Fm(69)p Fp(,)g(707-732,)i(2001.)21 3594 y([10])47 b(G.)28 b(P)m(op)s(o)m(v,)g (In)m(v)-5 b(arian)m(t)27 b(tori,)g(e\013ectiv)m(e)i(stabilit)m(y)d (and)g(quasimo)s(des)f(with)h(exp)s(onen)m(tially)f(small)h(error)208 3706 y(terms)k(I,)h(I)s(I,)f Fe(A)n(nn.)h(Henri)i(Poinc)-5 b(ar)n(\023)-44 b(e)31 b Fm(1)g Fp(\(2000\),)i(223-248,)g(249-279.)21 3894 y([11])47 b(L.)31 b(Ro)s(dino,)e(Linear)h(partial)f(di\013eren)m (tial)g(op)s(erators)h(in)g(Gevrey)h(spaces,)g(W)-8 b(orld)30 b(Scien)m(ti\014c,)g(Singa-)208 4007 y(p)s(ore,)g(1993.)21 4195 y([12])47 b(E.)23 b(Zehnder,)g(Generalized)g(implicit)d(function)i (theorems)h(with)f(applications)f(to)j(some)f(small)e(divisor)208 4308 y(problems)29 b(I.)h Fe(Comm.)k(Pur)-5 b(e)32 b(Appl.)h(Math.)p Fp(,)f Fm(28)e Fp(\(1975\),)j(91-140.)208 4576 y(G.)38 b(P)-8 b(.:)54 b(Univ)m(ersit)m(\023)-43 b(e)37 b(de)f(Nan)m(tes,)k(D)m (\023)-43 b(epartemen)m(t)40 b(de)d(Math)m(\023)-43 b(ematiques,)40 b(UMR)d(6629)i(du)c(CNRS,)i(2,)208 4689 y(rue)30 b(de)g(la)g(Houssini)m (\022)-43 b(ere,)30 b(BP)h(92208,)h(44072)h(Nan)m(tes)f(Cedex)e(03,)h (F)-8 b(rance,)208 4802 y(p)s(op)s(o)m(v@math.univ-nan)m(tes.fr)1820 5927 y(29)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0305180503368--