This is a multi-part message in MIME format. ---------------0212101300914 Content-Type: text/plain; name="02-512.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-512.keywords" Spectral shift function, long range scattering, Breit-Wigner formula, semiclassical analysis ---------------0212101300914 Content-Type: application/postscript; name="paper2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paper2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: paper2.dvi %%Pages: 33 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips paper2.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.05.22:1555 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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y Ff(1)1298 4847 y Ft(and,)28 b(if)g Fs(f)1600 4859 y Fp(z)1638 4847 y Ft(\()p Fs(\025)p Ft(\))c(=)f(\()p Fs(\025)c Fq(\000)f Fs(z)t Ft(\))2119 4817 y Fm(\000)p Fr(1)2207 4847 y Ft(,)28 b(w)n(e)f(get)871 5027 y Fs(<)22 b(u)1006 5039 y Fp(p)1044 5027 y Fs(;)14 b(f)1122 5039 y Fp(z)1183 5027 y Fs(>)p Ft(=)22 b(\()p Fq(\000)p Ft(1\))1506 4993 y Fp(p)1544 5027 y Ft(T)-7 b(r)1644 4960 y Fo(\000)1682 5027 y Ft(\()p Fs(H)1783 5039 y Fr(0)1839 5027 y Ft(+)18 b Fs(V)37 b Fq(\000)18 b Fs(z)t Ft(\))2165 4993 y Fm(\000)p Fr(1)2254 4960 y Fo(\000)2292 5027 y Fs(V)h Ft(\()p Fs(H)2460 5039 y Fr(0)2516 5027 y Fq(\000)f Fs(z)t Ft(\))2674 4993 y Fm(\000)p Fr(1)2763 4960 y Fo(\001)2801 4977 y Fp(p)2839 4960 y Fo(\001)2891 5027 y Fs(:)662 b Ft(\(0.6\))204 5191 y(Ho)n(w)n(ev)n(er,)29 b(as)h(men)n(tioned)g(in)h([25)o(],)h(one)e (cannot)g(use)g(the)h(in)n(v)-5 b(ariance)29 b(principle)h(an)n(y)g (longer)f(in)i(\(0)p Fs(:)p Ft(5\),)g(when)204 5291 y Fs(p)f Fq(\025)h Ft(2:)45 b(this)33 b(means)f(that,)h(if)g(w)n(e)f (only)g(assume)f(that)i(\(0)p Fs(:)p Ft(4\))f(holds,)h(w)n(e)f(cannot)f (de\014ned)i(a)f(distribution)g Fs(u)3709 5303 y Fp(p)1955 5540 y Ft(2)p eop %%Page: 3 3 3 2 bop 204 531 a Ft(simply)34 b(b)n(y)f(replacing)g(the)h(op)r (erators)d(b)n(y)j(some)f(p)r(o)n(w)n(er)f(of)i(their)g(resolv)n(en)n (ts.)53 b(In)34 b(particular,)g(Koplienk)n(o's)204 631 y(w)n(ork)24 b(cannot)i(b)r(e)g(applied)f(to)h(di\013eren)n(tial)g(op)r (erators)e(\(except)i(for)f(the)h(sp)r(ecial)g(case)e Fs(p)f Ft(=)g(2,)j(see)f([25)o(,)h(26)o(],)h(see)204 731 y(also)i(Rybkin)h([36)o(]\).)44 b(This)29 b(problem)h(has)f(b)r (een)h(considered)f(b)n(y)h(the)g(author)f(in)h([4,)g(7)o(],)h(where)e (the)h(follo)n(wing)204 830 y(result)d(is)h(pro)n(v)n(ed:)204 996 y Fg(Theorem)i(0.1.)40 b Fi(L)l(et)27 b Fs(H)1009 1008 y Fr(0)1070 996 y Ft(=)22 b Fs(p)p Ft(\()p Fs(x;)14 b(D)r Ft(\))29 b Fi(and)f Fs(H)1675 1008 y Fr(0)1726 996 y Ft(+)14 b Fs(V)41 b Ft(=)23 b Fs(H)2051 1008 y Fr(0)2102 996 y Ft(+)14 b Fs(v)s Ft(\()p Fs(x;)g(D)r Ft(\))28 b Fi(b)l(e)g(self-adjoint)h(di\013er)l(ential)g(op)l(er)l (ators)204 1096 y(on)c Fl(R)372 1066 y Fp(d)442 1096 y Fi(with)i(smo)l(oth)e(b)l(ounde)l(d)h(c)l(o)l(e\016cients,)i(to)l (gether)d(with)h(their)h(derivatives,)h(and)e(uniformly)h(el)t(liptic,) i(\(i.e.)204 1196 y(their)h(princip)l(al)i(symb)l(ols)e(ar)l(e)g(lower) h(b)l(ounde)l(d)f(on)g Fl(R)1902 1165 y Fp(d)1965 1196 y Fq(\002)18 b Fl(S)2100 1165 y Fp(d)p Fm(\000)p Fr(1)2248 1196 y Fi(by)30 b(some)g Fs(c)23 b(>)g Ft(0)p Fi(\).)38 b(Mor)l(e)l(over,)31 b(we)f(supp)l(ose)1192 1378 y Fq(j)p Fs(@)1264 1344 y Fp(\014)1309 1378 y Fs(v)1349 1390 y Fp(\013)1396 1378 y Ft(\()p Fs(x)p Ft(\))p Fq(j)24 b(\024)f Fs(C)6 b Fq(h)p Fs(x)p Fq(i)1818 1344 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\024)p Fm(j)p Fp(\013)p Fm(j)2083 1378 y Fs(;)184 b(\032)23 b(>)f(d=p;)983 b Ft(\(0.7\))204 1561 y Fi(with)30 b Fs(\024)23 b Fq(2)g Ft([0)p Fs(;)14 b Ft(1])p Fi(,)30 b(and)g Fs(v)s Ft(\()p Fs(x;)14 b(D)r Ft(\))24 b(=)1290 1499 y Fo(P)1377 1586 y Fm(j)p Fp(\013)p Fm(j\024)p Fr(2)p Fp(m)1622 1561 y Fs(v)1662 1573 y Fp(\013)1709 1561 y Ft(\()p Fs(x)p Ft(\))p Fs(D)1891 1531 y Fp(\013)1940 1561 y Fi(.)328 1671 y(Then)34 b(for)g(al)t(l)g Fs(f)j Fq(2)29 b(S)6 b Ft(\()p Fl(R)q Ft(\))p Fi(,)41 b(formula)33 b Ft(\(0)p Fs(:)p Ft(5\))g Fi(makes)h(sense)e(and)i(it)f(de\014nes)g(a) g(temp)l(er)l(ate)g(distribution)h Fs(u)3684 1683 y Fp(p)3722 1671 y Fi(,)204 1770 y(c)l(al)t(le)l(d)d(sp)l(e)l(ctr)l(al)f (distribution)g(of)h(or)l(der)g Fs(p)p Fi(.)204 1936 y Ft(Remark)c(that)g Fs(u)744 1948 y Fp(p)805 1936 y Fq(\021)c Ft(0)k(b)r(elo)n(w)g(the)h(sp)r(ectra)f(of)h(the)g(op)r (erators,)e(whic)n(h)h(is)h(ob)n(vious)e(b)n(y)h(form)n(ula)g(\(0)p Fs(:)p Ft(5\).)328 2036 y(W)-7 b(e)21 b(emphasize)f(that)h Fs(H)1096 2048 y Fr(0)1154 2036 y Ft(migh)n(t)g(ha)n(v)n(e)e(v)-5 b(ariable)20 b(co)r(e\016cien)n(ts)g(and)g(that)h Fs(V)40 b Ft(migh)n(t)20 b(b)r(e)h(of)g(the)g(same)f(order)f(as)204 2136 y Fs(H)273 2148 y Fr(0)310 2136 y Ft(.)36 b(This)26 b(theorem)f(is)h(actually)f(pro)n(v)n(ed)f(for)h(a)g(wider)h(class)e (of)i(pseudo-di\013eren)n(tial)f(op)r(erators)e(in)j([4,)g(7)o(],)h (but)204 2235 y(w)n(e)g(only)g(quote)h(it)g(in)f(this)h(less)f(tec)n (hnical)h(setting,)f(for)g(stating)h(our)e(main)i(results)f(in)h(the)g (next)g(subsection.)328 2360 y(Sev)n(eral)35 b(natural)f(questions)h (arise)g(when)g(w)n(e)h(consider)e(the)i(distributions)g Fs(u)2874 2372 y Fp(p)2947 2360 y Ft(for)f(pseudo-di\013eren)n(tial)204 2460 y(op)r(erators.)41 b(When)30 b(one)f(lo)r(oks)g(at)g(form)n(ula)g (\(0)p Fs(:)p Ft(3\),)h(it)g(is)f(natural)g(to)h(w)n(an)n(t)f(to)g (establish)g(a)h(relation)e(b)r(et)n(w)n(een)204 2559 y Fs(u)252 2571 y Fp(p)316 2559 y Ft(and)f(some)f(scattering)g (determinan)n(ts.)36 b(Moreo)n(v)n(er,)24 b(for)i(general)f Fs(\032)e(>)g Ft(0,)k(it)g(is)f(kno)n(wn)g(that)h(the)g(scattering)204 2659 y(matrix)k Fs(S)5 b Ft(\()p Fs(\025)p Ft(\))33 b(ma)n(y)f(not)g(b) 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b(imp)r(ortan)n(t)f(adv)-5 b(an)n(tage)30 b(of)h(\(0)p Fs(:)p Ft(8\))g(is)g(that)g(there)g(is)g (no)g(distinction)g(b)r(et)n(w)n(een)g(the)h(short-range,)d Fs(\032)g(>)g Ft(1,)204 3522 y(and)f(long-range)e(conditions,)i Fs(\032)c(>)g Ft(0.)38 b(But)29 b(recall)e(that)i(the)f(p)r (erturbations)g Fs(V)19 b Ft(,)28 b(that)h(w)n(e)f(consider)f(here,)h (are)204 3622 y(in)f(general)f Fs(H)655 3634 y Fr(0)692 3622 y Ft(-b)r(ounded)h(and)g(not)g Fs(H)1435 3634 y Fr(0)1473 3622 y Ft(-compact,)f(th)n(us)h(the)h(meaning)e(of)i Fs(D)2675 3634 y Fp(p)2713 3622 y Ft(\()p Fs(z)t Ft(\))f(is)g(not)g (clear.)36 b(T)-7 b(o)26 b(de\014ne)i(it,)204 3721 y(w)n(e)h(will)g(ha) n(v)n(e)f(to)i(use)f(the)g(meromorphic)f(con)n(tin)n(uation)h(of)g(a)g (so)f(called)h Fi(r)l(e)l(gularize)l(d)k(zeta)e(function)p Ft(,)f(as)f(it)g(is)204 3821 y(can)24 b(b)r(e)i(done)e(for)h(the)g(SSF) g Fs(s)1134 3833 y Fr(1)1171 3821 y Ft(.)36 b(This)25 b(is)g(a)g(rather)e(usual)i(pro)r(cess)f(and)g(w)n(e)h(refer)f(to)h (the)g(nice)g(pap)r(er)f(of)h(M)r(\177)-44 b(uller)204 3921 y([29)o(])28 b(on)f(this)h(sub)5 b(ject)28 b(\(see)f(also)g (Bruneau)f([8])i(and)f(Carron)f([12)o(]\).)328 4020 y(Finally)31 b(w)n(e)f(men)n(tion)g(that)h(another)e(kind)i(of)g(regularization,)d (inspired)j(b)n(y)f(T)-7 b(a)n(ylor's)28 b(form)n(ula)i(to)r(o,)h(has) 204 4120 y(b)r(een)g(considered)f(b)n(y)g(Melin)h(in)g([28)o(])g(and)f (recen)n(t)g(results)g(asso)r(ciated)g(to)g(a)g(related)g(regularized)f (scattering)204 4220 y(phase,)e(in)h(connection)f(with)h(some)f(heat)h (in)n(v)-5 b(arian)n(ts,)26 b(ha)n(v)n(e)g(b)r(een)i(announced)f(b)n(y) h(Hitrik-P)n(oltero)n(vic)n(h)c([23)o(].)204 4452 y Fu(0.2)112 b(Main)38 b(results)e(of)i(this)f(article)204 4605 y Ft(W)-7 b(e)36 b(are)f(going)f(to)i(pro)n(v)n(e)e(results)i(partially)e (announced)i(in)g([6].)61 b(In)36 b(all)g(what)g(follo)n(ws,)h Fs(\032)f(>)h Ft(0)e(is)h(a)f(\014xed)204 4705 y(p)r(ositiv)n(e)27 b(real)g(n)n(um)n(b)r(er)g(and)g Fs(p)c Fq(\025)g Ft(1)k(an)g(in)n (teger)g(suc)n(h)g(that)1824 4888 y Fs(\032)c(>)f(d=p:)204 5070 y Ft(Unless)27 b(something)g(else)h(is)f(sp)r(eci\014ed,)h(the)g (dimension)g Fs(d)f Ft(satis\014es)g Fs(d)d Fq(\025)e Ft(1.)328 5170 y(In)k(the)g(follo)n(wing)e(theorem,)i(w)n(e)f(de\014ne) h(the)g(determinan)n(t)f Fs(D)2298 5182 y Fp(p)2336 5170 y Ft(\()p Fs(z)t Ft(\))h(and)f(the)h(asso)r(ciated)e(scattering)h (phase)204 5269 y Fs(s)243 5281 y Fp(p)281 5269 y Ft(,)j(when)f(the)h (p)r(erturbation)f Fs(V)46 b Ft(is)27 b(only)g Fs(H)1606 5281 y Fr(0)1643 5269 y Ft(-b)r(ounded.)37 b(W)-7 b(e)28 b(consider)e(the)i(same)e(di\013eren)n(tial)i(op)r(erators)d Fs(H)3710 5281 y Fr(0)1955 5540 y Ft(3)p eop %%Page: 4 4 4 3 bop 204 531 a Ft(and)26 b Fs(H)433 543 y Fr(0)486 531 y Ft(+)15 b Fs(V)45 b Ft(as)26 b(in)g(the)h(theorem)f(0)p Fs(:)p Ft(1)f(with)i Fs(\024)c Ft(=)f(0.)36 b(W)-7 b(e)27 b(shall)f(use)g(the)g(so-called)f Fi(r)l(e)l(gularize)l(d)30 b(Zeta)e(function)204 631 y Ft(de\014ned)g(b)n(y)1522 731 y Fs(\020)1558 743 y Fp(s)1594 731 y Ft(\()p Fs(z)t Ft(\))23 b(:=)p Fs(<)f(u)1947 743 y Fp(p)1985 731 y Fs(;)14 b Ft(\()p Fs(:)19 b Fq(\000)f Fs(z)t Ft(\))2254 696 y Fm(\000)p Fp(s)2364 731 y Fs(>)204 871 y Ft(where)27 b(\()p Fs(:)19 b Fq(\000)f Fs(z)t Ft(\))676 841 y Fm(\000)p Fp(s)790 871 y Ft(is)28 b(the)g(function)g Fs(\025)23 b Fq(7!)g Ft(\()p Fs(\025)d Fq(\000)e Fs(z)t Ft(\))1777 841 y Fm(\000)p Fp(s)1864 871 y Ft(.)204 1025 y Fg(Theorem)30 b(0.2.)40 b Fi(i\))31 b Fs(\020)925 1037 y Fp(s)961 1025 y Ft(\()p Fs(z)t Ft(\))e Fi(is)i(wel)t(l)g(de\014ne)l(d)f(for)h Ft(Re)g Fs(s)23 b Fq(\035)h Ft(1)29 b Fi(and)i Fs(z)c Fq(2)d Fl(C)40 b Fq(n)18 b Ft([inf)j(supp)30 b Fs(u)2972 1037 y Fp(p)3010 1025 y Fs(;)14 b Fq(1)p Ft(\))p Fi(.)40 b(It)30 b(has)h(a)f(mer)l(o-)204 1124 y(morphic)35 b(c)l(ontinuation,)g (with)f(r)l(esp)l(e)l(ct)f(to)h Fs(s)p Fi(,)h(to)e(the)h(c)l(omplex)g (plane)g(and)g(this)g(c)l(ontinuation)g(is)f(r)l(e)l(gular)h(at)204 1224 y Fs(s)23 b Ft(=)f(0)p Fi(.)204 1323 y(ii\))34 b(The)g(function)f 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y Fp(p)1405 1936 y Ft(\()p Fs(z)t Ft(\))h(coincide)f(with)h(Det)2212 1948 y Fp(p)2251 1936 y Ft(\(1)25 b(+)h Fs(V)18 b Ft(\()p Fs(H)2608 1948 y Fr(0)2672 1936 y Fq(\000)25 b Fs(z)t Ft(\))2837 1906 y Fm(\000)p Fr(1)2926 1936 y Ft(\),)41 b(the)e(usual)f(F)-7 b(redholm)204 2036 y(determinan)n(t)27 b(of)h(order)e Fs(p)h Ft(when)h Fs(V)19 b Ft(\()p Fs(H)1438 2048 y Fr(0)1494 2036 y Fq(\000)f Fs(z)t Ft(\))1652 2006 y Fm(\000)p Fr(1)1764 2036 y Fq(2)23 b Fg(S)1895 2048 y Ff(p)1969 2036 y Ft(\(see)k(section)h(for)f(more)f(details)i(1\).)328 2157 y(W)-7 b(e)22 b(add)f(that)g(the)h(con)n(v)n(ergence)d(in)i(\(0)p Fs(:)p Ft(9\))g(cannot)g(b)r(e)g(p)r(oin)n(t)n(wise)g(in)h(general,)f (since)g(it)g(w)n(as)g(pro)n(v)n(ed)e(in)j([4)o(,)g(7)o(])204 2257 y(that)27 b Fs(u)431 2269 y Fr(2)491 2257 y Ft(=)c Fs(\021)623 2227 y Fm(00)692 2257 y Ft(\(in)28 b(the)f(distributions)g (sense\))g(for)f(some)g(lo)r(cally)h(in)n(tegrable)e(function)j Fs(\021)s Ft(,)f(called)g(Koplienk)n(o's)204 2356 y(function)j(\(see)g (also)f([25)o(]\).)44 b(F)-7 b(orm)n(ula)29 b(\(0)p Fs(:)p Ft(9\))h(can)g(b)r(e)g(view)n(ed)f(as)h(a)f(generalization)f(of)i(\(0)p Fs(:)p Ft(3\),)g(and)g(it)h(suggests)204 2456 y(the)d(follo)n(wing)e (de\014nition:)204 2610 y Fg(De\014nition)31 b(0.3.)40 b Fi(The)33 b(r)l(e)l(gularize)l(d)g(sc)l(attering)g(phase)g Fs(s)2078 2622 y Fp(p)2145 2610 y Fq(2)28 b(S)2284 2580 y Fm(0)2308 2610 y Ft(\()p Fl(R)p Ft(\))39 b Fi(is)32 b(the)h(unique)f(primitive)i(of)f Fs(u)3473 2622 y Fp(p)3543 2610 y Fi(which)204 2709 y(vanishes)e(ne)l(ar)f Fq(\0001)p Fi(.)204 2863 y Ft(Recall)19 b(that)h Fs(u)666 2875 y Fp(p)724 2863 y Ft(v)-5 b(anishes)19 b(b)r(elo)n(w)h(the)g(sp)r(ectra)f (of)h(the)g(op)r(erators,)g(so)f(this)h(de\014nition)g(mak)n(es)f (sense.)34 b(Moreo)n(v)n(er,)204 2963 y(if)28 b Fs(p)23 b Ft(=)f(1,)28 b(w)n(e)f(get)g(the)h(usual)f(scattering)g(phase)g Fi(i.e.)38 b Ft(the)28 b(Birman-Krein's)e(SSF.)328 3063 y(No)n(w)32 b(w)n(e)f(giv)n(e)g(results)g(in)h(Euclidean)f(scattering,) h(in)g(the)h(high)e(energy)g(and)h(semiclassical)e(limits.)50 b(F)-7 b(or)204 3162 y(all)27 b(these)h(results)f(w)n(e)g(no)n(w)g (assume)g(that)h Fs(d)23 b Fq(\025)g Ft(2.)204 3316 y Fg(Theorem)30 b(0.4.)40 b Fi(Assume)25 b(that)g Fs(H)1339 3328 y Fr(0)1400 3316 y Ft(=)d Fs(p)1529 3328 y Fr(0)1566 3316 y Ft(\()p Fs(D)r Ft(\))p Fi(,)28 b(with)e Fs(p)1972 3328 y Fr(0)2009 3316 y Ft(\()p Fs(\030)t Ft(\))e(=)2224 3254 y Fo(P)2312 3341 y Fm(j)p Fp(\013)p Fm(j)p Fr(=2)p Fp(m)2555 3316 y Fs(p)2597 3328 y Fp(\013)2645 3316 y Fs(\030)2685 3286 y Fp(\013)2757 3316 y Fi(c)l(onstant)h(c)l(o)l (e\016cients)h(el)t(liptic)204 3416 y(p)l(ositive)31 b(p)l(olynomial,)h(and)f(c)l(onsider)f Fs(V)49 b Fi(as)30 b(in)g(the)l(or)l(em)g Ft(0)p Fs(:)p Ft(1)f Fi(with)h Fs(\024)23 b Ft(=)g(1)p Fi(.)38 b(Then)204 3515 y(i\))30 b Fs(s)332 3527 y Fp(p)400 3515 y Fi(is)g Fs(C)554 3485 y Fm(1)654 3515 y Fi(on)g Ft(\(0)p Fs(;)14 b Fq(1)p Ft(\))19 b Fq(n)f 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Ft(\))2474 4138 y Fo(\001)2513 4205 y Fq(j)30 b(!)f(1)p Ft(,)j(as)f Fq(j)p Fs(t)p Fq(j)e(!)g(1)p Ft(,)j(uniformly)f(on)204 4305 y(eac)n(h)36 b(compact)g(subset)g(of)h Fs(T)1169 4275 y Fp(?)1206 4305 y Fl(R)1260 4275 y Fp(d)1329 4305 y Fq(n)24 b Ft(0,)39 b(where)d Fs(\036)1797 4275 y Fp(t)1863 4305 y Ft(is)h(the)g(Hamiltonian)f(\015o)n(w)g(of)h(the)g (principal)f(sym)n(b)r(ol)g(of)204 4404 y Fs(H)273 4416 y Fr(0)326 4404 y Ft(+)15 b Fs(V)k Ft(,)27 b(and)f(\005)d(:)g Fs(T)875 4374 y Fp(?)912 4404 y Fl(R)966 4374 y Fp(d)1034 4404 y Fq(!)g Fl(R)1194 4374 y Fp(d)1265 4404 y Ft(is)j(the)h(pro)5 b(jection.)35 b(Here)26 b(the)h(principal)e(sym)n(b)r(ol)h(is)g(the)h (classical)e(one,)h(that)204 4504 y(is)h(the)h(leading)f(homogeneous)f (part)h(of)h(order)e(2)p Fs(m)h Ft(of)h(the)g(full)g(sym)n(b)r(ol)f(of) h Fs(H)2689 4516 y Fr(0)2744 4504 y Ft(+)18 b Fs(V)h Ft(.)328 4604 y(This)35 b(theorem)g(is)g(w)n(ell)g(kno)n(wn)g(for)f Fs(p)i 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Ft(=)23 b Fs(V)c Ft(\()p Fs(x)p Ft(\))29 b(is)e(a)g(p)r(oten)n(tial)h (suc)n(h)f(that)1197 5291 y Fs(@)1246 5257 y Fp(\013)1293 5291 y Fs(V)19 b Ft(\()p Fs(x)p Ft(\))24 b(=)f Fq(O)r Ft(\()p Fq(h)p Fs(x)p Fq(i)1794 5257 y Fm(\000)p Fp(\032)p Fm(\000j)p Fp(\013)p Fm(j)2020 5291 y Ft(\))p Fs(;)236 b(\013)24 b Fq(2)f Fl(N)2520 5257 y Fp(d)2565 5291 y Fs(;)947 b Ft(\(0.10\))1955 5540 y(4)p eop %%Page: 5 5 5 4 bop 204 531 a Ft(and)25 b(let)h Fs(I)k Fq(\032)23 b Ft(\(0)p Fs(;)14 b Fq(1)p Ft(\))26 b(b)r(e)g(an)f(in)n(terv)-5 b(al.)36 b(This)26 b(in)n(terv)-5 b(al)25 b(is)g(said)h(to)f(b)r(e)h (non)g(trapping)f(if)h Fq(j)p Ft(\005)3044 464 y Fo(\000)3083 531 y Fs(\036)3132 501 y Fp(t)3161 531 y Ft(\()p Fs(x;)14 b(\030)t Ft(\))3349 464 y Fo(\001)3388 531 y Fq(j)24 b(!)f(1)p Ft(,)j(as)204 631 y Fq(j)p Fs(t)p Fq(j)d(!)g(1)p Ft(,)h(uniformly)f(on)g(eac)n(h)f(compact)h(subset)g(of)g Fs(p)1916 601 y Fm(\000)p Fr(1)2005 631 y Ft(\()p Fs(I)7 b Ft(\))24 b(where)f Fs(p)p Ft(\()p Fs(x;)14 b(\030)t Ft(\))23 b(=)g Fs(\030)2753 601 y Fr(2)2800 631 y Ft(+)10 b Fs(V)18 b Ft(\()p Fs(x)p Ft(\))24 b(is)f(the)h(semiclassical)204 731 y(principal)j(sym)n(b)r(ol)g(of)h Fs(H)998 743 y Fr(0)1053 731 y Ft(+)18 b Fs(V)h Ft(.)204 890 y Fg(Theorem)30 b(0.5.)40 b Fi(Assume)29 b(that)h Fs(I)37 b Fi(is)30 b(non)f(critic)l(al)i(for)g Fs(V)19 b Fi(.)204 989 y(i\))30 b(If)g Fs(I)37 b Fi(is)30 b(non)f(tr)l(apping,)j(we)e(have)h(the)e(fol) t(lowing)k(c)l(omplete)d(asymptotic)h(exp)l(ansion)f(in)g Fs(C)3171 959 y Fm(1)3242 989 y Ft(\()p Fs(I)7 b Ft(\))p Fi(:)748 1180 y Fs(s)787 1145 y Fm(0)787 1200 y Fp(p)825 1180 y Ft(\()p Fs(\025;)14 b(h)p Ft(\))24 b Fq(\030)e Fs(h)1181 1145 y Fm(\000)p Fp(d)1287 1101 y Fo(X)1286 1280 y Fp(k)q Fm(\025)p Fr(0)1421 1180 y Fs(h)1469 1145 y Fp(k)1510 1180 y Fs(a)1554 1145 y Fp(V)1554 1200 y(k)q(;p)1649 1180 y Ft(\()p Fs(\025)p Ft(\))p Fs(;)184 b(h)23 b Fq(&)g Ft(0)p Fs(;)183 b(a)2437 1145 y Fp(V)2437 1200 y(k)q(;p)2555 1180 y Fq(2)23 b Fs(C)2698 1145 y Fm(1)2769 1180 y Ft(\()p Fs(I)7 b Ft(\))p Fs(;)74 b(k)26 b Fq(2)d Fl(N)t Fs(;)204 1452 y Fi(wher)l(e,)31 b(if)f(we)g(set)g Fs(c)833 1464 y Fp(d)895 1452 y Ft(=)22 b(\(2)p Fs(\031)s Ft(\))1138 1422 y Fm(\000)p Fp(d)1229 1452 y Ft(v)n(ol)o(\()p Fl(S)1420 1422 y Fp(d)o Fm(\000)p Fr(1)1537 1452 y Ft(\))p Fs(=)p Ft(2)p Fi(,)30 b(the)g(le)l(ading)h(c)l(o)l(e\016cient)f(is)392 1709 y Fs(a)436 1674 y Fp(V)436 1729 y Fr(0)p Fp(;p)528 1709 y Ft(\()p Fs(\025)p Ft(\))24 b(=)e Fs(c)787 1721 y Fp(d)840 1596 y Fo(Z)923 1709 y Ft(\()p Fs(\025)d Fq(\000)f Fs(V)h Ft(\()p Fs(x)p Ft(\)\))1326 1634 y Fe(d)p Fd(\000)p Fh(2)p 1326 1647 104 3 v 1364 1680 a(2)1316 1729 y Fr(+)1463 1709 y Fq(\000)1546 1601 y Fp(p)p Fm(\000)p Fr(1)1546 1630 y Fo(X)1548 1806 y Fp(j)s Fr(=0)1665 1709 y Ft(\()p Fq(\000)p Ft(1\))1836 1674 y Fp(j)1871 1709 y Ft(\()p Fs(d=)p Ft(2)f Fq(\000)g Ft(1\))c Fq(\001)g(\001)g(\001)f Ft(\()p Fs(d=)p Ft(2)18 b Fq(\000)g Fs(j)5 b Ft(\))p Fs(\025)2718 1652 y Fe(d)p 2719 1661 31 3 v 2720 1694 a Fh(2)2760 1674 y Fm(\000)p Fr(1)p Fm(\000)p Fp(j)2932 1709 y Fs(V)18 b Ft(\()p Fs(x)p Ft(\))3109 1674 y Fp(j)3145 1709 y Fs(=j)5 b Ft(!)30 b Fs(dx;)143 b Ft(\(0.11\))204 1971 y Fi(with)29 b(the)g(c)l(onvention)f(that)h Ft(\()p Fs(d=)p Ft(2)15 b Fq(\000)h Ft(1\))e Fq(\001)g(\001)g(\001)f Ft(\()p Fs(d=)p Ft(2)i Fq(\000)h Fs(j)5 b Ft(\))23 b(=)g(1)28 b Fi(if)h Fs(j)f Ft(=)23 b(0)p Fi(.)38 b(Mor)l(e)l(over)30 b Ft(\()p Fs(t)p Ft(\))2857 1983 y Fr(+)2935 1971 y Ft(=)23 b(max)o(\()p Fs(t;)14 b Ft(0\))29 b Fi(and,)h(when)204 2087 y Fs(d)23 b Ft(=)g(2)p Fi(,)29 b Ft(\()p Fs(t)p Ft(\))548 2044 y Fp(d=)p Fr(2)p Fm(\000)p Fr(1)548 2107 y(+)763 2087 y Ft(=)22 b(1)30 b Fi(if)g Fs(t)23 b(>)g Ft(0)p Fi(,)30 b(and)g Ft(0)f Fi(otherwise)i(\(He)l(aviside)h (function\).)204 2186 y(ii\))e(If)h(the)e(fol)t(lowing)k(estimates)d (holds)h(for)f(some)h Fs(n)22 b Fq(\025)h Ft(1)29 b Fi(and)h Fs(s)23 b(>)g Ft(0)p Fi(,)30 b(lo)l(c)l(al)t(ly)h(uniformly)g(on)f Fs(I)7 b Fi(,)1028 2360 y Fq(h)p Fs(x)p Fq(i)1139 2326 y Fm(\000)p Fp(s)1228 2360 y Ft(\()p Fs(H)1329 2372 y Fr(0)1385 2360 y Ft(+)18 b Fs(V)37 b Fq(\000)18 b Fs(\025)h Fq(\006)f Fs(i)p Ft(0\))1889 2326 y Fm(\000)p Fr(1)1977 2360 y Fq(h)p Fs(x)p Fq(i)2088 2326 y Fm(\000)p Fp(s)2200 2360 y Ft(=)k Fq(O)r Ft(\(exp)q(\()p Fs(h)2595 2326 y Fm(\000)p Fp(n)2692 2360 y Ft(\)\))779 b(\(0.12\))204 2534 y Fi(then,)30 b(for)g(al)t(l)h Fs(\027)e Fq(\025)22 b Ft(0)p Fi(,)30 b(the)g(R)n(iesz)g(me)l(an)f(of)i(or)l(der)f Fs(\027)35 b Fi(has)c(the)f(fol)t(lowing)i(exp)l(ansion,)f(lo)l(c)l(al) t(ly)g(uniform)f(on)g Fs(I)818 2683 y Fo(Z)901 2704 y Fp(\025)864 2872 y Fm(\0001)1000 2796 y Fs(s)1039 2762 y Fm(0)1039 2817 y Fp(p)1078 2796 y Ft(\()p Fs(\026;)14 b(h)p Ft(\)\()p Fs(\025)19 b Fq(\000)f Fs(\026)p Ft(\))1541 2762 y Fp(\027)1583 2796 y Fs(d\026)23 b Ft(=)g Fs(h)1835 2762 y Fm(\000)p Fp(d)1939 2685 y Fr([)p Fp(\027)t Fr(])2014 2693 y Fh(+)1940 2717 y Fo(X)1939 2896 y Fp(k)q Fr(=0)2074 2796 y Fs(C)2139 2762 y Fp(\027)2133 2817 y(k)q(;V)2248 2796 y Ft(\()p Fs(\025)p Ft(\))p Fs(h)2408 2762 y Fp(k)2468 2796 y Ft(+)18 b Fq(O)r Ft(\()p Fs(h)2699 2762 y Fr(1+)p Fp(\027)t Fm(\000)p Fp(d)2911 2796 y Ft(\))p Fs(;)569 b Ft(\(0.13\))204 3049 y Fi(wher)l(e)30 b Ft([)p Fs(\027)5 b Ft(])530 3061 y Fr(+)615 3049 y Fi(is)31 b(the)e(smal)t(lest)i(inte)l (ger)f Fq(\025)22 b Fs(\027)5 b Fi(.)39 b(In)30 b(p)l(articular,)h(if)f Fs(\027)f Ft(=)22 b(0)p Fi(,)30 b(we)g(get)g(the)g(Weyl)g(formula)1357 3223 y Fs(s)1396 3235 y Fp(p)1435 3223 y Ft(\()p Fs(\025;)14 b(h)p Ft(\))24 b(=)e Fs(h)1791 3189 y Fm(\000)p Fp(d)1882 3223 y Fs(C)1947 3189 y Fr(0)1941 3244 y(0)p Fp(;V)2051 3223 y Ft(\()p Fs(\025)p Ft(\))e(+)e Fq(O)r Ft(\()p Fs(h)2414 3189 y Fr(1)p Fm(\000)p Fp(d)2538 3223 y Ft(\))p Fs(:)328 3397 y Ft(W)-7 b(e)31 b(precise)f(some)g(details)g(ab)r(out)g(the)h (de\014nition)g(of)f(the)h(Riesz)f(mean,)h(de\014ned)g(b)n(y)f(the)h (left)g(hand)f(side)204 3497 y(of)d(\(0)p Fs(:)p Ft(13\).)36 b(It)28 b(m)n(ust)g(b)r(e)g(understo)r(o)r(d)f(as)999 3733 y Fs(<)c(f)9 b(u)1185 3745 y 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Fs(H)2061 4072 y Fr(1)2099 4060 y Ft(\))29 b(b)n(y)g(theorem)g(0)p Fs(:)p Ft(4.)40 b(Moreo)n(v)n(er)26 b(it)k(is)f(kno)n(wn)f(that,)204 4160 y(in)g(this)g(case,)e Fs(\033)708 4172 y Fp(pp)781 4160 y Ft(\()p Fs(H)882 4172 y Fr(1)920 4160 y Ft(\))19 b Fq(\\)f Ft(\(0)p Fs(;)c Fq(1)p Ft(\))28 b(is)g(empt)n(y)f(\(see)h([1) o(,)g(34)o(]\).)328 4259 y(W)-7 b(e)25 b(add)g(t)n(w)n(o)e(others)h (remarks)f(on)h(this)h(theorem.)35 b(First,)25 b(when)g Fs(d)g Ft(is)f(ev)n(en,)h(all)f(the)h(terms)f(corresp)r(onding)204 4359 y(to)30 b Fs(j)j Fq(\025)28 b Fs(d=)p Ft(2)i(v)-5 b(anish)30 b(in)h(\(0)p Fs(:)p Ft(11\))f(\(if)h(ev)n(er)f Fs(d=)p Ft(2)d Fq(\024)g Fs(p)21 b Fq(\000)f Ft(1\).)45 b(In)31 b(particular,)f(if)h Fs(d)d Ft(=)g(2,)j(the)g(leading)f(term)g (is)h(the)204 4458 y(same)c(as)g(for)g(the)h(usual)f(SSF,)h(and)g(more) e(generally)-7 b(,)27 b(when)g Fs(d)h Ft(is)g(ev)n(en,)f Fs(a)2564 4428 y Fp(V)2564 4479 y Fr(0)p Fp(;p)2678 4458 y Ft(=)c Fs(a)2810 4428 y Fp(V)2810 4485 y Fr(0)p Fp(;d=)p Fr(2)2996 4458 y Ft(for)k(all)g Fs(p)c Fq(\025)g Fs(d=)p Ft(2.)328 4570 y(The)35 b(second)f(p)r(oin)n(t)h(is)g(that)g (su\016cien)n(t)f(conditions)h(for)f(the)h(v)-5 b(alidit)n(y)34 b(of)h(the)g(estimates)g(\(0)p Fs(:)p Ft(12\))e(can)i(b)r(e)204 4670 y(found)28 b(in)g(the)g(w)n(ork)e(of)h(Burq)g([10)o(],)h(com)n (bined)f(with)i(the)e(one)h(of)f(Bruneau-P)n(etk)n(o)n(v)e([9)o(].)328 4793 y(In)k(the)g(next)g(theorem,)g(w)n(e)f(pro)n(v)n(e)f(that)i(the)g (phase)f(shift)i Fq(\006)p Ft(1)p Fs(=)p Ft(2)d(whic)n(h)h(o)r(ccurs)g (for)g(the)h(SSF)g Fs(s)3407 4805 y Fr(1)3473 4793 y Ft(close)f(to)204 4892 y(a)i(resonance,)g(holds)h(for)f(all)h Fs(s)1191 4904 y Fp(p)1229 4892 y Ft(.)47 b(This)31 b(phenomenon)g(is)f (kno)n(wn)h(as)f(Breit-Wigner)f(form)n(ula)h(\(see)h(the)h(b)r(o)r(ok) 204 4992 y([39)o(])j(for)e(its)i(ph)n(ysical)f(in)n(terpretation\))g (and)g(has)g(already)f(b)r(een)i(considered)e(b)n(y)h(sev)n(eral)f (authors.)57 b(Let)34 b(us)204 5092 y(quote)27 b(the)h(pap)r(ers)e(of)i (Com)n(b)r(es-Duclos-Klein-Seiler)d([14)o(],)i(P)n(etk)n(o)n(v-Zw)n (orski)c([31)o(],)28 b(and)f(esp)r(ecially)g(the)h(ones)204 5191 y(of)e(G)n(\023)-39 b(erard-Martinez-Rob)r(ert)22 b([17)o(],)k(for)f Fs(s)1553 5203 y Fr(1)1616 5191 y Ft(and)h(the)g(scattering)f(cross)f(section,)i(and)g(Rob)r(ert)f([34)o (],)i(whic)n(h)f(w)n(e)204 5291 y(ha)n(v)n(e)g(follo)n(w)n(ed.)1955 5540 y(5)p eop %%Page: 6 6 6 5 bop 328 531 a Ft(W)-7 b(e)26 b(consider)d(an)i(energy)f(lev)n(el)g Fs(\025)1399 543 y Fr(0)1460 531 y Fq(2)f Fs(I)32 b Ft(\(whic)n(h)25 b(can)g(b)r(e)g(critical)g(for)f Fs(V)19 b Ft(\))25 b(and)g(the)g(set)g (of)g(resonances)e(\000\()p Fs(h)p Ft(\))204 631 y(close)k(to)g Fs(\025)552 643 y Fr(0)617 631 y Ft(de\014ned)h(b)n(y)g (Hel\013er-Sj\177)-42 b(ostrand)26 b(in)i([22)o(].)37 b(W)-7 b(e)28 b(supp)r(ose,)g(as)f(in)g([17],)g(that:)204 731 y Fq(\017)33 b Ft(there)g(exists)g(a)h(connected)f(op)r(en)h (subset)1687 710 y(^)1668 731 y Fs(O)i Fq(\032)c Fl(R)1918 700 y Fp(d)1997 731 y Ft(and)h(a)g(compact)g(connected)h(set)f Fs(U)42 b Fc(b)3321 710 y Ft(^)3302 731 y Fs(O)37 b Ft(suc)n(h)c(that) 204 830 y Fs(V)42 b(>)22 b(\025)429 842 y Fr(0)495 830 y Ft(in)611 809 y(^)592 830 y Fs(O)f Fq(n)d Fs(U)9 b Ft(,)27 b(and)h Fs(V)41 b(<)23 b(\025)1239 842 y Fr(0)1304 830 y Ft(in)28 b Fs(U)f Fq([)19 b Fl(R)1613 800 y Fp(d)1676 830 y Fq(n)1755 809 y Ft(^)1736 830 y Fs(O)s Ft(.)204 930 y Fq(\017)27 b Fs(V)47 b Ft(is)27 b(holomorphic)f(in)i Fq(f)5 b Ft(~)-47 b Fs(x)23 b Fq(2)h Fl(C)1268 900 y Fp(d)1340 930 y Fq(j)k(j)p Ft(Im)33 b(~)-47 b Fs(x)p Fq(j)23 b(\024)g Fs(")1761 942 y Fr(0)1798 930 y Fq(j)p Ft(Re)33 b(~)-47 b Fs(x)p Fq(j)p Fs(;)42 b Ft(Re)33 b(~)-48 b Fs(x)28 b Ft(in)g(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)g Fl(R)3118 900 y Fp(d)3182 930 y Fq(n)3260 909 y Ft(^)3242 930 y Fs(O)r Fq(g)p Ft(,)204 1029 y Fq(\017)i Ft(lim)390 1044 y Fm(j)p Fp(t)p Fm(j\0451)605 1029 y Fq(j)p Ft(\005)690 962 y Fo(\000)728 1029 y Fs(\036)777 999 y Fp(t)807 1029 y Ft(\()p Fs(x;)14 b(\030)t Ft(\))995 962 y Fo(\001)1034 1029 y Fq(j)26 b Ft(=)f Fq(1)p Ft(,)k(for)g(all)g(\()p Fs(x;)14 b(\030)t Ft(\))27 b Fq(2)f Fl(R)1904 999 y Fp(d)1968 1029 y Fq(n)2048 1008 y Ft(^)2029 1029 y Fs(O)c Fq(\002)d Fl(R)2252 999 y Fp(d)2326 1029 y Ft(suc)n(h)28 b(that)i Fs(p)p Ft(\()p Fs(x;)14 b(\030)t Ft(\))26 b(=)g Fs(\030)3083 999 y Fr(2)3139 1029 y Ft(+)19 b Fs(V)g Ft(\()p Fs(x)p Ft(\))27 b(=)e Fs(\025)3566 1041 y Fr(0)3604 1029 y Ft(.)41 b Fs(\036)3717 999 y Fp(t)204 1138 y Ft(is)27 b(the)h(Hamiltonian)g (\015o)n(w)f(of)g Fs(p)h Ft(and)f(\005)c(:)g Fs(T)1598 1108 y Fp(?)1636 1138 y Fl(R)1690 1108 y Fp(d)1757 1138 y Fq(!)h Fl(R)1918 1108 y Fp(d)1990 1138 y Ft(the)k(pro)5 b(jection.)204 1237 y Fq(\017)27 b Ft(There)g(exists)g(a)g(family)g(of) h(op)r(en)f(sets)h(\012\()p Fs(h)p Ft(\))23 b Fq(2)g Fl(C)15 b Ft(,)34 b(suc)n(h)27 b(that)h Fq(\\)2328 1249 y Fp(h>)p Fr(0)2456 1237 y Ft(\012\()p Fs(h)p Ft(\))c(=)e Fq(f)p Fs(\025)2829 1249 y Fr(0)2866 1237 y Fq(g)p Ft(,)28 b Fs(I)7 b Ft(\()p Fs(h)p Ft(\))23 b(:=)g(\012\()p Fs(h)p Ft(\))18 b Fq(\\)h Fl(R)29 b Fq(6)p Ft(=)23 b Fq(;)p Ft(,)204 1337 y(and)32 b(suc)n(h)f(that)h(\012\()p Fs(h)p Ft(\))22 b Fq(\\)g Ft(\000\()p Fs(h)p Ft(\))32 b(con)n(tains)f(only)g (one)h(resonance)e(that)i(w)n(e)g(note)f Fs(r)r Ft(\()p Fs(h)p Ft(\).)51 b(Moreo)n(v)n(er)29 b(w)n(e)j(assume)204 1437 y(that)c(there)f(exist)h Fs(")22 b(>)h Ft(0)k(and)h Fs(c)23 b(>)f Ft(0)27 b(suc)n(h)h(that)g(dist\(\000\()p Fs(h)p Ft(\))p Fs(;)14 b(@)5 b Ft(\012\()p Fs(h)p Ft(\)\))24 b Fq(\025)e Fs(ce)2562 1407 y Fm(\000)p Fp("=h)2722 1437 y Ft(.)328 1536 y(Under)28 b(these)g(assumptions)e(and)i(\(0)p Fs(:)p Ft(10\),)f(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n(wing)f(result:)204 1692 y Fg(Theorem)j(0.6)h(\(Breit-Wigner)g(t)m(yp)s(e)h(form)m(ula\).) 40 b Fi(L)l(et)j Fs(\016)s Ft(\()p Fs(h)p Ft(\))h Fi(b)l(e)g(a)g (family)i(of)f(p)l(ositive)g(r)l(e)l(al)f(numb)l(ers)204 1791 y(such)34 b(that,)j Ft(lim)714 1803 y Fp(h)p Fm(&)p Fr(0)870 1791 y Fs(h)918 1761 y Fm(\000)p Fp(d)1009 1791 y Fs(\016)s Ft(\()p Fs(h)p Ft(\))32 b(=)f(0)p Fi(,)36 b Ft(lim)1507 1803 y Fp(h)p Fm(&)p Fr(0)1663 1791 y Fq(j)p Ft(Im)f Fs(r)r Ft(\()p Fs(h)p Ft(\))p Fq(j)1994 1761 y Fm(\000)p Fr(1)2085 1791 y Fs(\016)s Ft(\()p Fs(h)p Ft(\))d(=)f(+)p Fq(1)p Fi(,)36 b(and)f(satisfying)h Ft(Re)f Fs(r)r Ft(\()p Fs(h)p Ft(\))23 b Fq(\006)e Fs(\016)s Ft(\()p Fs(h)p Ft(\))33 b Fq(2)204 1891 y Fs(I)7 b Ft(\()p Fs(h)p Ft(\))p Fi(,)30 b(for)h(al)t(l)g Fs(h)e Fi(smal)t(l)i(enough.)39 b(Then)1097 2099 y Ft(lim)1085 2153 y Fp(h)p Fm(&)p Fr(0)1237 2099 y Fs(s)1276 2111 y Fp(p)1315 2099 y Ft(\(Re)30 b Fs(r)r Ft(\()p Fs(h)p Ft(\))19 b Fq(\006)f Fs(\016)s Ft(\()p Fs(h)p Ft(\))p Fs(;)c(h)p Ft(\))19 b Fq(\000)f Fs(s)2138 2111 y Fp(p)2177 2099 y Ft(\(Re)30 b Fs(r)r Ft(\()p Fs(h)p Ft(\))p Fs(;)14 b(h)p Ft(\))24 b(=)e Fq(\006)2791 2043 y Ft(1)p 2791 2080 42 4 v 2791 2156 a(2)2842 2099 y Fs(:)204 2387 y Fv(1)135 b(F)-11 b(redholm)44 b(determinan)l(ts)j (and)d(sp)t(ectral)i(distributions)204 2569 y Ft(W)-7 b(e)28 b(are)f(going)g(to)h(use)g(the)h(Sc)n(hatten)f(ideals)f Fg(S)1726 2581 y Ff(p)1801 2569 y Ft(\(on)h Fs(L)2006 2539 y Fr(2)2042 2569 y Ft(\()p Fl(R)2129 2539 y Fp(d)2173 2569 y Ft(\)\))h(th)n(us)f(w)n(e)g(recall)f(some)h(basic)f (de\014nitions)h(and)204 2669 y(prop)r(erties)f(\(for)i(more)f(details) g(see)g(Goh)n(b)r(erg-Krein)f([19)o(],)i(Simon)f([38)o(])h(or)f(Y)-7 b(afaev)28 b([40)o(]\).)40 b(An)29 b(op)r(erator)e Fs(A)i Ft(is)204 2768 y(said)f(to)h(b)r(e)h(in)f Fg(S)745 2780 y Ff(p)820 2768 y Ft(if)g Fq(j)p Fs(A)p Fq(j)d Ft(=)f(\()p Fs(A)1215 2738 y Fp(?)1254 2768 y Fs(A)p Ft(\))1348 2738 y Fr(1)p Fp(=)p Fr(2)1482 2768 y Ft(is)j(compact)h(with)g(a)g(sp)r (ectrum)g(in)g Fs(l)2650 2738 y Fp(p)2688 2768 y Ft(\()p Fl(N)t Ft(\))q(.)47 b(If)29 b(this)g(is)g(true,)g(then)h Fs(A)f Ft(is)204 2868 y(compact)e(as)g(w)n(ell)g(and)h(its)f(sp)r (ectrum)h(\()p Fs(\025)1526 2880 y Fp(j)1562 2868 y Ft(\))1594 2880 y Fp(j)s Fm(2)p Fn(N)1739 2868 y Fq(2)23 b Fs(l)1844 2838 y Fp(p)1882 2868 y Ft(\()p 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y Ft(\()p Fl(R)q Ft(\),)42 b(the)36 b(space)d(of)i(compactly)f(supp)r(orted)h (distributions,)h(one)e(can)h(de\014ne)f(the)i(so)e(called)204 4265 y Fi(Stieltjes)c(tr)l(ansform)e Ft(of)f Fs(u)g Ft(as)g(the)h (follo)n(wing)f(holomorphic)f(function)i(on)g Fl(C)39 b Fq(n)18 b Ft(supp\()p Fs(u)p Ft(\))1544 4434 y Fs(U)9 b Ft(\()p Fs(z)t Ft(\))22 b(=)p Fs(<)h(u;)14 b Ft(\()p Fs(:)k Fq(\000)g Fs(z)t Ft(\))2208 4400 y Fm(\000)p Fr(1)2320 4434 y Fs(>;)204 4603 y Ft(and)27 b(one)g(can)h(reco)n(v)n(er)d Fs(u)i Ft(from)g Fs(U)37 b Ft(thanks)27 b(to)g(the)h(in)n(v)n(ersion)e (form)n(ula:)1048 4773 y(lim)1015 4827 y Fp(")p Fm(&)p Fr(+0)1211 4773 y Fs(U)9 b Ft(\()p Fs(\025)18 b Ft(+)g Fs(i")p Ft(\))h Fq(\000)f Fs(U)9 b Ft(\()p Fs(\025)18 b Fq(\000)g Fs(i")p Ft(\))23 b(=)g(2)p Fs(i\031)s(u)p Ft(\()p Fs(\025)p Ft(\))p Fs(;)180 b Ft(in)28 b Fq(D)2765 4738 y Fm(0)2788 4773 y Ft(\()p Fl(R)p Ft(\))q Fs(:)204 4992 y Ft(This)20 b(is)h(essen)n(tially)e(equiv)-5 b(alen)n(t)21 b(to)f(the)h(fact)g(that)g(2)p Fs(i\031)s(\016)1959 5004 y Fr(0)1995 4992 y Ft(\()p Fs(\025)p Ft(\))j(=)f(\()p Fs(\025)t Fq(\000)t Fs(i)p Ft(0\))2475 4962 y Fm(\000)p Fr(1)2569 4992 y Fq(\000)t Ft(\()p Fs(\025)t Ft(+)t Fs(i)p Ft(0\))2894 4962 y Fm(\000)p Fr(1)3004 4992 y Ft(and)e(v)n(ery)e (elemen)n(tary)-7 b(.)204 5092 y(When)33 b(considering)e(temp)r(erate)i (distributions)f(in)h Fq(S)1938 5061 y Fm(0)1962 5092 y Ft(\()p Fl(R)p Ft(\),)40 b(one)32 b(cannot)g(de\014ne)h(the)g (Stieltjes)h(transform)d(in)204 5191 y(general.)k(Nev)n(ertheless,)25 b(under)g(some)h(conditions)f(at)h(in\014nit)n(y)g(on)f Fs(u)e Fq(2)g(S)2559 5161 y Fm(0)2583 5191 y Ft(\()p Fl(R)p Ft(\))q(,)32 b(describ)r(ed)26 b(b)r(elo)n(w,)f(w)n(e'll)h(sho)n (w)204 5291 y(that)i(it)g(is)f(p)r(ossible.)1955 5540 y(6)p eop %%Page: 7 7 7 6 bop 328 531 a Ft(Assuming)28 b(that)g Fs(u)f Ft(is)g(temp)r(erate,) h(the)g(follo)n(wing)f(form)n(ula)f(mak)n(es)h(sense)1460 706 y Fs(U)1517 718 y Fp(s)1552 706 y Ft(\()p Fs(z)t Ft(\))c(=)p Fs(<)f(u;)14 b Ft(\()p Fs(:)k Fq(\000)g Fs(z)t Ft(\))2150 671 y Fm(\000)p Fp(s)2260 706 y Fs(>)1251 b Ft(\(1.1\))204 880 y(if)28 b(Re)g Fs(s)23 b Fq(\035)g Ft(1)k(and)g Fs(z)36 b(=)-51 b Fq(2)23 b Fl(R)34 b Ft(\(or)27 b Fs(z)f Fq(2)d Fl(C)40 b Fq(n)18 b Ft([inf)j(supp\()p Fs(u)p Ft(\))p Fs(;)14 b Fq(1)p Ft(\))28 b(if)g(the)g(inf)34 b(is)28 b(\014nite\).)38 b(Since)28 b(w)n(e)f(ha)n(v)n(e)1284 1054 y(\()p Fs(\025)20 b Fq(\000)e Fs(z)t Ft(\))1542 1020 y Fm(\000)p Fr(1)1653 1054 y Ft(=)23 b Fs(@)1785 1066 y Fp(z)1837 987 y Fo(\000)1875 1054 y Fs(@)1919 1066 y Fp(s)1955 1054 y Ft(\()p Fs(\025)c Fq(\000)f Fs(z)t Ft(\))2212 1020 y Fm(\000)p Fp(s)2299 987 y Fo(\001)2337 1088 y Fm(j)2357 1096 y Fe(s)p Fh(=0)2477 1054 y Fs(;)1076 b Ft(\(1.2\))204 1248 y(where)22 b Fs(z)k Fq(2)d Fl(C)g Fq(n)8 b Ft([)p Fs(\025;)14 b Fq(1)p Ft(\),)29 b(it)23 b(is)f(natural)g(to)g(consider)f(the)i(existence)f(of)g(a)g (meromorphic)e(con)n(tin)n(uation)i(of)g Fs(U)3582 1260 y Fp(s)3617 1248 y Ft(\()p Fs(z)t Ft(\),)204 1347 y(with)i(resp)r(ect)f (to)g Fs(s)p Ft(,)h(at)f(0.)35 b(Note)24 b(that)f(the)h(complex)f(p)r (o)n(w)n(ers)f(are)g(de\014ned)i(on)f Fl(C)i Fq(n)10 b Ft(\()p Fq(\0001)p Fs(;)k Ft(0])28 b(with)c(the)g(principal)204 1447 y(determination)j(of)h(the)g(log.)328 1546 y(W)-7 b(e)28 b(giv)n(e)f(no)n(w)f(su\016cien)n(t)i(conditions)f(to)g(get)h(a) f(meromorphic)f(con)n(tin)n(uation.)36 b(Assume)27 b(that)h Fs(u)22 b Fq(2)i(S)3576 1516 y Fm(0)3600 1546 y Ft(\()p Fl(R)p Ft(\),)204 1646 y(with)39 b(inf)7 b(\(supp\()p Fs(u)p Ft(\)\))42 b Fs(>)e Fq(\0001)p Ft(,)h(and)e(that)f(its)h (Laplace)e(transform,)j(de\014ned)f(as)f Fs(Lu)p Ft(\()p Fs(t)p Ft(\))j(=)p Fs(<)f(u;)14 b(f)3381 1658 y Fp(t)3450 1646 y Fs(>)38 b Ft(\(with)204 1746 y Fs(f)245 1758 y Fp(t)274 1746 y Ft(\()p Fs(\025)p Ft(\))24 b(=)e Fs(e)536 1716 y Fm(\000)p Fp(t\025)657 1746 y Ft(\))27 b(has)h(the)g(follo)n (wing)e(\014nite)i(expansion)974 2012 y Fs(Lu)p Ft(\()p Fs(t)p Ft(\))23 b(=)1323 1908 y Fp(n)1284 1933 y Fo(X)1286 2110 y Fp(j)s Fr(=0)1418 1901 y Fp(k)q Fr(\()p Fp(j)s Fr(\))1418 1933 y Fo(X)1417 2112 y Fp(k)q Fr(=0)1552 2012 y Fs(a)1596 2024 y Fp(j)s(k)1668 2012 y Fs(t)1698 1978 y Fp(\013)1741 1986 y Fe(j)1790 2012 y Ft(log)1897 1975 y Fp(k)1952 2012 y Fs(t)18 b Ft(+)g Fq(O)r Ft(\()p Fs(t)2213 1978 y Fp(\014)2251 1986 y Fe(n)2296 2012 y Ft(\))p Fs(;)181 b(t)23 b Fq(&)g Ft(0)2733 1978 y Fr(+)2787 2012 y Fs(;)766 b Ft(\(1.3\))204 2271 y(where)28 b(the)g Fs(a)632 2283 y Fp(j)s(k)732 2271 y Ft(are)g(complex)g(n)n(um)n(b)r (ers,)g Fs(\013)1610 2283 y Fr(0)1672 2271 y Fs(<)23 b(\013)1813 2283 y Fr(1)1875 2271 y Fs(<)h Fq(\001)14 b(\001)g(\001)24 b Fs(<)g(\013)2227 2283 y Fp(n)2301 2271 y Ft(are)j(real)h(n)n(um)n(b)r(ers)g(and)g Fs(\014)3148 2283 y Fp(n)3217 2271 y Fs(>)c Ft(0.)39 b(Moreo)n(v)n(er)204 2370 y(w)n(e)27 b(assume,)g(as)g(in)h([29)o(],)g(that)1362 2545 y Fq(\000)p Fs(\013)1480 2557 y Fp(j)1538 2545 y Fq(2)23 b Fl(N)33 b Fq(\))23 b Fs(a)1849 2557 y Fp(j)s(k)1944 2545 y Ft(=)f(0)55 b Fq(8)27 b Fs(k)f(>)c 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Ft(+)d(1)p Fs(;)14 b Ft(+)p Fq(1)p Ft(\);)40 b(in)d(that)f(case)g(w)n(e)g(kno)n(w,)h(thanks)f(to)g (the)h(pro)r(of)f(of)g Fs(i)p Ft(\),)i(that)f Fs(U)2905 842 y Fp(s)2940 830 y Ft(\()p Fs(z)t Ft(\))f(is)g(holomorphic)f(on)204 930 y Fs(V)19 b Ft(\(0\))i Fq(\002)g(f)p Ft(Re)32 b Fs(z)h(<)d(r)r Fq(g)p Ft(,)k(where)d Fs(s)f Fq(2)h Fs(V)19 b Ft(\(0\))32 b(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)h(0.)50 b(Hence)32 b Fs(@)2673 942 y Fp(z)2711 930 y Fs(@)2755 942 y Fp(s)2791 930 y Fs(U)9 b Ft(\(0)p Fs(;)14 b(z)t Ft(\))20 b Fq(\000)h Fs(@)3193 942 y Fp(z)3231 930 y Fs(@)3275 942 y Fp(s)3311 930 y Fs(U)9 b Ft(\(0)p Fs(;)18 b Ft(\026)-46 b Fs(z)s Ft(\))30 b Fq(!)h Ft(0)204 1029 y(when)d(Im)f(\()p Fs(z)t Ft(\))c Fq(&)g Ft(0,)28 b(lo)r(cally)f(uniformly)g(with)h(resp)r(ect)f (to)h(Re)g Fs(z)t Ft(,)f(and)g(this)h(completes)f(the)h(pro)r(of.)310 b Fc(\003)328 1179 y Ft(Remark)27 b(that,)h(if)g(w)n(e)f(replace)g(the) h(assumption)f(\(1)p Fs(:)p Ft(3\))g(b)n(y)h(the)g(follo)n(wing)1148 1452 y Fs(Lu)p Ft(\()p Fs(t)p Ft(\))23 b Fq(\030)1484 1349 y Fm(1)1457 1373 y Fo(X)1460 1550 y Fp(j)s Fr(=0)1592 1341 y Fp(k)q Fr(\()p Fp(j)s Fr(\))1591 1373 y Fo(X)1591 1552 y Fp(k)q Fr(=0)1726 1452 y Fs(a)1770 1464 y Fp(j)s(k)1841 1452 y Fs(t)1871 1418 y Fp(\013)1914 1426 y Fe(j)1963 1452 y Ft(log)2071 1415 y Fp(k)2125 1452 y Fs(t;)180 b(t)23 b Fq(&)g Ft(0)2559 1418 y Fr(+)2614 1452 y Fs(;)939 b Ft(\(1.5\))204 1718 y(with)35 b Fs(\013)453 1730 y Fp(j)523 1718 y Fs(<)g(\013)676 1730 y Fp(j)s Fr(+1)830 1718 y Ft(and)g Fs(\013)1052 1730 y Fp(j)1122 1718 y Fq(!)g(1)p Ft(,)i(w)n(e)e(get)f(a)h(meromorphic)e(con)n(tin)n(uation)h (on)h Fl(C)55 b Ft(and)35 b(the)g(p)r(oles)g(are)f(the)204 1817 y(p)r(oin)n(ts)1361 1917 y Fq(\000)p Fs(\013)1479 1929 y Fp(j)1533 1917 y Fq(\000)18 b Fs(l)r(;)179 b Ft(suc)n(h)27 b(that)h Fs(\013)2265 1929 y Fp(j)2319 1917 y Ft(+)18 b Fs(l)33 b(=)-51 b Fq(2)24 b Fl(N)204 2066 y Ft(where)f Fs(j;)14 b(l)25 b Fq(2)e Fl(N)t Ft(.)42 b(Moreo)n(v)n(er,)22 b(if)j(for)e(all)h Fs(j)29 b Ft(and)24 b Fs(l)r Ft(,)g Fs(\013)1799 2078 y Fp(j)1845 2066 y Ft(+)11 b Fs(l)34 b(=)-52 b Fq(2)24 b Fl(N)t Ft(,)31 b(then)24 b(the)g(meromorphic)f(con) n(tin)n(uation)g(of)h Fs(U)3605 2078 y Fp(s)3640 2066 y Ft(\()p Fs(z)t Ft(\))204 2166 y(v)-5 b(anishes)27 b(when)h Fq(\000)p Fs(s)22 b Fq(2)i Fl(N)37 b Ft(\(see)28 b([4)o(,)g(7])f(for)g (applications\).)328 2290 y(No)n(w)22 b(w)n(e)f(turn)h(to)g(an)f (example)h(of)g(computation)f(of)h(Stieltjes)g(transform.)34 b(Consider)21 b(a)h(family)f(of)h(abstract)204 2390 y(self-adjoin)n(t) 31 b(op)r(erators)e Fs(H)1075 2402 y Fp(\017)1136 2390 y Ft(=)g Fs(H)1299 2402 y Fr(0)1357 2390 y Ft(+)20 b Fs(\017V)f Ft(,)32 b(dep)r(ending)g(on)f Fs(\017)e Fq(2)g Ft([0)p Fs(;)14 b Ft(1].)47 b(Assume)31 b(moreo)n(v)n(er)e(that)i Fs(H)3429 2402 y Fr(0)3498 2390 y Ft(is)g(semi)204 2489 y(b)r(ounded)24 b(from)g(b)r(elo)n(w)f(and)h(that)g Fs(V)43 b Ft(is)24 b Fs(H)1537 2501 y Fr(0)1598 2489 y Ft(compact;)h(more)e (precisely)-7 b(,)24 b(w)n(e)g(assume)f(that,)i(for)f(one)f Fs(z)36 b(=)-52 b Fq(2)24 b Fs(\033)s Ft(\()p Fs(H)3677 2501 y Fr(0)3715 2489 y Ft(\))204 2589 y(\(and)k(hence)f(for)g(all\))h (w)n(e)f(ha)n(v)n(e)1182 2770 y Fs(V)19 b Ft(\()p Fs(H)1350 2782 y Fr(0)1406 2770 y Fq(\000)f Fs(z)t Ft(\))1564 2736 y Fm(\000)p Fp(\027)1680 2770 y Fq(2)23 b Fg(S)1811 2782 y Ff(p)1857 2770 y Fs(;)180 b Ft(for)27 b(some)55 b Fs(\027)28 b Fq(2)c Ft([0)p Fs(;)14 b Ft(1\))p Fs(:)204 2952 y Ft(Then)38 b(the)g(domain)g(of)g Fs(H)1064 2964 y Fp(\017)1133 2952 y Ft(is)g(indep)r(enden)n(t)h(of)f Fs(\017)p Ft(,)i Fs(H)1979 2964 y Fp(\017)2051 2952 y Fq(\025)g Ft(min)q(\(inf)21 b Fs(\033)s Ft(\()p Fs(H)2593 2964 y Fr(0)2631 2952 y Ft(\))p Fs(;)14 b Ft(inf)21 b Fs(\033)s Ft(\()p Fs(H)2966 2964 y Fr(0)3029 2952 y Ft(+)k Fs(V)19 b Ft(\)\))38 b(and)g(one)g(can) 204 3052 y(consider)26 b(the)i(functional)965 3312 y Fs(<)22 b(u;)14 b(f)31 b(>)p Ft(=)23 b(T)-7 b(r)1461 3145 y Fo(0)1461 3295 y(@)1534 3312 y Fs(f)9 b Ft(\()p Fs(H)1685 3324 y 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b(and)e(is)h(w)n(ell)g(de\014ned)204 3793 y(when)e Fs(f)j Ft(=)23 b Fs(f)622 3805 y Fp(z)687 3793 y Ft(with)29 b Fs(f)918 3805 y Fp(z)955 3793 y Ft(\()p Fs(\025)p Ft(\))24 b(=)f(\()p Fs(\025)c Fq(\000)f Fs(z)t Ft(\))1436 3762 y Fm(\000)p Fr(1)1525 3793 y Ft(.)37 b(In)28 b(that)g(case,)e(w)n(e)i(ha)n(v)n(e)973 3974 y Fs(<)23 b(u;)14 b(f)1187 3986 y Fp(z)1247 3974 y Fs(>)p Ft(=)22 b(\()p Fq(\000)p Ft(1\))1570 3940 y Fp(p)1608 3974 y Ft(T)-7 b(r)1708 3907 y Fo(\000)1746 3974 y Ft(\()p Fs(H)1847 3986 y Fr(0)1903 3974 y Ft(+)18 b Fs(V)37 b Fq(\000)18 b Fs(z)t Ft(\))2229 3940 y Fm(\000)p Fr(1)2318 3907 y Fo(\000)2356 3974 y Fs(V)h Ft(\()p Fs(H)2524 3986 y Fr(0)2580 3974 y Fq(\000)f Fs(z)t Ft(\))2738 3940 y Fm(\000)p Fr(1)2826 3907 y Fo(\001)2865 3924 y Fp(p)2903 3907 y Fo(\001)2955 3974 y Fs(;)204 4156 y Ft(since)32 b Fs(@)461 4126 y Fp(j)456 4176 y(\017)496 4156 y Fs(R)559 4168 y Fp(\017)591 4156 y Ft(\()p Fs(z)t Ft(\))f(=)g Fs(j)5 b Ft(!\()p Fq(\000)p Ft(1\))1058 4126 y Fp(j)1092 4156 y Fs(R)1155 4168 y Fp(\017)1187 4156 y Ft(\()p Fs(z)t Ft(\)\()p Fs(V)19 b(R)1456 4168 y Fp(\017)1488 4156 y Ft(\()p Fs(z)t Ft(\)\))1627 4126 y Fp(j)1662 4156 y Ft(.)52 b(Using)33 b(the)g(Cauc)n(h)n(y)e(form)n(ula,)i(w)n(e)f(can)g(de\014ne) h Fs(<)e(u;)14 b(f)39 b(>)33 b Ft(for)204 4255 y Fs(f)9 b Ft(\()p Fs(\025)p Ft(\))23 b(=)g(\()p Fs(\025)15 b Fq(\000)f Fs(E)5 b Ft(\))749 4225 y Fm(\000)p Fp(s)836 4255 y Ft(,)26 b(Re)g Fs(s)d(>)g Ft(1)i(and)g Fs(E)30 b Ft(real,)25 b(b)r(elo)n(w)g(the)h(sp)r(ectra.)36 b(Then,)26 b(b)n(y)f(the)h(Mellin)g(transform,)f(w)n(e)g(can)204 4355 y(extend)30 b Fs(u)f Ft(to)h(the)g(class)f(of)h(Sc)n(h)n(w)n(artz) f(functions)h(supp)r(orted)f(in)i(\()p Fs(E)5 b(;)14 b Fq(1)p Ft(\),)31 b(and)e(th)n(us)h Fs(u)g Ft(de\014nes)g(a)f(temp)r (erate)204 4455 y(distribution.)328 4554 y(Using)f(these)f (considerations,)f(w)n(e)h(can)h(de\014ne)f Fs(U)1909 4566 y Fp(s)1945 4554 y Ft(\()p Fs(z)t Ft(\))g(as)g(in)h(\(1)p Fs(:)p Ft(1\))f(and)h(w)n(e)f(get)g(the)h(follo)n(wing)f(result.)204 4720 y Fg(Prop)s(osition)j(1.2.)40 b Fi(F)-6 b(or)30 b(al)t(l)h(c)l(omplex)g(numb)l(er)e Fs(z)36 b(=)-51 b Fq(2)23 b Ft([min)2115 4652 y Fo(\000)2167 4720 y Ft(inf)d Fs(\033)s Ft(\()p Fs(H)2432 4732 y Fr(0)2470 4720 y Ft(\))p Fs(;)14 b Ft(inf)21 b Fs(\033)s Ft(\()p Fs(H)2805 4732 y Fr(0)2862 4720 y Ft(+)d Fs(V)h Ft(\))3044 4652 y Fo(\001)3082 4720 y Fs(;)14 b Fq(1)p Ft(\))p Fi(,)31 b Fs(s)23 b Fq(7!)h Fs(U)3516 4732 y Fp(s)3551 4720 y Ft(\()p Fs(z)t Ft(\))30 b Fi(is)204 4819 y(holomorphic)j(in)d(the)g(r)l(e)l(gion)g Ft(Re)g Fs(s)22 b(>)h Fq(\000)p Fs(p)p Ft(\(1)18 b Fq(\000)g Fs(\027)5 b Ft(\))30 b Fi(and)g(we)g(have)1274 5001 y Fs(@)1318 5013 y Fp(s)1354 5001 y Fs(U)1411 5013 y Fp(s)1446 5001 y Ft(\()p Fs(z)t Ft(\))1553 5016 y Fm(j)1573 5024 y Fe(s)p Fh(=0)1703 5001 y Ft(=)22 b Fq(\000)14 b Ft(log)g(Det)2122 5013 y Fp(p)2161 4934 y Fo(\000)2199 5001 y Ft(1)k(+)g Fs(V)h(R)2472 5013 y Fr(0)2509 5001 y Ft(\()p Fs(z)t Ft(\))2616 4934 y Fo(\001)2654 5001 y Fs(;)204 5191 y Fi(wher)l(e)35 b(the)g(br)l(anch)h(of)g Ft(log)14 b Fi(Det)1212 5203 y Fp(p)1251 5124 y Fo(\000)1289 5191 y Ft(1)21 b(+)h Fs(V)d(R)1569 5203 y Fr(0)1606 5191 y Ft(\()p Fs(z)t Ft(\))1713 5124 y Fo(\001)1786 5191 y Fi(is)35 b(determine)l(d)h(by)f (the)g(c)l(ondition)h(that)f(it)g(go)l(es)g(to)g Ft(0)g Fi(when)204 5291 y Ft(Re)30 b Fs(z)21 b Fq(\000)d(j)p Ft(Im)30 b Fs(z)t Fq(j)23 b(!)g(\0001)p Fi(.)1955 5540 y Ft(8)p eop %%Page: 9 9 9 8 bop 204 531 a Fg(Pro)s(of.)56 b Ft(Firstly)34 b(w)n(e)g(assume)g (that)g Fs(z)k Fq(2)c Fs(K)6 b Ft(,)36 b(with)e Fs(K)40 b Ft(compact.)56 b(Th)n(us)34 b(w)n(e)g(can)g(consider)f(a)h(con)n (tour)f Fq(C)3660 543 y Fp(K)3724 531 y Ft(,)204 631 y(surrounding)e([min)846 564 y Fo(\000)898 631 y Ft(inf)21 b Fs(\033)s Ft(\()p Fs(H)1164 643 y Fr(0)1202 631 y Ft(\))p Fs(;)14 b Ft(inf)21 b Fs(\033)s Ft(\()p Fs(H)1537 643 y Fr(0)1596 631 y Ft(+)h Fs(V)d Ft(\))1782 564 y Fo(\001)1820 631 y Fs(;)14 b Fq(1)p Ft(\),)34 b(suc)n(h)e(that)h Fs(k)24 b Fq(\000)e Fs(z)43 b(=)-51 b Fq(2)31 b Ft(\()p Fq(\0001)p Fs(;)14 b Ft(0],)34 b(for)d(all)i Fs(k)h Fq(2)d(C)3517 643 y Fp(K)3613 631 y Ft(and)204 731 y Fs(z)26 b Fq(2)e Fs(K)30 b Ft(\(roughly)24 b(sp)r(eaking,)h(the)g(sp)r(ectrum)g(lies)g (inside)g(the)g(con)n(tour)f(and)h Fs(K)30 b Ft(lies)25 b(outside\).)36 b(By)25 b(the)g(Cauc)n(h)n(y)204 830 y(form)n(ula,)i(if)h(w)n(e)f(note)g Fs(R)977 842 y Fp(\017)1009 830 y Ft(\()p Fs(k)s Ft(\))d(=)e(\()p Fs(H)1331 842 y Fp(\017)1382 830 y Fq(\000)c Fs(k)s Ft(\))1543 800 y Fm(\000)p Fr(1)1632 830 y Ft(,)28 b(w)n(e)f(ha)n(v)n(e)1188 1024 y(\()p Fs(H)1289 1036 y Fp(\017)1339 1024 y Fq(\000)18 b Fs(z)t Ft(\))1497 990 y Fm(\000)p Fp(s)1607 1024 y Ft(=)1737 968 y Fs(i)p 1705 1005 92 4 v 1705 1081 a Ft(2)p Fs(\031)1821 911 y Fo(Z)1867 1100 y Fm(C)1903 1108 y Fe(K)1961 1024 y Ft(\()p Fs(k)k Fq(\000)c Fs(z)t Ft(\))2216 990 y Fm(\000)p Fp(s)2303 1024 y Fs(R)2366 1036 y Fp(\017)2398 1024 y Ft(\()p Fs(k)s Ft(\))p 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y Ft(\()p Fs(k)s Ft(\))2351 1539 y Fo(\000)2390 1606 y Fs(V)h(R)2520 1618 y Fr(0)2557 1606 y Ft(\()p Fs(k)s Ft(\))2667 1539 y Fo(\001)2705 1556 y Fp(p)2744 1606 y Fs(dk)2833 1489 y Fo(\023)2908 1606 y Fs(:)645 b Ft(\(1.8\))204 1825 y(Since)33 b(the)h(trace)f(norm)f (of)i Fs(R)1175 1837 y Fr(1)1212 1825 y Ft(\()p Fs(k)s Ft(\))1322 1758 y Fo(\000)1360 1825 y Fs(V)19 b(R)1490 1837 y Fr(0)1528 1825 y Ft(\()p Fs(k)s Ft(\))1638 1758 y Fo(\001)1676 1775 y Fp(p)1748 1825 y Ft(is)33 b Fq(O)r Ft(\()p Fq(j)p Fs(k)s Fq(j)2029 1795 y Fm(\000)p Fr(1)p Fm(\000)p Fp(p)p Fr(\(1)p Fm(\000)p Fp(\027)t Fr(\))2379 1825 y Ft(\),)i(the)f(righ)n(t)f(hand)g(side)g(is)g(holomorphic)204 1925 y(with)27 b(resp)r(ect)f(to)g Fs(s)h Ft(when)f(Re)h Fs(s)c(>)g Fq(\000)p Fs(p)p Ft(\(1)15 b Fq(\000)h Fs(\027)5 b Ft(\),)27 b(th)n(us)g(it)g(is)f(regular)f(at)h Fs(s)d Ft(=)g(0.)36 b(Then)26 b(if)h(w)n(e)f(di\013eren)n(tiate)h(with)204 2025 y(resp)r(ect)g(to)h Fs(z)t Ft(,)f(w)n(e)g(get,)g(thanks)h(to)f 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Fs(U)1461 2548 y Fp(s)1496 2536 y Ft(\()p Fs(z)t Ft(\))1603 2551 y Fm(j)1623 2559 y Fe(s)p Fh(=0)1756 2536 y Fq(!)d Ft(0)i(when)h Fs(z)g Fq(!)d(\0001)p Ft(.)43 b(If)30 b Fs(z)j Ft(is)c(real)g(and)h Fs(z)g Fq(!)c(\0001)p Ft(,)k(w)n(e)g(can)204 2635 y(c)n(ho)r(ose)23 b(a)i(con)n(tour)e Fq(C)29 b Ft(as)24 b(ab)r(o)n(v)n(e)g(and)g(indep)r (endan)n(t)i(of)e Fs(z)t Ft(,)h(suc)n(h)f(that)h(Re)g(\()p Fs(k)16 b Fq(\000)d Fs(z)t Ft(\))22 b Fs(>)g Ft(0)j(for)f(all)g Fs(k)i Fq(2)e(C)5 b Ft(.)35 b(W)-7 b(e)25 b(ha)n(v)n(e)739 2837 y Fs(@)783 2849 y Fp(s)819 2837 y Fs(U)876 2849 y Fp(s)911 2837 y Ft(\()p Fs(z)t Ft(\))1018 2852 y Fm(j)1038 2860 y Fe(s)p Fh(=0)1167 2837 y Ft(=)e(\()p Fq(\000)p Ft(1\))1426 2802 y Fp(p)1464 2837 y Ft(T)-7 b(r)1563 2720 y Fo(\022)1666 2781 y Fs(i)p 1635 2818 V 1635 2894 a Ft(2)p Fs(\031)1750 2724 y Fo(Z)1796 2912 y Fm(C)1853 2837 y Fq(\000)14 b Ft(log\()p Fs(k)21 b Fq(\000)d Fs(z)t Ft(\))p Fs(R)2356 2849 y Fr(1)2393 2837 y Ft(\()p Fs(k)s Ft(\))2503 2770 y Fo(\000)2542 2837 y Fs(V)h(R)2672 2849 y Fr(0)2709 2837 y Ft(\()p Fs(k)s Ft(\))2819 2770 y Fo(\001)2857 2787 y Fp(p)2896 2837 y Fs(dk)2985 2720 y Fo(\023)3576 2837 y Ft(\(1.9\))204 3034 y(where)36 b(w)n(e)g(shall)g(do)g(an)h(in)n (tegration)e(b)n(y)h(part.)64 b(T)-7 b(o)36 b(justify)h(this)g(in)n (tegration)e(b)n(y)i(part,)h(w)n(e)e(consider)g(the)204 3133 y(function)28 b Fs(')p Ft(\()p Fs(k)s Ft(\))c(=)e Fs(T)12 b(r)904 3066 y Fo(\000)942 3133 y Fs(R)1005 3145 y Fr(1)1043 3133 y Ft(\()p Fs(k)s Ft(\))1153 3066 y Fo(\000)1191 3133 y Fs(V)19 b(R)1321 3145 y Fr(0)1358 3133 y Ft(\()p Fs(k)s Ft(\))1468 3066 y Fo(\001)1507 3083 y Fp(p)1545 3066 y Fo(\001)1611 3133 y Ft(and)27 b(its)h(primitiv)n(e)f(v)-5 b(anshing)27 b(when)h Fs(k)e Fq(!)d(\0001)k Ft(de\014ned)h(b)n(y)1331 3340 y(\010\()p Fs(k)s Ft(\))23 b(=)1612 3227 y Fo(Z)1695 3248 y Fp(c)1658 3416 y Fm(\0001)1794 3340 y Fs(')p Ft(\()p Fs(u)p Ft(\))p Fs(du)18 b Ft(+)2152 3227 y Fo(Z)2199 3416 y Fr([)p Fp(c;k)q Fr(])2340 3340 y Fs(')p Ft(\()p Fs(u)p Ft(\))p Fs(du;)204 3561 y Ft(where)27 b Fs(c)d Fq(2)f Fl(R)34 b Ft(is)28 b(suc)n(h)g(that)g Fs(c)23 b(<)g Ft(min)1422 3494 y Fo(\000)1474 3561 y Ft(inf)d Fs(\033)s Ft(\()p Fs(H)1739 3573 y Fr(0)1777 3561 y Ft(\))p Fs(;)14 b Ft(inf)21 b Fs(\033)s Ft(\()p Fs(H)2112 3573 y Fr(0)2169 3561 y Ft(+)d Fs(V)h Ft(\))2351 3494 y Fo(\001)2389 3561 y Ft(,)28 b(and)g([)p Fs(c;)14 b(k)s Ft(])28 b(is)g(the)g(line)g (segmen)n(t)f(joining)204 3672 y Fs(c)32 b Ft(to)g Fs(k)s Ft(.)50 b(W)-7 b(e)32 b(can)g(assume)f(that)h Fq(C)j Ft(=)c Fs(c)21 b Ft(+)1594 3605 y Fo(\000)1632 3672 y Fs(e)1671 3642 y Fp(i\031)r(=)p Fr(4)1806 3672 y Ft([0)p Fs(;)14 b Fq(1)p Ft(\))21 b Fq([)h Fs(e)2160 3642 y Fm(\000)p Fp(i\031)r(=)p Fr(4)2347 3672 y Ft([0)p Fs(;)14 b Fq(1)p Ft(\)\))33 b(and)e(then)i(it)f(is)g(easy)f(to)h(see)g(that)204 3783 y Fq(j)p Ft(\010\()p Fs(k)s Ft(\))p Fq(j)42 b Ft(=)f Fq(O)r Ft(\()p Fq(j)p Fs(k)s Fq(j)760 3753 y Fm(\000)p Fp(p)p Fr(\(1)p Fm(\000)p Fp(\027)t Fr(\))1025 3783 y Ft(\),)h(when)c Fs(k)45 b Fq(2)d(C)5 b Ft(,)41 b(b)n(y)d(writing)2070 3716 y Fo(R)2109 3813 y Fp(\015)2144 3822 y Fe(k)2199 3783 y Fs(')p Ft(\()p Fs(u)p Ft(\))p Fs(du)j Ft(=)g(0)e(com)n(bined)f (with)h(the)g(fact)g(that)204 3906 y Fq(j)p Fs(')p Ft(\()p Fs(k)s Ft(\))p Fq(j)28 b Ft(=)g Fq(O)r Ft(\()p Fq(j)p Fs(k)s Fq(j)727 3876 y Fm(\000)p Fr(1)p Fm(\000)p Fp(p)p Fr(\(1)p Fm(\000)p Fp(\027)t Fr(\))1077 3906 y Ft(\).)46 b(Here)30 b Fs(\015)1420 3918 y Fp(k)1491 3906 y Ft(is)g(the)h(union)g (of)f(the)h(segmen)n(ts)f([)p Fq(\000j)p Fs(k)s Fq(j)p Fs(;)14 b(c)p Ft(])30 b(and)g Fs(c)20 b Ft(+)g Fs(e)3207 3876 y Fp(i\031)r(=)p Fr(4)3343 3906 y Ft([0)p Fs(;)14 b Fq(j)p Fs(k)s Fq(j)p Ft(],)30 b(and)204 4006 y(of)e(the)i(p)r(ortion) e(of)g(circle)g(cen)n(tered)h(at)f Fs(c)h Ft(with)g(radius)f Fq(j)p Fs(k)s Fq(j)h Ft(where)f(the)h(argumen)n(t)f(ranges)f(b)r(et)n (w)n(een)h Fs(\031)s(=)p Ft(4)g(and)204 4105 y Fs(\031)h Ft(\(or)d(the)h(same)e(with)i Fq(\000)p Fs(\031)s(=)p Ft(4)e(ev)n(erywhere)f(and)i Fq(\000)p Fs(\031)s 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(the)h(construction)f(with)h(resp)r(ect)g(to)f(the)h(principal)f(sym)n (b)r(ol)g(of)h Fs(V)19 b Ft(.)328 930 y(W)-7 b(e)29 b(ha)n(v)n(e)f(to)g (precise)g(one)g(p)r(oin)n(t)h(on)g(what)f(w)n(e)g(shall)h(use)f(as)g (a)g(semiclassical)f(principal)i(sym)n(b)r(ol.)39 b(In)29 b(our)204 1029 y(constructions,)e Fs(q)h Ft(=)c Fs(q)s Ft(\()p Fs(x;)14 b(\030)t(;)g(h)p Ft(\))30 b(will)e(pla)n(y)g(the)h (role)f(of)g(the)h(principal)f(sym)n(b)r(ol)g(\(see)g(\(2)p Fs(:)p Ft(1\))h(for)e(the)i(notation\).)204 1129 y(It)22 b(is)h(more)e(con)n(v)n(enien)n(t,)i(for)e(tec)n(hnical)h(reasons,)g (to)g(do)g(so)g(since)28 b(~)-48 b Fs(q)s Ft(\()p Fs(h)p Ft(\))23 b(is)f(in)n(tegrable)f(with)i(resp)r(ect)f(to)g Fs(x)h Ft(whic)n(h)204 1229 y(is)k(useful)h(when)g(one)f(needs)h(to)f (write)h(traces.)328 1328 y(W)-7 b(e)29 b(are)e(going)f(to)i(appro)n (ximate)e(the)j(principal)e(sym)n(b)r(ol)h(of)f Fs(V)47 b Ft(b)n(y)28 b(a)f(fast)h(deca)n(ying)f(one)h(so)f(that)h(w)n(e)g(can) 204 1428 y(de\014ne)g(the)g(usual)f(SSF)h(for)f(the)h(pair)f Fs(H)1488 1440 y Fr(0)1525 1428 y Fs(;)14 b(H)1631 1440 y Fr(0)1687 1428 y Ft(+)k Fs(\017V)g Ft(.)37 b(Th)n(us)28 b(w)n(e)f(c)n(ho)r(ose)f(a)h(subset)1426 1604 y Fq(B)e(\032)d Fs(S)1644 1616 y Fr(1)1682 1604 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\026)p Ft(\))p Fs(;)179 b(\026)23 b(>)g(d)1217 b Ft(\(2.6\))204 1781 y(in)30 b(whic)n(h)f(all)h(sym)n(b)r(ols)f(are)g (real)g(and)g(satis\014e)g(\(2)p Fs(:)p Ft(3\))h(\(they)g(also)f(can)g (dep)r(end)h(on)g Fs(h)p Ft(\).)43 b(W)-7 b(e)31 b(can)e(c)n(ho)r(ose)f Fq(B)k Ft(to)204 1880 y(b)r(e)27 b(b)r(ounded)g(in)g Fs(S)801 1892 y Fr(1)838 1880 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\).)36 b(W)-7 b(e)27 b(also)f(assume,)g(whic)n (h)g(is)h(the)g(most)f(imp)r(ortan)n(t,)h(that)g(this)g(subset)f(con)n (tains)204 1980 y(a)32 b(sequence)g(whic)n(h)h(con)n(v)n(erges)d(to)i Fs(q)k Ft(in)d(the)g(top)r(ology)e(of)i Fs(S)2173 1992 y Fr(1)2210 1980 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)2442 1992 y Fr(1)2479 1980 y Ft(\),)34 b(for)e(all)g Fs(\032)2863 1992 y Fr(1)2932 1980 y Fs(<)f(\032)p Ft(.)52 b(The)33 b(existence)f(of)204 2080 y(suc)n(h)26 b(a)f(subset)h(is)g(clear)g(b)n (y)f(standard)h(prop)r(erties)f(of)h(these)g(classes)f(of)h(sym)n(b)r (ols.)36 b(Moreo)n(v)n(er)23 b(one)j(can)f(c)n(ho)r(ose)204 2179 y Fs(\032)247 2191 y Fr(1)312 2179 y Ft(suc)n(h)j(that)g Fs(\032)23 b(>)h(\032)878 2191 y Fr(1)938 2179 y Fs(>)g(d=p)p Ft(,)j(th)n(us)h(there)g(is)g(no)g(loss)f(of)h(generalit)n(y)e(if)j(w)n (e)e(c)n(ho)r(ose)g Fs(\032)h Ft(sligh)n(tly)f(smaller,)g(since)204 2279 y(the)j(only)g(prop)r(ert)n(y)e(that)i(w)n(e)g(use)g(is)f Fs(\032)e(>)g(d=p)p Ft(.)43 b(Then)30 b(w)n(e)g(can)f(assume)g(that)h (the)h(sequence)e(con)n(v)n(erges)e(to)j Fs(q)204 2379 y Ft(in)f Fs(S)353 2391 y Fr(1)390 2379 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\).)40 b(Finally)-7 b(,)29 b(w)n(e)f(can)g (also)g(assume)g(that)h Fs(\032)1973 2348 y Fm(\000)p Fr(1)2096 2379 y Fs(=)-51 b Fq(2)25 b Fl(N)39 b Ft(whic)n(h)28 b(will)h(b)r(e)g(useful)h(for)e(tec)n(hnical)g(reasons)204 2478 y(whic)n(h)f(app)r(ear)g(in)h(the)g(app)r(endix.)204 2710 y Fu(2.1)112 b(Tw)m(o)37 b(tec)m(hnical)f(lemmas)204 2863 y Ft(Let)28 b Fs(\037)23 b Fq(2)g Fs(C)571 2833 y Fm(1)565 2883 y Fr(0)642 2863 y Ft(\()p Fl(R)728 2833 y Fp(d)773 2863 y Ft(\))k(b)r(e)h(a)g(cuto\013)g(function)g(suc)n(h)f (that)h Fs(\037)23 b Fq(\021)g Ft(1)k(on)g(the)h(ball)f(of)h(radius)f Fs(R)d(>)e Ft(0.)37 b(\(2)p Fs(:)p Ft(3\).)204 3024 y Fg(Lemma)29 b(2.2.)40 b Fi(One)d(c)l(an)h(cho)l(ose)g Fs(R)g(>)f Ft(0)g Fi(lar)l(ge)h(enough)g(such)f(that,)j(for)e(al)t(l)h (functions)e Fs(f)3206 3036 y Fr(1)3243 3024 y Fs(;)14 b(f)3321 3036 y Fr(0)3395 3024 y Fq(2)37 b Fs(C)3552 2994 y Fm(1)3546 3045 y Fr(0)3623 3024 y Ft(\()p Fl(R)p Ft(\))204 3124 y Fi(satisfying)31 b Fs(f)614 3136 y Fr(1)674 3124 y Fq(\021)23 b Ft(1)29 b Fi(ne)l(ar)h Ft(supp)f Fs(f)1262 3136 y Fr(0)1299 3124 y Fi(,)h(we)g(have)h(the)f(fol)t (lowing)j(estimates)c(in)h Fg(S)2678 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y Fs(;)14 b Fq(\000)p Fs(n)p Ft(\),)37 b(for)e(some)g Fs(k)s Ft(\()p Fs(n)p Ft(\))h Fq(\025)g Ft(0.)60 b(If)36 b Fs(x)g Ft(is)f(large)f(enough,)j(and)e Fs(!)s Ft(\()p Fs(\030)t Ft(\))24 b(+)f Fs(\017q)s Ft(\()p Fs(x;)14 b(\030)t(;)g(h)p Ft(\))37 b Fq(2)g Ft(supp)e Fs(f)3488 4751 y Fr(0)3525 4739 y Ft(,)j(then)204 4838 y Fs(!)s Ft(\()p Fs(\030)t Ft(\))23 b Fq(2)h Fs(f)515 4803 y Fm(\000)p Fr(1)506 4860 y(1)603 4838 y Ft(\(1\),)k(and)g(then)g (the)g(comp)r(osition)f(form)n(ula)f(of)i(pseudo-di\013eren)n(tial)e (op)r(erators)g(yields)1122 5015 y(\(1)18 b Fq(\000)g Fs(\037)p Ft(\))1381 4948 y Fo(\000)1419 5015 y Ft(1)g Fq(\000)g Fs(f)1603 5027 y Fr(1)1640 5015 y Ft(\()p Fs(H)1741 5027 y Fr(0)1779 5015 y Ft(\))1811 4948 y Fo(\001)1849 5015 y Fs(a)1893 5027 y Fp(N)s(;\017)2000 5015 y Ft(\()p Fs(x;)c(hD)r(;)g(h)p Ft(\))23 b Fq(2)h Fs(h)2502 4980 y Fm(1)2572 5015 y Fs(O)r(p)2679 4948 y Fo(\000)2717 5015 y Fq(S)6 b Ft(\))p Fs(;)204 5191 y Ft(where)28 b Fq(S)36 b Ft(is)29 b(the)g(Sc)n(h)n(w)n(artz)f(class.)40 b(The)29 b(form)f(of)h(\(2)p Fs(:)p Ft(7\))g(is)g(simply)g(due)g(to)g (the)g(fact)g(that)h Fs(@)3164 5161 y Fp(j)3159 5212 y(\017)3198 5191 y Fs(f)9 b Ft(\()p Fs(H)3349 5203 y Fr(0)3406 5191 y Ft(+)19 b Fs(\017V)f Ft(\))3622 5206 y Fm(j)3642 5214 y Fe(\017)p Fh(=0)204 5291 y Ft(is)27 b(purely)g(pseudo-di\013eren)n(tial,)g(and)h(to)f(the)h(functional)g (calculus)f(of)g([7].)1109 b Fc(\003)1934 5540 y Ft(12)p eop %%Page: 13 13 13 12 bop 328 531 a Ft(In)38 b(all)g(what)f(follo)n(ws,)i(w)n(e)f(use)f (the)h(notation)f Fs(U)1931 543 y Fp(\017)1963 531 y Ft(\()p Fs(t)p Ft(\))j(=)g(exp\()p Fq(\000)p Fs(itH)2554 543 y Fp(\017)2585 531 y Fs(=h)p Ft(\),)g(and)d Fs(I)45 b Ft(is)38 b(an)f(op)r(en)h(in)n(terv)-5 b(al.)204 631 y(Consider)30 b(a)h(family)g(of)g(b)r(ounded)h(op)r(erators)d Fs(B)1759 643 y Fp(\017)1822 631 y Ft(whic)n(h)i(is)g Fs(C)2215 601 y Fp(p)2285 631 y Ft(with)h(resp)r(ect)f(to)g Fs(\017)d Fq(2)i Ft([0)p Fs(;)14 b Ft(1],)31 b(and)g(suc)n(h)g(that)204 731 y Fs(B)267 743 y Fp(\017)299 731 y Fq(h)p Fs(x)p Fq(i)421 685 y Fr(\026)410 700 y Fp(d)475 731 y Ft(is)26 b(still)g Fs(C)782 700 y Fp(p)847 731 y Ft(in)g(the)g(class)f(of)g(b)r (ounded)i(op)r(erators,)d(for)h(some)2441 709 y(\026)2426 731 y Fs(d)f(>)e(d)p Ft(.)37 b(In)26 b(the)g(applications)f Fs(B)3449 743 y Fp(\017)3507 731 y Ft(will)h(b)r(e)204 830 y(an)h(op)r(erator)f(with)i(a)f(rather)g(fast)g(deca)n(ying)g(k)n (ernel.)36 b(Then)27 b(w)n(e)h(ha)n(v)n(e)204 996 y Fg(Lemma)h(2.3.)40 b Fi(The)34 b(family)h(of)f(op)l(er)l(ators)g Fs(B)1689 1008 y Fp(\017)1721 996 y Fs(f)9 b Ft(\()p Fs(H)1872 1008 y Fp(\017)1903 996 y Ft(\))p Fs(U)1992 1008 y Fp(\017)2024 996 y Ft(\()p Fs(t)p Ft(\))33 b Fi(is)h Fs(C)2309 966 y Fp(p)2380 996 y Fi(in)f(the)h(tr)l(ac)l(e)e(class)i(\(with)g(r)l(esp) l(e)l(ct)e(to)h Fs(\017)c Fq(2)204 1096 y Ft([0)p Fs(;)14 b Ft(1])p Fi(\))28 b(for)j(al)t(l)g Fs(f)h Fq(2)23 b Fs(C)901 1066 y Fm(1)895 1116 y Fr(0)972 1096 y Ft(\()p Fs(I)7 b Ft(\))p Fi(,)30 b(and)h(al)t(l)f Fs(t)23 b Fq(2)h Fl(R)p Fi(.)44 b(Mor)l(e)l(over,)32 b(for)f(al)t(l)f Fs(j)e Fq(\025)23 b Ft(1)1544 1277 y Fs(d)1587 1246 y Fp(j)p 1528 1314 112 4 v 1528 1390 a Fs(d\017)1605 1366 y Fp(j)1650 1333 y Fi(T)-6 b(r)13 b Ft(\()q Fs(B)1847 1345 y Fp(\017)1878 1333 y Fs(f)c Ft(\()p Fs(H)2029 1345 y Fp(\017)2061 1333 y Ft(\))p Fs(U)2150 1345 y Fp(\017)2182 1333 y Ft(\()p Fs(t)p Ft(\)\))2309 1358 y Fm(j)2329 1366 y Fe(\017)p Fh(=0)204 1539 y Fi(c)l(an)30 b(b)l(e)f(written)h(as)g(a)g (line)l(ar)g(c)l(ombination)h(of)g(the)f(tr)l(ac)l(es)f(of)i(the)f(fol) t(lowing)i(op)l(er)l(ators)f(with)f Ft(0)23 b Fq(\024)f Fs(n)h Fq(\024)g Fs(j)5 b Fi(:)302 1727 y Fs(@)351 1697 y Fp(j)s Fm(\000)p Fp(n)346 1748 y(\017)478 1727 y Fs(B)541 1739 y Fp(\017)p 302 1764 272 4 v 391 1840 a Fs(h)439 1816 y Fp(n)583 1783 y Fq(j)606 1795 y Fp(\017)p Fr(=0)736 1670 y Fo(Z)819 1691 y Fp(t)782 1859 y Fr(0)848 1670 y Fo(Z)931 1691 y Fp(t)p Fm(\000)p Fp(t)1033 1699 y Fh(1)894 1859 y Fr(0)1015 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2327 y Fs(t)1642 2339 y Fr(0)1703 2327 y Ft(=)22 b Fs(t;)184 b(t)2057 2339 y Fp(n)p Fr(+1)2209 2327 y Ft(=)23 b(0)204 2510 y Fi(and)34 b(if)g Fs(n)c Ft(=)f(0)k Fi(ther)l(e)h(is)g(no)f(inte) l(gr)l(al)h(sign.)50 b(The)35 b(symb)l(ols)f Fs(a)2172 2522 y Fp(l)2231 2510 y Fi(ar)l(e)g(in)f Fs(S)2532 2522 y Fr(1)2570 2510 y Ft(\()p Fq(\0001)p Fs(;)14 b Fq(\000)p Fs(\032)p Ft(\))33 b Fi(if)h Ft(1)29 b Fq(\024)h Fs(l)h Fq(\024)f Fs(n)p Fi(,)k(wher)l(e)l(as)204 2610 y Fs(a)248 2622 y Fr(0)319 2610 y Fi(and)g Fs(a)528 2622 y Fp(n)p Fr(+1)691 2610 y Fi(ar)l(e)h(in)f Fs(S)994 2622 y Fr(1)1031 2610 y Ft(\()p Fq(\0001)p Fs(;)14 b Ft(0\))p Fi(.)51 b(They)35 b(al)t(l)g(have)g(an)f(exp)l(ansion)g(with)h(r)l(esp)l(e)l (ct)f(to)f Fs(h)h Fi(and)h(ar)l(e)f(c)l(omp)l(actly)204 2709 y(supp)l(orte)l(d)h(with)g(r)l(esp)l(e)l(ct)g(to)g Fs(\030)k Fi(in)c(a)g(smal)t(l)g(neighb)l(orho)l(o)l(d)j(of)d Fs(!)2285 2679 y Fm(\000)p Fr(1)2374 2709 y Ft(\(supp)g Fs(f)9 b Ft(\))p Fi(.)54 b(As)34 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b(transforms)g(of)i(traces)e(similar)h(to)g(the)204 4585 y(one)g(of)g(lemma)g(2)p Fs(:)p Ft(3.)35 b(Since)27 b(w)n(e)e(w)n(an)n(t)h(to)g(pro)n(v)n(e)e(that)j(the)f(traces)f(are)g (smo)r(oth,)i(w)n(e)e(need)i(to)f(pro)n(v)n(e)e(fast)i(deca)n(y)204 4684 y(in)31 b(time,)i(whic)n(h)e(will)g(b)r(e)h(obtained)f(b)n(y)f(a)h (non)g(stationary)f(phase)g(argumen)n(t.)47 b(If)31 b(one)g(lo)r(oks)f (at)h(the)h(critical)204 4784 y(p)r(oin)n(ts)h(of)f(the)h(phases)f (naturally)g(in)n(v)n(olv)n(ed)g(in)h(the)g(traces)f(of)g(lemma)h(2)p Fs(:)p Ft(3,)h(w)n(e)e(\014nd)h(that)g Fs(t)g Ft(m)n(ust)g(b)r(e)g (zero.)204 4883 y(Ho)n(w)n(ev)n(er)e(it)j(is)g(not)g(clear)e(ho)n(w)h (to)h(do)f(the)h(in)n(tegrations)e(b)n(y)h(parts,)i(to)e(get)g(the)h (fast)g(deca)n(y)f(a)n(w)n(a)n(y)e(from)j(0,)204 4983 y(that's)27 b(wh)n(y)h(w)n(e)f(need)h(to)f(establish)g(the)h(next)g (lemma.)328 5083 y(Let's)g(consider)e(the)i(set)g(of)g(all)f(complex)g (v)-5 b(alued)28 b(functions)g Fs(f)36 b Ft(suc)n(h)27 b(that,)h(for)f(all)858 5265 y Fs(\013)911 5277 y Fr(2)949 5265 y Fs(;)14 b Fq(\001)g(\001)g(\001)27 b Fs(;)14 b(\013)1200 5277 y Fp(k)1264 5265 y Fq(2)24 b Fl(N)1397 5231 y Fp(d)1441 5265 y Fs(;)42 b(\014)1553 5277 y Fr(1)1590 5265 y Fs(;)14 b Fq(\001)g(\001)g(\001)27 b Fs(;)14 b(\014)1835 5277 y Fp(k)q Fm(\000)p Fr(1)1984 5265 y Fq(2)24 b Fl(N)2117 5231 y Fp(d)2161 5265 y Fs(;)42 b(\014)2273 5277 y Fp(k)2337 5265 y Fq(2)23 b Fl(N)2469 5231 y Fp(d)2514 5265 y Fs(;)69 b Fq(j)p Fs(\014)2676 5277 y Fp(k)2717 5265 y Fq(j)23 b(\024)g Fs(N)1934 5540 y Ft(13)p eop %%Page: 14 14 14 13 bop 204 531 a Ft(one)27 b(can)g(\014nd)h(a)f(constan)n(t)g Fs(C)34 b Ft(suc)n(h)28 b(that)608 711 y Fq(j)p Fs(@)680 677 y Fp(\013)723 685 y Fh(2)756 677 y Fm(\001\001\001)o Fp(\013)858 686 y Fe(k)675 731 y Fp(x)713 739 y Fh(2)746 731 y Fm(\001\001\001)o Fp(x)843 740 y Fe(k)898 711 y Fs(@)947 667 y Fp(\014)985 675 y Fh(1)1017 667 y Fm(\001\001\001)o Fp(\014)1114 676 y Fe(k)q Fd(\000)p Fh(1)942 736 y Fp(\030)972 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1222 y Fs(;)14 b Fq(\001)g(\001)g(\001)28 b Fs(;)14 b(\030)2444 1234 y Fp(k)2508 1222 y Fq(2)23 b Fl(R)2640 1187 y Fp(d)2685 1222 y Fs(;)42 b(h)23 b Fq(2)g Ft(\(0)p Fs(;)14 b Ft(1])p Fs(:)204 1424 y Ft(W)-7 b(e)26 b(also)e(assume)h(that)h(these)f(functions)h Fs(f)34 b Ft(are)25 b(supp)r(orted)g(in)2233 1357 y Fo(\000)2271 1424 y Fs(!)2326 1394 y Fm(\000)p Fr(1)2415 1424 y Ft(\()p Fs(I)7 b Ft(\))2522 1357 y Fo(\001)2561 1374 y Fp(k)2627 1424 y Ft(with)26 b(resp)r(ect)f(to)h Fs(\030)3232 1436 y Fr(1)3269 1424 y Fs(;)14 b Fq(\001)g(\001)g(\001)28 b Fs(;)14 b(\030)3504 1436 y Fp(k)3545 1424 y Ft(.)36 b(The)204 1524 y(b)r(est)28 b(constan)n(ts)e Fs(C)34 b Ft(in)28 b(\(2)p Fs(:)p Ft(9\))g(are)e(semi-norms)h(and)g(de\014ne)h (the)g(top)r(ology)e(on)h(the)h(set.)204 1623 y Fg(Remark.)40 b Ft(This)29 b(t)n(yp)r(e)h(of)f(functions)g(is)g(what)h(w)n(e)f(get)f (as)h(amplitudes)g(if)h(w)n(e)f(expand)g(the)g(k)n(ernel)g(in)g(lemma) 204 1723 y(2)p Fs(:)p Ft(3,)37 b(except)f(that)h Fs(\016)i 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Fe(d)931 2658 y Fr(\))957 2641 y Fe(k)1011 2582 y Fs(e)1066 2525 y Fe(i)p 1060 2534 35 3 v 1060 2568 a(h)1104 2548 y Fp(\036)1144 2557 y Fe(k)1185 2582 y Fs(f)9 b(dx)1325 2594 y Fr(1)1376 2582 y Fq(\001)14 b(\001)g(\001)g Fs(dx)1577 2594 y Fp(k)1618 2582 y Fs(d\030)1697 2594 y Fr(1)1749 2582 y Fq(\001)g(\001)g(\001)f Fs(d\030)1938 2594 y Fp(k)1980 2511 y Fo(\014)1980 2561 y(\014)2031 2582 y Fq(\024)22 b Fs(C)6 b Fq(h)p Fs(x)2262 2594 y Fr(0)p Fp(;)p Fr(1)2353 2582 y Fq(i)2385 2548 y Fm(\000)p Fp(\016)2467 2556 y Fh(1)2518 2582 y Fq(\001)14 b(\001)g(\001)g(h)p Fs(x)2708 2594 y Fr(0)p Fp(;n)2806 2582 y Fq(i)2838 2548 y Fm(\000)p Fp(\016)2920 2556 y Fh(1)2971 2465 y Fo(\022)3042 2526 y Fs(h)p 3042 2563 48 4 v 3051 2639 a(t)3100 2465 y Fo(\023)3161 2482 y Fp(N)3238 2582 y Fs(:)274 b Ft(\(2.10\))204 2818 y Fi(Her)l(e)29 b(the)h(phase)h(function)f(is)g(given)h(by)982 3075 y Fs(\036)1031 3087 y Fp(k)1096 3075 y Ft(=)1225 2971 y Fp(k)1183 2996 y Fo(X)1186 3173 y Fp(j)s Fr(=1)1303 3075 y Fq(h)p 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3601 y Fs(=h)p Ft(\))g(in)n(v)-5 b(arian)n(t)695 3863 y Fs(P)748 3875 y Fp(l)797 3863 y Ft(=)23 b Fq(\000)960 3806 y Fs(h)p 960 3844 V 970 3920 a(i)1076 3732 y Fo(P)1163 3752 y Fp(l)1163 3819 y(j)s Fr(=1)1296 3794 y Fs(t)1326 3806 y Fp(j)1361 3794 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)1553 3806 y Fp(j)1589 3794 y Ft(\))p 1027 3844 643 4 v 1027 3936 a Fq(j)1064 3873 y Fo(P)1152 3894 y Fp(l)1152 3960 y(j)s Fr(=1)1285 3936 y Fs(t)1315 3948 y Fp(j)1350 3936 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)1542 3948 y Fp(j)1577 3936 y Ft(\))p Fq(j)1632 3912 y Fr(2)1680 3863 y Fs(:)1703 3795 y Fo(\000)1804 3759 y Fp(l)1755 3784 y Fo(X)1757 3961 y Fp(j)s Fr(=1)1888 3863 y Fq(r)1957 3875 y Fp(\030)1987 3883 y Fe(j)2023 3795 y Fo(\001)2079 3863 y Fq(\000)18 b Ft(\()p Fs(x)2241 3875 y Fp(l)p Fr(+1)2370 3863 y Fq(\000)g Fs(x)2500 3875 y Fr(1)2538 3863 y Ft(\))p Fs(:)2652 3732 y Fo(P)2739 3752 y Fp(l)2739 3819 y(j)s Fr(=1)2872 3794 y Fs(t)2902 3806 y Fp(j)2937 3794 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)3129 3806 y Fp(j)3165 3794 y Ft(\))p 2603 3844 V 2603 3936 a Fq(j)2640 3873 y Fo(P)2728 3894 y Fp(l)2728 3960 y(j)s Fr(=1)2861 3936 y Fs(t)2891 3948 y Fp(j)2925 3936 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)3117 3948 y Fp(j)3153 3936 y Ft(\))p Fq(j)3208 3912 y Fr(2)204 4126 y Ft(where)27 b(one)g(should)g(notice)h(that,)g(if)g Fs(l)c Ft(=)f Fs(k)s Ft(,)28 b Fs(x)1664 4138 y Fr(1)1720 4126 y Fq(\000)18 b Fs(x)1850 4138 y Fp(l)p Fr(+1)1983 4126 y Ft(=)k(0.)37 b(W)-7 b(e)28 b(shall)f(use)h(as)e(w)n(ell)1353 4347 y Fs(Q)1419 4359 y Fp(l)1467 4347 y Ft(=)1564 4291 y Fs(h)p 1564 4328 48 4 v 1574 4404 a(i)1674 4291 y(\030)1710 4303 y Fp(l)1754 4291 y Fq(\000)18 b Fs(\030)1873 4303 y Fp(l)p Fm(\000)p Fr(1)p 1632 4328 394 4 v 1632 4404 a Fq(j)p Fs(\030)1691 4416 y Fp(l)1735 4404 y Fq(\000)g Fs(\030)1854 4416 y Fp(l)p Fm(\000)p Fr(1)1965 4404 y Fq(j)1988 4380 y Fr(2)2036 4347 y Fs(:)p Fq(r)2128 4359 y Fp(x)2166 4368 y Fe(l)2194 4347 y Fs(;)180 b(l)24 b Fq(\025)f Ft(2)p Fs(:)204 4584 y Ft(Let's)37 b(start)g(with)h Fs(k)43 b Ft(=)c(2.)66 b(W)-7 b(e)38 b(c)n(ho)r(ose)f Fs(\037)i Fq(2)h Fs(C)1825 4554 y Fm(1)1819 4604 y Fr(0)1896 4584 y Ft(\()p Fl(R)1982 4554 y Fp(d)2027 4584 y Ft(\),)g Fs(\037)g Fq(\021)f Ft(1)e(near)g(0)g(,)j(and)e(w)n(e)f(write)g Fs(f)48 b Ft(=)40 b Fs(f)3495 4596 y Fr(1)3556 4584 y Ft(+)25 b Fs(f)3687 4596 y Fr(2)3724 4584 y Ft(,)204 4683 y(with)39 b Fs(f)445 4695 y Fr(1)524 4683 y Ft(=)i Fs(\037)p Ft(\()p Fs(\030)750 4695 y Fr(1)814 4683 y Fq(\000)25 b Fs(\030)940 4695 y Fr(2)978 4683 y Ft(\))p Fs(f)9 b Ft(.)71 b(By)38 b(c)n(ho)r(osing)f(the)j(supp)r(ort)e(of)h Fs(\037)g Ft(small)f(enough,)k(and)c(using)h(the)g(fact)g(that)204 4783 y Fq(jr)p Fs(!)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)24 b(\025)e Fs(")h(>)g Ft(0)k(on)g Fs(!)978 4753 y Fm(\000)p Fr(1)1067 4783 y Ft(\()p Fs(I)7 b Ft(\),)28 b(w)n(e)f(get)1471 4984 y Fq(j)p Fs(t)1524 4996 y Fr(1)1562 4984 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)1754 4996 y Fr(1)1791 4984 y Ft(\))19 b(+)f Fs(t)1955 4996 y Fr(2)1992 4984 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)2184 4996 y Fr(2)2222 4984 y Ft(\))p Fq(j)23 b(\025)2399 4928 y Fs(")p 2398 4965 42 4 v 2398 5041 a Ft(2)2450 4984 y Fs(t)204 5191 y Ft(on)29 b(the)h(supp)r(ort)g (of)f Fs(f)913 5203 y Fr(1)950 5191 y Ft(,)h(and)g(w)n(e)f(do)g Fs(N)39 b Ft(in)n(tegrations)28 b(b)n(y)h(parts)g(with)h Fs(P)2547 5203 y Fr(2)2614 5191 y Ft(\(whic)n(h)g(is)g(indep)r(enden)n (t)g(of)g Fs(x)3588 5203 y Fr(1)3626 5191 y Fs(;)14 b(x)3710 5203 y Fr(2)204 5291 y Ft(since)27 b Fs(l)e Ft(=)d Fs(k)s Ft(\).)37 b(This)28 b(pro)n(v)n(es)e(that)i(the)f(estimate)h(\(2)p Fs(:)p Ft(10\))f(holds)g(for)g Fs(f)2409 5303 y Fr(1)2446 5291 y Ft(.)1934 5540 y(14)p eop %%Page: 15 15 15 14 bop 328 531 a Ft(T)-7 b(o)25 b(estimate)h(the)g(same)f(in)n (tegral)f(with)i Fs(f)1654 543 y Fr(2)1691 531 y Ft(,)g(w)n(e)f (\014rst)g(do)g Fs(N)35 b Ft(in)n(tegrations)24 b(b)n(y)h(parts)g(with) h Fs(Q)3274 543 y Fr(2)3336 531 y Ft(whic)n(h)f(giv)n(es)204 631 y(a)i(gain)g(of)g Fs(h)595 601 y Fp(N)658 631 y Fq(h)p Fs(x)737 643 y Fr(2)775 631 y Fq(i)807 601 y Fm(\000)p Fp(N)922 631 y Ft(.)37 b(Then:)204 731 y Fq(\017)c Ft(either)g Fs(t)550 743 y Fr(1)620 731 y Fq(\025)f Fs(t=)p Ft(2,)i(and)g(w)n(e)f (do)g Fs(N)42 b Ft(in)n(tegrations)32 b(b)n(y)h(parts)g(with)h Fs(P)2463 743 y Fr(1)2500 731 y Ft(,)h(whic)n(h)f(giv)n(es)e(a)h(gain)g (of)g Fq(O)r Ft(\()p Fs(t)3502 695 y Fm(\000)p Fp(N)3502 753 y Fr(1)3617 731 y Ft(\))h(=)204 830 y Fq(O)r Ft(\()p Fs(t)334 800 y Fm(\000)p Fp(N)449 830 y Ft(\).)41 b(W)-7 b(e)30 b(get)e(also)g Fq(O)r Ft(\()p Fq(h)p Fs(x)1176 842 y Fr(1)1234 830 y Fq(\000)19 b Fs(x)1365 842 y Fr(2)1402 830 y Fq(i)1434 800 y Fp(N)1498 830 y Ft(\))29 b(but)g(this)g(do)r (esn't)g(cause)g(an)n(y)f(trouble)g(thanks)h(to)f(\(2)p Fs(:)p Ft(9\))h(and)g(to)f(to)204 930 y(the)g(gain)f(of)g Fq(h)p Fs(x)700 942 y Fr(2)738 930 y Fq(i)770 900 y Fm(\000)p Fp(N)885 930 y Ft(,)204 1029 y Fq(\017)k Ft(or)g Fs(t)413 1041 y Fr(2)480 1029 y Fq(\025)f Fs(t=)p Ft(2,)i(and)g(w)n(e)f(do)h Fs(N)40 b Ft(in)n(tegrations)30 b(b)n(y)i(part)f(with)2238 1008 y(~)2220 1029 y Fs(P)2273 1041 y Fr(1)2310 1029 y Ft(,)i(whic)n(h)f(is)g(the)g(same)f(op)r(erator)f(as)h Fs(P)3553 1041 y Fr(1)3623 1029 y Ft(but)204 1129 y(where)c(w)n(e)g(sw) n(ap)g Fs(x)819 1141 y Fr(1)884 1129 y Ft(and)h Fs(x)1093 1141 y Fr(2)1130 1129 y Ft(,)g(and)f Fs(\030)1378 1141 y Fr(1)1444 1129 y Ft(and)g Fs(\030)1641 1141 y Fr(2)1679 1129 y Ft(.)37 b(As)27 b(ab)r(o)n(v)n(e,)g(w)n(e)g(get)g(a)g(gain)g(of) h Fq(O)r Ft(\()p Fs(t)2854 1094 y Fm(\000)p Fp(N)2854 1151 y Fr(2)2969 1129 y Ft(\))c(=)e Fq(O)r Ft(\()p Fs(t)3242 1099 y Fm(\000)p Fp(N)3358 1129 y Ft(\).)204 1229 y(Th)n(us)27 b(the)h(result)f(is)h(pro)n(v)n(ed)e(when)i Fs(k)e Ft(=)c(2.)328 1328 y(No)n(w)40 b(assume)f(that)i Fs(k)47 b Fq(\025)c Ft(3)d(and)g(that)g(the)h(result)f(holds)f(to)i(all)e(orders)g Fq(\024)k Fs(k)30 b Fq(\000)c Ft(1.)75 b(W)-7 b(e)40 b(de\014ne,)k(for)204 1428 y(1)22 b Fq(\024)h Fs(l)h Fq(\024)f Fs(k)e Fq(\000)d Ft(1,)28 b(the)g(function)g Fs(\037)1253 1440 y Fp(l)1301 1428 y Ft(=)23 b Fs(\037)p Ft(\()p Fs(\030)1509 1440 y Fp(l)1553 1428 y Fq(\000)18 b Fs(\030)1672 1440 y Fp(l)p Fr(+1)1782 1428 y Ft(\))28 b(and)g(then)g(write)968 1682 y Fs(f)j Ft(=)23 b Fs(\037)1180 1694 y Fr(1)1231 1682 y Fq(\001)14 b(\001)g(\001)g Fs(\037)1394 1694 y Fp(k)q Fm(\000)p Fr(1)1520 1682 y Fs(f)27 b Ft(+)18 b(\(1)g Fq(\000)g Fs(\037)1898 1694 y Fr(1)1935 1682 y Ft(\))p Fs(f)27 b Ft(+)2118 1579 y Fp(k)q Fm(\000)p Fr(1)2119 1604 y Fo(X)2127 1782 y Fp(l)p Fr(=2)2254 1682 y Fs(\037)2306 1694 y Fr(1)2357 1682 y Fq(\001)14 b(\001)g(\001)g Fs(\037)2520 1694 y Fp(l)p Fm(\000)p Fr(1)2630 1682 y Ft(\(1)k Fq(\000)g Fs(\037)2857 1694 y Fp(l)2883 1682 y Ft(\))p Fs(f)t(:)204 1954 y Ft(By)30 b(c)n(ho)r(osing)e(the)j(supp)r (ort)f(of)g Fs(\037)g Ft(small)g(enough)f(\(and)i Fs(\037)c Fq(\021)g Ft(1)i(near)h(0\),)g(w)n(e)g(get)g Fq(j)2876 1892 y Fo(P)2964 1912 y Fp(k)2964 1979 y(j)s Fr(=1)3096 1954 y Fs(t)3126 1966 y Fp(j)3161 1954 y Fq(r)p Fs(!)s Ft(\()p Fs(\030)3353 1966 y Fp(j)3389 1954 y Ft(\))p Fq(j)d(\025)g Fs(ct)j Ft(on)204 2053 y(the)e(supp)r(ort)f(of)h Fs(\037)801 2065 y Fr(1)852 2053 y Fq(\001)14 b(\001)g(\001)f Fs(\037)1014 2065 y Fp(k)q Fm(\000)p Fr(1)1168 2053 y Ft(for)27 b(some)g Fs(c)c(>)g Ft(0;)k(then)h(doing)f Fs(N)36 b Ft(in)n(tegrations)26 b(b)n(y)i(parts)f(with)h Fs(P)3286 2065 y Fp(k)3355 2053 y Ft(sho)n(ws)e(that)586 2223 y Fo(\014)586 2273 y(\014)627 2181 y(Z)724 2294 y Fs(e)779 2237 y Fe(i)p 773 2246 35 3 v 773 2279 a(h)817 2259 y Fp(\036)857 2268 y Fe(k)897 2294 y Fs(\037)949 2306 y Fr(1)1000 2294 y Fq(\001)14 b(\001)g(\001)g Fs(\037)1163 2306 y Fp(k)q Fm(\000)p Fr(1)1289 2294 y Fs(f)9 b(dx)1429 2306 y Fr(1)1480 2294 y Fq(\001)14 b(\001)g(\001)g Fs(dx)1681 2306 y Fp(k)1722 2294 y Fs(d\030)1801 2306 y Fr(1)1853 2294 y Fq(\001)g(\001)g(\001)g Fs(d\030)2043 2306 y Fp(k)2084 2223 y Fo(\014)2084 2273 y(\014)2135 2294 y Fq(\024)22 b Fs(C)6 b Fq(h)p Fs(x)2366 2306 y Fr(0)p Fp(;)p Fr(1)2457 2294 y Fq(i)2489 2259 y Fm(\000)p Fp(\016)2571 2267 y Fh(1)2622 2294 y Fq(\001)14 b(\001)g(\001)g(h)p Fs(x)2812 2306 y Fr(0)p Fp(;n)2911 2294 y Fq(i)2943 2259 y Fm(\000)p Fp(\016)3025 2267 y Fh(1)3075 2177 y Fo(\022)3146 2237 y Fs(h)p 3146 2275 48 4 v 3155 2351 a(t)3204 2177 y Fo(\023)3265 2194 y Fp(N)3342 2294 y Fs(:)204 2516 y Ft(No)n(w)26 b(w)n(e)g(consider)f(the)i(in)n(tegral)e(with)i(\(1)16 b Fq(\000)f Fs(\037)1688 2528 y Fr(1)1725 2516 y Ft(\))p Fs(f)36 b Ft(in)26 b(whic)n(h)h(w)n(e)f(can)g(do)g Fs(N)35 b Ft(in)n(tegrations)25 b(b)n(y)h(parts)f(with)i Fs(Q)3687 2528 y Fr(2)3724 2516 y Ft(,)204 2616 y(from)g(whic)n(h)h(w)n(e)f(get)g (a)g(gain)g(of)h Fs(h)1290 2585 y Fp(N)1353 2616 y Fq(h)p Fs(x)1432 2628 y Fr(2)1470 2616 y Fq(i)1502 2585 y Fm(\000)p Fp(N)1617 2616 y Ft(.)37 b(There)27 b(are)f(again)h(t)n(w)n(o)g(cases:) 204 2715 y Fq(\017)g Ft(either)g Fs(t)538 2727 y Fr(1)599 2715 y Fq(\025)22 b Fs(t=)p Ft(2,)27 b(and)h(w)n(e)f(do)g(exactly)g(as) g(for)g Fs(k)f Ft(=)d(2,)204 2815 y Fq(\017)34 b Ft(or)f Fs(t)418 2827 y Fr(2)478 2815 y Ft(+)22 b Fq(\001)14 b(\001)g(\001)23 b Ft(+)f Fs(t)802 2827 y Fp(k)877 2815 y Fq(\025)34 b Fs(t=)p Ft(2,)h(and)f(w)n(e)f(use)h(the)h(induction)f (assumption.)56 b(Precisely)-7 b(,)35 b(the)f(function)h(that)g(w)n(e) 204 2914 y(in)n(tegrate)26 b(has)h(the)h(follo)n(wing)f(form)1394 3101 y Fs(e)1449 3044 y Fe(i)p 1443 3053 35 3 v 1443 3086 a(h)1496 3051 y Fr(~)1487 3066 y Fp(\036)1527 3075 y Fe(k)q Fd(\000)p Fh(1)1658 3079 y Ft(~)1641 3101 y Fs(f)8 b Ft(\()d(~)-47 b Fs(x)1769 3113 y Fr(0)1807 3101 y Fs(;)19 b Ft(~)-47 b Fs(x)1891 3113 y Fr(2)1929 3101 y Fs(;)14 b Fq(\001)g(\001)g(\001)27 b Fs(;)19 b Ft(~)-47 b Fs(x)2174 3113 y Fp(k)2215 3101 y Fs(;)2261 3079 y Ft(~)2252 3101 y Fs(\030)2288 3113 y Fr(2)2326 3101 y Fs(;)2371 3079 y Ft(~)2363 3101 y Fs(\030)2399 3113 y Fp(k)2440 3101 y Fs(;)14 b(h)p Ft(\))204 3273 y(where)27 b(the)g(new)h(v)-5 b(ariables)26 b(are)32 b(~)-48 b Fs(x)1286 3285 y Fr(0)1347 3273 y Ft(=)23 b(\()p Fs(x)1514 3285 y Fr(0)1552 3273 y Fs(;)14 b(x)1636 3285 y Fr(1)1673 3273 y Fs(;)g(\030)1746 3285 y Fr(1)1784 3273 y Ft(\),)28 b(\()5 b(~)-47 b Fs(x)1946 3285 y Fp(j)s Fm(\000)p Fr(1)2066 3273 y Fs(;)2112 3251 y Ft(~)2103 3273 y Fs(\030)2139 3285 y Fp(j)s Fm(\000)p Fr(1)2260 3273 y Ft(\))23 b(=)g(\()p Fs(x)2482 3285 y Fp(j)2517 3273 y Fs(;)14 b(\030)2590 3285 y Fp(j)2626 3273 y Ft(\),)27 b(when)h(2)23 b Fq(\024)f Fs(j)28 b Fq(\024)23 b Fs(k)30 b Ft(and)d(the)h(new)204 3373 y(phase)f(function)h(is)1046 3528 y(~)1035 3550 y Fs(\036)1084 3562 y Fp(k)q Fm(\000)p Fr(1)1233 3550 y Ft(=)1321 3446 y Fp(k)q Fm(\000)p Fr(1)1322 3471 y Fo(X)1324 3648 y Fp(j)s Fr(=1)1442 3550 y Fq(h)5 b Ft(~)-47 b Fs(x)1521 3562 y Fp(j)1576 3550 y Fq(\000)23 b Ft(~)-47 b Fs(x)1706 3562 y Fp(j)s Fr(+1)1825 3550 y Fs(;)1870 3528 y Ft(~)1862 3550 y Fs(\030)1898 3562 y Fp(j)1933 3550 y Fq(i)19 b(\000)f Fs(t)2097 3562 y Fp(j)s Fr(+1)2216 3550 y Fs(!)s Ft(\()2312 3528 y(~)2303 3550 y Fs(\030)2339 3562 y Fp(j)2374 3550 y Ft(\))p Fs(;)186 b Ft(~)-48 b Fs(x)2656 3562 y 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Ft(from)f(whic)n(h)h(w)n(e)f(ha)n(v)n(e)f(a)h(gain)g (of)204 4761 y Fs(h)252 4730 y Fp(N)315 4761 y Fq(h)p Fs(x)394 4773 y Fp(l)420 4761 y Fq(i)452 4730 y Fm(\000)p Fp(N)595 4761 y Ft(and)c(there)g(are)g(again)f(t)n(w)n(o)h(p)r (ossibilities:)204 4860 y Fq(\017)g Ft(either)g Fs(t)538 4872 y Fr(1)593 4860 y Ft(+)17 b Fq(\001)d(\001)g(\001)k Ft(+)f Fs(t)902 4872 y Fp(l)951 4860 y Fq(\025)22 b Fs(t=)p Ft(2,)27 b(and)g(then,)h(b)n(y)f(c)n(ho)r(osing)f(the)h(supp)r(ort)g (of)h Fs(\037)f Ft(small)g(enough)g(w)n(e)f(can)h(in)n(tegrate)204 4960 y(b)n(y)g(parts)g Fs(N)36 b Ft(times)28 b(with)g Fs(P)1099 4972 y Fp(l)p Fm(\000)p Fr(1)1210 4960 y Ft(.)37 b(Then)28 b(w)n(e)f(get)460 5142 y Fo(\014)460 5192 y(\014)501 5100 y(Z)598 5213 y Fs(e)653 5156 y Fe(i)p 647 5165 35 3 v 647 5199 a(h)691 5179 y Fp(\036)731 5188 y Fe(k)771 5213 y Fs(\037)823 5225 y Fr(1)875 5213 y Fq(\001)14 b(\001)g(\001)f Fs(\037)1037 5225 y Fp(l)p Fm(\000)p Fr(1)1148 5213 y Ft(\(1)18 b Fq(\000)g Fs(\037)1375 5225 y Fp(l)1400 5213 y Ft(\))p Fs(f)9 b(dx)1572 5225 y Fr(1)1624 5213 y Fq(\001)14 b(\001)g(\001)g Fs(dx)1825 5225 y Fp(l)1851 5213 y Fs(d\030)1930 5225 y Fr(1)1981 5213 y Fq(\001)g(\001)g(\001)g Fs(d\030)2171 5225 y Fp(l)2197 5142 y Fo(\014)2197 5192 y(\014)2248 5213 y Fq(\024)22 b Fs(C)2400 5146 y Fo(\000)2449 5157 y Fs(h)p 2449 5194 48 4 v 2458 5270 a(t)2507 5146 y Fo(\001)2545 5163 y Fp(N)2658 5109 y(n)2626 5134 y Fo(Y)2621 5311 y Fp(j)s Fr(=1)2736 5213 y Fq(h)p Fs(x)2815 5225 y Fr(0)p Fp(;j)2904 5213 y Fq(i)2936 5179 y Fm(\000)p Fp(\016)3018 5187 y Fh(1)3144 5109 y Fp(k)3109 5134 y Fo(Y)3069 5313 y Fp(j)s Fr(=)p Fp(l)p Fr(+1)3256 5213 y Fq(h)p Fs(x)3335 5225 y Fp(j)3370 5213 y Fq(i)3402 5179 y Fm(\000)p Fp(\016)1934 5540 y Ft(15)p eop %%Page: 16 16 16 15 bop 204 531 a Ft(from)27 b(whic)n(h)h(w)n(e)f(deduce)h(the)g (result)f(easily)-7 b(,)204 631 y Fq(\017)24 b Ft(or)g Fs(t)399 643 y Fp(l)p Fr(+1)521 631 y Ft(+)12 b Fq(\001)i(\001)g(\001)e Ft(+)g Fs(t)814 643 y Fp(k)878 631 y Fq(\025)22 b Fs(t=)p 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Fp(j)2156 974 y Ft(\))h(+)f Fq(h)p Fs(x)2369 986 y Fp(k)2429 974 y Fq(\000)g Fs(x)2559 986 y Fp(l)2585 974 y Fs(;)c(\030)2658 986 y Fp(k)2699 974 y Fq(i)k(\000)g Fs(t)2862 986 y Fp(k)2903 974 y Fs(!)s Ft(\()p Fs(\030)3026 986 y Fp(k)3067 974 y Ft(\))p Fs(;)204 1243 y Ft(and)27 b(w)n(e)g(remark)f(that)h Fs(\036)999 1255 y Fp(k)1058 1243 y Fq(\000)1152 1221 y Ft(~)1141 1243 y Fs(\036)1190 1255 y Fp(k)q Fm(\000)p Fp(l)p Fr(+1)1406 1243 y Fq(\000)17 b(h)p Fs(x)1567 1255 y Fp(l)1611 1243 y Fq(\000)h Fs(x)1741 1255 y Fr(1)1778 1243 y Fs(;)c(\030)1851 1255 y Fp(k)1893 1243 y Fq(i)k Ft(+)f Fq(h)p Fs(x)2104 1255 y Fp(l)2130 1243 y Fs(;)d(\030)2203 1255 y Fp(l)2229 1243 y Fq(i)28 b Ft(dep)r(ends)f(only)g(on)32 b(~)-47 b Fs(x)2953 1255 y Fr(0)3018 1243 y Ft(and)27 b(the)h(parameters,)204 1342 y(and)f(w)n(e)g(can)h(get)f(the)h(result)f(as)g(b)r(efore.)37 b(The)27 b(pro)r(of)h(is)f(no)n(w)g(complete.)1157 b Fc(\003)204 1571 y Fu(2.3)112 b(The)38 b(regular)f(terms)204 1724 y Ft(In)28 b(that)f(subsection)h(w)n(e)f(will)h(study)f(the)h (distributions)g(de\014ned,)g(when)38 b(~)-52 b Fs(\037)23 b Fq(2)g Fs(S)2756 1736 y Fr(1)2793 1724 y Ft(\(1)p Fs(;)14 b Fq(\0001)p Ft(\))28 b(and)f Fs(j)h Fq(\025)23 b Ft(0,)k(b)n(y)897 1942 y Fs(<)c(u)p Ft(\()p Fs(h)p Ft(\))p Fs(;)14 b(f)32 b(>)p Ft(=)22 b(T)-7 b(r)1506 1825 y Fo(\022)1577 1942 y Ft(~)-52 b Fs(\037)p Ft(\()p Fs(x;)14 b(hD)r Ft(\))1914 1885 y Fs(d)1957 1855 y Fp(j)p 1897 1923 112 4 v 1897 1999 a Fs(d\017)1974 1975 y Fp(j)2019 1942 y Fs(f)9 b Ft(\()p Fs(H)2170 1954 y Fp(\017)2202 1942 y Ft(\))2234 1957 y Fm(j)2254 1965 y Fe(\017)p Fh(=0)2358 1825 y Fo(\023)2433 1942 y Fs(;)180 b(f)32 b Fq(2)23 b Fs(C)2852 1907 y Fm(1)2846 1962 y Fr(0)2923 1942 y Ft(\()p Fs(I)7 b Ft(\))p Fs(:)204 2150 y Ft(It)23 b(is)f(already)f(kno)n(wn)h(that)g(these)h (distributions)f(ha)n(v)n(e)f(a)h(full)h(asymptotic)f(expansion)g(in)g Fs(h)h Ft(in)f(the)h(w)n(eak)f(sense)204 2249 y(\(see)30 b([4)o(,)h(7)o(]\);)h(this)e(means)g(that)g(there)g(exists)g(a)f (sequence)h(\()p Fs(u)2190 2261 y Fp(k)2230 2249 y Ft(\))2262 2261 y Fp(k)q Fm(2)p Fn(N)2420 2249 y Ft(of)g(elemen)n(ts)g(of)g Fq(D)3022 2219 y Fm(0)3046 2249 y Ft(\()p Fs(I)7 b Ft(\),)31 b(indep)r(enden)n(t)g(of)204 2349 y Fs(h)p Ft(,)c(suc)n(h)h(that)g(for) f(all)g Fs(N)32 b Fq(\025)22 b Ft(0)1000 2589 y Fs(u)p Ft(\()p Fs(h)p Ft(\))h(=)g Fs(h)1319 2554 y Fm(\000)p Fp(d)1454 2485 y(N)1423 2510 y Fo(X)1423 2689 y Fp(k)q Fr(=0)1558 2589 y Fs(h)1606 2554 y Fp(k)1646 2589 y Fs(u)1694 2601 y Fp(k)1753 2589 y Ft(+)18 b Fs(h)1884 2554 y Fp(N)6 b Fr(+1)p Fm(\000)p Fp(d)2123 2589 y Ft(~)-48 b Fs(u)2165 2601 y Fp(N)2228 2589 y Ft(\()p Fs(h)p Ft(\))p Fs(;)180 b(h)23 b Fq(&)g Ft(0)p Fs(;)750 b Ft(\(2.11\))204 2831 y(where)40 b(~)-47 b Fs(u)500 2843 y Fp(N)562 2831 y Ft(\()p Fs(h)p Ft(\))36 b(is)f(a)g(b)r(ounded)h(family)g(of)f Fq(D)1655 2801 y Fm(0)1679 2831 y Ft(\()p Fs(I)7 b Ft(\))36 b(\()p Fi(i.e.)62 b Fs(<)41 b Ft(~)-47 b Fs(u)2178 2843 y Fp(N)2240 2831 y Ft(\()p Fs(h)p Ft(\))p Fs(;)14 b(f)45 b(>)35 b Ft(is)h(b)r(ounded)f(for)g(all)g Fs(f)45 b Fq(2)37 b Fs(C)3514 2801 y Fm(1)3508 2851 y Fr(0)3584 2831 y Ft(\()p Fs(I)7 b Ft(\)\).)204 2931 y(Our)25 b(goal)g(is)i(to)f(pro)n(v) n(e)e(that)j(the)g(asymptotic)f(\(2)p Fs(:)p Ft(11\))f(holds)h(in)h (the)f(strong)f(sense,)i(that)f(is)g(in)h(the)g Fs(C)3442 2900 y Fm(1)3539 2931 y Ft(sense.)204 3030 y(This)g(is)g(the)h(purp)r (ose)f(of)g(the)h(follo)n(wing)e(prop)r(osition.)36 b(Recall)26 b(that)i Fs(I)i Fq(\032)23 b Ft(\(0)p Fs(;)14 b Fq(1)p Ft(\))27 b(is)g(an)g(op)r(en)h(an)f(non)g(critical)204 3130 y(in)n(terv)-5 b(al)27 b(for)g Fs(!)s Ft(.)204 3280 y Fg(Prop)s(osition)j(2.5.)40 b Fi(The)31 b(distributions)f Fs(u)1606 3292 y Fp(k)1676 3280 y Fi(ar)l(e)g Fs(C)1882 3250 y Fm(1)1983 3280 y Fi(on)f Fs(I)37 b Fi(and)f Ft(~)-48 b Fs(u)2383 3292 y Fp(N)2446 3280 y Ft(\()p Fs(h)p Ft(\))30 b Fi(is)g(a)g(b)l(ounde)l(d)g(family)i(in)e Fs(C)3485 3250 y Fm(1)3555 3280 y Ft(\()p Fs(I)7 b Ft(\))p Fi(.)204 3430 y Fg(Pro)s(of.)43 b Ft(Because)29 b(of)h(the)g(uniqueness)f(of)h (the)g(asymptotic)g(expansion,)f(it)i(is)e(su\016cien)n(t)h(to)g(sho)n (w)f(that)h Fs(u)p Ft(\()p Fs(h)p Ft(\))204 3530 y(has)25 b(a)g(full)i(asymptotic)e(expansion)g(in)h Fs(C)1530 3500 y Fm(1)1601 3530 y Ft(\()p Fs(J)8 b Ft(\))26 b(for)f(all)h Fs(J)31 b Fc(b)23 b Fs(I)7 b Ft(.)36 b(By)26 b(c)n(ho)r(osing)e(a)h (cuto\013)h(function)h Fs(g)e Fq(2)f Fs(C)3546 3500 y Fm(1)3540 3551 y Fr(0)3616 3530 y Ft(\()p Fs(I)7 b Ft(\),)204 3630 y Fs(g)25 b Fq(\021)e Ft(1)h(near)g Fs(J)8 b Ft(,)26 b(w)n(e)e(only)g(ha)n(v)n(e)g(to)g(study)h Fs(g)s(u)p Ft(\()p Fs(h)p Ft(\),)g(and)f(w)n(e)h(can)f(consider)g(its)h (semiclassical)e(F)-7 b(ourier)23 b(transform)204 3729 y(whic)n(h)k(is)1066 3926 y Fs(w)r Ft(\()p Fs(h;)14 b(t)p Ft(\))24 b(=)e(T)-7 b(r)1517 3809 y Fo(\022)1588 3926 y Ft(~)-52 b Fs(\037)p Ft(\()p Fs(x;)14 b(hD)r Ft(\))1924 3870 y Fs(d)1967 3839 y Fp(j)p 1908 3907 V 1908 3983 a Fs(d\017)1985 3959 y Fp(j)2030 3858 y Fo(\000)2068 3926 y Fs(g)s Ft(\()p Fs(H)2212 3938 y Fp(\017)2243 3926 y Ft(\))p Fs(U)2332 3938 y Fp(\017)2364 3926 y Ft(\()p Fs(t)p Ft(\))2458 3858 y Fo(\001)2497 3809 y(\023)2558 4009 y Fm(j)2578 4017 y Fe(\017)p Fh(=0)2696 3926 y Fs(:)204 4157 y Ft(By)33 b(lemma)h(2)p Fs(:)p Ft(3,)g Fs(w)r Ft(\()p Fs(h;)14 b(t)p Ft(\))35 b(is)f(a)f(linear)g(com)n(bination)g(\(with)h (co)r(e\016cien)n(ts)g(indep)r(enden)n(t)g(on)g Fs(h)f Ft(and)h Fs(t)p Ft(\))g(of)g(the)204 4257 y(follo)n(wing)26 b(t)n(yp)r(e)i(of)g(functions,)g(with)g(0)22 b Fq(\024)h Fs(k)j Fq(\024)d Fs(j)5 b Ft(:)540 4473 y Fs(h)588 4439 y Fm(\000)p Fp(k)694 4360 y Fo(Z)777 4380 y Fp(t)740 4548 y Fr(0)792 4360 y Fo(Z)876 4380 y Fp(t)901 4388 y Fh(1)839 4548 y Fr(0)909 4473 y Fq(\001)14 b(\001)g(\001)1006 4360 y Fo(Z)1089 4380 y Fp(t)1114 4389 y Fe(k)q Fd(\000)p Fh(1)1052 4548 y Fr(0)1242 4473 y Ft(T)-7 b(r)1328 4406 y Fo(\000)1366 4473 y Fq(K)1429 4485 y Fp(k)1470 4473 y Ft(\()p Fs(t)18 b Fq(\000)g Fs(t)1663 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4877 y Fp(k)q Fr(+1)959 4865 y Fs(;)g(h)p Ft(\))23 b(=)f Fs(a)1230 4877 y Fr(1)1267 4865 y Ft(\()p Fs(x;)14 b(hD)r(;)g(h)p Ft(\))p Fs(U)1676 4877 y Fr(0)1714 4865 y Ft(\()p Fs(t)1776 4877 y Fr(1)1813 4865 y Ft(\))p Fs(a)1889 4877 y Fr(2)1927 4865 y Ft(\()p Fs(x;)g(hD)r(;)g(h)p Ft(\))p Fs(U)2336 4877 y Fr(0)2373 4865 y Ft(\()p Fs(t)2435 4877 y Fr(2)2473 4865 y Ft(\))g Fq(\001)g(\001)g(\001)g Fs(a)2674 4877 y Fp(k)q Fr(+1)2798 4865 y Ft(\()p Fs(x;)g(hD)r(;)g(h)p Ft(\))p Fs(U)3207 4877 y Fr(0)3245 4865 y Ft(\()p Fs(t)3307 4877 y Fp(k)q Fr(+1)3432 4865 y Ft(\))p Fs(;)48 b Ft(\(2.13\))204 5028 y(with)32 b(sym)n(b)r(ols)f Fs(a)762 5040 y Fr(1)799 5028 y Fs(;)14 b Fq(\001)g(\001)g(\001)28 b Fs(;)14 b(a)1042 5040 y Fp(k)q Fr(+1)1198 5028 y Ft(all)32 b(supp)r(orted)f(in)h(a)g (small)f(neigh)n(b)r(ourho)r(o)r(d)g(of)h Fs(!)2834 4998 y Fm(\000)p Fr(1)2922 5028 y Ft(\(supp)q(\()p Fs(g)s Ft(\)\))g(with)g(resp)r(ect)204 5128 y(to)27 b(the)h Fs(\030)k Ft(v)-5 b(ariable,)27 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722 y Fp(\036)778 731 y Fe(k)q Fh(+1)890 756 y Fs(a)934 768 y Fr(1)971 756 y Ft(\()p Fs(x)1050 768 y Fr(1)1088 756 y Fs(;)14 b(\030)1161 768 y Fr(1)1198 756 y Ft(\))p Fs(a)1274 768 y Fr(2)1311 756 y Ft(\()p Fs(x)1390 768 y Fr(2)1428 756 y Fs(;)g(\030)1501 768 y Fr(2)1539 756 y Ft(\))g Fq(\001)g(\001)g(\001)g Fs(a)1740 768 y Fp(k)q Fr(+1)1865 756 y Ft(\()p Fs(x)1944 768 y Fp(k)q Fr(+1)2069 756 y Fs(;)g(\030)2142 768 y Fp(k)q Fr(+1)2267 756 y Ft(\))p Fs(dx)2389 768 y Fr(2)2441 756 y Fq(\001)g(\001)g(\001)g Fs(dx)2642 768 y Fp(k)q Fr(+1)2768 756 y Fs(d\030)2847 768 y Fr(1)2898 756 y Fq(\001)g(\001)g(\001)g Fs(d\030)3088 768 y Fp(k)q Fr(+1)3213 756 y Fs(;)299 b Ft(\(2.14\))204 982 y(where)27 b(the)h(phase)f (function)h Fs(\036)1192 994 y Fp(k)q Fr(+1)1341 982 y Ft(=)22 b Fs(\036)1477 994 y Fp(k)q Fr(+1)1603 982 y Ft(\()p Fs(x)1682 994 y Fr(1)1720 982 y Fs(;)14 b Fq(\001)g(\001)g (\001)27 b Fs(;)14 b(x)1965 994 y Fp(k)q Fr(+2)2090 982 y Fs(;)g(\030)2163 994 y Fr(1)2201 982 y Fs(;)g 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1816 y Ft(.)56 b(By)34 b(rep)r(eating)f(this)i(op)r(eration)e(with)i (the)f(other)g Fs(x)3712 1828 y Fp(j)204 1916 y Ft(v)-5 b(ariables,)37 b(w)n(e)f(can)h(assume)e(that)i(the)g(sym)n(b)r(ol)f(is) g(in)n(tegrable)g(with)h(resp)r(ect)f(to)g(eac)n(h)g Fs(x)3135 1928 y Fp(j)3207 1916 y Ft(since)g(it)h(deca)n(ys)204 2015 y(lik)n(e)d Fq(h)p Fs(x)442 2027 y Fp(j)478 2015 y Fq(i)510 1985 y Fm(\000)p Fr(\()p Fp(k)q Fr(+2)p Fm(\000)p Fp(j)s Fr(\))p Fp(d)p Fm(\000)p Fp(\032)977 2015 y Ft(with)h(resp)r (ect)g(to)f(eac)n(h)g Fs(x)1814 2027 y Fp(j)1849 2015 y Ft(,)j(with)e Fs(k)26 b Ft(+)d(2)f Fq(\000)h Fs(j)40 b Fq(\025)34 b Ft(1.)58 b(Notice)35 b(that)g(this)f(generates)g(a)204 2115 y(factor)j Fq(O)r Ft(\()p Fs(t)582 2085 y Fp(k)q Fr(\()p Fp(k)q Fr(+1\))p Fp(d=)p Fr(2)897 2115 y Ft(\))h(but)h(an)n(yw) n(a)n(y)-7 b(,)38 b(w)n(e)f(are)g(no)n(w)g(in)h(p)r(osition)f(to)h(use) f(the)h(lemma)g(2)p Fs(:)p Ft(4:)56 b(it)38 b(pro)n(v)n(es)e(that)204 2215 y Fs(w)r Ft(\()p Fs(t;)14 b(h)p Ft(\))24 b(=)g Fq(O)r Ft(\(\()p Fs(h=t)p Ft(\))841 2184 y Fm(1)912 2215 y Ft(\),)k(when)g Fq(j)p Fs(t)p Fq(j)c(\025)g Fs(c)k Ft(and)f Fs(h)d Fq(2)g Ft(\(0)p Fs(;)14 b(h)1936 2227 y Fr(0)1973 2215 y Ft(],)28 b(th)n(us,)g(if)h Fs(')24 b Fq(2)g Fs(C)2552 2184 y Fm(1)2546 2235 y Fr(0)2623 2215 y Ft(\()p Fq(\000)p Ft(2)p Fs(c;)14 b Ft(2)p Fs(c)p Ft(\))27 b(is)h(suc)n(h)f(that)i Fs(')24 b Fq(\021)f Ft(1)k(on)204 2314 y(\()p Fq(\000)p Fs(c;)14 b(c)p Ft(\),)27 b(w)n(e)h(ha)n(v)n(e)932 2539 y Fs(\025)c Fq(7!)f Ft(\(2)p Fs(\031)s(h)p Ft(\))1314 2505 y Fm(\000)p Fr(1)1417 2426 y Fo(Z)1514 2539 y Fs(e)1569 2483 y Fe(i)p 1562 2492 V 1562 2525 a(h)1607 2505 y Fp(t\025)1675 2539 y Ft(\(1)c Fq(\000)f Fs(')p Ft(\)\()p Fs(t)p Ft(\))p Fs(w)r Ft(\()p Fs(t;)c(h)p Ft(\))p Fs(dt)25 b Ft(=)e Fq(O)r Ft(\()p Fs(h)2605 2505 y Fm(1)2676 2539 y Ft(\))55 b(in)28 b Fs(C)2925 2505 y Fm(1)2996 2539 y Fs(:)328 2764 y Ft(No)n(w)35 b(w)n(e)h(consider)e(the)i(region)e Fq(j)p Fs(t)p Fq(j)j(\024)f Ft(2)p Fs(c)p Ft(.)60 b(The)36 b(op)r(erator)e Fq(K)2355 2776 y Fp(k)2432 2764 y Ft(app)r(ears)g(in)i (\(2)p Fs(:)p Ft(12\))f(as)g(the)h(pro)r(duct)f(of)204 2864 y Fs(a)248 2876 y Fr(0)285 2864 y Ft(\()p Fs(x;)14 b(hD)r(;)g(h)p Ft(\))p Fs(U)694 2876 y Fr(0)731 2864 y Ft(\()p Fs(t)p Ft(\))41 b(and)e(of)h(the)g(op)r(erators)e Fs(U)1738 2876 y Fr(0)1775 2864 y Ft(\()p Fq(\000)p Fs(t)1902 2876 y Fp(j)1937 2864 y Ft(\))p Fs(a)2013 2876 y Fp(j)2048 2864 y Ft(\()p Fs(x;)14 b(hD)r(;)g(h)p Ft(\))p Fs(U)2457 2876 y Fr(0)2494 2864 y Ft(\()p Fs(t)2556 2876 y Fp(j)2592 2864 y Ft(\),)43 b(1)g Fq(\024)f Fs(j)49 b Fq(\024)43 b Fs(k)s Ft(,)f(with)f Fq(j)p Fs(t)3439 2876 y Fp(j)3474 2864 y Fq(j)i(\024)g(j)p Fs(t)p Fq(j)p Ft(,)204 2964 y(whic)n(h)30 b(are,)h(b)n(y)f(the)h(Egoro)n(v's)c(theorem,)k (pseudo-di\013eren)n(tial)f(op)r(erators.)43 b(Their)30 b(pro)r(duct)h(can)f(b)r(e)h(written)205 3063 y(~)-43 b Fs(a)248 3075 y Fp(k)289 3063 y Ft(\()p Fs(x;)14 b(hD)r(;)g(t;)g(k)s Ft(\),)28 b(with)h(~)-43 b Fs(a)990 3075 y Fp(k)1059 3063 y 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y(phase)27 b(theorem,)g(and)g(this)h(completes)g(the)g(pro)r(of.)1839 b Fc(\003)204 3940 y Fu(2.4)112 b(Pro)s(of)37 b(of)h(theorem)f(2)p Fb(:)p Fu(1)204 4094 y Ft(W)-7 b(e)28 b(ha)n(v)n(e)e(to)i(in)n(v)n (estigate,)e(for)h Fs(\025)d Fq(2)f Fs(J)31 b Fc(b)23 b Fs(I)7 b Ft(,)27 b(the)h(follo)n(wing)f(distribution)1176 4355 y Fs(u)1224 4367 y Fp(p)1262 4355 y Ft(\()p Fs(\025;)14 b(h)p Ft(\))23 b(=)g Fs(u)1618 4367 y Fr(1)p Fp(;)p Fr(1)1708 4355 y Ft(\()p Fs(\025;)14 b(h)p Ft(\))19 b Fq(\000)2007 4247 y Fp(p)p Fm(\000)p Fr(1)2007 4276 y Fo(X)2009 4453 y Fp(j)s Fr(=0)2161 4298 y Ft(1)p 2151 4336 63 4 v 2151 4412 a Fs(j)5 b Ft(!)2223 4355 y Fs(@)2272 4320 y Fp(j)2267 4375 y(\017)2306 4355 y Fs(u)2354 4367 y Fr(1)p Fp(;\017)2438 4355 y Ft(\()p Fs(\025;)14 b(h)p Ft(\))p Fq(j)2658 4367 y Fp(\017)p Fr(=0)204 4621 y Ft(where,)27 b(in)h(the)g(w)n(eak)e (sense,)h(w)n(e)h(ha)n(v)n(e)1165 4849 y Fs(u)1213 4861 y Fr(1)p Fp(;\017)1297 4849 y Ft(\()p Fs(\025;)14 b(h)p Ft(\))24 b(=)e(T)-7 b(r)1705 4732 y 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Fm(\006)3454 1229 y Ft(dep)r(ends)204 1328 y(smo)r(othly)j(on)h Fs(\017)g Ft(and)g(that)g(w)n(e)g(get)g(a)f (p)r(o)n(w)n(er)g Fq(h)p Fs(x)p Fq(i)1805 1298 y Fm(\000)p Fp(\032)1929 1328 y Ft(at)h(eac)n(h)f(di\013eren)n(tiation)h(with)g (resp)r(ect)g(to)g Fs(\017)p Ft(.)50 b(W)-7 b(e)32 b(also)204 1428 y(need)d(to)g(con)n(trol)e(the)i(constructions)f(with)i(resp)r (ect)e(to)h(the)g(principal)g(sym)n(b)r(ol)f(of)h(the)g(p)r (erturbation,)g Fs(q)f Fq(2)e(B)204 1528 y Ft(\(see)h(the)h(previous)f (section\).)328 1649 y(W)-7 b(e)31 b(only)e(treat)g(the)i(outgoing)d (case,)i(since)f(the)i(incoming)e(is)h(similar)f(and)h(w)n(e)f(drop)g (the)i(supscrit)e(+)g(on)204 1748 y(the)f(phases)f(and)g(the)h(sym)n(b) r(ols.)328 1848 y(Let)d Fs(\033)521 1860 y Fr(0)582 1848 y Fq(2)e Ft(\()p Fq(\000)p Ft(1)p Fs(;)14 b Ft(1\),)24 b(and)g Fs(J)1161 1860 y Fr(0)1223 1848 y Ft(an)g(op)r(en)g(in)n(terv) -5 b(al)24 b(suc)n(h)g(that)g Fs(J)2235 1860 y Fr(0)2296 1848 y Fc(b)e Fs(I)7 b Ft(.)36 b(W)-7 b(e)25 b(shall)f(use)g(the)g (follo)n(wing)g(notation)1367 2014 y Fs(H)1443 1980 y Fp(cl)1436 2034 y(\017)1498 2014 y Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;) g(h)p Ft(\))23 b(=)g Fs(!)s Ft(\()p Fs(\030)t Ft(\))c(+)f Fs(\017q)s Ft(\()p Fs(x;)c(\030)t(;)g(h)p Ft(\))p Fs(:)204 2180 y Ft(This)37 b(\014rst)g(prop)r(osition)g(solv)n(es)e(the)j (Hamilton-Jacobi)e(equation)h(\(3)p Fs(:)p Ft(3\))g(and)g(lists)g(the)h (prop)r(erties)e(of)i(the)204 2279 y(phase.)204 2432 y Fg(Prop)s(osition)30 b(3.1.)40 b Fi(Ther)l(e)g(exists)g Fs(R)1459 2444 y Fr(0)1537 2432 y Fs(>)g Ft(0)f Fi(such)h(that)f(for)i (al)t(l)f Fs(q)k Fq(2)d(B)s Fi(,)h(one)d(c)l(an)h(\014nd)f(a)h (function)g Fs(')h Fq(2)204 2532 y Fs(C)269 2501 y Fm(1)339 2532 y Ft(\()p Fl(R)426 2501 y Fr(2)p Fp(d)522 2532 y Fq(\002)18 b Ft([0)p Fs(;)c Ft(1])j Fq(\002)h Ft([0)p Fs(;)c(h)1022 2544 y Fr(0)1059 2532 y Ft(])p Fs(;)g Fl(R)p Ft(\))36 b Fi(satisfying)1321 2697 y Fs(H)1397 2663 y Fp(cl)1390 2718 y(\017)1452 2697 y Ft(\()p Fs(x;)14 b(@)1612 2709 y Fp(x)1655 2697 y Fs(')p Ft(\()p Fs(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)g(h)p Ft(\))23 b(=)g Fs(!)s Ft(\()p Fs(\030)t Ft(\))p Fs(;)1113 b Ft(\(3.3\))204 2863 y Fi(for)30 b(al)t(l)h Ft(\()p Fs(x;)14 b(\030)t Ft(\))24 b Fq(2)g Ft(\000)798 2833 y Fr(+)853 2863 y Ft(\()p Fs(R)948 2875 y Fr(0)985 2863 y Fs(;)14 b(J)1068 2875 y Fr(0)1106 2863 y Fs(;)g(\033)1190 2875 y Fr(0)1227 2863 y Ft(\))p Fi(,)31 b Fs(\017)22 b Fq(2)i Ft([0)p Fs(;)14 b Ft(1])28 b Fi(and)j Fs(h)23 b Fq(2)g Ft([0)p Fs(;)14 b(h)2106 2875 y Fr(0)2143 2863 y Ft(])p Fi(.)38 b Fs(')30 b Fi(satis\014es)g(the)g(additional)i(c)l (onditions:)1611 3029 y Fs(')p Ft(\()p Fs(x;)14 b(\030)t(;)g Ft(0)p Fs(;)g(h)p Ft(\))23 b(=)g Fq(h)p Fs(x;)14 b(\030)t Fq(i)p Fs(;)1237 b Ft(\(3.4\))1425 3154 y Fq(jr)1517 3124 y Fp(t)1517 3174 y(x)1559 3154 y Fq(r)1628 3166 y Fp(\030)1664 3154 y Fs(')p Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))20 b Fq(\000)e Ft(1)2207 3166 y Fp(d)2245 3154 y Fq(j)23 b(\024)g Ft(1)p Fs(=)p Ft(2)p Fs(;)1048 b Ft(\(3.5\))878 3291 y Fq(j)p Fs(@)950 3261 y Fp(\013)945 3311 y(x)997 3291 y Fs(@)1046 3251 y Fp(\014)1041 3316 y(\030)1090 3291 y Fs(@)1139 3261 y Fp(n)1134 3311 y(\017)1184 3291 y Fs(@)1233 3261 y Fp(l)1228 3314 y(h)1271 3224 y Fo(\000)1309 3291 y Fs(')p Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))20 b Fq(\000)e(h)p Fs(x;)c(\030)t Fq(i)1998 3224 y Fo(\001)2037 3291 y Fq(j)23 b(\024)f Fs(C)6 b Ft(\()p Fs(\013;)14 b(\014)t(;)g(n;)g(l)r Ft(\))p Fq(h)p Fs(x)p Fq(i)2702 3261 y Fr(1)p Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2997 3269 y Fh(+)3050 3291 y Fs(;)503 b Ft(\(3.6\))204 3472 y Fi(for)30 b(al)t(l)g Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))23 b Fq(2)g Fl(R)953 3442 y Fr(2)q Fp(d)1048 3472 y Fq(\002)16 b Ft([0)p Fs(;)e Ft(1])i Fq(\002)g Ft([0)p Fs(;)e(h)1543 3484 y Fr(0)1580 3472 y Ft(])29 b Fi(and)g Fs(\013;)14 b(\014)t(;)g(n;)g(l)r Fi(,)29 b(with)h(the)f(notation)g Fs(n)2835 3484 y Fr(+)2913 3472 y Ft(=)23 b(max)o(\(1)p Fs(;)14 b(n)p Ft(\))p Fi(.)39 b(Mor)l(e)l(over)204 3572 y(we)30 b(c)l(an)g(cho)l(ose)h(the)f(function)f Fs(')h Fi(dep)l(ending)h(c)l(ontinuously)f(on)g Fs(q)s Fi(,)g(that)g(is)452 3738 y Fq(j)p Fs(@)524 3704 y Fp(\013)519 3758 y(x)571 3738 y Fs(@)620 3698 y Fp(\014)615 3763 y(\030)665 3738 y Fs(@)714 3704 y Fp(n)709 3758 y(\017)759 3738 y Fs(@)808 3704 y Fp(l)803 3758 y(h)846 3671 y Fo(\000)884 3738 y Fs(')938 3750 y Fr(1)975 3738 y Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g (h)p Ft(\))19 b Fq(\000)f Fs(')1475 3750 y Fr(2)1513 3738 y Ft(\()p Fs(x;)c(\030)t(;)g(\017;)g(h)p Ft(\))1857 3671 y Fo(\001)1895 3738 y Fq(j)23 b(\024)g Fs(C)2094 3704 y Fm(0)2117 3738 y Ft(\()p Fs(\013;)14 b(\014)t(;)g(n;)g(l)r Ft(\))p Fq(N)e Ft(\()p Fs(q)2625 3704 y Fr(1)2682 3738 y Fq(\000)18 b Fs(q)2805 3704 y Fr(2)2842 3738 y Ft(\))p Fq(h)p Fs(x)p Fq(i)2985 3704 y Fr(1)p Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)3280 3712 y Fh(+)3576 3738 y Ft(\(3.7\))204 3917 y Fi(for)30 b(some)h(semi-norm)e Fq(N)43 b Fi(on)29 b Fs(S)1245 3929 y Fr(1)1282 3917 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\))p Fi(,)30 b(when)g Fs(')1867 3929 y Fp(j)1932 3917 y Fi(is)h(asso)l(ciate)l(d)g(to)f Fs(q)2550 3887 y Fp(j)2585 3917 y Fi(,)g Fs(j)e Ft(=)22 b(1)p Fs(;)14 b Ft(2)p Fi(.)328 4070 y Ft(The)27 b(pro)r(of)f(of)h (this)g(prop)r(osition)e(will)i(b)r(e)g(established)g(in)g(the)g(next)g (subsection.)36 b(In)27 b(the)g(presen)n(t)f(subsec-)204 4170 y(tion,)f(w)n(e)g(are)e(only)i(in)n(terested)f(in)h(the)g (construction)f(of)h(the)g(parametrix)e(and)h(w)n(e)h(quote)f(the)h (results)g(that)g(w)n(e)204 4269 y(are)h(going)h(to)g(use.)328 4369 y(Once)f(the)h(phase)f(function)i(is)e(de\014ned,)h(w)n(e)f(ha)n (v)n(e)g(to)g(consider)g Fs(X)j Ft(=)23 b Fs(X)7 b Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\),)26 b(the)h(solution)f(of) h(the)204 4468 y(follo)n(wing)f(di\013eren)n(tial)i(equation)1492 4562 y Fo(\()1592 4627 y Ft(_)1559 4648 y Fs(X)h Ft(=)23 b(\002\()p Fs(X)r(;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)1559 4767 y(X)1628 4782 y Fm(j)1648 4790 y Fe(t)p Fh(=0)1773 4767 y Ft(=)23 b Fs(x;)204 4959 y Ft(where)477 4938 y(_)444 4959 y Fs(X)29 b Ft(=)23 b Fs(@)674 4971 y Fp(t)703 4959 y Fs(X)34 b Ft(and)28 b(\002)f(is)g(the)h(v)n(ector)f (\014eld)g(de\014ned)h(b)n(y)1055 5125 y(\002\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(:)p Fq(r)1556 5137 y Fp(x)1621 5125 y Ft(=)22 b(\()p Fq(r)1809 5137 y Fp(\030)1846 5125 y Fs(H)1922 5091 y Fp(cl)1915 5146 y(\017)1977 5125 y Ft(\))2009 5058 y Fo(\000)2047 5125 y Fs(x;)14 b(@)2175 5137 y Fp(x)2218 5125 y Fs(')p Ft(\()p Fs(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)g(h)2701 5058 y Fo(\001)2739 5125 y Fs(:)p Fq(r)2831 5137 y Fp(x)2873 5125 y Fs(:)204 5291 y Ft(Let)28 b Fs(J)399 5303 y Fr(1)463 5291 y Ft(b)r(e)h(an)e(op)r(en)g(in)n(terv) -5 b(al)27 b(suc)n(h)h(that)g Fs(J)1606 5303 y Fr(1)1666 5291 y Fc(b)23 b Fs(J)1800 5303 y Fr(0)1837 5291 y Ft(,)28 b(and)f(let)h Fs(\033)2216 5303 y Fr(1)2281 5291 y Ft(b)r(e)g(suc)n(h)g (that)f Fq(\000)p Ft(1)c Fs(<)f(\033)3025 5303 y Fr(1)3086 5291 y Fs(<)h(\033)3221 5303 y Fr(0)3258 5291 y Ft(.)1934 5540 y(20)p eop %%Page: 21 21 21 20 bop 204 531 a Fg(Prop)s(osition)30 b(3.2.)40 b Fi(Ther)l(e)h(exists)e Fs(R)1459 543 y Fr(1)1538 531 y Fs(>)i Ft(0)f Fi(and)g Fs(e)1936 543 y Fr(1)2014 531 y Fs(>)i Ft(0)d Fi(such)h(that,)j(for)e(al)t(l)g Ft(\()p Fs(x;)14 b(\030)t Ft(\))42 b Fq(2)g Ft(\000)3260 501 y Fr(+)3315 531 y Ft(\()p Fs(R)3410 543 y Fr(1)3448 531 y Fs(;)14 b(J)3531 543 y Fr(1)3568 531 y Fs(;)g(\033)3652 543 y Fr(1)3689 531 y Ft(\))p Fi(,)204 631 y Ft(\()p Fs(\017;)g(h)p Ft(\))23 b Fq(2)g Ft([0)p Fs(;)14 b Ft(1])k Fq(\002)g Ft([0)p Fs(;)c(h)906 643 y Fr(0)942 631 y Ft(])p Fi(,)31 b(and)f Fs(q)c Fq(2)d(B)s Fi(,)30 b(the)f(solution)h Fs(X)7 b Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))30 b Fi(is)g(de\014ne)l(d)g(for)g Fs(t)24 b Fq(\025)e Ft(0)p Fi(,)30 b(and)g(satis\014es)1486 808 y Fq(j)p Fs(X)7 b Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(j)22 b(\025)h Fs(e)2168 820 y Fr(1)2205 808 y Ft(\()p Fs(t)c Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))p Fs(;)1059 b Ft(\(3.8\))1253 866 y Fo(\000)1292 933 y Fs(X)7 b Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)g(\030)1856 866 y Fo(\001)1916 933 y Fq(2)24 b Ft(\000)2047 899 y Fr(+)2102 933 y Ft(\()p Fs(R)2197 945 y Fr(0)2234 933 y Fs(;)14 b(J)2317 945 y Fr(0)2354 933 y Fs(;)g(\033)2438 945 y Fr(0)2476 933 y Ft(\))p Fs(:)1045 b Ft(\(3.9\))204 1110 y Fi(Mor)l(e)l(over,)31 b(for)g(al)t(l)g Fs(\013;)14 b(\014)t(;)g(n;)g(l)31 b Fi(ther)l(e)f(exists)f Fs(C)h(>)22 b Ft(0)30 b Fi(such)f(that)829 1288 y Fq(j)p Fs(@)901 1253 y Fp(\013)896 1308 y(x)948 1288 y Fs(@)997 1248 y Fp(\014)992 1313 y(\030)1042 1288 y Fs(@)1091 1253 y Fp(n)1086 1308 y(\017)1136 1288 y Fs(@)1185 1253 y Fp(l)1180 1308 y(h)1223 1220 y Fo(\000)1261 1288 y Fs(X)7 b Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))k Fq(\000)g Fs(x)h Fq(\000)f Fs(tv)s Ft(\()p Fs(\030)t Ft(\))2175 1220 y Fo(\001)2214 1288 y Fq(j)23 b(\024)g Fs(C)6 b Fq(h)p Fs(t)p Fq(ih)p Fs(x)p Fq(i)2618 1253 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2880 1261 y Fh(+)2932 1288 y Fs(;)580 b Ft(\(3.10\))204 1476 y Fi(for)30 b(al)t(l)h Ft(\()p Fs(x;)14 b(\030)t Ft(\))24 b Fq(2)g Ft(\000)798 1446 y Fr(+)853 1476 y Ft(\()p Fs(R)948 1488 y Fr(1)985 1476 y Fs(;)14 b(J)1068 1488 y Fr(1)1106 1476 y Fs(;)g(\033)1190 1488 y Fr(1)1227 1476 y Ft(\))p Fi(,)31 b Ft(\()p Fs(\017;)14 b(h)p Ft(\))23 b Fq(2)g Ft([0)p Fs(;)14 b Ft(1])k Fq(\002)g Ft([0)p Fs(;)c(h)2017 1488 y Fr(0)2053 1476 y Ft(])p Fi(,)30 b Fs(t)23 b Fq(\025)g Ft(0)29 b Fi(and)h Fs(q)d Fq(2)c(B)s Fi(.)204 1576 y(Mor)l(e)h(gener)l (al)t(ly,)i(if)e(the)f(supscripts)g Fs(j)28 b Fi(r)l(efer)c(to)f (functions)g(asso)l(ciate)l(d)h(to)f Fs(q)2583 1546 y Fp(j)2618 1576 y Fi(,)i Fs(j)j Ft(=)23 b(1)p Fs(;)14 b Ft(2)p Fi(,)23 b(then,)i(for)f(al)t(l)g Fs(\013;)14 b(\014)t(;)g(n;)g(l)r Fi(,)204 1675 y(ther)l(e)30 b(exists)f(a)h(c)l (onstant)f Fs(C)36 b Fi(and)30 b(a)g(semi-norm)g Fq(N)42 b Fi(in)30 b Fs(S)2049 1687 y Fr(1)2086 1675 y Ft(\()p Fs(!)s(;)14 b 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b(form)n(ula,)g(w)n(e)i(write)1091 1788 y Fs(')p Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))19 b Fq(\000)f Fs(')p Ft(\()p Fs(y)s(;)c(\030)t(;)g(\017;)g(h)p Ft(\))23 b(=)g Fq(h)p Fs(x)c Fq(\000)f Fs(y)s(;)h Ft(~)-47 b Fs(\021)s Ft(\()p Fs(x;)14 b(y)s(;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(i)204 1973 y Ft(where)34 b(~)-48 b Fs(\021)t Ft(\()p Fs(x;)14 b(y)s(;)g(\030)t(;)g(\017;)g(h)p Ft(\))25 b(=)1030 1907 y Fo(R)1086 1927 y Fr(1)1069 2003 y(0)1137 1973 y Fq(r)1206 1985 y Fp(x)1248 1973 y Fs(')p Ft(\()p Fs(y)d Ft(+)d Fs(u)p Ft(\()p Fs(x)h Fq(\000)f Fs(y)s Ft(\))p Fs(;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(du)29 b Ft(satis\014es)f(the)h(prop)r(erties)g(whic)n(h)g(w)n(e)f(sum)i(up)f (in)204 2073 y(the)f(follo)n(wing)e(lemma.)204 2231 y Fg(Lemma)j(3.3.)40 b Fi(i\))30 b(The)h(map)g Fs(\030)c Fq(7!)i Ft(~)-48 b Fs(\021)s Ft(\()p Fs(x;)14 b(y)s(;)g(\030)t(;)g (\017;)g(h)p Ft(\))30 b Fi(is)g(a)g(glob)l(al)h(di\013e)l(omorphism)i (of)d Fl(R)2966 2201 y Fp(d)3011 2231 y Fi(.)204 2331 y(ii\))g(If)h(we)f(c)l(al)t(l)690 2309 y Ft(~)681 2331 y Fs(\030)t Ft(\()p Fs(x;)14 b(y)s(;)g(:;)g(\017;)g(h)p Ft(\))30 b Fi(its)g(inverse,)h(we)f(have)h(the)f(fol)t(lowing)i (estimates)1311 2503 y Fs(C)1376 2467 y Fm(\000)p Fr(1)1370 2525 y(0)1465 2503 y Fq(h)p Fs(\021)s Fq(i)24 b(\024)f(h)1726 2481 y Ft(~)1717 2503 y Fs(\030)t Ft(\()p Fs(x;)14 b(y)s(;)g(\021)s(;)g (\017;)g(h)p Ft(\))p Fq(i)24 b(\024)e Fs(C)2388 2515 y Fr(0)2426 2503 y Fq(h)p Fs(\021)s Fq(i)p Fs(;)184 b Fi(for)31 b(some)f Fs(C)3145 2515 y Fr(0)3206 2503 y Fs(>)22 b Ft(0)p Fs(;)177 b Ft(\(3.16\))413 2641 y Fq(j)p Fs(@)485 2606 y Fp(\013)480 2661 y(x)532 2641 y Fs(@)581 2606 y Fp(\013)624 2581 y Fd(0)576 2661 y Fp(y)651 2641 y Fs(@)700 2606 y Fp(\014)695 2661 y(\021)744 2641 y Fs(@)793 2606 y Fp(n)788 2661 y(\017)838 2641 y Fs(@)887 2606 y Fp(n)882 2661 y(h)932 2641 y Ft(\()973 2619 y(~)964 2641 y Fs(\030)t Ft(\()p Fs(x;)14 b(y)s(;)g(\021)s(;)g(\017;)g(h)p Ft(\))19 b Fq(\000)f Fs(\021)s Ft(\))p Fq(j)23 b(\024)g Fs(C)1804 2653 y Fp(\013;\013)1910 2637 y Fd(0)1933 2653 y Fp(;\014)s(;n;l)2099 2641 y Fq(h)p Fs(x)p Fq(i)2210 2606 y Fm(\000)p Fp(m)2326 2641 y Fq(h)p Fs(y)s Fq(i)2434 2606 y Fm(\000)p Fp(m)2545 2581 y Fd(0)2568 2606 y Fm(\000)p Fp(\032n)2695 2614 y Fh(+)2746 2641 y Fq(h)p Fs(x)c Fq(\000)f Fs(y)s Fq(i)3003 2606 y Fp(m)p Fr(+)p Fp(m)3172 2581 y Fd(0)3194 2606 y Fr(+)p Fp(\032n)3320 2614 y Fh(+)3535 2641 y Ft(\(3.17\))204 2824 y Fi(for)34 b(al)t(l)g Fs(x;)14 b(y)s(;)g(\021)33 b Fq(2)c Fl(R)839 2794 y Fp(d)884 2824 y Fi(,)35 b Fs(\017)29 b Fq(2)g Ft([0)p Fs(;)14 b Ft(1])p Fi(,)34 b Fs(h)29 b Fq(2)h Ft([0)p Fs(;)14 b(h)1629 2836 y Fr(0)1666 2824 y Ft(])p Fi(,)34 b(and)g Fs(m;)14 b(m)2096 2794 y Fm(0)2152 2824 y Fi(such)33 b(that)h Fs(m)29 b Fq(\024)g(j)p Fs(\013)p Fq(j)p Fi(,)35 b Fs(m)2946 2794 y Fm(0)2998 2824 y Fq(\024)29 b(j)p Fs(\013)3168 2794 y Fm(0)3192 2824 y Fq(j)p Fi(.)49 b(Mor)l(e)l(over)35 b(al)t(l)204 2924 y(these)26 b(estimates)h(ar)l(e)f(uniform)h(with)g(r) l(esp)l(e)l(ct)f(to)g Fs(q)h Fq(2)c(B)28 b Fi(and)f(we)g(have)g(c)l (ontinuity)f(estimates)g(for)3339 2902 y Ft(~)3331 2924 y Fs(\030)k Fi(analo)l(gous)204 3023 y(to)g Ft(\(3)p Fs(:)p Ft(7\))p Fi(.)204 3181 y Fg(Pro)s(of.)36 b Ft(The)26 b(fact)g(that)g(w)n(e)g(get)g(a)f(global)g(di\013eomorphism)h(is)g (just)g(due)h(to)f(\(3)p Fs(:)p Ft(5\))f(and)h(to)g(a)g(global)f(in)n (v)n(ersion)204 3281 y(theorem)i(of)g([37].)37 b(Then)27 b(the)h(estimates)g(are)e(simple)i(consequences)e(of)i(prop)r(osition)e (3)p Fs(:)p Ft(1.)588 b Fc(\003)204 3403 y Ft(After)31 b(the)f(c)n(hange)f(of)i(v)-5 b(ariable)29 b Fs(\030)j Ft(=)1427 3381 y(~)1418 3403 y Fs(\030)t Ft(\()p Fs(x;)14 b(y)s(;)g(\021)s(;)g(\017;)g(h)p Ft(\),)31 b(\(3)p Fs(:)p Ft(15\))f(can)g(b)r(e)h(written)f(as)g(the)h(sum,)g(for)f Fs(k)3374 3415 y Fr(1)3411 3403 y Fs(;)14 b(k)3491 3415 y Fr(2)3556 3403 y Fq(\024)27 b Fs(N)9 b Ft(,)204 3503 y(of)27 b(the)h(k)n(ernels)f(of)g(\(2)p Fs(\031)s(h)p Ft(\))1017 3473 y Fm(\000)p Fp(d)1108 3503 y Fs(h)1156 3473 y Fp(k)1191 3481 y Fh(1)1224 3473 y Fr(+)p Fp(k)1310 3481 y Fh(2)1346 3503 y Fs(c)1382 3515 y Fp(k)1417 3523 y Fh(1)1450 3515 y Fp(;k)1505 3523 y Fh(2)1542 3503 y Ft(\()p Fs(x;)14 b(hD)r(;)g(\017;)g(h)p Ft(\),)28 b(where)597 3753 y Fs(c)633 3765 y Fp(k)668 3773 y Fh(1)701 3765 y Fp(;k)756 3773 y Fh(2)792 3753 y Ft(\()p Fs(x;)14 b(\021)s(;)g(\017;) g(h)p Ft(\))24 b Fq(\030)1252 3674 y Fo(X)1290 3849 y Fp(\013)1385 3753 y Fs(h)1433 3719 y Fm(j)p Fp(\013)p Fm(j)1547 3697 y Ft(1)p 1530 3734 77 4 v 1530 3810 a Fs(\013)p Ft(!)1616 3753 y Fs(@)1665 3719 y Fp(\013)1660 3774 y(y)1712 3753 y Fs(D)1783 3719 y Fp(\013)1781 3774 y(\021)1844 3611 y Fo( )1910 3753 y Fs(a)1954 3765 y Fp(k)1989 3773 y Fh(1)2026 3753 y Ft(\()p Fs(x;)2151 3731 y Ft(~)2142 3753 y Fs(\030)5 b(;)14 b(\017;)g(h)p Ft(\))p 2371 3666 449 4 v Fs(b)2407 3765 y Fp(k)2442 3773 y Fh(2)2478 3753 y Ft(\()p Fs(y)s(;)2599 3731 y Ft(~)2591 3753 y Fs(\030)t(;)g(\017;)g(h)p Ft(\))2818 3683 y Fo(\014)2818 3732 y(\014)2858 3697 y Fs(@)2915 3675 y Ft(~)2907 3697 y Fs(\030)p 2856 3734 93 4 v 2856 3810 a(@)5 b(\021)2959 3683 y Fo(\014)2959 3732 y(\014)2987 3611 y(!)3052 3861 y Fm(j)3072 3869 y Fe(y)q Fh(=)p Fe(x)3535 3753 y Ft(\(3.18\))204 4036 y(in)19 b(whic)n(h)g(one)g(m)n(ust)h(k)n (eep)e(in)i(mind)g(that)1516 4014 y(~)1507 4036 y Fs(\030)27 b Ft(=)1667 4014 y(~)1658 4036 y Fs(\030)t Ft(\()p Fs(x;)14 b(y)s(;)g(\021)s(;)g(\017;)g(h)p Ft(\).)34 b(This)20 b(allo)n(ws)d(to)j(compute)f(explicitly)g Fs(b)3413 4048 y Fr(0)3450 4036 y Fs(;)14 b Fq(\001)g(\001)g(\001)28 b Fs(;)14 b(b)3685 4048 y Fp(N)204 4135 y Ft(b)n(y)27 b(sa)n(ying)f(that)i(the)g(righ)n(t)f(hand)h(side)f(of)h(\(3)p Fs(:)p Ft(18\))e(m)n(ust)i(b)r(e)g(equal)f(to)h Fs(\037)2497 4147 y Fr(+)2552 4135 y Ft(\()p Fs(x;)14 b(\021)s Ft(\).)38 b(This)27 b(yields)p 502 4327 418 4 v 502 4400 a Fs(b)538 4412 y Fr(0)575 4400 y Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))24 b(=)e Fs(\037)1082 4412 y Fr(+)1137 4400 y Ft(\()p Fs(x;)f Ft(~)-49 b Fs(\021)t Ft(\()p Fs(x;)14 b(x;)g(\030)t(;)g(\017;)g (h)p Ft(\)\))1772 4258 y Fo( )1839 4400 y Fs(a)1883 4412 y Fr(0)1920 4400 y Ft(\()p Fs(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))2264 4329 y Fo(\014)2264 4379 y(\014)2304 4343 y Fs(@)2361 4321 y Ft(~)2353 4343 y Fs(\030)p 2302 4380 93 4 v 2302 4456 a(@)5 b(\021)2405 4400 y Ft(\()p Fs(x;)14 b(x;)21 b Ft(~)-49 b Fs(\021)t Ft(\()p Fs(x;)14 b(x;)g(\030)t(;)g (\017;)g(h)p Ft(\))p Fs(;)g(\017;)g(h)p Ft(\))3266 4329 y Fo(\014)3266 4379 y(\014)3294 4258 y(!)3360 4275 y Fm(\000)p Fr(1)204 4652 y Ft(whic)n(h)26 b(is)h(w)n(ell)f(de\014ned)h (since)f(the)h(term)f(in)h(\()p Fq(\001)14 b(\001)g(\001)g Ft(\))1787 4622 y Fm(\000)p Fr(1)1903 4652 y Ft(cannot)26 b(v)-5 b(anish)26 b(on)h(the)g(supp)r(ort)f(of)g Fs(\037)3137 4664 y Fr(+)3192 4652 y Ft(\()p Fs(:;)21 b Ft(~)-49 b Fs(\021)t Ft(\()p Fs(:;)14 b(:;)g(:;)g(\017;)g(h)p Ft(\)\).)204 4752 y(W)-7 b(e)30 b(don't)g(go)f(an)n(y)g(further)h(in)n(to)f(details) h(and)g(summarize)f(the)h(results)f(on)h(the)g(sym)n(b)r(ols)f Fs(b)3169 4764 y Fr(0)3206 4752 y Fs(;)14 b Fq(\001)g(\001)g(\001)27 b Fs(;)14 b(b)3440 4764 y Fp(N)3533 4752 y Ft(in)30 b(the)204 4851 y(follo)n(wing)c(prop)r(osition.)36 b(Let)28 b Fs(J)1218 4863 y Fr(3)1278 4851 y Fc(b)23 b Fs(J)1412 4863 y Fr(2)1477 4851 y Ft(and)k Fq(\000)p Ft(1)22 b Fs(<)h(\033)1902 4863 y Fr(3)1963 4851 y Fs(<)f(\033)2097 4863 y Fr(2)2135 4851 y Ft(.)204 5009 y Fg(Prop)s(osition)30 b(3.4.)40 b Fi(If)26 b Fs(\037)1045 5021 y Fr(+)1125 5009 y Fi(is)f(supp)l(orte)l (d)h(in)f Ft(\000)1724 4979 y Fr(+)1779 5009 y Ft(\()p Fs(R)1874 5021 y Fr(3)1912 5009 y Fs(;)14 b(J)1995 5021 y Fr(3)2032 5009 y Fs(;)g(\033)2116 5021 y Fr(3)2154 5009 y Ft(\))25 b Fi(with)h Fs(R)2450 5021 y Fr(3)2512 5009 y Fi(lar)l(ge)g(enough,)h(one)e(c)l(an)h(\014nd)e(symb)l(ols)204 5109 y Fs(b)240 5121 y Fr(0)277 5109 y Fs(;)14 b Fq(\001)g(\001)g(\001) 27 b Fs(;)14 b(b)511 5121 y Fp(N)603 5109 y Fi(supp)l(orte)l(d)31 b(in)e Ft(\000)1127 5079 y Fr(+)1182 5109 y Ft(\()p Fs(R)1277 5121 y Fr(2)1315 5109 y Fs(;)14 b(J)1398 5121 y Fr(2)1435 5109 y Fs(;)g(\033)1519 5121 y Fr(2)1557 5109 y Ft(\))p Fi(,)30 b(dep)l(ending)h(smo)l(othly)g(on)f Fs(\017;)14 b(h)29 b Fi(such)h(that:)848 5291 y Fq(j)p Fs(@)920 5257 y Fp(\013)915 5311 y(x)967 5291 y Fs(@)1016 5251 y Fp(\014)1011 5316 y(\030)1060 5291 y Fs(@)1109 5257 y Fp(n)1104 5311 y(\017)1154 5291 y Fs(@)1203 5257 y Fp(l)1198 5311 y(h)1241 5291 y Fs(b)1277 5303 y Fp(k)1318 5291 y Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(j)23 b(\024)g Fs(C)6 b Fq(h)p Fs(x)p Fq(i)1972 5257 y Fm(\000)p Fp(k)q Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2328 5291 y Fs(;)183 b Ft(0)23 b Fq(\024)f Fs(k)k Fq(\024)d Fs(N)t(;)1934 5540 y Ft(22)p eop %%Page: 23 23 23 22 bop 204 531 a Fi(and)28 b Fs(b)399 543 y Fp(k)466 531 y Fi(dep)l(ends)h(c)l(ontinuously,)f(in)f Fs(S)1427 543 y Fr(1)1464 531 y Ft(\()p Fq(\0001)p Fs(;)14 b Fq(\000)p Fs(k)s Ft(\))p Fi(,)28 b(on)f Fs(q)f Fq(2)e(B)s Fi(.)37 b(Mor)l(e)l(over)28 b Ft(\(3)p Fs(:)p Ft(14\))f Fi(holds)h(with)g Fs(r)3281 543 y Fp(N)3345 531 y Ft(\()p Fs(:;)14 b(:;)g(\017;)g(h)p Ft(\))27 b Fi(in)204 631 y(a)j(b)l(ounde)l(d)g(subset)f(of)i Fs(S)987 643 y Fr(1)1024 631 y Ft(\()p Fq(\0001)p Fs(;)14 b Fq(\000)p Fs(N)9 b Ft(\))p Fi(,)29 b(dep)l(ending)j(c)l(ontinuously)d (on)h Fs(q)j Fi(and)d(such)g(that)1161 813 y Fq(j)p Fs(@)1233 778 y Fp(\013)1228 833 y(x)1280 813 y Fs(@)1329 773 y Fp(\014)1324 838 y(\030)1374 813 y Fs(@)1423 778 y Fp(n)1418 833 y(\017)1468 813 y Fs(r)1505 825 y Fp(N)1568 813 y Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(j)24 b(\024)e Fs(C)6 b Fq(h)p Fs(x)p Fq(i)2222 778 y Fm(\000)p Fp(N)g Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2600 813 y Fs(;)912 b Ft(\(3.19\))328 985 y(All)28 b(what)g(w)n(e)f(ha)n(v)n (e)g(done)g(leads)g(to)g(the)h(follo)n(wing)f(theorem.)204 1142 y Fg(Theorem)j(3.5.)40 b Fi(F)-6 b(or)36 b(al)t(l)g Fs(J)41 b Fc(b)32 b Fs(I)7 b Fi(,)37 b(and)f Fq(\000)p Ft(1)31 b Fs(<)i(\033)1823 1154 y Fr(+)1911 1142 y Fs(<)g Ft(1)p Fi(,)j(ther)l(e)f(exists)g Fs(R)e(>)g Ft(0)p Fi(,)j(such)g (that,)g(if)g Ft(supp)g Fs(\037)3595 1154 y Fr(+)3683 1142 y Fq(\032)204 1242 y Ft(\000)256 1212 y Fr(+)311 1242 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)573 1254 y Fr(+)628 1242 y Ft(\))p Fi(,)30 b(then)g Fs(U)957 1254 y Fp(\017)988 1242 y Ft(\()p Fs(t)p Ft(\))p Fs(\037)1134 1254 y Fr(+)1190 1242 y Ft(\()p Fs(x;)14 b(hD)r Ft(\))19 b Fq(\000)f Fs(A)1621 1254 y Fp(N)1685 1242 y Ft(\()p Fs(\017;)c(h)p Ft(\))p Fs(U)1925 1254 y Fr(0)1962 1242 y Ft(\()p Fs(t)p Ft(\))p Fs(B)2119 1254 y Fp(N)2182 1242 y Ft(\()p Fs(\017;)g(h)p Ft(\))2365 1212 y Fp(?)2433 1242 y Fi(is)30 b(e)l(qual)g(to)855 1471 y Fs(h)903 1437 y Fp(N)980 1358 y Fo(Z)1063 1379 y Fp(t)1026 1547 y Fr(0)1106 1471 y Fs(U)1163 1483 y Fp(\017)1194 1471 y Ft(\()p Fs(t)19 b Fq(\000)f Fs(s)p Ft(\))p Fs(K)1500 1483 y Fp(N)1563 1471 y Ft(\()p Fs(h;)c(\017;)g(s)p Ft(\))p Fs(ds)k Ft(+)g Fs(h)2053 1437 y Fp(N)6 b Fr(+1)2200 1471 y Fs(U)2257 1483 y Fp(\017)2289 1471 y Ft(\()p Fs(t)p Ft(\))p Fs(r)2420 1483 y Fp(N)2484 1471 y Ft(\()p Fs(x;)14 b(hD)r(;)g(\017;)g(h)p Ft(\))p Fs(;)605 b Ft(\(3.20\))204 1690 y Fi(for)37 b(al)t(l)h Fs(t)e Fq(\025)f Ft(0)p Fi(,)k Fs(h)c Fq(2)h Ft([0)p Fs(;)14 b(h)1065 1702 y Fr(0)1102 1690 y Ft(])p Fi(,)39 b Fs(\017)c Fq(2)h Ft([0)p Fs(;)14 b Ft(1])36 b Fi(and)h Fs(q)i Fq(2)d(B)s Fi(.)59 b Fs(r)2066 1702 y Fp(N)2129 1690 y Ft(\()p Fs(:;)14 b(:;)g(\017;)g(h)p Ft(\))37 b Fi(is)g(a)g(symb)l(ol)g(in)g Fs(S)3080 1702 y Fr(1)3117 1690 y Ft(\()p Fq(\0001)p Fs(;)14 b Fq(\000)p Fs(N)9 b Ft(\))36 b Fi(which)204 1790 y(dep)l(ends)28 b(smo)l(othly)h(on)e Fs(\017)p Fi(,)h(and)g(is)g(b)l(ounde)l(d)g(with)g(r)l(esp)l(e)l(ct)f (to)g Fs(h)p Fi(.)38 b(The)28 b(kernel)g Fq(K)2745 1802 y Fp(N)2808 1790 y Ft(\()p Fs(x;)14 b(y)s(;)g(h;)g(\017;)g(s)p Ft(\))28 b Fi(of)g Fs(K)3426 1802 y Fp(N)3488 1790 y Ft(\()p Fs(h;)14 b(\017;)g(s)p Ft(\))204 1889 y Fi(satis\014es)30 b(the)g(fol)t(lowing)i(estimates:)809 2061 y Fs(@)858 2027 y Fp(\013)853 2082 y(x)905 2061 y Fs(@)954 2027 y Fp(\014)949 2082 y(y)998 2061 y Fs(@)1047 2027 y Fp(n)1042 2082 y(\017)1092 2061 y Fq(K)1155 2073 y Fp(N)1219 2061 y Ft(\()p Fs(x;)14 b(y)s(;)g(h;)g(\017;)g(s)p Ft(\))22 b(=)h Fq(O)r Ft(\()p Fq(h)p Fs(x)p Fq(i)1964 2027 y Fm(\000)p Fr([)p Fp(N)q(=)p Fr(4])p Fm(\000j)p Fp(\013)p Fm(j)2314 2061 y Fq(h)p Fs(y)s Fq(i)2422 2027 y Fm(\000)p Fr([)p Fp(N)q(=)p Fr(4])p Fm(\000j)p Fp(\014)s Fm(j)2769 2061 y Fq(h)p Fs(s)p Fq(i)2872 2027 y Fm(\000)p Fr([)p Fp(N)q(=)p Fr(4])3087 2061 y Ft(\))p Fs(;)204 2244 y Fi(for)30 b(al)t(l)h Fs(s)23 b Fq(\025)g Ft(0)p Fs(;)43 b(x;)14 b(y)26 b Fq(2)d Fl(R)996 2214 y Fp(d)1041 2244 y Fs(;)44 b(h)23 b Fq(2)g Ft(\(0)p Fs(;)14 b(h)1416 2256 y Fr(0)1453 2244 y Ft(])p Fs(;)43 b(\017)23 b Fq(2)h Ft([0)p Fs(;)14 b Ft(1])28 b Fi(and)j(uniformly)f(with)h(r)l(esp)l(e)l(ct)e(to)h Fs(q)c Fq(2)d(B)s Fi(.)204 2475 y Fu(3.2)112 b(Pro)s(of)37 b(of)h(prop)s(osition)e(3)p Fb(:)p Fu(1)204 2628 y Ft(F)-7 b(ollo)n(wing)26 b([24)o(,)i(16)o(,)g(34)o(],)g(w)n(e)f(\014rst)g (recall)g(the)h(form)n(ula)e(whic)n(h)i(de\014nes)g Fs(')g Ft(in)f(some)g(suitable)h(outgoing)e(area)1074 2858 y Fs(')p Ft(\()p Fs(x;)14 b(\030)t(;)g(\017;)g(h)p Ft(\))24 b(=)f Fq(h)p Fs(x;)14 b(\030)t Fq(i)19 b Ft(+)1874 2745 y Fo(Z)1957 2765 y Fr(+)p Fm(1)1920 2933 y Fr(0)2102 2802 y Fs(@)5 b(F)p 2102 2839 114 4 v 2120 2915 a(@)g(t)2226 2858 y Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(dt)825 b Ft(\(3.21\))204 3076 y(where)27 b(w)n(e)g(ha)n(v)n (e,)g(for)g(some)g Fs(R)c(>)g Ft(0)k(big)h(enough)f(\(whose)g (existence)g(will)h(b)r(e)g(pro)n(v)n(ed)e(b)r(elo)n(w\))636 3235 y Fs(@)5 b(F)p 636 3273 V 653 3349 a(@)g(t)759 3292 y Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))84 b(=)e Fs(H)1477 3257 y Fp(cl)1470 3312 y(\017)1532 3292 y Ft(\()5 b(~)-47 b Fs(z)t Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g (h)p Ft(\))p Fs(;)g(\030)t(;)g(h)p Ft(\))k Fq(\000)g Fs(H)2389 3257 y Fp(cl)2382 3312 y(\017)2444 3292 y Ft(\()5 b(~)-47 b Fs(z)t Ft(\()p Fs(t;)14 b(R)q(v)s Ft(\()p Fs(\030)t Ft(\))p Fs(;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)g(\030)t(;)g(h)p Ft(\))p Fs(;)224 b Ft(\(3.22\))722 3444 y(~)-47 b Fs(z)s Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))84 b(=)p 1401 3398 48 4 v 82 w Fs(x)q Ft(\()p Fs(t;)14 b(x;)p 1632 3398 45 4 v 14 w(\021)s Ft(\()p Fs(t;)g(x;)g(\030)t(;)g(\017;)g(h) p Ft(\))p Fs(;)g(\017;)g(h)p Ft(\))1260 b(\(3.23\))204 3616 y(with)p 393 3570 V 28 w Fs(\021)s Ft(\()p Fs(t;)14 b(x;)g(:;)g(\017;)g(h)p Ft(\))28 b(the)g(in)n(v)n(erse)e(of)i(the)g (di\013eomorphism)1286 3788 y(\000)1338 3754 y Fr(+)1338 3809 y Fp(x)1393 3788 y Ft(\()p Fs(R)1489 3754 y Fm(0)1512 3788 y Fs(;)14 b(J)1603 3754 y Fm(0)1626 3788 y Fs(;)g(\033)1713 3754 y Fm(0)1737 3788 y Ft(\))23 b Fq(3)g Fs(\021)k Fq(7!)p 2044 3720 41 4 v 23 w Fs(\030)t Ft(\()p Fs(t;)14 b(x;)g(\021)s(;)g (\017;)g(h)p Ft(\))1036 b(\(3.24\))204 3960 y(whic)n(h)27 b(is)h(w)n(ell)f(de\014ned)h(on)1004 4132 y(\000)1056 4098 y Fr(+)1056 4152 y Fp(x)1111 4132 y Ft(\()p Fs(R)1207 4098 y Fm(0)1230 4132 y Fs(;)14 b(J)1321 4098 y Fm(0)1344 4132 y Fs(;)g(\033)1431 4098 y Fm(0)1455 4132 y Ft(\))23 b(=)g Fq(f)p Fs(\021)i Fq(2)f Fl(R)1839 4098 y Fp(d)1911 4132 y Fq(j)k Ft(\()p Fs(x;)14 b(\021)s Ft(\))24 b Fq(2)f Ft(\000)2308 4098 y Fr(+)2363 4132 y Ft(\()p Fs(R)2459 4098 y Fm(0)2483 4132 y Fs(;)14 b(J)2574 4098 y Fm(0)2597 4132 y Fs(;)g(\033)2684 4098 y Fm(0)2708 4132 y Ft(\))p Fq(g)753 b Ft(\(3.25\))204 4304 y(for)39 b(a)h(suitable)g(outgoing)f (area)f(\000)1345 4274 y Fr(+)1400 4304 y Ft(\()p Fs(R)1496 4274 y Fm(0)1520 4304 y Fs(;)14 b(J)1611 4274 y Fm(0)1634 4304 y Fs(;)g(\033)1721 4274 y Fm(0)1744 4304 y Ft(\))41 b(\(see)f(lemma)g(3)p Fs(:)p Ft(7)f(b)r(elo)n(w\).)74 b(Ev)n(erywhere,)41 b(\()p 3281 4258 48 4 v Fs(x)q Ft(\()p Fs(t;)14 b(:)p Ft(\))p Fs(;)p 3520 4236 41 4 v 14 w(\030)t Ft(\()p Fs(t;)g(:)p Ft(\)\))204 4404 y(denotes)27 b(the)h(classical)e (\015o)n(w,)h(that)h(is)g(the)g(solution)f(to)1011 4504 y Fo(\()1078 4589 y Fs(@)1122 4601 y Fp(t)p 1151 4543 48 4 v 1151 4589 a Fs(x)d Ft(=)82 b Fs(@)1413 4601 y Fp(\030)1450 4589 y Fs(H)1526 4559 y Fp(cl)1519 4610 y(\017)1580 4589 y Ft(\()p 1612 4543 V Fs(x)q(;)p 1697 4521 41 4 v 14 w(\030)t(;)14 b(h)p Ft(\))p Fs(;)p 2057 4543 48 4 v 180 w(x)q Ft(\(0)p Fs(;)g(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))23 b(=)f Fs(x;)1078 4709 y(@)1122 4721 y Fp(t)p 1151 4641 41 4 v 1151 4709 a Fs(\030)27 b Ft(=)90 b Fq(\000)p Fs(@)1478 4721 y Fp(x)1520 4709 y Fs(H)1596 4678 y Fp(cl)1589 4729 y(\017)1650 4709 y Ft(\()p 1682 4663 48 4 v Fs(x)q(;)p 1767 4641 41 4 v 14 w(\030)t(;)14 b(h)p Ft(\))p Fs(;)p 2127 4641 V 180 w(\030)t Ft(\(0)p Fs(;)g(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))23 b(=)g Fs(\030)t(:)204 4892 y Ft(The)29 b(pro)r(of)f(just)i (consists)e(in)h(a)g(tec)n(hnical)g(review)f(of)h(pro)r(ofs)f(of)h([16) o(,)g(34)o(])h(to)e(sho)n(w)h(that)g(all)g(the)g(ob)5 b(jects)29 b(can)204 4992 y(b)r(e)f(constructed)f(and)g(estimated)h (uniformly)f(with)i(resp)r(ect)e(to)g(the)h(parameters)e Fs(\017;)14 b(h)27 b Ft(and)h Fs(q)s Ft(.)328 5092 y(Let)23 b Fs(J)32 b Fc(b)22 b Fs(I)7 b Ft(,)24 b(and)f Fs(\033)k Fq(2)c Ft(\()p Fq(\000)p Ft(1)p Fs(;)14 b Ft(1\).)34 b(The)23 b(\014rst)g(step)g(is)g(to)g(estimate)g(the)g(classical)f (\015o)n(w;)i(this)f(is)g(the)g(goal)f(of)h(the)204 5191 y(the)k(next)f(lemma)h(where,)f(if)h Fs(q)1172 5203 y Fr(1)1209 5191 y Fs(;)14 b(q)1283 5203 y Fr(2)1343 5191 y Fq(2)24 b(B)s Ft(,)i(w)n(e)g(denote)g(b)n(y)g(\()p 2062 5146 48 4 v Fs(x)2110 5203 y Fr(1)2147 5191 y Fs(;)p 2184 5124 41 4 v 14 w(\030)2224 5212 y Fr(1)2262 5191 y Ft(\))g(and)h(\()p 2513 5146 48 4 v Fs(x)2561 5203 y Fr(2)2598 5191 y Fs(;)p 2635 5124 41 4 v 14 w(\030)2675 5212 y Fr(2)2712 5191 y Ft(\))g(the)g(resp)r(ectiv)n(e)e(Hamiltonian) 204 5291 y(\015o)n(ws)i(of)g Fs(!)21 b Ft(+)d Fs(\017q)731 5303 y Fr(1)796 5291 y Ft(and)27 b Fs(!)21 b Ft(+)d Fs(\017q)1184 5303 y Fr(2)1221 5291 y Ft(.)1934 5540 y(23)p eop %%Page: 24 24 24 23 bop 204 531 a Fg(Lemma)29 b(3.6.)40 b Fi(Ther)l(e)d(exists)f Fs(R)f(>)f Ft(0)h Fi(such)h(that,)i(for)f(al)t(l)g Fs(\015)k Fi(de\014ne)l(d)36 b(by)g Fs(@)2680 501 y Fp(\015)2757 531 y Ft(=)e Fs(@)2905 501 y Fp(\013)2900 552 y(x)2952 531 y Fs(@)3001 491 y Fp(\014)2996 556 y(\030)3045 531 y Fs(@)3094 501 y Fp(n)3089 552 y(\017)3139 531 y Fs(@)3188 501 y Fp(l)3183 555 y(h)3226 531 y Fi(,)k(one)e(c)l(an)g(\014nd)204 631 y Fs(C)29 b(>)23 b Ft(0)29 b Fi(and)h Fq(N)42 b Fi(semi-norm)30 b(in)g Fq(S)1291 643 y Fr(1)1329 631 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\))29 b Fi(such)h(that)904 808 y Fq(j)p Fs(@)976 773 y Fp(\015)1018 740 y Fo(\000)p 1056 762 48 4 v 68 x Fs(x)1104 820 y Fr(1)1160 808 y Fq(\000)p 1243 762 V 18 w Fs(x)1290 820 y Fr(2)1327 740 y Fo(\001)1365 808 y Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(j)84 b(\024)e Fs(C)6 b Fq(N)12 b Ft(\()p Fs(q)2244 820 y Fr(1)2301 808 y Fq(\000)18 b Fs(q)2421 820 y Fr(2)2458 808 y Ft(\))p Fq(h)p Fs(t)p Fq(ih)p Fs(x)p Fq(i)2695 773 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2957 781 y Fh(+)3010 808 y Fs(;)919 948 y Fq(j)p Fs(@)991 913 y Fp(\015)1033 880 y Fo(\000)p 1071 880 41 4 v 68 x Fs(\030)1111 968 y Fr(1)1167 948 y Fq(\000)p 1250 880 V 18 w Fs(\030)1290 968 y Fr(2)1327 880 y Fo(\001)1365 948 y Ft(\()p Fs(t;)c(x;)g(\030)t(;) g(\017;)g(h)p Ft(\))p Fq(j)84 b(\024)e Fs(C)6 b Fq(N)12 b Ft(\()p Fs(q)2244 960 y Fr(1)2301 948 y Fq(\000)18 b Fs(q)2421 960 y Fr(2)2458 948 y Ft(\))p Fq(h)p Fs(x)p Fq(i)2601 913 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2863 921 y Fh(+)2915 948 y Fs(;)204 1124 y Fi(for)30 b(al)t(l)h Ft(\()p Fs(x;)14 b(\030)t Ft(\))24 b Fq(2)g Ft(\000)798 1094 y Fr(+)853 1124 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))p Fi(,)30 b Ft(\()p Fs(\017;)14 b(h)p Ft(\))23 b Fq(2)h Ft([0)p Fs(;)14 b Ft(1])j Fq(\002)h Ft([0)p Fs(;)c(h)1907 1136 y Fr(0)1944 1124 y Ft(])p Fi(,)30 b Fs(t)23 b Fq(\025)g Ft(0)29 b Fi(and)h Fs(q)2432 1136 y Fr(1)2469 1124 y Fs(;)14 b(q)2543 1136 y Fr(2)2604 1124 y Fq(2)23 b(B)s Fi(.)204 1286 y Fg(Pro)s(of.)43 b Ft(By)30 b(reviewing)f(the)h(pro)r(of)f(of)h(lemma)g(2)p Fs(:)p Ft(2)f(of)h([16)o(])g(\(see)g(also)f(prop)r(osition)g Fs(A:)p Ft(5)h(of)g(this)g(pap)r(er\),)h(one)204 1385 y(get)c(easily)g(the)h(existence)f(of)h Fs(R)c(>)e Ft(0)27 b(and)h Fs(c)23 b(>)f Ft(0)28 b(suc)n(h)f(that)1186 1562 y Fq(j)p 1209 1516 48 4 v Fs(x)q Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g (\017;)g(h)p Ft(\))k Fq(\000)g Fs(x)h Fq(\000)f Fs(tv)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)84 b(\024)f Fs(c)p Fq(h)p Fs(t)p Fq(ih)p Fs(x)p Fq(i)2591 1527 y Fm(\000)p Fp(\032)2683 1562 y Fs(;)829 b Ft(\(3.26\))1480 1686 y Fq(j)p 1503 1619 41 4 v Fs(\030)t Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))19 b Fq(\000)f Fs(\030)t Fq(j)83 b(\024)g Fs(c)p Fq(h)p Fs(x)p Fq(i)2497 1652 y Fm(\000)p Fp(\032)2588 1686 y Fs(;)924 b Ft(\(3.27\))1615 1811 y Fq(j)p 1638 1765 48 4 v Fs(x)p Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(j)83 b(\025)g Fs(c)2386 1777 y Fm(\000)p Fr(1)2475 1811 y Ft(\()p Fq(j)p Fs(x)p Fq(j)19 b Ft(+)f Fs(t)p Ft(\))771 b(\(3.28\))204 1987 y(for)29 b(all)h Fs(t)e Fq(\025)f Ft(0,)j Fs(q)h Fq(2)d(B)s Ft(,)i Fs(\017)d Fq(2)h Ft([0)p Fs(;)14 b Ft(1],)30 b Fs(h)d Fq(2)h Ft([0)p Fs(;)14 b(h)1630 1999 y Fr(0)1666 1987 y Ft(])31 b(and)f(\()p Fs(x;)14 b(\030)t Ft(\))28 b Fq(2)g Ft(\000)2235 1957 y Fr(+)2290 1987 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\).)45 b(F)-7 b(rom)30 b(this)g(w)n(e)g(deduce)g(that)h(our)204 2087 y(lemma)h(is)h(true)f(when)h Fs(q)1004 2099 y Fr(2)1072 2087 y Fq(\021)e Ft(0;)k(indeed,)f(b)n(y)e(considering)f Fs(y)j Ft(=)d Fs(@)2334 2099 y Fp(\017)p 2366 2041 V 2366 2087 a Fs(x)i Ft(and)f Fs(\021)j Ft(=)c Fs(@)2828 2099 y Fp(\017)p 2859 2019 41 4 v 2859 2087 a Fs(\030)5 b Ft(,)33 b(w)n(ee)g(see)f(that)g(they)h(are)204 2187 y(solutions)23 b(of)g(the)h(system)f(\()p Fs(A:)p Ft(1\))h(with)g Fs(y)1488 2199 y Fr(0)1548 2187 y Ft(=)f Fs(@)1680 2199 y Fp(\030)1716 2187 y Fs(q)s Ft(\()p 1788 2141 48 4 v Fs(x)q(;)p 1873 2119 41 4 v 14 w(\030)t Ft(\),)i Fs(\021)2034 2199 y Fr(0)2095 2187 y Ft(=)d Fq(\000)p Fs(@)2291 2199 y Fp(x)2333 2187 y Fs(q)s Ft(\()p 2405 2141 48 4 v Fs(x;)p 2489 2119 41 4 v 14 w(\030)t Ft(\))i(\(v)n(erifying)f(\()p Fs(A:)p Ft(4\))h(with)g Fs(\026)f Ft(=)g(0\))g(and)1129 2363 y Fs(A)g Ft(=)g Fs(\017@)1385 2329 y Fr(2)1380 2384 y Fp(x;\030)1474 2363 y Fs(q)s Ft(\()p 1546 2318 48 4 v Fs(x;)p 1630 2296 41 4 v 14 w(\030)t Ft(\))p Fs(;)167 b(B)27 b Ft(=)c Fs(@)2119 2329 y Fr(2)2114 2384 y Fp(\030)r(;\030)2202 2363 y Fs(!)s Ft(\()p 2289 2296 V Fs(\030)t Ft(\))c(+)f Fs(\017@)2546 2329 y Fr(2)2541 2384 y Fp(\030)r(;\030)2629 2363 y Fs(q)s Ft(\()p 2701 2318 48 4 v Fs(x)q(;)p 2786 2296 41 4 v 14 w(\030)t Ft(\))p Fs(;)1056 2503 y(C)29 b Ft(=)23 b Fq(\000)p Fs(\017@)1380 2468 y Fr(2)1375 2523 y Fp(x;x)1474 2503 y Fs(q)s Ft(\()p 1546 2457 48 4 v Fs(x;)p 1630 2435 41 4 v 14 w(\030)t Ft(\))p Fs(;)167 b(D)25 b Ft(=)e Fq(\000)p Fs(\017@)2222 2468 y Fr(2)2217 2523 y Fp(x;\030)2310 2503 y Fs(q)s Ft(\()p 2382 2457 48 4 v Fs(x;)p 2466 2435 41 4 v 14 w(\030)t Ft(\))p Fs(;)204 2679 y Ft(whic)n(h)31 b(satis\014es)f(\()p Fs(A:)p Ft(2\),)j(\()p Fs(A:)p Ft(3\),)f(b)r(ecause)f Fs(q)h Fq(2)e(B)j Ft(and)e(thanks)g(to)g (the)g(estimates)g(\(3)p Fs(:)p Ft(26\),)h(\(3)p Fs(:)p Ft(27\))e(and)h(\(3)p Fs(:)p Ft(28\))204 2779 y(.)46 b(More)30 b(generally)-7 b(,)30 b(b)n(y)g(induction)i(on)e Fq(j)p Fs(\015)5 b Fq(j)28 b(\025)g Ft(0,)j(w)n(e)f(get)h(the)g (estimates)f(on)h Fs(@)2733 2749 y Fp(\015)2775 2779 y Fs(@)2819 2791 y Fp(\017)p 2851 2733 48 4 v 2851 2779 a Fs(x)p Ft(,)h Fs(@)3002 2749 y Fp(\015)3044 2779 y Fs(@)3088 2791 y Fp(\017)p 3120 2711 41 4 v 3120 2779 a Fs(\030)j Ft(b)n(y)c(mean)f(of)h(the)204 2878 y(prop)r(osition)22 b Fs(A:)p Ft(1.)35 b(This)23 b(pro)n(v)n(es)e(the)i(result)g(when)g Fs(q)1872 2890 y Fr(2)1932 2878 y Fq(\021)g Ft(0.)35 b(Then)23 b(w)n(e)g(get)f(the)i(general)d(result)i(b)n(y)g(considering) 204 2978 y(the)28 b(system)f(\()p Fs(A:)p Ft(1\))h(with)g Fs(y)e Ft(=)p 1185 2932 48 4 v 23 w Fs(x)1232 2990 y Fr(2)1288 2978 y Fq(\000)p 1371 2932 V 18 w Fs(x)1418 2990 y Fr(1)1456 2978 y Ft(,)h Fs(\021)g Ft(=)p 1661 2910 41 4 v 22 w Fs(\030)1701 2998 y Fr(2)1757 2978 y Fq(\000)p 1840 2910 V 18 w Fs(\030)1880 2998 y Fr(1)1945 2978 y Ft(and)766 3214 y Fs(A)83 b Ft(=)1058 3101 y Fo(Z)1141 3122 y Fr(1)1105 3290 y(0)1193 3214 y Fs(\017@)1276 3180 y Fr(2)1271 3235 y Fp(x;\030)1364 3214 y Fs(q)1401 3226 y Fr(2)1438 3214 y Ft(\()p 1470 3168 48 4 v Fs(x)1518 3226 y Fr(1)1574 3214 y Ft(+)18 b Fs(u)p Ft(\()p 1737 3168 V Fs(x)1784 3226 y Fr(2)1840 3214 y Fq(\000)p 1923 3168 V 18 w Fs(x)1970 3226 y Fr(1)2007 3214 y Ft(\))p Fs(;)p 2076 3146 41 4 v 14 w(\030)2117 3234 y Fr(1)2154 3214 y Ft(\))p Fs(du;)761 3453 y(B)87 b Ft(=)1058 3340 y Fo(Z)1141 3360 y Fr(1)1105 3528 y(0)1193 3453 y Fs(@)1242 3419 y Fr(2)1237 3473 y Fp(\030)r(;\030)1325 3453 y Fs(!)s Ft(\()p 1412 3385 V Fs(\030)1452 3473 y Fr(1)1508 3453 y Ft(+)18 b Fs(u)p Ft(\()p 1671 3385 V Fs(\030)1711 3473 y Fr(2)1766 3453 y Fq(\000)p 1849 3385 V 18 w Fs(\030)1889 3473 y Fr(1)1927 3453 y Ft(\)\))h(+)f Fs(\017@)2176 3419 y Fr(2)2171 3473 y Fp(\030)r(;\030)2259 3453 y Fs(q)2296 3465 y Fr(2)2333 3453 y Ft(\()p 2365 3407 48 4 v Fs(x)2413 3465 y Fr(2)2450 3453 y Fs(;)p 2487 3385 41 4 v 14 w(\030)2527 3473 y Fr(1)2583 3453 y Ft(+)g Fs(u)p Ft(\()p 2746 3385 V Fs(\030)2786 3473 y Fr(2)2841 3453 y Fq(\000)p 2925 3385 V 19 w Fs(\030)2965 3473 y Fr(1)3002 3453 y Ft(\)\))p Fs(du;)763 3692 y(C)89 b Ft(=)82 b Fq(\000)1137 3579 y Fo(Z)1220 3599 y Fr(1)1183 3767 y(0)1271 3692 y Fs(\017@)1354 3657 y Fr(2)1349 3712 y Fp(x;x)1448 3692 y Fs(q)1485 3704 y Fr(2)1522 3692 y Ft(\()p 1554 3646 48 4 v Fs(x)1602 3704 y Fr(1)1658 3692 y Ft(+)18 b Fs(u)p Ft(\()p 1821 3646 V Fs(x)1868 3704 y Fr(2)1924 3692 y Fq(\000)p 2007 3646 V 18 w Fs(x)2054 3704 y Fr(1)2091 3692 y Ft(\))p Fs(;)p 2160 3624 41 4 v 14 w(\030)2201 3712 y Fr(2)2238 3692 y Ft(\))p Fs(du;)757 3930 y(D)85 b Ft(=)d Fq(\000)1137 3817 y Fo(Z)1220 3838 y Fr(1)1183 4006 y(0)1271 3930 y Fs(\017@)1354 3896 y Fr(2)1349 3951 y Fp(x;\030)1442 3930 y Fs(q)1479 3942 y Fr(2)1517 3930 y Ft(\()p 1549 3885 48 4 v Fs(x)1596 3942 y Fr(1)1634 3930 y Fs(;)p 1671 3863 41 4 v 14 w(\030)1711 3951 y Fr(1)1766 3930 y Ft(+)18 b Fs(u)p Ft(\()p 1929 3863 V Fs(\030)1969 3951 y Fr(2)2025 3930 y Fq(\000)p 2108 3863 V 18 w Fs(\030)2148 3951 y Fr(1)2186 3930 y Ft(\)\))p Fs(du;)204 4149 y Ft(and)27 b(with)h(the)g(second)f(terms)h Fs(y)1238 4161 y Fr(0)1302 4149 y Ft(and)g Fs(\021)1505 4161 y Fr(0)1570 4149 y Ft(giv)n(en)f(b)n(y)1265 4326 y Fs(y)1306 4338 y Fr(0)1426 4326 y Ft(=)83 b Fs(\017)1622 4258 y Fo(\000)1659 4326 y Fs(@)1703 4338 y Fp(\030)1740 4326 y Fs(q)1777 4338 y Fr(2)1814 4326 y Ft(\()p 1846 4280 48 4 v Fs(x)1894 4338 y Fr(1)1931 4326 y Fs(;)p 1968 4258 41 4 v 14 w(\030)2008 4346 y Fr(1)2045 4326 y Ft(\))19 b Fq(\000)f Fs(@)2223 4338 y Fp(\030)2260 4326 y Fs(q)2297 4338 y Fr(1)2334 4326 y Ft(\()p 2366 4280 48 4 v Fs(x)2414 4338 y Fr(1)2451 4326 y Fs(;)p 2488 4258 41 4 v 14 w(\030)2528 4346 y Fr(1)2565 4326 y Ft(\))2597 4258 y Fo(\001)2649 4326 y Fs(;)1265 4462 y(\021)1306 4474 y Fr(0)1426 4462 y Ft(=)83 b Fs(\017)1622 4395 y Fo(\000)1659 4462 y Fs(@)1703 4474 y Fp(x)1745 4462 y Fs(q)1782 4474 y Fr(1)1820 4462 y Ft(\()p 1852 4416 48 4 v Fs(x)1899 4474 y Fr(1)1937 4462 y Fs(;)p 1974 4395 41 4 v 14 w(\030)2014 4482 y Fr(1)2051 4462 y Ft(\))19 b Fq(\000)f Fs(@)2229 4474 y Fp(x)2270 4462 y Fs(q)2307 4474 y Fr(2)2345 4462 y Ft(\()p 2377 4416 48 4 v Fs(x)2425 4474 y Fr(1)2462 4462 y Fs(;)p 2499 4395 41 4 v 14 w(\030)2539 4482 y Fr(1)2576 4462 y Ft(\))2608 4395 y Fo(\001)2660 4462 y Fs(:)204 4639 y Ft(By)27 b(increasing)f Fs(R)j Ft(if)f(necessary)e(\(uniformly)h (with)h(resp)r(ect)g(to)f Fs(\017;)14 b(h;)g(q)s Ft(\))28 b(so)e(that,)i(for)f(some)g Fs(c)3174 4609 y Fm(0)3221 4639 y Fs(>)22 b Ft(0)1131 4815 y Fq(j)p 1154 4770 48 4 v Fs(x)1201 4827 y Fr(1)1257 4815 y Ft(+)c Fs(u)p Ft(\()p 1420 4770 V Fs(x)1467 4827 y Fr(2)1523 4815 y Fq(\000)p 1606 4770 V 18 w Fs(x)1653 4827 y Fr(1)1691 4815 y Ft(\))p Fq(j)23 b(\025)g Fs(c)1893 4781 y Fm(0)1916 4815 y Ft(\()p Fq(j)p Fs(x)p Fq(j)c Ft(+)f Fs(t)p Ft(\))p Fs(;)180 b Fq(8)27 b Fs(u)c Fq(2)g Ft([0)p Fs(;)14 b Ft(1])p Fs(;)204 4992 y Ft(w)n(e)39 b(see)f(that)i Fs(A;)14 b(B)t(;)g(C)q(;)g(D)r(;)g(y) 1124 5004 y Fr(0)1200 4992 y Ft(and)39 b Fs(\021)1414 5004 y Fr(0)1491 4992 y Ft(satisfy)g(\()p Fs(A:)p Ft(2\),)j(\()p Fs(A:)p Ft(3\))e(and)f(\()p Fs(A:)p Ft(4\).)72 b(This)39 b(allo)n(ws)e(us)i(to)g(use)g(again)204 5092 y(prop)r(osition)f Fs(A:)p Ft(1)h(from)g(whic)n(h)g(w)n(e)g(get)g(the)h(result)f(b)n(y)g (applying)g Fs(@)2484 5061 y Fp(\015)2566 5092 y Ft(to)g(this)g(new)h (system,)i(arguing)37 b(b)n(y)204 5191 y(induction)h(on)f Fq(j)p Fs(\015)5 b Fq(j)39 b(\025)g Ft(0)e(\(remark)g(that)g(b)r (ecause)g(of)h(the)g(forms)f(of)g Fs(y)2499 5203 y Fr(0)2574 5191 y Ft(and)g Fs(\021)2786 5203 y Fr(0)2861 5191 y Ft(w)n(e'll)g(get)g(estimates)h(with)204 5291 y Fq(N)12 b Ft(\()p Fs(q)353 5303 y Fr(2)409 5291 y Fq(\000)18 b Fs(q)529 5303 y Fr(1)567 5291 y Ft(\)\).)3029 b Fc(\003)1934 5540 y Ft(24)p eop %%Page: 25 25 25 24 bop 204 531 a Fg(Lemma)29 b(3.7.)40 b Fi(Ther)l(e)d(exists)1229 510 y Ft(~)1211 531 y Fs(R)d Fq(\025)g Fs(R)i Fi(such)g(that,)i(for)f (al)t(l)g Fq(j)p Fs(x)p Fq(j)d Fs(>)2412 510 y Ft(~)2394 531 y Fs(R)q Fi(,)j Fs(t)d Fq(\025)g Ft(0)p Fi(,)j Ft(\()p Fs(\017;)14 b(h)p Ft(\))34 b Fq(2)h Ft([0)p Fs(;)14 b Ft(1])22 b Fq(\002)g Ft([0)p Fs(;)14 b(h)3520 543 y Fr(0)3557 531 y Ft(])36 b Fi(and)204 631 y Fs(q)26 b Fq(2)d(B)1408 731 y Ft(\000)1460 696 y Fr(+)1460 751 y Fp(x)1515 731 y Ft(\()1565 710 y(~)1547 731 y Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))24 b Fq(3)f Fs(\021)j Fq(7!)p 2087 663 41 4 v 23 w Fs(\030)t Ft(\()p Fs(t;)14 b(x;)g(\021)s(;)g(\017;)g(h)p Ft(\))204 874 y Fi(is)30 b(a)g(di\013e)l(omorphism)i(onto)e(its)g(r)l (ange.)39 b(We)30 b(c)l(al)t(l)p 1816 828 45 4 v 30 w Fs(\021)t Ft(\()p Fs(t;)14 b(x;)g(:;)g(\017;)g(h)p Ft(\))30 b Fi(its)f(inverse.)204 1032 y Fg(Pro)s(of:)k Ft(W)-7 b(e)22 b(use)g(the)g(same)f(metho)r(d)i(as)e(G)n(\023)-39 b(erard-Martinez)18 b(in)k([16]:)34 b(thanks)21 b(to)h(lemma)g(3)p Fs(:)p Ft(6,)g Fs(@)3228 1044 y Fp(\021)p 3268 965 41 4 v 3268 1032 a Fs(\030)k Ft(is)c(in)n(v)n(ertible)204 1141 y(when)29 b Fq(j)p Fs(x)p Fq(j)c(\025)648 1120 y Ft(~)630 1141 y Fs(R)g(>>)f Ft(1,)29 b(so)f(w)n(e)g(ha)n(v)n(e)g(a)g (lo)r(cal)g(di\013eomorphism,)h(and)f(then)h(it)h(is)e(su\016cien)n(t)h (to)g(sho)n(w)e(that)p 3622 1073 V 29 w Fs(\030)33 b Ft(is)204 1240 y(injectiv)n(e.)k(This)27 b(is)h(a)f(consequence)g(of)g (the)h(fact)g(that,)g(when)g Fs(\021)s(;)14 b(\021)2323 1210 y Fm(0)2369 1240 y Fq(2)24 b Ft(\000)2500 1210 y Fr(+)2500 1261 y Fp(x)2555 1240 y Ft(\()2605 1219 y(~)2587 1240 y Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))306 1382 y Fo(\014)306 1431 y(\014)p 334 1384 V 21 x Fs(\030)t Ft(\()p Fs(t;)g(x;)g(\021)s(;)g(\017;)g(h)p Ft(\))19 b Fq(\000)p 891 1384 V 18 w Fs(\030)t Ft(\()p Fs(t;)14 b(x;)g(\021)1158 1418 y Fm(0)1182 1452 y Fs(;)g(\017;)g(h)p Ft(\))1370 1382 y Fo(\014)1370 1431 y(\014)1420 1452 y Fq(\025)1518 1396 y Ft(1)p 1518 1433 42 4 v 1518 1509 a(2)1569 1452 y Fq(j)p Fs(\021)22 b Fq(\000)c Fs(\021)1782 1418 y Fm(0)1806 1452 y Fq(j)p Fs(;)180 b Fq(8)26 b(j)p Fs(x)p Fq(j)e(\025)2328 1431 y Ft(~)2310 1452 y Fs(R;)42 b(t)23 b Fq(\025)g Ft(0)p Fs(;)14 b(\017)22 b Fq(2)h Ft([0)p Fs(;)14 b Ft(1])p Fs(;)41 b(h)23 b Fq(2)g Ft([0)p Fs(;)14 b(h)3322 1464 y Fr(0)3359 1452 y Ft(])p Fs(;)41 b(q)26 b Fq(2)e(B)204 1667 y Ft(if)300 1646 y(~)281 1667 y Fs(R)30 b Ft(is)g(big)f(enough,)g (b)n(y)f(application)h(of)g(the)h(T)-7 b(a)n(ylor)27 b(form)n(ula)h(to)h(the)h(second)f(order,)f(and)h(using)g(again)f(the) 204 1767 y(previous)e(lemma.)37 b(The)28 b(pro)r(of)f(is)g(complete.) 2032 b Fc(\003)328 1866 y Ft(The)39 b(follo)n(wing)e(prop)r(osition,)j (whic)n(h)e(\\con)n(trols")e(the)i(range)f(of)i(\()p Fs(x;)14 b(\021)s Ft(\))42 b Fq(7!)f Ft(\()p Fs(x;)p 2968 1799 41 4 v 14 w(\030)t Ft(\()p Fs(t;)14 b(x;)g(\021)s(;)g(\017;)g (h)p Ft(\)\))39 b(will)g(b)r(e)204 1966 y(useful)28 b(to)f(de\014ne)h (\(3)p Fs(:)p Ft(23\))f(\(and)h(hence)f(\(3)p Fs(:)p Ft(21\)\))g(in)h(an)g(outgoing)e(area)g(indep)r(enden)n(t)i(on)g Fs(\017)p Ft(,)f Fs(h)h Ft(and)f Fs(q)s Ft(.)204 2138 y Fg(Prop)s(osition)j(3.8.)40 b Fi(F)-6 b(or)31 b(al)t(l)g(op)l(en)g (interval)g Fs(J)1740 2108 y Fm(0)1764 2138 y Fi(,)g(with)p 2001 2072 55 4 v 31 w Fs(J)2055 2086 y Fm(0)2102 2138 y Fq(\032)24 b Fs(J)39 b Fi(and)31 b(al)t(l)g Fq(\000)p Ft(1)23 b Fs(<)h(\033)2826 2108 y Fm(0)2874 2138 y Fs(<)g(\033)s Fi(,)32 b(ther)l(e)e(exists)g Fs(R)3569 2108 y Fm(0)3617 2138 y Fs(>)24 b Ft(0)204 2238 y Fi(such)30 b(that)1232 2411 y Ft(\000)1284 2376 y Fr(+)1339 2411 y Ft(\()p Fs(R)1435 2376 y Fm(0)1458 2411 y Fs(;)14 b(J)1549 2376 y Fm(0)1572 2411 y Fs(;)g(\033)1659 2376 y Fm(0)1683 2411 y Ft(\))23 b Fq(\032)g Ft(\010)1886 2423 y Fp(t;\017;h;q)2073 2343 y Fo(\000)2111 2411 y Ft(\000)2163 2376 y Fr(+)2218 2411 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))2515 2343 y Fo(\001)3535 2411 y Ft(\(3.29\))204 2583 y Fi(for)30 b(al)t(l)h Fs(t)23 b Fq(\025)g Ft(0)p Fi(,)30 b Ft(\()p Fs(\017;)14 b(h)p Ft(\))23 b Fq(2)g Ft([0)p Fs(;)14 b Ft(1])k Fq(\002)g Ft([0)p Fs(;)c(h)1395 2595 y Fr(0)1431 2583 y Ft(])30 b Fi(and)g Fs(q)c Fq(2)e(B)s Fi(,)29 b(wher)l(e)h Ft(\010)2193 2595 y Fp(t;\017;h;q)2410 2583 y Fi(denotes)h(the)e(fol)t (lowing)k(map)1253 2756 y Ft(\000)1305 2722 y Fr(+)1360 2756 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))24 b Fq(3)f Ft(\()p Fs(x;)14 b(\021)s Ft(\))24 b Fq(7!)2081 2689 y Fo(\000)2120 2756 y Fs(x;)p 2204 2689 41 4 v 14 w(\030)t Ft(\()p Fs(t;)14 b(x;)g(\021)s(;)g(\017;)g(h)p Ft(\))2659 2689 y Fo(\001)204 2929 y Fi(which)31 b(is)f(a)g(di\013e)l (omorphism)j(onto)d(its)f(\(op)l(en\))h(r)l(ange.)204 3029 y(Mor)l(e)l(over,)g(we)d(c)l(an)h(cho)l(ose)h Fs(R)1189 2998 y Fm(0)1239 3029 y Fi(such)e(that,)i(for)f(al)t(l)h Fs(\015)j Fi(ther)l(e)27 b(exists)g Fs(C)j Fq(\025)22 b Ft(0)27 b Fi(and)h Fq(N)40 b Fi(semi-norm)27 b(in)h Fq(S)3446 3041 y Fr(1)3483 3029 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\))204 3128 y Fi(such)30 b(that,)g(for)g(al)t (l)h Ft(\()p Fs(x;)14 b(\030)t Ft(\))24 b Fq(2)g Ft(\000)1182 3098 y Fr(+)1237 3128 y Ft(\()p Fs(R)1333 3098 y Fm(0)1356 3128 y Fs(;)14 b(J)1447 3098 y Fm(0)1470 3128 y Fs(;)g(\033)1557 3098 y Fm(0)1581 3128 y Ft(\))p Fi(,)30 b Fs(\017)23 b Fq(2)g Ft([0)p Fs(;)14 b Ft(1])p Fi(,)30 b Fs(h)23 b Fq(2)g Ft([0)p Fs(;)14 b(h)2324 3140 y Fr(0)2361 3128 y Ft(])29 b Fi(and)i Fs(q)2612 3140 y Fp(j)2670 3128 y Fq(2)23 b(B)32 b Fi(we)e(have)938 3311 y Fq(j)p Fs(@)1010 3277 y Fp(\015)1053 3244 y Fo(\000)p 1091 3265 45 4 v 67 x Fs(\021)1135 3331 y Fr(1)1191 3311 y Fq(\000)p 1274 3265 V 18 w Fs(\021)1318 3331 y Fr(2)1355 3244 y Fo(\001)1393 3311 y Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fq(j)24 b(\024)e Fs(C)6 b Fq(N)12 b Ft(\()p Fs(q)2152 3323 y Fr(1)2209 3311 y Fq(\000)18 b Fs(q)2329 3323 y Fr(2)2366 3311 y Ft(\))p Fq(h)p Fs(x)p Fq(i)2509 3277 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2771 3285 y Fh(+)2823 3311 y Fs(;)689 b Ft(\(3.30\))204 3484 y Fi(wher)l(e)p 438 3438 V 30 w Fs(\021)482 3504 y Fp(j)547 3484 y Fi(is)30 b(asso)l(ciate)l(d)h(to)f Fs(q)1161 3496 y Fp(j)1196 3484 y Fi(,)g Fs(j)e Ft(=)23 b(1)p Fs(;)14 b Ft(2)p Fi(.)204 3642 y Fg(Pro)s(of.)59 b Ft(\(3)p Fs(:)p Ft(29\))35 b(is)g(based)g(on)g(the)h(follo)n(wing)e(basic)h(top)r (ological)f(fact)h(\(see)h([33)o(]\):)53 b(if)35 b Fs(A;)14 b(B)41 b Fq(\032)35 b Fl(R)3401 3612 y Fp(n)3488 3642 y Ft(with)h Fs(A)204 3742 y Ft(connected,)27 b(and)h(if)g Fs(f)j Ft(:)p 971 3675 68 4 v 24 w Fs(B)c Fq(!)c Fl(R)1221 3711 y Fp(n)1300 3742 y Ft(is)k(an)g(homeomorphism)g(on)n(to)g(its)g (range,)g(then)1046 3914 y Fs(A)19 b Fq(\\)g Fs(f)9 b Ft(\()p Fs(B)t Ft(\))23 b Fq(6)p Ft(=)g Fq(;)82 b Ft(and)h Fs(A)19 b Fq(\\)f Fs(f)9 b Ft(\()p Fs(@)c(B)t Ft(\))23 b(=)g Fq(;)50 b(\))23 b Fs(A)h Fq(\032)e Fs(f)9 b Ft(\()p Fs(B)t Ft(\))p Fs(:)204 4087 y Ft(T)-7 b(o)27 b(use)g(this)g(prop)r (ert)n(y)-7 b(,)27 b(w)n(e)g(write)g Fs(!)1375 4057 y Fm(\000)p Fr(1)1463 4087 y Ft(\()p Fs(J)1549 4057 y Fm(0)1573 4087 y Ft(\))c(=)1716 4025 y Fo(`)1794 4112 y Fp(j)1843 4087 y Ft(\012)1903 4099 y Fp(j)1938 4087 y Ft(,)28 b(where)e(the)i (\012)2431 4099 y Fp(j)2466 4087 y Ft('s)f(are)f(the)i(\(op)r(en\))g (connected)f(comp)r(o-)204 4203 y(nen)n(ts)e(of)h Fs(!)569 4173 y Fm(\000)p Fr(1)658 4203 y Ft(\()p Fs(J)744 4173 y Fm(0)767 4203 y Ft(\).)37 b(F)-7 b(or)25 b(all)g Fs(R)1183 4173 y Fm(0)1206 4203 y Ft(,)h(\000)1307 4168 y Fr(+)1307 4226 y Fp(j)1362 4203 y Ft(\()p Fs(R)1458 4173 y Fm(0)1481 4203 y Ft(\))e(=)e Fq(f)p Ft(\()p Fs(x;)14 b(\030)t Ft(\))26 b Fq(j)g(j)p Fs(x)p Fq(j)e Fs(>)e(R)2197 4173 y Fm(0)2220 4203 y Fs(;)40 b(\030)27 b Fq(2)c Ft(\012)2484 4215 y Fp(j)2519 4203 y Fs(;)40 b Ft(cos)o(\()p Fs(x;)14 b(v)s Ft(\()p Fs(\030)t Ft(\)\))25 b Fs(>)e Fq(\000)p Fs(\033)3216 4173 y Fm(0)3239 4203 y Fq(g)i Ft(is)g(connected)204 4303 y(\(simple)j(exercise\),)f(th)n(us)g(it)h(is)g(su\016cien)n(t)f 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Fs(:)p Ft(31\))31 b(is)h(a)f(simple)h (consequence)f(of)h(the)h(fact)f(that)p 1944 4737 41 4 v 32 w Fs(\030)t Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))31 b Fq(!)f Fs(\030)36 b Ft(as)c Fq(j)p Fs(x)p Fq(j)e(!)h(1)p Ft(,)i(b)r(ecause)f(it)g(implies)204 4904 y(that)i(\010)450 4916 y Fp(t;\017;h;q)637 4904 y Ft(\()p Fs(x;)14 b(\030)t Ft(\))34 b Fq(2)g Ft(\000)1000 4869 y Fr(+)1000 4927 y Fp(j)1055 4904 y Ft(\()p Fs(R)1151 4874 y Fm(0)1174 4904 y Ft(\))g(if)g(\()p Fs(x;)14 b(\030)t Ft(\))35 b Fq(2)e Ft(\000)1685 4869 y Fr(+)1685 4927 y Fp(j)1740 4904 y Ft(\()p Fs(R)1836 4874 y Fm(0)1859 4904 y Ft(\))1891 4837 y Fo(\000)1963 4904 y Fq(\032)g Ft(\000)2113 4874 y Fr(+)2168 4904 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))2465 4837 y Fo(\001)2537 4904 y Ft(with)34 b Fq(j)p Fs(x)p Fq(j)g Ft(big)g(enough.)54 b(In)34 b(order)e(to)204 5018 y(pro)n(v)n(e)c(\(3)p Fs(:)p Ft(32\),)j(w)n(e)f(remark)e(that,)k(if)f(\()p Fs(x;)14 b(\030)t Ft(\))28 b Fq(2)g Ft(\000)1745 4983 y Fr(+)1745 5042 y Fp(j)1800 5018 y Ft(\()p Fs(R)1896 4988 y Fm(0)1919 5018 y Ft(\))21 b Fq(\\)f Ft(\010)2107 5030 y Fp(t;\017;h;q)2295 4951 y Fo(\000)2333 5018 y Fs(@)5 b Ft(\000)2434 4988 y Fr(+)2489 5018 y Ft(\()p Fs(R)q(;)14 b(J)o(;)g(\033)s Ft(\))2786 4951 y Fo(\001)2824 5018 y Ft(,)31 b(with)g Fs(R)3134 4988 y Fm(0)3184 5018 y Fs(>)d(R)j Ft(then)f(there)204 5118 y(exists)d Fs(\030)473 5088 y Fm(0)524 5118 y Ft(suc)n(h)g(that) 731 5291 y Fs(\030)h Ft(=)p 882 5223 V 22 w Fs(\030)t Ft(\()p Fs(t;)14 b(x;)g(\030)1145 5257 y Fm(0)1169 5291 y Fs(;)g(\017;)g(h)p Ft(\))83 b(and)1657 5224 y Fo(\000)1695 5291 y Fs(!)s Ft(\()p Fs(\030)1822 5257 y Fm(0)1845 5291 y Ft(\))23 b Fq(2)h Fs(@)5 b(J)91 b Ft(or)k(cos\()p Fs(x;)14 b(v)s Ft(\()p Fs(\030)2678 5257 y Fm(0)2702 5291 y Ft(\)\))24 b(=)e Fq(\000)p Fs(\033)2992 5224 y Fo(\001)3030 5291 y Fs(:)482 b Ft(\(3.33\))1934 5540 y(25)p eop %%Page: 26 26 26 25 bop 204 531 a Ft(By)30 b(lemma)g(3)p Fs(:)p Ft(6,)h(it)g(is)f (clear)g(that,)h(giv)n(en)f(an)n(y)f Fs(\016)i(>)d Ft(0,)j(w)n(e)f(ha)n (v)n(e)f Fq(j)p Fs(\030)c Fq(\000)20 b Fs(\030)2512 501 y Fm(0)2535 531 y Fq(j)28 b Fs(<)g(\016)33 b Ft(if)e Fs(R)2892 501 y Fm(0)2943 531 y Fs(>>)c Ft(1)j(\(uniformly)h(with)204 631 y(resp)r(ect)36 b(to)g Fs(t;)14 b(\017;)g(h;)g(q)s Ft(\);)41 b(as)36 b Fs(J)44 b Ft(is)36 b(a)g(neigh)n(b)r(orho)r(o)r(d)g (of)p 1971 564 78 4 v 36 w Fs(J)2025 607 y Fm(0)2085 631 y Ft(and)g Fs(\033)41 b Fq(6)p Ft(=)d Fs(\033)2496 601 y Fm(0)2520 631 y Ft(,)g(\(3)p Fs(:)p Ft(33\))e(is)g(imp)r(ossible) h(if)g Fs(R)3485 601 y Fm(0)3544 631 y Ft(is)f(big)204 731 y(enough.)g(This)28 b(completes)f(the)h(pro)r(of)f(of)g(the)h (\014rst)g(p)r(oin)n(t.)204 830 y(T)-7 b(o)27 b(pro)n(v)n(e)f(the)i (estimate)f(\(3)p Fs(:)p Ft(30\),)g(w)n(e)h(use)f(the)h(trivial)f (relation)510 989 y(0)22 b(=)662 922 y Fo(\000)p 700 922 41 4 v 67 x Fs(\030)740 1010 y Fr(1)777 989 y Ft(\()p Fs(t;)14 b(x;)p 960 944 45 4 v 14 w(\021)1005 1010 y Fr(1)1042 989 y Fs(;)g(\017;)g(h)p Ft(\))k Fq(\000)p 1331 922 41 4 v 18 w Fs(\030)1371 1010 y Fr(1)1409 989 y Ft(\()p Fs(t;)c(x;)p 1592 944 45 4 v 14 w(\021)1636 1010 y Fr(2)1674 989 y Fs(;)g(\017;)g(h)p Ft(\))1862 922 y Fo(\001)1918 989 y Ft(+)2001 922 y Fo(\000)p 2039 922 41 4 v 67 x Fs(\030)2079 1010 y Fr(1)2116 989 y Ft(\()p Fs(t;)g(x;)p 2299 944 45 4 v 14 w(\021)2344 1010 y Fr(2)2381 989 y Fs(;)g(\017;)g(h)p Ft(\))k Fq(\000)p 2670 922 41 4 v 18 w Fs(\030)2710 1010 y Fr(2)2748 989 y Ft(\()p Fs(t;)c(x;)p 2931 944 45 4 v 14 w(\021)2975 1010 y Fr(2)3013 989 y Fs(;)g(\017;)g(h)p Ft(\))3201 922 y Fo(\001)3252 989 y Fs(:)260 b Ft(\(3.34\))204 1149 y(Since,)36 b(for)d(an)n(y)h Fs(\016)j(>)c Ft(0,)j(w)n(e)d(can)h(c)n(ho)r(ose)f Fs(R)1643 1118 y Fm(0)1700 1149 y Fs(>)g Ft(0)h(big)g(enough,)h(suc)n(h)f(that)g Fq(j)p 2739 1103 V Fs(\021)2784 1169 y Fp(j)2841 1149 y Fq(\000)23 b Fs(\030)t Fq(j)34 b(\024)f Fs(\016)s Ft(,)j(\()p Fs(j)j Ft(=)34 b(1)p Fs(;)14 b Ft(2\))33 b(and)204 1263 y Fq(j)p Fs(@)271 1275 y Fp(\030)p 307 1195 41 4 v 307 1263 a Fs(\030)347 1283 y Fr(1)385 1263 y Ft(\()p Fs(t;)14 b(x;)g(u)p 616 1217 45 4 v(\021)660 1283 y Fr(1)722 1263 y Ft(+)25 b(\(1)g Fq(\000)f Fs(u)p Ft(\))p 1080 1217 V Fs(\021)1124 1283 y Fr(2)1162 1263 y Fs(;)14 b(\017;)g(h)p Ft(\))24 b Fq(\000)h Ft(1)1506 1275 y Fp(d)1544 1263 y Fq(j)40 b(\024)f Fs(\016)s Ft(,)h(for)d(all)g Fs(u)i Fq(2)h Ft([0)p Fs(;)14 b Ft(1],)39 b(\()p Fs(x;)14 b(\030)t Ft(\))41 b Fq(2)f Ft(\000)2863 1233 y Fr(+)2918 1263 y Ft(\()p Fs(R)3014 1233 y Fm(0)3037 1263 y Fs(;)14 b(J)3128 1233 y Fm(0)3151 1263 y Fs(;)g(\033)3238 1233 y Fm(0)3262 1263 y Ft(\)\),)41 b Fs(\017)e Fq(2)h Ft([0)p Fs(;)14 b Ft(1],)204 1362 y Fs(h)23 b Fq(2)g Ft([0)p Fs(;)14 b(h)503 1374 y Fr(0)540 1362 y Ft(])27 b(and)h Fs(q)789 1374 y Fp(j)847 1362 y Fq(2)23 b(B)s Ft(,)k(w)n(e)g(get)h(b)n(y)f(T)-7 b(a)n(ylor)26 b(form)n(ula:)724 1471 y Fo(Z)807 1491 y Fr(1)770 1659 y(0)858 1584 y Fs(@)902 1596 y Fp(\030)p 939 1516 41 4 v 939 1584 a Fs(\030)979 1604 y Fr(1)1016 1584 y Ft(\()p Fs(t;)14 b(x;)g(u)p 1247 1538 45 4 v(\021)1291 1604 y Fr(1)1347 1584 y Ft(+)k(\(1)g Fq(\000)g Fs(u)p Ft(\))p 1685 1538 V Fs(\021)1729 1604 y Fr(2)1767 1584 y Fs(;)c(\017;)g(h)p Ft(\))p Fs(du)p Ft(\()p 2078 1538 V Fs(\021)2121 1604 y Fr(2)2177 1584 y Fq(\000)p 2260 1538 V 18 w Fs(\021)2304 1604 y Fr(1)2342 1584 y Ft(\))23 b(=)g Fq(O)2553 1517 y Fo(\000)2591 1584 y Fq(N)12 b Ft(\()p Fs(q)2740 1596 y Fr(1)2796 1584 y Fq(\000)18 b Fs(q)2916 1596 y Fr(2)2954 1584 y Ft(\))p Fq(h)p Fs(x)p Fq(i)3097 1550 y Fm(\000)p Fp(\032)3189 1517 y Fo(\001)204 1790 y Ft(whic)n(h)29 b(pro)n(v)n(es)d(the)k(result)e(when)h Fq(j)p Fs(\015)5 b Fq(j)25 b Ft(=)f(0,)29 b(b)n(y)g(using)f(the)h (lemma)g(3)p Fs(:)p Ft(6.)39 b(Then)29 b(w)n(e)g(get)f(the)i(general)d (result)h(b)n(y)204 1889 y(induction)g(on)f Fq(j)p Fs(\015)5 b Fq(j)27 b Ft(b)n(y)h(di\013eren)n(tiating)f(the)h(relation)f(\(3)p Fs(:)p Ft(34\).)36 b(W)-7 b(e)28 b(don't)g(go)e(further)i(in)n(to)f (details.)326 b Fc(\003)204 1989 y Fg(Pro)s(of)34 b(of)g(the)h(prop)s (osition)d Ft(3)p Fs(:)p Ft(1)p Fg(.)42 b Ft(First)30 b(w)n(e)f(men)n(tion)h(that)g(the)g(fact)g(that)g Fs(')p Ft(,)h(giv)n(en)e(b)n(y)g(\(3)p Fs(:)p Ft(21\),)h(satis\014es)204 2088 y(the)24 b(Hamilton-Jacobi)f(equation)g(\(3)p Fs(:)p Ft(3\),)h(is)g(standard)f(\(see)h(again)f([16)o(,)h(34)o(]\))g(and)g(w) n(e)f(only)h(insist)g(on)f(the)i(pro)r(of)204 2188 y(of)i(the)h (estimates.)328 2288 y(Let)k Fs(J)527 2300 y Fr(0)596 2288 y Ft(and)g Fs(\033)809 2300 y Fr(0)878 2288 y Ft(b)r(e)g(\014xed)g (and)f(let's)h(c)n(ho)r(ose)f(op)r(en)g(in)n(terv)-5 b(als)31 b Fs(J)o(;)14 b(J)2490 2258 y Fm(0)2545 2288 y Ft(and)31 b(real)g(n)n(um)n(b)r(ers)g Fs(\033)n(;)14 b(\033)3347 2258 y Fm(0)3404 2288 y Ft(suc)n(h)31 b(that)204 2387 y Fs(J)250 2399 y Fr(0)310 2387 y Fc(b)23 b Fs(J)452 2357 y Fm(0)498 2387 y Fc(b)g Fs(J)31 b Fc(b)22 b Fs(I)35 b Ft(and)28 b Fq(\000)p Ft(1)22 b Fs(<)g(\033)1246 2399 y Fr(0)1307 2387 y Fs(<)h(\033)1445 2357 y Fm(0)1492 2387 y Fs(<)f(\033)27 b(<)22 b Ft(1.)204 2487 y(According)j(to)h(the)g (form)n(ula)g(\(3)p Fs(:)p Ft(21\))f(and)h(follo)n(wing)f([16)o(],)i (the)f(estimate)g(\(3)p Fs(:)p Ft(6\))g(will)g(b)r(e)h(a)f(consequence) f(of)h(the)204 2587 y(fact)i(that)1118 2746 y Fq(j)p Fs(@)1190 2711 y Fp(\015)1232 2678 y Fo(\000)1270 2746 y Fs(@)1314 2758 y Fp(x)1361 2746 y Ft(~)-47 b Fs(z)t Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))k Fq(\000)g Ft(1)1953 2758 y Fp(d)1992 2678 y Fo(\001)2030 2746 y Fq(j)23 b(\024)f Fs(c)2199 2758 y Fp(\015)2242 2746 y Fq(h)p Fs(x)p Fq(i)2353 2711 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2615 2719 y Fh(+)3535 2746 y Ft(\(3.35\))204 2923 y(where)35 b Fs(@)501 2893 y Fp(\015)580 2923 y Ft(=)i Fs(@)731 2893 y Fp(\013)726 2943 y(x)778 2923 y Fs(@)827 2883 y Fp(\014)822 2948 y(\030)871 2923 y Fs(@)920 2893 y Fp(n)915 2943 y(\017)965 2923 y Fs(@)1014 2893 y Fp(l)1009 2946 y(h)1088 2923 y Ft(and)f Fs(c)1294 2935 y Fp(\015)1372 2923 y Ft(is)g(a)g(constan)n(t)f(indep)r(enden)n(t) i(of)e Fs(t)i Fq(\025)g Ft(0,)g(\()p Fs(x;)14 b(\030)t Ft(\))38 b Fq(2)g Ft(\000)3108 2893 y Fr(+)3163 2923 y Ft(\()p Fs(R)3259 2893 y Fm(0)3282 2923 y Fs(;)14 b(J)3373 2893 y Fm(0)3396 2923 y Fs(;)g(R)3497 2893 y Fm(0)3520 2923 y Ft(\),)38 b(and)204 3022 y Fs(\017;)14 b(h;)g(q)s Ft(.)36 b(T)-7 b(o)27 b(pro)n(v)n(e)f(this)i(estimate,)g(w)n(e)f(study) -7 b(,)28 b(when)g(0)22 b Fq(\024)h Fs(s)g Fq(\024)g Fs(t)p Ft(,)p 1066 3136 48 4 v 1066 3182 a Fs(x)p Ft(\()p Fs(s;)p 1221 3136 45 4 v 14 w(\021)s Ft(\))h(=)p 1408 3136 48 4 v 22 w Fs(x)q Ft(\()p Fs(s;)p 1564 3136 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))84 b(=)p 1965 3136 48 4 v 82 w Fs(x)q Ft(\()p Fs(s;)14 b(x;)p 2205 3136 45 4 v 14 w(\021)t Ft(\()p Fs(t;)g(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)g(\017;)g(h)p Ft(\))p Fs(;)p 1080 3238 41 4 v 1080 3306 a(\030)t Ft(\()p Fs(s;)p 1228 3260 45 4 v 14 w(\021)t Ft(\))23 b(=)p 1416 3238 41 4 v 23 w Fs(\030)t Ft(\()p Fs(s;)p 1564 3260 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))84 b(=)p 1965 3238 41 4 v 82 w Fs(\030)5 b Ft(\()p Fs(s;)14 b(x;)p 2198 3260 45 4 v 14 w(\021)s Ft(\()p Fs(t;)g(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))p Fs(;)g(\017;)g(h)p Ft(\))204 3465 y(whic)n(h)27 b(are)g(solutions)g(of)g(the)h(follo)n (wing)f(equations)p 818 3621 48 4 v 818 3667 a Fs(x)p Ft(\()p Fs(s;)p 973 3621 45 4 v 14 w(\021)t Ft(\()p Fs(t)p Ft(\)\))84 b(=)e Fs(x)19 b Ft(+)1524 3554 y Fo(Z)1607 3574 y Fp(s)1570 3742 y Fr(0)1656 3667 y Fs(@)1700 3679 y Fp(\030)1737 3667 y Fs(!)s Ft(\()p 1824 3599 41 4 v Fs(\030)t Ft(\()p Fs(u;)p 1981 3621 45 4 v 14 w(\021)s Ft(\)\))g(+)f Fs(\017@)2269 3679 y Fp(\030)2305 3667 y Fs(q)s Ft(\()p 2377 3621 48 4 v Fs(x)q Ft(\()p Fs(u;)p 2542 3621 45 4 v 14 w(\021)s Ft(\))p Fs(;)p 2655 3599 41 4 v 14 w(\030)t Ft(\()p Fs(u;)p 2812 3621 45 4 v 14 w(\021)s Ft(\))p Fs(;)c(h)p Ft(\))p Fs(du;)416 b Ft(\(3.36\))p 825 3836 41 4 v 825 3904 a Fs(\030)t Ft(\()p Fs(s;)p 973 3858 45 4 v 14 w(\021)t Ft(\()p Fs(t)p Ft(\)\))84 b(=)e Fs(\030)23 b Ft(+)1517 3791 y Fo(Z)1600 3811 y Fp(t)1563 3979 y(s)1643 3904 y Fs(\017@)1721 3916 y Fp(x)1762 3904 y Fs(q)s Ft(\()p 1834 3858 48 4 v Fs(x)q Ft(\()p Fs(u;)p 1999 3858 45 4 v 14 w(\021)s Ft(\))p Fs(;)p 2112 3836 41 4 v 14 w(\030)t Ft(\()p Fs(u;)p 2269 3858 45 4 v 14 w(\021)s Ft(\))p Fs(;)14 b(h)p Ft(\))p Fs(du:)959 b Ft(\(3.37\))204 4113 y(W)-7 b(e)35 b(remark)e(that)h(the)h(follo)n (wing)f(three)g(estimates)g(hold,)i(when)f(\()p Fs(x;)14 b(\030)t Ft(\))35 b Fq(2)g Ft(\000)2735 4083 y Fr(+)2790 4113 y Ft(\()p Fs(R)2886 4083 y Fm(0)2910 4113 y Fs(;)14 b(J)3001 4083 y Fm(0)3024 4113 y Fs(;)g(\033)3111 4083 y Fm(0)3134 4113 y Ft(\),)37 b Fs(\017)d Fq(2)h Ft([0)p Fs(;)14 b Ft(1],)35 b Fs(h)g Fq(2)204 4213 y Ft([0)p Fs(;)14 b(h)354 4225 y Fr(0)390 4213 y Ft(],)28 b Fs(q)e Fq(2)e(B)29 b Ft(and)f Fs(t)23 b Fq(\025)f Ft(0,)28 b(if)g(w)n(e)f(c)n (ho)r(ose)f Fs(R)1610 4183 y Fm(0)1656 4213 y Fs(>)d Ft(0)k(large)f(enough)1654 4372 y Fq(j)p 1677 4326 48 4 v Fs(x)q Ft(\()p Fs(s;)p 1833 4326 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))p Fq(j)84 b(\025)f Fs(e)2297 4384 y Fr(0)2333 4372 y Ft(\()p Fs(s)19 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))p Fs(;)881 b Ft(\(3.38\))1050 4497 y Fq(j)p Fs(@)1122 4462 y Fp(\015)1164 4429 y Fo(\000)p 1202 4451 48 4 v 68 x Fs(x)p Ft(\()p Fs(s;)p 1357 4451 45 4 v 14 w(\021)t Ft(\()p Fs(t)p Ft(\)\))19 b Fq(\000)f Fs(x)h Fq(\000)f Fs(sv)s Ft(\()p Fs(\030)t Ft(\))1965 4429 y Fo(\001)2004 4497 y Fq(j)83 b(\024)g Fs(c)2294 4509 y Fp(\015)2336 4497 y Fq(h)p Fs(s)p Fq(ih)p Fs(x)p Fq(i)2550 4462 y Fm(\000)p Fp(\032n)2677 4470 y Fh(+)2726 4462 y Fm(\000j)p Fp(\013)p Fm(j)2864 4497 y Fs(;)648 b Ft(\(3.39\))1353 4637 y Fq(j)p Fs(@)1425 4602 y Fp(\015)1467 4570 y Fo(\000)p 1505 4569 41 4 v 67 x Fs(\030)t Ft(\()p Fs(s;)p 1653 4591 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))20 b Fq(\000)e Fs(\030)1966 4570 y Fo(\001)2004 4637 y Fq(j)83 b(\024)g Fs(c)2294 4649 y Fp(\015)2336 4637 y Fq(h)p Fs(x)p Fq(i)2447 4602 y Fm(\000)p Fp(\032n)2574 4610 y Fh(+)2622 4602 y Fm(\000j)p Fp(\013)p Fm(j)2761 4637 y Fs(:)751 b Ft(\(3.40\))204 4796 y(\(3)p Fs(:)p Ft(38\))27 b(is)h(a)g(consequence)f(of)h(\(3)p Fs(:)p Ft(39\))f(whic)n(h,)i(as)e(\(3)p Fs(:)p Ft(40\),)h(is)g(a)f (consequence)h(of)g(\(3)p Fs(:)p Ft(30\))f(and)h(lemma)g(3)p Fs(:)p Ft(6.)38 b(No)n(w)204 4896 y(w)n(e)27 b(can)f(pro)n(v)n(e)g(\(3) p Fs(:)p Ft(35\))g(when)i Fq(j)p Fs(\015)5 b Fq(j)23 b Ft(=)f(0)27 b(b)r(ecause,)g(b)n(y)g(applying)f Fs(@)2257 4908 y Fp(x)2327 4896 y Ft(to)h(\(3)p Fs(:)p Ft(36\))f(and)h(\(3)p Fs(:)p Ft(37\),)g(w)n(e)g(\014nd)g(that)h(\()p Fs(y)s(;)14 b(\021)s Ft(\))204 4995 y(de\014ned)28 b(as)1501 5154 y Fs(y)86 b Ft(=)d Fs(@)1820 5166 y Fp(x)1862 5087 y Fo(\000)p 1900 5109 48 4 v 67 x Fs(x)p Ft(\()p Fs(s;)p 2055 5109 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))20 b Fq(\000)e Fs(x)2375 5087 y Fo(\001)2413 5154 y Fs(;)1099 b Ft(\(3.41\))1501 5291 y Fs(\021)86 b Ft(=)d Fs(@)1820 5303 y Fp(x)1862 5224 y Fo(\000)p 1900 5223 41 4 v 67 x Fs(\030)t Ft(\()p Fs(s;)p 2048 5245 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))2218 5224 y Fo(\001)2257 5291 y Fs(;)1255 b Ft(\(3.42\))1934 5540 y(26)p eop %%Page: 27 27 27 26 bop 204 531 a Ft(is)26 b(solution)g(of)h(a)f(system)g(as)g(\()p Fs(A:)p Ft(10\))g(\(notice)h(that)g(the)g(fact)f(that)h Fs(y)i Ft(and)e Fs(\021)i Ft(are)d(matrices)g(and)g(not)g(v)n(ectors)f (is)204 631 y(irrelev)-5 b(an)n(t\),)27 b(so)g(prop)r(osition)f Fs(A:)p Ft(3)i(pro)n(v)n(es)d(that,)j(uniformly)g(with)g(resp)r(ect)f (to)h(the)g(parameters,)e(w)n(e)h(ha)n(v)n(e)1177 825 y Fq(j)p Fs(y)s Fq(j)22 b(\024)h Fs(c)p Fq(j)p Fs(x)p Fq(j)1506 791 y Fm(\000)p Fp(\032)1597 825 y Fs(;)180 b Fq(j)p Fs(\021)s Fq(j)23 b(\024)g Fs(c)p Ft(\()p Fs(s)c Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2335 791 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)2506 766 y Fd(0)2533 825 y Fq(j)p Fs(x)p Fq(j)2626 791 y Fp(\032)2660 766 y Fd(0)2684 791 y Fm(\000)p Fp(\032)204 1008 y Ft(\(see)27 b(prop)r(osition)g Fs(A:)p Ft(3)g(for)g(the)h(de\014nition)g(of)g Fs(\032)1738 978 y Fm(0)1761 1008 y Ft(\).)37 b(Then)28 b(w)n(e)f(get,)h(b)n(y)f (induction)h(on)f Fq(j)p Fs(\015)5 b Fq(j)23 b(\025)g Ft(0,)k(that)1239 1191 y Fq(j)p Fs(@)1311 1156 y Fp(\015)1354 1191 y Fs(y)s Fq(j)82 b(\024)h Fs(c)1687 1203 y Fp(\015)1729 1191 y Fq(j)p Fs(x)p Fq(j)1822 1156 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2084 1164 y Fh(+)2136 1191 y Fs(;)1376 b Ft(\(3.43\))1239 1330 y Fq(j)p Fs(@)1311 1296 y Fp(\015)1353 1330 y Fs(\021)s Fq(j)83 b(\024)g Fs(c)1687 1342 y Fp(\015)1729 1330 y Ft(\()p Fs(s)19 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2027 1296 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)2198 1271 y Fd(0)2226 1330 y Fq(j)p Fs(x)p Fq(j)2319 1296 y Fp(\032)2353 1271 y Fd(0)2376 1296 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2638 1304 y Fh(+)2689 1330 y Fs(;)823 b Ft(\(3.44\))204 1513 y(again)26 b(b)n(y)h(application)g(of)h(prop)r (osition)f Fs(A:)p Ft(3)g(and)h(using)f(the)h(fact)f(that,)h(when)g Fs(a)23 b Fq(2)h Fs(S)2957 1525 y Fr(1)2994 1513 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\026)p Ft(\))27 b(with)h Fs(\026)23 b Fq(\025)g Ft(0,)980 1695 y Fs(@)1029 1661 y Fp(\015)1071 1628 y Fo(\000)1109 1695 y Fs(a)p Ft(\()p 1185 1650 48 4 v Fs(x)q Ft(\()p Fs(s;)p 1341 1650 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\))p Fs(;)p 1548 1628 41 4 v 14 w(\030)5 b Ft(\()p Fs(s;)p 1697 1650 45 4 v 14 w(\021)s Ft(\()p Fs(t)p Ft(\)\)\))1899 1628 y Fo(\001)1961 1695 y Ft(=)23 b Fq(O)2117 1628 y Fo(\000)2156 1695 y Ft(\()p Fs(s)18 b Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))2453 1661 y Fm(\000)p Fp(\026)2550 1695 y Fq(j)p Fs(x)p Fq(j)2643 1661 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)2910 1628 y Fo(\001)2948 1695 y Fs(;)204 1878 y Ft(so)23 b(the)h(estimate)f(\(3)p Fs(:)p Ft(35\))g(is)g(pro)n(v)n(ed.)34 b(In)24 b(the)g(same)f(w)n(a)n (y)-7 b(,)23 b(\(3)p Fs(:)p Ft(7\))g(is)h(essen)n(tially)e(a)h (consequence)g(of)g(the)h(fact)g(that)740 2061 y Fq(j)p Fs(@)812 2026 y Fp(\015)855 1993 y Fo(\000)893 2061 y Fs(@)937 2073 y Fp(x)984 2061 y Ft(~)-47 b Fs(z)1018 2073 y Fr(1)1055 2061 y Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))k Fq(\000)g Fs(@)1611 2073 y Fp(x)1658 2061 y Ft(~)-47 b Fs(z)1692 2073 y Fr(2)1729 2061 y Ft(\()p Fs(t;)14 b(x;)g(\030)t(;)g(\017;)g(h)p Ft(\))2140 1993 y Fo(\001)2178 2061 y Fq(j)23 b(\024)g Fs(c)2348 2073 y Fp(\015)2391 2061 y Fq(N)12 b Ft(\()p Fs(q)2540 2073 y Fr(1)2596 2061 y Fq(\000)18 b Fs(q)2716 2073 y Fr(2)2753 2061 y Ft(\))p Fq(h)p Fs(x)p Fq(i)2896 2026 y Fm(\000j)p Fp(\013)p Fm(j\000)p Fp(\032n)3158 2034 y Fh(+)204 2243 y Ft(with)29 b(~)-47 b Fs(z)428 2255 y Fp(j)487 2243 y Ft(asso)r(ciated)23 b(to)h Fs(q)1014 2255 y Fp(j)1049 2243 y Ft(.)35 b(The)24 b(pro)r(of)g(of)g(this)g(new)g(estimate)g(is)g(done)g(as)f(b)r(efore)h (applying)f(prop)r(osition)g Fs(A:)p Ft(3)204 2343 y(\(with)28 b Fs(C)484 2355 y Fr(1)545 2343 y Ft(=)23 b Fq(O)r Ft(\()p Fq(N)12 b Ft(\()p Fs(q)882 2355 y Fr(1)939 2343 y Fq(\000)18 b Fs(q)1059 2355 y Fr(2)1096 2343 y Ft(\)\)\))29 b(to)e Fs(y)f Ft(=)d Fs(y)1518 2355 y Fr(1)1574 2343 y Fq(\000)18 b Fs(y)1698 2355 y Fr(2)1762 2343 y Ft(and)28 b Fs(\021)e Ft(=)d Fs(\021)2120 2355 y Fr(1)2176 2343 y Fq(\000)18 b Fs(\021)2300 2355 y Fr(2)2337 2343 y Ft(,)28 b(if)g Fs(y)2505 2355 y Fp(j)2568 2343 y Ft(and)f Fs(\021)2770 2355 y Fp(j)2833 2343 y Ft(are)g(de\014ned)h(b)n(y)f(\(3)p Fs(:)p Ft(41\))g(and)204 2443 y(\(3)p Fs(:)p Ft(42\))g(with)p 633 2397 48 4 v 28 w Fs(x)680 2455 y Fp(j)743 2443 y Ft(and)p 904 2397 45 4 v 27 w Fs(\021)948 2463 y Fp(j)983 2443 y Ft(,)h(instead)f(of)p 1414 2397 48 4 v 27 w Fs(x)i Ft(and)p 1651 2397 45 4 v 27 w Fs(\021)s Ft(.)37 b(W)-7 b(e)28 b(can)f(apply)g(prop)r(osition)g Fs(A:)p Ft(3,)g(once)g(w)n(e)g (ha)n(v)n(e)g(c)n(hosen)f Fs(R)3724 2412 y Fm(0)204 2542 y Ft(large)g(enough)h(so)g(that)1142 2642 y Fq(j)p 1165 2596 48 4 v Fs(x)1213 2654 y Fr(1)1268 2642 y Ft(+)18 b Fs(u)p Ft(\()p 1431 2596 V Fs(x)1479 2654 y Fr(2)1534 2642 y Fq(\000)p 1617 2596 V 18 w Fs(x)1665 2654 y Fr(1)1702 2642 y Ft(\))p Fq(j)23 b(\025)g Fs(c)1904 2608 y Fm(0)1927 2642 y Ft(\()p Fq(j)p Fs(x)p Fq(j)d Ft(+)e Fs(t)p Ft(\))p Fs(;)180 b Fq(8)27 b Fs(u)22 b Fq(2)i Ft([0)p Fs(;)14 b Ft(1])204 2791 y(with)30 b Fs(c)431 2761 y Fm(0)482 2791 y Fs(>)c Ft(0,)k(whic)n(h)g(allo)n(ws)f(to)h(get)f(the)i(righ)n(t) e(estimates)h(for)f(the)h(matrices)g(in)n(v)n(olv)n(ed)e(in)i(the)h (system)f(from)204 2891 y(whic)n(h)d(\()p Fs(y)s(;)14 b(\021)s Ft(\))28 b(is)g(solution.)204 2991 y(Finally)-7 b(,)27 b(c)n(ho)r(osing)f Fs(R)902 3003 y Fr(0)962 2991 y Fs(>)c(R)1113 2960 y Fm(0)1136 2991 y Ft(,)28 b(and)f(m)n(ultiplying) g Fs(@)5 b(F)i(=@)e(t)26 b Ft(b)n(y)h(a)f(cuto\013)i(function)f(in)g Fs(S)2933 3003 y Fr(1)2970 2991 y Ft(\()p Fq(\0001)p Fs(;)14 b Ft(0\))27 b(supp)r(orted)g(in)204 3090 y(\000)256 3060 y Fr(+)311 3090 y Ft(\()p Fs(R)407 3060 y Fm(0)430 3090 y Fs(;)14 b(J)521 3060 y Fm(0)544 3090 y Fs(;)g(\033)631 3060 y Fm(0)655 3090 y Ft(\))28 b(and)f(iden)n(tically)h(1)f(in)h(a)f (neigh)n(b)r(orho)r(o)r(d)f(of)i(\000)2181 3060 y Fr(+)2236 3090 y Ft(\()p Fs(R)2331 3102 y Fr(0)2368 3090 y Fs(;)14 b(J)2451 3102 y Fr(0)2489 3090 y Fs(;)g(\033)2573 3102 y Fr(0)2610 3090 y Ft(\),)28 b(w)n(e)f(get)h(the)g(result.)359 b Fc(\003)204 3365 y Fv(A)134 b(Estimates)47 b(on)e(solutions)h(of)f (some)g(ODE)204 3547 y Ft(In)31 b(this)g(app)r(endix,)g(w)n(e)f(are)g (going)f(to)i(giv)n(e)f(estimates)g(on)g Fs(y)h Ft(=)d Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))31 b(and)g Fs(\021)g Ft(=)c Fs(\021)s Ft(\()p Fs(t;)14 b(x)p Ft(\))32 b(solutions)e(of)h (some)204 3646 y(ordinary)26 b(di\013eren)n(tial)h(equations,)g(or)g (in)n(tegral)f(equations.)328 3746 y(The)34 b(\014rst)g(prop)r(osition) f(will)h(pro)n(v)n(e)e(estimates)i(on)g(the)g(solutions)f(of)h(the)g (follo)n(wing)f(linear)g(di\013eren)n(tial)204 3845 y(system)27 b(\(where)42 b(_)-37 b Fs(z)26 b Ft(=)c Fs(@)948 3857 y Fp(t)978 3845 y Fs(z)t Ft(\))1471 3961 y Fo(\()1553 4046 y Ft(_)-38 b Fs(y)25 b Ft(=)e Fs(Ay)e Ft(+)d Fs(B)t(\021)k Ft(+)c Fs(y)2153 4058 y Fr(0)2190 4046 y Fs(;)1553 4166 y Ft(_)-38 b Fs(\021)26 b Ft(=)c Fs(C)6 b(y)22 b Ft(+)c Fs(D)r(\021)k Ft(+)c Fs(\021)2161 4178 y Fr(0)2198 4166 y Fs(;)3556 4103 y Ft(\(A.1\))204 4368 y(where)34 b Fs(A;)14 b(B)t(;)g(C)q(;)g(D)37 b Ft(are)d Fs(d)24 b Fq(\002)e Fs(d)35 b Ft(matrices)f(and)h Fs(y)1785 4380 y Fr(0)1822 4368 y Fs(;)14 b(\021)1900 4380 y Fr(0)1972 4368 y Ft(are)33 b(v)n(ectors)g(of)i Fl(R)2562 4338 y Fp(d)2642 4368 y Ft(all)f(dep)r(ending)h(con)n(tin)n(uously)e(on)204 4468 y(\()p Fs(t;)14 b(x)p Ft(\))41 b Fq(2)f Ft([0)p Fs(;)14 b Fq(1)p Ft(\))25 b Fq(\002)g(E)7 b Ft(\()p Fs(R)q Ft(\),)41 b(with)e Fq(E)7 b Ft(\()p Fs(R)q Ft(\))40 b Fq(\032)g Fs(B)t Ft(\(0)p Fs(;)14 b(R)q Ft(\))1891 4438 y Fp(c)1925 4468 y Ft(,)40 b(and)e(satisfying)f(the)h(follo)n(wing)f(estimates)g (for)h(some)204 4568 y Fs(C)263 4580 y Fr(0)323 4568 y Fq(\025)23 b Ft(0)p Fs(;)k(C)562 4580 y Fr(1)623 4568 y Fq(\025)c Ft(0)k(and)g Fs(\026)c Fq(\025)g Ft(0)820 4750 y Fq(j)p Fs(A)p Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)24 b(\024)f Fs(C)1277 4762 y Fr(0)1314 4750 y Ft(\()p Fs(t)c Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))1603 4716 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)1779 4750 y Fs(;)166 b Fq(j)p Fs(B)t Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)25 b(\024)d Fs(C)2430 4762 y Fr(0)2468 4750 y Fs(;)1065 b Ft(\(A.2\))817 4875 y Fq(j)p Fs(C)6 b Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)24 b(\024)f Fs(C)1277 4887 y Fr(0)1314 4875 y Ft(\()p Fs(t)c Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))1603 4841 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(2)1779 4875 y Fs(;)166 b Fq(j)p Fs(D)r Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)25 b(\024)d Fs(C)2434 4887 y Fr(0)2472 4875 y Ft(\()p Fs(t)c Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))2760 4841 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)2937 4875 y Fs(;)596 b Ft(\(A.3\))993 4999 y Fq(j)p Fs(y)1057 5011 y Fr(0)1094 4999 y Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)24 b(\024)e Fs(C)1465 5011 y Fr(1)1503 4999 y Fq(j)p Fs(x)p Fq(j)1596 4965 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)1779 4999 y Fs(;)166 b Fq(j)p Fs(\021)2032 5011 y Fr(0)2070 4999 y Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)24 b(\024)f Fs(C)2442 5011 y Fr(1)2479 4999 y Fq(j)p Fs(x)p Fq(j)2572 4965 y Fm(\000)p Fp(\026)2669 4999 y Ft(\()p Fs(t)c Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2958 4965 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)3556 4999 y Ft(\(A.4\))204 5182 y(for)27 b(all)g Fs(t)c Fq(\025)g Ft(0)k(and)g Fs(x)d Fq(2)f(E)7 b Ft(\()p Fs(R)q Ft(\))29 b(\(the)f(matrix)f(norm,)g(also)f(denoted)i(b)n(y)f Fq(j)p Fs(:)p Fq(j)h Ft(is)g(suc)n(h)f(that)h Fq(j)p Fs(E)5 b(z)t Fq(j)22 b(\024)h(j)p Fs(E)5 b Fq(j)p Fs(:)p Fq(j)p Fs(z)t Fq(j)p Ft(\).)1934 5540 y(27)p eop %%Page: 28 28 28 27 bop 204 531 a Fg(Prop)s(osition)30 b(A.1.)40 b Fi(Ther)l(e)29 b(exists)d Fs(R)e Fq(\025)f Ft(1)j Fi(lar)l(ge)i (enough,)h(and)f Fs(C)2360 543 y Fr(2)2424 531 y Fi(which)h(b)l(oth)f (dep)l(end)g(only)g(on)f Fs(C)3455 543 y Fr(0)3520 531 y Fi(and)h Fs(\032)p Fi(,)204 631 y(such)i(that)f(for)i(al)t(l)g Fs(x)23 b Fq(2)h(E)7 b Ft(\()p Fs(R)q Ft(\))p Fi(,)30 b(the)g(solution)g Ft(\()p Fs(y)s(;)14 b(\021)s Ft(\))30 b Fi(which)i(vanishes)f(at)e Fs(t)23 b Ft(=)g(0)29 b Fi(satis\014es)1426 808 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)83 b(\024)g Fs(C)1984 820 y Fr(2)2021 808 y Fs(C)2080 820 y Fr(1)2118 808 y Fq(h)p Fs(t)p Fq(ij)p Fs(x)p Fq(j)2305 773 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2489 808 y Fs(;)1044 b Ft(\(A.5\))1425 932 y Fq(j)p Fs(\021)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)84 b(\024)f Fs(C)1984 944 y Fr(2)2021 932 y Fs(C)2080 944 y Fr(1)2118 932 y Fq(j)p Fs(x)p Fq(j)2211 898 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2394 932 y Fs(:)1139 b Ft(\(A.6\))328 1109 y(The)28 b(pro)r(of)f(is)g(a)h(consequence)e(of)i(the)g(follo)n (wing)e(lemma.)204 1270 y Fg(Lemma)j(A.2.)41 b Fi(Assume)29 b(that)h Fs(C)1296 1282 y Fr(1)1356 1270 y Ft(=)23 b(1)p Fi(.)39 b(Then)30 b(for)h(al)t(l)g Ft(0)22 b Fs(<)h(")g Fq(\024)g Ft(1)29 b Fi(ther)l(e)h(exists)f Fs(R)24 b Ft(=)f Fs(R)q Ft(\()p Fs(C)3155 1282 y Fr(0)3193 1270 y Fs(;)14 b(\032;)g(")p Ft(\))22 b Fq(\025)h Ft(1)p Fi(,)30 b(such)204 1370 y(that)g(for)g(al)t(l)h Fq(j)p Fs(x)p Fq(j)24 b(\025)e Fs(R)31 b Fi(and)f Fs(t)23 b Fq(\025)g Ft(0)986 1555 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)23 b(\024)g Fs(")p Fq(h)p Fs(t)p Fq(ij)p Fs(x)p Fq(j)1601 1494 y Fe(\032)p 1602 1508 31 3 v 1603 1540 a Fh(2)1642 1520 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)1825 1555 y Fs(;)183 b Fq(j)p Fs(\021)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)25 b(\024)d Fs(")p Fq(j)p Fs(x)p Fq(j)2553 1494 y Fe(\032)p 2553 1508 V 2554 1540 a Fh(2)2594 1520 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2776 1555 y Fs(:)328 1731 y Fg(Pro)s(of.)37 b Ft(Let's)27 b(c)n(ho)r(ose)g Fs(R)h Ft(suc)n(h)f(that)1706 1908 y(2)p Fs(e)1787 1878 y Fr(2)p Fp(C)1868 1886 y Fh(0)1903 1908 y Fs(R)1967 1878 y Fm(\000)2029 1851 y Fe(\032)p 2029 1865 V 2030 1898 a Fh(2)2096 1908 y Fq(\024)c Fs(";)1310 b Ft(\(A.7\))547 2033 y(2)p Fs(C)648 2045 y Fr(0)699 1966 y Fo(R)754 1986 y Fm(1)738 2062 y Fr(0)825 2033 y Ft(\()p Fq(j)p Fs(x)p Fq(j)19 b Ft(+)f Fs(s)p Ft(\))1123 2002 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)1299 2033 y Fs(ds)g Ft(+)g Fq(j)p Fs(x)p Fq(j)1575 2002 y Fp(\032)p Fm(\000)1672 1976 y Fe(\032)p 1672 1990 V 1673 2022 a Fh(2)1730 1966 y Fo(R)1786 1986 y Fm(1)1770 2062 y Fr(0)1856 2033 y Ft(\()p Fq(j)p Fs(x)p Fq(j)h Ft(+)f Fs(s)p Ft(\))2154 2002 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)2330 2033 y Fs(ds)23 b Fq(\024)2534 2000 y Fp(")p 2533 2014 34 4 v 2533 2061 a Fr(4)2576 2033 y Fq(j)p Fs(x)p Fq(j)2669 2002 y Fm(\000)2731 1976 y Fe(\032)p 2731 1990 31 3 v 2732 2022 a Fh(4)2776 2033 y Fs(;)179 b Fq(8)27 b Fs(x)d Fq(2)f(E)7 b Ft(\()p Fs(R)q Ft(\))p Fs(;)153 b Ft(\(A.8\))1181 2170 y Fs(C)1240 2182 y Fr(0)1292 2103 y Fo(R)1347 2124 y Fm(1)1331 2200 y Fr(0)1417 2170 y Ft(\()p Fs(R)20 b Ft(+)e Fs(s)p Ft(\))1686 2140 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)1862 2170 y Fs(ds)g Ft(+)g Fs(C)2104 2182 y Fr(0)2142 2170 y Fs(R)2206 2140 y Fm(\000)2267 2113 y Fe(\032)p 2267 2127 V 2268 2160 a Fh(4)2330 2170 y Ft(+)g Fs(R)2477 2140 y Fm(\000)2539 2113 y Fe(\032)p 2539 2127 V 2540 2160 a Fh(2)2606 2170 y Fq(\024)2704 2137 y Fp(")p 2703 2151 34 4 v 2703 2199 a Fr(2)2746 2170 y Fs(:)787 b Ft(\(A.9\))204 2347 y(W)-7 b(e)28 b(shall)f(pro)n(v)n(e)f(that)i(for)f(all)g Fs(x)d Fq(2)f(E)7 b Ft(\()p Fs(R)q Ft(\),)28 b(w)n(e)g(ha)n(v)n(e)e Fs(I)7 b Ft(\()p Fs(x)p Ft(\))24 b(=)f([0)p Fs(;)14 b Fq(1)p Ft(\))27 b(where)530 2532 y Fs(I)7 b Ft(\()p Fs(x)p Ft(\))25 b(=)d Fq(f)p Fs(T)34 b Fq(\025)22 b Ft(0)28 b Fq(j)f(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)24 b(\024)f Fs(")p Fq(h)p Fs(t)p Fq(ij)p Fs(x)p Fq(j)1744 2471 y Fe(\032)p 1745 2485 31 3 v 1746 2518 a Fh(2)1785 2498 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)1995 2532 y Ft(and)k Fq(j)p Fs(\021)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)25 b(\024)d Fs(")p Fq(j)p Fs(x)p Fq(j)2678 2471 y Fe(\032)p 2679 2485 V 2680 2518 a Fh(2)2719 2498 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2901 2532 y Fs(;)42 b Fq(8)27 b Fs(t)c Fq(2)g Ft([0)p Fs(;)14 b(T)e Ft(])p Fq(g)p Fs(:)204 2708 y Ft(F)-7 b(or)27 b Fs(t)c Fq(2)g Ft([0)p Fs(;)14 b Ft(1])27 b(w)n(e)g(ha)n(v)n(e)g(easily)871 2943 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)19 b Ft(+)f Fq(j)p Fs(\021)s Ft(\()p Fs(t;)c(x)p Ft(\))p Fq(j)24 b(\024)f Ft(2)p Fs(C)1722 2955 y Fr(0)1773 2830 y Fo(Z)1856 2850 y Fp(t)1819 3018 y Fr(0)1899 2943 y Fq(j)p Fs(y)s Ft(\()p Fs(s;)14 b(x)p Ft(\))p Fq(j)19 b Ft(+)f Fq(j)p Fs(\021)s Ft(\()p Fs(s;)c(x)p Ft(\))p Fq(j)p Fs(ds)19 b Ft(+)g(2)p Fq(j)p Fs(x)p Fq(j)2875 2909 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)3057 2943 y Fs(;)204 3161 y Ft(so)27 b(b)n(y)g(the)h(Gron)n(w)n(all's)e(lemma)h (w)n(e)h(get)1076 3338 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)20 b Ft(+)e Fq(j)p Fs(\021)s Ft(\()p Fs(t;)c(x)p Ft(\))p Fq(j)24 b(\024)f Ft(2)p Fq(j)p Fs(x)p Fq(j)1962 3304 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2144 3338 y Fs(e)2183 3304 y Fr(2)p Fp(C)2264 3312 y Fh(0)2300 3338 y Fs(;)180 b Fq(8)27 b Fs(t)c Fq(2)g Ft([0)p Fs(;)14 b Ft(1])204 3515 y(and)38 b(the)g(righ)n(t-hand)f(side)h(is)g(lo)n(w)n (er)e(than)i Fs(")p Fq(j)p Fs(x)p Fq(j)1792 3458 y Fe(\032)p 1793 3472 V 1794 3505 a Fh(2)1833 3485 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2054 3515 y Ft(if)g(\()p Fs(A:)p Ft(7\))h(is)f (satis\014ed,)i(hence)e Fs(I)7 b Ft(\()p Fs(x)p Ft(\))39 b(is)f(not)g(empt)n(y)-7 b(.)204 3614 y(Moreo)n(v)n(er)30 b Fs(I)7 b Ft(\()p Fs(x)p Ft(\))33 b(is)g(clearly)e(an)i(in)n(terv)-5 b(al)32 b(so)f(w)n(e)i(just)g(ha)n(v)n(e)e(to)i(pro)n(v)n(e)e(that)i Fs(T)12 b Ft(\()p Fs(x)p Ft(\))31 b(:=)g(sup)14 b Fs(I)7 b Ft(\()p Fs(x)p Ft(\))32 b(=)f(+)p Fq(1)p Ft(.)52 b(W)-7 b(e)204 3714 y(argue)39 b(b)n(y)h(con)n(tradiction)f(so)h(w)n(e)g (assume)g(that)g Fs(T)12 b Ft(\()p Fs(x)p Ft(\))45 b Fs(<)f Fq(1)p Ft(.)75 b(Then)41 b(for)f(all)g Fs(t)k Fq(2)h Ft([0)p Fs(;)14 b(T)e Ft(\()p Fs(x)p Ft(\)])40 b(w)n(e)g(get,)j(b)n(y)204 3814 y(in)n(tegrating)26 b(the)i(second)f (equation)g(of)h(\()p Fs(A:)p Ft(1\))g(b)r(et)n(w)n(een)f(0)g(and)h Fs(t)646 4048 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)84 b(\024)f Ft(2)p Fs(C)1247 4060 y Fr(0)1298 3935 y Fo(Z)1381 3955 y Fp(t)1344 4124 y Fr(0)1410 4048 y Ft(\()p Fq(j)p Fs(x)p Fq(j)19 b Ft(+)f Fs(s)p Ft(\))1708 4014 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)1884 4048 y Fs(")p Fq(j)p Fs(x)p Fq(j)2026 3987 y Fe(\032)p 2026 4001 V 2027 4034 a Fh(2)2066 4014 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2249 4048 y Fs(ds)g Ft(+)g Fq(j)p Fs(x)p Fq(j)2525 4014 y Fm(\000)p Fp(\026)2636 3935 y Fo(Z)2719 3955 y Fp(t)2682 4124 y Fr(0)2749 4048 y Ft(\()p Fs(s)g Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))3046 4014 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)3223 4048 y Fs(ds)998 4249 y Fq(\024)1157 4193 y Fs(")p 1156 4230 42 4 v 1156 4306 a Ft(4)1207 4249 y Fq(j)p Fs(x)p Fq(j)1300 4214 y Fm(\000)1362 4188 y Fe(\032)p 1362 4202 31 3 v 1363 4234 a Fh(4)1403 4214 y Fr(+)1463 4188 y Fe(\032)p 1463 4202 V 1464 4234 a Fh(2)1504 4214 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)204 4453 y Ft(if)35 b(\()p Fs(A:)p Ft(8\))g(is)f(satis\014ed;)j(then)e (using)f(this)h(new)g(estimate)f(and)g(b)n(y)g(in)n(tegrating)f(the)i (\014rst)f(equation)g(of)h(\()p Fs(A:)p Ft(1\))204 4553 y(b)r(et)n(w)n(een)27 b(0)h(and)f Fs(t)h Ft(w)n(e)f(get)663 4783 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fq(j)84 b(\024)e(h)p Fs(t)p Fq(ij)p Fs(x)p Fq(j)1359 4721 y Fe(\032)p 1360 4735 V 1361 4768 a Fh(2)1400 4748 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)1597 4665 y Fo(\022)1658 4783 y Fs("C)1756 4795 y Fr(0)1807 4670 y Fo(Z)1890 4690 y Fp(t)1853 4858 y Fr(0)1919 4783 y Ft(\()p Fs(s)19 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2217 4748 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)2393 4783 y Fs(ds)h Ft(+)f Fs("C)2675 4795 y Fr(0)2712 4783 y Fq(j)p Fs(x)p Fq(j)2805 4748 y Fm(\000)2868 4721 y Fe(\032)p 2868 4735 V 2869 4768 a Fh(4)2930 4783 y Ft(+)g Fq(j)p Fs(x)p Fq(j)3106 4748 y Fm(\000)3169 4721 y Fe(\032)p 3169 4735 V 3170 4768 a Fh(2)3213 4665 y Fo(\023)1015 4987 y Fq(\024)1174 4930 y Fs(")p 1172 4967 42 4 v 1172 5043 a Ft(2)1224 4987 y Fq(h)p Fs(t)p Fq(ij)p Fs(x)p Fq(j)1421 4925 y Fe(\032)p 1422 4939 31 3 v 1423 4972 a Fh(2)1462 4952 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)204 5191 y Ft(if)29 b(\()p Fs(A:)p Ft(9\))h(is)f(satis\014ed.)40 b(This)29 b(sho)n(ws)f(that)h(the)g(estimates)g(on)g Fs(y)i Ft(and)e Fs(\021)j Ft(are)c(v)-5 b(alid)29 b(on)g([0)p Fs(;)14 b(T)3112 5161 y Fm(0)3134 5191 y Ft(])29 b(with)g Fs(T)3437 5161 y Fm(0)3485 5191 y Fs(>)c(T)12 b Ft(\()p Fs(x)p Ft(\))204 5291 y(whic)n(h)27 b(is)h(a)f(con)n(tradiction.)2587 b Fc(\003)1934 5540 y Ft(28)p eop %%Page: 29 29 29 28 bop 204 531 a Fg(Pro)s(of)41 b(of)g(the)g(prop)s(osition.)60 b Ft(If)36 b Fs(C)1481 543 y Fr(1)1555 531 y Ft(=)h(0,)g Fs(y)j Ft(=)c Fs(\021)k Ft(=)d(0)e(and)h(the)g(result)f(is)h(trivial.) 61 b(If)36 b Fs(C)3307 543 y Fr(1)3382 531 y Fs(>)g Ft(0,)i(then)204 631 y Fs(y)s(=C)349 643 y Fr(1)386 631 y Fs(;)14 b(\021)s(=C)568 643 y Fr(1)642 631 y Ft(is)38 b(solution)f(of)h(the)g(same)f(system)g (with)h Fs(y)2061 643 y Fr(0)2098 631 y Fs(;)14 b(\021)2176 643 y Fr(0)2251 631 y Ft(replaced)37 b(b)n(y)g Fs(y)2755 643 y Fr(0)2792 631 y Fs(=C)2893 643 y Fr(1)2930 631 y Fs(;)14 b(\021)3008 643 y Fr(0)3046 631 y Fs(=C)3147 643 y Fr(1)3221 631 y Ft(whic)n(h)38 b(satis\014es)204 731 y Fq(j)p Fs(y)268 743 y Fr(0)305 731 y Fs(=C)406 743 y Fr(1)443 731 y Fq(j)h(\024)f(j)p Fs(x)p Fq(j)701 700 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)921 731 y Ft(and)f Fq(j)p Fs(\021)1156 743 y Fr(0)1194 731 y Fs(=C)1295 743 y Fr(1)1332 731 y Fq(j)h(\024)h(j)p Fs(x)p Fq(j)1590 700 y Fm(\000)p Fp(\026)1687 731 y Ft(\()p Fs(t)25 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))1988 700 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)2164 731 y Ft(,)40 b(so)c(w)n(e)h(ma)n(y)f (use)h(the)g(previous)f(lemma.)65 b(W)-7 b(e)204 830 y(c)n(ho)r(ose)26 b Fs(")d Ft(=)g(1)k(so,)g(for)g(some)g Fs(R)d Fq(\025)e Ft(1)27 b(dep)r(ending)h(only)g(on)f Fs(C)2145 842 y Fr(0)2210 830 y Ft(and)h Fs(\032)p Ft(,)f(w)n(e)g(ha)n (v)n(e)536 989 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fs(=C)882 1001 y Fr(1)920 989 y Fq(j)23 b(\024)g(h)p Fs(t)p Fq(ij)p Fs(x)p Fq(j)1251 928 y Fe(\032)p 1252 942 31 3 v 1253 975 a Fh(2)1292 955 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)1475 989 y Fs(;)69 b Ft(and)55 b Fq(j)p Fs(\021)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fs(=C)2102 1001 y Fr(1)2140 989 y Fq(j)23 b(\024)g(j)p Fs(x)p Fq(j)2377 928 y Fe(\032)p 2377 942 V 2378 975 a Fh(2)2418 955 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2600 989 y Fs(;)180 b Fq(8)27 b Fs(t)c Fq(\025)g Ft(0)p Fs(;)41 b Fq(j)p Fs(x)p Fq(j)23 b(\025)g Fs(R)q(:)204 1148 y Ft(Then)28 b(b)n(y)f(in)n(tegrating)f(the)i(second)f(line)h(of)f(\()p Fs(A:)p Ft(1\))i(w)n(e)e(get)576 1354 y Fq(j)p Fs(\021)s Ft(\()p Fs(t;)14 b(x)p Ft(\))p Fs(=C)922 1366 y Fr(1)960 1354 y Fq(j)83 b(\024)g Fs(C)1273 1366 y Fr(0)1324 1241 y Fo(Z)1407 1261 y Fm(1)1370 1429 y Fr(0)1478 1354 y Ft(\()p Fs(s)18 b Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))1775 1319 y Fm(\000)p Fr(2)p Fm(\000)p Fp(\032)1952 1354 y Ft(\(1)g(+)g Fs(s)p Ft(\))p Fq(j)p Fs(x)p Fq(j)2301 1292 y Fe(\032)p 2302 1306 V 2303 1339 a Fh(2)2342 1319 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2525 1354 y Fs(ds)g Ft(+)1214 1580 y Fs(C)1273 1592 y Fr(0)1324 1467 y Fo(Z)1407 1487 y Fm(1)1370 1656 y Fr(0)1478 1580 y Ft(\()p Fs(s)g Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))1775 1546 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)1952 1580 y Fq(j)p Fs(x)p Fq(j)2055 1519 y Fe(\032)p 2055 1533 V 2056 1566 a Fh(2)2095 1546 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)2278 1580 y Fs(ds)h Ft(+)f Fq(j)p Fs(x)p Fq(j)2555 1546 y Fm(\000)p Fp(\026)2665 1467 y Fo(Z)2748 1487 y Fm(1)2712 1656 y Fr(0)2819 1580 y Ft(\()p Fs(s)g Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))3116 1546 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)3293 1580 y Fs(ds)1066 1759 y Fq(\024)83 b Fs(C)1273 1771 y Fr(2)1310 1759 y Fq(j)p Fs(x)p Fq(j)1403 1725 y Fm(\000)p Fp(\026)p Fm(\000)p Fp(\032)204 1918 y Ft(with)27 b Fs(C)451 1930 y Fr(2)515 1918 y Ft(whic)n(h)g(dep)r(ends)g(only)f(on)h Fs(C)1427 1930 y Fr(0)1491 1918 y Ft(and)g Fs(\032)p Ft(.)36 b(In)27 b(the)g(same)f(w)n(a)n(y)g(w)n(e)g(get)g(the)h(exp)r (ected)g(estimate)g(on)f Fs(y)s(=C)3710 1930 y Fr(1)204 2017 y Ft(\(b)n(y)h(using)h(the)g(\014nal)f(estimate)h(on)f Fs(\021)s(=C)1489 2029 y Fr(1)1526 2017 y Ft(\).)38 b(The)27 b(pro)r(of)g(is)h(complete.)1247 b Fc(\003)328 2157 y Ft(Our)34 b(second)f(prop)r(osition)g(pro)n(v)n(es)f(estimates)i(on)g (the)g(solutions)f(of)h(the)h(follo)n(wing)e(system)g(of)h(in)n(tegral) 204 2257 y(equations)948 2346 y Fo(8)948 2420 y(>)948 2445 y(<)948 2595 y(>)948 2620 y(:)1022 2424 y Fs(y)s Ft(\()p Fs(s)p Ft(\))23 b(=)g Fs(y)1321 2436 y Fr(0)1358 2424 y Ft(\()p Fs(s)p Ft(\))c(+)1563 2357 y Fo(R)1618 2378 y Fp(s)1602 2454 y Fr(0)1667 2424 y Fs(A)p Ft(\()p Fs(u)p Ft(\))p Fs(y)s Ft(\()p Fs(u)p Ft(\))p Fs(du)g Ft(+)2190 2357 y Fo(R)2245 2378 y Fp(s)2229 2454 y Fr(0)2294 2424 y Fs(B)t Ft(\()p Fs(u)p Ft(\))p Fs(\021)s Ft(\()p Fs(u)p Ft(\))p Fs(du;)1022 2663 y(\021)s Ft(\()p Fs(s)p Ft(\))24 b(=)e Fs(\021)1321 2675 y Fr(0)1359 2663 y Ft(\()p Fs(s)p Ft(\))d(+)1564 2597 y Fo(R)1619 2617 y Fp(t)1603 2693 y(s)1662 2663 y Fs(C)6 b Ft(\()p Fs(u)p Ft(\))p Fs(y)s Ft(\()p Fs(u)p Ft(\))p Fs(du)19 b Ft(+)2188 2597 y Fo(R)2243 2617 y Fp(t)2227 2693 y(s)2286 2663 y Fs(D)r Ft(\()p Fs(u)p Ft(\))p Fs(\021)s Ft(\()p Fs(u)p Ft(\))p Fs(du;)3514 2541 y Ft(\(A.10\))204 2829 y(where)33 b(0)f Fq(\024)h Fs(s)f Fq(\024)h Fs(t)p Ft(,)i(and)e Fs(y)j Ft(=)c Fs(y)s Ft(\()p Fs(s)p Ft(\))h(=)g Fs(y)s Ft(\()p Fs(s;)14 b(x)p Ft(\))34 b(\(as)f Fs(\021)s Ft(,)i Fs(y)2046 2841 y Fr(0)2083 2829 y Ft(,)g Fs(\021)2182 2841 y Fr(0)2220 2829 y Ft(,)g Fs(A)p Ft(,)g Fs(B)t Ft(,)g Fs(C)40 b Ft(and)34 b Fs(D)r Ft(\),)h(with)f Fs(A;)14 b(B)t(;)g(C)q(;)g(D)36 b(d)23 b Fq(\002)f Fs(d)204 2929 y Ft(matrices)28 b(satisfying)h(\()p Fs(A:)p Ft(2\))p Fs(;)14 b Ft(\()p Fs(A:)p Ft(3\))30 b(and)f Fs(y)1563 2941 y Fr(0)1600 2929 y Fs(;)14 b(\021)1678 2941 y Fr(0)1745 2929 y Ft(v)n(ectors)28 b(satisfying)h(\()p Fs(A:)p Ft(4\).)42 b(W)-7 b(e)30 b(assume)e(moreo)n(v)n(er)f(that)j (the)204 3028 y(solution)d Fs(y)s(;)14 b(\021)30 b Ft(of)e(the)g (system)f(satis\014es)g(the)h(follo)n(wing)f Fi(a)j(priori)f Ft(estimates:)1516 3187 y Fq(j)p Fs(y)s Fq(j)83 b(\024)g Fs(C)1896 3199 y Fr(1)1933 3187 y Fq(h)p Fs(s)p Fq(ij)p Fs(x)p Fq(j)2129 3153 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2398 3187 y Fs(;)1093 b Ft(\(A.11\))1516 3312 y Fq(j)p Fs(\021)s Fq(j)83 b(\024)g Fs(C)1896 3324 y Fr(1)1933 3312 y Fq(j)p Fs(x)p Fq(j)2026 3277 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2295 3312 y Fs(:)1196 b Ft(\(A.12\))204 3470 y(The)27 b(fact)f(that)h(w)n(e)g(in) n(tegrate)e(on)i([)p Fs(s;)14 b(t)p Ft(])26 b(on)h(the)g(second)f(line) h(of)f(\()p Fs(A:)p Ft(10\))h(will)g(pro)n(vides)e(estimates)i(on)f Fs(y)j Ft(and)e Fs(\021)204 3570 y Ft(indep)r(enden)n(t)h(of)g Fs(s)p Ft(.)37 b(Precisely)26 b(w)n(e)h(ha)n(v)n(e)204 3717 y Fg(Prop)s(osition)j(A.3.)40 b Fi(Assume)29 b(that)h Fs(\032)1457 3687 y Fm(\000)p Fr(1)1578 3717 y Fs(=)-51 b Fq(2)23 b Fl(N)t Fi(.)45 b(Ther)l(e)30 b(exists)f Fs(C)2295 3729 y Fr(2)2362 3717 y Fi(which)i(dep)l(ends)g(only)f(on)f Fs(\032)h Fi(and)f Fs(C)3495 3729 y Fr(0)3533 3717 y Fi(,)h(such)204 3817 y(that)1293 3976 y Fq(j)p Fs(y)s Fq(j)82 b(\024)h Fs(C)1672 3988 y Fr(2)1710 3976 y Fs(C)1769 3988 y Fr(1)1806 3976 y Fq(j)p Fs(x)p Fq(j)1899 3941 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)2082 3976 y Fs(;)1409 b Ft(\(A.13\))1292 4112 y Fq(j)p Fs(\021)s Fq(j)83 b(\024)g Fs(C)1672 4124 y Fr(2)1710 4112 y Fs(C)1769 4124 y Fr(1)1806 4112 y Ft(\()p Fs(s)19 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2104 4078 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\032)2275 4053 y Fd(0)2303 4112 y Fq(j)p Fs(x)p Fq(j)2396 4078 y Fp(\032)2430 4053 y Fd(0)2453 4078 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)2636 4112 y Fs(;)855 b Ft(\(A.14\))204 4271 y Fi(wher)l(e)30 b Fs(\032)481 4241 y Fm(0)528 4271 y Fs(>)22 b Ft(0)29 b Fi(is)h(de\014ne)l(d)h(by)f Ft(1)18 b(+)g Fs(\032)1350 4241 y Fm(0)1396 4271 y Ft(=)23 b(\()p Fs(k)e Ft(+)d(1\))p Fs(\032)p Fi(,)30 b(if)h Fs(k)26 b Fq(\025)c Ft(0)29 b Fi(is)i(the)e(lowest)i(inte)l(ger)e(such)h(that)g Ft(\()p Fs(k)22 b Ft(+)c(1\))p Fs(\032)k(>)h Ft(1)p Fi(.)328 4418 y Ft(This)j(prop)r(osition)f(giv)n(es)f(a)i Fs(L)1277 4388 y Fm(1)1372 4418 y Ft(estimate)g(for)f Fs(y)k Ft(with)d(resp)r (ect)g(to)f Fs(s)p Ft(,)h(and)g(a)f Fs(L)2839 4388 y Fr(1)2902 4418 y Ft(one)g(for)h Fs(\021)s Ft(,)g(whic)n(h)g(will)g(b)r (e)204 4517 y(crucial)h(for)g(the)h(pro)r(of)f(of)g(prop)r(osition)g(3) p Fs(:)p Ft(1.)204 4617 y Fg(Pro)s(of.)36 b Ft(By)27 b(putting)h(the)f Fi(a)j(priori)f Ft(estimates)e(\()p Fs(A:)p Ft(11\))g(and)g(\()p Fs(A:)p Ft(12\))g(in)h(the)f(second)g (equation)g(of)g(the)g(system)204 4717 y(\()p Fs(A:)p Ft(10\),)g(w)n(e)h(get)f(easily)g(that)1394 4816 y Fs(\021)s Ft(\()p Fs(s)p Ft(\))c(=)g Fq(O)1720 4749 y Fo(\000)1759 4816 y Fs(C)1818 4828 y Fr(1)1855 4816 y Ft(\()p Fs(s)c Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2153 4782 y Fm(\000)p Fp(\032)2244 4816 y Fq(j)p Fs(x)p Fq(j)2337 4782 y Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2519 4749 y Fo(\001)204 4951 y Ft(whic)n(h)31 b(means)g(that)g Fq(j)p Fs(\021)s Ft(\()p Fs(s)p Ft(\))p Fq(j)f(\024)e Fs(cC)1297 4963 y Fr(1)1335 4951 y Ft(\()p Fs(s)21 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))1637 4921 y Fm(\000)p Fp(\032)1729 4951 y Fq(j)p Fs(x)p Fq(j)1822 4921 y Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2035 4951 y Ft(for)31 b(some)f Fs(c)h Ft(dep)r(ending)h(only)f(on) g Fs(\032)g Ft(and)g Fs(C)3450 4963 y Fr(0)3487 4951 y Ft(.)48 b(Then)204 5051 y(using)27 b(this)h(new)g(estimate)f (together)g(with)h(\()p Fs(A:)p Ft(11\))g(in)f(the)h(\014rst)g (equation)f(of)g(\()p Fs(A:)p Ft(10\))h(w)n(e)f(get)969 5257 y Fs(y)s Ft(\()p Fs(s)p Ft(\))c(=)g Fq(O)1295 5189 y Fo(\000)1333 5257 y Fs(C)1392 5269 y Fr(1)1430 5257 y Fq(j)p Fs(x)p Fq(j)1523 5222 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)1706 5189 y Fo(\001)1762 5257 y Ft(+)18 b Fq(O)1913 5189 y Fo(\000)1952 5257 y Fs(C)2011 5269 y Fr(1)2062 5144 y Fo(Z)2145 5164 y Fp(s)2108 5332 y Fr(0)2181 5257 y Ft(\()p Fs(u)g Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))2487 5222 y Fm(\000)p Fp(\032)2578 5257 y Fq(j)p Fs(x)p Fq(j)2671 5222 y Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2853 5257 y Fs(du)2944 5189 y Fo(\001)1934 5540 y Ft(29)p eop %%Page: 30 30 30 29 bop 204 531 a Ft(and)27 b(there)h(are)e(t)n(w)n(o)h(cases;)g(if)h Fs(\032)23 b(>)f Ft(1:)37 b(then)28 b(w)n(e)f(get)g(the)h(result)g(and) f(if)h Fs(\032)23 b(<)g Ft(1,)k(w)n(e)g(ha)n(v)n(e)g(the)h(estimate) 1068 714 y Fs(y)s Ft(\()p Fs(s)p Ft(\))23 b(=)g Fq(O)1394 647 y Fo(\000)1432 714 y Fs(C)1491 726 y Fr(1)1529 714 y Fq(j)p Fs(x)p Fq(j)1622 680 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)1805 647 y Fo(\001)1862 714 y Ft(+)18 b Fq(O)2013 647 y Fo(\000)2051 714 y Fs(C)2110 726 y Fr(1)2148 714 y Ft(\()p Fs(s)g Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))2445 680 y Fr(1)p Fm(\000)p Fp(\032)2570 714 y Fq(j)p Fs(x)p Fq(j)2663 680 y Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2845 647 y Fo(\001)204 897 y Ft(so)31 b(w)n(e)h(are)f(in)i(p)r(osition)f(to) g(apply)g(the)h(follo)n(wing)e(lemma)h(from)g(whic)n(h)g(the)h(pro)r (of)e(of)i(the)f(prop)r(osition)f(is)i(a)204 996 y(trivial)27 b(consequence)f(b)n(y)i(a)f(\014nite)h(induction.)2003 b Fc(\003)204 1162 y Fg(Lemma)29 b(A.4.)41 b Fi(Assume)29 b(that)g(for)i(some)f Fs(n)23 b Fq(\025)g Ft(1)p Fi(,)29 b(we)i(have)792 1345 y Fs(\021)s Ft(\()p Fs(s)p Ft(\))83 b(=)g Fq(O)1238 1278 y Fo(\000)1276 1345 y Fs(C)1335 1357 y Fr(1)1373 1345 y Ft(\()p Fs(s)19 b Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))1671 1311 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)1847 1345 y Fq(j)p Fs(x)p Fq(j)1940 1311 y Fm(\000)p Fp(\026)2037 1278 y Fo(\001)2093 1345 y Ft(+)g Fq(O)2244 1278 y Fo(\000)2283 1345 y Fs(C)2342 1357 y Fr(1)2379 1345 y Ft(\()p Fs(s)h Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2677 1311 y Fm(\000)p Fp(n\032)2809 1345 y Fq(j)p Fs(x)p Fq(j)2902 1311 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\026)3084 1278 y Fo(\001)3122 1345 y Fs(;)369 b Ft(\(A.15\))792 1479 y Fs(y)s Ft(\()p Fs(s)p Ft(\))83 b(=)g Fq(O)1238 1412 y Fo(\000)1276 1479 y Fs(C)1335 1491 y Fr(1)1373 1479 y Fq(j)p Fs(x)p Fq(j)1466 1445 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)1649 1412 y Fo(\001)1706 1479 y Ft(+)18 b Fq(O)1857 1412 y Fo(\000)1895 1479 y Fs(C)1954 1491 y Fr(1)1992 1479 y Ft(\()p Fs(s)g Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))2289 1445 y Fr(1)p Fm(\000)p Fp(n\032)2455 1479 y Fq(j)p Fs(x)p Fq(j)2548 1445 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\026)2730 1412 y Fo(\001)2768 1479 y Fs(;)723 b Ft(\(A.16\))204 1662 y Fi(then)29 b(we)h(have)730 1844 y Fs(\021)s Ft(\()p Fs(s)p Ft(\))84 b(=)f Fq(O)1177 1777 y Fo(\000)1215 1844 y Fs(C)1274 1856 y Fr(1)1312 1844 y Ft(\()p Fs(s)18 b Ft(+)g Fq(j)p Fs(x)p Fq(j)p Ft(\))1609 1810 y Fm(\000)p Fp(\032)p Fm(\000)p Fr(1)1786 1844 y Fq(j)p Fs(x)p Fq(j)1879 1810 y Fm(\000)p Fp(\026)1976 1777 y Fo(\001)2032 1844 y Ft(+)g Fq(O)2183 1777 y Fo(\000)2222 1844 y Fs(C)2281 1856 y Fr(1)2318 1844 y Ft(\()p Fs(s)h Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2616 1810 y Fm(\000)p Fr(\()p Fp(n)p Fr(+1\))p Fp(\032)2884 1844 y Fq(j)p Fs(x)p Fq(j)2977 1810 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\026)3159 1777 y Fo(\001)3197 1844 y Fs(;)294 b Ft(\(A.17\))204 2027 y Fi(and)500 2264 y Fs(y)s Ft(\()p Fs(s)p Ft(\))24 b(=)758 2122 y Fo(\()825 2207 y Fq(O)893 2140 y Fo(\000)931 2207 y Fs(C)990 2219 y Fr(1)1028 2207 y Fq(j)p Fs(x)p Fq(j)1121 2177 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)1304 2140 y Fo(\001)1361 2207 y Ft(+)18 b Fq(O)1512 2140 y Fo(\000)1550 2207 y Fs(C)1609 2219 y Fr(1)1647 2207 y Ft(\()p Fs(s)h Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))1945 2177 y Fr(1)p Fm(\000)p Fr(\()p Fp(n)p Fr(+1\))p Fp(\032)2246 2207 y Fq(j)p Fs(x)p Fq(j)2339 2177 y Fm(\000)p Fr(1)p Fm(\000)p Fp(\026)2521 2140 y Fo(\001)2559 2207 y Fs(;)73 b Fi(if)31 b Ft(1)18 b Fq(\000)g Ft(\()p Fs(n)h Ft(+)f(1\))p Fs(\032)k(>)h Ft(0)825 2326 y Fq(O)893 2259 y Fo(\000)931 2326 y Fs(C)990 2338 y Fr(1)1028 2326 y Fq(j)p Fs(x)p Fq(j)1121 2296 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)1304 2259 y Fo(\001)1342 2326 y Fs(;)1292 b Fi(if)30 b Ft(1)18 b Fq(\000)g Ft(\()p Fs(n)h Ft(+)f(1\))p Fs(\032)23 b(<)f Ft(0)p Fs(:)328 2521 y Fg(Pro)s(of.)43 b Ft(Putting)30 b(the)g(estimates)f(\()p Fs(A:)p Ft(16\))h(and)g(\()p Fs(A:)p Ft(15\))f(in)h(the)h(second)e (equation)g(of)h(the)g(system)f(\()p Fs(A:)p Ft(10\))204 2621 y(leads)g(directly)g(to)h(the)g(estimate)f(\()p Fs(A:)p Ft(17\).)43 b(Then)30 b(b)n(y)f(putting)h(\()p Fs(A:)p Ft(17\))g(and)f(\()p Fs(A:)p Ft(16\))h(in)g(the)g(\014rst)f (equation)g(of)204 2720 y(the)f(system,)f(one)g(pro)n(v)n(es)f(that)880 2945 y Fs(y)s Ft(\()p Fs(s)p Ft(\))d(=)g Fq(O)1206 2878 y Fo(\000)1245 2945 y Fs(C)1304 2957 y Fr(1)1341 2945 y Fq(j)p Fs(x)p Fq(j)1434 2911 y Fm(\000)p Fp(\032)p Fm(\000)p Fp(\026)1617 2878 y Fo(\001)1674 2945 y Ft(+)18 b Fq(O)1825 2878 y Fo(\000)1863 2945 y Fs(C)1922 2957 y Fr(1)1974 2832 y Fo(Z)2057 2853 y Fp(s)2020 3021 y Fr(0)2092 2945 y Ft(\()p Fs(u)h Ft(+)f Fq(j)p Fs(x)p Fq(j)p Ft(\))2399 2911 y Fm(\000)p Fr(\()p Fp(n)p Fr(+1\))p Fp(\032)2667 2945 y Fq(j)p Fs(x)p Fq(j)2760 2911 y Fm(\000)p Fp(\026)p Fm(\000)p Fr(1)2942 2945 y Fs(du)3033 2878 y Fo(\001)204 3174 y Ft(from)27 b(whic)n(h)h(w)n(e)f(deduce)h(the)g (estimates)f(on)g Fs(y)s Ft(\()p Fs(s)p Ft(\))h(in)g(eac)n(h)f(case,)f (and)i(this)g(completes)f(the)h(pro)r(of.)303 b Fc(\003)328 3324 y Ft(Our)27 b(\014nal)h(result)f(giv)n(es)f(estimates)i(on)f Fs(y)f Ft(=)c Fs(y)s Ft(\()p Fs(t)p Ft(\))i(=)e Fs(y)s Ft(\()p Fs(t;)14 b(x;)g(\030)t Ft(\),)29 b(the)f(solution)f(of)1476 3439 y Fo(\()1557 3525 y Ft(_)-38 b Fs(y)26 b Ft(=)d Fs(v)s Ft(\()p Fs(\030)t Ft(\))c(+)f Fs(w)r Ft(\()p Fs(y)s(;)c(\030)t Ft(\))p Fs(;)1542 3644 y(y)1583 3659 y Fm(j)1603 3667 y Fe(t)p Fh(=0)1729 3644 y Ft(=)22 b Fs(x;)3514 3581 y Ft(\(A.18\))204 3847 y(when)34 b(\()p Fs(x;)14 b(\030)t Ft(\))34 b Fq(2)g Ft(\000)790 3817 y Fr(+)845 3847 y Ft(\()p Fs(R)940 3859 y Fr(0)978 3847 y Fs(;)14 b(J)1061 3859 y Fr(0)1098 3847 y Fs(;)g(\033)1182 3859 y Fr(0)1219 3847 y Ft(\).)56 b(W)-7 b(e)34 b(assume)f(that)i Fs(w)h Fq(2)d Fs(S)2193 3859 y Fr(1)2230 3847 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\))34 b(\(with)h(v)-5 b(alues)33 b(in)h Fl(R)3166 3817 y Fp(d)3211 3847 y Ft(\))g(whic)n(h)g(means)204 3946 y(that,)28 b(for)f(all)g Fs(\013;)14 b(\014)1240 4046 y Fq(j)p Fs(@)1312 4012 y Fp(\013)1307 4067 y(x)1359 4046 y Fs(@)1408 4006 y Fp(\014)1403 4071 y(\030)1452 4046 y Fs(w)r Ft(\()p Fs(x;)g(\030)t Ft(\))p Fq(j)25 b(\024)e Fs(C)1896 4058 y Fp(\013\014)1984 4046 y Ft(\(1)18 b(+)g Fs(!)s Ft(\()p Fs(\030)t Ft(\)\))p Fq(h)p Fs(x)p Fq(i)2461 4012 y Fm(\000)p Fp(\032)p Fm(\000j)p Fp(\013)p Fm(j)2688 4046 y Fs(:)204 4195 y Ft(Then)28 b(w)n(e)f(ha)n(v)n(e)204 4361 y Fg(Prop)s(osition)j(A.5.)40 b Fi(Ther)l(e)i(exists)1439 4340 y Ft(~)1421 4361 y Fs(R)1484 4373 y Fr(0)1563 4361 y Fq(\025)g Fs(R)1733 4373 y Fr(0)1810 4361 y Fi(and)f Fs(c)2018 4373 y Fr(0)2095 4361 y Fi(\(b)l(oth)g(dep)l(ending)g(only)g (on)f Fs(J)3078 4373 y Fr(0)3115 4361 y Fi(,)k Fs(\033)3231 4373 y Fr(0)3308 4361 y Fi(and)d(a)g(\014nite)204 4461 y(numb)l(er)35 b(of)i(semi-norms)f(of)h Fs(w)i Fi(in)d Fs(S)1428 4473 y Fr(1)1465 4461 y Ft(\()p Fs(!)s(;)14 b Fq(\000)p Fs(\032)p Ft(\))p Fi(\))36 b(such)g(that,)i(for)f(al)t(l)h Ft(\()p Fs(x;)14 b(\030)t Ft(\))35 b Fq(2)g Ft(\000)2827 4431 y Fr(+)2882 4461 y Ft(\()2933 4440 y(~)2914 4461 y Fs(R)2977 4473 y Fr(0)3015 4461 y Fs(;)14 b(J)3098 4473 y Fr(0)3135 4461 y Fs(;)g(\033)3219 4473 y Fr(0)3257 4461 y Ft(\))p Fi(,)38 b Fs(y)s Ft(\()p Fs(t;)14 b(x;)g(\030)t Ft(\))37 b Fi(is)204 4561 y(de\014ne)l(d)30 b(for)g(al)t(l)h Fs(t)23 b Fq(\025)g Ft(0)29 b Fi(and)1104 4743 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)14 b(x;)g(\030)t Ft(\))19 b Fq(\000)f Fs(x)h Fq(\000)f Fs(tv)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)24 b(\024)f Fs(c)2025 4755 y Fr(0)2062 4743 y Fq(h)p Fs(t)p Fq(ih)p Fs(x)p Fq(i)2267 4709 y Fm(\000)p Fp(\032)2359 4743 y Fs(;)184 b Fq(8)29 b Fs(t)23 b Fq(\025)f Ft(0)p Fs(:)204 4926 y Fg(Pro)s(of.)36 b Ft(By)27 b(di\013eren)n(tiating)h (the)g(expression)e Fq(h)15 b Ft(_)-38 b Fs(y)s(;)14 b(y)s Fq(i)27 b Ft(with)h(resp)r(ect)g(to)f Fs(t)p Ft(,)h(w)n(e)f(get)h (easily)1464 5106 y Fs(@)1513 5076 y Fr(2)p 1449 5143 116 4 v 1449 5219 a Fs(@)5 b(t)1528 5195 y Fr(2)1574 5162 y Fq(j)p Fs(y)s Fq(j)1664 5128 y Fr(2)1725 5162 y Fq(\025)22 b Ft(2)p Fq(j)15 b Ft(_)-38 b Fs(y)r Fq(j)1943 5128 y Fr(2)1999 5162 y Fq(\000)18 b Fs(C)6 b Fq(h)p Fs(y)s Fq(i)2255 5128 y Fm(\000)p Fp(\032)3514 5162 y Ft(\(A.19\))1934 5540 y(30)p eop %%Page: 31 31 31 30 bop 204 531 a Ft(where)36 b Fs(C)44 b Ft(dep)r(ends)37 b(only)f(on)h Fs(J)1248 543 y Fr(0)1322 531 y Ft(and)g(some)f (semi-norms)g(of)h Fs(w)r Ft(.)66 b(Indep)r(enden)n(tly)-7 b(,)40 b(let's)d(remark)e(that)j(w)n(e)204 631 y(can)30 b(c)n(ho)r(ose)g Fs(\016)663 643 y Fr(0)700 631 y Fs(;)14 b(e)776 643 y Fr(0)842 631 y Fs(>)28 b Ft(0)i(small)h(enough)f(and)1700 610 y(~)1682 631 y Fs(R)1745 643 y Fr(0)1813 631 y Ft(large)g(enough)g (\(all)h(dep)r(ending)g(only)g(on)f Fs(J)3212 643 y Fr(0)3249 631 y Fs(;)14 b(\033)3333 643 y Fr(0)3402 631 y Ft(and)31 b(some)204 731 y(semi-norms)26 b(of)i Fs(w)r Ft(\))g(suc)n(h)g(that)818 903 y(\(1)19 b Fq(\000)f Fs(\016)1031 915 y Fr(0)1068 903 y Ft(\))p Fs(t)1130 869 y Fr(2)1167 903 y Fq(j)p Fs(v)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)1360 869 y Fr(2)1417 903 y Ft(+)g(2)p Fs(t)p Fq(h)p Fs(x;)c(v)s Ft(\()p Fs(\030)t Ft(\))20 b(+)e Fs(w)r Ft(\()p Fs(x;)c(\030)t Ft(\))p Fq(i)20 b Ft(+)e Fq(j)p Fs(x)p Fq(j)2415 869 y Fr(2)2476 903 y Fq(\025)23 b Fs(e)2603 915 y Fr(0)2640 903 y Ft(\()p Fq(j)p Fs(x)p Fq(j)c Ft(+)f Fs(t)p Ft(\))2929 869 y Fr(2)3514 903 y Ft(\(A.20\))204 1076 y(for)27 b(all)g Fs(t)c Fq(\025)g Ft(0)k(and)g(\()p Fs(x;)14 b(\030)t Ft(\))25 b Fq(2)e Ft(\000)1160 1045 y Fr(+)1215 1076 y Ft(\()1265 1055 y(~)1247 1076 y Fs(R)1310 1088 y Fr(0)1348 1076 y Fs(;)14 b(J)1431 1088 y Fr(0)1468 1076 y Fs(;)g(\033)1552 1088 y Fr(0)1589 1076 y Ft(\).)204 1175 y(No)n(w)27 b(w)n(e)g(in)n(tro)r (duce)g(the)h(set)713 1348 y Fs(I)7 b Ft(\()p Fs(x;)14 b(\030)t Ft(\))24 b(=)e Fq(f)p Fs(t)h Fq(\025)f Ft(0)28 b Fq(j)f Ft(2)p Fq(j)15 b Ft(_)-38 b Fs(y)s Ft(\()p Fs(s)p Ft(\))p Fq(j)1592 1313 y Fr(2)1648 1348 y Fq(\000)18 b Fs(C)6 b Fq(h)p Fs(y)s Ft(\()p Fs(s)p Ft(\))p Fq(i)2007 1313 y Fm(\000)p Fp(\032)2121 1348 y Fq(\025)23 b Ft(2\(1)18 b Fq(\000)g Fs(\016)2463 1360 y Fr(0)2500 1348 y Ft(\))p Fq(j)p Fs(v)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)2725 1313 y Fr(2)2764 1348 y Fs(;)41 b Fq(8)27 b Fs(s)c Fq(2)g Ft([0)p Fs(;)14 b(t)p Ft(])p Fq(g)204 1520 y Ft(and)29 b(w)n(e)h(shall)f(pro)n(v)n(e)f(that)i Fs(T)12 b Ft(\()p Fs(x;)i(\030)t Ft(\))27 b(:=)f(sup)14 b Fs(I)7 b Ft(\()p Fs(x;)14 b(\030)t Ft(\))31 b(is)f(+)p Fq(1)p Ft(,)g(when)f(\()p Fs(x;)14 b(\030)t Ft(\))28 b Fq(2)f Ft(\000)2742 1490 y Fr(+)2797 1520 y Ft(\()2847 1499 y(~)2829 1520 y Fs(R)2892 1532 y Fr(0)2930 1520 y Fs(;)14 b(J)3013 1532 y Fr(0)3050 1520 y Fs(;)g(\033)3134 1532 y Fr(0)3171 1520 y Ft(\),)31 b(p)r(ossibly)f(after)204 1620 y(increasing)608 1599 y(~)590 1620 y Fs(R)653 1632 y Fr(0)691 1620 y Ft(.)36 b(First,)27 b(w)n(e)e(remark)g(that,)i(when)f Fs(t)d Ft(=)g(0,)j(2)p Fq(j)14 b Ft(_)-37 b Fs(y)r Fq(j)2158 1590 y Fr(2)2211 1620 y Fq(\000)15 b Fs(C)6 b Fq(h)p Fs(y)s Fq(i)2464 1590 y Fm(\000)p Fp(\032)2578 1620 y Fq(\025)23 b Ft(2\(1)15 b Fq(\000)g Fs(\016)2914 1632 y Fr(0)2951 1620 y Fs(=)p Ft(2\))p Fq(j)p Fs(v)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)3260 1590 y Fr(2)3324 1620 y Ft(b)n(y)26 b(c)n(ho)r(osing)222 1699 y(~)204 1720 y Fs(R)267 1732 y Fr(0)328 1720 y Ft(large)d(enough,)h(whic)n(h)g(pro)n(v)n(es)f(that)h Fs(T)12 b Ft(\()p Fs(x;)i(\030)t Ft(\))23 b Fs(>)g Ft(0.)35 b(Th)n(us,)25 b(for)f(all)g Fs(t)f Fq(2)g Ft([0)p Fs(;)14 b(T)e Ft(\()p Fs(x;)i(\030)t Ft(\)\))24 b(w)n(e)g(get,)h(b)n(y)f(in)n (tegrating)204 1819 y(t)n(w)n(o)j(times)g(\()p Fs(A:)p Ft(19\))h(on)f([0)p Fs(;)14 b(t)p Ft(],)1444 1992 y Fq(j)p Fs(y)s Ft(\()p Fs(t;)g(x;)g(\030)t Ft(\))p Fq(j)1789 1957 y Fr(2)1850 1992 y Fq(\025)23 b Fs(e)1977 2004 y Fr(0)2014 1992 y Ft(\()p Fq(j)p Fs(x)p Fq(j)c Ft(+)f Fs(t)p Ft(\))2303 1957 y Fr(2)3514 1992 y Ft(\(A.21\))204 2164 y(using)25 b(\()p Fs(A:)p Ft(20\))h(and)g(the)g(de\014nition)h(of) f Fs(I)7 b Ft(\()p Fs(x;)14 b(\030)t Ft(\).)37 b(This)26 b(pro)n(v)n(es)e(that)i Fq(j)15 b Ft(_)-38 b Fs(y)s Ft(\()p Fs(t)p Ft(\))p Fq(j)2536 2134 y Fr(2)2597 2164 y Ft(=)23 b Fs(v)s Ft(\()p Fs(\030)t Ft(\))16 b(+)e Fq(O)r Ft(\()p Fq(h)p Fs(x)p Fq(i)3138 2134 y Fm(\000)p Fp(\032)3231 2164 y Ft(\),)26 b(and)g(then)g(b)n(y)204 2264 y(increasing)611 2243 y(~)593 2264 y Fs(R)656 2276 y Fr(0)722 2264 y Ft(\(in)k(a)e(w)n (a)n(y)g(dep)r(ending)h(only)g(on)f Fs(J)1839 2276 y Fr(0)1876 2264 y Fs(;)14 b(C)36 b Ft(and)28 b(a)h(\014nite)g(n)n(um)n (b)r(er)g(of)g(semi-norms)e(of)i Fs(w)r Ft(\))h(w)n(e)e(get)204 2363 y(for)f(all)g Fs(t)c Fq(2)h Ft([0)p Fs(;)14 b(T)e Ft(\()p Fs(x;)i(\030)t Ft(\)\))1261 2463 y Fq(j)g Ft(_)-37 b Fs(y)r Ft(\()p Fs(t)p Ft(\))p Fq(j)1444 2429 y Fr(2)1501 2463 y Fq(\000)18 b Fs(C)6 b Fq(h)p Fs(y)s Ft(\()p Fs(t)p Ft(\))p Fq(i)1851 2429 y Fm(\000)p Fp(\032)1965 2463 y Fq(\025)23 b Ft(2\(1)18 b Fq(\000)g Fs(\016)2307 2475 y Fr(0)2344 2463 y Fs(=)p Ft(2\))p Fq(j)p Fs(v)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)2653 2429 y Fr(2)204 2606 y Ft(whic)n(h)34 b(pro)n(v)n(es)f(that)i Fq(j)15 b Ft(_)-38 b Fs(y)r Fq(j)988 2576 y Fr(2)1049 2606 y Fq(\000)22 b Fs(C)6 b Fq(h)p Fs(y)s Fq(i)1309 2576 y Fm(\000)p Fp(\032)1435 2606 y Fq(\025)34 b Ft(\(1)23 b Fq(\000)f Fs(\016)1755 2618 y Fr(0)1792 2606 y Ft(\))p Fq(j)p Fs(v)s Ft(\()p Fs(\030)t Ft(\))p Fq(j)2017 2576 y Fr(2)2091 2606 y Ft(on)34 b([0)p Fs(;)14 b(T)2376 2576 y Fm(0)2398 2606 y Ft(\()p Fs(x;)g(\030)t Ft(\)\))35 b(with)g Fs(T)2910 2576 y Fm(0)2933 2606 y Ft(\()p Fs(x;)14 b(\030)t Ft(\))35 b Fs(>)g(T)12 b Ft(\()p Fs(x;)i(\030)t Ft(\).)57 b(This)204 2706 y(pro)n(v)n(es)28 b(that)i Fs(T)12 b Ft(\()p Fs(x;)i(\030)t Ft(\))28 b(=)f(+)p Fq(1)i Ft(and)h(that)h(\()p Fs(A:)p Ft(21\))f(holds)g(on)g Fl(R)2193 2718 y Fr(+)2254 2706 y Ft(.)44 b(The)30 b(result)g(follo)n (ws)f(easily)h(b)n(y)g(in)n(tegrating)204 2806 y(\()p Fs(A:)p Ft(18\))d(on)h([0)p Fs(;)14 b(t)p Ft(],)27 b(taking)g(\()p Fs(A:)p Ft(21\))h(in)n(to)f(accoun)n(t.)1910 b Fc(\003)204 3078 y Fv(References)245 3260 y Ft([1])41 b Fa(M.)26 b(Arai,)h(J.)f(Uchiy)-6 b(ama)p Ft(,)23 b Fi(Gr)l(owth)j(or)l(der)g(of) g(eigenfunctions)g(of)g(Schr\177)-42 b(odinger)27 b(op)l(er)l(ators)f (with)g(p)l(oten-)374 3360 y(tials)i(admitting)g(some)g(inte)l(gr)l(al) f(c)l(onditions)h(I,)g(II)p Ft(,)d(Publ.)g(Res.)f(Math.)h(Sci.,)h(32)e (N.)h(4,)g(581-616,)d(617-631)374 3460 y(\(1996\).)245 3622 y([2])41 b Fa(M.)36 b(Sh.)g(Birman,)h(M.)e(G.)h(Krein)p Ft(,)e Fi(On)f(the)h(the)l(ory)h(of)g(wave)g(op)l(er)l(ators)g(and)f (sc)l(attering)g(op)l(er)l(ators)p Ft(,)374 3721 y(Dokl.)26 b(Ak)-5 b(ad.)27 b(Nauk)e(SSSR,)i(144,)e(475-478)e(\(1962\).)h(English) i(transl.)f(in)h(So)n(viet)g(Math.)g(Dokl.)g Fg(3)f Ft(\(1962\).)245 3883 y([3])41 b Fa(M.)33 b(Sh.)g(Birman,)g(D.)g(R.)g(Y)-7 b(af)g(aev)p Ft(,)29 b Fi(The)k(sp)l(e)l(ctr)l(al)e(shift)h(function.)g (The)h(work)f(of)g(M.)g(G.)g(Kr)l(ein)g(and)374 3983 y(its)e(further)g(development)p Ft(,)f(St.)f(P)n(etersburg)e(Math.)i (J.)f(4,)g(833-870)e(\(1993\).)245 4145 y([4])41 b Fa(J.M.)d(Bouclet)p Ft(,)d Fi(Distributions)h(sp)l(e)l(ctr)l(ales)g(p)l(our)g(des)g(op)n (\023)-40 b(er)l(ateurs)37 b(p)l(erturb)n(\023)-40 b(es,)36 b Ft(PhD)e(thesis,)i(Nan)n(tes)374 4244 y(Univ)n(ersit)n(y)27 b(\(2000\).)245 4406 y([5])p 374 4419 368 4 v 409 w(,)22 b Fi(F)-6 b(onction)24 b(sp)l(e)l(ctr)l(ale)g(p)l(our)f(des)h(p)l (erturb)l(ations)g(r)l(elativement)g(Hilb)l(ert-Schmidt,)f Ft(C.R.)e(Acad.)374 4506 y(Sci.)28 b(P)n(aris,)e(t.)i(332,)e(S)n(\023) -39 b(erie)27 b(I,)g(887-892)e(\(2001\).)245 4668 y([6])p 374 4681 V 409 w(,)d Fi(Gener)l(alize)l(d)k(sc)l(attering)e(phase)i (for)f(long)f(r)l(ange)h(p)l(erturb)l(ations)f(of)h(el)t(liptic)h(op)l (er)l(ators,)e Ft(C.R.)374 4768 y(Acad.)k(Sci.)g(P)n(aris,)e(t.)i(333,) e(S)n(\023)-39 b(erie)26 b(I,)i(845-850)c(\(2001\).)245 4930 y([7])p 374 4943 V 409 w(,)i Fi(T)-6 b(r)l(ac)l(es)29 b(formulae)g(for)h(r)l(elatively)g(Hilb)l(ert-Schmidt)e(p)l(erturb)l (ations,)f Ft(Asymptotic)g(Analysis)374 5029 y(\(to)h(app)r(ear\).)245 5191 y([8])41 b Fa(V.)35 b(Br)n(unea)n(u)p Ft(,)d Fi(F)-6 b(onctions)33 b(zeta)g(et)g(eta)g(en)g(pr)l(esenc)l(e)g(de)h(sp)l(e)l (ctr)l(e)f(c)l(ontinu.)f(\(Zeta)h(and)h(eta)f(functions)374 5291 y(for)e(c)l(ontinuous)e(sp)l(e)l(ctrum\),)e Ft(C.)h(R.)g(Acad.)f (Sci.)h(P)n(aris,)e(S)n(\023)-39 b(erie)26 b(I)i(323,)e(No.)i(5,)f (475-480)d(\(1996\).)1934 5540 y(31)p eop %%Page: 32 32 32 31 bop 245 531 a Ft([9])41 b Fa(V.)46 b(Br)n(unea)n(u,)51 b(V.)46 b(Petk)n(o)n(v)p Ft(,)g Fi(Semiclassic)l(al)f(estimates)f(for)g (tr)l(apping)h(p)l(erturb)l(ations,)h Ft(Comm)n(un.)374 631 y(Math.)28 b(Ph)n(ys.)f(213,)f(No.)i(2,)f(413-432)d(\(2000\).)204 797 y([10])40 b Fa(N.)h(Bur)n(q)p Ft(,)e Fi(L)l(ower)g(b)l(ounds)g(for) g(shap)l(e)h(r)l(esonanc)l(es)e(widths)i(of)f(long)g(r)l(ange)g (Schr\177)-42 b(odinger)40 b(op)l(er)l(ators,)374 897 y Ft(Amer.)28 b(J.)f(Math,)h(\(to)g(app)r(ear\).)204 1063 y([11])40 b Fa(V.)33 b(S.)g(Buslaev,)g(L.)f(D.)h(F)-10 b(addeev)p Ft(,)31 b Fi(F)-6 b(ormulas)32 b(for)g(tr)l(ac)l(es)f(for)i (a)f(singular)f(Sturm-Liouvil)t(le)h(di\013er-)374 1162 y(ential)f(op)l(er)l(ator,)e Ft(So)n(v.)e(Math.)h(Dokl.)f(1,)g(451-454) e(\(1960\).)204 1328 y([12])40 b Fa(G.)34 b(Carr)n(on)p Ft(,)d Fi(D)n(\023)-40 b(eterminant)32 b(r)l(elatif)h(et)g(la)g (fonction)g(Xi,)f Ft(Amer.)e(J.)g(Math,)i(v)n(ol.)d(124,)h(No.)h(2,)f (307-352)374 1428 y(\(2002\).)204 1594 y([13])40 b Fa(Y.)23 b(Colin-de-Verdi)1130 1587 y(\022)1130 1594 y(ere)p Ft(,)g Fi(Une)g(formule)g(de)h(tr)l(ac)l(e)f(p)l(our)g(l'op)n(\023)-40 b(er)l(ateur)24 b(de)g(Schr\177)-42 b(odinger)25 b(dans)e Fl(R)3479 1564 y Fr(3)3522 1594 y Fi(,)g Ft(Ann.)374 1694 y(Sci.)28 b(E.N.S.,)g(IV)g(s)n(\023)-39 b(er.)26 b(14,)h(27-39)e(\(1981\).)204 1860 y([14])40 b Fa(J.M.)34 b(Combes,)h(P.)f(Duclos,)g(M.)g(Klein,)h(R.)f(Seiler)p Ft(,)d Fi(The)j(shap)l(e)f(r)l(esonanc)l(e,)g Ft(Comm)n(un.)d(Math.)374 1959 y(Ph)n(ys.)d(110,)f(215-236)f(\(1987\).)204 2125 y([15])40 b Fa(R.)g(G.)f(Fr)n(oese,)j(P.)d(D.)h(Hislop)p Ft(,)d Fi(On)f(the)i(distribution)g(of)g(r)l(esonanc)l(es)f(for)h(some) g(asymptotic)l(al)t(ly)374 2225 y(hyp)l(erb)l(olic)33 b(manifolds,)d Ft(Journ)n(\023)-39 b(ees)25 b(EDP)-7 b(,)27 b(exp.)h(VI)r(I)g(\(2000\).)204 2391 y([16])40 b Fa(C.)32 b(G)569 2384 y(\023)569 2391 y(erard,)f(A.)g(Mar)-6 b(tinez)p Ft(,)29 b Fi(Pr)l(olongement)h(m)n(\023)-40 b(er)l(omorphe)32 b(de)e(la)h(matric)l(e)g(de)f(sc)l(attering)g(p)l (our)h(des)374 2491 y(pr)l(obl)n(\022)-40 b(emes)31 b(\022)-42 b(a)31 b(deux)e(c)l(orps)h(\022)-42 b(a)31 b(longue)f(p)l(ort)n(\023) -40 b(ee,)28 b Ft(Ann.)g(I.H.P)-7 b(.,)28 b(Ph)n(ys.)f(Th)n(\023)-39 b(eor.)26 b(51,)g(No.1,)h(81-110,)e(\(1989\).)204 2657 y([17])40 b Fa(C.)28 b(G)565 2650 y(\023)565 2657 y(erard,)f(A.)f(Mar) -6 b(tinez,)29 b(D.)d(R)n(ober)-6 b(t)p Ft(,)26 b Fi(Br)l(eit-Wigner)g (formulas)h(for)g(the)g(sc)l(attering)f(phase)i(and)374 2756 y(the)j(total)f(sc)l(attering)h(cr)l(oss)f(se)l(ction)h(in)f(the)h (semiclassic)l(al)h(limit,)e Ft(Comm)n(un.)e(Math.)g(Ph)n(ys.)g(121,)f (No.2,)374 2856 y(323-336,)e(\(1989\).)204 3022 y([18])40 b Fa(F.)32 b(Gesztesy,)e(K.)h(A.)g(Makar)n(o)n(v,)f(S.)h(N.)g(Nabok)n (o)p Ft(,)26 b Fi(The)31 b(sp)l(e)l(ctr)l(al)f(shift)h(op)l(er)l(ator,) e Ft(Op)r(er.)e(Theory)-7 b(,)374 3122 y(Adv.)28 b(Appl.)h(108,)d (59-90)f(\(1999\).)204 3288 y([19])40 b Fa(I.)31 b(C.)h(Gohber)n(g,)g (M.)f(G.)g(Krein)d Fi(Intr)l(o)l(duction)i(to)g(the)g(the)l(ory)h(of)g (line)l(ar)g(nonselfadjoint)h(op)l(er)l(ators,)374 3387 y Ft(T)-7 b(ranslations)26 b(of)i(Mathematical)f(Monographs,)f(V)-7 b(ol.)27 b(18.)g(AMS.)h(Pro)n(vidence,)e(R.I.)i(\(1969\).)204 3553 y([20])40 b Fa(L.)33 b(Guillop)823 3546 y(\023)823 3553 y(e)p Ft(,)d Fi(Une)h(formule)h(de)g(tr)l(ac)l(e)f(p)l(our)h(l'op) n(\023)-40 b(er)l(ateur)33 b(de)f(Schr\177)-42 b(odinger)33 b(dans)f Fl(R)3159 3523 y Fp(n)3210 3553 y Fi(,)e Ft(th)n(\022)-39 b(ese)29 b(de)h(do)r(c-)374 3653 y(torat,)d(Univ)n(ersit)n(\023)-39 b(e)26 b(de)i(Grenoble)f(\(1981\).)204 3819 y([21])40 b Fa(L.)e(Guillop)828 3812 y(\023)828 3819 y(e,)h(M.)f(Zw)n(orski)p Ft(,)d Fi(Sc)l(attering)h(asymptotics)h(for)g(R)n(iemann)e(surfac)l (es,)i Ft(Ann.)e(of)f(Math.,)374 3919 y(145,)27 b(597-660)d(\(1997\).) 204 4085 y([22])40 b Fa(B.)28 b(Helffer,)h(J.)e(Sj)1080 4078 y(\177)1077 4085 y(ostrand)p Ft(,)d Fi(R)n(\023)-40 b(esonanc)l(es)27 b(en)g(limite)g(semi-classique,)g Ft(Bull.)d(S.M.F.,) i(m)n(\023)-39 b(emoire)22 b(No)374 4184 y(24/25)k(\(1986\).)204 4350 y([23])40 b Fa(M.)f(Hitrik,)g(I.)f(Pol)-6 b(ter)n(o)n(vich)p Ft(,)37 b Fi(R)l(e)l(gularize)l(d)f(tr)l(ac)l(es)g(and)h(T)-6 b(aylor)37 b(exp)l(ansions)g(for)g(the)f(he)l(at)h(semi-)374 4450 y(gr)l(oup,)29 b Ft(preprin)n(t,)e(\(2001\).)204 4616 y([24])40 b Fa(H.)34 b(Isozaki,)g(H.)g(Kit)-6 b(ad)n(a)35 b Ft(,)c Fi(Mo)l(di\014e)l(d)k(wave)e(op)l(er)l(ators)h(with)f(time)g (indep)l(endant)g(mo)l(di\014ers,)g Ft(J.)d(F)-7 b(ac.)374 4716 y(Sci.,)28 b(Univ.)g(T)-7 b(oky)n(o,)26 b(Sect.)i(I)g(A)g(32,)f (77-104)e(\(1985\).)204 4882 y([25])40 b Fa(L.S.)32 b(K)n(oplienk)n(o)p Ft(,)c Fi(T)-6 b(r)l(ac)l(e)32 b(formula)g(for)g(non)e(tr)l(ac)l (e-class)i(p)l(erturb)l(ations,)e Ft(Sib.)f(Math.)g(J.)g(25,)f(735-743) 374 4981 y(\(1984\).)204 5147 y([26])p 374 5160 368 4 v 408 w(,)i Fi(R)l(e)l(gularize)l(d)i(sp)l(e)l(ctr)l(al)g(shift)h (function)e(for)i(one)f(dimensional)h(Schr\177)-42 b(odinger)34 b(op)l(er)l(ator)e(with)374 5247 y(slow)t(ly)g(de)l(cr)l(e)l(asing)e(p) l(otential,)f Ft(Sib.)f(Math.)g(J.)f(26,)g(365-369)e(\(1985\).)1934 5540 y(32)p eop %%Page: 33 33 33 32 bop 204 531 a Ft([27])40 b Fa(M.)22 b(G.)g(Krein)p Ft(,)f Fi(Perturb)l(ation)h(determinants)g(and)g(a)g(formula)h(for)g (the)e(tr)l(ac)l(e)h(of)h(unitary)f(and)g(selfadjoint)374 631 y(op)l(er)l(ators)p Ft(,)29 b(So)n(viet)e(Math.)h(Dokl.)g(3,)f (707-710)d(\(1962\).)204 797 y([28])40 b Fa(A.)j(Melin)p Ft(,)f Fi(T)-6 b(r)l(ac)l(e)40 b(distributions)h(asso)l(ciate)l(d)h(to) e(the)g(Schr\177)-42 b(odinger)42 b(op)l(er)l(ator,)h Ft(J.)c(Anal.)g(Math.)g(59,)374 897 y(133-160)25 b(\(1992\).)204 1063 y([29])40 b Fa(W.)33 b(M)611 1056 y(\177)608 1063 y(uller)p Ft(,)d Fi(R)l(elative)i(zeta)g(functions,)h(r)l(elative)f (determinants)g(and)g(sc)l(attering)g(the)l(ory,)f Ft(Comm)n(un.)374 1162 y(Math.)d(Ph)n(ys.)f(192)f(\(2\),)i(309-347)c(\(1998\).)204 1328 y([30])40 b Fa(V.)35 b(Petk)n(o)n(v,)h(G.)f(Popo)n(v)p Ft(,)e Fi(Asymptotic)h(b)l(ehaviour)h(of)g(the)f(sc)l(attering)f(phase) i(for)g(non-tr)l(apping)f(ob-)374 1428 y(stacles,)29 b Ft(Ann.)f(Inst,)g(F)-7 b(ourier)27 b(32,)g(No.3,)g(111-149)d (\(1982\).)204 1594 y([31])40 b Fa(V.)35 b(Petk)n(o)n(v,)h(M.)f(Zw)n (orski)p Ft(,)d Fi(Br)l(eit-Wigner)i(appr)l(oximation)i(and)e(the)f (distribution)h(of)g(r)l(esonanc)l(es,)374 1694 y Ft(Comm)n(un.)25 b(Math.)h(Ph)n(ys.)e(204)f(\(2\),)j(329-351)c(\(1999\),)i(and)h Fi(Err)l(atum,)g Ft(Comm)n(un.)g(Math.)g(Ph)n(ys.)f(214)g(\(3\),)374 1793 y(733-735)h(\(2000\).)204 1959 y([32])40 b Fa(A.)k(Pushnitski)p Ft(,)e Fi(Sp)l(e)l(ctr)l(al)f(shift)h(function)f(for)h(the)f(Schr\177) -42 b(odinger)43 b(op)l(er)l(ator)f(in)f(the)g(lar)l(ge)h(c)l(oupling) 374 2059 y(c)l(onstant)29 b(limit,)g Ft(Comm.)f(in)g(P)-7 b(.D.E.,)27 b(25)g(\(3)g(&)g(4\),)h(703-736)c(\(2000\).)204 2225 y([33])40 b Fa(D.)30 b(R)n(ober)-6 b(t)p Ft(,)27 b Fi(Asymptotique)i(de)g(la)g(phase)h(de)f(di\013usion)g(\022)-42 b(a)29 b(haute)d(\023)-40 b(ener)l(gie)29 b(p)l(our)g(des)g(p)l(erturb) l(ations)f(du)374 2325 y(se)l(c)l(ond)j(or)l(dr)l(e)f(du)g(L)l (aplacien,)g Ft(Ann.)e(Sci.)1736 2304 y(\023)1729 2325 y(Ecole)e(Norm.)h(Sup.,)i(4)n(\022)-39 b(eme)26 b(s)n(\023)-39 b(erie,)26 b(No.)h(25,)g(107-134)d(\(1992\).)204 2491 y([34])p 374 2504 368 4 v 408 w(,)g Fi(R)l(elative)j(time)g(delay)h (for)f(p)l(erturb)l(ations)f(of)h(el)t(liptic)i(op)l(er)l(ators)e(and)g (semiclassic)l(al)h(asymp-)374 2590 y(totics,)h Ft(J.)e(F)-7 b(unct.)29 b(Anal.)e(126,)g(No.1,)g(36-82)e(\(1994\).)204 2756 y([35])p 374 2769 V 408 w(,)31 b Fi(Semiclassic)l(al)k (asymptotics)f(for)f(the)g(sp)l(e)l(ctr)l(al)g(shift)g(function)g(,)f Ft(Buslaev,)f(V.)g(\(ed\))g(et)g(al.,)374 2856 y(Di\013eren)n(tial)25 b(op)r(erators)e(and)i(sp)r(ectral)f(theory)-7 b(.)25 b(M.)g(Sh.)g(Birman's)f(70th)h(anniv)n(ersary)d(collection.)i(Pro)n(v-) 374 2956 y(idence,)k(RI.)g(AMS)g(T)-7 b(ransl,)27 b(Ser.2,)g(189)f (\(41\),)h(187-203)e(\(1999\).)204 3122 y([36])40 b Fa(A.)f(R)-7 b(ybkin)p Ft(,)38 b Fi(On)f(a)g(tr)l(ac)l(e)g(formula)h(of)g(the)f (Buslaev-F)-6 b(adde)l(ev)40 b(typ)l(e)d(for)h(a)f(long)h(r)l(ange)f(p) l(otential,)i Ft(J.)374 3221 y(Math.)28 b(Ph)n(ys)f(40,)g(No.3,)g (1334-1343)c(\(1999\).)204 3387 y([37])40 b Fa(J.)31 b(T.)h(Schw)-7 b(ar)h(tz)p Ft(,)27 b Fi(Non)j(line)l(ar)g(functional)g (analysis,)g Ft(Gordon)d(&)g(Breac)n(h,)f(\(1969\).)204 3553 y([38])40 b Fa(B.)26 b(Simon)c Fi(T)-6 b(r)l(ac)l(e)26 b(ide)l(als)g(and)g(their)f(applic)l(ations)p Ft(,)h(Cam)n(bridge)c (Univ)n(ersit)n(y)f(Press,)h(Cam)n(bridge)f(\(1979\).)204 3719 y([39])40 b Fa(J.)31 b(R.)h(T)-7 b(a)h(ylor)p Ft(,)27 b Fi(Sc)l(attering)j(the)l(ory,)e Ft(John)f(Wiley)h(&)g(Sons,)f(Inc.)h (\(1972\).)204 3885 y([40])40 b Fa(D.)35 b(Y)-7 b(af)g(aev)p Ft(,)33 b Fi(Mathematic)l(al)i(sc)l(attering)f(the)l(ory.)g(Gener)l(al) g(the)l(ory,)g Ft(v)n(ol.)d(105,)g(A.M.S.)i(Rho)r(de)f(Island)374 3985 y(\(1992\).)204 4151 y([41])p 374 4164 V 408 w(,)25 b Fi(The)j(sc)l(attering)f(amplitude)h(for)g(the)f(Schr\177)-42 b(odinger)29 b(e)l(quation)e(with)h(a)f(long-r)l(ange)h(p)l(otential,) 374 4251 y Ft(Comm)n(un.)g(Math.)g(Ph)n(ys.)e(191)h(\(1\),)g(183-218)e (\(1998\).)1934 5540 y(33)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0212101300914--