Content-Type: multipart/mixed; boundary="-------------0306200736502" This is a multi-part message in MIME format. ---------------0306200736502 Content-Type: text/plain; name="03-294.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-294.keywords" Schr\"odinger operators, scattering theory ---------------0306200736502 Content-Type: application/postscript; name="lap_final_20_63.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lap_final_20_63.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: lap_final_20_63.dvi %%Pages: 45 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o lap_final_20_63.ps lap_final_20_63 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2003.06.20:1433 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 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Ft(B)s Fu(\()p Ft(z)746 782 y Fs(2)765 775 y Fu(\))p Fp(k)j(\024)f Ft(C)t Fp(j)p Ft(z)951 782 y Fs(1)981 775 y Fp(\000)e Ft(z)1054 782 y Fs(2)1073 775 y Fp(j)1087 754 y Fm(\015)1123 775 y Fu(for)j(all)g Ft(z)1284 782 y Fs(1)1303 775 y Ft(;)8 b(z)1348 782 y Fs(2)1381 775 y Fp(2)14 b Fu(\000)1458 782 y Fm(\022)1478 775 y Ft(:)278 863 y Fu(A)d(main)f(result)h(of)g(the)h(presen)o(t)e(pap)q(er) i(is)g(the)f(follo)o(wing)g Fh(limiting)j(absorption)228 921 y(principle)20 b Fu(at)f(zero)g(energy)g(for)g(a)h(Sc)o(hr\177)-24 b(odinger)19 b(op)q(erator)h Ft(H)j Fu(=)c Fp(\000)p Fu(\001)12 b(+)h Ft(V)31 b Fu(on)228 979 y Fp(H)14 b Fu(=)g Ft(L)369 961 y Fs(2)389 979 y Fu(\()p Fn(R)447 961 y Fm(d)465 979 y Fu(\).)21 b(W)l(e)16 b(recall)f(the)i(notation)g Ft(R)p Fu(\()p Ft(\020)t Fu(\))d(=)g(\()p Ft(H)i Fp(\000)11 b Ft(\020)t Fu(\))1348 961 y Fo(\000)p Fs(1)1395 979 y Fu(.)22 b(Notice)15 b(that)i(it)228 1037 y(is)h(enough)i(to)f(imp)q 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Fm(\026)1366 1381 y Fu(;)22 b Fp(h)p Ft(x)p Fp(i)14 b Fu(=)1534 1340 y Fp(p)p 1575 1340 132 2 v 1575 1381 a Fu(1)e(+)f Ft(x)1688 1367 y Fs(2)1707 1381 y Fh(.)278 1439 y Fu(\(2\))21 b Fh(F)l(or)c(al)r(l)h Ft(\013)d Fp(2)f Fu(\()p Fn(N)e Fp([)f(f)p Fu(0)p Fp(g)p Fu(\))817 1421 y Fm(d)855 1439 y Fh(ther)n(e)17 b(exists)h Ft(C)1144 1446 y Fm(\013)1183 1439 y Ft(>)c Fu(0)j Fh(such)h(that)728 1532 y Fp(h)p Ft(x)p Fp(i)794 1511 y Fm(\026)p Fs(+)p Fo(j)p Fm(\013)p Fo(j)888 1532 y Fp(j)p Ft(@)931 1511 y Fm(\013)955 1532 y Ft(V)983 1539 y Fs(1)1003 1532 y Fu(\()p Ft(x)p Fu(\))p Fp(j)13 b(\024)h Ft(C)1184 1539 y Fm(\013)1208 1532 y Ft(:)278 1620 y Fu(\(3\))21 b Fh(Ther)n(e)c(exists)h Ft(\017)654 1627 y Fs(2)687 1620 y Ft(>)c Fu(0)k Fh(such)g(that)f Fp(\000j)p Ft(x)p Fp(j)1086 1602 y Fo(\000)p Fs(2)1141 1620 y Fu(\()p Ft(x)11 b Fp(\001)g(r)p Fu(\()p Fp(j)p Ft(x)p Fp(j)1341 1602 y Fs(2)1359 1620 y Ft(V)1387 1627 y Fs(1)1407 1620 y Fu(\)\))j Fp(\025)g(\000)p Ft(\017)1571 1627 y Fs(2)1590 1620 y Ft(V)1618 1627 y Fs(1)1638 1620 y Fh(.)278 1678 y Fu(\(4\))21 b Ft(V)389 1685 y Fs(2)409 1678 y Fu(\()p Fp(\000)p Fu(\001)10 b(+)h Ft(i)p Fu(\))603 1660 y Fo(\000)p Fs(1)667 1678 y Fh(is)18 b(a)f(c)n(omp)n(act)g(op)n (er)n(ator)f(on)i Ft(L)1245 1660 y Fs(2)1265 1678 y Fu(\()p Fn(R)1323 1660 y Fm(d)1340 1678 y Fu(\))p Fh(.)278 1736 y Fu(\(5\))j Fh(Ther)n(e)c(exists)h Ft(\016)o(;)8 b(C)q(;)g(R)14 b(>)f Fu(0)18 b Fh(such)g(that)730 1828 y Fp(j)p Ft(V)772 1835 y Fs(2)792 1828 y Fu(\()p Ft(x)p Fu(\))p Fp(j)13 b(\024)h Ft(C)t Fp(j)p Ft(x)p Fp(j)1033 1808 y Fo(\000)p Fs(1)p Fo(\000)p Fm(\026=)p Fs(2)p Fo(\000)p Fm(\016)1207 1828 y Ft(;)361 1917 y Fh(for)j Fp(j)p Ft(x)p Fp(j)c Ft(>)h(R)p Fh(.)278 1975 y Fu(\(6\))21 b Ft(V)31 b Fh(satis\014es)21 b(unique)h(c)n(ontinuation)f(at)f(in\014nity)h(\(se)n(e)g(Assumption)f (2.1)361 2033 y(in)d(Se)n(ction)i(2\).)278 2106 y(Then)i(for)g(al)r(l)h Ft(s)f Fp(2)f Fu(\()684 2087 y Fs(1)p 684 2095 18 2 v 684 2123 a(2)721 2106 y Fu(+)777 2084 y Fm(\026)p 777 2095 22 2 v 779 2123 a Fs(4)803 2106 y Ft(;)830 2087 y Fs(1)p 830 2095 18 2 v 830 2123 a(2)867 2106 y Fu(+)923 2084 y Fm(\026)p 923 2095 22 2 v 925 2123 a Fs(4)963 2106 y Fu(+)14 b Ft(\016)r Fu(\))20 b Fh(and)i(al)r(l)g Fu(0)f Ft(<)f(\022)i(<)e(\031)j Fh(the)e(family)228 2166 y(of)i(op)n(er)n(ators)e Ft(B)s Fu(\()p Ft(\020)t Fu(\))i(=)h Fp(h)p Ft(x)p Fp(i)762 2148 y Fo(\000)p Fm(s)809 2166 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Fp(h)p Ft(x)p Fp(i)975 2148 y Fo(\000)p Fm(s)1044 2166 y Fh(is)f(uniformly)g(H\177)-25 b(older)23 b(c)n(ontinuous)228 2224 y(in)d Fu(\000)320 2231 y Fm(\022)340 2224 y Fh(.)31 b(F)l(or)20 b Ft(s)f(>)g Fu(3)p Ft(=)p Fu(2\(1)c(+)e Ft(\026=)p Fu(2\))22 b Fh(\(in)e(the)h(c)n (ase)f Ft(\016)h(>)e Fu(\(1)14 b(+)f Ft(\026=)p Fu(2)p Fh(\)\))21 b(the)g(H\177)-25 b(older)228 2282 y(exp)n(onent)21 b(may)e(b)n(e)h(chosen)h(to)e(b)n(e)h Ft(\015)h Fu(=)d(1)p Fh(,)i(and)g(for)f Ft(s)f Fp(\024)g Fu(3)p Ft(=)p Fu(2\(1)c(+)e Ft(\026=)p Fu(2\))21 b Fh(to)f(b)n(e)228 2340 y(any)f Ft(\015)i(<)d(s)p Fu(\(1)13 b(+)g Ft(\026=)p Fu(2\))651 2322 y Fo(\000)p Fs(1)712 2340 y Fp(\000)f Fu(1)p Ft(=)p Fu(2)p Fh(.)30 b(In)20 b(p)n(articular)f(ther)n(e)h(exists)g Ft(C)1465 2347 y Fm(s;\022)1529 2340 y Ft(>)e Fu(0)i Fh(such)228 2398 y(that)659 2486 y Fu(sup)656 2527 y Fm(\020)r Fo(2)p Fs(\000)720 2533 y Fi(\022)745 2444 y Fl(\015)745 2474 y(\015)773 2486 y Fp(h)p Ft(x)p Fp(i)839 2466 y Fo(\000)p Fm(s)885 2486 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Fp(h)p Ft(x)p Fp(i)1051 2466 y Fo(\000)p Fm(s)1098 2444 y Fl(\015)1098 2474 y(\015)1139 2486 y Fp(\024)14 b Ft(C)1227 2493 y Fm(s;\022)1272 2486 y Ft(;)336 b Fu(\(1.4\))228 2608 y Fh(and)17 b(the)h(limits)965 2658 y Fq(3)p eop %%Page: 4 4 4 3 bop 479 291 a Fp(h)p Ft(x)p Fp(i)545 270 y Fo(\000)p Fm(s)591 291 y Ft(R)p Fu(\(0)12 b(+)f Ft(i)p Fu(0\))p Fp(h)p Ft(x)p Fp(i)858 270 y Fo(\000)p Fm(s)918 291 y Fp(\021)61 b Fu(lim)971 322 y Fm(\020)r Fo(!)p Fs(0)p Fm(;\020)r Fo(2)p Fs(\000)1116 328 y Fi(\022)1132 291 y Fp(h)p Ft(x)p Fp(i)1198 270 y Fo(\000)p Fm(s)1245 291 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Fp(h)p Ft(x)p Fp(i)1411 270 y Fo(\000)p Fm(s)1457 291 y Ft(;)464 421 y Fp(h)p Ft(x)p Fp(i)530 400 y Fo(\000)p Fm(s)577 421 y Ft(R)p Fu(\(0)12 b Fp(\000)f Ft(i)p Fu(0\))17 b Fp(h)p Ft(x)p Fp(i)862 400 y Fo(\000)p Fm(s)922 421 y Fp(\021)61 b Fu(lim)975 452 y Fm(\020)r Fo(!)p Fs(0)p Fm(;\020)r Fo(2)p Fs(\000)1120 458 y Fi(\022)1136 421 y Fp(h)p Ft(x)p Fp(i)1202 400 y Fo(\000)p Fm(s)1248 421 y Ft(R)1294 381 y Fl(\000)1321 408 y Fu(\026)1317 421 y Ft(\020)1342 381 y Fl(\001)1373 421 y Fp(h)p Ft(x)p Fp(i)1439 400 y Fo(\000)p Fm(s)228 529 y Fh(exist)18 b(in)g Fp(B)r Fu(\()p Ft(L)488 511 y Fs(2)507 529 y Fu(\()p Fn(R)565 511 y Fm(d)582 529 y Fu(\)\))p Fh(.)278 612 y Fu(F)l(rom)c(the)i(facts)f(that)h Ft(V)27 b Fu(is)16 b(negativ)o(e)f(at)h(in\014nit)o(y)e(and)i(deca)o (ys)g(slo)o(w)o(er)f(than)228 670 y Ft(r)251 652 y Fo(\000)p Fs(2)317 670 y Fu(it)i(follo)o(ws,)i(cf.)27 b([ReSi78)o(,)19 b(Theorem)e(XI)q(I)q(I.6)g(\(V)l(ol.)27 b(IV,)17 b(p.)27 b(87\)],)19 b(that)g Ft(H)228 728 y Fu(has)d(in\014nitely)d(man)o(y)h (negativ)o(e)g(eigen)o(v)m(alues)g(accum)o(ulating)f(at)j(zero.)k (Clearly)228 786 y(b)o(y)h(\(1.4\))q(,)h(zero)f(is)g(not)h(an)g(eigen)o (v)m(alue)e(of)i Ft(H)t Fu(.)37 b(\(See)20 b(Section)h(2)h(for)g(a)g (more)228 844 y(general)16 b(result.\))278 902 y(W)l(e)j(also)g(get)h (existence)d(of)j(limits)d(for)i(p)q(o)o(w)o(ers)h(of)f(the)g(resolv)o (en)o(t.)29 b(The)20 b(a-)228 960 y(symptotic)9 b(expansion)j(\(1.2\))f (is)g(an)h(easy)f(consequence)g(of)g(Theorem)f(1.2)i(b)q(elo)o(w.)228 1042 y Fy(Theorem)k(1.2)j Fu(\(Iterated)c(resolv)o(en)o(ts\))p Fy(.)23 b Fh(L)n(et)18 b Ft(V)27 b Fu(=)16 b Ft(V)1249 1049 y Fs(1)1281 1042 y Fu(+)c Ft(V)1359 1049 y Fs(2)1397 1042 y Fh(satisfy)19 b(the)g(c)n(on-)228 1101 y(ditions)e(in)h(The)n (or)n(em)f(1.1)g(with)h Fu(\(5\))f Fh(r)n(eplac)n(e)n(d)g(by)240 1169 y Fu(\(5'\))j(supp)q(\()p Ft(V)484 1176 y Fs(2)504 1169 y Fu(\))d Fh(is)h(c)n(omp)n(act.)228 1237 y(L)n(et)e Ft(H)i Fu(=)c Fp(\000)p Fu(\001)8 b(+)h Ft(V)28 b Fh(and)16 b Ft(R)p Fu(\()p Ft(\020)t Fu(\))f(=)e(\()p Ft(H)h Fp(\000)8 b Ft(\020)t Fu(\))1035 1219 y Fo(\000)p Fs(1)1099 1237 y Fh(b)n(e)17 b(given)h(as)e(b)n(efor)n(e)g(and)h Ft(k)r Fu(\()p Ft(x)p Fu(\))c(=)228 1297 y Fp(h)p Ft(x)p Fp(i)294 1279 y Fs(1+)p Fm(\026=)p Fs(2)398 1297 y Fh(,)19 b Ft(x)c Fp(2)h Fn(R)563 1279 y Fm(d)581 1297 y Fh(.)25 b(L)n(et)19 b Ft(m)c Fp(2)h Fn(N)p Fh(,)k Ft(\022)d Fp(2)f Fu(\(0)p Ft(;)8 b(\031)r Fu(\))18 b Fh(and)h Ft(\017)d(>)g Fu(0)p Fh(.)26 b(Then)19 b(ther)n(e)f(exists)228 1355 y Ft(C)f(>)d Fu(0)k Fh(such)g(that)581 1391 y Fl(\015)581 1421 y(\015)609 1433 y Ft(k)636 1413 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)846 1433 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))946 1413 y Fm(m)980 1433 y Ft(k)1007 1413 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1217 1391 y Fl(\015)1217 1421 y(\015)1259 1433 y Fp(\024)13 b Ft(C)q(;)261 b Fu(\(1.5\))228 1512 y Fh(for)17 b(al)r(l)h Ft(\020)g Fp(2)c Fu(\000)491 1519 y Fm(\022)511 1512 y Fh(.)228 1570 y(F)l(urthermor)n(e,)i(the)i (function)613 1650 y Ft(\020)g Fp(7!)13 b Ft(k)742 1629 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)953 1650 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))1053 1629 y Fm(m)1086 1650 y Ft(k)1113 1629 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1324 1650 y Ft(;)228 1731 y Fh(extends)19 b(to)e(a)h(c)n (ontinuous)g(function)h(on)p 1009 1691 31 2 v 18 w Fu(\000)1039 1738 y Fm(\022)1076 1731 y Fh(\(the)f(closur)n(e)g(of)f Fu(\000)1428 1738 y Fm(\022)1448 1731 y Fh(\).)228 1813 y Fy(Remarks)f(1.3.)24 b Fu(1\))d(F)l(or)h(\014xed)g Ft(m)i Fp(2)g Fn(N)f Fu(the)f(b)q(ound)h Ft(V)1311 1820 y Fs(2)1355 1813 y Fu(=)h Ft(O)q Fu(\()p Ft(k)1501 1795 y Fo(\000)p Fm(m)p Fo(\000)p Fm(\017)1604 1813 y Fu(\))e(suf-)292 1871 y(\014ces)16 b(for)h(\(1.5\).)22 b(\(This)16 b(follo)o(ws)h (readily)e(from)h(our)g(pro)q(of)i(b)o(y)e(a)h(little)e(more)292 1929 y(b)q(o)q(okk)o(eeping.\))228 1987 y(2\))21 b(In)12 b(dimension)e Ft(d)k Fu(=)g(1)f(one)f(ma)o(y)e(sho)o(w)j(b)o(y)f (WKB-analysis,)g(see)g(for)g(example)292 2045 y([Y)l(af82],)21 b(that)h(\(1.5\))f(is)f(optimal)g(in)g(the)h(follo)o(wing)g(sense:)30 b(There)21 b(exists)292 2103 y Ft(\036)14 b Fp(2)g Ft(L)415 2085 y Fm(d)435 2103 y Fu(\()p Fn(R)493 2085 y Fm(d)510 2103 y Fu(\))i(suc)o(h)g(that)663 2186 y(sup)659 2227 y Fm(\020)r Fo(2)p Fs(\000)723 2233 y Fi(\022)748 2143 y Fl(\015)748 2173 y(\015)776 2186 y Ft(k)803 2165 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))972 2186 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))1072 2165 y Fm(m)1105 2186 y Ft(\036)1134 2143 y Fl(\015)1134 2173 y(\015)1176 2186 y Fu(=)e Fp(1)p Ft(:)292 2310 y Fu(\(This)26 b(ma)o(y)f(b)q(e)i (done)g(using)g Ft(R)p Fu(\()p Ft(\020)t Fu(\))990 2292 y Fm(m)1055 2310 y Fu(=)k(\(\()p Ft(m)17 b Fp(\000)h Fu(1\)!\))1355 2292 y Fo(\000)p Fs(1)1416 2291 y Fm(d)1434 2279 y Fi(m)p Fg(\000)p Fk(1)p 1407 2299 105 2 v 1407 2327 a Fm(d\020)1443 2318 y Fi(m)p Fg(\000)p Fk(1)1517 2310 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))27 b(and)292 2375 y(the)21 b(standard)h(form)o(ula)e(for)i Ft(R)p Fu(\()p Ft(\020)t Fu(\))f(in)g(terms)f(of)i(outgoing/incoming)f(ge-)292 2433 y(neralized)f(eigenfunctions;)i(analysing)f(the)g(large)g Ft(x)p Fu(-asymptotics)f(of)h(the)292 2492 y(\()p Ft(m)12 b Fp(\000)g Fu(1\)'th)18 b(deriv)m(ativ)o(e)e(w.r.t.)26 b Ft(\020)c Fu(of)c(those)h(functions)f(yields)e(the)i(result.\))292 2550 y(W)l(e)c(w)o(ould)g(exp)q(ect)f(that)i(the)f(same)f(result)h(is)f (true)h(in)g(an)o(y)g(dimension,)f(also)292 2608 y(for)j(p)q(oten)o (tials)h(that)f(are)h(not)f(radially)g(symmetri)o(c.)965 2658 y Fq(4)p eop %%Page: 5 5 5 4 bop 278 213 a Fu(Due)16 b(to)h(Theorem)d(1.1)j(w)o(e)f(ma)o(y)f (de\014ne)391 301 y Ft(E)430 281 y Fo(0)441 301 y Fu(\(+0\))g(=)o(\(2)p Ft(\031)r(i)p Fu(\))702 281 y Fo(\000)p Fs(1)757 301 y Fu(\()p Ft(R)p Fu(\(0)d(+)f Ft(i)p Fu(0\))g Fp(\000)g Ft(R)p Fu(\(0)h Fp(\000)f Ft(i)p Fu(0\)\))j Fp(2)g(B)r Fu(\()p Fp(H)1416 308 y Fs(1)1435 301 y Ft(;)8 b Fp(H)1499 308 y Fs(2)1518 301 y Fu(\);)593 385 y Fp(H)635 392 y Fs(1)669 385 y Fu(=)14 b Ft(k)748 364 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)872 385 y Ft(L)905 364 y Fs(2)925 385 y Fu(\()p Fn(R)983 364 y Fm(d)1000 385 y Fu(\))p Ft(;)22 b Fp(H)1097 392 y Fs(2)1130 385 y Fu(=)14 b Ft(k)1209 364 y Fs(1)p Fm(=)p Fs(2+)p Fm(\017)1306 385 y Ft(L)1339 364 y Fs(2)1359 385 y Fu(\()p Fn(R)1417 364 y Fm(d)1434 385 y Fu(\))p Ft(:)228 477 y Fu(Motiv)m(ated)i(b)o(y)g (the)g(next)f(result)h(w)o(e)g(shall)g(pro)o(v)o(e)f(\(in)h(Section)g (5\))h(that)806 565 y Ft(E)845 545 y Fo(0)857 565 y Fu(\(+0\))d Fp(6)p Fu(=)g(0)p Ft(:)561 b Fu(\(1.6\))228 654 y(Let)12 b Ft(F)7 b Fu(\()p Fp(j)p Ft(x)p Fp(j)13 b Ft(<)h(C)t Fu(\))e(denote)g(the)g(m)o(ultiplic)o(ation)e(op)q(erator)k(b)o(y)e (the)g(c)o(haracteristic)228 712 y(function)k(of)g Fp(f)p Ft(x)p Fp(j)e(j)p Ft(x)p Fp(j)f 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1467 y Fp(k)14 b Fu(=)f Ft(O)q Fu(\()p Ft(t)1283 1446 y Fo(\000)p Fs(\(1+)p Fm(\017)1383 1435 y Fg(0)1395 1446 y Fs(\))1414 1433 y Fk(1)p 1414 1439 16 2 v 1414 1459 a(2)1436 1467 y Fu(\);)153 b(\(1.7\))373 1547 y Ft(\024)14 b Fu(=)f(\(1)f Fp(\000)f Ft(\017)p Fu(\)\(1)g(+)g Ft(\026=)p Fu(2\))809 1527 y Fo(\000)p Fs(1)857 1547 y Ft(:)228 1643 y Fy(Remarks)16 b(1.5.)24 b Fu(1\))d(By)11 b(time)e(rev)o(ersal)i(in)o(v)m(ariance)g (there)g(are)g(similar)f(b)q(ounds)292 1701 y(for)16 b Ft(t)e Fp(!)g(\0001)p Fu(.)228 1759 y(2\))21 b(Due)15 b(to)g(\(1.6\))f(and)i(Theorem)d(1.4)i(\(i\))f(the)g(b)q(est)h(one)g (could)f(hop)q(e)i(for)e(to)h(the)292 1817 y(righ)o(t)i(in)f(\(1.7\))h (w)o(ould)g(b)q(e)g(the)g(b)q(ound)h Ft(O)q Fu(\()p Ft(t)1106 1799 y Fo(\000)p Fs(1)1154 1817 y Fu(\))f(\(for)g Ft(f)5 b Fu(\(0\))16 b Fp(6)p Fu(=)f(0\).)23 b(Moreo)o(v)o(er)292 1875 y(w)o(e)17 b(w)o(ould)h(exp)q(ect)f(that)h Ft(\024)f Fu(=)f(\(1)d(+)f Ft(\026=)p Fu(2\))1073 1857 y Fo(\000)p Fs(1)1139 1875 y Fu(is)17 b(indeed)g(the)h(b)q(orderline)f(for)292 1934 y(this)e(kind)g(of)g(lo)o(w)g(energy)l(,)g(minim)o(al)e(v)o(elo)q (cit)o(y)g(estimate;)g(see)i(Theorem)f(4.7)292 1992 y(for)i(an)h (analogous)h(b)q(ound)g(in)e(classical)f(mec)o(hanics.)228 2050 y(3\))21 b(If)e Ft(V)372 2057 y Fs(2)413 2050 y Fp(2)h Ft(C)505 2032 y Fo(1)501 2062 y Fs(0)542 2050 y Fu(\()p Fn(R)600 2032 y Fm(d)617 2050 y Fu(\))g(one)h(ma)o(y)d(tak)o (e)h Ft(f)26 b Fu(=)20 b(1)h(in)e(Theorem)g(1.4)h(\(i\))g(and)h (\(ii\).)292 2108 y(This)16 b(follo)o(ws)g(readily)f(from)g(the)h(giv)o (en)f(statemen)o(ts)f(and)j(w)o(ell-established)292 2166 y(high)e(energy)f(estimates,)f(see)h([Kit84,)g(Theorem)g(1.1],)g([CyP)o (e84,)h(Theorem)292 2224 y(1])23 b(or)g([Jen89,)h(Theorem)d(1.2)i (\(ii\)].)40 b(Some)21 b(lo)q(cal)i(singularities)f(ma)o(y)f(b)q(e)292 2282 y(included)14 b(\(see)g(for)h(example)e([Jen89,)i(Section)f(6])h (for)g(a)g(sp)q(eci\014c)f(treatmen)o(t)292 2340 y(of)i(the)g(purely)g 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y(in)k(the)h(high)g (energy)f(regime\),)f(cf.)28 b(the)19 b(seminal)e(w)o(ork)i([IsKi85)o (].)29 b(W)l(e)18 b(w)o(ould)228 503 y(hop)q(e)i(that)f(our)h (estimates)d(w)o(ould)j(op)q(en)f(up)h(for)f(results)g(on)h (asymptotics)e(of)228 561 y(scattering)e(quan)o(tities;)f(this)h(needs) g(elab)q(oration)h(elsewhere.)278 619 y(Let)23 b(us)h(also)h(men)o (tion)c(the)j(p)q(ersp)q(ectiv)o(e)e(of)i(asymptotics)f(at)h (thresholds)228 678 y(for)18 b(the)g(man)o(y-b)q(o)q(dy)g(problem.)25 b(This)18 b(app)q(ears,)h(although)g(v)o(ery)e(in)o(teresting,)228 736 y(far)c(more)f(am)o(bitious.)19 b(In)13 b(the)g(time-dep)q(enden)o (t)e(approac)o(h)j(to)g(the)f(asymptotic)228 794 y(completeness)d (problem)h(con)o(trol)i(of)f(the)h(dynamical)d(b)q(eha)o(viour)j(at)g (thresholds)228 852 y(is)i(imp)q(ortan)o(t.)20 b(W)l(e)c(w)o(ere)f (motiv)m(ated)f(to)i(lo)q(ok)g(at)g(the)g(ab)q(o)o(v)o(e)g(t)o(w)o(o-b) q(o)q(dy)g(prob-)228 910 y(lems)e(b)o(y)h(the)g(pap)q(ers)i([G)o(\023) -23 b(er93])15 b(and)i([Sk03])e(on)h(three-b)q(o)q(dy)h(asymptotic)d (com-)228 968 y(pleteness)g(with)i(long-range)g(p)q(oten)o(tials;)g (some)e(estimates,)g(similar)f(in)i(nature)228 1026 y(to)h(\(1.7\))q(,) f(pla)o(y)h(an)h(imp)q(ortan)o(t)e(role)h(in)g(these)g(pap)q(ers.)278 1084 y(T)l(o)f(minimi)o(ze)c(confusion,)16 b(let)e(us)h(men)o(tion)e(a) j(n)o(um)o(b)q(er)d(of)i(\(standard\))h(nota-)228 1142 y(tions)g(and)h(con)o(v)o(en)o(tions)e(that)i(will)e(b)q(e)h(used)h(in) f(the)g(pap)q(er.)278 1201 y(W)l(e)e(will)g(write)h Fp(<)p Ft(z)g Fu(=)f Fp(<)p Fu(\()p Ft(z)r Fu(\))p Ft(;)8 b Fp(=)p Ft(z)15 b Fu(=)f Fp(=)p Fu(\()p Ft(z)r Fu(\))h(for)g(the)g(real) f(and)i(imaginary)e(part)228 1259 y(of)f Ft(z)r Fu(,)g(b)q(oth)h(in)e (the)h(case)g(where)f Ft(z)j Fu(is)e(a)g(complex)d(n)o(um)o(b)q(er)h (and,)j(more)e(generally)l(,)228 1317 y(when)k(it)g(is)g(an)h(op)q (erator.)278 1375 y(W)l(e)f(de\014ne)g(for)g(an)o(y)g(op)q(en)h Ft(U)i Fp(\022)14 b Fn(R)931 1357 y Fm(d)304 1471 y Fp(B)339 1450 y Fo(1)376 1471 y Fu(\()p Ft(U)5 b Fu(\))14 b(=)517 1415 y Fl(n)551 1471 y Ft(f)19 b Fp(2)14 b Ft(C)680 1450 y Fo(1)717 1471 y Fu(\()p Ft(U)5 b Fu(\))801 1413 y Fl(\014)801 1443 y(\014)801 1473 y(\014)826 1471 y Ft(@)855 1450 y Fm(\013)879 1471 y Ft(f)20 b Fp(2)14 b Ft(L)1003 1450 y Fo(1)1040 1471 y Fu(\()p Ft(U)5 b Fu(\))17 b(for)f(all)g Ft(\013)e Fp(2)g Fu(\()p Fn(N)e Fp([)g(f)p Fu(0)p Fp(g)p Fu(\))1571 1450 y Fm(d)1591 1415 y Fl(o)1632 1471 y Ft(:)278 1577 y Fu(Apart)21 b(from)f(the)h(notation)h Fp(h)p Ft(x)p Fp(i)h Fu(=)983 1536 y Fp(p)p 1025 1536 132 2 v 41 x Fu(1)11 b(+)g Ft(x)1137 1562 y Fs(2)1156 1577 y Fu(,)22 b(used)g(ab)q(o)o(v)o(e,)g(w)o(e)f(will)f(also)228 1635 y(need)c(the)g(notation)h Ft(p)d Fu(=)g Fp(\000)p Ft(i)p Fp(r)h Fu(and)i Ft(A)c Fu(=)h(\()p Ft(x)d Fp(\001)g Ft(p)g Fu(+)g Ft(p)h Fp(\001)f Ft(x)p Fu(\))p Ft(=)p Fu(2.)278 1693 y(The)j(virial)e Ft(W)21 b Fu(of)15 b(the)e(p)q(oten)o(tial)h Ft(V)25 b Fu(will)13 b(pla)o(y)h(an)g(imp)q(ortan)o(t)f(role)h (through-)228 1751 y(out)j(the)f(pap)q(er.)21 b(It)16 b(is)g(de\014ned)g(b)o(y)759 1834 y Ft(W)21 b Fu(=)14 b Fp(\000)p Fu(2)p Ft(V)22 b Fp(\000)11 b Ft(x)g Fp(\001)g(r)p Ft(V)s(:)430 b Fu(\(1.8\))228 1919 y(W)l(e)19 b(recall)e(the)i (\(formal\))f(iden)o(tit)o(y)f Ft(i)p Fu([)p Ft(H)q(;)8 b(A)p Fu(])17 b(=)h(2)p Ft(H)g Fu(+)13 b Ft(W)7 b Fu(.)29 b(The)19 b(assumptions)228 1977 y(\(1\))d(and)h(\(3\))g(of)f(Theorem)f (1.1)i(in)f(the)g(case)g Ft(V)1092 1984 y Fs(2)1126 1977 y Fu(=)e(0)i(yield)774 2060 y Ft(W)7 b Fu(\()p Ft(x)p Fu(\))13 b Fp(\025)h Ft(\017)979 2067 y Fs(1)998 2060 y Ft(\017)1018 2067 y Fs(2)1037 2060 y Fp(h)p Ft(x)p Fp(i)1103 2040 y Fo(\000)p Fm(\026)1155 2060 y Ft(;)453 b Fu(\(1.9\))228 2143 y(so)17 b(in)f(particular,)f Ft(W)23 b Fu(is)16 b(p)q(ositiv)o(e)g(in)g(this)g(case.)278 2201 y(In)k(Section)g(2)i(w)o(e)e(will)g(pro)o(v)o(e)g(a)h(result)f(of)h (indep)q(enden)o(t)f(in)o(terest)g(\(Theo-)228 2259 y(rem)13 b(2.4\),)i(namely)e(the)h(non-existence)g(of)h(eigen)o(v)m(alue)f(at)h (zero)g(under)f(sligh)o(tly)228 2317 y(w)o(eak)o(er)i(assumptions)h (than)h(those)g(of)f(Theorem)f(1.1.)25 b(This)17 b(is)g(a)h(basic)f (result)228 2375 y(whic)o(h)i(will)f(allo)o(w)h(us)h(to)g(pro)o(v)o(e)e (Theorem)g(1.1)i(b)o(y)f(a)h(p)q(erturbativ)o(e)e(metho)q(d:)228 2433 y(First)f(w)o(e)g(shall)g(pro)o(v)o(e)g(\(1.4\))h(in)f(the)g(case) h Ft(V)1063 2440 y Fs(2)1099 2433 y Fu(=)e(0,)i(and)g(then)g(later)f (on)h(in)f(ge-)228 2492 y(neral)d(b)o(y)g(p)q(erturbation)h(theory)f (and)i(Theorem)d(2.4.)21 b(W)l(e)14 b(pro)o(v)o(e)g(Theorem)f(2.4)228 2550 y(b)o(y)g(extending)f(the)h(metho)q(d)g(of)g(pro)q(of)i(of)e ([ReSi78,)g(Theorem)f(XI)q(I)q(I.58])g(dealing)228 2608 y(with)k(absence)g(of)h(p)q(ositiv)o(e)e(eigen)o(v)m(alues)g(to)i (incorp)q(orate)g(the)f(case)g Ft(E)h Fu(=)d(0.)965 2658 y Fq(6)p eop %%Page: 7 7 7 6 bop 278 213 a Fu(The)22 b(b)q(ound)i(\(1.4\))f(of)h(Theorem)d(1.1)i (in)g(the)f(case)h Ft(V)1316 220 y Fs(2)1361 213 y Fu(=)i(0)e(is)g(sho) o(wn)h(in)228 271 y(Section)16 b(3|this)g(is)g(done)g(using)h(a)g (non-standard)h(Mourre)e(theory)l(.)278 329 y(In)g(Section)g(4)h(w)o(e) f(pro)o(v)o(e)g(Theorem)f(1.2)i(in)f(the)h(case)f Ft(V)1313 336 y Fs(2)1348 329 y Fu(=)e(0.)23 b(The)17 b(pro)q(of)g(is)228 387 y(accomplished)10 b(using)j(the)f(strategy)g(from)f([GIS96)q(])g (and)i(an)g(energy-dep)q(enden)o(t)228 445 y(pseudo)q(di\013eren)o (tial)25 b(calculus.)49 b(Some)24 b(tec)o(hnical)g(v)o(eri\014cations)h (concerning)228 503 y(this)20 b(calculus)f(will)g(b)q(e)i(giv)o(en)e (in)h(App)q(endix)f(B.)32 b(In)20 b(App)q(endix)g(A)f(a)i(certain)228 561 y(`algebraic)14 b(v)o(eri\014cation')g(is)h(giv)o(en.)20 b(Using)15 b(Theorems)f(1.2)h(\(with)g Ft(V)1499 568 y Fs(2)1533 561 y Fu(=)f(0\))i(and)228 619 y(2.4)e(w)o(e)g(\014nally)g (pro)o(v)o(e)f(Theorem)g(1.1)i(in)f(the)g(general)g(case)g(in)g (Section)g(5.)21 b(That)228 678 y(section)14 b(also)i(con)o(tains)e (the)h(pro)q(of)h(of)f(Theorem)e(1.2,)i(\(1.6\))g(and)h(Theorem)d(1.4.) 550 778 y(2.)27 b Fv(Absence)18 b(of)g(eigenv)l(alue)d(a)m(t)k(zer)o(o) 278 865 y Fu(In)11 b(this)h(section)f(w)o(e)g(will)g(pro)o(v)o(e)g(a)h (basic)f(result,)h(namely)e(that)i(for)g(long-range)228 923 y(p)q(oten)o(tials)17 b Ft(V)28 b Fu(that)18 b(are)f Fh(ne)n(gative)i Fu(at)e(in\014nit)o(y)l(,)f(there)g(is)h(no)h Ft(L)1403 905 y Fs(2)1423 923 y Fu(-eigenfunction)228 981 y(with)d(energy)g(zero.)20 b(That)c(is)f(the)g(result)f(of)i (Theorem)e(2.4.)21 b(In)15 b(Subsection)g(5.1)228 1039 y(w)o(e)j(will)g(use)h(this)g(insigh)o(t)g(to)g(obtain)g(the)g (limiting)d(absorption)21 b(principle)c(at)228 1098 y(zero)f(energy)l (,)f(Theorem)g(1.1.)278 1156 y(W)l(e)g(in)o(tend)f(to)i(generalize)e (the)h(`Kato-Agmon-Simon)f(Theorem')g([ReSi78,)228 1214 y(Theorem)k(XI)q(I)q(I.58])g(b)o(y)g(using)i(a)f(mo)q(di\014cation)g (of)g(its)g(pro)q(of.)31 b(In)19 b(particular,)228 1272 y(w)o(e)13 b(shall)h(apply)g(unique)g(con)o(tin)o(uation.)20 b(The)14 b(pro)q(of)h(uses)f(ODE)h(tec)o(hniques)e(in)228 1330 y(the)i(radial)h(co)q(ordinate,)g(so)g(let)f(us)h(sp)q(ecify)f (the)h(notation)g Ft(x)e Fu(=)f Ft(r)q(!)k Fp(2)d Fn(R)1581 1312 y Fm(d)1598 1330 y Fu(,)h(with)228 1388 y Ft(!)h Fp(2)e Fn(S)352 1370 y Fm(d)p Fo(\000)p Fs(1)414 1388 y Fu(.)278 1446 y(The)20 b(conditions)g(whic)o(h)f(exclude)g (zero-energy)g(eigenfunctions)h(are)g(giv)o(en)228 1504 y(in)c(Assumptions)f(2.1)i(and)g(2.3.)228 1592 y Fy(Assumption)g(2.1)i Fu(\(Unique)c(con)o(tin)o(uation)h(at)g(in\014nit)o(y\))p Fy(.)23 b Fu(The)14 b(function)h Ft(V)25 b Fu(:)228 1650 y Fn(R)267 1632 y Fm(d)300 1650 y Fp(!)16 b Fn(R)f Fu(is)i(measurable,) f(and)i(if)f Ft(u)f Fp(2)h Ft(H)1020 1632 y Fs(2)1039 1650 y Fu(\()p Fn(R)1097 1632 y Fm(d)1115 1650 y Fu(\),)g Ft(u)f Fu(=)g(0)i(in)f(a)h(neigh)o(b)q(ourho)q(o)q(d)228 1708 y(of)e Fp(1)p Fu(,)g(the)g(pro)q(duct)h Ft(V)11 b( )k Fp(2)g Ft(L)797 1690 y Fs(2)816 1708 y Fu(\()p Fn(R)874 1690 y Fm(d)892 1708 y Fu(\))h(and)h Ft(u)f Fu(is)g(a)g(distributional)g(solution)h(to)806 1787 y Fp(\000)p Fu(\001)p Ft(u)10 b Fu(+)h Ft(V)h(u)h Fu(=)h(0)p Ft(;)228 1866 y Fu(then)i Ft(u)d Fu(=)h(0.)228 1954 y Fy(Remark)i(2.2.)24 b Fu(Di\013eren)o(t)18 b(results)h(exist)f(in)h (this)g(direction.)28 b(The)19 b(condition)228 2012 y Ft(V)25 b Fp(2)14 b Ft(L)361 1986 y Fm(d=)p Fs(2)361 2026 y(lo)q(c)417 2012 y Fu(\()p Fn(R)475 1994 y Fm(d)492 2012 y Fu(\))i(su\016ces)g(for)g Ft(d)f Fp(\025)e Fu(3,)j(see)g ([JerKen85].)228 2099 y Fy(Assumption)h(2.3.)24 b Fu(The)12 b(function)h Ft(V)23 b Fu(can)13 b(b)q(e)f(written)g(as)h Ft(V)25 b Fu(=)14 b Ft(V)1489 2106 y Fs(1)1512 2099 y Fu(+)s Ft(V)1581 2106 y Fs(2)1602 2099 y Fu(,)e(suc)o(h)228 2157 y(that:)24 b(F)l(or)18 b(some)f Ft(s)f Fp(2)h Fu([0)p Ft(;)8 b Fu(1\),)17 b(some)g Ft(R;)8 b(C)20 b(>)c Fu(0)j(and)f(a)g(p)q (ositiv)o(e)f(di\013eren)o(tiable)228 2215 y(function)f Ft(h)e Fu(=)g Ft(h)p Fu(\()p Ft(r)q Fu(\))i(de\014ned)g(on)h([)p Ft(R;)8 b Fp(1)p Fu(\))16 b(w)o(e)g(ha)o(v)o(e)277 2284 y(\(1\))21 b Ft(V)388 2291 y Fs(1)421 2284 y Fu(and)13 b Ft(V)540 2291 y Fs(2)574 2284 y Fu(are)g(b)q(ounded)h(on)f Fp(j)p Ft(x)p Fp(j)g Ft(>)h(R)p Fu(,)g(and)f Ft(V)1217 2291 y Fs(1)1251 2284 y Fu(is)f(negativ)o(e)h(on)g Fp(j)p Ft(x)p Fp(j)g Ft(>)h(R)p Fu(.)277 2343 y(\(2\))21 b(sup)433 2355 y Fm(!)q Fo(2)p Fm(S)503 2346 y Fi(d)p Fg(\000)p Fk(1)584 2324 y Fm(d)p 576 2332 36 2 v 576 2360 a(dr)616 2343 y Fu(\()p Ft(r)658 2325 y Fm(s)p Fs(+1)722 2343 y Ft(V)750 2350 y Fs(1)770 2343 y Fu(\()p Ft(r)q(!)r Fu(\)\))14 b Fp(\024)f(\000)p Ft(r)1010 2325 y Fm(s)1029 2343 y Ft(h)1057 2325 y Fs(2)1077 2343 y Fu(\()p Ft(r)q Fu(\))j(when)h Ft(r)e(>)f(R)p Fu(.)277 2404 y(\(3\))21 b Ft(r)383 2386 y Fo(\000)p Fs(1)441 2404 y Fu(+)11 b Ft(r)f Fu(sup)595 2416 y Fm(!)q Fo(2)p Fm(S)665 2406 y Fi(d)p Fg(\000)p Fk(1)733 2404 y Fp(j)p Ft(V)775 2411 y Fs(2)795 2404 y Fu(\()p Ft(r)q(!)r Fu(\))p Fp(j)k Fu(=)g Ft(o)p Fu(\()p Ft(h)p Fu(\))j(as)f Ft(r)g Fp(!)d(1)p Fu(.)277 2462 y(\(4\))21 b Ft(h)388 2444 y Fo(0)399 2462 y Fu(\()p Ft(r)q Fu(\))14 b Fp(\024)g Ft(C)t(h)594 2444 y Fs(2)613 2462 y Fu(\()p Ft(r)q Fu(\))j(on)g Fp(j)p Ft(x)p Fp(j)c Ft(>)h(R)p Fu(.)278 2550 y(With)e(the)h(ab)q(o)o(v)o(e)f (assumptions)h(w)o(e)f(can)h(pro)o(v)o(e)f(the)h(absence)g(of)g (zero-energy)228 2608 y(eigenstates.)965 2658 y Fq(7)p eop %%Page: 8 8 8 7 bop 228 213 a Fy(Theorem)16 b(2.4.)24 b Fh(Supp)n(ose)e Ft(V)35 b Fu(=)23 b Ft(V)919 220 y Fs(1)954 213 y Fu(+)14 b Ft(V)1034 220 y Fs(2)1077 213 y Fh(satis\014es)23 b(Assumptions)g (2.1)f(and)228 271 y(2.3.)g(Supp)n(ose)17 b(furthermor)n(e)g(that)g Ft( )f Fp(2)e Ft(H)1022 253 y Fs(2)1018 284 y(lo)q(c)1064 271 y Fu(\()p Fn(R)1122 253 y Fm(d)1140 271 y Fu(\))j Fh(satis\014es)h Fu(\(1\){\(3\))h Fh(b)n(elow.)278 344 y Fu(\(1\))361 304 y Fl(R)384 361 y Fo(j)p Fm(x)p Fo(j)p Fm(>R)488 344 y Ft(h)516 326 y Fs(2)536 344 y Fu(\()p Ft(r)q Fu(\))p Fp(j)p Ft( )r Fu(\()p Ft(x)p Fu(\))p Fp(j)725 326 y Fs(2)752 344 y Ft(dx)14 b(<)g Fp(1)j Fh(and)1033 304 y Fl(R)1056 361 y Fo(j)p Fm(x)p Fo(j)p Fm(>R)1160 344 y Fp(j)p Ft(V)1202 351 y Fs(1)1222 344 y Fu(\()p Ft(x)p Fu(\))p Fp(jj)p Ft( )r Fu(\()p Ft(x)p Fu(\))p Fp(j)1430 326 y Fs(2)1457 344 y Ft(dx)d(<)f Fp(1)p Fh(.)278 415 y Fu(\(2\))21 b Ft(p)385 422 y Fm(j)403 415 y Ft( )16 b Fp(2)e Ft(L)531 397 y Fs(2)551 415 y Fu(\()p Fn(R)609 397 y Fm(d)626 415 y Fu(\);)22 b Ft(j)16 b Fu(=)e(1)p Ft(;)8 b Fp(\001)g(\001)g(\001)17 b Ft(;)8 b(d:)278 474 y Fu(\(3\))21 b Fh(The)16 b(pr)n(o)n(duct)g Ft(V)11 b( )k Fp(2)f Ft(L)797 456 y Fs(2)797 487 y(lo)q(c)844 474 y Fu(\()p Fn(R)902 456 y Fm(d)919 474 y Fu(\))p Fh(,)i(and)h Fu(\()p Fp(\000)p Fu(\001)8 b(+)h Ft(V)i Fu(\))p Ft( )16 b Fu(=)d(0)k Fh(in)g(the)g(sense)g(of)361 532 y(distributions.)278 605 y(Then)h Ft( )d Fu(=)f(0)p Fh(.)228 697 y Fy(Remarks)i(2.5.)24 b Fu(1\))d(The)16 b(assumptions)f(in)h(Theorem)e(1.1)i(are)g(stronger)h (than)292 757 y(Assumption)d(2.3:)21 b(T)l(ak)o(e)15 b Ft(h)f Fu(=)g Ft(\017r)912 739 y Fo(\000)p Fm(\026=)p Fs(2)1013 757 y Fu(for)i(a)f(small)f Ft(\017)f(>)h Fu(0)i(and)g Ft(s)f Fu(close)g(to)g(1.)228 815 y(2\))21 b(Supp)q(ose)d(Assumption)e (2.3)i(and)g(that)g Ft( )f Fp(2)f Ft(H)1180 797 y Fs(2)1176 828 y(lo)q(c)1222 815 y Fu(\()p Fn(R)1280 797 y Fm(d)1297 815 y Fu(\))i(ob)q(eys)g(\(1\),)f(\(2\))h(and)292 873 y(\(3\))d(except)f(that)h(the)g(condition)g(\()p Fp(\000)p Fu(\001)8 b(+)g Ft(V)j Fu(\))p Ft( )k Fu(=)f(0)i(is)e(only)h(required)f (to)h(b)q(e)292 931 y(ful\014lled)g(outside)h(some)f(su\016cien)o(tly)g (large)h(ball)g Ft(B)1253 938 y Fm(\032)1287 931 y Fu(=)e Fp(f)p Ft(x)f Fp(2)i Fn(R)1492 913 y Fm(d)1509 931 y Fp(j)f(j)p Ft(x)p Fp(j)f(\024)h Ft(\032)p Fp(g)p Fu(,)292 990 y(then)k(our)g(pro)q(of)h(yields)e(the)h(conclusion)g(that)h Ft( )f Fu(=)f(0)h(outside)g Ft(B)1547 997 y Fm(\032)1567 990 y Fu(.)27 b(\(This)292 1048 y(remark)15 b(will)g(b)q(e)h(used)g(in) g(Subsection)g(5.3.\))228 1106 y(3\))21 b(Using)16 b(the)g(negativit)o (y)f(of)i Ft(V)826 1113 y Fs(1)862 1106 y Fu(w)o(e)f(see)g(that)567 1220 y(sup)540 1265 y Fm(!)q Fo(2)p Fm(S)610 1256 y Fi(d)p Fg(\000)p Fk(1)693 1187 y Ft(d)p 681 1209 49 2 v 681 1255 a(dr)735 1220 y(r)758 1200 y Fs(2)778 1220 y Ft(V)806 1227 y Fs(1)826 1220 y Fu(\()p Ft(r)q(!)r Fu(\))e Fp(\024)g(\000)p Ft(r)q(h)1076 1200 y Fs(2)1096 1220 y Fu(\()p Ft(r)q Fu(\);)24 b(for)16 b Ft(r)g(>)d(R:)213 b Fu(\(2.1\))228 1344 y(4\))21 b(By)15 b(in)o(tegrating)i(\(2.1\))g(w)o(e)e(get)i(the)f (b)q(ound)597 1459 y(lim)8 b(sup)628 1498 y Fm(\032)p Fo(!1)755 1459 y Ft(\032)780 1438 y Fo(\000)p Fs(2)836 1391 y Fl(Z)885 1404 y Fm(\032)863 1504 y(R)914 1459 y Ft(r)q(h)965 1438 y Fs(2)985 1459 y Fu(\()p Ft(r)q Fu(\))p Ft(dr)16 b Fp(\024)24 b Fu(sup)1161 1502 y Fo(j)p Fm(x)p Fo(j)p Fm(>R)1263 1459 y Fp(j)p Ft(V)1305 1466 y Fs(1)1325 1459 y Fp(j)p Ft(;)292 1584 y Fu(whic)o(h)15 b(essen)o(tially)g(is)h(a)h(b)q(oundedness)g(condition)f(on)h Ft(h)p Fu(.)228 1642 y(5\))k(A)13 b(sligh)o(tly)e(more)h(general)h (result)f(ma)o(y)g(b)q(e)h(obtained)g(in)g(terms)f(of)h(an)h Ft(h)f Fu(that)292 1700 y(dep)q(ends)k(on)f(the)g(angle)h Ft(!)h Fu(as)f(w)o(ell.)228 1792 y Fy(Example)f(2.6.)24 b Fu(Supp)q(ose)c(that)f(for)h(some)e Ft(\026)h Fp(2)g Fu([0)p Ft(;)8 b Fu(2\),)19 b Ft(V)1340 1799 y Fs(1)1361 1792 y Fu(\()p Ft(x)p Fu(\))f(=)h Ft(v)r Fu(\()p Ft(!)r Fu(\))p Ft(r)1621 1774 y Fo(\000)p Fm(\026)1684 1792 y Fu(+)228 1852 y Ft(R)p Fu(\()p Ft(x)p Fu(\))j(where)g(sup)9 b Ft(v)r Fu(\()p Ft(!)r Fu(\))23 b Ft(<)i Fu(0,)e Ft(R)p Fu(\()p Ft(x)p Fu(\))h(=)h Ft(o)p Fu(\()p Ft(r)1080 1834 y Fo(\000)p Fm(\026)1131 1852 y Fu(\))d(and)h(sup)1347 1863 y Fm(!)q Fo(2)p Fm(S)1417 1854 y Fi(d)p Fg(\000)p Fk(1)1498 1832 y Fm(d)p 1489 1840 36 2 v 1489 1869 a(dr)1529 1852 y Ft(R)p Fu(\()p Ft(r)q(!)r Fu(\))i(=)228 1916 y Ft(o)p Fu(\()p Ft(r)293 1898 y Fo(\000)p Fm(\026)p Fo(\000)p Fs(1)389 1916 y Fu(\),)c(and)g Ft(V)570 1923 y Fs(2)610 1916 y Fu(=)g Ft(o)p Fu(\()p Ft(r)734 1898 y Fo(\000)p Fs(1)p Fo(\000)p Fm(\026=)p Fs(2)865 1916 y Fu(\).)33 b(Then)21 b(Assumption)e(2.3)h(holds,)h(cf.)32 b(Re-)228 1974 y(marks)13 b(2.5)h(1.)21 b(The)13 b(particular)h(case)g(giv)o(en)f (b)o(y)g(putting)h Ft(\026)g Fu(=)g(0)g(and)g Ft(v)i Fu(equal)d(to)228 2032 y(a)g(negativ)o(e)f(constan)o(t)h(\(in)g(that)g (case)g(w)o(e)f(ma)o(y)g(tak)o(e)g Ft(s)i Fu(=)f(0)h(in)e(Assumption)g (2.3\))228 2090 y(yields)g(the)h(Kato-Agmon-Simon)f(theorem)g(on)i (absence)f(of)g(p)q(ositiv)o(e)g(eigen)o(v)m(al-)228 2148 y(ues)i(for)g(Sc)o(hr\177)-24 b(o)q(dinger)15 b(op)q(erators)h ([ReSi78,)f(Theorem)e(XI)q(I)q(I.58].)20 b(\(Notice)14 b(that)228 2207 y(an)o(y)i(p)q(ositiv)o(e)f(energy)h(ma)o(y)f(b)q(e)h (absorb)q(ed)i(in)o(to)e Ft(V)1163 2214 y Fs(1)1183 2207 y Fu(.\))228 2299 y Fh(Pr)n(o)n(of)g(of)h(The)n(or)n(em)f(2.4.)21 b Fu(The)f(pro)q(of)g(is)g(based)g(on)h(unique)e(con)o(tin)o(uation,)h (so)228 2357 y(w)o(e)e(aim)f(at)i(pro)o(ving)g(that)g Ft( )h Fu(v)m(anishes)f(iden)o(tically)d(on)j Fn(R)1341 2339 y Fm(d)1371 2357 y Fp(n)13 b(fj)p Ft(x)p Fp(j)k(\024)h Ft(R)1601 2364 y Fs(1)1620 2357 y Fp(g)h Fu(for)228 2415 y(some)e(\(su\016cien)o(tly)g(big\))i Ft(R)758 2422 y Fs(1)778 2415 y Fu(.)28 b(Let)19 b Ft(B)i Fu(b)q(e)e(the)g(\(negativ)o (e\))e(Laplace-Beltrami)228 2473 y(op)q(erator)g(on)g Fn(S)523 2455 y Fm(d)p Fo(\000)p Fs(1)601 2473 y Fu(suc)o(h)f(that)637 2590 y Fp(\000)p Fu(\001)e(=)f Fp(\000)838 2557 y Ft(@)867 2538 y Fs(2)p 826 2579 72 2 v 826 2624 a Ft(@)s(r)878 2610 y Fs(2)914 2590 y Fp(\000)968 2557 y Ft(d)f Fp(\000)f Fu(1)p 968 2579 111 2 v 1012 2624 a Ft(r)1101 2557 y(@)p 1089 2579 52 2 v 1089 2624 a(@)s(r)1157 2590 y Fp(\000)1221 2557 y Fu(1)p 1212 2579 44 2 v 1212 2624 a Ft(r)1235 2610 y Fs(2)1260 2590 y Ft(B)s(:)965 2658 y Fq(8)p eop %%Page: 9 9 9 8 bop 228 213 a Fu(W)l(e)20 b(will)f(need)h(the)g(inner)f(pro)q(duct) i Fp(h\001)p Ft(;)8 b Fp(\001i)1021 222 y Fm(L)1045 213 y Fk(2)1063 222 y Fs(\()p Ff(S)1098 213 y Fi(d)p Fg(\000)p Fk(1)1152 222 y Fs(\))1188 213 y Fu(and)21 b(the)f(asso)q(ciated)h (norm)228 272 y Fp(k)12 b(\001)f(k)315 282 y Fm(L)339 272 y Fk(2)357 282 y Fs(\()p Ff(S)392 272 y Fi(d)p Fg(\000)p Fk(1)446 282 y Fs(\))479 272 y Fu(rep)q(eatedly)l(.)24 b(F)l(or)18 b(shortness)g(w)o(e)f(will)g(therefore)f(just)i(write)f Fp(h\001)p Ft(;)8 b Fp(\001i)228 331 y Fu(and)16 b Fp(k)9 b(\001)g(k)p Fu(.)21 b(Unless)14 b(otherwise)i(stated,)f(all)g(inner)g (pro)q(ducts)h(and)g(norms)e(in)h(the)228 390 y(pro)q(of)h(will)d (refer)h(to)h(these|this)f(is)g(con)o(trary)h(to)f(the)h(con)o(v)o(en)o (tion)e(in)h(the)h(later)228 448 y(sections)k(of)g(the)g(pap)q(er)g (where)g(inner)g(pro)q(ducts)h(and)f(norms)g(are)g(on)g Ft(L)1608 430 y Fs(2)1628 448 y Fu(\()p Fn(R)1686 430 y Fm(d)1703 448 y Fu(\))228 506 y(unless)d(otherwise)g(stated.)278 564 y(Let)h Ft( )i Fu(b)q(e)f(the)f(function)h(from)e(the)i(statemen)o (t)e(of)h(the)h(theorem.)23 b(W)l(e)18 b(ma)o(y)228 622 y(assume)g(that)g Ft( )i Fu(is)f(real-v)m(alued.)27 b(W)l(e)18 b(de\014ne,)g(for)h Ft(r)g Fp(2)f Fu(\(0)p Ft(;)8 b Fp(1)p Fu(\),)18 b(the)g(function)228 680 y Ft(w)263 687 y Fm( )289 680 y Fu(\()p Ft(r)o(;)8 b Fp(\001)p Fu(\))13 b(=)h Ft(w)q Fu(\()p Ft(r)o(;)8 b Fp(\001)p Fu(\))13 b Fp(2)h Ft(L)673 662 y Fs(2)693 680 y Fu(\()p Fn(S)743 662 y Fm(d)p Fo(\000)p Fs(1)805 680 y Fu(\))i(b)o(y)722 776 y Ft(w)q Fu(\()p Ft(r)o(;)8 b(!)r Fu(\))13 b(=)h Ft(r)959 755 y Fs(\()p Fm(d)p Fo(\000)p Fs(1\))p Fm(=)p Fs(2)1088 776 y Ft( )r Fu(\()p Ft(r)q(!)r Fu(\))p Ft(:)228 866 y Fu(Notice,)g(that)j(if)f Ft(\036)e Fp(2)g Ft(C)674 848 y Fo(1)670 878 y Fs(0)710 866 y Fu(\()p Fn(R)768 848 y Fm(d)786 866 y Fu(\),)h(then)343 957 y Ft(w)378 964 y Fm(H)s(\036)446 957 y Fu(=)f Fp(\000)p Ft(w)573 936 y Fo(0)o(0)572 969 y Fm(\036)605 957 y Fp(\000)d Ft(r)678 936 y Fo(\000)p Fs(2)726 957 y Ft(B)s(w)801 964 y Fm(\036)834 957 y Fu(+)888 937 y Fs(1)p 888 945 18 2 v 888 974 a(4)911 957 y Fu(\()p Ft(d)g Fp(\000)g Fu(1\)\()p Ft(d)h Fp(\000)f Fu(3\))p Ft(r)1231 936 y Fo(\000)p Fs(2)1278 957 y Ft(w)1313 964 y Fm(\036)1347 957 y Fu(+)g Ft(V)h(w)1471 964 y Fm(\036)1494 957 y Ft(:)114 b Fu(\(2.2\))228 1047 y(W)l(e)16 b(de\014ne)g(furthermore,)355 1137 y Ft(F)7 b Fu(\()p Ft(r)q Fu(\))13 b(=)h Fp(k)p Ft(w)581 1116 y Fo(0)593 1137 y Fp(k)618 1116 y Fs(2)648 1137 y Fu(+)d Ft(r)720 1116 y Fo(\000)p Fs(2)768 1137 y Fp(h)p Ft(w)q(;)d(B)s(w)q Fp(i)k(\000)e(h)p Ft(w)q(;)e(V)1106 1144 y Fs(1)1127 1137 y Ft(w)q Fp(i)k(\000)e Ft(sr)1289 1116 y Fo(\000)p Fs(1)1337 1137 y Fp(h)p Ft(w)q(;)e(w)1450 1116 y Fo(0)1462 1137 y Fp(i)p Ft(;)135 b Fu(\(2.3\))228 1235 y(where)16 b Ft(w)405 1217 y Fo(0)417 1235 y Fu(\()p Ft(r)o(;)8 b(!)r Fu(\))13 b(=)608 1216 y Fm(@)p 600 1224 38 2 v 600 1253 a(@)r(r)642 1235 y Ft(w)q Fu(\()p Ft(r)o(;)8 b(!)r Fu(\))p Ft(:)278 1294 y Fu(One)16 b(ma)o(y)g(note)h(that)g(since) f Ft(V)28 b Fu(is)17 b(lo)q(cally)f(b)q(ounded)i(outside)f(a)g (su\016cien)o(tly)228 1352 y(large)f(ball)f(cen)o(tered)f(at)j(the)e (origin,)h(one)g(gets)g(b)o(y)f(elliptic)f(regularit)o(y)h(\(see)g(for) 228 1413 y(instance)k([GiT)l(r01)q(]\))g(that)h Ft( )i Fp(2)e Ft(C)893 1390 y Fs(1)p Fm(;\013)889 1427 y Fs(lo)q(c)945 1413 y Fu(\()p Fn(R)1003 1395 y Fm(d)1034 1413 y Fp(n)13 b(fj)p Ft(x)p Fp(j)19 b(\024)h Ft(R)p Fp(g)p Fu(\))g(\(for)h Ft(R)f Fu(giv)o(en)f(as)i(in)228 1471 y(Assumption)15 b(2.3)i(and)g(for)g(all)f Ft(\013)f Fp(2)f Fu([0)p Ft(;)8 b Fu(1\)\).)22 b(Th)o(us)17 b Ft(F)23 b Fu(is)17 b(a)g(H\177)-24 b(older)15 b(con)o(tin)o(uous)228 1529 y(function.)278 1588 y(The)j(\014rst)h(imp)q(ortan)o(t)e(step)h(in)g(the)g(pro)q(of)i (is)e(to)g(establish)h(that)f Ft(F)25 b Fu(is)18 b(in)o(te-)228 1646 y(grable)f(at)h(in\014nit)o(y)l(,)e(i.e.)24 b Ft(F)e Fp(2)16 b Ft(L)839 1628 y Fs(1)859 1646 y Fu(\(\()p Ft(R;)8 b Fp(1)p Fu(\);)g Ft(dr)q Fu(\).)25 b(In)17 b(order)h(to)g(see)f(this,) g(notice)228 1704 y(that)f(for)h Ft(\036)d Fp(2)g Ft(C)537 1686 y Fo(1)533 1716 y Fs(0)574 1704 y Fu(\()p Fn(R)632 1686 y Fm(d)649 1704 y Fu(\))i(w)o(e)g(ha)o(v)o(e)320 1757 y Fl(Z)369 1770 y Fo(1)347 1870 y Fs(0)415 1754 y Fl(\032)452 1825 y Fp(k)p Ft(r)500 1804 y Fs(\()p Fm(d)p Fo(\000)p Fs(1\))p Fm(=)p Fs(2)645 1791 y Ft(@)p 634 1813 52 2 v 634 1859 a(@)s(r)691 1825 y(\036)p Fu(\()p Ft(r)q(!)r Fu(\))p Fp(k)838 1804 y Fs(2)869 1825 y Fp(\000)10 b Ft(r)941 1804 y Fo(\000)p Fs(2)989 1825 y Fp(h)p Ft(w)1043 1832 y Fm(\036)1066 1825 y Ft(;)e(B)s(w)1163 1832 y Fm(\036)1185 1825 y Fp(i)1204 1754 y Fl(\033)1259 1825 y Ft(dr)15 b Fu(=)f Fp(kj)p Ft(p)p Fp(j)p Ft(\036)p Fp(k)1504 1804 y Fs(2)1504 1839 y Fm(L)1528 1829 y Fk(2)1545 1839 y Fs(\()p Ff(R)1586 1829 y Fi(d)1601 1839 y Fs(\))1617 1825 y Ft(:)228 1943 y Fu(This)i(implies,)d(since)j Fp(\000)p Ft(B)g Fp(\025)e Fu(0,)330 1993 y Fl(Z)380 2006 y Fo(1)358 2106 y Fm(R)425 2061 y Fp(k)p Ft(w)486 2040 y Fo(0)485 2073 y Fm(\036)508 2061 y Fp(k)533 2040 y Fs(2)561 2061 y Ft(dr)i Fp(\024)d Ft(C)723 1993 y Fl(Z)773 2006 y Fo(1)751 2106 y Fm(R)818 1991 y Fl(\022)855 2061 y Fp(k)p Ft(r)903 2040 y Fs(\()p Fm(d)p Fo(\000)p Fs(1\))p Fm(=)p Fs(2)1048 2027 y Ft(@)p 1036 2050 V 1036 2095 a(@)s(r)1093 2061 y(\036)p Fu(\()p Ft(r)q(!)r Fu(\))p Fp(k)1240 2040 y Fs(2)1271 2061 y Fu(+)e Ft(r)1343 2040 y Fo(\000)p Fs(2)1391 2061 y Fp(k)p Ft(w)1451 2068 y Fm(\036)1474 2061 y Fp(k)1499 2040 y Fs(2)1518 1991 y Fl(\023)1571 2061 y Ft(dr)624 2199 y Fp(\024)i Ft(C)715 2179 y Fo(0)735 2129 y Fl(\022)771 2199 y Fp(kj)p Ft(p)p Fp(j)p Ft(\036)p Fp(k)902 2179 y Fs(2)902 2213 y Fm(L)926 2204 y Fk(2)943 2213 y Fs(\()p Ff(R)984 2204 y Fi(d)999 2213 y Fs(\))1026 2199 y Fu(+)1075 2131 y Fl(Z)1103 2244 y Fo(j)p Fm(x)p Fo(j\025)p Fm(R)1207 2199 y Ft(h)p Fu(\()p Ft(r)q Fu(\))1296 2179 y Fs(2)1316 2199 y Fp(j)p Ft(\036)p Fu(\()p Ft(x)p Fu(\))p Fp(j)1439 2179 y Fs(2)1466 2199 y Ft(dx)1519 2129 y Fl(\023)1564 2199 y Ft(:)278 2323 y Fu(This,)j(and)h(an)g(appro)o(ximation)e (argumen)o(t,)g(pro)o(v)o(es)h(that)h(the)f(\014rst)h(term)e(in)228 2381 y(\(2.3\))22 b(is)f(in)o(tegrable.)37 b(Similar)19 b(considerations)j(apply)g(to)g(the)f(other)h(terms.)228 2440 y(Th)o(us)16 b Ft(F)21 b Fp(2)14 b Ft(L)484 2421 y Fs(1)503 2440 y Fu(\(\()p Ft(R;)8 b Fp(1)p Fu(\);)g Ft(dr)q Fu(\).)278 2498 y(W)l(e)16 b(will)f(next)g(compute)g(\()p Ft(r)q(F)7 b Fu(\))861 2480 y Fo(0)889 2498 y Fu(using)16 b(that)h(formally)329 2588 y(0)d(=)g Ft(w)454 2595 y Fm(H)s( )525 2588 y Fu(=)g Fp(\000)p Ft(w)652 2568 y Fo(00)684 2588 y Fp(\000)d Ft(r)757 2568 y Fo(\000)p Fs(2)805 2588 y Ft(B)s(w)g Fu(+)945 2569 y Fs(1)p 945 2577 18 2 v 945 2605 a(4)968 2588 y Fu(\()p Ft(d)g Fp(\000)g Fu(1\)\()p Ft(d)h Fp(\000)f Fu(3\))p Ft(r)1288 2568 y Fo(\000)p Fs(2)1336 2588 y Ft(w)h Fu(+)f Ft(V)g(w)q(:)101 b Fu(\(2.4\))965 2658 y Fq(9)p eop %%Page: 10 10 10 9 bop 228 213 a Fu(Using)17 b(\(2.4\),)f(a)g(formal)f(computation)h (giv)o(es)333 303 y(\()p Ft(r)q(F)7 b Fu(\()p Ft(r)q Fu(\)\))494 282 y Fo(0)519 303 y Fu(=)14 b(2)p Fp(h)p Ft(w)650 282 y Fo(0)662 303 y Ft(;)8 b Fu([)p Ft(r)q(V)749 310 y Fs(2)780 303 y Fu(+)834 283 y Fs(1)p 834 291 18 2 v 834 320 a(4)857 303 y Fu(\()p Ft(d)j Fp(\000)g Fu(1\)\()p Ft(d)h Fp(\000)f Fu(3\))p Ft(r)1177 282 y Fo(\000)p Fs(1)1225 303 y Fu(])p Ft(w)q Fp(i)g Fu(+)g(\(1)g Fp(\000)g Ft(s)p Fu(\))p Fp(k)p Ft(w)1561 282 y Fo(0)1573 303 y Fp(k)1598 282 y Fs(2)516 383 y Fp(\000)g(h)p Ft(w)q(;)d Fu(\()p Ft(r)q(V)713 390 y Fs(1)734 383 y Fu(\))753 362 y Fo(0)765 383 y Ft(w)q Fp(i)j(\000)g Fu(\(1)g Fp(\000)g Ft(s)p Fu(\))p Ft(r)1050 362 y Fo(\000)p Fs(2)1098 383 y Fp(h)p Ft(w)q(;)d(B)s(w)q Fp(i)516 463 y(\000)j Ft(s)p Fp(h)p Ft(w)q(;)d Fu([)685 443 y Fs(1)p 685 452 V 685 480 a(4)708 463 y Fu(\()p Ft(d)j Fp(\000)g Fu(1\)\()p Ft(d)h Fp(\000)e Fu(3\))p Ft(r)1027 443 y Fo(\000)p Fs(2)1086 463 y Fu(+)h Ft(V)h Fu(])p Ft(w)q Fp(i)p Ft(:)364 b Fu(\(2.5\))228 553 y(Let)17 b Ft(\016)g Fu(=)f(1)c Fp(\000)f Ft(s)16 b(>)f Fu(0.)25 b(Using)17 b(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz,)f(w)o(e)h (estimate)e(the)i(\014rst)h(term)228 611 y(in)e(\(2.5\))g(b)q(elo)o(w)g (b)o(y)727 701 y Fp(\000)p Ft(\016)r Fp(k)p Ft(w)851 681 y Fo(0)862 701 y Fp(k)887 681 y Fs(2)918 701 y Fp(\000)10 b(h)p Ft(w)q(;)e(o)p Fu(\()p Ft(h)1114 681 y Fs(2)1135 701 y Fu(\))p Ft(w)q Fp(i)p Ft(:)399 b Fu(\(2.6\))228 791 y(F)l(or)16 b(the)g(third)g(term)f(w)o(e)g(estimate)497 895 y Fp(\000)p Fu(\()p Ft(r)q(V)606 902 y Fs(1)627 895 y Fu(\))646 874 y Fo(0)671 895 y Fu(=)f Fp(\000)p Ft(r)785 874 y Fo(\000)p Fm(s)847 861 y Ft(d)p 836 884 49 2 v 836 929 a(dr)889 895 y Fu(\()p Ft(r)931 874 y Fm(s)p Fs(+1)995 895 y Ft(V)1023 902 y Fs(1)1043 895 y Fu(\))d(+)g Ft(sV)1173 902 y Fs(1)1207 895 y Fp(\025)j Ft(h)1288 874 y Fs(2)1319 895 y Fu(+)d Ft(sV)1419 902 y Fs(1)1439 895 y Ft(:)169 b Fu(\(2.7\))228 1000 y(Notice)12 b(that)i(the)f(con)o (tribution)g(from)g Ft(sV)1001 1007 y Fs(1)1034 1000 y Fu(to)h(the)f(righ)o(t)g(and)h(the)g(con)o(tribution)228 1058 y(from)i(the)i(term)e Ft(V)576 1065 y Fs(1)613 1058 y Fu(in)i(the)f(\014fth)h(term)d(of)22 b(\(2.5\))c(in)f(fact)g(cancel.) 25 b(Clearly)16 b(the)228 1116 y(fourth)i(term)e(is)h(non-negativ)o(e.) 26 b(Hence)16 b(w)o(e)h(conclude)g(from)g(\(2.5\),)h(\(2.6\))g(and)228 1174 y(\(2.7\))e(that)h(for)g(a)f(su\016cien)o(tly)e(large)j Ft(R)966 1181 y Fs(1)999 1174 y Fp(\025)d Ft(R)458 1257 y(d)p 446 1279 V 446 1325 a(dr)500 1290 y Fu(\()p Ft(r)q(F)7 b Fu(\()p Ft(r)q Fu(\)\))14 b Fp(\025)f(h)p Ft(w)q(;)8 b Fu(\()p Ft(h)851 1270 y Fs(2)882 1290 y Fu(+)j Ft(o)p Fu(\()p Ft(h)1001 1270 y Fs(2)1022 1290 y Fu(\)\))p Ft(w)q Fp(i)j(\025)g Fu(0)i(when)h Ft(r)e(>)e(R)1475 1297 y Fs(1)1495 1290 y Ft(:)113 b Fu(\(2.8\))228 1395 y(Therefore)16 b(w)o(e)f(ha)o(v)o(e,)g(for)i(all)f Ft(r)f(>)f(r)900 1402 y Fs(1)933 1395 y Ft(>)g(R)1022 1402 y Fs(1)1042 1395 y Fu(,)800 1511 y Ft(F)7 b Fu(\()p Ft(r)q Fu(\))14 b Fp(\025)971 1478 y Ft(r)993 1485 y Fs(1)1013 1478 y Ft(F)7 b Fu(\()p Ft(r)1093 1485 y Fs(1)1112 1478 y Fu(\))p 971 1500 160 2 v 1040 1546 a Ft(r)1136 1511 y(:)472 b Fu(\(2.9\))278 1619 y(The)12 b(deduction)f(of)16 b(\(2.9\))c(ab)q(o)o (v)o(e)g(w)o(as)g(based)h(on)f(formal)f(calculations.)19 b(Ho)o(w-)228 1677 y(ev)o(er,)e(it)g(is)h(standard)i(to)e(use)h(appro)o (ximation)e(argumen)o(ts)g(\(see)h(the)g(pro)q(of)h(of)228 1735 y([ReSi78)o(,)24 b(Thm.)38 b(XI)q(I)q(I.58]\))21 b(to)h(justify)g(the)g(inequalit)o(y)e(rigorously)l(.)39 b(Belo)o(w)228 1793 y(w)o(e)14 b(will)g(do)h(a)h(similar)c(formal)i (calculation)g(without)h(commen)o(t|again)e(it)h(can)228 1851 y(easily)h(b)q(e)i(justi\014ed)e(using)i(appro)o(ximations)f(b)o (y)f(smo)q(oth)i(functions.)278 1909 y(Since)e Ft(F)20 b Fp(2)14 b Ft(L)537 1891 y Fs(1)557 1909 y Fu(\()p Ft(dr)q Fu(\))j(it)f(follo)o(ws)g(from)f(\(2.9\))i(that)721 1999 y Ft(F)7 b Fu(\()p Ft(r)q Fu(\))13 b Fp(\024)h Fu(0)j(for)f(all)g Ft(r)f(>)f(R)1196 2006 y Fs(1)1216 1999 y Ft(:)368 b Fu(\(2.10\))278 2089 y(Next)15 b(w)o(e)h(de\014ne,)f(with)h Ft(w)769 2096 y Fm(m)816 2089 y Fu(=)e Ft(r)891 2071 y Fm(m)925 2089 y Ft(w)j Fu(for)g Ft(m)c(>)h Fu(0)p Ft(;)269 2193 y(G)p Fu(\()p Ft(m;)8 b(r)q Fu(\))14 b(=)g Fp(k)p Ft(w)560 2172 y Fo(0)559 2205 y Fm(m)592 2193 y Fp(k)617 2172 y Fs(2)648 2193 y Fu(+)d Ft(r)720 2172 y Fo(\000)p Fs(2)767 2193 y Fp(h)p Ft(w)821 2200 y Fm(m)855 2193 y Ft(;)d(B)s(w)952 2200 y Fm(m)985 2193 y Fp(i)777 2273 y Fu(+)j Fp(h)p Ft(w)880 2280 y Fm(m)914 2273 y Ft(;)d Fu(\()p Ft(m)p Fu(\()p Ft(m)i Fu(+)h(1\))p Ft(r)1185 2253 y Fo(\000)p Fs(2)1244 2273 y Fp(\000)g Ft(g)i Fp(\000)e Ft(V)1408 2280 y Fs(1)1428 2273 y Fu(\))p Ft(w)1482 2280 y Fm(m)1515 2273 y Fp(i)p Ft(;)50 b Fu(\(2.11\))228 2368 y(where)21 b Ft(g)j Fu(=)e Ft(\017r)524 2350 y Fo(\000)p Fs(1)572 2368 y Ft(h)p Fu(\()p Ft(r)q Fu(\))f(with)g Ft(\017)h Fu(=)h(\(2)p Ft(C)t Fu(\))1002 2350 y Fo(\000)p Fs(1)1049 2368 y Fu(.)36 b(Here)20 b Ft(C)25 b Fu(is)c(the)g(constan)o (t)g(from)228 2426 y(Assumption)15 b(2.3.)21 b(The)c(function)f Ft(w)922 2433 y Fm(m)971 2426 y Fu(satis\014es)247 2516 y Ft(w)283 2495 y Fo(0)o(0)282 2528 y Fm(m)326 2516 y Fp(\000)11 b Fu(2)p Ft(mr)466 2495 y Fo(\000)p Fs(1)513 2516 y Ft(w)549 2495 y Fo(0)548 2528 y Fm(m)592 2516 y Fu(+)g Ft(r)664 2495 y Fo(\000)p Fs(2)720 2475 y Fl(\000)743 2516 y Ft(m)p Fu(\()p Ft(m)f Fu(+)h(1\))h Fp(\000)1017 2496 y Fs(1)p 1017 2504 18 2 v 1017 2533 a(4)1039 2516 y Fu(\()p Ft(d)g Fp(\000)e Fu(1\)\()p Ft(d)i Fp(\000)f Fu(3\))g(+)g Ft(B)1436 2475 y Fl(\001)1467 2516 y Ft(w)1502 2523 y Fm(m)1546 2516 y Fp(\000)g Ft(V)g(w)1670 2523 y Fm(m)260 2588 y Fu(=)j(0)p Ft(:)956 2658 y Fq(10)p eop %%Page: 11 11 11 10 bop 228 217 a Fu(W)l(e)20 b(no)o(w)g(compute,)g(using)h(\(2.1\))g (in)f(the)g(\014rst)g(inequalit)o(y)l(,)f(for)i Ft(r)h(>)1577 204 y Fu(~)1566 217 y Ft(R)1603 224 y Fs(1)1644 217 y Fu(and)228 279 y Ft(m)13 b(>)h(M)5 b Fu(,)16 b(for)h(some)e(large)h(p)q (ositiv)o(e)925 266 y(~)915 279 y Ft(R)952 286 y Fs(1)988 279 y Fu(and)h Ft(M)5 b Fu(:)358 383 y(\(2)p Ft(r)q Fu(\))443 362 y Fo(\000)p Fs(1)507 349 y Ft(d)p 495 371 49 2 v 495 417 a(dr)549 383 y Fu(\()p Ft(r)591 362 y Fs(2)611 383 y Ft(G)p Fu(\()p Ft(m;)j(r)q Fu(\)\))371 477 y(=)14 b Fp(h)p Ft(w)477 484 y Fm(m)511 477 y Ft(;)8 b Fu(\()p Ft(r)q(V)603 484 y Fs(2)634 477 y Fp(\000)j Ft(r)q(g)j Fu(+)798 458 y Fs(1)p 798 466 18 2 v 798 495 a(4)820 477 y Fu(\()p Ft(d)e Fp(\000)e Fu(1\)\()p Ft(d)i Fp(\000)f Fu(3\))p Ft(r)1140 457 y Fo(\000)p Fs(1)1188 477 y Fu(\))p Ft(w)1243 457 y Fo(0)1242 490 y Fm(m)1275 477 y Fp(i)417 558 y Fu(+)g(\(2)p Ft(m)h Fu(+)f(1\))p Fp(k)p Ft(w)717 537 y Fo(0)716 570 y Fm(m)749 558 y Fp(k)774 537 y Fs(2)805 558 y Fp(\000)g(h)p Ft(w)909 565 y Fm(m)942 558 y Ft(;)d Fu(\(2)p Ft(r)q Fu(\))1049 537 y Fo(\000)p Fs(1)1097 558 y Fu(\()p Ft(r)1139 537 y Fs(2)1159 558 y Fu(\()p Ft(g)13 b Fu(+)e Ft(V)1291 565 y Fs(1)1311 558 y Fu(\)\))1349 537 y Fo(0)1361 558 y Ft(w)1396 565 y Fm(m)1429 558 y Fp(i)371 638 y(\025)j Fu(\(2)p Ft(m)d Fu(+)g(1\))p Fp(k)p Ft(w)674 617 y Fo(0)673 650 y Fm(m)707 638 y Fp(k)732 617 y Fs(2)762 638 y Fu(+)g(\()835 618 y Fs(1)p 835 626 V 835 655 a(2)858 638 y Ft(h)886 617 y Fs(2)917 638 y Fp(\000)g Fu(\(2)p Ft(r)q Fu(\))1052 617 y Fo(\000)p Fs(1)1099 638 y Fu(\()p Ft(r)1141 617 y Fs(2)1161 638 y Ft(g)r Fu(\))1205 617 y Fo(0)1217 638 y Fu(\))p Fp(k)p Ft(w)1296 645 y Fm(m)1329 638 y Fp(k)1354 617 y Fs(2)417 715 y Fu(+)g Fp(h)p Ft(w)520 722 y Fm(m)554 715 y Ft(;)d Fu(\()p Ft(o)p Fu(\()p Ft(h)p Fu(\))j Fp(\000)g Ft(r)q(g)r Fu(\))p Ft(w)848 694 y Fo(0)847 727 y Fm(m)881 715 y Fp(i)371 795 y Fu(=)j(\(2)p Ft(m)d Fu(+)g(1\))p Fp(k)p Ft(w)673 775 y Fo(0)672 807 y Fm(m)706 795 y Fp(k)731 775 y Fs(2)761 795 y Fu(+)g(\()834 776 y Fs(1)p 834 784 V 834 812 a(4)857 795 y Ft(h)885 775 y Fs(2)916 795 y Fu(+)g Ft(o)p Fu(\()p Ft(h)1035 775 y Fs(2)1055 795 y Fu(\)\))p Fp(k)p Ft(w)1153 802 y Fm(m)1186 795 y Fp(k)1211 775 y Fs(2)1242 795 y Fu(+)g Fp(h)p Ft(w)1345 802 y Fm(m)1378 795 y Ft(;)d(O)q Fu(\()p Ft(h)p Fu(\))p Ft(w)1540 775 y Fo(0)1539 807 y Fm(m)1573 795 y Fp(i)371 879 y(\025)14 b Fu(0;)25 b(for)16 b Ft(r)f(>)661 866 y Fu(~)650 879 y Ft(R)687 886 y Fs(1)707 879 y Ft(;)8 b(m)13 b(>)h(M)r(:)698 b Fu(\(2.12\))228 970 y(F)l(or)19 b(con)o(v)o(enience)d(w)o(e)i(assume) g(henceforth)g(that)1193 957 y(~)1182 970 y Ft(R)1219 977 y Fs(1)1257 970 y Fu(=)g Ft(R)1350 977 y Fs(1)1370 970 y Fu(.)29 b(W)l(e)18 b(learn)g(from)228 1028 y(\(2.12\))f(that)h Ft(r)499 1010 y Fs(2)519 1028 y Ft(G)p Fu(\()p Ft(m;)8 b(r)q Fu(\))17 b(is)g(non-decreasing)h(in)e Ft(r)h(>)e(R)1268 1035 y Fs(1)1288 1028 y Fu(,)i(pro)o(vided)f(that)h Ft(m)e(>)228 1086 y(M)5 b Fu(.)278 1144 y(W)l(e)11 b(will)g(no)o(w)h(com)o(bine)e (\(2.10\))j(and)f(\(2.12\))h(to)f(pro)o(v)o(e)f(that)h Ft(w)h Fu(\(and)g(therefore)228 1202 y Ft( )r Fu(\))j(v)m(anishes)g(in) g(a)h(neigh)o(b)q(ourho)q(o)q(d)h(of)e(in\014nit)o(y)l(.)k(Clearly)301 1282 y Ft(r)324 1262 y Fo(\000)p Fs(2)p Fm(m)403 1282 y Ft(G)p Fu(\()p Ft(m;)8 b(r)q Fu(\))14 b(=)979 b(\(2.13\))301 1363 y Fp(k)p Ft(w)362 1342 y Fo(0)385 1363 y Fu(+)11 b Ft(mr)500 1342 y Fo(\000)p Fs(1)547 1363 y Ft(w)q Fp(k)608 1342 y Fs(2)639 1363 y Fu(+)g Ft(r)711 1342 y Fo(\000)p Fs(2)758 1363 y Fp(h)p Ft(w)q(;)d(B)s(w)q Fp(i)k Fu(+)f Fp(h)p Ft(w)q(;)d Fu(\()p Ft(m)p Fu(\()p Ft(m)j Fu(+)g(1\))p Ft(r)1318 1342 y Fo(\000)p Fs(2)1377 1363 y Fp(\000)g Ft(g)i Fp(\000)e Ft(V)1541 1370 y Fs(1)1561 1363 y Fu(\))p Ft(w)q Fp(i)p Ft(:)228 1445 y Fu(Supp)q(ose)21 b Ft(w)q Fu(\()p Ft(r)500 1452 y Fs(1)520 1445 y Fu(\))f Fp(6)p Fu(=)h(0)f(for)h(some)e Ft(r)889 1452 y Fs(1)929 1445 y Ft(>)i(R)1025 1452 y Fs(1)1044 1445 y Fu(.)33 b(Then)21 b(w)o(e)e(ha)o(v)o(e)h(for)g(some)f(large)228 1503 y(enough)e Ft(m)439 1510 y Fs(1)472 1503 y Ft(>)d(M)5 b Fu(,)285 1583 y Ft(r)308 1562 y Fo(\000)p Fs(2)307 1595 y(1)356 1583 y Fp(h)p Ft(w)q Fu(\()p Ft(r)452 1590 y Fs(1)472 1583 y Fu(\))p Ft(;)j(B)s(w)q Fu(\()p Ft(r)630 1590 y Fs(1)649 1583 y Fu(\))p Fp(i)k Fu(+)f Fp(h)p Ft(w)q Fu(\()p Ft(r)844 1590 y Fs(1)864 1583 y Fu(\))p Ft(;)d Fu(\()p Ft(m)967 1590 y Fs(1)986 1583 y Fu(\()p Ft(m)1048 1590 y Fs(1)1079 1583 y Fu(+)j(1\))p Ft(r)1194 1562 y Fo(\000)p Fs(2)1193 1595 y(1)1252 1583 y Fp(\000)g Ft(g)r Fu(\()p Ft(r)1368 1590 y Fs(1)1388 1583 y Fu(\)\))p Ft(w)q Fu(\()p Ft(r)1503 1590 y Fs(1)1523 1583 y Fu(\))p Fp(i)j Ft(>)g Fu(0)p Ft(:)1598 1641 y Fu(\(2.14\))228 1723 y(Clearly)i(\(2.12\),)g (\(2.13\))h(and)g(\(2.14\))g(yield)686 1803 y Ft(G)p Fu(\()p Ft(m)786 1810 y Fs(1)806 1803 y Ft(;)8 b(r)q Fu(\))14 b Ft(>)g Fu(0)i(for)h(all)f Ft(r)f Fp(\025)e Ft(r)1230 1810 y Fs(1)1250 1803 y Ft(:)334 b Fu(\(2.15\))228 1888 y(Next,)12 b(w)o(e)h(get)g(from)f(Assumption)f(2.3)j(\(3\))f(that) h(lim)1204 1895 y Fm(r)q Fo(!1)1302 1888 y Ft(r)1325 1870 y Fs(2)1345 1888 y Ft(g)r Fu(\()p Ft(r)q Fu(\))h(=)e Fp(1)p Fu(.)20 b(There-)228 1946 y(fore,)c(for)g(some)f Ft(R)571 1953 y Fs(2)605 1946 y Ft(>)f(r)679 1953 y Fs(1)698 1946 y Fu(,)i(w)o(e)g(ha)o(v)o(e)313 1983 y Fl(\024)345 2020 y Fu(2)p Ft(m)412 2027 y Fs(1)442 2020 y Fu(+)11 b Ft(s)p 345 2042 170 2 v 417 2088 a Fu(2)530 2053 y(+)g Ft(m)622 2060 y Fs(1)642 2053 y Fu(\(2)p Ft(m)728 2060 y Fs(1)759 2053 y Fu(+)g(1\))851 1983 y Fl(\025)886 2053 y Ft(r)909 2033 y Fo(\000)p Fs(2)967 2053 y Fp(\000)g Ft(g)r Fu(\()p Ft(r)q Fu(\))j Fp(\024)g Fu(0)i(for)h(all)f Ft(r)f(>)f(R)1479 2060 y Fs(2)1498 2053 y Ft(:)86 b Fu(\(2.16\))228 2161 y(Since)15 b(b)o(y)h(assumption)700 2196 y Fl(Z)728 2309 y Fm(r)q Fo(\025)p Fm(R)799 2314 y Fk(2)826 2264 y Ft(r)849 2244 y Fo(\000)p Fs(2)897 2264 y Fp(k)p Ft(w)q Fu(\()p Ft(r)q Fu(\))p Fp(k)1044 2244 y Fs(2)1072 2264 y Ft(dr)g(<)e Fp(1)p Ft(;)228 2380 y Fu(the)f(function)f Ft(r)k Fp(7!)d Ft(r)620 2362 y Fo(\000)p Fs(1)668 2380 y Fp(k)p Ft(w)q Fu(\()p Ft(r)q Fu(\))p Fp(k)815 2362 y Fs(2)848 2380 y Fu(cannot)g(b)q(e)g(strictly)f(increasing)h(near)g (in\014nit)o(y)l(.)228 2441 y(W)l(e)i(pic)o(k)g Ft(r)436 2448 y Fs(2)470 2441 y Ft(>)e(R)558 2448 y Fs(2)578 2441 y Fu(,)j(suc)o(h)g(that)g(\()p Ft(r)865 2423 y Fo(\000)p Fs(1)912 2441 y Fp(k)p Ft(w)q Fu(\()p Ft(r)q Fu(\))p Fp(k)1059 2423 y Fs(2)1079 2441 y Fu(\))1098 2415 y Fo(0)1118 2399 y Fl(\014)1118 2429 y(\014)1135 2461 y Fm(r)q Fs(=)p Fm(r)1195 2466 y Fk(2)1228 2441 y Fp(\024)e Fu(0)i(and)h(estimate)d (using)228 2507 y(\(2.15\))j(and)g(\(2.16\),)670 2588 y(0)e Ft(<)e(r)783 2567 y Fo(\000)p Fs(2)p Fm(m)859 2572 y Fk(1)782 2600 y Fs(2)879 2588 y Ft(G)p Fu(\()p Ft(m)979 2595 y Fs(1)999 2588 y Ft(;)8 b(r)1043 2595 y Fs(2)1063 2588 y Fu(\))14 b Fp(\024)f Ft(F)7 b Fu(\()p Ft(r)1228 2595 y Fs(2)1247 2588 y Fu(\))p Ft(;)956 2658 y Fq(11)p eop %%Page: 12 12 12 11 bop 228 213 a Fu(whic)o(h)15 b(b)o(y)i(\(2.10\))g(is)f(imp)q (ossible.)278 271 y(W)l(e)j(conclude)g(that)h Ft(w)q Fu(\()p Ft(r)q Fu(\))g(=)g(0)g(for)g(all)f Ft(r)i(>)e(R)1183 278 y Fs(1)1203 271 y Fu(.)31 b(Since)19 b(therefore)g Ft( )i Fu(=)f(0)228 329 y(on)e(a)h(neigh)o(b)q(ourho)q(o)q(d)g(of)g (in\014nit)o(y)l(,)d Ft( )i Fp(2)f Ft(H)1055 311 y Fs(2)1075 329 y Fu(\()p Fn(R)1133 311 y Fm(d)1150 329 y Fu(\))h(and)h Ft(V)11 b( )18 b Fp(2)f Ft(L)1456 311 y Fs(2)1476 329 y Fu(\()p Fn(R)1534 311 y Fm(d)1551 329 y Fu(\).)26 b(F)l(rom)228 387 y(Assumption)15 b(2.1)i(w)o(e)e(conclude)h(that)h Ft( )e Fu(=)f(0.)p 1687 387 2 33 v 1689 356 30 2 v 1689 387 V 1718 387 2 33 v 479 499 a(3.)28 b Fv(Limiting)19 b(absorption)e(principle)h(f)o(or)g Ft(V)1451 506 y Fs(1)278 587 y Fu(In)c(this)h(section)g(w)o(e)g(will)f(pro)o(v)o(e)g(the)h (limiting)d(absorption)17 b(principle)c(b)q(ound)228 645 y(\(1.4\))21 b(in)f(the)g(case)g Ft(V)633 652 y Fs(2)675 645 y Fu(=)h(0)f(in)h(the)f(set)g(of)h(conditions)f(of)h(Theorem)e (1.1.)35 b(A)228 703 y(somewhat)21 b(more)e(general)i(result)g(will)f (b)q(e)h(stated)h(in)f(Corollary)g(3.5)h(b)q(elo)o(w.)228 761 y(Explicitly)l(,)13 b(w)o(e)i(study)h(the)f(op)q(erator)i Ft(H)984 768 y Fs(1)1018 761 y Fu(=)d Ft(p)1094 743 y Fs(2)1124 761 y Fu(+)c Ft(V)1200 768 y Fs(1)1220 761 y Fu(,)16 b(where)f Ft(V)25 b Fu(=)14 b Ft(V)1523 768 y Fs(1)1559 761 y Fu(satis\014es)228 819 y(the)22 b(assumptions)g(of)h (Theorem)e(1.1.)39 b(In)22 b(Section)g(5)h(w)o(e)f(will)f(mak)o(e)f(a)j (p)q(er-)228 877 y(turbativ)o(e)e(argumen)o(t)f(to)j(include)d Ft(V)935 884 y Fs(2)977 877 y Fu(\(the)i(short-range,)i(non)f (sign-de\014nite)228 935 y(part)d(of)g(the)g(p)q(oten)o(tial\).)31 b(F)l(or)20 b(simplicit)o(y)c(w)o(e)k(will)e(skip)i(the)f(indices)g(in) g(this)228 993 y(section)h(and)h(write)f Ft(H)25 b Fu(and)d Ft(V)32 b Fu(instead)20 b(of)h Ft(H)1120 1000 y Fs(1)1161 993 y Fu(and)h Ft(V)1289 1000 y Fs(1)1309 993 y Fu(,)f(resp)q(ectiv)o (ely)l(.)32 b(Our)228 1052 y(approac)o(h)23 b(is)f(v)o(ery)g (di\013eren)o(t)f(from)h([Nak94],)h(where)f(a)h(similar)e(\(but)h(m)o (uc)o(h)228 1110 y(w)o(eak)o(er\))c(result)h(is)h(pro)o(v)o(ed.)30 b(There)19 b(an)h(auxiliary)f(op)q(erator)i(\(the)e(in)o(v)o(erse)f(of) 228 1168 y(the)h(Birman-Sc)o(h)o(winger)e(k)o(ernel\))g(is)i(studied,)h (whereas)f(w)o(e)g(do)h(Mourre)f(the-)228 1226 y(ory)d(directly)e(on)j Ft(H)t Fu(.)k(Since)15 b(w)o(e)h(do)g(not)h(ha)o(v)o(e)e(a)i(p)q (ositiv)o(e)e(comm)o(utator)f(in)h(the)228 1284 y(sense)21 b(of)g(Mourre)g([Mou81])g(at)g(zero-energy)l(,)h(this)f(will)f(b)q(e)h (a)g(non-standard)228 1342 y(Mourre)f(theory)l(.)32 b(\(F)l(or)21 b(another)f(non-standard)i(Mourre)e(theory)l(,)h(used)f(in)g(a)228 1400 y(v)o(ery)15 b(di\013eren)o(t)g(setting,)h(w)o(e)g(refer)f(to)i ([Her91)o(].\))278 1458 y(T)l(o)23 b(motiv)m(ate)e(our)h(strategy)h(of) g(pro)q(of)g(of)g(Theorem)e(1.1)i(w)o(e)f(notice)g(that)228 1516 y(if)d Ft(V)32 b Fu(=)21 b Ft(V)423 1523 y Fs(1)457 1516 y Fu(+)13 b Ft(V)536 1523 y Fs(2)577 1516 y Fu(only)20 b(satis\014es)g(the)g(b)q(ounds)i(of)e(the)g(assumptions)g(\(1\))h(and) 228 1575 y(\(3\))h(for)g Fp(j)p Ft(x)p Fp(j)g Ft(>)i(R)e Fu(\(and)g(in)g(addition)g(\(2\),)h(\(4\))f(and)g(\(5\)\))g(then)g (there)f(exists)228 1637 y(another)d(decomp)q(osition)f Ft(V)28 b Fu(=)849 1624 y(~)841 1637 y Ft(V)869 1644 y Fs(1)902 1637 y Fu(+)959 1624 y(~)952 1637 y Ft(V)980 1644 y Fs(2)1018 1637 y Fu(for)18 b(whic)o(h)f(\(1\){\(5\))i(hold.)26 b(T)l(o)18 b(pro)o(v)o(e)228 1699 y(this)g(w)o(e)g(de\014ne)550 1686 y(~)542 1699 y Ft(V)570 1706 y Fs(1)608 1699 y Fu(=)g Ft(V)692 1706 y Fs(1)712 1699 y Ft(\037)743 1706 y Fs(+)772 1699 y Fu(\()p Ft(r)q(=n)p Fu(\))13 b Fp(\000)g Ft(\017)971 1706 y Fs(1)990 1699 y Fp(h)p Ft(x)p Fp(i)1056 1681 y Fo(\000)p Fm(\026)1107 1699 y Ft(\037)1138 1706 y Fo(\000)1167 1699 y Fu(\()p Ft(r)q(=n)p Fu(\).)29 b(Here)17 b(1)h(=)g Ft(\037)1570 1706 y Fs(+)1612 1699 y Fu(+)12 b Ft(\037)1693 1706 y Fo(\000)228 1757 y Fu(is)22 b(a)h(standard,)i(smo)q(oth)d (partition)g(of)h(unit)o(y)f(on)h Fn(R)1251 1764 y Fs(+)1277 1757 y Fu(:)34 b(supp)8 b Ft(\037)1464 1764 y Fs(+)1518 1757 y Fu(=)24 b([1)p Ft(;)8 b Fp(1)p Fu(\),)228 1815 y Ft(\037)259 1822 y Fs(+)288 1815 y Fu(\()p Ft(t)p Fu(\))13 b(=)h(1)h(for)g Ft(t)f Fp(\025)f Fu(2.)21 b(Clearly)14 b(w)o(e)g(ma)o(y)g(assume)g(that)h Ft(\017)1298 1822 y Fs(2)1331 1815 y Ft(<)f Fu(2)8 b Fp(\000)g Ft(\026)p Fu(.)21 b(W)l(e)14 b(claim)228 1873 y(that)i(for)h(an)o(y)f Ft(n)e(>)g(R)734 1958 y Fp(\000)p Ft(x)c Fp(\001)h(r)885 1945 y Fu(~)878 1958 y Ft(V)906 1965 y Fs(1)940 1958 y Fp(\025)i Fu(\(2)f Fp(\000)e Ft(\017)1116 1965 y Fs(2)1136 1958 y Fu(\))1162 1945 y(~)1155 1958 y Ft(V)1183 1965 y Fs(1)1203 1958 y Ft(:)405 b Fu(\(3.1\))228 2044 y(Using)16 b(that)h(\(3.1\))f(is)g(satis\014ed)h(for)f Ft(V)926 2051 y Fs(1)963 2044 y Fu(on)g Fp(j)p Ft(x)p Fp(j)e Ft(>)f(R)p Fu(,)k(w)o(e)e(\014nd)366 2132 y Fp(\000)p Ft(x)10 b Fp(\001)h(r)517 2119 y Fu(~)510 2132 y Ft(V)538 2139 y Fs(1)572 2132 y Fp(\025)i Fu(\(2)f Fp(\000)e Ft(\017)748 2139 y Fs(2)768 2132 y Fu(\))794 2119 y(~)787 2132 y Ft(V)815 2139 y Fs(1)569 2212 y Fu(+)h Ft(\017)638 2219 y Fs(1)657 2212 y Fp(h)p Ft(x)p Fp(i)723 2192 y Fo(\000)p Fm(\026)774 2212 y Ft(\037)805 2219 y Fo(\000)835 2212 y Fu(\()p Ft(r)q(=n)p Fu(\))957 2172 y Fl(\000)980 2212 y Fu(\(2)h Fp(\000)f Ft(\017)1105 2219 y Fs(2)1124 2212 y Fu(\))g(+)g Fp(h)p Ft(x)p Fp(i)1269 2192 y Fm(\026)1293 2212 y Fu(\()p Ft(x)g Fp(\001)g(rh)p Ft(x)p Fp(i)1484 2192 y Fo(\000)p Fm(\026)1534 2212 y Fu(\))1553 2172 y Fl(\001)569 2305 y Fp(\000)626 2272 y Ft(r)p 624 2294 30 2 v 624 2340 a(n)658 2305 y(\037)689 2285 y Fo(0)689 2318 y(\000)718 2305 y Fu(\()p Ft(r)q(=n)p Fu(\))p Fp(h)p Ft(x)p Fp(i)898 2285 y Fo(\000)p Fm(\026)958 2305 y Fu(\()p Fp(\000h)p Ft(x)p Fp(i)1082 2285 y Fm(\026)1106 2305 y Ft(V)1134 2312 y Fs(1)1165 2305 y Fp(\000)g Ft(\017)1235 2312 y Fs(1)1254 2305 y Fu(\))572 2404 y Fp(\025)i Fu(\(2)f Fp(\000)e Ft(\017)748 2411 y Fs(2)768 2404 y Fu(\))794 2391 y(~)787 2404 y Ft(V)815 2411 y Fs(1)835 2404 y Ft(;)228 2490 y Fu(pro)o(ving)17 b(\(3.1\).)278 2550 y(W)l(e)c(de\014ne)f(the)h (op)q(erator)i Fp(h)p Ft(A)p Fp(i)f Fu(=)g(\()p Ft(C)8 b Fu(+)d Ft(A)1054 2532 y Fs(2)1073 2550 y Fu(\))1092 2532 y Fs(1)p Fm(=)p Fs(2)1160 2550 y Fu(with)13 b Ft(C)18 b Fp(\035)13 b Fu(1)h(\014xed)f(\(yielding)228 2608 y(b)q(etter)i(comm) o(utation)f(prop)q(erties)h(than)i(with)e Ft(C)i Fu(=)d(1,)i(cf.)21 b([M\034Sk02,)15 b(Lemma)956 2658 y Fq(12)p eop %%Page: 13 13 13 12 bop 228 213 a Fu(2.5]\).)21 b(F)l(or)16 b Ft(E)h Fp(\025)d Fu(0,)i(let)f Ft(f)699 220 y Fm(E)745 213 y Fu(b)q(e)i(the)f(function)746 305 y Ft(f)770 312 y Fm(E)814 305 y Fu(=)865 265 y Fl(\000)888 305 y Ft(E)e Fu(+)d Fp(h)p Ft(x)p Fp(i)1053 285 y Fo(\000)p Fm(\026)1104 265 y Fl(\001)1127 276 y Fs(1)p Fm(=)p Fs(2)1191 305 y Ft(:)417 b Fu(\(3.2\))228 394 y(F)l(or)21 b(the)g(purp)q(ose)i(of)e (pro)o(ving)h(\(1.4\))g(one)g(could)f(replace)f Ft(f)1386 401 y Fm(E)1437 394 y Fu(b)o(y)h Fp(h)p Ft(x)p Fp(i)1576 376 y Fo(\000)p Fm(\026=)p Fs(2)1684 394 y Fu(e-)228 452 y(v)o(erywhere)c(in)i(this)g(section.)29 b(But)19 b(w)o(e)f(include)g(it)g(here)h(since)f(the)h(estimates)228 510 y(w)o(e)d(establish)g(\(with)g Ft(f)655 517 y Fm(E)685 510 y Fu(\))g(will)f(b)q(e)h(imp)q(ortan)o(t)g(in)g(Section)f(4.)278 568 y(It)f(is)g(easy)h(to)g(see)g(that)g Ft(p)746 550 y Fs(2)774 568 y Fu(+)8 b Ft(f)849 550 y Fs(2)844 581 y Fm(E)888 568 y Fu(de\014nes)15 b(a)g(p)q(ositiv)o(e,)f(self-adjoin)o (t)g(op)q(erator,)228 626 y(so)j(w)o(e)e(can)i(use)f(the)g(sp)q(ectral) g(theorem)f(to)i(de\014ne,)e(for)i Ft(\020)h Fp(2)c Fn(C)9 b Fu(,)18 b(the)e(op)q(erator)726 707 y Ft(\015)h Fu(=)c Ft(\015)844 715 y Fo(j)p Fm(\020)r Fo(j)898 707 y Fu(=)h(\()p Ft(p)993 687 y Fs(2)1024 707 y Fu(+)d Ft(f)1102 687 y Fs(2)1097 719 y Fo(j)p Fm(\020)r Fo(j)1137 707 y Fu(\))1156 687 y Fs(1)p Fm(=)p Fs(2)1211 707 y Ft(;)405 b Fu(\(3.3\))278 791 y(Recall)15 b(from)g(\(1.8\))h(and)h(\(1.9\))g(that)653 871 y Ft(W)j Fu(=)14 b Fp(\000)p Fu(2)p Ft(V)23 b Fp(\000)11 b Ft(x)f Fp(\001)h(r)p Ft(V)25 b Fp(\025)13 b Ft(c)p Fp(h)p Ft(x)p Fp(i)1232 850 y Fo(\000)p Fm(\026)1284 871 y Ft(:)228 951 y Fu(Consider)646 1031 y Ft(R)683 1038 y Fm(\020)703 1031 y Fu(\()p Ft(\017)p Fu(\))g(=)h(\()p Ft(H)h Fp(\000)c Ft(i\017i)p Fu([)p Ft(H)q(;)d(A)p Fu(])h Fp(\000)i Ft(\020)t Fu(\))1235 1010 y Fo(\000)p Fs(1)1282 1031 y Ft(;)326 b Fu(\(3.4\))228 1111 y(in)16 b(a)g(range)h(of)g(the)f (form)461 1191 y(0)e Ft(<)g Fp(=)p Ft(\020)t(;)22 b Fp(j)p Ft(\020)t Fp(j)14 b(\024)f Fu(1)p Ft(;)22 b Fp(\000)p Fu(2)p Ft(\017)910 1198 y Fs(0)930 1191 y Fp(<)p Ft(\020)c Fp(\024)13 b(=)p Ft(\020)t Fu(;)22 b(0)14 b Ft(<)g(\017)f Fp(\024)h Ft(\017)1350 1170 y Fo(0)1350 1203 y Fs(0)1383 1191 y Fp(\024)g Ft(\017)1456 1198 y Fs(0)1475 1191 y Ft(:)133 b Fu(\(3.5\))228 1274 y(\(The)22 b(existence)e(of)i(the)g(in)o (v)o(erse)e(in)i(the)g(de\014nition)f(of)i Ft(R)1358 1281 y Fm(\020)1378 1274 y Fu(\()p Ft(\017)p Fu(\))e(follo)o(ws)h(from) 228 1332 y(the)13 b(calculation)g(\(3.7\))h(and)h(the)e(theory)h(of)g (n)o(umerical)d(range)j([Kat95)q(,)g(Section)228 1390 y(V.3.1].\))33 b(The)20 b(results)g(giv)o(en)g(b)q(elo)o(w)g(ha)o(v)o (e)g(completely)d(analogous)23 b(v)o(ersions)228 1448 y(with)14 b(the)g(signs)h(of)g Fp(=)p Ft(\020)j Fu(and)d Ft(\017)f Fu(b)q(oth)i(negativ)o(e;)d(for)i(con)o(v)o(enience)d(w)o(e)i (shall)g(only)228 1506 y(explicitly)f(state)k(the)f(results)g(for)g (the)g(case)g(of)h(p)q(ositiv)o(e)e(signs.)278 1564 y(W)l(e)i(\014rst)h (form)o(ulate)e(and)i(pro)o(v)o(e)f(a)h(v)o(ersion)f(of)h(the)g (quadratic)f(estimate)f(of)228 1622 y([Mou81].)228 1710 y Fy(Lemm)o(a)g(3.1.)24 b Fh(F)l(or)13 b(al)r(l)h Ft(\017)710 1717 y Fs(0)744 1710 y Ft(>)f Fu(0)h Fh(ther)n(e)f(exists)i Ft(\017)1100 1692 y Fo(0)1100 1723 y Fs(0)1133 1710 y Ft(>)f Fu(0)f Fh(such)h(that)g(for)e(al)r(l)j(p)n(ositive)228 1769 y Ft(\017)h Fp(\024)g Ft(\017)339 1751 y Fo(0)339 1781 y Fs(0)377 1769 y Fh(and)j(al)r(l)i Ft(\020)h Fh(as)d(in)k Fu(\(3.5\))p Fh(,)c(we)h(have)g(with)f Ft(\015)g Fu(=)e Ft(\015)1270 1776 y Fo(j)p Fm(\020)r Fo(j)1328 1769 y Fh(given)k(by)i Fu(\(3.3\))c Fh(and)228 1827 y(for)e(any)g(b)n(ounde)n (d)h(op)n(er)n(ator)d Ft(B)625 1907 y Fp(k)p Ft(\015)s(R)715 1914 y Fm(\020)735 1907 y Fu(\()p Ft(\017)p Fu(\))p Ft(B)s Fp(k)858 1886 y Fs(2)890 1907 y Fp(\024)f Ft(C)t(\017)1002 1886 y Fo(\000)p Fs(1)1048 1907 y Fp(k)p Ft(B)1113 1886 y Fo(\003)1133 1907 y Ft(R)1170 1914 y Fm(\020)1189 1907 y Fu(\()p Ft(\017)p Fu(\))p Ft(B)s Fp(k)p Ft(:)296 b Fu(\(3.6\))228 1998 y Fy(Remark)16 b(3.2.)24 b Fu(Note)16 b(that)h Ft(\020)k Fu(app)q(ears)d(t)o(wice)e(in)g(\(3.6\):)23 b(In)16 b(the)h(de\014nition)f(of)228 2056 y Ft(\015)h Fu(=)c Ft(\015)346 2063 y Fo(j)p Fm(\020)r Fo(j)402 2056 y Fu(and)k(in)f Ft(R)591 2063 y Fm(\020)611 2056 y Fu(\()p Ft(\017)p Fu(\).)228 2144 y Fh(Pr)n(o)n(of.)j Fu(Let)d Ft(T)k Fu(=)14 b Ft(\015)s(R)630 2151 y Fm(\020)650 2144 y Fu(\()p Ft(\017)p Fu(\))p Ft(B)s Fu(.)21 b(Then)16 b(b)o(y)g(the)g(de\014nition)g(of)g Ft(\015)521 2226 y(T)557 2205 y Fo(\003)576 2226 y Ft(T)k Fu(=)14 b(\()p Ft(R)733 2233 y Fm(\020)753 2226 y Fu(\()p Ft(\017)p Fu(\))p Ft(B)s Fu(\))870 2205 y Fo(\003)889 2226 y Fu(\()p Ft(p)932 2205 y Fs(2)963 2226 y Fu(+)d Fp(j)p Ft(\020)t Fp(j)g Fu(+)g Fp(h)p Ft(x)p Fp(i)1191 2205 y Fo(\000)p Fm(\026)1242 2226 y Fu(\))p Ft(R)1298 2233 y Fm(\020)1318 2226 y Fu(\()p Ft(\017)p Fu(\))p Ft(B)s(:)228 2305 y Fu(Notice)k(that)i(with)f Ft(C)633 2312 y Fs(1)666 2305 y Fu(=)e(2)p Ft(C)777 2312 y Fs(2)808 2305 y Fp(\000)d Fu(1)16 b(w)o(e)g(ha)o(v)o(e,)f(using)i Ft(i)p Fu([)p Ft(H)q(;)8 b(A)p Fu(])k(=)i(2)p Ft(H)i Fu(+)11 b Ft(W)c Fu(,)380 2386 y Fp(\000)j Ft(C)464 2393 y Fs(1)484 2386 y Fp(<)e Fu(\()p Ft(H)16 b Fp(\000)10 b Ft(i\017i)p Fu([)p Ft(H)q(;)e(A)p Fu(])i Fp(\000)g Ft(\020)t Fu(\))h Fp(\000)g Ft(\017)1018 2365 y Fo(\000)p Fs(1)1065 2386 y Ft(C)1100 2393 y Fs(2)1120 2386 y Fp(=)d Fu(\()p Ft(H)15 b Fp(\000)c Ft(i\017i)p Fu([)p Ft(H)q(;)d(A)p Fu(])h Fp(\000)i Ft(\020)t Fu(\))382 2466 y(=)j Ft(p)458 2445 y Fs(2)489 2466 y Fu(+)d Ft(C)573 2473 y Fs(2)593 2466 y Ft(W)18 b Fu(+)11 b Ft(V)22 b Fu(+)11 b Ft(g)828 2473 y Fm(\020)849 2466 y Fu(\()p Ft(\017)p Fu(\);)21 b Ft(g)965 2473 y Fm(\020)985 2466 y Fu(\()p Ft(\017)p Fu(\))14 b(=)f Ft(C)1143 2473 y Fs(1)1163 2466 y Fp(<)p Ft(\020)i Fu(+)c Ft(\017)1304 2445 y Fo(\000)p Fs(1)1351 2466 y Ft(C)1386 2473 y Fs(2)1406 2466 y Fp(=)p Ft(\020)t(:)141 b Fu(\(3.7\))228 2550 y(F)l(rom)16 b(the)h(h)o(yp)q(othesis)h(on)g Ft(W)25 b Fu(and)18 b Ft(V)11 b Fu(,)18 b(w)o(e)f(ha)o(v)o(e)f Ft(C)1212 2557 y Fs(2)1232 2550 y Ft(W)j Fu(+)12 b Ft(V)27 b Fp(\025)16 b(h)p Ft(x)p Fp(i)1523 2532 y Fo(\000)p Fm(\026)1592 2550 y Fu(for)i Ft(C)1703 2557 y Fs(2)228 2608 y Fu(su\016cien)o(tly)d (big.)25 b(Fix)16 b(suc)o(h)h Ft(C)811 2615 y Fs(2)831 2608 y Fu(.)24 b(Next,)16 b(notice)h(that)h(there)f(exists)f(a)i(p)q (ositiv)o(e)956 2658 y Fq(13)p eop %%Page: 14 14 14 13 bop 228 213 a Ft(\017)248 195 y Fo(0)248 225 y Fs(0)289 213 y Fu(suc)o(h)22 b(that)g(for)g(all)g Ft(\020)k Fu(ob)q(eying)c(\(3.5\))h(and)f(for)g(all)g Ft(\017)h Fp(\024)g Ft(\017)1406 195 y Fo(0)1406 225 y Fs(0)1447 213 y Fu(indeed)e Fp(j)p Ft(\020)t Fp(j)i(\024)228 271 y(j<)p Ft(\020)t Fp(j)11 b Fu(+)g Fp(j=)p Ft(\020)t Fp(j)i(\024)h Ft(g)555 278 y Fm(\020)575 271 y Fu(\()p Ft(\017)p Fu(\).)21 b(Therefore,)15 b(w)o(e)h(can)g(estimate)445 351 y Ft(T)481 331 y Fo(\003)500 351 y Ft(T)k Fp(\024)14 b(\000)p Ft(C)676 358 y Fs(1)695 351 y Fp(<)p Fu(\()p Ft(B)790 331 y Fo(\003)809 351 y Ft(R)846 358 y Fm(\020)866 351 y Fu(\()p Ft(\017)p Fu(\))924 331 y Fo(\003)944 351 y Ft(B)s Fu(\))c Fp(\000)h Ft(C)1098 358 y Fs(2)1118 351 y Ft(\017)1138 331 y Fo(\000)p Fs(1)1185 351 y Fp(=)p Fu(\()p Ft(B)1280 331 y Fo(\003)1299 351 y Ft(R)1336 358 y Fm(\020)1356 351 y Fu(\()p Ft(\017)p Fu(\))1414 331 y Fo(\003)1433 351 y Ft(B)s Fu(\))p Ft(;)228 432 y Fu(yielding)k(\(3.6\))q(.)p 1687 432 2 33 v 1689 401 30 2 v 1689 432 V 1718 432 2 33 v 278 517 a(Notice)f(that)h(Lemma)e (3.1)j(also)g(holds)f(with)g Ft(\015)k Fu(replaced)14 b(b)o(y)h Ft(f)1444 525 y Fo(j)p Fm(\020)r Fo(j)1483 517 y Ft(;)8 b Fp(h)p Ft(x)p Fp(i)1571 499 y Fo(\000)p Fm(\026=)p Fs(2)1658 517 y Ft(;)g(p)1704 524 y Fm(j)228 577 y Fu(or)16 b Fp(j)p Ft(p)p Fp(j)p Fu(.)22 b(This)16 b(follo)o(ws)g(from)f(the)h(pro)q(of)i(since,)d(for)h(instance)g Fp(h)p Ft(x)p Fp(i)1436 559 y Fo(\000)p Fm(\026)1501 577 y Fp(\024)e Ft(\015)1582 559 y Fs(2)1602 577 y Fu(.)228 660 y Fy(Lemm)o(a)i(3.3.)24 b Fh(The)c(op)n(er)n(ator)e Ft(R)871 667 y Fm(\020)891 660 y Fu(\()p Ft(\017)p Fu(\))h Fh(\(de\014ne)n(d)i(in)i Fu(\(3.4\))q Fh(\))c(has)h(the)g(fol)r(lowing) 228 719 y(derivative,)274 787 y Ft(d)p 264 810 46 2 v 264 855 a(d\017)314 821 y(R)351 828 y Fm(\020)371 821 y Fu(\()p Ft(\017)p Fu(\))14 b(=)g(\(1)d Fp(\000)g Fu(2)p Ft(i\017)p Fu(\))679 800 y Fo(\000)p Fs(1)734 821 y Fp(f)p Ft(R)796 828 y Fm(\020)816 821 y Fu(\()p Ft(\017)p Fu(\))p Ft(A)f Fp(\000)h Ft(AR)1045 828 y Fm(\020)1065 821 y Fu(\()p Ft(\017)p Fu(\))f(+)h Ft(i\017R)1256 828 y Fm(\020)1276 821 y Fu(\()p Ft(\017)p Fu(\)\()p Ft(x)f Fp(\001)h(r)p Ft(W)c Fu(\))p Ft(R)1567 828 y Fm(\020)1586 821 y Fu(\()p Ft(\017)p Fu(\))p Fp(g)h Ft(:)228 920 y Fh(Pr)n(o)n(of.)19 b Fu(W)l(e)d(compute)269 988 y Ft(d)p 259 1011 V 259 1056 a(d\017)309 1022 y(R)346 1029 y Fm(\020)366 1022 y Fu(\()p Ft(\017)p Fu(\))d(=)h Fp(\000)p Ft(R)565 1029 y Fm(\020)585 1022 y Fu(\()p Ft(\017)p Fu(\)[)p Ft(H)q(;)8 b(A)p Fu(])p Ft(R)808 1029 y Fm(\020)827 1022 y Fu(\()p Ft(\017)p Fu(\))268 1112 y(=)14 b Ft(R)357 1119 y Fm(\020)377 1112 y Fu(\()p Ft(\017)p Fu(\))p Ft(A)c Fp(\000)h Ft(AR)606 1119 y Fm(\020)625 1112 y Fu(\()p Ft(\017)p Fu(\))g(+)g Ft(\017R)800 1119 y Fm(\020)820 1112 y Fu(\()p Ft(\017)p Fu(\)[[)p Ft(H)q(;)d(A)p Fu(])p Ft(;)g(A)p Fu(])p Ft(R)1130 1119 y Fm(\020)1147 1112 y Fu(\()p Ft(\017)p Fu(\))268 1188 y(=)14 b Ft(R)357 1195 y Fm(\020)377 1188 y Fu(\()p Ft(\017)p Fu(\))p Ft(A)c Fp(\000)h Ft(AR)606 1195 y Fm(\020)625 1188 y Fu(\()p Ft(\017)p Fu(\))g Fp(\000)g Fu(2)p Ft(i\017R)842 1195 y Fm(\020)861 1188 y Fu(\()p Ft(\017)p Fu(\)[)p Ft(H)q(;)d(A)p Fu(])p Ft(R)1084 1195 y Fm(\020)1103 1188 y Fu(\()p Ft(\017)p Fu(\))j(+)g Ft(i\017R)1295 1195 y Fm(\020)1314 1188 y Fu(\()p Ft(\017)p Fu(\)\()p Ft(x)g Fp(\001)f(r)p Ft(W)d Fu(\))p Ft(R)1605 1195 y Fm(\020)1625 1188 y Fu(\()p Ft(\017)p Fu(\))p Ft(:)228 1267 y Fu(W)l(e)20 b(no)o(w)i(insert)e(the)h(\014rst)g(line)f(of)h(the)f(ab)q(o)o(v)o(e)h (calculation)f(in)h(the)g(last)g(and)228 1325 y(isolate)392 1305 y Fm(d)p 385 1313 33 2 v 385 1342 a(d\017)422 1325 y Ft(R)459 1332 y Fm(\020)479 1325 y Fu(\()p Ft(\017)p Fu(\))16 b(in)g(the)g(resulting)g(equation.)p 1687 1325 2 33 v 1689 1293 30 2 v 1689 1325 V 1718 1325 2 33 v 278 1408 a(Using)k(Mourre's)h(tec)o(hnique)e(of)i(di\013eren)o(tial)f (inequalities,)g(w)o(e)g(can)h(\(with)228 1466 y(some)15 b(w)o(ork\))h(pro)o(v)o(e)g(limiti)o(ng)e(absorption)k(principle)c (estimates.)228 1550 y Fy(Lemm)o(a)i(3.4.)24 b Fh(F)l(or)13 b(al)r(l)h Ft(\017)710 1557 y Fs(0)744 1550 y Ft(>)f Fu(0)h Fh(ther)n(e)f(exists)i Ft(\017)1100 1532 y Fo(0)1100 1563 y Fs(0)1133 1550 y Ft(>)f Fu(0)f Fh(such)h(that)g(for)e(al)r(l)j (p)n(ositive)228 1608 y Ft(\017)f Fp(\024)g Ft(\017)335 1590 y Fo(0)335 1621 y Fs(0)355 1608 y Fh(,)j(al)r(l)i(p)n(ositive)f Ft(\016)e(<)f Fu(1)p Ft(=)p Fu(2)j Fh(and)g(al)r(l)h Ft(\020)j Fh(as)c(in)k Fu(\(3.5\))c Fh(we)g(have)h(the)f(fol)r(lowing) 228 1666 y(estimate)g(uniformly)f(in)h Ft(\020)k Fh(and)17 b Ft(\017)p Fh(,)g(with)h Ft(f)h Fu(=)14 b Ft(f)1118 1674 y Fo(j)p Fm(\020)r Fo(j)1175 1666 y Fh(given)19 b(by)j Fu(\(3.2\))p Fh(,)326 1707 y Fl(\015)326 1737 y(\015)326 1767 y(\015)353 1764 y Fp(h)p Ft(\017A)p Fp(i)448 1744 y Fo(\000)p Fs(1)496 1764 y Ft(f)525 1739 y Fs(1)p Fm(=)p Fs(2)520 1780 y Fo(j)p Fm(\020)r Fo(j)580 1764 y Fp(h)p Ft(x)p Fp(i)646 1744 y Fo(\000)p Fs(\(1)p Fm(=)p Fs(2+)p Fm(\016)q Fs(\))801 1764 y Ft(R)838 1771 y Fm(\020)858 1764 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)982 1744 y Fo(\000)p Fs(\(1)p Fm(=)p Fs(2+)p Fm(\016)q Fs(\))1136 1764 y Ft(f)1165 1739 y Fs(1)p Fm(=)p Fs(2)1160 1780 y Fo(j)p Fm(\020)r Fo(j)1220 1764 y Fp(h)p Ft(\017A)p Fp(i)1315 1744 y Fo(\000)p Fs(1)1362 1707 y Fl(\015)1362 1737 y(\015)1362 1767 y(\015)1404 1764 y Fp(\024)14 b Ft(C)1492 1771 y Fm(\016)1510 1764 y Ft(:)98 b Fu(\(3.8\))228 1861 y Fh(Pr)n(o)n(of.)19 b Fu(W)l(e)d(study)g(the)g(function)346 1941 y Ft(F)378 1948 y Fm(\020)397 1941 y Fu(\()p Ft(\017)p Fu(\))e(=)f Fp(h)p Ft(\017A)p Fp(i)615 1920 y Fo(\000)p Fs(1)662 1941 y Ft(f)691 1920 y Fs(1)p Fm(=)p Fs(2)747 1941 y Fp(h)p Ft(x)p Fp(i)813 1920 y Fo(\000)p Fm(s)859 1941 y Ft(R)896 1948 y Fm(\020)916 1941 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)1040 1920 y Fo(\000)p Fm(s)1086 1941 y Ft(f)1115 1920 y Fs(1)p Fm(=)p Fs(2)1170 1941 y Fp(h)p Ft(\017A)p Fp(i)1265 1920 y Fo(\000)p Fs(1)1313 1941 y Fu(;)21 b Ft(s)14 b Fu(=)g(1)p Ft(=)p Fu(2)e(+)f Ft(\016)o(:)228 2019 y Fu(F)l(rom)k(Lemma)f(3.1)j(w)o(e)e(get)803 2098 y Fp(k)p Ft(F)860 2105 y Fm(\020)879 2098 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)f(\024)f Ft(C)t(\017)1087 2077 y Fo(\000)p Fs(1)1134 2098 y Ft(:)474 b Fu(\(3.9\))228 2176 y(W)l(e)16 b(will)f(pro)o(v)o(e)486 2279 y Fp(k)525 2245 y Ft(d)p 516 2267 46 2 v 516 2313 a(d\017)565 2279 y(F)597 2286 y Fm(\020)617 2279 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)e(\024)h Ft(C)813 2223 y Fl(n)846 2279 y Fp(k)p Ft(F)903 2286 y Fm(\020)922 2279 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)d Fu(+)g Ft(\017)1085 2258 y Fo(\000)p Fs(1+)p Fm(\016)1174 2246 y Fg(0)1187 2279 y Fp(k)p Ft(F)1244 2286 y Fm(\020)1264 2279 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)1347 2258 y Fs(1)p Fm(=)p Fs(2)1401 2223 y Fl(o)1443 2279 y Ft(;)141 b Fu(\(3.10\))228 2375 y(for)14 b(an)o(y)g(p)q(ositiv)o (e)f Ft(\016)592 2357 y Fo(0)617 2375 y Fp(\024)h Ft(\016)r Fu(.)20 b(Com)o(bining)13 b(\(3.9\))h(and)h(\(3.10\))g(w)o(e)e(clearly) g(get)h(\(3.8\))228 2433 y(b)o(y)i(rep)q(eated)g(in)o(tegration.)278 2491 y(T)l(o)j(pro)o(v)o(e)f(\(3.10\))h(w)o(e)f(calculate)921 2471 y Fm(d)p 913 2480 33 2 v 913 2508 a(d\017)951 2491 y Ft(F)983 2498 y Fm(\020)1002 2491 y Fu(\()p Ft(\017)p Fu(\))h(using)g(Lemma)d(3.3.)29 b(After)17 b(com-)228 2550 y(m)o(utation)e(w)o(e)i(need)f(to)h(estimate)e(the)i(follo)o(wing) g(terms)e(\(and)i(some)f(`adjoin)o(t')228 2608 y(expressions\):)956 2658 y Fq(14)p eop %%Page: 15 15 15 14 bop 379 255 a Fl(\015)379 285 y(\015)407 297 y Fp(h)p Ft(\017A)p Fp(i)502 276 y Fo(\000)p Fs(1)549 297 y Ft(Af)615 276 y Fs(1)p Fm(=)p Fs(2)670 297 y Fp(h)p Ft(x)p Fp(i)736 276 y Fo(\000)p Fm(s)782 297 y Ft(R)819 304 y Fm(\020)839 297 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)963 276 y Fo(\000)p Fm(s)1009 297 y Ft(f)1038 276 y Fs(1)p Fm(=)p Fs(2)1094 297 y Fp(h)p Ft(\017A)p Fp(i)1189 276 y Fo(\000)p Fs(1)1236 255 y Fl(\015)1236 285 y(\015)1272 297 y Ft(;)312 b Fu(\(3.11\))379 338 y Fl(\015)379 368 y(\015)407 381 y Fp(h)p Ft(\017A)p Fp(i)502 360 y Fo(\000)p Fs(3)549 381 y Ft(\017A)606 360 y Fs(2)625 381 y Ft(f)654 360 y Fs(1)p Fm(=)p Fs(2)709 381 y Fp(h)p Ft(x)p Fp(i)775 360 y Fo(\000)p Fm(s)822 381 y Ft(R)859 388 y Fm(\020)879 381 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)1003 360 y Fo(\000)p Fm(s)1049 381 y Ft(f)1078 360 y Fs(1)p Fm(=)p Fs(2)1133 381 y Fp(h)p Ft(\017A)p Fp(i)1228 360 y Fo(\000)p Fs(1)1275 338 y Fl(\015)1275 368 y(\015)1311 381 y Ft(;)273 b Fu(\(3.12\))385 476 y Ft(\017)405 419 y Fl(\015)405 449 y(\015)405 479 y(\015)432 476 y Fp(h)p Ft(\017A)p Fp(i)527 456 y Fo(\000)p Fs(1)574 476 y Ft(f)603 456 y Fs(1)p Fm(=)p Fs(2)659 476 y Fp(h)p Ft(x)p Fp(i)725 456 y Fo(\000)p Fm(s)771 476 y Ft(R)808 483 y Fm(\020)828 476 y Fu(\()p Ft(\017)p Fu(\))714 585 y Fp(\002)11 b Fu(\()p Ft(x)g Fp(\001)g(r)p Ft(W)c Fu(\))p Ft(R)998 592 y Fm(\020)1017 585 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)1141 564 y Fo(\000)p Fm(s)1187 585 y Ft(f)1216 564 y Fs(1)p Fm(=)p Fs(2)1271 585 y Fp(h)p Ft(\017A)p Fp(i)1366 564 y Fo(\000)p Fs(1)1413 527 y Fl(\015)1413 557 y(\015)1413 587 y(\015)1441 585 y Ft(:)143 b Fu(\(3.13\))228 685 y(Here)18 b(and)h(b)q(elo)o(w)g(it)f (is)h(useful)g(to)g(note)g(the)f(uniform)g(b)q(ound)i(\(4.9\))f(whic)o (h)f(is)228 743 y(also)f(v)m(alid)e(with)i(the)f(presen)o(t)f (de\014nition)h(of)h Ft(f)1105 750 y Fm(E)1134 743 y Fu(.)278 801 y(The)i(term)f Ft(x)13 b Fp(\001)g(r)p Ft(W)26 b Fu(of)d(\(3.13\))d(satis\014es)g Fp(j)p Ft(x)12 b Fp(\001)h(r)p Ft(W)7 b Fp(j)19 b(\024)g Ft(C)t Fp(h)p Ft(x)p Fp(i)1444 783 y Fo(\000)p Fm(\026)1495 801 y Fu(.)30 b(Whence,)228 859 y(using)13 b(Lemma)f(3.1,)h(the)g(expression)g(\(3.13\))h(ma)o(y)e (b)q(e)i(estimated)d(in)i(agreemen)o(t)228 917 y(with)j(\(3.10\))q(.) 278 975 y(So)g(it)g(su\016ces)g(to)h(pro)o(v)o(e)e(that)650 1062 y Fp(k)p Ft(AF)744 1069 y Fm(\020)763 1062 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)e(\024)h Ft(C)t(\017)971 1041 y Fo(\000)p Fs(1+)p Fm(\016)1060 1029 y Fg(0)1073 1062 y Fp(k)p Ft(F)1130 1069 y Fm(\020)1149 1062 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)1232 1041 y Fs(1)p Fm(=)p Fs(2)1287 1062 y Ft(:)297 b Fu(\(3.14\))228 1146 y(Using)16 b(that)648 1212 y Fp(kh)p Ft(\017A)p Fp(i)768 1191 y Fo(\000)p Fs(1)815 1212 y Fp(h)p Ft(A)p Fp(i)890 1191 y Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\016)988 1180 y Fg(0)1001 1212 y Fp(k)e(\024)f Ft(C)t(\017)1151 1191 y Fo(\000)p Fs(1)p Fm(=)p Fs(2+)p Fm(\016)1276 1180 y Fg(0)1288 1212 y Ft(;)228 1286 y Fu(\(3.14\))k(will)e(follo)o(w)h(from)f(the)h(b)q(ound)499 1375 y Fp(kh)p Ft(A)p Fp(i)599 1355 y Fs(1)p Fm(=)p Fs(2+)p Fm(\016)697 1343 y Fg(0)709 1375 y Ft(f)738 1355 y Fs(1)p Fm(=)p Fs(2)794 1375 y Fp(h)p Ft(x)p Fp(i)860 1355 y Fo(\000)p Fm(s)906 1375 y Ft(R)943 1382 y Fm(\020)963 1375 y Fu(\()p Ft(\017)p Fu(\))p Ft( )r Fp(k)1080 1355 y Fs(2)1113 1375 y Fp(\024)d Ft(C)t(\017)1224 1355 y Fo(\000)p Fs(1)1271 1375 y Fp(k)p Ft(F)1328 1382 y Fm(\020)1347 1375 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)p Ft(;)154 b Fu(\(3.15\))228 1468 y(where)16 b Ft( )f Fu(=)f Ft(f)497 1450 y Fs(1)p Fm(=)p Fs(2)552 1468 y Fp(h)p Ft(x)p Fp(i)618 1450 y Fo(\000)p Fm(s)664 1468 y Fp(h)p Ft(\017A)p Fp(i)759 1450 y Fo(\000)p Fs(1)807 1468 y Ft(\036)p Fu(;)h Fp(k)p Ft(\036)p Fp(k)f Fu(=)g(1.)21 b(F)l(or)c(this)f(b)q(ound)h(w)o(e)f (estimate)341 1516 y Fl(\020)371 1571 y Fp(h)p Ft(A)p Fp(i)446 1550 y Fs(1)p Fm(=)p Fs(2+)p Fm(\016)544 1539 y Fg(0)557 1571 y Ft(f)586 1550 y Fs(1)p Fm(=)p Fs(2)641 1571 y Fp(h)p Ft(x)p Fp(i)707 1550 y Fo(\000)p Fm(s)753 1516 y Fl(\021)783 1527 y Fo(\003)811 1571 y Fp(h)p Ft(A)p Fp(i)886 1550 y Fs(1)p Fm(=)p Fs(2+)p Fm(\016)984 1539 y Fg(0)997 1571 y Fp(h)p Ft(x)p Fp(i)1063 1550 y Fo(\000)p Fm(s)1109 1571 y Ft(f)1138 1550 y Fs(1)p Fm(=)p Fs(2)347 1668 y Fp(\024)d Ft(C)434 1675 y Fs(1)454 1668 y Ft(f)5 b Fp(h)p Ft(x)p Fp(i)549 1647 y Fo(\000)p Fs(2)p Fm(s)624 1668 y Fu(+)11 b Ft(f)702 1647 y Fs(1)p Fm(=)p Fs(2)758 1668 y Fp(h)p Ft(x)p Fp(i)824 1647 y Fo(\000)p Fm(s)870 1668 y Ft(A)907 1647 y Fs(2)926 1668 y Fp(h)p Ft(A)p Fp(i)1001 1647 y Fs(2)p Fm(\016)1036 1636 y Fg(0)1047 1647 y Fo(\000)p Fs(1)1094 1668 y Fp(h)p Ft(x)p Fp(i)1160 1647 y Fo(\000)p Fm(s)1207 1668 y Ft(f)1236 1647 y Fs(1)p Fm(=)p Fs(2)347 1796 y Fp(\024)426 1734 y Fm(d)399 1749 y Fl(X)405 1854 y Fm(j)r Fs(=1)480 1796 y Fp(<)524 1741 y Fl(\020)554 1796 y Ft(p)578 1803 y Fm(j)596 1796 y Ft(B)633 1803 y Fm(j)652 1796 y Fp(h)p Ft(A)p Fp(i)727 1776 y Fs(2)p Fm(\016)762 1764 y Fg(0)775 1796 y Ft(f)5 b Fp(h)p Ft(x)p Fp(i)870 1776 y Fo(\000)p Fs(2)p Fm(\016)934 1741 y Fl(\021)975 1796 y Fu(+)11 b Ft(C)1059 1803 y Fs(2)1079 1796 y Ft(f)1108 1776 y Fs(1)p Fm(=)p Fs(2)1163 1796 y Fp(h)p Ft(x)p Fp(i)1229 1776 y Fo(\000)p Fm(s)1276 1796 y Fp(h)p Ft(A)p Fp(i)1351 1776 y Fs(2)p Fm(\016)1386 1764 y Fg(0)1399 1796 y Fp(h)p Ft(x)p Fp(i)1465 1776 y Fo(\000)p Fm(s)1511 1796 y Ft(f)1540 1776 y Fs(1)p Fm(=)p Fs(2)1595 1796 y Fu(;)333 1930 y Ft(B)370 1937 y Fm(j)402 1930 y Fu(=)j Ft(f)483 1910 y Fs(1)p Fm(=)p Fs(2)538 1930 y Fp(h)p Ft(x)p Fp(i)604 1910 y Fo(\000)p Fm(s)650 1930 y Ft(x)678 1937 y Fm(j)696 1930 y Fp(h)p Ft(A)p Fp(i)771 1910 y Fs(2)p Fm(\016)806 1898 y Fg(0)817 1910 y Fo(\000)p Fs(1)865 1930 y Ft(Af)931 1910 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)1013 1930 y Fp(h)p Ft(x)p Fp(i)1079 1910 y Fm(s)p Fo(\000)p Fs(1)1143 1930 y Fp(h)p Ft(A)p Fp(i)1218 1910 y Fo(\000)p Fs(2)p Fm(\016)1280 1898 y Fg(0)1293 1930 y Ft(:)278 2015 y Fu(The)j(op)q(erators)h Ft(B)632 2022 y Fm(j)667 2015 y Fu(are)f(b)q(ounded)h(uniformly)d(in)h Ft(\020)t Fu(.)24 b(This)17 b(ma)o(y)e(readily)h(b)q(e)228 2073 y(pro)o(v)o(ed)21 b(b)o(y)g(the)h(tec)o(hniques)f(of)h(the)g(pro)q (of)h(of)f([M\034Sk02,)h(Lemma)d(2.5])i(and)228 2133 y(will)d(not)h(b)q(e)h(done)f(here.)32 b(\(W)l(e)20 b(mo)o(v)o(e)e(the) i(factor)g Fp(h)p Ft(A)p Fp(i)1284 2115 y Fs(2)p Fm(\016)1319 2103 y Fg(0)1330 2115 y Fo(\000)p Fs(1)1398 2133 y Fu(to)g(the)g(righ)o (t)g(b)o(y)228 2191 y(comm)o(utation)14 b(using)i(the)g(represen)o (tation)g(form)o(ula)f([M\034Sk02,)h(\(2.5\)].\))278 2250 y(The)22 b(t)o(w)o(o)h(terms)e(on)i(the)f(righ)o(t)h(hand)g(side)f (are)h(estimated)e(in)h(the)g(same)228 2308 y(fashion)c(in)f(the)g(exp) q(ectation)f(in)h(the)g(state)h Ft(R)1103 2315 y Fm(\020)1123 2308 y Fu(\()p Ft(\017)p Fu(\))p Ft( )r Fu(,)e(so)i(let)e(us)i(only)f (consider)228 2367 y(the)f(\014rst)g(one)h(in)f(details.)k(W)l(e)c(get) h(with)f Ft(g)g Fu(=)d Fp(h)p Ft(x)p Fp(i)1165 2349 y Fo(\000)p Fs(2)p Fm(\016)1230 2367 y Ft(f)294 2410 y Fl(\014)294 2440 y(\014)294 2470 y(\014)310 2467 y Fp(h)p Ft(p)353 2474 y Fm(j)372 2467 y Ft(R)409 2474 y Fm(\020)429 2467 y Fu(\()p Ft(\017)p Fu(\))p Ft( )r(;)8 b(B)580 2474 y Fm(j)598 2467 y Fp(h)p Ft(A)p Fp(i)673 2446 y Fs(2)p Fm(\016)708 2435 y Fg(0)721 2467 y Ft(g)r(R)783 2474 y Fm(\020)803 2467 y Fu(\()p Ft(\017)p Fu(\))p Ft( )r Fp(i)914 2410 y Fl(\014)914 2440 y(\014)914 2470 y(\014)299 2575 y Fp(\024)14 b Ft(C)398 2533 y Fl(\015)398 2563 y(\015)426 2575 y Ft(p)450 2582 y Fm(j)469 2575 y Ft(R)506 2582 y Fm(\020)526 2575 y Fu(\()p Ft(\017)p Fu(\))p Ft(f)613 2555 y Fs(1)p Fm(=)p Fs(2)668 2575 y Fp(h)p Ft(x)p Fp(i)734 2555 y Fo(\000)p Fm(s)780 2575 y Fp(h)p Ft(\017A)p Fp(i)875 2555 y Fo(\000)p Fs(1)922 2533 y Fl(\015)922 2563 y(\015)961 2575 y Fp(\001)986 2518 y Fl(\015)986 2548 y(\015)986 2578 y(\015)1014 2575 y Fp(h)p Ft(A)p Fp(i)1089 2555 y Fs(2)p Fm(\016)1124 2543 y Fg(0)1137 2575 y Ft(g)r(R)1199 2582 y Fm(\020)1219 2575 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)1343 2555 y Fo(\000)p Fm(s)1389 2575 y Ft(f)1418 2555 y Fs(1)p Fm(=)p Fs(2)1473 2575 y Fp(h)p Ft(\017A)p Fp(i)1568 2555 y Fo(\000)p Fs(1)1615 2518 y Fl(\015)1615 2548 y(\015)1615 2578 y(\015)1651 2575 y Ft(:)956 2658 y Fq(15)p eop %%Page: 16 16 16 15 bop 228 213 a Fu(An)16 b(application)g(of)g(Lemma)e(3.1)j(to)g (the)f(\014rst)g(factor)h(giv)o(es)482 263 y Fl(\015)482 293 y(\015)510 305 y Ft(p)534 312 y Fm(j)552 305 y Ft(R)589 312 y Fm(\020)609 305 y Fu(\()p Ft(\017)p Fu(\))p Ft(f)696 285 y Fs(1)p Fm(=)p Fs(2)751 305 y Fp(h)p Ft(x)p Fp(i)817 285 y Fo(\000)p Fm(s)864 305 y Fp(h)p Ft(\017A)p Fp(i)959 285 y Fo(\000)p Fs(1)1006 263 y Fl(\015)1006 293 y(\015)1047 305 y Fp(\024)d Ft(C)t(\017)1159 285 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)1241 305 y Fp(k)p Ft(F)1298 312 y Fm(\020)1317 305 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)1400 285 y Fs(1)p Fm(=)p Fs(2)1454 305 y Ft(:)278 397 y Fu(T)l(o)19 b(estimate)d(the)i (second)h(factor)g(w)o(e)f(use)g([M\034Sk02)q(,)g(Lemma)e(7.1],)j(whic) o(h)228 456 y(sa)o(ys)d(that)h(for)g(all)e Ft(s)602 437 y Fo(0)628 456 y Ft(>)e Fu(0)k(there)f(exists)f(a)i(constan)o(t)g Ft(C)g(>)d Fu(0)j(suc)o(h)f(that)718 549 y Fp(kh)p Ft(A)p Fp(i)818 528 y Fm(s)834 517 y Fg(0)848 549 y Fp(h)p Ft(x)p Fp(i)914 528 y Fo(\000)p Fm(s)957 517 y Fg(0)971 549 y Fp(h)p Ft(p)p Fp(i)1033 528 y Fo(\000)p Fm(s)1076 517 y Fg(0)1091 549 y Fp(k)e(\024)g Ft(C)q(;)228 640 y Fu(to)i(write)560 678 y Fl(\015)560 708 y(\015)560 737 y(\015)588 735 y Fp(h)p Ft(A)p Fp(i)663 714 y Fs(2)p Fm(\016)698 703 y Fg(0)711 735 y Ft(g)734 742 y Fm(j)752 735 y Ft(R)789 742 y Fm(\020)809 735 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)933 714 y Fo(\000)p Fm(s)979 735 y Ft(f)1008 714 y Fs(1)p Fm(=)p Fs(2)1064 735 y Fp(h)p Ft(\017A)p Fp(i)1159 714 y Fo(\000)p Fs(1)1206 678 y Fl(\015)1206 708 y(\015)1206 737 y(\015)566 843 y Fp(\024)d Ft(C)665 786 y Fl(\015)665 816 y(\015)665 846 y(\015)693 843 y Fp(h)p Ft(p)p Fp(ih)p Ft(x)p Fp(i)821 823 y Fs(2)p Fm(\016)856 811 y Fg(0)870 843 y Ft(g)r(R)932 850 y Fm(\020)952 843 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)1076 823 y Fo(\000)p Fm(s)1122 843 y Ft(f)1151 823 y Fs(1)p Fm(=)p Fs(2)1207 843 y Fp(h)p Ft(\017A)p Fp(i)1302 823 y Fo(\000)p Fs(1)1349 786 y Fl(\015)1349 816 y(\015)1349 846 y(\015)1385 843 y Ft(:)228 947 y Fu(By)i(Lemma)f(3.1)434 994 y Fl(\015)434 1024 y(\015)434 1054 y(\015)461 1052 y Fp(h)p Ft(x)p Fp(i)527 1031 y Fs(2)p Fm(\016)562 1019 y Fg(0)576 1052 y Ft(g)r(R)638 1059 y Fm(\020)658 1052 y Fu(\()p Ft(\017)p Fu(\))p Ft(f)745 1031 y Fs(1)p Fm(=)p Fs(2)800 1052 y Fp(h)p Ft(x)p Fp(i)866 1031 y Fo(\000)p Fm(s)912 1052 y Fp(h)p Ft(\017A)p Fp(i)1007 1031 y Fo(\000)p Fs(1)1054 994 y Fl(\015)1054 1024 y(\015)1054 1054 y(\015)1096 1052 y Fp(\024)f Ft(C)t(\017)1207 1031 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)1289 1052 y Fp(k)p Ft(F)1346 1059 y Fm(\020)1365 1052 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)1448 1031 y Fs(1)p Fm(=)p Fs(2)1503 1052 y Ft(:)228 1155 y Fu(Moreo)o(v)o(er)400 1247 y(\()p Ft(p)443 1254 y Fm(i)458 1247 y Fp(h)p Ft(x)p Fp(i)524 1226 y Fs(2)p Fm(\016)559 1215 y Fg(0)572 1247 y Ft(g)r Fu(\))616 1226 y Fo(\003)636 1247 y Ft(p)660 1254 y Fm(i)674 1247 y Fp(h)p Ft(x)p Fp(i)740 1226 y Fs(2)p Fm(\016)775 1215 y Fg(0)788 1247 y Ft(g)j Fu(=)e Ft(p)903 1254 y Fm(i)918 1247 y Ft(g)943 1226 y Fs(2)963 1247 y Fp(h)p Ft(x)p Fp(i)1029 1226 y Fs(4)p Fm(\016)1064 1215 y Fg(0)1077 1247 y Ft(p)1101 1254 y Fm(i)1127 1247 y Fu(+)d Ft(O)q Fu(\()p Fp(h)p Ft(x)p Fp(i)1299 1226 y Fo(\000)p Fs(2+4)p Fm(\016)1406 1215 y Fg(0)1418 1226 y Fo(\000)p Fs(4)p Fm(\016)1482 1247 y Ft(f)1511 1226 y Fs(2)1531 1247 y Fu(\))414 1327 y(=)j Ft(p)490 1334 y Fm(i)504 1327 y Ft(O)q Fu(\(1\))p Ft(p)628 1334 y Fm(i)655 1327 y Fu(+)d Ft(O)q Fu(\()p Ft(f)790 1306 y Fs(2)811 1327 y Fu(\))p Ft(;)228 1419 y Fu(from)k(whic)o(h)h(w)o(e)f(obtain)i (the)f(\014nal)g(estimate)436 1456 y Fl(\015)436 1486 y(\015)436 1516 y(\015)464 1513 y Fp(h)p Ft(A)p Fp(i)539 1493 y Fs(2)p Fm(\016)574 1481 y Fg(0)587 1513 y Ft(g)r(R)649 1520 y Fm(\020)669 1513 y Fu(\()p Ft(\017)p Fu(\))p Fp(h)p Ft(x)p Fp(i)793 1493 y Fo(\000)p Fm(s)839 1513 y Ft(f)868 1493 y Fs(1)p Fm(=)p Fs(2)923 1513 y Fp(h)p Ft(\017A)p Fp(i)1018 1493 y Fo(\000)p Fs(1)1065 1456 y Fl(\015)1065 1486 y(\015)1065 1516 y(\015)1107 1513 y Fp(\024)d Ft(C)t(\017)1218 1493 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)1300 1513 y Fp(k)p Ft(F)1357 1520 y Fm(\020)1376 1513 y Fu(\()p Ft(\017)p Fu(\))p Fp(k)1459 1493 y Fs(1)p Fm(=)p Fs(2)228 1618 y Fu(b)o(y)k(another)h(application)f(of)h(Lemma)e(3.1.)25 b(Therefore)17 b(w)o(e)g(ha)o(v)o(e)g(pro)o(v)o(ed)g(that)228 1676 y(the)11 b(con)o(tribution)h(from)e(the)i(\014rst)g(term)e(agrees) i(with)f(the)h(b)q(ound)h(\(3.15\).)20 b(Since)228 1734 y(the)c(other)g(term)f(can)h(b)q(e)g(treated)g(similarly)e(the)i(pro)q (of)h(of)j(\(3.8\))d(is)f(done.)p 1687 1734 2 33 v 1689 1703 30 2 v 1689 1734 V 1718 1734 2 33 v 278 1833 a(By)f(taking)i Ft(\017)c Fp(!)h Fu(0)i(w)o(e)g(obtain)228 1927 y Fy(Corollary)j(3.5.)k Fh(F)l(or)17 b(any)g Ft(\016)f(>)d Fu(0)18 b Fh(and)g Ft(\022)d Fp(2)f Fu(\(0)p Ft(;)8 b(\031)r Fu(\))17 b Fh(the)h(op)n(er)n(ators)505 2023 y Ft(\020)g Fp(7!)13 b(h)p Ft(x)p Fp(i)673 2002 y Fo(\000)p Fs(\(1)p Fm(=)p Fs(2+)p Fm(\026=)p Fs(4+)p Fm(\016)q Fs(\))912 2023 y Fu(\()p Ft(H)i Fp(\000)c Ft(\020)t Fu(\))1080 2002 y Fo(\000)p Fs(1)1127 2023 y Fp(h)p Ft(x)p Fp(i)1193 2002 y Fo(\000)p Fs(\(1)p Fm(=)p Fs(2+)p Fm(\026=)p Fs(4+)p Fm(\016)q Fs(\))1432 2023 y Ft(;)228 2114 y Fh(and)511 2185 y Ft(\020)18 b Fp(7!)c(h)p Ft(x)p Fp(i)680 2165 y Fo(\000)p Fs(\(1)p Fm(=)p Fs(2+)p Fm(\016)q Fs(\))834 2185 y Ft(f)863 2160 y Fs(1)p Fm(=)p Fs(2)858 2201 y Fo(j)p Fm(\020)r Fo(j)919 2185 y Fu(\()p Ft(H)h Fp(\000)c Ft(\020)t Fu(\))1087 2165 y Fo(\000)p Fs(1)1134 2185 y Ft(f)1163 2160 y Fs(1)p Fm(=)p Fs(2)1158 2201 y Fo(j)p Fm(\020)r Fo(j)1218 2185 y Fp(h)p Ft(x)p Fp(i)1284 2165 y Fo(\000)p Fs(\(1)p Fm(=)p Fs(2+)p Fm(\016)q Fs(\))228 2271 y Fh(ar)n(e)17 b(b)n(ounde)n(d)g(uniformly)g(in)h(the)g(se)n(ctor) g Fu(\000)1031 2278 y Fm(\022)1051 2271 y Fh(.)228 2366 y Fy(Remark)e(3.6.)24 b Fu(One)18 b(ma)o(y)g(sho)o(w)h(uniform)e(H\177) -24 b(older)18 b(con)o(tin)o(uit)o(y)f(in)i(\000)1558 2373 y Fm(\022)1596 2366 y Fu(of)g(the)228 2424 y(\014rst)e(function)f (in)h(Corollary)g(3.5)g(b)o(y)f(a)h(certain)f(mo)q(di\014cation)g(of)h (the)f(Mourre)228 2482 y(t)o(yp)q(e)j(metho)q(d)g(of)h([PSS81])g(using) f(b)q(ounds)i(from)e(the)g(presen)o(t)g(section.)31 b(The)228 2540 y(pro)q(of)18 b(w)o(e)f(giv)o(e)g(in)g(Section)g(5)h(is)f(more)f (e\016cien)o(t)g(and)i(yields)e(a)i(b)q(etter)f(H\177)-24 b(older)228 2598 y(exp)q(onen)o(t)16 b(although)h(it)f(requires)f(more) g(smo)q(othness)i(of)f(the)g(p)q(oten)o(tial.)956 2658 y Fq(16)p eop %%Page: 17 17 17 16 bop 598 213 a Fu(4.)27 b Fv(Itera)m(ted)19 b(resol)l(vents)d(f)o (or)i Ft(V)1332 220 y Fs(1)228 300 y Fu(4.1.)28 b Fy(Discussion)20 b(and)h(statemen)n(t)e(of)i(results.)i Fu(In)18 b(this)g(section)g(w)o (e)f(will)228 358 y(use)g(a)h(strategy)g(similar)d(to)j(the)f(one)g (from)g([GIS96])g(to)h(pro)o(v)o(e)e(w)o(eigh)o(ted)h(esti-)228 416 y(mates)12 b(for)h(iterated)g(resolv)o(en)o(ts.)19 b(In)13 b([GIS96])g(this)g(strategy)g(w)o(as)h(dev)o(elop)q(ed)e(to)228 474 y(pro)o(v)o(e)i(w)o(eigh)o(ted)g(estimates)f(for)j(p)q(o)o(w)o(ers) f(of)g(resolv)o(en)o(ts)f(at)h(energies)g(for)g(whic)o(h)228 532 y(a)k(Mourre)g(estimate)d(holds.)30 b(Although)19 b(the)f(di\013eren)o(t)g(\(m)o(ultiple)d(comm)o(uta-)228 590 y(tor\))i(approac)o(h)h(of)g([JMP84])f(can)g(b)q(e)g(extended)g(to) g(giv)o(e)f(some)h(results)g(in)f(the)228 648 y(presen)o(t)e(con)o (text,)g(it)h(seems)e(imp)q(ossible)g(to)j(deriv)o(e)d Fh(optimal)i Fu(w)o(eigh)o(ts)g(b)o(y)f(some)228 707 y([JMP84]-t)o(yp)q(e)21 b(tec)o(hnique.)34 b(Only)20 b(the)h(case)g Ft(V)34 b Fu(=)22 b Ft(V)1263 714 y Fs(1)1304 707 y Fu(is)f(considered,)h(as)g(in)228 765 y(Section)16 b(3.)278 823 y(The)g(follo)o(wing)g(sym)o(b)q(ols)f(will)g(pla)o(y)h(a) g(prominen)o(t)f(role:)538 932 y Ft(a)564 939 y Fs(0)583 932 y Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b(=)813 898 y Ft(\030)836 880 y Fs(2)p 765 921 140 2 v 765 966 a Ft(f)789 973 y Fm(E)819 966 y Fu(\()p Ft(x)p Fu(\))885 952 y Fs(2)909 932 y Ft(;)36 b(b)p Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b(=)1181 898 y Ft(x)p 1161 921 67 2 v 1161 966 a Fp(h)p Ft(x)p Fp(i)1244 932 y(\001)1322 898 y Ft(\030)p 1274 921 120 2 v 1274 966 a(f)1298 973 y Fm(E)1328 966 y Fu(\()p Ft(x)p Fu(\))1398 932 y Ft(:)210 b Fu(\(4.1\))228 1040 y(Here)15 b(and)i(in)f(the)g(rest)g(of)g(the)g(pap)q(er)573 1142 y Ft(f)j Fu(=)14 b Ft(f)692 1149 y Fm(E)736 1142 y Fu(=)787 1080 y Fl(q)p 837 1080 541 2 v 62 x Ft(\024)865 1122 y Fo(\000)p Fs(2)865 1154 y(0)912 1142 y Ft(E)g Fu(+)d(\(1)h Fp(\000)f Ft(\026=)p Fu(2\))1212 1128 y Fo(\000)p Fs(1)1260 1142 y Fp(h)p Ft(x)p Fp(i)1326 1128 y Fo(\000)p Fm(\026)1622 1142 y Fu(\(4.2\))228 1231 y(is)g(essen)o (tially)f(the)i(function)f(from)g(\(3.2\)|w)o(e)g(ha)o(v)o(e)g(only)h (in)o(tro)q(duced)f(a)h(dep)q(en-)228 1290 y(dence)18 b(on)i(the)f(constan)o(ts)i Ft(\026;)8 b(\024)823 1297 y Fs(0)862 1290 y Ft(>)19 b Fu(0)h(that)f(app)q(ear)i(in)e(Theorem)f (4.1)i(b)q(elo)o(w.)228 1348 y(These)i(constan)o(ts)i(are)e(imp)q(osed) g(on)h(us,)h(due)e(to)h(the)f(need)h(for)f(\(4.35\))i(and)228 1406 y(\(4.36\))15 b(\(and)h(their)e(quan)o(tum)f(analogues\))j(to)g (hold.)21 b(In)14 b(a)h(part)h(of)f(this)g(section)228 1464 y(the)k(simpler)f(de\014nition)h(\(3.2\))i(of)f Ft(f)25 b Fu(w)o(ould)20 b(b)q(e)g(su\016cien)o(t,)f(but)h(in)g(the)f (criti-)228 1522 y(cal)d(Subsection)h(4.5)g(the)g(de\014nition)f (\(4.2\))h(ab)q(o)o(v)o(e)g(will)f(b)q(e)h(imp)q(ortan)o(t.)22 b(These)228 1580 y(sym)o(b)q(ols)17 b(will)h(b)q(e)g(studied)g (together)h(with)f(the)h(resolv)o(en)o(t)e Ft(R)p Fu(\()p Ft(\020)t Fu(\).)28 b(W)l(e)18 b(alw)o(a)o(ys)228 1638 y(tak)o(e)g Ft(E)j Fu(=)d Fp(j)p Ft(\020)t Fp(j)h Fu(with)f Ft(\020)23 b Fp(2)18 b Fu(\000)759 1645 y Fm(\022)798 1638 y Fu(in)g(\(4.1\).)29 b(Th)o(us)19 b Ft(a)1152 1645 y Fs(0)1190 1638 y Fu(and)h Ft(b)e Fu(dep)q(end)h(on)g Ft(\020)t Fu(,)g(ev)o(en)228 1696 y(though)14 b(w)o(e)f(do)h(not)g (include)e(this)h(explicitly)d(in)j(the)g(notation.)22 b(W)l(e)13 b(also)h(de\014ne)228 1755 y(a)j(useful)f(auxiliary)f(w)o (eigh)o(t)h(function)g Ft(w)i Fu(and)f(the)f(sym)o(b)q(ol)f Ft(h)i Fu(of)g(the)f(op)q(erator)228 1813 y Ft(H)k Fu(b)o(y)448 1893 y Ft(w)15 b Fu(=)f Ft(w)585 1900 y Fm(E)615 1893 y Fu(\()p Ft(x)p Fu(\))f(=)h Fp(h)p Ft(x)p Fp(i)p Ft(f)836 1900 y Fm(E)866 1893 y Fu(\()p Ft(x)p Fu(\))p Ft(;)35 b(h)14 b Fu(=)g Ft(h)p Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b(=)g Ft(\030)1303 1872 y Fs(2)1334 1893 y Fu(+)d Ft(V)h Fu(\()p Ft(x)p Fu(\))p Ft(:)119 b Fu(\(4.3\))278 1976 y(W)l(e)15 b(will)g(let)g(Op)580 1958 y Fs(w)600 1976 y Fu(\()p Ft(a)p Fu(\))g(denote)h(the)g(W)l(eyl)f(quan)o(tization)g(of) h(a)h(sym)o(b)q(ol)d Ft(a)p Fu(.)21 b(Ex-)228 2035 y(plicitly)14 b(Op)449 2016 y Fs(w)468 2035 y Fu(\()p Ft(a)p Fu(\))i(acts)h(as)f (follo)o(ws)351 2141 y(\(Op)427 2121 y Fs(w)447 2141 y Fu(\()p Ft(a)p Fu(\))p Ft(\036)p Fu(\)\()p Ft(x)p Fu(\))d(=)g(\(2)p Ft(\031)r Fu(\))781 2121 y Fo(\000)p Fm(d)837 2073 y Fl(Z)5 b(Z)928 2141 y Ft(e)951 2121 y Fm(i)p Fs(\()p Fm(x)p Fo(\000)p Fm(y)q Fs(\))p Fm(\030)1075 2141 y Ft(a)p Fu(\(\()p Ft(x)11 b Fu(+)g Ft(y)r Fu(\))p Ft(=)p Fu(2)p Ft(;)d(\030)r Fu(\))p Ft(\036)p Fu(\()p Ft(y)r Fu(\))g Ft(dy)r(d\030)r(;)228 2245 y Fu(with)16 b(mapping)f(prop)q(erties)i (dep)q(ending,)f(of)g(course,)g(on)h(the)f(sym)o(b)q(ol)f Ft(a)p Fu(.)278 2303 y(W)l(e)h(will)f(b)q(e)h(concerned)g(with)g(pro)o (ving)g(the)g(follo)o(wing)g(result:)228 2392 y Fy(Theorem)g(4.1.)24 b Fh(L)n(et)e Ft(V)12 b Fu(\()p Ft(x)p Fu(\))22 b Fh(satisfy)g(the)i(c) n(onditions)f(of)g(The)n(or)n(em)e(1.1)i(with)228 2450 y Ft(V)256 2457 y Fs(2)292 2450 y Fu(=)15 b(0)p Fh(.)26 b(We)18 b(r)n(eformulate)h(the)g(assumption)f(\(3\))g(as:)25 b(F)l(or)18 b(some)g Ft(\024)1513 2457 y Fs(0)1549 2450 y Ft(>)d Fu(0)k Fh(and)228 2508 y Fu(2)14 b Ft(>)g(\026)g(>)g Fu(0)p Fh(,)529 2588 y Ft(W)7 b Fu(\()p Ft(x)p Fu(\))13 b(=)h Fp(\000)p Fu(2)p Ft(V)d Fu(\()p Ft(x)p Fu(\))g Fp(\000)g Ft(x)g Fp(\001)f(r)p Ft(V)h Fu(\()p Ft(x)p Fu(\))j Fp(\025)f Fu(2)p Ft(\024)1270 2568 y Fs(2)1270 2601 y(0)1291 2588 y Fp(h)p Ft(x)p Fp(i)1357 2568 y Fo(\000)p Fm(\026)1408 2588 y Ft(:)200 b Fu(\(4.4\))956 2658 y Fq(17)p eop %%Page: 18 18 18 17 bop 228 213 a Fh(L)n(et)17 b Ft(\022)e Fp(2)f Fu(\(0)p Ft(;)8 b(\031)r Fu(\))16 b Fh(and)i Fu(\000)652 220 y Fm(\022)689 213 y Fh(b)n(e)f(as)g(de\014ne)n(d)h(in)k Fu(\(1.3\))p Fh(.)g(L)n(et)17 b Ft(a)1285 220 y Fs(0)1322 213 y Fh(and)g Ft(b)g Fh(b)n(e)g(as)g(de\014ne)n(d)228 272 y(in)k Fu(\(4.1\))d Fh(with)g Ft(E)f Fu(=)d Fp(j)p Ft(\020)t Fp(j)p Fh(.)23 b(De\014ne)c Ft(k)d Fu(=)e Ft(k)r Fu(\()p Ft(x)p Fu(\))g(=)g Fp(h)p Ft(x)p Fp(i)1183 254 y Fs(1+)p Fm(\026=)p Fs(2)1287 272 y Fh(.)23 b(Then)18 b(the)g(fol)r(lowing)228 330 y(c)n(onclusions,)d Fu(\(i\))10 b(-)i(\(iv\))p Fh(,)h(hold)g(for)f Ft(H)18 b Fu(=)c Ft(p)1000 312 y Fs(2)1021 330 y Fu(+)q Ft(V)23 b Fh(with)14 b(al)r(l)g(b)n(ounds) f(b)n(eing)g(uniform)228 388 y(in)18 b Ft(\020)f Fp(2)e Fu(\000)404 395 y Fm(\022)424 388 y Fh(:)267 460 y Fu(\(i\))20 b Fh(F)l(or)d(al)r(l)i Ft(\017)14 b(>)f Fu(0)18 b Fh(ther)n(e)g(exists) g Ft(C)f(>)d Fu(0)k Fh(such)g(that)691 549 y Fp(k)p Ft(k)743 529 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)867 549 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(k)994 529 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1119 549 y Fp(k)13 b(\024)h Ft(C)q(:)338 b Fu(\(4.5a\))254 637 y(\(ii\))19 b Fh(Ther)n(e)k(exists)h Ft(C)659 644 y Fs(0)703 637 y Ft(>)h Fu(0)p Fh(,)g(dep)n(ending)f(only)g(on)f Ft(V)12 b Fh(,)24 b(such)g(that)f(if)g Ft(F)1635 644 y Fs(+)1689 637 y Fp(2)339 695 y(B)374 677 y Fo(1)411 695 y Fu(\()p Fn(R)p Fu(\))p Fh(,)13 b Fu(supp)q(\()p Ft(F)668 702 y Fs(+)697 695 y Fu(\))h Fp(\032)f Fu(\()p Ft(C)836 702 y Fs(0)856 695 y Ft(;)8 b Fp(1)p Fu(\))16 b Fh(and)h Ft(F)1096 677 y Fo(0)1089 708 y Fs(+)1131 695 y Fp(2)d Ft(C)1217 677 y Fo(1)1213 708 y Fs(0)1254 695 y Fu(\()p Fn(R)p Fu(\))p Fh(,)g(then)j(for)f(al)r(l)i Ft(\017)13 b(>)h Fu(0)339 754 y Fh(and)k(al)r(l)h Ft(t)13 b(>)h Fu(0)k Fh(ther)n(e)f(exists)h Ft(C)g(>)c Fu(0)j Fh(such)h(that)531 839 y Fp(k)p Ft(k)583 818 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)720 839 y Fu(Op)777 818 y Fs(w)796 839 y Fu(\()p Ft(F)847 846 y Fs(+)876 839 y Fu(\()p Ft(a)921 846 y Fs(0)940 839 y Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(k)1105 818 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1270 839 y Fp(k)c(\024)g Ft(C)q(;)183 b Fu(\(4.5b\))531 922 y Fp(k)p Ft(k)583 902 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)747 922 y Ft(R)p Fu(\()p Ft(\020)t Fu(\)Op)904 902 y Fs(w)924 922 y Fu(\()p Ft(F)975 929 y Fs(+)1004 922 y Fu(\()p Ft(a)1049 929 y Fs(0)1068 922 y Fu(\)\))p Ft(k)1133 902 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1270 922 y Fp(k)14 b(\024)g Ft(C)q(:)188 b Fu(\(4.5c\))240 1019 y(\(iii\))19 b Fh(L)n(et)434 1006 y Fu(~)423 1019 y Ft(F)455 1026 y Fs(+)484 1019 y Ft(;)517 1006 y Fu(~)506 1019 y Ft(F)538 1026 y Fo(\000)581 1019 y Fp(2)14 b(B)663 1001 y Fo(1)700 1019 y Fu(\()p Fn(R)o Fu(\))g Fh(satisfy)j(\(with)g Ft(\024)1095 1026 y Fs(0)1132 1019 y Fh(fr)n(om)j Fu(\(4.4\))p Fh(\))d(for)f(some)h Ft(\024)d(>)g Fu(0)p Fh(,)377 1081 y Fp(\017)21 b Fu(inf)11 b(supp)q(\()621 1068 y(~)610 1081 y Ft(F)642 1088 y Fs(+)671 1081 y Fu(\))j Ft(>)f Fp(\000)p Ft(\024)h(>)g Fp(\000)p Ft(\024)955 1088 y Fs(0)974 1081 y Ft(;)j Fu(sup)9 b(supp)q(\()1218 1068 y(~)1207 1081 y Ft(F)1239 1088 y Fo(\000)1268 1081 y Fu(\))14 b Ft(<)f(\024)h(<)g(\024)1474 1088 y Fs(0)1494 1081 y Fh(.)377 1143 y Fp(\017)434 1131 y Fu(~)423 1143 y Ft(F)462 1125 y Fo(0)455 1156 y(\000)484 1143 y Ft(;)517 1131 y Fu(~)506 1143 y Ft(F)545 1125 y Fo(0)538 1156 y Fs(+)581 1143 y Fp(2)g Ft(C)667 1125 y Fo(1)663 1156 y Fs(0)704 1143 y Fu(\()p Fn(R)p Fu(\))p Fh(.)339 1201 y(L)n(et)20 b Ft(F)458 1208 y Fo(\000)505 1201 y Fp(2)f Ft(C)596 1183 y Fo(1)592 1214 y Fs(0)633 1201 y Fu(\()p Fn(R)p Fu(\))p Fh(.)27 b(Then)20 b(for)f(al)r(l)j Ft(\017;)8 b(t)17 b(>)h Fu(0)j Fh(ther)n(e)f(exists)g Ft(C)i(>)d Fu(0)h Fh(such)339 1260 y(that)407 1345 y Fp(k)p Ft(k)459 1324 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)596 1345 y Fu(Op)653 1324 y Fs(w)673 1345 y Fu(\()p Ft(F)724 1352 y Fo(\000)753 1345 y Fu(\()p Ft(a)798 1352 y Fs(0)817 1345 y Fu(\))847 1332 y(~)836 1345 y Ft(F)868 1352 y Fo(\000)897 1345 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(k)1102 1324 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1266 1345 y Fp(k)14 b(\024)g Ft(C)q(;)187 b Fu(\(4.5d\))407 1429 y Fp(k)p Ft(k)459 1408 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)623 1429 y Ft(R)p Fu(\()p Ft(\020)t Fu(\)Op)780 1408 y Fs(w)801 1429 y Fu(\()p Ft(F)852 1436 y Fo(\000)881 1429 y Fu(\()p Ft(a)926 1436 y Fs(0)945 1429 y Fu(\))975 1416 y(~)964 1429 y Ft(F)996 1436 y Fs(+)1025 1429 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1130 1408 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1266 1429 y Fp(k)14 b(\024)g Ft(C)q(:)192 b Fu(\(4.5e\))241 1525 y(\(iv\))20 b Fh(Supp)n(ose)538 1512 y Fu(~)527 1525 y Ft(F)559 1532 y Fs(+)607 1525 y Fh(and)714 1512 y Fu(~)703 1525 y Ft(F)735 1532 y Fo(\000)783 1525 y Fh(satisfy)f(the)h(assumptions)f(fr)n(om)f Fu(\(iii\))p Fh(,)g Ft(F)1567 1507 y Fs(1)1560 1537 y Fo(\000)1589 1525 y Ft(;)8 b(F)1650 1507 y Fs(2)1643 1537 y Fo(\000)1689 1525 y Fp(2)339 1583 y Ft(C)378 1565 y Fo(1)374 1595 y Fs(0)415 1583 y Fu(\()p Fn(R)p Fu(\))15 b Fh(and)654 1672 y Fu(dist\(supp)q(\()883 1660 y(~)872 1672 y Ft(F)904 1679 y Fs(+)933 1672 y Fu(\))p Ft(;)8 b Fu(supp\()1104 1660 y(~)1093 1672 y Ft(F)1125 1679 y Fo(\000)1154 1672 y Fu(\)\))14 b Ft(>)g Fu(0)p Ft(:)339 1758 y Fh(Then)k(for)f(al)r(l)i Ft(t)13 b(>)h Fu(0)k Fh(ther)n(e)f(exists)i Ft(C)e(>)d Fu(0)k Fh(such)f(that)362 1844 y Fp(k)p Ft(k)414 1824 y Fm(t)429 1844 y Fu(Op)486 1824 y Fs(w)505 1844 y Fu(\()p Ft(F)563 1824 y Fs(1)556 1857 y Fo(\000)586 1844 y Fu(\()p Ft(a)631 1851 y Fs(0)650 1844 y Fu(\))680 1832 y(~)669 1844 y Ft(F)701 1851 y Fo(\000)730 1844 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\)Op)965 1824 y Fs(w)985 1844 y Fu(\()p Ft(F)1043 1824 y Fs(2)1036 1857 y Fo(\000)1065 1844 y Fu(\()p Ft(a)1110 1851 y Fs(0)1129 1844 y Fu(\))1159 1832 y(~)1148 1844 y Ft(F)1180 1851 y Fs(+)1209 1844 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1314 1824 y Fm(t)1328 1844 y Fp(k)d(\024)g Ft(C)q(:)134 b Fu(\(4.5f)s(\))339 1941 y Fh(Supp)n(ose)25 b Ft(F)564 1948 y Fs(+)618 1941 y Fh(is)f(given)j(as)d(in)h Fu(\(ii\))p Fh(,)h(some)e(functions)1416 1928 y Fu(~)1405 1941 y Ft(F)1437 1948 y Fs(+)1466 1941 y Ft(;)1499 1928 y Fu(~)1488 1941 y Ft(F)1520 1948 y Fo(\000)1549 1941 y Ft(;)8 b(F)1603 1948 y Fo(\000)1657 1941 y Fh(ar)n(e)339 1999 y(given)19 b(as)f(in)f Fu(\(iii\))f Fh(and)i(supp)n(ose)654 2084 y Fu(dist\(supp)q(\()p Ft(F)904 2091 y Fo(\000)933 2084 y Fu(\))p Ft(;)8 b Fu(supp\()p Ft(F)1125 2091 y Fs(+)1154 2084 y Fu(\)\))14 b Ft(>)g Fu(0)p Ft(:)339 2169 y Fh(Then)k(for)f(al)r (l)i Ft(t)13 b(>)h Fu(0)k Fh(ther)n(e)f(exists)i Ft(C)e(>)d Fu(0)k Fh(such)f(that)415 2256 y Fp(k)p Ft(k)467 2235 y Fm(t)481 2256 y Fu(Op)538 2235 y Fs(w)558 2256 y Fu(\()p Ft(F)609 2263 y Fs(+)638 2256 y Fu(\()p Ft(a)683 2263 y Fs(0)702 2256 y Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\)Op)897 2235 y Fs(w)917 2256 y Fu(\()p Ft(F)968 2263 y Fo(\000)997 2256 y Fu(\()p Ft(a)1042 2263 y Fs(0)1062 2256 y Fu(\))1092 2243 y(~)1081 2256 y Ft(F)1113 2263 y Fs(+)1142 2256 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1247 2235 y Fm(t)1261 2256 y Fp(k)d(\024)f Ft(C)q(;)195 b Fu(\(4.5g)q(\))415 2340 y Fp(k)p Ft(k)467 2319 y Fm(t)481 2340 y Fu(Op)538 2319 y Fs(w)558 2340 y Fu(\()p Ft(F)609 2347 y Fo(\000)638 2340 y Fu(\()p Ft(a)683 2347 y Fs(0)702 2340 y Fu(\))732 2327 y(~)721 2340 y Ft(F)753 2347 y Fo(\000)782 2340 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\)Op)1017 2319 y Fs(w)1037 2340 y Fu(\()p Ft(F)1088 2347 y Fs(+)1117 2340 y Fu(\()p Ft(a)1162 2347 y Fs(0)1181 2340 y Fu(\)\))p Ft(k)1246 2319 y Fm(t)1261 2340 y Fp(k)14 b(\024)f Ft(C)q(:)193 b Fu(\(4.5h\))278 2433 y(Using)22 b(purely)g(algebraic)g(argumen)o(ts,)h(cf.)40 b([Jen85])22 b(or)h([Iso85)q(],)g(one)g(can)228 2492 y(get)17 b(the)h(follo)o(wing)f(result)g(from)f(Theorem)g(4.1.)25 b(An)17 b(elab)q(oration)i(is)e(giv)o(en)g(in)228 2550 y(App)q(endix)12 b(A.)19 b(\(See)12 b(\(5.8\))h(for)g(another)g (application)f(of)h(the)g(partition)f(of)h(unit)o(y)228 2608 y(needed)i(in)h(our)h(case.\))956 2658 y Fq(18)p eop %%Page: 19 19 19 18 bop 228 213 a Fy(Theorem)16 b(4.2.)24 b Fh(L)n(et)d(the)g (assumptions)g(and)g(notations)h(b)n(e)f(as)g(in)g(The)n(or)n(em)228 271 y(4.1.)g(Then)c(the)g(fol)r(lowing)h(c)n(onclusions)f(hold)f(for)g Ft(H)i Fu(=)c Ft(p)1310 253 y Fs(2)1338 271 y Fu(+)8 b Ft(V)27 b Fh(\(uniformly)16 b(in)228 329 y Ft(\020)i Fp(2)c Fu(\000)344 336 y Fm(\022)364 329 y Fh(\):)267 398 y Fu(\(i\))20 b Fh(L)n(et)g Ft(m)e Fp(2)h Fn(N)i Fh(and)f(let)h Ft(\017)d(>)g Fu(0)j Fh(b)n(e)f(arbitr)n(ary.)28 b(Then)21 b(ther)n(e)f(exists)g Ft(C)i(>)d Fu(0)339 456 y Fh(such)f(that)588 537 y Fp(k)p Ft(k)640 517 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)850 537 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))950 517 y Fm(m)984 537 y Ft(k)1011 517 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1221 537 y Fp(k)c(\024)g Ft(C)q(:)235 b Fu(\(4.6a\))254 621 y(\(ii\))19 b Fh(L)n(et)e Ft(C)458 628 y Fs(0)494 621 y Fh(b)n(e)g(the)g(numb)n(er)g(fr)n(om)e Fu(\(ii\))g Fh(in)i(The)n(or)n(em)f(4.1)g(and)h(supp)n(ose)f Ft(F)1646 628 y Fs(+)1689 621 y Fp(2)339 679 y(B)374 661 y Fo(1)411 679 y Fu(\()p Fn(R)p Fu(\))p Fh(,)22 b Fu(supp\()p Ft(F)676 686 y Fs(+)706 679 y Fu(\))j Fp(\032)g Fu(\()p Ft(C)868 686 y Fs(0)887 679 y Ft(;)8 b Fp(1)p Fu(\))24 b Fh(and)f Ft(F)1141 661 y Fo(0)1134 691 y Fs(+)1189 679 y Fp(2)i Ft(C)1286 661 y Fo(1)1282 691 y Fs(0)1323 679 y Fu(\()p Fn(R)p Fu(\))p Fh(.)38 b(Then)24 b(for)f(al)r(l)339 737 y Ft(m)14 b Fp(2)g Fn(N)k Fh(and)g(al)r(l)h Ft(\017;)8 b(t)13 b(>)g Fu(0)18 b Fh(ther)n(e)g(exists)g Ft(C)f(>)d Fu(0)k Fh(such)g(that)485 819 y Fp(k)p Ft(k)537 798 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)674 819 y Fu(Op)731 798 y Fs(w)750 819 y Fu(\()p Ft(F)801 826 y Fs(+)830 819 y Fu(\()p Ft(a)875 826 y Fs(0)894 819 y Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))1032 798 y Fm(m)1066 819 y Ft(k)1093 798 y Fo(\000)p Fm(t)p Fo(\000)p Fm(m)p Fs(+1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1316 819 y Fp(k)c(\024)g Ft(C)q(;)137 b Fu(\(4.6b\))485 903 y Fp(k)p Ft(k)537 882 y Fo(\000)p Fm(t)p Fo(\000)p Fm(m)p Fs(+1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)760 903 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))860 882 y Fm(m)894 903 y Fu(Op)951 882 y Fs(w)970 903 y Fu(\()p Ft(F)1021 910 y Fs(+)1050 903 y Fu(\()p Ft(a)1095 910 y Fs(0)1114 903 y Fu(\)\))p Ft(k)1179 882 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1316 903 y Fp(k)14 b(\024)g Ft(C)q(:)142 b Fu(\(4.6c\))240 995 y(\(iii\))19 b Fh(L)n(et)434 982 y Fu(~)423 995 y Ft(F)455 1002 y Fs(+)484 995 y Ft(;)517 982 y Fu(~)506 995 y Ft(F)538 1002 y Fo(\000)567 995 y Ft(;)8 b(F)621 1002 y Fo(\000)666 995 y Fh(satisfy)16 b(the)h(assumptions)g(fr)n(om)e Fu(\(iii\))g Fh(in)i(The)n(or)n(em)e (4.1.)339 1053 y(Then)j(for)f(al)r(l)i Ft(m)13 b Fp(2)h Fn(N)19 b Fh(and)f(al)r(l)g Ft(\017;)8 b(t)13 b(>)h Fu(0)k Fh(ther)n(e)f(exists)i Ft(C)e(>)d Fu(0)j Fh(such)h(that)361 1135 y Fp(k)p Ft(k)413 1114 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)550 1135 y Fu(Op)607 1114 y Fs(w)627 1135 y Fu(\()p Ft(F)678 1142 y Fo(\000)707 1135 y Fu(\()p Ft(a)752 1142 y Fs(0)771 1135 y Fu(\))801 1122 y(~)790 1135 y Ft(F)822 1142 y Fo(\000)851 1135 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))1029 1114 y Fm(m)1062 1135 y Ft(k)1089 1114 y Fo(\000)p Fm(t)p Fo(\000)p Fm(m)p Fs(+1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1312 1135 y Fp(k)c(\024)g Ft(C)q(;)141 b Fu(\(4.6d\))361 1219 y Fp(k)p Ft(k)413 1198 y Fo(\000)p Fm(t)p Fo(\000)p Fm(m)p Fs(+1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)636 1219 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))736 1198 y Fm(m)770 1219 y Fu(Op)827 1198 y Fs(w)846 1219 y Fu(\()p Ft(F)897 1226 y Fo(\000)927 1219 y Fu(\()p Ft(a)972 1226 y Fs(0)991 1219 y Fu(\))1021 1206 y(~)1010 1219 y Ft(F)1042 1226 y Fs(+)1071 1219 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1176 1198 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1312 1219 y Fp(k)14 b(\024)g Ft(C)q(:)146 b Fu(\(4.6e\))241 1311 y(\(iv\))20 b Fh(Supp)n(ose)538 1298 y Fu(~)527 1311 y Ft(F)559 1318 y Fs(+)607 1311 y Fh(and)714 1298 y Fu(~)703 1311 y Ft(F)735 1318 y Fo(\000)783 1311 y Fh(satisfy)f(the)h(assumptions)f(fr)n(om)f Fu(\(iii\))p Fh(,)g Ft(F)1567 1293 y Fs(1)1560 1323 y Fo(\000)1589 1311 y Ft(;)8 b(F)1650 1293 y Fs(2)1643 1323 y Fo(\000)1689 1311 y Fp(2)339 1369 y Ft(C)378 1351 y Fo(1)374 1381 y Fs(0)415 1369 y Fu(\()p Fn(R)p Fu(\))15 b Fh(and)654 1454 y Fu(dist\(supp)q(\()883 1441 y(~)872 1454 y Ft(F)904 1461 y Fs(+)933 1454 y Fu(\))p Ft(;)8 b Fu(supp\()1104 1441 y(~)1093 1454 y Ft(F)1125 1461 y Fo(\000)1154 1454 y Fu(\)\))14 b Ft(>)g Fu(0)p Ft(:)339 1534 y Fh(Then)k(for)f(al)r(l)i Ft(m)13 b Fp(2)h Fn(N)19 b Fh(and)f(al)r(l)g Ft(t)c(>)f Fu(0)18 b Fh(ther)n(e)g(exists)g Ft(C)f(>)d Fu(0)k Fh(such)g(that)346 1616 y Fp(k)p Ft(k)398 1596 y Fm(t)412 1616 y Fu(Op)469 1596 y Fs(w)489 1616 y Fu(\()p Ft(F)547 1596 y Fs(1)540 1629 y Fo(\000)569 1616 y Fu(\()p Ft(a)614 1623 y Fs(0)633 1616 y Fu(\))663 1604 y(~)652 1616 y Ft(F)684 1623 y Fo(\000)713 1616 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))891 1596 y Fm(m)925 1616 y Fu(Op)982 1596 y Fs(w)1001 1616 y Fu(\()p Ft(F)1059 1596 y Fs(2)1052 1629 y Fo(\000)1081 1616 y Fu(\()p Ft(a)1126 1623 y Fs(0)1146 1616 y Fu(\))1176 1604 y(~)1165 1616 y Ft(F)1197 1623 y Fs(+)1226 1616 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1331 1596 y Fm(t)1345 1616 y Fp(k)c(\024)f Ft(C)q(:)118 b Fu(\(4.6f)s(\))339 1708 y Fh(Supp)n(ose)25 b Ft(F)564 1715 y Fs(+)618 1708 y Fh(is)f(given)j(as)d(in)h Fu(\(ii\))p Fh(,)h(some)e(functions)1416 1696 y Fu(~)1405 1708 y Ft(F)1437 1715 y Fs(+)1466 1708 y Ft(;)1499 1696 y Fu(~)1488 1708 y Ft(F)1520 1715 y Fo(\000)1549 1708 y Ft(;)8 b(F)1603 1715 y Fo(\000)1657 1708 y Fh(ar)n(e)339 1767 y(given)19 b(as)f(in)f Fu(\(iii\))f Fh(and)i(supp)n(ose)654 1847 y Fu(dist\(supp)q(\()p Ft(F)904 1854 y Fo(\000)933 1847 y Fu(\))p Ft(;)8 b Fu(supp\()p Ft(F)1125 1854 y Fs(+)1154 1847 y Fu(\)\))14 b Ft(>)g Fu(0)p Ft(:)339 1928 y Fh(Then)k(for)f(al)r(l)i Ft(m)13 b Fp(2)h Fn(N)19 b Fh(and)f(al)r(l)g Ft(t)c(>)f Fu(0)18 b Fh(ther)n(e)g(exists)g Ft(C)f(>)d Fu(0)k Fh(such)g(that)401 2010 y Fp(k)p Ft(k)453 1990 y Fm(t)468 2010 y Fu(Op)525 1990 y Fs(w)544 2010 y Fu(\()p Ft(F)595 2017 y Fs(+)624 2010 y Fu(\()p Ft(a)669 2017 y Fs(0)689 2010 y Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))827 1990 y Fm(m)860 2010 y Fu(Op)917 1990 y Fs(w)937 2010 y Fu(\()p Ft(F)988 2017 y Fo(\000)1017 2010 y Fu(\()p Ft(a)1062 2017 y Fs(0)1081 2010 y Fu(\))1111 1998 y(~)1100 2010 y Ft(F)1132 2017 y Fs(+)1161 2010 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1266 1990 y Fm(t)1281 2010 y Fp(k)c(\024)f Ft(C)q(;)175 b Fu(\(4.6g)q(\))401 2094 y Fp(k)p Ft(k)453 2073 y Fm(t)468 2094 y Fu(Op)525 2073 y Fs(w)544 2094 y Fu(\()p Ft(F)595 2101 y Fo(\000)624 2094 y Fu(\()p Ft(a)669 2101 y Fs(0)689 2094 y Fu(\))719 2081 y(~)708 2094 y Ft(F)740 2101 y Fo(\000)769 2094 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))947 2073 y Fm(m)980 2094 y Fu(Op)1037 2073 y Fs(w)1057 2094 y Fu(\()p Ft(F)1108 2101 y Fs(+)1137 2094 y Fu(\()p Ft(a)1182 2101 y Fs(0)1201 2094 y Fu(\)\))p Ft(k)1266 2073 y Fm(t)1281 2094 y Fp(k)14 b(\024)f Ft(C)q(:)173 b Fu(\(4.6h\))278 2183 y(The)13 b(remaining)f(part)j(of)f(the)f(presen) o(t)g(section)h(is)f(dev)o(oted)g(to)h(pro)o(ving)g(The-)228 2241 y(orem)h(4.1.)278 2299 y(The)f(\014rst)h(statemen)o(t)e(\(4.5a\))i (is)g(a)g(restatemen)o(t)e(of)18 b(\(1.4\))d(\(in)f(the)h(case)f Ft(V)1650 2306 y Fs(2)1684 2299 y Fu(=)228 2357 y(0\))i(whic)o(h)g(w)o (as)h(pro)o(v)o(ed)e(in)h(Section)g(3,)g(cf.)21 b(Corollary)16 b(3.5.)278 2415 y(In)e(Subsection)g(4.2)h(w)o(e)f(shall)g(discuss)h(a)f (pseudo)q(di\013eren)o(tial)g(calculus)g(giv)o(en)228 2479 y(in)i(terms)e(of)j(the)f(metric)721 2459 y Fm(dx)759 2447 y Fk(2)p 717 2467 65 2 v 717 2496 a Fo(h)p Fm(x)p Fo(i)765 2486 y Fk(2)797 2479 y Fu(+)851 2457 y Fm(d\030)886 2445 y Fk(2)p 851 2467 53 2 v 856 2497 a Fm(f)877 2485 y Fk(2)873 2508 y Fi(E)909 2479 y Fu(,)f(where)h Ft(f)1103 2486 y Fm(E)1149 2479 y Fu(is)g(the)g(function)h(from)e(\(4.2\).)278 2550 y(The)20 b(lo)q(calization)h Ft(F)678 2557 y Fs(+)727 2550 y Fu(in)g(the)f(second)h(equation)g(\(4.5b\))g(can)g(b)q(e)g (though)o(t)228 2608 y(of)c(as)g(an)g(energy)f(lo)q(calization)g (\(uniform)g(in)g(energy\).)22 b(In)16 b(Subsection)h(4.3)g(w)o(e)956 2658 y Fq(19)p eop %%Page: 20 20 20 19 bop 228 213 a Fu(use)14 b(the)f(pseudo)q(di\013eren)o(tial)h (calculus)f(to)i(deal)e(with)h(\(4.5b\),)g(and)h(with)f(\(4.5c\),)228 271 y(\(4.5g\))j(and)g(\(4.5h\).)278 333 y(The)e(sym)o(b)q(ol)g Ft(F)576 340 y Fo(\000)605 333 y Fu(\()p Ft(a)650 340 y Fs(0)669 333 y Fu(\))699 320 y(~)688 333 y Ft(F)720 340 y Fo(\000)749 333 y Fu(\()p Ft(b)p Fu(\))g(lies)g(in)g(a)h(go)q(o)q (d)i(sym)o(b)q(ol)c(class;)i(this)f(w)o(ould)h(not)228 391 y(ha)o(v)o(e)j(b)q(een)i(the)f(case)g(without)h(the)f(factor)g Ft(F)1108 398 y Fo(\000)1137 391 y Fu(\()p Ft(a)1182 398 y Fs(0)1202 391 y Fu(\).)33 b(Th)o(us)21 b(to)f(pro)o(v)o(e)g (\(4.5d\))228 449 y(w)o(e)14 b(can)h(use)f(the)g(pseudo)q(di\013eren)o (tial)g(calculus.)20 b(A)14 b(p)q(ositiv)o(e)g(comm)o(utator)e(will)228 507 y(pla)o(y)g(a)i(ma)s(jor)e(role)g(in)h(the)g(analysis,)g(whic)o(h)f (is)h(carried)f(out)i(in)e(Subsection)h(4.5.)228 565 y(The)j(remaining)f(estimates,)f(\(4.5e\))i(and)h(\(4.5f\),)f(can)g(b)q (e)h(pro)o(v)o(ed)e(similarly)l(.)228 684 y(4.2.)28 b Fy(Pseudo)r(di\013eren)n(tial)12 b(calculus.)23 b Fu(W)l(e)12 b(will)f(use)h(pseudo)q(di\013eren)o(tial)f(op-)228 747 y(erators)16 b(in)f(the)g(W)l(eyl)f(calculus)h(asso)q(ciated)h(with)f (the)g(metric)e Ft(g)1433 754 y Fm(E)1477 747 y Fu(=)1539 728 y Fm(dx)1577 716 y Fk(2)p 1534 736 65 2 v 1534 765 a Fo(h)p Fm(x)p Fo(i)1582 755 y Fk(2)1613 747 y Fu(+)1665 725 y Fm(d\030)1700 714 y Fk(2)p 1665 736 53 2 v 1672 765 a Fm(f)1693 755 y Fk(2)228 814 y Fu(\(cf.)32 b([H\177)-24 b(or94,)20 b(Chapt.)34 b(XVI)q(I)q(I]\),)17 b(where)j Ft(f)25 b Fu(=)c Ft(f)1157 821 y Fm(E)1206 814 y Fu(is)f(the)g (energy-dep)q(enden)o(t)228 872 y(function)d(giv)o(en)g(in)h(\(4.2\).) 25 b(F)l(or)18 b(a)g(part)h(of)f(the)f(argumen)o(t)g(one)h(could)f (instead)228 939 y(use)g(the)g(\(energy-indep)q(enden)o(t\))f(metric) 1032 919 y Fm(dx)1070 907 y Fk(2)p 1027 927 65 2 v 1027 956 a Fo(h)p Fm(x)p Fo(i)1075 946 y Fk(2)1109 939 y Fu(+)1183 917 y Fm(d\030)1218 905 y Fk(2)p 1163 927 93 2 v 1163 956 a Fo(h)p Fm(x)p Fo(i)1211 946 y Fg(\000)p Fi(\026)1260 939 y Fu(.)25 b(Ho)o(w)o(ev)o(er)15 b(the)i(crucial)228 1006 y(p)q(ositivit)o(y)k(argumen)o(ts)g(in)h(Subsections)h(4.3)f(and)h (4.5)g(\(applications)f(of)g(the)228 1064 y(F)l(e\013erman-Phong)e (inequalit)o(y\))e(rely)h(indeed)g(on)h(the)g(more)f(precise)f(energy-) 228 1123 y(dep)q(enden)o(t)13 b(estimates.)19 b(It)13 b(is)h(clear)f(that)i(for)f Ft(E)j Fu(=)c(0,)i Ft(f)1262 1130 y Fm(E)1305 1123 y Fu(=)f Ft(C)t Fp(h)p Ft(x)p Fp(i)1462 1105 y Fo(\000)p Fm(\026=)p Fs(2)1562 1123 y Fu(and)g(the)228 1182 y(t)o(w)o(o)i(metrics)e(are)i(essen)o(tially)f(equal.)278 1240 y(Since)g Ft(\026)f(<)g Fu(2)j(w)o(e)e(ha)o(v)o(e)h(a)g(`Planc)o (k's)f(constan)o(t')i(of)f(size)558 1345 y Ft(w)594 1324 y Fo(\000)p Fs(1)655 1345 y Fu(=)e Fp(h)p Ft(x)p Fp(i)773 1324 y Fo(\000)p Fs(1)821 1345 y Ft(f)850 1324 y Fo(\000)p Fs(1)911 1345 y Fp(\024)f Fu(\(1)f Fp(\000)1073 1311 y Ft(\026)p 1073 1333 30 2 v 1075 1379 a Fu(2)1107 1345 y(\))1126 1324 y Fs(1)p Fm(=)p Fs(2)1181 1345 y Fp(h)p Ft(x)p Fp(i)1247 1324 y Fo(\000)p Fs(1+)p Fm(\026=)p Fs(2)1379 1345 y Ft(:)278 1469 y Fu(W)l(e)k(will)f(pro)o(v)o(e)h(that) 701 1449 y Fm(dx)739 1437 y Fk(2)p 696 1457 65 2 v 696 1486 a Fo(h)p Fm(x)p Fo(i)744 1476 y Fk(2)777 1469 y Fu(+)831 1447 y Fm(d\030)866 1435 y Fk(2)p 831 1457 53 2 v 838 1486 a Fm(f)859 1476 y Fk(2)905 1469 y Fu(satis\014es)h(the)f (de\014nition)g(of)h(a)f(H\177)-24 b(ormander)228 1536 y(metric)19 b(with)i(estimates)f(that)i(are)f(uniform)g(in)g Ft(E)26 b Fp(2)c Fu(\(0)p Ft(;)8 b Fu(1])q(.)37 b(Therefore)21 b(w)o(e)228 1594 y(get)14 b(uniform)e(\(in)i Ft(E)s Fu(\))g(con)o(trol) f(of)h(the)g(constan)o(ts)g(app)q(earing)h(in)f(the)f(pseudo)q(dif-)228 1652 y(feren)o(tial)i(calculus.)23 b(In)17 b(particular)g(the)g(b)q (oundedness)h(\(on)f Ft(L)1404 1634 y Fs(2)1424 1652 y Fu(\()p Fn(R)1482 1634 y Fm(d)1499 1652 y Fu(\)\))g(of)h(pseu-)228 1710 y(do)q(di\013eren)o(tial)i(op)q(erators)i(\([H\177)-24 b(or94,)22 b(Theorem)d(18.6.3]\))i(and)g(the)f(p)q(ositivit)o(y)228 1768 y(\(to)i(highest)g(order\))g(of)g(pseudo)q(di\013eren)o(tial)f(op) q(erators)j(with)d(p)q(ositiv)o(e)g(sym-)228 1826 y(b)q(ols)h(\(the)g (F)l(e\013erman-Phong)g(inequalit)o(y)e([H\177)-24 b(or94,)23 b(Theorem)d(18.6.8]\))i(hold)228 1884 y(uniformly)14 b(in)i Ft(E)s Fu(.)228 1980 y Fy(Lemm)o(a)g(4.3.)267 2115 y Fu(\(i\))k Fh(\(A)e(uniform)f(H\177)-25 b(ormander)17 b(metric)h Ft(g)1035 2122 y Fm(E)1065 2115 y Fh(.\))339 2173 y(F)l(or)h(p)n(oints)g Ft(v)f Fp(2)g Fn(R)708 2155 y Fm(d)738 2173 y Fp(\002)12 b Fn(R)828 2155 y Fm(d)864 2173 y Fh(write)20 b Ft(v)e Fu(=)f(\()p Ft(v)1130 2180 y Fm(x)1152 2173 y Ft(;)8 b(v)1198 2180 y Fm(\030)1216 2173 y Fu(\))p Fh(.)28 b(De\014ne)20 b(for)f Ft(E)h Fp(2)d Fu(\(0)p Ft(;)8 b Fu(1])339 2232 y Fh(and)18 b Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b Fp(2)g Fn(R)645 2214 y Fm(d)673 2232 y Fp(\002)d Fn(R)762 2214 y Fm(d)797 2232 y Fh(the)18 b(metric)g Ft(g)r Fu(\()p Ft(v)r Fu(\))13 b(=)h Ft(g)1209 2239 y Fm(E)1239 2232 y Fu(\()p Ft(v)r Fu(\))f(=)h Ft(g)1391 2240 y Fs(\()p Fm(x;\030)q Fs(\))1467 2232 y Fu(\()p Ft(v)r Fu(\))j Fh(by)711 2366 y Ft(g)734 2374 y Fs(\()p Fm(x;\030)q Fs(\))811 2366 y Fu(\()p Ft(v)r Fu(\))c(=)965 2333 y Ft(v)991 2315 y Fs(2)989 2345 y Fm(x)p 945 2355 87 2 v 945 2401 a Fp(h)p Ft(x)p Fp(i)1011 2386 y Fs(2)1047 2366 y Fu(+)1138 2328 y Ft(v)1164 2309 y Fs(2)1162 2341 y Fm(\030)p 1101 2355 120 2 v 1101 2401 a Ft(f)1130 2383 y Fs(2)1125 2414 y Fm(E)1155 2401 y Fu(\()p Ft(x)p Fu(\))1225 2366 y Ft(:)339 2492 y Fh(Then)18 b Ft(g)h Fh(is)d(a)h(H\177)-25 b(ormander)17 b(metric)g(uniformly)g(in)g Ft(E)g Fp(2)d Fu(\(0)p Ft(;)8 b Fu(1])p Fh(,)17 b(i.e.)22 b(ther)n(e)339 2550 y(exist)d(c)n(onstants)g Ft(C)703 2557 y Fs(1)723 2550 y Ft(;)8 b(C)780 2557 y Fs(2)799 2550 y Ft(;)g(N)20 b(>)15 b Fu(0)k Fh(indep)n(endent)h(of)e Ft(E)g Fp(2)d Fu(\(0)p Ft(;)8 b Fu(1])18 b Fh(such)h(that)339 2608 y Ft(g)h Fh(satis\014es)956 2658 y Fq(20)p eop %%Page: 21 21 21 20 bop 377 213 a Fp(\017)21 b Fu(\(slo)o(w)16 b(v)m(ariation\))i Fh(If)f Ft(g)848 220 y Fs(\()p Fm(x;\030)q Fs(\))924 213 y Fu(\(\()p Ft(y)r(;)8 b(\021)r Fu(\)\))13 b Fp(\024)h Fu(1)p Ft(=C)1223 220 y Fs(1)1261 213 y Fh(then)482 298 y Ft(g)505 306 y Fs(\()p Fm(x;\030)q Fs(\)+\()p Fm(y)q(;\021)q Fs(\))683 298 y Fu(\()p Ft(v)r Fu(\))f Fp(\024)h Ft(C)848 305 y Fs(1)868 298 y Ft(g)891 306 y Fs(\()p Fm(x;\030)q Fs(\))967 298 y Fu(\()p Ft(v)r Fu(\))j Fh(for)f(al)r(l)j Ft(v)c Fp(2)f Fn(R)1320 277 y Fm(d)1349 298 y Fp(\002)d Fn(R)1438 277 y Fm(d)1455 298 y Ft(:)377 377 y Fp(\017)21 b Fu(\(uncertain)o(t)o(y)e(principle\))g Ft(g)952 384 y Fm(E)1004 377 y Fp(\024)h Ft(g)1088 359 y Fm(\033)1086 390 y(E)1116 377 y Fh(,)j(wher)n(e)e Ft(g)1320 359 y Fm(\033)1318 390 y(E)1370 377 y Fh(denotes)h(the)g(dual)423 435 y(metric)16 b(of)f Ft(g)653 442 y Fm(E)699 435 y Fh(with)h(r)n(esp)n(e)n(ct)f(to)h(the)g(standar)n(d)f(symple)n(ctic)h (form)f Ft(\033)g Fu(=)423 494 y Ft(dx)c Fp(^)g Ft(d\030)r Fh(.)377 552 y Fp(\017)21 b Fu(\(temp)q(erateness\))11 b Fh(F)l(or)h(al)r(l)i Ft(v)h Fp(2)f Fn(R)1052 534 y Fm(d)1069 552 y Fp(\002)p Fn(R)1147 534 y Fm(d)1165 552 y Fp(nf)p Fu(0)p Fp(g)e Fh(and)h(al)r(l)h Fu(\()p Ft(x;)8 b(\030)r Fu(\))p Ft(;)g Fu(\()p Ft(y)r(;)g(\021)r Fu(\))13 b Fp(2)423 610 y Fn(R)462 592 y Fm(d)490 610 y Fp(\002)e Fn(R)579 592 y Fm(d)614 610 y Fh(we)18 b(have)444 691 y Ft(g)467 699 y Fs(\()p Fm(x;\030)q Fs(\))544 691 y Fu(\()p Ft(v)r Fu(\))13 b Fp(\024)g Ft(C)708 698 y Fs(2)728 691 y Ft(g)751 699 y Fs(\()p Fm(y)q(;\021)q Fs(\))828 691 y Fu(\()p Ft(v)r Fu(\))900 651 y Fl(\000)922 691 y Fu(1)f(+)f Ft(g)1032 671 y Fm(\033)1030 704 y Fs(\()p Fm(x;\030)q Fs(\))1106 691 y Fu(\(\()p Ft(y)r(;)d(\021)r Fu(\))j Fp(\000)f Fu(\()p Ft(x;)e(\030)r Fu(\)\))1427 651 y Fl(\001)1450 662 y Fm(N)1492 691 y Ft(:)254 775 y Fu(\(ii\))19 b Fh(\(Uniform)e(weight)i(functions.\))339 833 y(A)g(p)n(ositive)h(function)g Ft(m)c Fu(=)h Ft(m)922 840 y Fm(E)968 833 y Fu(=)f Ft(m)p Fu(\()p Ft(x;)8 b(\030)r Fu(\))19 b Fh(is)g(said)f(to)h(b)n(e)h(a)e(uniformly)339 891 y(temp)n(er)n(ate)h(weight)i(\(w.r.t.)27 b Ft(\033)o(;)8 b(g)r Fh(\))19 b(if)g(the)g(fol)r(lowing)j(two)d(c)n(onditions)h(ar)n (e)339 949 y(satis\014e)n(d)e(with)g(c)n(onstants)g(indep)n(endent)h (of)e Ft(E)g Fp(2)d Fu(\(0)p Ft(;)8 b Fu(1])377 1007 y Fp(\017)21 b Fh(Ther)n(e)c(exists)h Ft(c;)8 b(C)17 b(>)d Fu(0)k Fh(such)g(that)f(for)g(al)r(l)i Ft(v)r(;)8 b(v)1315 1014 y Fs(1)1347 1007 y Fp(2)14 b Fn(R)1433 989 y Fs(2)p Fm(d)1468 1007 y Fh(:)496 1087 y Ft(g)519 1094 y Fm(v)539 1087 y Fu(\()p Ft(v)582 1094 y Fs(1)602 1087 y Fu(\))f Fp(\024)h Ft(c)g Fp(\))f Ft(m)p Fu(\()p Ft(v)r Fu(\))p Ft(=C)k Fp(\024)d Ft(m)p Fu(\()p Ft(v)e Fu(+)f Ft(v)1192 1094 y Fs(1)1211 1087 y Fu(\))j Fp(\024)f Ft(C)t(m)p Fu(\()p Ft(v)r Fu(\))p Ft(:)377 1166 y Fp(\017)21 b Fh(Ther)n(e)c(exists)h Ft(C)q(;)8 b(N)19 b(>)14 b Fu(0)k Fh(such)f(that)h(for)f(al)r(l)i Ft(v)r(;)8 b(v)1336 1173 y Fs(1)1368 1166 y Fp(2)14 b Fn(R)1454 1148 y Fs(2)p Fm(d)1489 1166 y Fh(:)597 1254 y Ft(m)p Fu(\()p Ft(v)683 1261 y Fs(1)702 1254 y Fu(\))g Fp(\024)f Ft(C)t(m)p Fu(\()p Ft(v)r Fu(\))941 1213 y Fl(\000)963 1254 y Fu(1)e(+)g Ft(g)1072 1233 y Fm(\033)1070 1266 y(v)1087 1271 y Fk(1)1107 1254 y Fu(\()p Ft(v)i Fp(\000)d Ft(v)1236 1261 y Fs(1)1256 1254 y Fu(\))1275 1213 y Fl(\001)1298 1224 y Fm(N)1339 1254 y Ft(:)339 1342 y Fh(With)20 b(this)h(de\014nition,)h(any)f(of)f (the)h(functions)g Fp(h)p Ft(x)p Fp(i)p Ft(;)28 b(f)1384 1349 y Fm(E)1414 1342 y Ft(;)f Fp(h)p Ft(\030)r Fp(i)p Ft(;)20 b Fh(or)g Fp(h)1653 1320 y Fm(\030)p 1640 1330 43 2 v 1640 1359 a(f)1657 1365 y Fi(E)1688 1342 y Fp(i)p Fh(,)339 1404 y(as)k(wel)r(l)h(as)f(any)f(c)n(ombination)i(of)e(pr)n(o) n(ducts)g(of)g(r)n(e)n(al)g(p)n(owers)g(of)g(these)339 1462 y(functions,)17 b(is)e(a)g(uniformly)g(temp)n(er)n(ate)g(weight)h (function)h(for)d(the)i(metric)339 1520 y Ft(g)362 1527 y Fm(E)392 1520 y Fh(.)278 1608 y Fu(The)g(pro)q(of)h(of)g(Lemma)d(4.3) i(is)g(giv)o(en)g(in)g(App)q(endix)f(B.)278 1666 y(F)l(or)k(a)h (uniformly)d(temp)q(erate)h(w)o(eigh)o(t)h(function)g Ft(m)g Fu(=)g Ft(m)1386 1673 y Fm(E)1435 1666 y Fu(w)o(e)g(denote)h(b)o (y)228 1724 y Ft(S)258 1731 y Fe(unif)336 1724 y Fu(\()p Ft(m;)8 b(g)443 1731 y Fm(E)473 1724 y Fu(\))16 b(the)g(space)h(of)f Ft(C)817 1706 y Fo(1)870 1724 y Fu(functions)g(\(`sym)o(b)q(ols'\))f Ft(a)e Fu(=)h Ft(a)1449 1731 y Fm(\020)1485 1724 y Fu(satisfying)559 1814 y Fp(j)p Ft(@)602 1793 y Fm(\013)599 1826 y(x)626 1814 y Ft(@)655 1790 y Fm(\014)652 1828 y(\030)678 1814 y Ft(a)p Fu(\()p Ft(x;)8 b(\030)r Fu(\))p Fp(j)13 b(\024)h Ft(C)930 1821 y Fm(\013;\014)986 1814 y Ft(m)p Fu(\()p Ft(x;)8 b(\030)r Fu(\))p Fp(h)p Ft(x)p Fp(i)1206 1793 y Fo(\000j)p Fm(\013)p Fo(j)1278 1814 y Ft(f)1307 1789 y Fo(\000j)p Fm(\014)r Fo(j)1302 1828 y Fm(E)1378 1814 y Ft(;)238 b Fu(\(4.7\))228 1901 y(for)14 b(all)f Ft(\013;)8 b(\014)16 b Fp(2)e Fu(\()p Fn(N)6 b Fp([)g(f)p Fu(0)p Fp(g)p Fu(\))701 1883 y Fm(d)735 1901 y Fu(with)14 b(constan)o(ts)g Ft(C)1092 1908 y Fm(\013;\014)1161 1901 y Fu(indep)q(enden)o(t)f(of)h Ft(\020)k Fu(=)13 b Ft(E)s(e)1640 1883 y Fm(i\036)1689 1901 y Fp(2)228 1959 y Fu(\000)258 1966 y Fm(\022)278 1959 y Fu(.)37 b(W)l(e)21 b(let)g(\011)532 1966 y Fe(unif)610 1959 y Fu(\()p Ft(m;)8 b(g)717 1966 y Fm(E)747 1959 y Fu(\))21 b(denote)h(the)f(space)h(of)g(op)q(erators)h(giv)o(en)d(as)i (the)228 2017 y(W)l(eyl)14 b(quan)o(tization)h(of)h(sym)o(b)q(ols)e (from)h Ft(S)1016 2024 y Fe(unif)1094 2017 y Fu(\()p Ft(m;)8 b(g)1201 2024 y Fm(E)1231 2017 y Fu(\).)21 b(Th)o(us,)15 b(for)h(instance,)f(a)228 2075 y(short)i(v)o(eri\014cation)e(giv)o(es)g (\(with)h Ft(h)h Fu(from)e(\(4.3\)\))618 2186 y Ft(h;)8 b(h)j Fp(\000)f Ft(\020)18 b Fp(2)c Ft(S)872 2193 y Fm(unif)957 2115 y Fl(\022)994 2186 y Ft(f)1023 2165 y Fs(2)1018 2198 y Fo(j)p Fm(\020)r Fo(j)1057 2186 y Fp(h)1102 2152 y Ft(\030)p 1081 2174 64 2 v 1081 2220 a(f)1105 2228 y Fo(j)p Fm(\020)r Fo(j)1150 2186 y Fp(i)1169 2165 y Fs(2)1189 2186 y Ft(;)8 b(g)1234 2193 y Fo(j)p Fm(\020)r Fo(j)1274 2115 y Fl(\023)1319 2186 y Ft(:)289 b Fu(\(4.8\))278 2299 y(Notice)19 b(that)i(for)g(an)o(y)f Ft(C)759 2281 y Fo(1)755 2311 y Fs(0)796 2299 y Fu({function)g Ft(G)h Fu(w)o(e)g(ha)o(v)o(e)e(with)i Ft(b)f Fu(from)f(\(4.1\))i(for)228 2357 y Ft(E)c Fu(=)c(0,)518 2418 y Ft(@)547 2398 y Fm(\013)544 2431 y(x)571 2418 y Ft(@)600 2395 y Fm(\014)597 2432 y(\030)623 2418 y Ft(G)p Fu(\()p Ft(b)p Fu(\))h Fp(\031)g Fu(\()p Fp(h)p Ft(x)p Fp(i)872 2398 y Fs(\()p Fm(\026=)p Fs(2)p Fo(\000)p Fs(1\))1003 2418 y Fp(j)p Ft(\030)r Fp(j)p Fu(\))1073 2398 y Fo(j)p Fm(\013)p Fo(j)1118 2418 y Fp(h)p Ft(x)p Fp(i)1184 2398 y Fm(\026)p Fo(j)p Fm(\014)r Fo(j)p Fm(=)p Fs(2)1295 2418 y Fu(+)d Fp(\001)d(\001)g(\001)17 b Ft(:)228 2492 y Fu(Consequen)o(tly)h Ft(G)p Fu(\()p Ft(b)p Fu(\))g(is)h(not)g(a)g(go)q(o)q(d)h(sym)o(b)q(ol)e(\(not)h(ev)o (en)e(with)h Ft(\030)k Fu(considered)228 2550 y(as)15 b(b)q(ounded)g(since)e Ft(\026)i Fu(can)f(b)q(e)h(greater)f(than)h (1\).)21 b(The)14 b(remedy)e(for)j(this)f(has)h(al-)228 2608 y(ready)d(b)q(een)f(in)o(tro)q(duced)h(in)f(\(4.5\))q(:)19 b(Let)12 b Ft(F)1009 2615 y Fo(\000)1052 2608 y Fp(2)i Ft(C)1138 2590 y Fo(1)1134 2620 y Fs(0)1174 2608 y Fu(\()p Fn(R)p Fu(\))9 b(and)k(study)f Ft(F)1513 2615 y Fo(\000)1542 2608 y Fu(\()p Ft(a)1587 2615 y Fs(0)1606 2608 y Fu(\))p Ft(G)p Fu(\()p Ft(b)p Fu(\))956 2658 y Fq(21)p eop %%Page: 22 22 22 21 bop 228 213 a Fu(for)22 b(an)o(y)g Ft(C)445 195 y Fo(1)482 213 y Fu({function)g Ft(G)p Fu(.)40 b(Notice)21 b(that)h Fp(j)p Ft(b)p Fp(j)1114 195 y Fs(2)1157 213 y Fp(\024)i Ft(a)1246 220 y Fs(0)1265 213 y Fu(,)f(so)g Ft(b)f Fu(is)g(b)q(ounded)h(on)228 271 y(supp\()p Ft(F)379 278 y Fo(\000)408 271 y Fu(\()p Ft(a)453 278 y Fs(0)473 271 y Fu(\)\).)e(Using)16 b(the)g(elemen)o(tary)d(b)q(ound,)k(v)m(alid) f(for)h(an)o(y)f Ft(s)e Fp(2)g Fn(R)p Fu(,)738 359 y Fp(j)p Ft(@)781 339 y Fm(\013)778 371 y(x)805 359 y Ft(f)834 339 y Fm(s)829 371 y(E)859 359 y Fp(j)f(\024)h Ft(C)974 366 y Fm(\013)998 359 y Ft(f)1027 339 y Fm(s)1022 371 y(E)1052 359 y Fp(h)p Ft(x)p Fp(i)1118 339 y Fo(\000j)p Fm(\013)p Fo(j)1190 359 y Ft(;)418 b Fu(\(4.9\))228 446 y(with)15 b Ft(C)373 453 y Fm(\013)413 446 y Fu(indep)q(enden)o(t)f(of) i Ft(E)i Fu(\(or)e(in)f(short)h Ft(f)1079 453 y Fm(E)1123 446 y Fp(2)e Ft(S)1200 453 y Fe(unif)1278 446 y Fu(\()p Ft(f)1321 453 y Fm(E)1351 446 y Ft(;)8 b(g)1396 453 y Fm(E)1426 446 y Fu(\))15 b(recalling)f(the)228 505 y(con)o(v)o(en)o (tion)h Ft(E)i Fu(=)c Fp(j)p Ft(\020)t Fp(j)p Fu(\),)j(w)o(e)g(readily) f(infer)g(that)i(indeed)690 589 y Ft(F)722 596 y Fo(\000)751 589 y Fu(\()p Ft(a)796 596 y Fs(0)815 589 y Fu(\))p Ft(G)p Fu(\()p Ft(b)p Fu(\))d Fp(2)g Ft(S)1022 596 y Fe(unif)1101 589 y Fu(\(1)p Ft(;)8 b(g)1189 596 y Fm(E)1219 589 y Fu(\))g Ft(:)278 673 y Fu(Once)k(Lemma)e(4.3)k(is)e(established)g(w)o (e)h(ha)o(v)o(e)f(from)g([H\177)-24 b(or94,)13 b(Sections)f(18.4-6])228 731 y(\(cf.)21 b(in)15 b(particular)h([H\177)-24 b(or94)q(,)15 b(Theorems)h(18.6.3)h(and)f(18.6.8]\):)228 822 y Fy(Theorem)g(4.4.)267 951 y Fu(\(i\))k Fh(L)n(et)g Ft(a)e Fp(2)g Ft(S)551 958 y Fe(unif)630 951 y Fu(\(1)p Ft(;)8 b(g)718 958 y Fm(E)748 951 y Fu(\))p Fh(.)30 b(Then)20 b(ther)n(e)g(exists)h(a)f(c)n(onstant)g Ft(C)i(>)c Fu(0)p Fh(,)j(inde-)339 1009 y(p)n(endent)e(of)e Ft(\020)h Fp(2)c Fu(\000)693 1016 y Fm(\022)713 1009 y Fh(,)k(such)f(that)822 1094 y Fp(k)p Fu(Op)904 1073 y Fs(w)924 1094 y Fu(\()p Ft(a)p Fu(\))p Fp(k)c(\024)g Ft(C)q(:)254 1178 y Fu(\(ii\))19 b Fh(L)n(et)j Ft(a)f Fp(2)h Ft(S)560 1185 y Fe(unif)639 1178 y Fu(\()p Ft(w)694 1160 y Fs(2)693 1191 y Fm(E)723 1178 y Ft(;)8 b(g)768 1185 y Fm(E)798 1178 y Fu(\))21 b Fh(and)h(supp)n(ose)g Ft(a)f Fp(\025)h Fu(0)p Fh(.)36 b(Then)22 b(ther)n(e)g(exists)g(a)339 1236 y(c)n(onstant)d Ft(C)e(>)d Fu(0)k Fh(indep)n(endent)h(of)e Ft(\020)h Fp(2)c Fu(\000)1123 1243 y Fm(\022)1160 1236 y Fh(such)k(that)828 1321 y Fu(Op)885 1300 y Fs(w)904 1321 y Fu(\()p Ft(a)p Fu(\))c Fp(\025)f(\000)p Ft(C)q(;)339 1405 y Fh(as)18 b(a)f(quadr)n(atic)g(form)g(on)h Fp(S)t Fu(\()p Fn(R)936 1387 y Fm(d)953 1405 y Fu(\))p Fh(.)278 1496 y Fu(Another)13 b(useful)g(to)q(ol)i(is)e(the)h(comp)q(osition)f (rule)g(for)h(pseudo)q(di\013eren)o(tial)f(op-)228 1554 y(erators)24 b(\(see)g([H\177)-24 b(or94,)26 b(Theorem)c(18.5.4]\):)37 b(Supp)q(ose)25 b Ft(a)1332 1561 y Fs(1)1378 1554 y Fp(2)i Ft(S)1468 1561 y Fe(unif)1547 1554 y Fu(\()p Ft(m)1609 1561 y Fs(1)1628 1554 y Ft(;)8 b(g)1673 1561 y Fm(E)1703 1554 y Fu(\))228 1612 y(and)25 b Ft(a)357 1619 y Fs(2)404 1612 y Fp(2)j Ft(S)495 1619 y Fe(unif)573 1612 y Fu(\()p Ft(m)635 1619 y Fs(2)655 1612 y Ft(;)8 b(g)700 1619 y Fm(E)730 1612 y Fu(\).)46 b(Then)24 b(Op)1001 1594 y Fs(w)1021 1612 y Fu(\()p Ft(a)1066 1619 y Fs(1)1085 1612 y Fu(\)Op)1161 1594 y Fs(w)1180 1612 y Fu(\()p Ft(a)1225 1619 y Fs(2)1245 1612 y Fu(\))j(=)h(Op)1414 1594 y Fs(w)1434 1612 y Fu(\()p Ft(s)p Fu(\))c(with)g Ft(s)k Fp(2)228 1670 y Ft(S)258 1677 y Fe(unif)336 1670 y Fu(\()p Ft(m)398 1677 y Fs(1)418 1670 y Ft(m)461 1677 y Fs(2)480 1670 y Ft(;)8 b(g)525 1677 y Fm(E)555 1670 y Fu(\),)16 b(and)h (asymptotically)d Ft(s)g Fu(=)1115 1633 y Fl(P)1168 1646 y Fo(1)1168 1685 y Fm(j)r Fs(=0)1240 1670 y Ft(s)1263 1677 y Fm(j)1281 1670 y Fu(\()p Ft(x;)8 b(\030)r Fu(\))16 b(with)650 1767 y Ft(s)673 1774 y Fm(j)692 1767 y Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b Fp(2)g Ft(S)894 1774 y Fe(unif)972 1727 y Fl(\000)995 1767 y Ft(w)1031 1744 y Fo(\000)p Fm(j)1030 1781 y(E)1077 1767 y Ft(m)1120 1774 y Fs(1)1139 1767 y Ft(m)1182 1774 y Fs(2)1202 1767 y Ft(;)8 b(g)1247 1774 y Fm(E)1277 1727 y Fl(\001)228 1852 y Fu(giv)o(en)15 b(b)o(y)402 1968 y Ft(s)425 1975 y Fm(j)444 1968 y Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b(=)g(2)645 1948 y Fo(\000)p Fm(j)691 1968 y Ft(i)708 1948 y Fm(j)765 1921 y Fl(X)734 2029 y Fo(j)p Fm(\013)p Fs(+)p Fm(\014)r Fo(j)p Fs(=)p Fm(j)882 1935 y Fu(\()p Fp(\000)p Fu(1\))983 1917 y Fo(j)p Fm(\013)p Fo(j)p 882 1957 146 2 v 910 2003 a Ft(\013)p Fu(!)p Ft(\014)s Fu(!)1032 1968 y(\()p Ft(@)1080 1948 y Fm(\013)1077 1981 y(\030)1105 1968 y Ft(@)1134 1948 y Fm(\014)1131 1981 y(x)1157 1968 y Ft(a)1183 1975 y Fs(1)1202 1968 y Fu(\)\()p Ft(@)1269 1945 y Fm(\014)1266 1982 y(\030)1292 1968 y Ft(@)1321 1948 y Fm(\013)1318 1981 y(x)1345 1968 y Ft(a)1371 1975 y Fs(2)1390 1968 y Fu(\))p Ft(:)175 b Fu(\(4.10\))228 2107 y(Here)15 b(`asymptotically') e(means)i(that)i(for)g(all)e Ft(N)20 b Fp(2)14 b Fn(N)j Fu(w)o(e)f(ha)o(v)o(e)625 2237 y Ft(s)11 b Fp(\000)729 2175 y Fm(N)708 2190 y Fl(X)714 2295 y Fm(j)r Fs(=0)789 2237 y Ft(s)812 2244 y Fm(j)844 2237 y Fp(2)j Ft(S)921 2244 y Fe(unif)999 2182 y Fl(\020)1029 2237 y Ft(w)1065 2212 y Fo(\000)p Fs(\()p Fm(N)t Fs(+1\))1064 2251 y Fm(E)1199 2237 y Ft(;)8 b(g)1244 2244 y Fm(E)1274 2182 y Fl(\021)1312 2237 y Ft(:)278 2374 y Fu(Finally)h(w)o(e)i(shall)f(discuss)h(some)f (further)g(results)h(for)g(the)g(class)g Ft(S)1483 2381 y Fe(unif)1561 2374 y Fu(\()p Ft(m;)d(g)1668 2381 y Fm(E)1698 2374 y Fu(\).)228 2432 y(Although)20 b(they)g(will)f(not)i(b)q(e)f (used)h(in)f(this)g(pap)q(er,)h(w)o(e)f(feel)f(that)i(including)228 2490 y(them)14 b(migh)o(t)h(b)q(e)h(clarifying)f(for)i(the)f(reader.) 278 2549 y(Let)c(us)h(top)q(ologize)g Ft(S)678 2556 y Fe(unif)756 2549 y Fu(\()p Ft(m;)8 b(g)863 2556 y Fm(E)893 2549 y Fu(\))k(b)o(y)g(the)g(seminorms)e Fp(k)p Ft(a)p Fp(k)1382 2556 y Fm(\013;\014)1438 2549 y Fu(,)j(eac)o(h)e(de\014ned) 228 2608 y(as)18 b(the)e(smallest)g(constan)o(t)h Ft(C)794 2615 y Fm(\013;\014)867 2608 y Fu(indep)q(enden)o(t)g(of)g Ft(\020)i Fu(=)c Ft(E)s(e)1356 2590 y Fm(i\036)1406 2608 y Fp(2)h Fu(\000)1485 2615 y Fm(\022)1522 2608 y Fu(suc)o(h)h(that)956 2658 y Fq(22)p eop %%Page: 23 23 23 22 bop 228 213 a Fu(\(4.7\))19 b(holds.)31 b(F)l(rom)18 b([H\177)-24 b(or94,)20 b(Theorem)e(18.5.10])i(\(and)g(its)f(pro)q(of)s (\))i(w)o(e)e(learn)228 271 y(that)557 360 y Ft(e)580 340 y Fm(i\024)p Fo(h)p Fm(p)644 344 y Fi(x)663 340 y Fm(;p)691 346 y Fi(\030)708 340 y Fo(i)737 360 y Fu(:)14 b Ft(S)795 367 y Fe(unif)873 360 y Fu(\()p Ft(m;)8 b(g)980 367 y Fm(E)1010 360 y Fu(\))14 b Fp(!)f Ft(S)1136 367 y Fe(unif)1215 360 y Fu(\()p Ft(m;)8 b(g)1322 367 y Fm(E)1352 360 y Fu(\))557 436 y(is)14 b(a)g(top)q(ological)h(isomorphism)m(;)22 b Ft(\024)14 b Fp(2)g Fn(R)p Ft(:)264 b Fu(\(4.11\))228 532 y(Of)24 b(particular)h(in)o(terest)e(are)i(the)g(v)m(alues)g Ft(\024)j Fu(=)g Fp(\000)p Fu(1)p Ft(;)8 b Fp(\000)1310 512 y Fs(1)p 1310 521 18 2 v 1310 549 a(2)1333 532 y Ft(;)1360 512 y Fs(1)p 1360 521 V 1360 549 a(2)1382 532 y Ft(;)g Fu(1,)27 b(cf.)46 b([H\177)-24 b(or94)q(,)228 591 y(\(18.5.20\)],)19 b(linking)f(W)l(eyl)g(quan)o(tization)h(with)f (left)g(and)i(righ)o(t)e(Kohn-Niren-)228 649 y(b)q(erg)f(quan)o (tization.)278 707 y(Y)l(et)h(another)i(basic)f(result)g(is)f(a)i (uniform)e(\(partial\))h(v)o(ersion)f(of)i(the)f(Beals)228 765 y(criterion)d([Bea81,)h(Theorem)g(4.4])g(\(see)g(also)h([BoCh94)q (]\).)24 b(It)17 b(is)h(a)f(useful)h(to)q(ol)228 823 y(for)g(linking)e(pseudo)q(di\013eren)o(tial)i(and)g(functional)f (calculi,)f(cf.)25 b([DeG)o(\023)-23 b(e97)q(,)17 b(Ap-)228 881 y(p)q(endix)f(D].)f(W)l(e)h(in)o(tro)q(duce)g(ad)821 888 y Fs(B)848 881 y Fu(\(A\))d(=)h(AB)c Fp(\000)h Fu(BA)k(and)i (similarly)c(for)k(v)o(ector-)228 945 y(v)m(alued)f(op)q(erators,)i(ad) 662 921 y Fm(\014)662 958 y Fs(B)689 945 y Fu(\(A\),)e(de\014ned)g(as)i Fp(j)p Ft(\014)s Fp(j)d Fu(comp)q(ositions)h(of)h(op)q(erations)h(of) 228 1003 y(the)i(previous)h(t)o(yp)q(e.)33 b(The)21 b(Beals)f (criterion)f(in)i(our)g(case)f(is)h(the)f(c)o(haracteri-)228 1061 y(sation)h(of)h(the)e(space)h(\011)698 1068 y Fe(unif)777 1061 y Fu(\(1)p Ft(;)8 b(g)865 1068 y Fm(E)895 1061 y Fu(\))21 b(as)g(the)g(set)g(of)g Fp(B)r Fu(\()p Ft(L)1316 1043 y Fs(2)1335 1061 y Fu(\)-v)m(alued)g(functions)228 1120 y Ft(A)13 b Fu(=)h Ft(A)367 1127 y Fm(\020)403 1120 y Fu(on)i(\000)500 1127 y Fm(\022)537 1120 y Fu(for)g(whic)o(h)g(for)g (all)g(m)o(ultiindic)o(es)e Ft(\013;)8 b(\014)16 b Fp(2)e Fn(N)1345 1102 y Fm(d)653 1220 y Fp(kh)p Ft(x)p Fp(i)744 1199 y Fo(j)p Fm(\013)p Fo(j)789 1220 y Ft(f)818 1194 y Fo(j)p Fm(\014)r Fo(j)813 1236 y(j)p Fm(\020)r Fo(j)861 1220 y Fu(ad)913 1199 y Fm(\013)913 1232 y(p)938 1220 y Fu(ad)989 1199 y Fm(\014)989 1232 y(x)1013 1220 y Fu(\()p Ft(A)p Fu(\))p Fp(k)f(\024)g Ft(D)1218 1227 y Fm(\013;\014)1275 1220 y Ft(;)228 1314 y Fu(where)k(the)g(constan)o(ts)i Ft(D)713 1321 y Fm(\013;\014)787 1314 y Fu(are)e(indep)q(enden)o(t)g (of)h Ft(\020)i Fp(2)c Fu(\000)1324 1321 y Fm(\022)1344 1314 y Fu(.)25 b(This)18 b(c)o(haracteri-)228 1372 y(sation)d(ma)o(y)d (b)q(e)i(pro)o(v)o(ed)g(using)h(\(4.9\),)f(\(4.11\))q(,)g(the)g(pro)q (of)h(of)f([Bea77,)h(Theorem)228 1430 y(1.4])c(and)h(conjugation)g(b)o (y)e(the)h(F)l(ourier)g(transform,)g(see)g(the)g(pro)q(of)h(of)f([DeG)o (\023)-23 b(e97)q(,)228 1489 y(Theorem)15 b(D.8.2].)228 1598 y(4.3.)28 b Fy(Energy)19 b(estimate.)j Fu(In)17 b(this)h(subsection)g(w)o(e)g(will)e(pro)o(v)o(e)i(\(4.5b\).)26 b(The)228 1656 y(pro)q(of)16 b(of)j(\(4.5c\))d(is)f(v)o(ery)f(similar)f (\(w)o(e)i(ma)o(y)f(pro)q(ceed)h(in)g(the)h(same)e(fashion)i(for)228 1714 y(the)f(`adjoin)o(t')f(expression\),)h(and)h(the)f(statemen)o(ts)f (\(4.5g\))i(and)g(\(4.5h\))f(ma)o(y)f(b)q(e)228 1772 y(pro)o(v)o(ed)g(along)i(the)f(same)e(pattern)j(using)f(in)g(addition)g (\(4.5d\))g(and)h(\(4.5e\))e(\(the)228 1830 y(latter)i(estimates)e (will)h(b)q(e)i(pro)o(v)o(ed)e(indep)q(enden)o(tly)g(in)h(Subsection)g (4.5\).)278 1889 y(W)l(e)g(will)g(start)i(b)o(y)e(pro)o(ving)h(the)g (follo)o(wing)g(lemm)o(a)e(in)h(whic)o(h)h Ft(f)j Fu(=)15 b Ft(f)1587 1896 y Fo(j)p Fm(\020)r Fo(j)1644 1889 y Fu(and)228 1947 y Ft(w)g Fu(=)f Ft(w)365 1954 y Fo(j)p Fm(\020)r Fo(j)420 1947 y Fu(are)i(giv)o(en)g(b)o(y)g(\(4.2\))h(and)g (\(4.3\),)f(resp)q(ectiv)o(ely)l(.)228 2040 y Fy(Lemm)o(a)g(4.5.)24 b Fh(F)l(or)19 b(al)r(l)i Ft(\017)d(>)h Fu(0)p Fh(,)h Ft(s)f Fp(\025)f Fu(0)i Fh(and)g(al)r(l)h(functions)g Ft(F)1415 2047 y Fs(+)1464 2040 y Fh(as)f(in)k Fu(\(4.5b\))228 2098 y Fh(the)18 b(op)n(er)n(ator)379 2189 y Fp(h)p Ft(x)p Fp(i)445 2169 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)569 2189 y Ft(f)598 2169 y Fs(1)p Fm(=)p Fs(2)654 2189 y Ft(w)690 2169 y Fm(s)708 2189 y Fu(Op)765 2169 y Fs(w)785 2189 y Fu(\()p Ft(F)836 2196 y Fs(+)865 2189 y Fu(\()p Ft(a)910 2196 y Fs(0)929 2189 y Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(w)1103 2169 y Fo(\000)p Fm(s)1150 2189 y Ft(f)1179 2169 y Fs(1)p Fm(=)p Fs(2)1234 2189 y Fp(h)p Ft(x)p Fp(i)1300 2169 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1425 2189 y Ft(;)159 b Fu(\(4.12\))228 2279 y Fh(is)17 b(b)n(ounde)n(d)h(uniformly)f(in)h Ft(\020)g Fp(2)c Fu(\000)864 2286 y Fm(\022)884 2279 y Fh(.)228 2372 y(Pr)n(o)n(of.)19 b Fu(W)l(e)d(will)f(use)h(the)g (inequalit)o(y)f(\(with)h Ft(h)g Fu(from)f(\(4.3\)\),)500 2460 y Ft(\030)523 2442 y Fs(2)p 497 2482 50 2 v 497 2528 a Ft(f)526 2513 y Fs(2)565 2494 y Fu(=)622 2460 y Fp(<)p Fu(\()p Ft(h)c Fp(\000)f Ft(\020)t Fu(\))p 622 2482 188 2 v 691 2528 a Ft(f)720 2513 y Fs(2)826 2494 y Fp(\000)880 2460 y(<)p Fu(\()p Ft(V)23 b Fp(\000)10 b Ft(\020)t Fu(\))p 880 2482 200 2 v 955 2528 a Ft(f)984 2513 y Fs(2)1098 2494 y Fp(\024)1156 2460 y(<)p Fu(\()p Ft(h)h Fp(\000)g Ft(\020)t Fu(\))p 1156 2482 188 2 v 1225 2528 a Ft(f)1254 2513 y Fs(2)1360 2494 y Fu(+)g Ft(C)q(;)147 b Fu(\(4.13\))228 2608 y(where)16 b(the)g(constan)o(t)g Ft(C)k Fu(only)c(dep)q(ends)h(on)g Ft(V)11 b Fu(.)956 2658 y Fq(23)p eop %%Page: 24 24 24 23 bop 278 213 a Fu(By)16 b(\(4.13\))412 333 y Ft(F)444 340 y Fs(+)473 333 y Fu(\()p Ft(a)518 340 y Fs(0)538 333 y Fu(\))557 312 y Fs(2)590 333 y Fp(\024)e Ft(F)675 340 y Fs(+)704 333 y Fu(\()p Ft(a)749 340 y Fs(0)768 333 y Fu(\))787 312 y Fs(2)842 299 y Ft(\030)865 281 y Fs(2)p 812 321 104 2 v 812 367 a Ft(f)841 353 y Fs(2)861 367 y Ft(C)896 374 y Fs(0)934 333 y Fp(\024)g Ft(F)1019 340 y Fs(+)1048 333 y Fu(\()p Ft(a)1093 340 y Fs(0)1112 333 y Fu(\))1131 312 y Fs(2)1156 299 y Ft(f)1185 281 y Fo(\000)p Fs(2)1233 299 y Fp(<)p Fu(\()p Ft(h)d Fp(\000)f Ft(\020)t Fu(\))i(+)f Ft(C)p 1156 321 364 2 v 1310 367 a(C)1345 374 y Fs(0)1524 333 y Ft(:)228 449 y Fu(Th)o(us,)16 b(if)g Ft(C)r(=C)506 456 y Fs(0)540 449 y Fp(\024)d Fu(1)p Ft(=)p Fu(2,)k(i.e.)j(if)c Ft(C)859 456 y Fs(0)895 449 y Fu(has)h(b)q(een)f(c)o(hosen)g(su\016cien)o(tly)e(large,)640 567 y Ft(F)672 574 y Fs(+)701 567 y Fu(\()p Ft(a)746 574 y Fs(0)765 567 y Fu(\))784 546 y Fs(2)818 567 y Fp(\024)890 533 y Fu(2)p 875 555 55 2 v 875 601 a Ft(C)910 608 y Fs(0)935 567 y Ft(F)967 574 y Fs(+)996 567 y Fu(\()p Ft(a)1041 574 y Fs(0)1060 567 y Fu(\))1079 546 y Fs(2)1104 533 y Fp(<)p Fu(\()p Ft(h)d Fp(\000)g Ft(\020)t Fu(\))p 1104 555 188 2 v 1173 601 a Ft(f)1202 587 y Fs(2)1297 567 y Ft(:)287 b Fu(\(4.14\))228 690 y(By)16 b(\(4.8\))h(and)g(\(4.9\)) g(this)f(is)g(an)h(inequalit)o(y)d(for)j(sym)o(b)q(ols)e(in)h Ft(S)1420 697 y Fm(unif)1497 690 y Fu(\()p Fp(h)1542 668 y Fm(\030)p 1540 679 21 2 v 1540 707 a(f)1566 690 y Fp(i)1585 672 y Fs(2)1605 690 y Ft(;)8 b(g)1650 698 y Fo(j)p Fm(\020)r Fo(j)1690 690 y Fu(\).)228 753 y(The)16 b(constan)o(t)h Ft(C)560 760 y Fs(0)593 753 y Ft(>)d Fu(0)j(only)f(dep)q(ends)g(on)h Ft(V)27 b Fu(as)17 b(demanded)e(in)h (Theorem)f(4.1)228 811 y(\(ii\).)278 869 y(F)l(or)k Ft(s)h Fu(=)f(0,)h(the)g(b)q(ounds)h(of)i(\(4.12\))d(follo)o(w)f(from)g (Corollary)g(3.5)h(and)g(the)228 927 y(pseudo)q(di\013eren)o(tial)c (calculus.)278 985 y(Supp)q(ose)f(w)o(e)g(ha)o(v)o(e)f(pro)o(v)o(ed)g (uniform)f(b)q(oundedness)j(\(in)f Ft(\020)t Fu(\))f(of)h(the)g(op)q (erator)228 1043 y(\(4.12\))21 b(for)g(all)f Ft(\017)i(>)f Fu(0,)h(functions)f Ft(F)932 1050 y Fs(+)981 1043 y Fu(as)h(in)e (\(4.5b\),)i(and)f Ft(s)h Fp(\024)f Ft(s)1497 1050 y Fs(0)1537 1043 y Fu(for)g(some)228 1101 y Ft(s)251 1108 y Fs(0)284 1101 y Fp(\025)14 b Fu(0.)22 b(W)l(e)16 b(will)f(then)h(pro) o(v)o(e)f(that)i(the)f(op)q(erator)596 1191 y(Op)653 1171 y Fs(w)673 1191 y Fu(\()p Ft(f)5 b(w)757 1171 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)856 1191 y Ft(F)888 1198 y Fs(+)917 1191 y Fu(\()p Ft(a)962 1198 y Fs(0)982 1191 y Fu(\)\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(f)g(w)1185 1171 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)228 1280 y Fu(is)20 b(also)h(uniformly)d(b)q(ounded)j (for)g(all)f Ft(\017)h(>)f Fu(0,)i(suc)o(h)e(functions)g Ft(F)1482 1287 y Fs(+)1511 1280 y Fu(,)h(and)g Ft(s)g Fp(2)228 1338 y Fu(\()p Ft(s)270 1345 y Fs(0)289 1338 y Ft(;)8 b(s)334 1345 y Fs(0)365 1338 y Fu(+)j(1\).)22 b(Since)627 1430 y Ft(f)5 b(w)692 1410 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)805 1430 y Fu(=)14 b(\()p Fp(h)p Ft(x)p Fp(i)942 1410 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)1025 1430 y Ft(w)1061 1410 y Fo(\000)p Fm(\016)1107 1430 y Fu(\))p Ft(f)1155 1410 y Fs(1)p Fm(=)p Fs(2)1211 1430 y Ft(w)1247 1410 y Fm(s)p Fs(+)p Fm(\016)1310 1430 y Ft(;)228 1521 y Fu(and)i Ft(w)f Fu(=)e Fp(h)p Ft(x)p Fp(i)p Ft(f)20 b Fp(\025)14 b(h)p Ft(x)p Fp(i)652 1502 y Fs(1)p Fo(\000)p Fm(\026=)p Fs(2)756 1521 y Fu(,)h(this)g(is)g(equiv)m(alen)o(t)e(to)j (uniform)e(b)q(oundedness)i(of)228 1579 y(\(4.12\))h(for)f Ft(s)g Fu(is)h(the)f(same)f(range.)278 1637 y(Th)o(us)h(our)h(goal)g (will)e(b)q(e)h(to)h(pro)o(v)o(e)e(a)i(uniform)e(b)q(ound)i(on)672 1727 y Fp(k)p Fu(Op)754 1706 y Fs(w)773 1727 y Fu(\()p Ft(f)5 b(w)857 1706 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)956 1727 y Ft(F)988 1734 y Fs(+)1018 1727 y Fu(\()p Ft(a)1063 1734 y Fs(0)1082 1727 y Fu(\)\))p Ft(RT)1193 1706 y Fo(\000)p Fs(1)1240 1727 y Fp(k)p Ft(;)228 1817 y Fu(where)16 b Ft(R)e Fu(=)g Ft(R)p Fu(\()p Ft(\020)t Fu(\))j(and)f Ft(T)719 1799 y Fo(\000)p Fs(1)780 1817 y Fu(=)e Ft(f)5 b(w)897 1799 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1065 1817 y Fu(.)278 1875 y(Using)19 b(Theorem)g(4.4,)h(\(4.14\))h(and)f(the)f(sym)o(b)q (olic)f(calculus)h(w)o(e)g(ma)o(y)g(esti-)228 1933 y(mate)492 2021 y(Op)549 2001 y Fs(w)569 2021 y Fu(\()p Ft(a)p Fu(\))633 2001 y Fo(\003)652 2021 y Ft(P)7 b Fu(Op)747 2001 y Fs(w)767 2021 y Fu(\()p Ft(a)p Fu(\))13 b Fp(\025)h(\000)p Ft(C)t Fu(;)609 b(\(4.15\))492 2128 y Ft(a)14 b Fu(=)f Fp(h)610 2094 y Ft(\030)p 607 2116 30 2 v 607 2162 a(f)642 2128 y Fp(i)661 2107 y Fo(\000)p Fs(1)709 2128 y Ft(f)738 2107 y Fo(\000)p Fs(1)785 2128 y Ft(w)826 2094 y Fk(3)p 826 2100 16 2 v 826 2120 a(2)847 2107 y Fo(\000)p Fm(s)906 2128 y Fu(\()p Fp(2)h Ft(S)1002 2135 y Fm(unif)1079 2128 y Fu(\()p Ft(a;)8 b(g)1169 2136 y Fo(j)p Fm(\020)r Fo(j)1208 2128 y Fu(\)\))p Ft(;)492 2263 y(P)21 b Fu(=)14 b(Op)653 2242 y Fs(w)681 2192 y Fl(\022)717 2263 y Ft(w)753 2242 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)835 2263 y Ft(F)867 2270 y Fs(+)896 2263 y Fu(\()p Ft(a)941 2270 y Fs(0)960 2263 y Fu(\))979 2242 y Fs(2)999 2263 y Fu(\()1038 2229 y(2)p 1023 2251 55 2 v 1023 2297 a Ft(C)1058 2304 y Fs(0)1082 2263 y Fp(<)p Fu(\()p Ft(h)e Fp(\000)e Ft(\020)t Fu(\))h Fp(\000)g Ft(f)1360 2242 y Fs(2)1380 2263 y Fu(\))1399 2192 y Fl(\023)1444 2263 y Ft(:)278 2382 y Fu(T)l(o)17 b(write)f(\(4.15\))h(in)f(a)h(more)e(con)o(v)o(enien)o(t)g(form)g(w)o (e)h(\014rst)h(use)g(the)f(standard)228 2442 y(parametrix)i (construction)j(to)f(\014nd)h(a)f(sym)o(b)q(ol)f Ft(a)1172 2423 y Fs(\()p Fm(m)p Fs(\))1253 2442 y Fp(2)h Ft(S)1336 2449 y Fm(unif)1413 2442 y Fu(\()p Ft(a)1458 2423 y Fo(\000)p Fs(1)1505 2442 y Ft(;)8 b(g)1550 2449 y Fo(j)p Fm(\020)r Fo(j)1589 2442 y Fu(\))20 b(suc)o(h)228 2500 y(that)264 2588 y(Op)321 2568 y Fs(w)340 2588 y Fu(\()p Ft(a)p Fu(\)Op)461 2568 y Fs(w)480 2588 y Fu(\()p Ft(a)525 2568 y Fs(\()p Fm(m)p Fs(\))585 2588 y Fu(\))11 b Fp(\000)g Ft(I)17 b Fu(=)d(Op)813 2568 y Fs(w)833 2588 y Fu(\()p Ft(r)875 2568 y Fs(\()p Fm(m)p Fs(\))936 2588 y Fu(\))g Fp(2)g Fu(\011)1054 2595 y Fm(unif)1130 2588 y Fu(\()p Fp(h)p Ft(x)p Fp(i)1215 2568 y Fo(\000)p Fm(m)1276 2588 y Ft(;)8 b(g)1321 2596 y Fo(j)p Fm(\020)r Fo(j)1361 2588 y Fu(\);)22 b Ft(m)13 b(>)h Fu(0)p Ft(:)956 2658 y Fq(24)p eop %%Page: 25 25 25 24 bop 228 213 a Fu(Pic)o(k)12 b(a)h(function)g Ft(F)598 195 y Fs(1)591 225 y(+)633 213 y Fu(satisfying)g(the)g(same)f (assumptions)h(as)h Ft(F)1410 220 y Fs(+)1439 213 y Fu(,)g(and)f (further-)228 272 y(more,)j Ft(F)404 254 y Fs(1)397 284 y(+)442 272 y Fu(=)g(1)i(on)g(a)g(neigh)o(b)q(ourho)q(o)q(d)h(of)f (supp)9 b Ft(F)1181 279 y Fs(+)1210 272 y Fu(.)25 b(W)l(e)17 b(readily)g(sho)o(w,)h(that)228 330 y(for)e(an)o(y)g Ft(m)e(>)g Fu(0)306 432 y Ft(P)21 b Fp(\025)13 b(\000)p Ft(D)490 411 y Fo(\003)511 432 y Ft(D)g Fp(\000)d Ft(C)648 439 y Fs(1)668 432 y Fu(Op)725 411 y Fs(w)744 432 y Fu(\()p Ft(r)786 411 y Fs(1)785 444 y Fm(m)819 432 y Fu(\);)746 b(\(4.16\))306 538 y Ft(D)15 b Fu(=)f(Op)470 518 y Fs(w)490 538 y Fu(\()p Fp(h)536 504 y Ft(\030)p 533 527 30 2 v 533 572 a(f)567 538 y Fp(i)p Fu(\))p Ft(B)642 545 y Fs(1)662 538 y Ft(;)22 b(B)735 545 y Fs(1)769 538 y Fu(=)13 b(Op)877 518 y Fs(w)897 538 y Fu(\()p Ft(b)937 545 y Fs(1)956 538 y Ft(F)995 518 y Fs(1)988 550 y(+)1017 538 y Fu(\()p Ft(a)1062 545 y Fs(0)1082 538 y Fu(\)\))p Ft(;)21 b(b)1176 545 y Fs(1)1210 538 y Fp(2)14 b Ft(S)1287 545 y Fm(unif)1363 538 y Fu(\()p Ft(f)5 b(w)1447 518 y Fm(s)p Fo(\000)1495 504 y Fk(3)p 1497 510 16 2 v 1497 531 a(2)1519 538 y Ft(;)j(g)1564 546 y Fo(j)p Fm(\020)r Fo(j)1603 538 y Fu(\))p Ft(;)306 669 y(r)329 648 y Fs(1)328 681 y Fm(m)375 669 y Fp(2)14 b Ft(S)452 676 y Fm(unif)528 669 y Fu(\()p Fp(h)574 635 y Ft(\030)p 571 658 30 2 v 571 703 a(f)606 669 y Fp(i)625 648 y Fs(2)645 669 y Fp(h)p Ft(x)p Fp(i)711 648 y Fo(\000)p Fm(m)772 669 y Ft(;)8 b(g)817 677 y Fo(j)p Fm(\020)r Fo(j)857 669 y Fu(\))p Ft(:)278 798 y Fu(Using)16 b(\(4.16\))h(w)o(e)f(ha)o(v)o(e)g(to)g(b)q(ound)269 938 y Ft(T)305 918 y Fo(\000)p Fs(1)352 938 y Ft(R)389 918 y Fo(\003)409 883 y Fl(\020)459 904 y Fu(2)p 444 927 55 2 v 444 972 a Ft(C)479 979 y Fs(0)503 938 y Fu(Op)560 918 y Fs(w)588 898 y Fl(\000)611 938 y Ft(w)647 918 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)728 938 y Ft(F)760 945 y Fs(+)789 938 y Fu(\()p Ft(a)834 945 y Fs(0)854 938 y Fu(\))873 918 y Fs(2)892 938 y Fp(<)p Fu(\()p Ft(h)c Fp(\000)e Ft(\020)t Fu(\))1080 898 y Fl(\001)956 1056 y Fu(+)h Ft(D)1046 1035 y Fo(\003)1066 1056 y Ft(D)i Fu(+)e Ft(C)1203 1063 y Fs(1)1223 1056 y Fu(Op)1280 1035 y Fs(w)1299 1056 y Fu(\()p Ft(r)1341 1035 y Fs(1)1340 1068 y Fm(m)1374 1056 y Fu(\))1393 1000 y Fl(\021)1422 1056 y Ft(RT)1495 1035 y Fo(\000)p Fs(1)1542 1056 y Ft(:)42 b Fu(\(4.17\))228 1173 y(F)l(or)16 b(the)g(con)o(tribution)g(from)f (the)h(\014rst)h(term)d(in)i(\(4.17\))q(,)f(w)o(e)h(write)269 1347 y(Op)326 1327 y Fs(w)354 1307 y Fl(\000)377 1347 y Ft(w)413 1327 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)494 1347 y Ft(F)526 1354 y Fs(+)555 1347 y Fu(\()p Ft(a)600 1354 y Fs(0)620 1347 y Fu(\))639 1327 y Fs(2)658 1347 y Fp(<)p Fu(\()p Ft(h)11 b Fp(\000)g Ft(\020)t Fu(\))846 1307 y Fl(\001)883 1347 y Fp(\024)j(<)980 1307 y Fl(\000)1003 1347 y Fu(Op)1060 1327 y Fs(w)1087 1307 y Fl(\000)1110 1347 y Ft(w)1146 1327 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1228 1347 y Ft(F)1260 1354 y Fs(+)1289 1347 y Fu(\()p Ft(a)1334 1354 y Fs(0)1353 1347 y Fu(\))1372 1327 y Fs(2)1392 1307 y Fl(\001)1423 1347 y Fu(\()p Ft(H)h Fp(\000)c Ft(\020)t Fu(\))1591 1307 y Fl(\001)873 1454 y Fu(+)933 1441 y(^)922 1454 y Ft(B)962 1433 y Fo(\003)959 1466 y Fs(1)981 1454 y Fu(Op)1038 1433 y Fs(w)1057 1454 y Fu(\()p Fp(h)1103 1420 y Ft(\030)p 1100 1442 30 2 v 1100 1488 a(f)1135 1454 y Fp(i)p Fu(\))1173 1433 y Fs(2)1205 1441 y Fu(^)1193 1454 y Ft(B)1230 1461 y Fs(1)1261 1454 y Fu(+)g Ft(C)1345 1461 y Fs(2)1365 1454 y Fu(Op)1422 1433 y Fs(w)1441 1454 y Fu(\()r(^)-26 b Ft(r)1483 1433 y Fs(1)1482 1466 y Fm(m)1515 1454 y Fu(\))p Ft(;)50 b Fu(\(4.18\))228 1592 y(where)389 1580 y(^)378 1592 y Ft(B)415 1599 y Fs(1)459 1592 y Fu(and)28 b(^)-26 b Ft(r)586 1574 y Fs(1)585 1605 y Fm(m)643 1592 y Fu(ha)o(v)o(e)25 b(the)g(same)f(form)g(as)h Ft(B)1218 1599 y Fs(1)1263 1592 y Fu(and)h Ft(r)1390 1574 y Fs(1)1389 1605 y Fm(m)1422 1592 y Fu(,)h(resp)q(ectiv)o(ely)l(.)228 1650 y(Clearly)19 b(w)o(e)g(ma)o(y)f(estimate)g(the)i(con)o(tribution)f (from)g(the)g(\014rst)h(term)e(on)i(the)228 1709 y(righ)o(t)c(hand)h (side)e(of)20 b(\(4.18\))d(b)o(y)f(the)g(induction)g(h)o(yp)q(othesis)g (since)280 1811 y Ft(T)316 1790 y Fo(\000)p Fs(1)362 1811 y Ft(R)399 1790 y Fo(\003)420 1811 y Fp(<)464 1771 y Fl(\000)487 1811 y Fu(Op)544 1790 y Fs(w)571 1771 y Fl(\000)594 1811 y Ft(w)630 1790 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)712 1811 y Ft(F)744 1818 y Fs(+)773 1811 y Fu(\()p Ft(a)818 1818 y Fs(0)837 1811 y Fu(\))856 1790 y Fs(2)876 1771 y Fl(\001)907 1811 y Fu(\()p Ft(H)f Fp(\000)c Ft(\020)t Fu(\))1075 1771 y Fl(\001)1106 1811 y Ft(RT)1179 1790 y Fo(\000)p Fs(1)294 1891 y Fu(=)i Fp(<)389 1851 y Fl(\000)412 1891 y Ft(T)448 1871 y Fo(\000)p Fs(1)495 1891 y Ft(R)532 1871 y Fo(\003)552 1891 y Fu(Op)609 1871 y Fs(w)629 1891 y Fu(\()p Ft(w)684 1871 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)765 1891 y Ft(F)797 1898 y Fs(+)826 1891 y Fu(\()p Ft(a)871 1898 y Fs(0)890 1891 y Fu(\))909 1871 y Fs(2)929 1891 y Ft(T)965 1871 y Fo(\000)p Fs(1)1012 1851 y Fl(\001)294 1975 y Fu(=)g Fp(<)389 1935 y Fl(\000)412 1975 y Fu(\()p Fp(f)p Fu(Op)513 1955 y Fs(w)533 1975 y Fu(\()p Ft(a)p Fu(\))p Ft(f)5 b(w)662 1955 y Fm(s)p Fo(\000)p Fs(3)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)802 1975 y Fu(Op)859 1955 y Fs(w)879 1975 y Fu(\()p Ft(F)937 1955 y Fs(1)930 1987 y(+)959 1975 y Fu(\()p Ft(a)1004 1982 y Fs(0)1023 1975 y Fu(\)\))11 b(+)g(Op)1178 1955 y Fs(w)1198 1975 y Fu(\()p Ft(r)1239 1982 y Fm(m)1272 1975 y Fu(\))p Fp(g)p Ft(Rf)5 b(w)1418 1955 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1587 1975 y Fu(\))1606 1955 y Fo(\003)1626 1935 y Fl(\001)1657 1975 y Ft(;)228 2080 y Fu(where)16 b Ft(a)d Fp(2)h Ft(S)485 2087 y Fm(unif)562 2080 y Fu(\(1)p Ft(;)8 b(g)650 2088 y Fo(j)p Fm(\020)r Fo(j)690 2080 y Fu(\))16 b(and)h Ft(r)842 2087 y Fm(m)889 2080 y Fp(2)d Ft(S)966 2087 y Fm(unif)1043 2080 y Fu(\()p Fp(h)p Ft(x)p Fp(i)1128 2062 y Fo(\000)p Fm(m)1189 2080 y Ft(;)8 b(g)1234 2088 y Fo(j)p Fm(\020)r Fo(j)1273 2080 y Fu(\).)278 2138 y(So)13 b(in)g(order)g(to)g(\014nish)g(the)g(pro)q (of)h(w)o(e)f(only)f(ha)o(v)o(e)h(to)g(tak)o(e)f(care)h(of)g(the)g (second)228 2197 y(and)k(third)f(terms)e(from)h(\(4.17\))q(.)21 b(W)l(e)16 b(write)g(for)h(the)f(second)g(one)550 2329 y Ft(D)591 2308 y Fo(\003)612 2329 y Ft(D)g Fu(=)d Fp(<)763 2259 y Fl(\022)800 2329 y Ft(B)840 2308 y Fo(\003)837 2341 y Fs(1)871 2316 y Fu(\024)860 2329 y Ft(B)r Fu(Op)956 2308 y Fs(w)976 2329 y Fu(\()p Fp(h)1022 2295 y Ft(\030)p 1019 2317 V 1019 2363 a(f)1053 2329 y Fp(i)1072 2308 y Fs(2)1092 2329 y Fu(\))1111 2259 y Fl(\023)1159 2329 y Fu(+)e(Op)1265 2308 y Fs(w)1285 2329 y Fu(\()r(\024)-26 b Ft(r)1326 2336 y Fm(m)1359 2329 y Fu(\))p Ft(;)206 b Fu(\(4.19\))228 2472 y(where)384 2459 y(\024)372 2472 y Ft(B)22 b Fu(=)d(Op)545 2454 y Fs(w)565 2472 y Fu(\()582 2459 y(\024)584 2472 y Ft(b)616 2459 y Fu(\024)605 2472 y Ft(F)644 2454 y Fs(1)637 2484 y(+)666 2472 y Fu(\()p Ft(a)711 2479 y Fs(0)730 2472 y Fu(\)\))g(with)900 2459 y(\024)902 2472 y Ft(b)g Fp(2)h Ft(S)1025 2479 y Fm(unif)1101 2472 y Fu(\()p Ft(f)5 b(w)1185 2453 y Fm(s)p Fo(\000)1233 2440 y Fk(3)p 1235 2446 16 2 v 1235 2466 a(2)1257 2472 y Ft(;)j(g)1302 2480 y Fo(j)p Fm(\020)r Fo(j)1341 2472 y Fu(\))20 b(and)1489 2459 y(\024)1478 2472 y Ft(F)1517 2454 y Fs(1)1510 2484 y(+)1559 2472 y Fu(is)f(giv)o(en)228 2536 y(as)24 b Ft(F)334 2518 y Fs(1)327 2548 y(+)380 2536 y Fu(but)g(suc)o(h)f(that)718 2523 y(\024)707 2536 y Ft(F)746 2518 y Fs(1)739 2548 y(+)795 2536 y Fu(=)j(1)f(on)f(a)g (neigh)o(b)q(ourho)q(o)q(d)i(of)e(supp)8 b Ft(F)1582 2518 y Fs(1)1575 2548 y(+)1604 2536 y Fu(,)26 b(and)230 2601 y(\024)-26 b Ft(r)250 2608 y Fm(m)297 2601 y Fp(2)14 b Ft(S)374 2608 y Fm(unif)450 2601 y Fu(\()p Fp(h)495 2579 y Fm(\030)p 493 2589 21 2 v 493 2618 a(f)519 2601 y Fp(i)538 2583 y Fs(2)558 2601 y Fp(h)p Ft(x)p Fp(i)624 2583 y Fo(\000)p Fm(m)686 2601 y Ft(;)8 b(g)731 2609 y Fo(j)p Fm(\020)r Fo(j)770 2601 y Fu(\).)956 2658 y Fq(25)p eop %%Page: 26 26 26 25 bop 278 213 a Fu(W)l(e)16 b(write)474 309 y(Op)531 288 y Fs(w)551 309 y Fu(\()p Fp(h)597 275 y Ft(\030)p 594 298 30 2 v 594 343 a(f)629 309 y Fp(i)648 288 y Fs(2)668 309 y Fu(\))d(=)h Ft(f)781 288 y Fo(\000)p Fs(2)829 309 y Ft(p)853 288 y Fs(2)884 309 y Fu(+)d(Op)990 288 y Fs(w)1009 309 y Fu(\()p Ft(a)p Fu(\);)474 440 y Ft(a)j Fu(=)g(1)d Fp(\000)g Ft(i\030)i Fp(\001)e(r)p Ft(f)798 419 y Fo(\000)p Fs(2)856 440 y Fu(+)g(4)929 419 y Fo(\000)p Fs(1)977 440 y Fu(\001)p Ft(f)1047 419 y Fo(\000)p Fs(2)1108 440 y Fp(2)j Ft(S)1185 447 y Fm(unif)1261 440 y Fu(\()p Fp(h)1307 406 y Ft(\030)p 1304 428 V 1304 474 a(f)1339 440 y Fp(i)p Ft(;)8 b(g)1403 448 y Fo(j)p Fm(\020)r Fo(j)1443 440 y Fu(\))p Ft(:)278 549 y Fu(Substituted)16 b(in)o(to)g(\(4.19\))h(this) f(yields)500 636 y Ft(D)541 615 y Fo(\003)562 636 y Ft(D)g Fu(=)d Fp(<)p Fu(\()p Ft(B)764 615 y Fo(\003)761 648 y Fs(1)795 623 y Fu(\024)784 636 y Ft(B)r(f)852 615 y Fo(\000)p Fs(2)900 636 y Ft(p)924 615 y Fs(2)944 636 y Fu(\))e(+)g Fp(<)p Fu(\()p Ft(D)1119 615 y Fo(\003)1151 623 y Fu(~)1139 636 y Ft(B)s Fu(\))g(+)g(Op)1315 615 y Fs(w)1335 636 y Fu(\()r(~)-26 b Ft(r)1376 643 y Fm(m)1409 636 y Fu(\))p Ft(;)156 b Fu(\(4.20\))228 729 y(where)382 716 y(~)370 729 y Ft(B)21 b Fu(and)f(~)-26 b Ft(r)546 736 y Fm(m)597 729 y Fu(are)18 b(de\014ned)g(similarly)l(.)23 b(Next)17 b(w)o(e)g(substitute)h Ft(p)1517 711 y Fs(2)1554 729 y Fu(=)e(\()p Ft(H)g Fp(\000)228 787 y Ft(\020)t Fu(\))6 b(+)g(\()p Ft(\020)k Fp(\000)c Ft(V)11 b Fu(\))j(and)h(apply)e (Cauc)o(h)o(y-Sc)o(h)o(w)o(arz)g(to)h(the)g(second)g(term)e(to)i(the)g (righ)o(t)228 845 y(in)i(\(4.20\).)22 b(After)15 b(a)i(subtraction)f(w) o(e)g(conclude)g(that)394 930 y Ft(D)435 909 y Fo(\003)455 930 y Ft(D)g Fp(\024)d Fu(2)p Fp(<)631 889 y Fl(\000)655 930 y Ft(B)695 909 y Fo(\003)692 942 y Fs(1)726 917 y Fu(\024)714 930 y Ft(B)s(f)783 909 y Fo(\000)p Fs(2)830 930 y Fu(\()p Ft(H)j Fp(\000)10 b Ft(\020)t Fu(\))998 889 y Fl(\001)1032 930 y Fu(+)1093 917 y(\026)1081 930 y Ft(B)1121 909 y Fo(\003)1152 917 y Fu(\026)1141 930 y Ft(B)j Fu(+)e(Op)1297 909 y Fs(w)1317 930 y Fu(\()r(\026)-26 b Ft(r)1358 937 y Fm(m)1391 930 y Fu(\))p Ft(;)174 b Fu(\(4.21\))228 1018 y(for)16 b(y)o(et)g(another)g(couple)g(of)h (similar)d(op)q(erators)1157 1005 y(\026)1146 1018 y Ft(B)k Fu(and)f(Op)1353 1000 y Fs(w)1373 1018 y Fu(\()r(\026)-26 b Ft(r)1414 1025 y Fm(m)1447 1018 y Fu(\).)278 1076 y(W)l(e)14 b(ma)o(y)f(treat)i(the)g(con)o(tribution)f(from)g(the)g(\014rst)h(term) e(on)j(the)e(righ)o(t)h(hand)228 1134 y(side)i(of)k(\(4.21\))e(as)f(ab) q(o)o(v)o(e.)26 b(The)17 b(second)h(term)e(is)h(handled)h(b)o(y)f(the)h (induction)228 1192 y(h)o(yp)q(othesis.)i(The)12 b(third)g(term)f(is)h (similar)e(to)j(the)f(third)h(term)d(from)h(\(4.17\))q(.)20 b(F)l(or)228 1251 y(these)13 b(terms)f(w)o(e)h(use)g(the)h(resolv)o(en) o(t)e(iden)o(tit)o(y)f Ft(R)p Fu(\()p Ft(z)r Fu(\))j(=)g Ft(R)p Fu(\()p Ft(i)p Fu(\))5 b(+)g(\()p Ft(z)j Fp(\000)d Ft(i)p Fu(\))p Ft(R)p Fu(\()p Ft(i)p Fu(\))p Ft(R)p Fu(\()p Ft(z)r Fu(\))228 1309 y(and)17 b(the)f(fact)g(that)h Fp(h)p Ft(x)p Fp(i)675 1291 y Fo(\000)p Fm(m)736 1309 y Ft(p)760 1291 y Fs(2)780 1309 y Ft(R)p Fu(\()p Ft(i)p Fu(\))p Fp(h)p Ft(x)p Fp(i)938 1291 y Fm(m)988 1309 y Fu(is)f(b)q(ounded;)h(w)o(e)f(need)f Ft(m)h Fu(su\016cien)o(tly)228 1367 y(large.)p 1687 1367 2 33 v 1689 1335 30 2 v 1689 1367 V 1718 1367 2 33 v 278 1454 a(The)11 b(estimate)e(\(4.5b\))j(is)f (an)h(easy)f(consequence)f(of)i(Lemma)d(4.5)j(and)f(Lemma)228 1512 y(4.6)16 b(b)q(elo)o(w.)22 b(Lemma)14 b(4.6)i(will)f(also)i(b)q(e) g(useful)e(in)h(Subsection)g(4.5.)228 1602 y Fy(Lemm)o(a)g(4.6.)24 b Fh(Supp)n(ose)d Ft(A)f Fu(=)g Ft(A)879 1609 y Fm(\020)919 1602 y Fh(is)h(a)g Fp(B)r Fu(\()p Ft(L)1108 1584 y Fs(2)1127 1602 y Fu(\))p Fh(-value)n(d)h(function)g(in)f Ft(\020)k Fp(2)20 b Fu(\000)1702 1609 y Fm(\022)228 1660 y Fh(such)e(that)f(for)g (al)r(l)i Ft(\017)13 b(>)h Fu(0)p Fh(,)k Ft(s)c Fp(\025)f Fu(0)18 b Fh(the)g(op)n(er)n(ator)618 1743 y Fp(h)p Ft(x)p Fp(i)684 1723 y Fo(\000)p Fm(\017)728 1743 y Ft(f)5 b(w)793 1723 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)893 1743 y Ft(AR)p Fu(\()p Ft(\020)t Fu(\))p Ft(w)1066 1723 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1192 1743 y Ft(f)g Fp(h)p Ft(x)p Fp(i)1287 1723 y Fo(\000)p Fm(\017)1598 1743 y Fu(\(4.22\))228 1825 y Fh(is)17 b(uniformly)g(b)n(ounde)n(d.)23 b(Then)18 b(also)722 1909 y Ft(k)749 1888 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)886 1909 y Ft(AR)p Fu(\()p Ft(\020)t Fu(\))p Ft(k)1050 1888 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1214 1909 y Ft(;)228 1991 y Fh(is)f(b)n(ounde)n(d)h(uniformly)f(in)h Ft(\020)g Fp(2)c Fu(\000)864 1998 y Fm(\022)901 1991 y Fh(for)j(al)r(l)i Ft(\017)13 b(>)h Fu(0)p Fh(,)k Ft(t)13 b Fp(\025)h Fu(0)p Fh(.)228 2080 y(Pr)n(o)n(of)i(of)h(L)n(emma)f(4.6.) 21 b Fu(Let)16 b(1)e(=)g Ft(F)898 2087 y Fs(1)928 2080 y Fu(+)d Ft(F)1009 2087 y Fs(2)1045 2080 y Fu(b)q(e)16 b(a)h(sharp)g(partition)f(of)g(unit)o(y)g(on)228 2138 y Fn(R)p Fu(,)h(supp)q(\()p Ft(F)450 2145 y Fs(1)469 2138 y Fu(\))j(=)g(\()p Fp(\0001)p Ft(;)8 b Fu(0],)20 b(supp\()p Ft(F)919 2145 y Fs(2)938 2138 y Fu(\))h(=)f([0)p Ft(;)8 b Fu(+)p Fp(1)p Fu(\))19 b(and)h(let)g(us)g(consider)f(the)228 2196 y(t)o(w)o(o)d(terms)507 2279 y Fp(h)p Ft(x)p Fp(i)573 2258 y Fo(\000)p Fm(\017)617 2279 y Ft(k)644 2258 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)739 2279 y Ft(AR)p Fu(\()p Ft(\020)t Fu(\))p Ft(k)903 2258 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1026 2279 y Fp(h)p Ft(x)p Fp(i)1092 2258 y Fo(\000)p Fm(\017)1136 2279 y Ft(F)1168 2286 y Fs(1)1188 2279 y Fu(\()p Fp(j)p Ft(\020)t Fp(j)11 b(\000)f(h)p Ft(x)p Fp(i)1386 2258 y Fo(\000)p Fm(\026)1438 2279 y Fu(\))141 b(\(4.23\))493 2362 y Fp(h)p Ft(x)p Fp(i)559 2342 y Fo(\000)p Fm(\017)604 2362 y Ft(k)631 2342 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)726 2362 y Ft(AR)p Fu(\()p Ft(\020)t Fu(\))p Ft(k)890 2342 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1013 2362 y Fp(h)p Ft(x)p Fp(i)1079 2342 y Fo(\000)p Fm(\017)1123 2362 y Ft(F)1155 2369 y Fs(2)1174 2362 y Fu(\()p Fp(j)p Ft(\020)t Fp(j)11 b(\000)g(h)p Ft(x)p Fp(i)1373 2342 y Fo(\000)p Fm(\026)1424 2362 y Fu(\))p Ft(:)141 b Fu(\(4.24\))278 2448 y(First)14 b(w)o(e)g(study)i (\(4.23\).)21 b(On)15 b(supp\()p Ft(F)989 2455 y Fs(1)1009 2448 y Fu(\()p Fp(j)p Ft(\020)t Fp(j)8 b(\000)g(h)p Ft(x)p Fp(i)1202 2430 y Fo(\000)p Fm(\026)1252 2448 y Fu(\)\))15 b(w)o(e)f(ha)o(v)o(e)g Fp(j)p Ft(\020)t Fp(j)f(\024)h(h)p Ft(x)p Fp(i)1671 2430 y Fo(\000)p Fm(\026)228 2506 y Fu(and)j(therefore)666 2588 y Fp(h)p Ft(x)p Fp(i)732 2568 y Fo(\000)p Fm(\026=)p Fs(2)832 2588 y Fp(\024)c Ft(f)908 2596 y Fo(j)p Fm(\020)r Fo(j)948 2588 y Fu(\()p Ft(x)p Fu(\))g Fp(\024)h Ft(C)t Fp(h)p Ft(x)p Fp(i)1185 2568 y Fo(\000)p Fm(\026=)p Fs(2)1271 2588 y Ft(:)313 b Fu(\(4.25\))956 2658 y Fq(26)p eop %%Page: 27 27 27 26 bop 228 213 a Fu(Let)16 b(us)h(c)o(ho)q(ose)g Ft(s)f Fu(suc)o(h)g(that)375 318 y(\()p Fp(\000)p Ft(t)10 b Fp(\000)h Fu(1)p Ft(=)p Fu(2\)\(1)i(+)e Ft(\026=)p Fu(2\))k(=)e Fp(\000)p Ft(s)e Fp(\000)g Fu(1)p Ft(=)p Fu(2)h Fp(\000)f Fu(\()p Fp(\000)p Ft(s)g Fu(+)g(1)p Ft(=)p Fu(2\))p Ft(\026=)p Fu(2)p Ft(;)157 b Fu(\(4.26\))228 427 y(that)16 b(is)h Ft(s)c Fu(=)h Ft(t)p Fu(\(1)d(+)g Ft(\026=)p Fu(2\))p Ft(=)p Fu(\(1)i Fp(\000)e Ft(\026=)p Fu(2\))17 b(\(notice)f(that)g Ft(s)e Fp(\025)g Fu(0\),)i(and)h(therefore)414 533 y(\()p Ft(t)10 b Fp(\000)h Fu(1)p Ft(=)p Fu(2\)\(1)i(+)e Ft(\026=)p Fu(2\))k(=)e Ft(s)e Fp(\000)g Fu(1)p Ft(=)p Fu(2)h(+)f(\()p Fp(\000)p Ft(s)g Fp(\000)g Fu(1)p Ft(=)p Fu(2\))p Ft(\026=)p Fu(2)p Ft(:)188 b Fu(\(4.27\))228 642 y(So)17 b(\(using)g(\(4.27\)\))g (to)f(the)g(left)g(of)g(the)g(resolv)o(en)o(t)f(in)h(\(4.23\))h(w)o(e)f (can)g(write)617 795 y Ft(k)644 774 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)753 795 y Fu(=)d Ft(w)840 774 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)939 795 y Ft(f)977 725 y Fl(\022)1019 761 y Fp(h)p Ft(x)p Fp(i)1085 743 y Fo(\000)p Fm(\026=)p Fs(2)p 1019 783 153 2 v 1080 829 a Ft(f)1176 725 y Fl(\023)1213 731 y Fm(s)p Fs(+1)p Fm(=)p Fs(2)1320 795 y Ft(:)228 937 y Fu(W)l(e)j(kno)o(w,)g(since)f Ft(f)k Fp(\025)14 b(h)p Ft(x)p Fp(i)734 919 y Fo(\000)p Fm(\026=)p Fs(2)820 937 y Fu(,)i(that)h(the)f(\()p Fp(\001)p Fu(\))1092 919 y Fm(s)p Fs(+1)p Fm(=)p Fs(2)1190 937 y Fu(-term)f(is)h(b)q (ounded.)278 995 y(T)l(o)g(the)g(righ)o(t)g(in)g(\(4.23\))h(w)o(e)f (can)g(write)g(using)h(\(4.26\))575 1148 y Ft(k)602 1127 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)739 1148 y Fu(=)d Ft(w)827 1127 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)953 1148 y Ft(f)991 1078 y Fl(\022)1032 1114 y Fp(h)p Ft(x)p Fp(i)1098 1096 y Fo(\000)p Fm(\026=)p Fs(2)p 1032 1137 V 1094 1182 a Ft(f)1190 1078 y Fl(\023)1227 1084 y Fo(\000)p Fm(s)p Fs(+1)p Fm(=)p Fs(2)1361 1148 y Ft(:)228 1285 y Fu(No)o(w)i(w)o(e)g(infer)f(uniform)g(b)q(oundedness) i(of)j(\(4.23\))d(from)f(\(4.22\))g(and)h(\(4.25\))q(.)278 1343 y(The)h(b)q(oundedness)h(of)j(\(4.24\))c(is)g(more)f(subtle.)26 b(Here)17 b(w)o(e)h(cannot)g(con)o(v)o(ert)228 1402 y(all)d Ft(f)5 b Fu('s)16 b(to)g Fp(h)p Ft(x)p Fp(i)498 1384 y Fo(\000)p Fm(\026=)p Fs(2)585 1402 y Fu('s.)k(Instead)c(w)o(e)g(ha)o (v)o(e)f(to)h(compare)e(some)h Ft(f)5 b Fu('s)16 b(to)g Fp(j)p Ft(\020)t Fp(j)1573 1384 y Fs(1)p Fm(=)p Fs(2)1644 1402 y Fu(and)228 1462 y(others)g(to)h Fp(h)p Ft(x)p Fp(i)500 1444 y Fo(\000)p Fm(\026=)p Fs(2)587 1462 y Fu(.)k(Notice)15 b(that)h(on)h(supp)q(\()p Ft(F)1100 1469 y Fs(2)1119 1462 y Fu(\()p Fp(j)p Ft(\020)t Fp(j)11 b(\000)g(h)p Ft(x)p Fp(i)1318 1444 y Fo(\000)p Fm(\026)1369 1462 y Fu(\)\))16 b(w)o(e)g(ha)o(v)o(e)688 1572 y Fp(h)p Ft(x)p Fp(i)754 1551 y Fo(\000)p Fm(\026=)p Fs(2)854 1572 y Fp(\024)e Ft(f)931 1579 y Fo(j)p Fm(\020)r Fo(j)970 1572 y Fu(\()p Ft(x)p Fu(\))g Fp(\024)f Ft(C)t Fp(j)p Ft(\020)t Fp(j)1194 1551 y Fs(1)p Fm(=)p Fs(2)1249 1572 y Ft(:)335 b Fu(\(4.28\))228 1681 y(F)l(or)16 b Ft(s)e Fu(=)g(\(1)d(+)g Ft(\026=)p Fu(2\))p Ft(t)17 b Fu(w)o(e)f(ha)o(v)o(e) 569 1786 y(\()p Ft(t)11 b Fp(\000)g Fu(1)p Ft(=)p Fu(2\)\(1)h(+)f Ft(\026=)p Fu(2\))k(=)f Fp(\000)p Ft(\026=)p Fu(4)e(+)f Ft(s)g Fp(\000)g Fu(1)p Ft(=)p Fu(2)p Ft(;)226 b Fu(\(4.29\))570 1862 y(\()p Ft(t)11 b Fu(+)g(1)p Ft(=)p Fu(2\)\(1)h(+)f Ft(\026=)p Fu(2\))k(=)f Ft(s)d Fu(+)g(1)p Ft(=)p Fu(2)h(+)f Ft(\026=)p Fu(4)p Ft(:)266 b Fu(\(4.30\))228 1971 y(T)l(o)17 b(the)f(left)f(in)h(\(4.24\))h(w)o(e)f(write,)f(using)i(\(4.29\))q(,) 467 2124 y Ft(k)494 2103 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)603 2124 y Fu(=)655 2053 y Fl(\022)696 2090 y Fp(j)p Ft(\020)t Fp(j)749 2072 y Fs(1)p Fm(=)p Fs(2)p 696 2112 108 2 v 735 2158 a Ft(f)809 2053 y Fl(\023)846 2060 y Fm(s)872 2053 y Fl(\022)914 2090 y Fp(h)p Ft(x)p Fp(i)980 2072 y Fo(\000)p Fm(\026=)p Fs(2)p 914 2112 153 2 v 976 2158 a Ft(f)1071 2053 y Fl(\023)1108 2060 y Fs(1)p Fm(=)p Fs(2)1171 2124 y Fp(j)p Ft(\020)t Fp(j)1224 2103 y Fo(\000)p Fm(s=)p Fs(2)1305 2124 y Ft(f)5 b(w)1370 2103 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1470 2124 y Ft(:)228 2260 y Fu(T)l(o)17 b(the)f(righ)o(t)f(in)h(\(4.24\))h(w)o(e)f(write)453 2413 y Ft(k)480 2392 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)617 2413 y Fu(=)668 2343 y Fl(\022)749 2379 y Ft(f)p 710 2402 108 2 v 710 2447 a Fp(j)p Ft(\020)t Fp(j)763 2433 y Fs(1)p Fm(=)p Fs(2)823 2343 y Fl(\023)859 2354 y Fm(s)886 2343 y Fl(\022)928 2379 y Fp(h)p Ft(x)p Fp(i)994 2361 y Fo(\000)p Fm(\026=)p Fs(2)p 928 2402 153 2 v 989 2447 a Ft(f)1085 2343 y Fl(\023)1122 2350 y Fs(1)p Fm(=)p Fs(2)1185 2413 y Fp(j)p Ft(\020)t Fp(j)1238 2392 y Fm(s=)p Fs(2)1292 2413 y Ft(f)5 b(w)1357 2392 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1483 2413 y Ft(:)278 2550 y Fu(No)o(w)15 b(w)o(e)g(infer)f(uniform)g(b)q (oundedness)j(of)i(\(4.24\))d(from)e(\(4.22\))i(and)g(\(4.28\))q(.)p 1687 2608 2 33 v 1689 2577 30 2 v 1689 2608 V 1718 2608 2 33 v 956 2658 a Fq(27)p eop %%Page: 28 28 28 27 bop 228 213 a Fu(4.4.)28 b Fy(Classical)16 b(mec)n(hanics.)22 b Fu(The)14 b(purp)q(ose)h(of)g(this)f(subsection)g(is)g(t)o(w)o (ofold:)228 271 y(W)l(e)g(presen)o(t)f(v)m(arious)i(notation)f(and)h (computations)e(needed)h(in)f(the)h(next)g(sub-)228 329 y(section.)37 b(Secondly)21 b(w)o(e)g(sho)o(w)h(ho)o(w)g(one)g(can)g (construct)g(a)g(classical)f(`prop-)228 387 y(agation)f(observ)m(able') f(whic)o(h)f(yields)g(a)h(classical)f(analogue)i(of)g(Theorem)d(1.4.) 228 445 y(The)e(more)f(tec)o(hnical)g(material)f(of)j(Subsection)f(4.5) h(ma)o(y)e(b)q(e)h(view)o(ed)f(as)i(b)q(eing)228 503 y(based)h(on)f(the)g(classical)g(pro)q(of)h(presen)o(ted)f(here.)278 561 y(Recall)e(the)h(de\014nitions)g(of)g Ft(a)822 568 y Fs(0)842 561 y Fu(,)g Ft(b)p Fu(,)f Ft(f)5 b Fu(,)16 b Ft(h)p Fu(,)f Ft(w)h Fu(and)g Ft(W)22 b Fu(from)14 b(\(4.1\))q(,)h(\(4.2\),)g(\(4.3\))228 619 y(and)i(\(4.4\).)k(F)l(or)c (an)o(y)f Ft(E)h Fp(\025)c Fu(0)k(w)o(e)f(de\014ne)685 704 y Ft(v)f Fu(=)f Ft(v)800 711 y Fm(E)829 704 y Fu(\()p Ft(x)p Fu(\))f(=)h Ft(\024)988 683 y Fo(\000)p Fs(2)988 716 y(0)1035 704 y Ft(E)h Fu(+)c Fp(h)p Ft(x)p Fp(i)1201 683 y Fo(\000)p Fm(\026)1252 704 y Ft(;)228 787 y Fu(and)17 b(compute)614 890 y Fp(r)p Ft(f)5 b Fu(\()p Ft(x)p Fu(\))14 b(=)845 856 y Fp(\000)p Ft(\026)p 821 878 115 2 v 821 924 a Fu(2)e Fp(\000)f Ft(\026)941 890 y(f)970 869 y Fo(\000)p Fs(1)1017 890 y Fp(h)p Ft(x)p Fp(i)1083 869 y Fo(\000)p Fm(\026)p Fo(\000)p Fs(1)1204 856 y Ft(x)p 1185 878 67 2 v 1185 924 a Fp(h)p Ft(x)p Fp(i)1256 890 y Ft(;)607 1010 y Fp(r)p Ft(w)q Fu(\()p Ft(x)p Fu(\))j(=)f(\()p Ft(\024)863 990 y Fo(\000)p Fs(2)863 1022 y(0)911 1010 y Ft(E)h Fu(+)d Fp(h)p Ft(x)p Fp(i)1076 990 y Fo(\000)p Fm(\026)1127 1010 y Fu(\))1155 976 y Ft(x)p 1151 999 37 2 v 1151 1044 a(w)1206 1010 y Fu(=)j Ft(v)1292 976 y(x)p 1289 999 V 1289 1044 a(w)1329 1010 y(:)255 b Fu(\(4.31\))228 1109 y(Recall)15 b(also)i(the)f(de\014nition)f(of)i(the)f(P)o(oisson)h (brac)o(k)o(et:)625 1192 y Fp(f)p Ft(a;)8 b(b)p Fp(g)744 1199 y Fm(P)786 1192 y Fu(=)14 b Fp(r)880 1199 y Fm(\030)898 1192 y Ft(a)d Fp(\001)g(r)1002 1199 y Fm(x)1024 1192 y Ft(b)f Fp(\000)h(r)1147 1199 y Fm(x)1169 1192 y Ft(a)f Fp(\001)h(r)1272 1199 y Fm(\030)1291 1192 y Ft(b:)228 1279 y Fu(With)k(our)h(de\014nition)e(\(4.3\))i(of)g Ft(h)p Fu(\()p Ft(x;)8 b(\030)r Fu(\))15 b(w)o(e)g(get)g(for)h(an)o(y)f (sym)o(b)q(ol)f Ft(a)p Fu(,)h Fp(f)p Ft(h;)8 b(a)p Fp(g)1642 1286 y Fm(P)1684 1279 y Fu(=)228 1337 y(2)p Ft(\030)h Fp(\001)d(r)344 1344 y Fm(x)365 1337 y Ft(a)g Fp(\000)g(r)p Ft(V)16 b Fp(\001)6 b(r)590 1344 y Fm(\030)609 1337 y Ft(a:)13 b Fu(Th)o(us,)h(with)f Ft(b)h Fu(as)g(in)g(\(4.1\),)g(an)g (elemen)o(tary)d(calculation)228 1395 y(yields)564 1479 y Fp(f)p Ft(h;)d(b)p Fp(g)685 1486 y Fm(P)727 1479 y Fu(=)14 b Ft(w)815 1458 y Fo(\000)p Fs(1)871 1439 y Fl(\000)894 1479 y Fu(2)p Ft(h)d Fu(+)g Ft(W)c Fu(\()p Ft(x)p Fu(\))k Fp(\000)g Fu(2)p Ft(b)1231 1458 y Fs(2)1250 1479 y Ft(v)r Fu(\()p Ft(x)p Fu(\))1342 1439 y Fl(\001)1373 1479 y Ft(:)211 b Fu(\(4.32\))228 1572 y Fy(Theorem)16 b(4.7.)24 b Fh(L)n(et)15 b Ft(\024)687 1579 y Fs(0)721 1572 y Ft(>)f Fu(0)i Fh(b)n(e)g(given)i(as)e(in)k Fu(\(4.4\))p Fh(.)i(Then)16 b(for)f(any)h(classic)n(al)228 1630 y(orbit)h Ft(x)p Fu(\()p Ft(t)p Fu(\))g Fh(with)h(ener)n(gy)g Ft(E)e Fp(\025)e Fu(0)478 1738 y(lim)8 b(inf)494 1771 y Fo(j)p Fm(t)p Fo(j!1)622 1738 y Fp(j)p Ft(tC)t Fp(j)707 1717 y Fo(\000)p Fs(\(1+)798 1702 y Fi(\026)p 797 1710 19 2 v 799 1731 a Fk(2)820 1717 y Fs(\))834 1706 y Fg(\000)p Fk(1)877 1738 y Fp(j)p Ft(x)p Fu(\()p Ft(t)p Fu(\))p Fp(j)13 b(\025)h Fu(1;)22 b Ft(C)17 b Fu(=)d Ft(\024)1247 1745 y Fs(0)1306 1704 y Fu(2)d(+)g Ft(\026)p 1272 1727 182 2 v 1272 1781 a Fu(\(1)g Fp(\000)1381 1759 y Fm(\026)p 1381 1769 22 2 v 1383 1798 a Fs(2)1407 1781 y Fu(\))1431 1749 y Fk(1)p 1431 1755 16 2 v 1431 1775 a(2)1458 1738 y Ft(:)126 b Fu(\(4.33\))228 1869 y Fh(Pr)n(o)n(of.)19 b Fu(W)l(e)c(shall)h(only)f (pro)o(v)o(e)g(the)g(b)q(ound)i(for)e Ft(t)f Fp(!)g Fu(+)p Fp(1)p Fu(.)20 b(Let)c(us)g(\014x)f(0)g Ft(<)g Fu(~)-26 b Ft(\024)14 b(<)228 1928 y(\024)k(<)h(\024)359 1910 y Fo(0)389 1928 y Fp(\024)g Ft(\024)475 1935 y Fs(0)494 1928 y Fu(.)30 b(W)l(e)19 b(pic)o(k)f(a)h(real-v)m(alued,)g (decreasing,)g(smo)q(oth)g(function)1672 1915 y(~)1661 1928 y Ft(F)1693 1935 y Fo(\000)228 1986 y Fu(suc)o(h)d(that)496 2057 y(~)485 2070 y Ft(F)517 2077 y Fo(\000)546 2070 y Fu(\()r(~)-26 b Ft(\024)p Fu(\))14 b(=)g(1)p Ft(;)749 2057 y Fu(~)738 2070 y Ft(F)770 2077 y Fo(\000)799 2070 y Fu(\()p Ft(\024)p Fu(\))g(=)g(0)g(and)22 b(supp)q(\()1200 2057 y(~)1189 2070 y Ft(F)1228 2049 y Fo(0)1221 2082 y(\000)1250 2070 y Fu(\))14 b Fp(\032)f Fu(\()r(~)-26 b Ft(\024)q(;)8 b(\024)p Fu(\))p Ft(;)228 2154 y Fu(and)17 b(consider)f(the)g(observ)m(able)666 2237 y Ft(q)f Fu(=)f Ft(q)r Fu(\()p Ft(x;)8 b(\030)r Fu(\))13 b(=)h Ft(w)q Fu(\()p Ft(\024)1038 2217 y Fo(0)1061 2237 y Fp(\000)d Ft(b)p Fu(\))1162 2225 y(~)1151 2237 y Ft(F)1183 2244 y Fo(\000)1212 2237 y Fu(\()p Ft(b)p Fu(\))p Ft(:)313 b Fu(\(4.34\))228 2321 y(W)l(e)15 b(claim)e(that)j Ft(q)g Fu(has)g(a)g(non-p)q(ositiv)o(e)f(deriv)m(ativ)o(e:)k(First)c(w)o(e)g (compute)f(using)228 2379 y(\(4.32\):)577 2456 y Ft(d)p 569 2478 43 2 v 569 2524 a(dt)628 2477 y Fu(~)617 2490 y Ft(F)649 2497 y Fo(\000)678 2490 y Fu(\()p Ft(b)p Fu(\))f(=)802 2449 y Fl(\000)825 2490 y Fu(2)p Ft(E)h Fu(+)d Ft(W)18 b Fp(\000)11 b Fu(2)p Ft(b)1107 2469 y Fs(2)1127 2490 y Ft(v)1153 2449 y Fl(\001)1183 2490 y Ft(w)1219 2469 y Fo(\000)p Fs(1)1278 2477 y Fu(~)1267 2490 y Ft(F)1306 2469 y Fo(0)1299 2502 y(\000)1328 2490 y Fu(\()p Ft(b)p Fu(\))750 2588 y Fp(\024)j Fu(2\()p Ft(\024)874 2568 y Fs(2)874 2601 y(0)905 2588 y Fp(\000)d Ft(\024)983 2568 y Fs(2)1003 2588 y Fu(\))p Ft(v)r(w)1084 2568 y Fo(\000)p Fs(1)1141 2576 y Fu(~)1130 2588 y Ft(F)1169 2568 y Fo(0)1162 2601 y(\000)1191 2588 y Fu(\()p Ft(b)p Fu(\))j Fp(\024)f Fu(0)p Ft(:)244 b Fu(\(4.35\))956 2658 y Fq(28)p eop %%Page: 29 29 29 28 bop 228 213 a Fu(The)17 b(con)o(tribution)g(from)g(the)g(deriv)m (ativ)o(e)f(of)i(the)f(other)h(factors)g(on)g(the)g(righ)o(t)228 271 y(hand)e(side)g(of)j(\(4.34\))e(is)e(computed)g(and)h(estimated)e (using)j(\(4.31\))f(and)h(\(4.32\))228 329 y(as)298 367 y Fl(\022)349 404 y Ft(d)p 340 426 43 2 v 340 471 a(dt)396 437 y Fu(\()p Ft(w)q Fu(\()p Ft(\024)498 417 y Fo(0)521 437 y Fp(\000)11 b Ft(b)p Fu(\)\))630 367 y Fl(\023)686 425 y Fu(~)675 437 y Ft(F)707 444 y Fo(\000)736 437 y Fu(\()p Ft(b)p Fu(\))i(=)860 397 y Fl(\000)883 437 y Fu(2)p Ft(v)r(b)p Fu(\()p Ft(\024)1001 417 y Fo(0)1023 437 y Fp(\000)e Ft(b)p Fu(\))g Fp(\000)f Fu(2)p Ft(E)15 b Fp(\000)c Ft(W)18 b Fu(+)11 b(2)p Ft(v)r(b)1482 417 y Fs(2)1501 397 y Fl(\001)1543 425 y Fu(~)1532 437 y Ft(F)1564 444 y Fo(\000)1593 437 y Fu(\()p Ft(b)p Fu(\))808 549 y Fp(\024)j(\000)8 b Fu(\(2)p Ft(E)14 b Fu(+)d Ft(W)18 b Fp(\000)11 b Fu(2)p Ft(v)r(\024\024)1270 528 y Fo(0)1282 549 y Fu(\))1320 536 y(~)1309 549 y Ft(F)1341 556 y Fo(\000)1370 549 y Fu(\()p Ft(b)p Fu(\))808 633 y Fp(\024)j(\000)p Fu(2\()p Ft(\024)971 612 y Fs(2)971 645 y(0)1002 633 y Fp(\000)d Ft(\024\024)1108 612 y Fo(0)1119 633 y Fu(\))p Ft(v)1175 620 y Fu(~)1164 633 y Ft(F)1196 640 y Fo(\000)1225 633 y Fu(\()p Ft(b)p Fu(\))i Fp(\024)h Fu(0)p Ft(:)210 b Fu(\(4.36\))278 723 y(In)16 b(particular,)f(w)o(e)h(infer)f(that)608 804 y Ft(d)p 599 826 V 599 872 a(dt)647 837 y(q)g Fp(\024)f(\000)p Fu(2\()p Ft(\024)847 817 y Fs(2)847 850 y(0)878 837 y Fp(\000)d Ft(\024\024)984 817 y Fo(0)996 837 y Fu(\))p Fp(h)p Ft(x)p Fp(i)1081 817 y Fo(\000)p Fm(\026)1143 825 y Fu(~)1132 837 y Ft(F)1164 844 y Fo(\000)1193 837 y Fu(\()p Ft(b)p Fu(\))i Fp(\024)h Fu(0)p Ft(:)242 b Fu(\(4.37\))278 945 y(By)15 b(in)o(tegrating)i(\(4.37\))g(w)o(e)e (obtain)i(the)f(uniform)f(b)q(ound)605 1071 y Ft(q)r Fu(\()p Ft(T)7 b Fu(\))i(+)761 1003 y Fl(Z)811 1017 y Fm(T)789 1116 y Fs(0)839 1071 y Fp(h)p Ft(x)p Fu(\()p Ft(t)p Fu(\))p Fp(i)961 1051 y Fo(\000)p Fm(\026)1022 1059 y Fu(~)1011 1071 y Ft(F)1043 1078 y Fo(\000)1073 1071 y Fu(\()p Ft(b)p Fu(\()p Ft(t)p Fu(\)\))p Ft(dt)k Fp(\024)g Ft(C)q(;)228 1187 y Fu(b)o(y)j(whic)o(h)f(w)o(e)h(will)f(no)o (w)i(pro)o(v)o(e)e(that)725 1266 y(~)714 1279 y Ft(F)746 1286 y Fo(\000)775 1279 y Fu(\()p Ft(b)p Fu(\()p Ft(t)p Fu(\)\))e Fp(!)h Fu(0)g(for)g Ft(t)g Fp(!)f(1)p Ft(:)362 b Fu(\(4.38\))228 1380 y(Notice)21 b(that)h(b)o(y)h(\(4.35\),)746 1368 y(~)735 1380 y Ft(F)767 1387 y Fo(\000)796 1380 y Fu(\()p Ft(b)p Fu(\()p Ft(t)p Fu(\)\))g Fp(!)h Ft(c)p Fu(;)d(so)i(w)o(e)f(need)g(only)f(to)i(sho)o(w)g(that)228 1438 y Ft(c)17 b Fu(=)g(0.)26 b(There)18 b(are)g(t)o(w)o(o)g(cases:)25 b(1\))19 b Ft(x)e Fu(is)h(b)q(ounded,)h(or)g(2\))f Fp(j)p Ft(x)p Fu(\()p Ft(t)p Fu(\))p Fp(j)e(!)h(1)g Fu(along)228 1500 y(some)11 b(sequence)g Ft(t)j Fu(=)g Ft(t)646 1507 y Fm(n)683 1500 y Fp(!)f(1)p Fu(.)20 b(In)12 b(Case)h(1\))g(w)o(e)f(ha) o(v)o(e)1244 1488 y(~)1232 1500 y Ft(F)1264 1507 y Fo(\000)1294 1500 y Fu(\()p Ft(b)p Fu(\()p Ft(t)1371 1507 y Fm(n)1393 1500 y Fu(\)\))i Fp(!)g Fu(0)f(for)f(some)228 1559 y(sequence)j Ft(t)448 1566 y Fm(n)486 1559 y Fp(!)f(1)j Fu(\(b)o(y)f(the)h(b)q (oundedness)g(of)g(the)g(in)o(tegral\))f(yielding)f Ft(c)g Fu(=)f(0.)228 1621 y(In)21 b(Case)h(2\))g(w)o(e)f(learn)g(from)g(the)g (b)q(oundedness)i(of)f Ft(q)h Fu(that)1415 1608 y(~)1404 1621 y Ft(F)1436 1628 y Fo(\000)1465 1621 y Fu(\()p Ft(b)p Fu(\()p Ft(t)1542 1628 y Fm(n)1564 1621 y Fu(\)\))g Fp(!)g Fu(0)228 1679 y(yielding)15 b Ft(c)f Fu(=)f(0)k(in)f(this)g(case)g(to)q (o.)278 1737 y(Finally)l(,)e(de\014ne)531 1852 y Ft(F)7 b Fu(\()p Ft(r)q Fu(\))13 b(=)701 1818 y(1)p 701 1841 25 2 v 701 1886 a(2)738 1784 y Fl(Z)788 1797 y Fm(r)766 1897 y Fs(1)807 1852 y Fu(\()p Ft(\024)854 1831 y Fo(\000)p Fs(2)854 1864 y(0)901 1852 y Ft(E)i Fu(+)c(\(1)g Fp(\000)1110 1818 y Ft(\026)p 1110 1841 30 2 v 1112 1886 a Fu(2)1144 1852 y(\))1163 1831 y Fo(\000)p Fs(1)1210 1852 y Ft(s)1233 1831 y Fo(\000)p Fm(\026)1284 1852 y Fu(\))1303 1831 y Fo(\000)1335 1818 y Fk(1)p 1335 1824 16 2 v 1335 1844 a(2)1358 1852 y Ft(ds;)228 1968 y Fu(and)17 b(compute)841 2028 y Ft(d)p 832 2050 43 2 v 832 2096 a(dt)880 2062 y(F)7 b Fu(\()p Fp(h)p Ft(x)p Fp(i)p Fu(\))14 b(=)f Ft(b:)228 2159 y Fu(Com)o(bined)h(with)j(\(4.38\))f(this)h(yields)732 2243 y Ft(d)p 723 2265 V 723 2310 a(dt)771 2276 y(F)7 b Fu(\()p Fp(h)p Ft(x)p Fp(i)p Fu(\))14 b Fp(\025)h Fu(~)-26 b Ft(\024)14 b Fu(for)g Ft(t)g Fp(\025)f Ft(t)1197 2283 y Fs(~)-19 b Fm(\024)1218 2276 y Ft(:)228 2381 y Fu(Whence,)15 b(b)o(y)h(in)o(tegrating,)515 2459 y(1)p 515 2482 25 2 v 515 2527 a(2)544 2493 y(\(1)11 b Fp(\000)g Ft(\026=)p Fu(2\))749 2459 y Fk(1)p 750 2465 16 2 v 750 2486 a(2)773 2493 y Fu(\(1)g(+)g Ft(\026=)p Fu(2\))972 2473 y Fo(\000)p Fs(1)1020 2493 y Fp(h)p Ft(x)p Fu(\()p Ft(t)p Fu(\))p Fp(i)1142 2473 y Fs(1+)1192 2457 y Fi(\026)p 1192 2465 19 2 v 1194 2486 a Fk(2)1232 2493 y Fp(\025)k Fu(~)-26 b Ft(\024)q(t)10 b Fp(\000)h Ft(C)q(:)228 2598 y Fu(Since)17 b(~)-26 b Ft(\024)16 b Fu(can)h(b)q(e)f(tak)o(en)g(arbitrarily)f(close) h(to)h Ft(\024)1127 2605 y Fs(0)1147 2598 y Fu(,)e(w)o(e)h(are)h(done.) p 1687 2598 2 33 v 1689 2567 30 2 v 1689 2598 V 1718 2598 2 33 v 956 2658 a Fq(29)p eop %%Page: 30 30 30 29 bop 228 213 a Fy(Remarks)16 b(4.8.)24 b Fu(1\))d(The)f(explicit)e (b)q(ounding)j(constan)o(t)g(of)f(Theorem)f(4.7)h(is)292 271 y(optimal.)f(This)13 b(ma)o(y)e(readily)h(b)q(e)h(seen)f(b)o(y)h (examining)e(an)i(almost-b)q(ounded)292 329 y(orbit)23 b(for)h(the)f(p)q(oten)o(tial)h Ft(V)11 b Fu(\()p Ft(x)p Fu(\))26 b(=)g Fp(\000)p Ft(C)t Fp(j)p Ft(x)p Fp(j)1134 311 y Fo(\000)p Fm(\026)1183 329 y Fu(,)f(cf.)43 b([DeG)o(\023)-23 b(e97,)25 b(Example)292 387 y(2.2.4],)16 b([G)o(\023)-23 b(er93])16 b(and)g([Sk03)q(].)228 445 y(2\))21 b(The)c(zero-energy)f (orbits)h(can)f(b)q(eha)o(v)o(e)g(somewhat)h(unexp)q(ected)e(lik)o(e)g (loga-)292 503 y(rithmic)10 b(spirals.)20 b(\(W)l(e)12 b(encoun)o(tered)g(\014rst)g(suc)o(h)h(example)d(in)i(a)h(preliminary) 292 561 y(v)o(ersion)i(of)i(the)f(b)q(o)q(ok)i([DeG)o(\023)-23 b(e97].\))228 652 y(4.5.)28 b Fy(Phase)12 b(space)h(lo)r(calization.)22 b Fu(In)10 b(this)h(subsection)g(w)o(e)f(will)g(pro)o(v)o(e)g (\(4.5d\).)228 711 y(This)j(is)f(the)h(main)e(di\016cult)o(y)g(in)h (pro)o(ving)h(Theorem)e(4.1.)21 b(The)13 b(pro)q(ofs)h(of)i(\(4.5e\)) 228 769 y(and)k(\(4.5f\))f(are)h(essen)o(tially)e(iden)o(tical)f(to)j (the)f(presen)o(t)g(pro)q(of)i(of)i(\(4.5d\))d(and)228 827 y(will)15 b(b)q(e)h(omitted.)278 885 y(Let)j(us)h(\014x)f(a)h(real) f Ft(\024)676 867 y Fo(0)707 885 y Fu(with)g Ft(\024)g(<)g(\024)953 867 y Fo(0)984 885 y Ft(<)g(\024)1069 892 y Fs(0)1089 885 y Fu(,)h(and)g(consider)f(the)g(observ)m(able)228 943 y Ft(Q)267 950 y Fm(s)299 943 y Fu(=)13 b(Op)407 925 y Fs(w)427 943 y Fu(\()p Ft(q)468 950 y Fm(s)486 943 y Fu(\),)j(where)478 1033 y Ft(q)500 1040 y Fm(s)532 1033 y Fu(=)e Ft(q)606 1040 y Fm(s)624 1033 y Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b(=)g(\()p Ft(w)q Fu(\()p Ft(\024)903 1012 y Fo(0)926 1033 y Fp(\000)c Ft(b)p Fu(\)\))1034 1012 y Fm(s)1064 1020 y Fu(~)1052 1033 y Ft(F)1084 1040 y Fo(\000)1114 1033 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1205 1040 y Fo(\000)1233 1033 y Fu(\()p Ft(a)1278 1040 y Fs(0)1298 1033 y Fu(\);)e Ft(s)13 b Fp(2)h Fn(R)p Ft(:)123 b Fu(\(4.39\))228 1121 y(W)l(e)16 b(recall)g(from)f(\(4.3\))i(that)g Ft(w)e Fu(=)g Fp(h)p Ft(x)p Fp(i)p Ft(f)5 b Fu(;)17 b(here)f(and)h(henceforth) f Ft(f)k Fu(=)14 b Ft(f)1571 1129 y Fo(j)p Fm(\020)r Fo(j)1627 1121 y Fu(with)228 1179 y Ft(\020)22 b Fp(2)c Fu(\000)352 1186 y Fm(\022)391 1179 y Fu(\(as)h(for)g Ft(a)575 1186 y Fs(0)614 1179 y Fu(and)g Ft(b)p Fu(\).)28 b(On)19 b(supp)q(\()p Ft(F)1029 1186 y Fo(\000)1058 1179 y Fu(\()p Ft(a)1103 1186 y Fs(0)1122 1179 y Fu(\)\))g(the)f(sym)o(b)q (ol)g Ft(b)g Fu(is)h(b)q(ounded,)228 1238 y(and)e(therefore)e Ft(q)549 1245 y Fm(s)581 1238 y Fp(2)f Ft(S)658 1245 y Fe(unif)737 1198 y Fl(\000)759 1238 y Ft(w)795 1220 y Fm(s)814 1238 y Ft(;)8 b(g)859 1246 y Fo(j)p Fm(\020)r Fo(j)899 1198 y Fl(\001)922 1238 y Fu(.)21 b(W)l(e)16 b(will)f(pro)o(v)o(e)g(the)h(follo)o(wing)g(result.)228 1329 y Fy(Lemm)o(a)g(4.9.)24 b Fh(F)l(or)17 b(al)r(l)i Ft(\017)13 b(>)h Fu(0)p Fh(,)k Ft(s)c Fp(\025)f Fu(0)18 b Fh(the)g(op)n(er)n(ator)580 1416 y Fp(h)p Ft(x)p Fp(i)646 1395 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)771 1416 y Ft(f)800 1395 y Fs(1)p Fm(=)p Fs(2)855 1416 y Ft(Q)894 1423 y Fm(s)912 1416 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(w)1048 1395 y Fo(\000)p Fm(s)1095 1416 y Ft(f)1124 1395 y Fs(1)p Fm(=)p Fs(2)1179 1416 y Fp(h)p Ft(x)p Fp(i)1245 1395 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)228 1501 y Fh(is)f(b)n(ounde)n(d)h(uniformly)f(in)h Ft(\020)g Fp(2)c Fu(\000)864 1508 y Fm(\022)884 1501 y Fh(.)228 1593 y(Pr)n(o)n(of.)19 b Fu(Notice)c(\014rst)h(that)g(the)g(statemen)o (t)e(for)i Ft(s)e Fu(=)g(0)i(follo)o(ws)g(from)f(Corollary)228 1651 y(3.5.)21 b(W)l(e)16 b(will)f(pro)o(v)o(e)g(the)h(lemm)o(a)e(b)o (y)h(induction)h(in)f Ft(s)h Fu(using)g(some)f(ideas)h(from)228 1709 y(the)g(pro)q(of)h(of)g([GIS96,)f(Lemma)e(2.6].)278 1767 y(Supp)q(ose)k(w)o(e)f(ha)o(v)o(e)g(pro)o(v)o(ed)g(Lemma)e(4.9)j (for)g(all)f Ft(s)f Fp(\024)g Ft(s)1334 1774 y Fs(0)1371 1767 y Fu(for)i(some)f Ft(s)1594 1774 y Fs(0)1629 1767 y Fp(\025)f Fu(0.)228 1825 y(W)l(e)d(will)e(then)i(pro)o(v)o(e)f(that)i (for)f(all)g Ft(\017)g(>)h Fu(0)f(and)h Ft(s)g Fp(2)g Fu(\()p Ft(s)1212 1832 y Fs(0)1231 1825 y Ft(;)8 b(s)1276 1832 y Fs(0)1307 1825 y Fu(+)j(1)p Ft(=)p Fu(2\))j(the)f(op)q(erator) 486 1914 y Ft(f)5 b(Q)554 1922 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)653 1914 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(f)g(w)818 1894 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1001 1914 y Fu(is)13 b(uniformly)g(b)q(ounded)q Ft(:)133 b Fu(\(4.40\))228 2000 y(By)17 b(the)h(calculus)g(of)h(pseudo)q(di\013eren)o(tial)f(op)q (erators)h(and)g(Corollary)g(3.5)f(this)228 2058 y(is)c(equiv)m(alen)o (t)f(to)h(the)g(statemen)o(t)f(of)i(the)f(lemm)o(a)e(for)j Ft(s)f Fu(in)g(the)g(same)f(range,)i(cf.)228 2116 y(the)h(pro)q(of)h (of)g(Lemma)d(4.5.)278 2176 y(W)l(e)i(ma)o(y)e(without)j(loss)g(of)f (generalit)o(y)f(assume)h(that)g Ft(F)1328 2183 y Fo(\000)1374 2176 y Fu(and)1480 2163 y(~)1469 2176 y Ft(F)1501 2183 y Fo(\000)1546 2176 y Fu(are)g(real-)228 2234 y(v)m(alued,)f(and)i(\(b) o(y)f(the)g(\011DO)g(calculus\))g(that)525 2319 y Ft(F)557 2326 y Fo(\000)600 2319 y Fu(=)e(1)i(on)h(a)g(neigh)o(b)q(ourho)q(o)q (d)h(of)f([0)p Ft(;)8 b(C)1285 2326 y Fs(0)1304 2319 y Fu(])p Ft(;)274 b Fu(\(4.41\))536 2391 y(~)525 2403 y Ft(F)557 2410 y Fo(\000)602 2403 y Fu(is)17 b(decreasing)f(and)25 b(supp\()1121 2391 y(~)1110 2403 y Ft(F)1149 2383 y Fo(0)1142 2415 y(\000)1171 2403 y Fu(\))14 b Fp(\032)g Fu(\()p Fp(\000)p Ft(\024;)8 b(\024)p Fu(\))p Ft(;)180 b Fu(\(4.42\))228 2492 y(where)16 b Ft(C)404 2499 y Fs(0)440 2492 y Fu(is)g(the)g (constan)o(t)g(from)g(Theorem)e(4.1)j(\(ii\).)278 2550 y(W)l(e)g(will)g(place)g(us)h(in)g(a)g(situation,)g(where)g(w)o(e)f (can)h(apply)g(the)f(F)l(e\013erman-)228 2608 y(Phong)27 b(inequalit)o(y)c(\(i.e.)49 b(Theorem)24 b(4.4)i(\(ii\)\))f(as)h(follo) o(ws:)40 b(Clearly)25 b Ft(q)1640 2590 y Fs(2)1638 2620 y Fm(s)1689 2608 y Fp(2)956 2658 y Fq(30)p eop %%Page: 31 31 31 30 bop 228 216 a Ft(S)258 223 y Fe(unif)336 176 y Fl(\000)359 216 y Ft(w)395 198 y Fs(2)p Fm(s)431 216 y Ft(;)8 b(g)476 224 y Fo(j)p Fm(\020)r Fo(j)516 176 y Fl(\001)539 216 y Fu(,)22 b(and)g(therefore,)f(since)f(\(b)o(y)h (\(4.8\))q(\))g Ft(h)h Fp(2)g Ft(S)1391 223 y Fe(unif)1461 216 y Fu(\()p Ft(f)1509 198 y Fs(2)1530 216 y Fp(h)1556 194 y Fm(\030)p 1554 205 21 2 v 1554 233 a(f)1579 216 y Fp(i)1598 198 y Fs(2)1619 216 y Ft(;)8 b(g)1664 224 y Fo(j)p Fm(\020)r Fo(j)1703 216 y Fu(\))228 289 y(and)17 b Fp(h)349 267 y Fm(\030)p 347 277 V 347 306 a(f)373 289 y Fp(i)f Fu(is)g(b)q(ounded)h(on)g(supp)q(\()p Ft(F)876 296 y Fo(\000)905 289 y Fu(\()p Ft(a)950 296 y Fs(0)969 289 y Fu(\)\),)645 392 y Fp(f)p Ft(h;)8 b(q)744 372 y Fs(2)742 405 y Fm(s)763 392 y Fp(g)788 399 y Fm(P)831 392 y Fp(2)14 b Ft(S)908 399 y Fe(unif)987 352 y Fl(\000)1009 392 y Ft(f)1038 372 y Fs(2)1058 392 y Ft(w)1094 372 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1176 392 y Ft(;)8 b(g)1221 400 y Fo(j)p Fm(\020)r Fo(j)1260 352 y Fl(\001)1292 392 y Ft(:)228 487 y Fu(W)l(e)17 b(will)f(estimate)f(the)i(brac)o(k)o(et)f (from)g(ab)q(o)o(v)o(e)h(b)o(y)g(a)h Ft(\033)f Fp(2)e Ft(S)1349 494 y Fe(unif)1420 487 y Fu(\()p Ft(f)1468 469 y Fs(2)1488 487 y Ft(w)1524 469 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1605 487 y Ft(;)8 b(g)1650 495 y Fo(j)p Fm(\020)r Fo(j)1690 487 y Fu(\).)228 548 y(W)l(e)16 b(then)g(get)g(as)h(an)g(op)q (erator)g(inequalit)o(y)e(on)h Ft(L)1155 530 y Fs(2)1175 548 y Fu(\()p Fn(R)1233 530 y Fm(d)1250 548 y Fu(\))443 644 y Ft(w)479 623 y Fo(\000)p Fs(\()p Fm(s)p Fo(\000)p Fs(3)p Fm(=)p Fs(2\))633 644 y Ft(f)662 623 y Fo(\000)p Fs(1)709 644 y Fu(Op)766 623 y Fs(w)786 644 y Fu(\()p Ft(\033)c Fp(\000)f(f)p Ft(h;)d(q)994 623 y Fs(2)992 656 y Fm(s)1013 644 y Fp(g)1038 651 y Fm(P)1067 644 y Fu(\))p Ft(f)1115 623 y Fo(\000)p Fs(1)1163 644 y Ft(w)1199 623 y Fo(\000)p Fs(\()p Fm(s)p Fo(\000)p Fs(3)p Fm(=)p Fs(2\))1366 644 y Fp(\025)14 b(\000)p Ft(C)q(:)228 735 y Fu(The)j(e\013ectiv)o(e)d(form)i(of)h(this)g(estimate)d(suited)j(for) g(implem)o(en)n(ting)d(the)i(induc-)228 793 y(tion)g(h)o(yp)q(othesis)g (will)g(b)q(e:)21 b(F)l(or)16 b(an)o(y)g Ft(m)e(>)g Fu(0)530 885 y(Op)587 864 y Fs(w)607 885 y Fu(\()p Ft(\033)e Fp(\000)f(f)p Ft(h;)d(q)815 864 y Fs(2)813 897 y Fm(s)834 885 y Fp(g)859 892 y Fm(P)888 885 y Fu(\))14 b Fp(\025)g(\000)p Ft(C)t(B)1092 864 y Fo(\003)1110 885 y Ft(B)g Fp(\000)d Ft(C)1246 892 y Fm(m)1279 885 y Fp(h)p Ft(x)p Fp(i)1345 864 y Fo(\000)p Fm(m)1406 885 y Ft(;)186 b Fu(\(4.43\))228 985 y(where)12 b Ft(B)k Fu(=)e Fp(h)p Ft(x)p Fp(i)536 967 y Fo(\000)p Fs(1)p Fm(=)p Fs(2)619 985 y Ft(w)655 967 y Fm(s)p Fo(\000)p Fm(s)714 971 y Fi(o)733 967 y Fo(\000)p Fs(1)780 985 y Ft(f)809 967 y Fs(1)p Fm(=)p Fs(2)864 985 y Ft(Q)903 967 y Fs(1)903 997 y Fm(s)919 1002 y Fk(0)951 985 y Fu(with)e Ft(Q)1097 967 y Fs(1)1097 997 y Fm(s)1113 1002 y Fk(0)1145 985 y Fu(giv)o(en)g(as)h(the)f(quan)o(tization)g(of)228 1052 y(a)j(sym)o(b)q(ol)f Ft(q)457 1034 y Fs(1)455 1064 y Fm(s)471 1069 y Fk(0)506 1052 y Fu(of)h(the)g(form)f(\(4.39\))i(with) f(the)g(functions)h Ft(F)1332 1059 y Fo(\000)1376 1052 y Fu(and)1481 1039 y(~)1470 1052 y Ft(F)1502 1059 y Fo(\000)1546 1052 y Fu(replaced)228 1118 y(b)o(y)e(sa)o(y)i Ft(F)417 1100 y Fs(1)410 1131 y Fo(\000)454 1118 y Fu(and)559 1106 y(~)548 1118 y Ft(F)587 1100 y Fs(1)580 1131 y Fo(\000)609 1118 y Fu(;)f(these)g(functions)g(ob)q(ey)h(\(4.41\))g(and)f(\(4.42\))h (but)g(they)e(are)228 1176 y(`larger')h(than)i(the)f(previous)g(ones.) 278 1235 y(Let)21 b(us)g(\014rst)g(calculate)f(the)h(principal)f(sym)o (b)q(ol)f Fp(f)p Ft(h;)8 b(q)1315 1242 y Fs(2)p Fm(s)1351 1235 y Fp(g)1376 1242 y Fm(P)1426 1235 y Fu(of)21 b Ft(i)p Fu([)p Ft(H)q(;)8 b(Q)1619 1217 y Fo(\003)1619 1247 y Fm(s)1638 1235 y Ft(Q)1677 1242 y Fm(s)1695 1235 y Fu(].)228 1293 y(Here)15 b(and)j(henceforth)e(w)o(e)g(put,)h(with)f(a)h(sligh)o (t)f(abuse)i(of)f(notation,)g Ft(q)1563 1300 y Fs(2)p Fm(s)1613 1293 y Fu(=)e Ft(q)1690 1275 y Fs(2)1688 1305 y Fm(s)1709 1293 y Fu(.)228 1351 y(W)l(e)h(decomp)q(ose)354 1442 y Fp(f)p Ft(h;)8 b(q)451 1449 y Fs(2)p Fm(s)487 1442 y Fp(g)512 1449 y Fm(P)555 1442 y Fu(=)14 b Ft(T)636 1449 y Fs(1)666 1442 y Fu(+)d Ft(T)744 1449 y Fs(2)774 1442 y Fu(+)g Ft(T)852 1449 y Fs(3)872 1442 y Fu(;)712 b(\(4.44\))493 1528 y Ft(T)522 1535 y Fs(1)555 1528 y Fu(=)14 b(2)p Ft(s)8 b Fu(\()p Ft(w)q Fu(\()p Ft(\024)764 1507 y Fo(0)787 1528 y Fp(\000)j Ft(b)p Fu(\)\))896 1502 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)996 1515 y Fu(~)985 1528 y Ft(F)1024 1507 y Fs(2)1017 1540 y Fo(\000)1046 1528 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1144 1507 y Fs(2)1137 1540 y Fo(\000)1166 1528 y Fu(\()p Ft(a)1211 1535 y Fs(0)1230 1528 y Fu(\))p Fp(f)p Ft(h;)d(w)q Fu(\()p Ft(\024)1407 1507 y Fo(0)1430 1528 y Fp(\000)j Ft(b)p Fu(\))p Fp(g)1545 1535 y Fm(P)1574 1528 y Ft(;)493 1613 y(T)522 1620 y Fs(2)555 1613 y Fu(=)j(2)8 b(\()p Ft(w)q Fu(\()p Ft(\024)741 1593 y Fo(0)764 1613 y Fp(\000)j Ft(b)p Fu(\)\))873 1587 y Fs(2)p Fm(s)928 1601 y Fu(~)917 1613 y Ft(F)949 1620 y Fo(\000)978 1613 y Fu(\()p Ft(b)p Fu(\))1048 1601 y(~)1037 1613 y Ft(F)1076 1593 y Fo(0)1069 1626 y(\000)1098 1613 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1196 1593 y Fs(2)1189 1626 y Fo(\000)1218 1613 y Fu(\()p Ft(a)1263 1620 y Fs(0)1282 1613 y Fu(\))p Fp(f)p Ft(h;)d(b)p Fp(g)1422 1620 y Fm(P)1451 1613 y Ft(;)493 1699 y(T)522 1706 y Fs(3)555 1699 y Fu(=)14 b(2)8 b(\()p Ft(w)q Fu(\()p Ft(\024)741 1678 y Fo(0)764 1699 y Fp(\000)j Ft(b)p Fu(\)\))873 1673 y Fs(2)p Fm(s)928 1686 y Fu(~)917 1699 y Ft(F)956 1678 y Fs(2)949 1711 y Fo(\000)978 1699 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1069 1706 y Fo(\000)1098 1699 y Fu(\()p Ft(a)1143 1706 y Fs(0)1162 1699 y Fu(\))p Ft(F)1220 1678 y Fo(0)1213 1711 y(\000)1242 1699 y Fu(\()p Ft(a)1287 1706 y Fs(0)1307 1699 y Fu(\))p Fp(f)p Ft(h;)d(a)1427 1706 y Fs(0)1446 1699 y Fp(g)1471 1706 y Fm(P)1500 1699 y Ft(:)278 1801 y Fu(De\014ning)19 b Ft(q)497 1808 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)597 1801 y Fu(=)f(\()p Ft(w)q Fu(\()p Ft(\024)755 1783 y Fo(0)780 1801 y Fp(\000)13 b Ft(b)p Fu(\)\))891 1783 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)983 1789 y Fu(~)972 1801 y Ft(F)1011 1783 y Fs(2)1004 1814 y Fo(\000)1033 1801 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1131 1783 y Fs(2)1124 1814 y Fo(\000)1153 1801 y Fu(\()p Ft(a)1198 1808 y Fs(0)1217 1801 y Fu(\))19 b(w)o(e)g(get)g(from)f(the)h(com-)228 1860 y(putation)e(in)f(\(4.36\)) 649 1951 y Ft(T)678 1958 y Fs(1)711 1951 y Fu(=)e(2)p Ft(sq)832 1958 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)913 1951 y Fu(\(2)p Ft(\024)984 1931 y Fo(0)996 1951 y Ft(v)r(b)c Fp(\000)h Fu(2)p Ft(h)g Fp(\000)g Ft(W)c Fu(\))p Ft(:)296 b Fu(\(4.45\))228 2046 y(By)16 b(\(4.32\))h(the)f(second)g(term)f(in)h (\(4.44\))h(b)q(ecomes,)d(cf.)21 b(\(4.35\))q(,)611 2141 y Ft(T)640 2148 y Fs(2)673 2141 y Fu(=)14 b(2)t(^)-28 b Ft(q)771 2148 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)863 2129 y Fu(~)852 2141 y Ft(F)891 2121 y Fo(0)884 2154 y(\000)913 2141 y Fu(\()p Ft(b)p Fu(\)\(2)p Ft(h)11 b Fu(+)g Ft(W)18 b Fp(\000)11 b Fu(2)p Ft(b)1262 2121 y Fs(2)1281 2141 y Ft(v)r Fu(\))p Ft(;)266 b Fu(\(4.46\))228 2244 y(where)19 b(^)-27 b Ft(q)391 2251 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)485 2244 y Fu(=)14 b Ft(w)573 2226 y Fo(\000)p Fs(1)629 2244 y Fu(\()p Ft(w)q Fu(\()p Ft(\024)731 2226 y Fo(0)754 2244 y Fp(\000)c Ft(b)p Fu(\)\))862 2220 y Fs(2)p Fm(s)918 2231 y Fu(~)906 2244 y Ft(F)938 2251 y Fo(\000)968 2244 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1066 2226 y Fs(2)1059 2256 y Fo(\000)1087 2244 y Fu(\()p Ft(a)1132 2251 y Fs(0)1152 2244 y Fu(\).)278 2302 y(Finally)k(for)j(the)f(third)f(term)g(in)g (\(4.44\))i(w)o(e)f(ma)o(y)e(write)i(with)g(some)f(sym)o(b)q(ol)231 2361 y(\024)-27 b Ft(q)250 2368 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)344 2361 y Fp(2)14 b Ft(S)421 2368 y Fe(unif)500 2321 y Fl(\000)523 2361 y Ft(w)559 2343 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)640 2361 y Ft(;)8 b(g)685 2369 y Fo(j)p Fm(\020)r Fo(j)725 2321 y Fl(\001)763 2458 y Ft(T)792 2465 y Fs(3)825 2458 y Fu(=)14 b Ft(f)906 2437 y Fs(2)930 2458 y Fu(\024)-28 b Ft(q)948 2465 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1029 2458 y Ft(F)1068 2437 y Fo(0)1061 2470 y(\000)1090 2458 y Fu(\()p Ft(a)1135 2465 y Fs(0)1154 2458 y Fu(\))p Ft(:)411 b Fu(\(4.47\))278 2550 y(The)17 b(rest)g(of)h(the)f (calculation)g(is)g(split)f(in)h(t)o(w)o(o)h(dep)q(ending)f(on)h(the)f (relativ)o(e)228 2608 y(size)e(of)i Fp(<)p Ft(\020)j Fu(and)d Fp(j)p Ft(\020)t Fp(j)p Fu(.)956 2658 y Fq(31)p eop %%Page: 32 32 32 31 bop 278 223 a Fy(Case)14 b(1.)23 b Fp(<)p Ft(\020)18 b Fp(\025)607 199 y Fs(\()p Fm(\024)641 187 y Fg(0)652 199 y Fs(\))666 187 y Fk(2)p 607 211 77 2 v 626 241 a Fm(\024)646 229 y Fk(2)646 251 y(0)688 223 y Fp(j)p Ft(\020)t Fp(j)p Fy(.)i Fu(Using)12 b(this,)h(the)f(fact)g(that)h Ft(b)h(<)g(\024)e Fu(on)h(supp)q(\()p Ft(q)1623 230 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1703 223 y Fu(\))228 293 y(and)k(\(4.4\))f(w)o(e)g(ma)o(y)f(estimate)f(the)i(righ)o(t)g(hand)h (side)f(of)k(\(4.45\))278 385 y(2)p Ft(sq)347 392 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)428 385 y Fu(\(2)p Ft(\024)499 364 y Fo(0)512 385 y Ft(v)r(b)10 b Fp(\000)g Fu(2)p Ft(h)i Fp(\000)f Ft(W)c Fu(\))292 465 y Fp(\024)14 b Fu(2)p Ft(sq)414 472 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)503 425 y Fl(\010)532 465 y Fu(2)p Ft(\024\024)612 445 y Fo(0)624 465 y Fu(\()p Fp(j)p Ft(\020)t Fp(j)p Ft(=\024)748 445 y Fs(2)748 477 y(0)779 465 y Fu(+)d Fp(h)p Ft(x)p Fp(i)894 445 y Fo(\000)p Fm(\026)946 465 y Fu(\))g Fp(\000)f Fu(2)p Fp(<)p Ft(\020)16 b Fp(\000)11 b Fu(2)p Fp(<)p Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))g Fp(\000)g Fu(2)p Ft(\024)1497 445 y Fs(2)1497 477 y(0)1517 465 y Fp(h)p Ft(x)p Fp(i)1583 445 y Fo(\000)p Fm(\026)1634 425 y Fl(\011)292 548 y Fp(\024)j(\000)p Ft(\016)r(q)430 556 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)527 548 y Ft(f)556 527 y Fs(2)576 548 y Ft(q)598 556 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)708 548 y Fp(\000)d Fu(2)p Ft(s)813 508 y Fl(\010)842 548 y Fu(\()p Ft(h)h Fp(\000)p 950 507 26 2 v 10 w Ft(\020)t Fu(\))p Ft(q)1016 555 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1108 548 y Fu(+)f Ft(q)1179 555 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1260 548 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1412 508 y Fl(\011)1449 548 y Ft(;)135 b Fu(\(4.48\))228 643 y(here)15 b(with)i Ft(\016)e Fu(=)f Ft(\016)r Fu(\()p Ft(\024;)8 b(\024)655 625 y Fo(0)666 643 y Ft(;)g(\024)716 650 y Fs(0)735 643 y Ft(;)g(s;)g(\026)p Fu(\))14 b Ft(>)g Fu(0)j(and)521 739 y Ft(q)543 747 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)642 739 y Fu(\()p Ft(x;)8 b(\030)r Fu(\))14 b(=)g(\()p Ft(w)q Fu(\()p Ft(\024)921 718 y Fo(0)944 739 y Fp(\000)c Ft(b)p Fu(\)\))1052 718 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1162 726 y Fu(~)1151 739 y Ft(F)1183 746 y Fo(\000)1212 739 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1303 746 y Fo(\000)1332 739 y Fu(\()p Ft(a)1377 746 y Fs(0)1396 739 y Fu(\))p Ft(:)228 832 y Fu(Since)21 b Ft(f)390 814 y Fo(\000)p Fs(2)438 832 y Ft(h)h Fu(is)g(b)q(ounded)i (on)f(supp\()p Ft(q)964 839 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1045 832 y Fu(\))f(the)g(righ)o(t)g(hand)h(side)f(of)27 b(\(4.48\))228 891 y(clearly)15 b(lies)g(in)h Ft(S)555 898 y Fe(unif)633 851 y Fl(\000)656 891 y Ft(f)685 873 y Fs(2)705 891 y Ft(w)741 873 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)823 891 y Ft(;)8 b(g)868 899 y Fo(j)p Fm(\020)r Fo(j)907 851 y Fl(\001)930 891 y Fu(.)278 950 y(T)l(o)15 b(estimate)d(the)i(righ)o(t)g(hand)h(side)f(of)k(\(4.46\))e(w)o(e)d (use)i(the)f(prop)q(ert)o(y)h(\(4.42\))q(.)228 1012 y(Th)o(us)362 1000 y(~)351 1012 y Ft(F)390 994 y Fo(0)383 1025 y(\000)426 1012 y Fp(\024)e Fu(0)j(and)g(w)o(e)f(ha)o(v)o(e)g Ft(b)816 994 y Fs(2)849 1012 y Fp(\024)f Ft(\024)930 994 y Fs(2)965 1012 y Fu(on)i(supp\()1162 1000 y(~)1151 1012 y Ft(F)1190 994 y Fo(0)1183 1025 y(\000)1212 1012 y Fu(\()p Ft(b)p Fu(\)\).)21 b(So)16 b(w)o(e)f(see)g(as)h(ab)q(o)o(v)o(e)228 1071 y(that)355 1162 y(2)t(^)-28 b Ft(q)401 1169 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)493 1150 y Fu(~)482 1162 y Ft(F)521 1142 y Fo(0)514 1175 y(\000)543 1162 y Fu(\()p Ft(b)p Fu(\)\(2)p Ft(h)11 b Fu(+)g Ft(W)18 b Fp(\000)11 b Fu(2)p Ft(b)892 1142 y Fs(2)912 1162 y Ft(v)r Fu(\))i Fp(\024)g Fu(4)t(^)-28 b Ft(q)1068 1169 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1161 1150 y Fu(~)1149 1162 y Ft(F)1188 1142 y Fo(0)1181 1175 y(\000)1211 1162 y Fu(\()p Ft(b)p Fu(\))p Fp(<)p Fu(\()p Ft(h)10 b Fp(\000)h Ft(\020)t Fu(\))p Ft(:)127 b Fu(\(4.49\))228 1262 y(Clearly)13 b(the)g(righ)o(t)g(hand)h(side)g (of)j(\(4.49\))d(is)g(a)g(sym)o(b)q(ol)e(in)h Ft(S)1333 1269 y Fe(unif)1412 1222 y Fl(\000)1435 1262 y Ft(f)1464 1244 y Fs(2)1484 1262 y Ft(w)1520 1244 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1601 1262 y Ft(;)8 b(g)1646 1270 y Fo(j)p Fm(\020)r Fo(j)1686 1222 y Fl(\001)1709 1262 y Fu(.)278 1320 y(The)20 b(input)g(to)h(our)g(application)f(of)h(the)f(F)l (e\013erman-Phong)h(inequalit)o(y)d(is)228 1379 y(therefore)d(the)h (estimate)f(\(com)o(bining)g(\(4.44\))i(-)f(\(4.49\))q(\))313 1474 y Ft(\016)r Fu(\()p Ft(f)5 b(q)407 1481 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)505 1474 y Fu(\))524 1453 y Fs(2)557 1474 y Fp(\024)14 b(\000f)p Ft(h;)8 b(q)746 1481 y Fs(2)p Fm(s)781 1474 y Fp(g)806 1481 y Fm(P)847 1474 y Fp(\000)i Fu(2)p Ft(s)951 1433 y Fl(\010)981 1474 y Fu(\()p Ft(h)h Fp(\000)p 1089 1433 V 11 w Ft(\020)t Fu(\))p Ft(q)1155 1481 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1247 1474 y Fu(+)g Ft(q)1318 1481 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1399 1474 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1551 1433 y Fl(\011)324 1569 y Fu(+)g(2)405 1514 y Fl(n)439 1569 y Fu(\()p Ft(h)g Fp(\000)p 547 1528 V 11 w Ft(\020)t Fu(\))t(^)-28 b Ft(q)613 1576 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)704 1556 y Fu(~)693 1569 y Ft(F)732 1548 y Fo(0)725 1581 y(\000)755 1569 y Fu(\()p Ft(b)p Fu(\))10 b(+)15 b(^)-28 b Ft(q)895 1576 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)987 1556 y Fu(~)976 1569 y Ft(F)1015 1548 y Fo(0)1008 1581 y(\000)1037 1569 y Fu(\()p Ft(b)p Fu(\)\()p Ft(h)11 b Fp(\000)g Ft(\020)t Fu(\))1248 1514 y Fl(o)1292 1569 y Fu(+)g Ft(f)1370 1548 y Fs(2)1394 1569 y Fu(\024)-28 b Ft(q)1412 1576 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1493 1569 y Ft(F)1532 1548 y Fo(0)1525 1581 y(\000)1554 1569 y Fu(\()p Ft(a)1599 1576 y Fs(0)1618 1569 y Fu(\))327 1657 y(=)13 b Fp(\000f)p Ft(h;)8 b(q)514 1664 y Fs(2)p Fm(s)550 1657 y Fp(g)575 1664 y Fm(P)1598 1657 y Fu(\(4.50\))531 1740 y(+)580 1700 y Fl(\010)609 1740 y Fu(\()p Ft(h)k Fp(\000)p 717 1700 V 10 w Ft(\020)t Fu(\))p Ft(q)785 1720 y Fs(\014nal)783 1752 y(2)p Fm(s)p Fo(\000)p Fs(1)875 1740 y Fu(+)f Ft(q)948 1720 y Fs(\014nal)946 1752 y(2)p Fm(s)p Fo(\000)p Fs(1)1027 1740 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1179 1700 y Fl(\011)1219 1740 y Fu(+)g Ft(f)1297 1720 y Fs(2)1321 1740 y Fu(\024)-28 b Ft(q)1339 1747 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1420 1740 y Ft(F)1459 1720 y Fo(0)1452 1752 y(\000)1481 1740 y Fu(\()p Ft(a)1526 1747 y Fs(0)1545 1740 y Fu(\))p Ft(;)228 1832 y Fu(with)655 1904 y Ft(q)679 1883 y Fs(\014nal)677 1916 y(2)p Fm(s)p Fo(\000)p Fs(1)772 1904 y Fu(=)13 b(2)t(^)-28 b Ft(q)869 1911 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)962 1891 y Fu(~)951 1904 y Ft(F)990 1883 y Fo(0)983 1916 y(\000)1012 1904 y Fu(\()p Ft(b)p Fu(\))10 b Fp(\000)h Fu(2)p Ft(sq)1200 1911 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1281 1904 y Ft(:)278 1986 y Fu(T)l(o)k(sho)o(w)i(\(4.40\))f(w)o(e)e(apply)i(\(4.43\))g(and)g (\(4.50\).)21 b(W)l(e)15 b(in)o(tro)q(duce)g Ft(R)f Fu(=)g Ft(R)p Fu(\()p Ft(\020)t Fu(\),)228 2045 y Ft(T)264 2027 y Fo(\000)p Fs(1)333 2045 y Fu(=)22 b Ft(f)5 b(w)458 2027 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)626 2045 y Fu(,)22 b Ft(\036)g Fp(2)h Ft(L)802 2027 y Fs(2)822 2045 y Fu(,)f Fp(k)p Ft(\036)p Fp(k)g Fu(=)g(1)g(and)f Ft( )j Fu(=)e Ft(RT)1353 2027 y Fo(\000)p Fs(1)1401 2045 y Ft(\036)p Fu(,)f(and)h(use)f(the)228 2103 y(induction)16 b(h)o(yp)q(othesis)g(and)h(Corollary)f(3.5)h(to)f (obtain)333 2195 y Ft(\016)r Fp(k)p Ft(f)5 b(Q)450 2203 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)549 2195 y Ft( )r Fp(k)608 2175 y Fs(2)638 2195 y Fp(\000)11 b Ft(C)723 2202 y Fs(1)756 2195 y Fp(\024)j Ft(\016)840 2155 y Fl(\012)864 2195 y Fu(Op)921 2175 y Fs(w)940 2195 y Fu(\(\()p Ft(q)1000 2203 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1099 2195 y Ft(f)5 b Fu(\))1147 2175 y Fs(2)1167 2195 y Fu(\))1186 2155 y Fl(\013)1209 2215 y Fm( )756 2277 y Fp(\024)14 b(\000)8 b(h)p Fu(Op)932 2257 y Fs(w)952 2277 y Fu(\()p Fp(f)p Ft(h;)g(q)1068 2284 y Fs(2)p Fm(s)1103 2277 y Fp(g)1128 2284 y Fm(P)1157 2277 y Fu(\))p Fp(i)1196 2292 y Fm( )1598 2277 y Fu(\(4.51\))753 2361 y(+)j(2)834 2321 y Fl(\012)858 2361 y Fu(Op)915 2341 y Fs(w)943 2321 y Fl(\000)966 2361 y Fu(\()p Ft(h)g Fp(\000)p 1074 2321 V 11 w Ft(\020)t Fu(\))p Ft(q)1142 2341 y Fs(\014nal)1140 2374 y(2)p Fm(s)p Fo(\000)p Fs(1)1232 2361 y Fu(+)g Ft(q)1305 2341 y Fs(\014nal)1303 2374 y(2)p Fm(s)p Fo(\000)p Fs(1)1384 2361 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1536 2321 y Fl(\001)o(\013)1582 2381 y Fm( )753 2448 y Fu(+)802 2408 y Fl(\012)826 2448 y Fu(Op)883 2428 y Fs(w)902 2448 y Fu(\()p Ft(f)950 2428 y Fs(2)974 2448 y Fu(\024)-28 b Ft(q)992 2455 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1073 2448 y Ft(F)1112 2428 y Fo(0)1105 2461 y(\000)1134 2448 y Fu(\()p Ft(a)1179 2455 y Fs(0)1198 2448 y Fu(\)\))1236 2408 y Fl(\013)1260 2468 y Fm( )1297 2448 y Fu(+)11 b Ft(C)1381 2455 y Fs(2)1401 2448 y Ft(:)228 2550 y Fu(On)k(the)g(righ)o (t)f(hand)i(side)e(of)19 b(\(4.51\))d(w)o(e)e(consider)h(eac)o(h)f (term)g(separately)l(.)20 b(F)l(or)228 2608 y(the)h(\014rst)g(term)f(w) o(e)g(kno)o(w)i(that)f Fp(f)p Ft(h;)8 b(q)960 2615 y Fs(2)p Fm(s)996 2608 y Fp(g)1021 2615 y Fm(P)1071 2608 y Fu(is)21 b(the)g(principal)f(sym)o(b)q(ol)g(of)h(the)956 2658 y Fq(32)p eop %%Page: 33 33 33 32 bop 228 213 a Fu(pseudo)q(di\013eren)o(tial)17 b(op)q(erator)i Ft(i)p Fu([)p Ft(H)q(;)8 b(Q)952 220 y Fs(2)p Fm(s)988 213 y Fu(];)17 b Ft(Q)1072 220 y Fs(2)p Fm(s)1125 213 y Fu(=)f Ft(Q)1218 195 y Fo(\003)1218 225 y Fm(s)1237 213 y Ft(Q)1276 220 y Fm(s)1295 213 y Fu(.)26 b(Th)o(us,)18 b(the)f(pseudo-)228 271 y(di\013eren)o(tial)e(calculus)g (and)i(the)f(induction)g(h)o(yp)q(othesis)g(yield)532 354 y Fp(\000h)p Fu(Op)647 333 y Fs(w)667 354 y Fu(\()p Fp(f)p Ft(h;)8 b(q)783 361 y Fs(2)p Fm(s)818 354 y Fp(g)843 361 y Fm(P)873 354 y Fu(\))p Fp(i)911 361 y Fm( )951 354 y Fp(\024)14 b(\000h)p Ft(i)p Fu([)p Ft(H)q(;)8 b(Q)1195 361 y Fs(2)p Fm(s)1230 354 y Fu(])p Fp(i)1263 361 y Fm( )1300 354 y Fu(+)j Ft(C)1384 361 y Fs(3)1404 354 y Ft(:)180 b Fu(\(4.52\))228 438 y(Clearly)450 521 y Fp(\000h)p Ft(i)p Fu([)p Ft(H)q(;)8 b(Q)641 528 y Fs(2)p Fm(s)677 521 y Fu(])p Fp(i)710 528 y Fm( )750 521 y Fu(=)13 b(2)p Fp(=h)p Ft(T)916 500 y Fo(\000)p Fs(1)964 521 y Ft(\036;)8 b(Q)1054 528 y Fs(2)p Fm(s)1089 521 y Ft( )r Fp(i)j(\000)g Fu(2)p Fp(=)p Fu(\()p Ft(\020)t Fu(\))p Fp(k)p Ft(Q)1390 528 y Fm(s)1408 521 y Ft( )r Fp(k)1467 500 y Fs(2)1486 521 y Ft(:)228 604 y Fu(Since)17 b Fp(=)p Ft(\020)22 b(>)c Fu(0)h(w)o(e)f(ma)o(y)f(drop)i(the)f(second)g(term.)27 b(Th)o(us)18 b(w)o(e)h(conclude)e(from)228 662 y(\(4.52\))j(\(after)f (an)h(application)f(of)h(the)f(pseudo)q(di\013eren)o(tial)g(calculus,)g (the)g(in-)228 720 y(duction)d(h)o(yp)q(othesis)g(and)h(Cauc)o(h)o (y-Sc)o(h)o(w)o(arz\))e(that)315 804 y Fp(\000)c(h)p Fu(Op)441 783 y Fs(w)461 804 y Fu(\()p Fp(f)p Ft(h;)d(q)577 811 y Fs(2)p Fm(s)612 804 y Fp(g)637 811 y Fm(P)666 804 y Fu(\))p Fp(i)704 811 y Fm( )318 884 y Fp(\024)13 b Ft(\021)r Fp(k)p Ft(f)5 b(Q)489 892 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)588 884 y Ft( )r Fp(k)647 863 y Fs(2)677 884 y Fu(+)11 b Ft(\021)752 863 y Fo(\000)p Fs(1)799 884 y Fp(k)p Ft(f)853 863 y Fo(\000)p Fs(1)900 884 y Ft(Q)939 892 y Fm(s)p Fs(+1)p Fm(=)p Fs(2)1038 884 y Ft(T)1074 863 y Fo(\000)p Fs(1)1120 884 y Ft(\036)p Fp(k)1174 863 y Fs(2)1205 884 y Fu(+)g Ft(C)1289 891 y Fs(4)1309 884 y Fu(;)21 b Ft(\021)16 b Fu(=)e Ft(\016)r(=)p Fu(2)p Ft(:)76 b Fu(\(4.53\))228 976 y(Notice)14 b(that)i(with)g Ft(T)632 958 y Fo(\000)p Fs(1)692 976 y Fu(=)e Ft(f)5 b(w)809 958 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)993 976 y Fu(the)15 b(second)h(term)e(on)i (the)f(righ)o(t)h(hand)228 1034 y(side)g(is)g(clearly)e(b)q(ounded)k (since)d(it)h(do)q(es)h(not)g(con)o(tain)f(a)g(resolv)o(en)o(t.)278 1092 y(The)23 b(second)g(term)e(in)i(\(4.51\))h(is)e(similar)f(but)j (easier.)41 b(W)l(e)23 b(calculate)f(as)228 1150 y(ab)q(o)o(v)o(e,)381 1193 y Fl(\012)405 1234 y Fu(Op)462 1213 y Fs(w)490 1193 y Fl(\000)512 1234 y Fu(\()p Ft(h)12 b Fp(\000)p 620 1193 26 2 v 10 w Ft(\020)t Fu(\))p Ft(q)688 1213 y Fs(\014nal)686 1246 y(2)p Fm(s)p Fo(\000)p Fs(1)778 1234 y Fu(+)f Ft(q)851 1213 y Fs(\014nal)849 1246 y(2)p Fm(s)p Fo(\000)p Fs(1)930 1234 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1082 1193 y Fl(\001\013)1128 1253 y Fm( )1598 1234 y Fu(\(4.54\))387 1322 y Fp(\024)i Fu(2)p Fp()13 b Fu(0.)22 b(Using)16 b(that)h Fp(<)p Ft(\020)h(<)840 2074 y Fs(\()p Fm(\024)874 2063 y Fg(0)886 2074 y Fs(\))900 2063 y Fk(2)p 840 2087 77 2 v 860 2116 a Fm(\024)880 2105 y Fk(2)880 2127 y(0)922 2098 y Fp(j)p Ft(\020)t Fp(j)p Fu(,)d(w)o(e)h(c)o(ho)q (ose)h Ft(C)i Fu(so)e(big)g(that)597 2226 y(2)p Fp(<)p Ft(\020)e Fu(+)c Ft(C)t Fp(=)p Ft(\020)k Fp(\000)908 2193 y Fu(2)p Ft(\024\024)988 2175 y Fo(0)p 908 2215 93 2 v 930 2261 a Ft(\024)958 2243 y Fs(2)958 2273 y(0)1005 2226 y Fp(j)p Ft(\020)t Fp(j)f(\025)f(=)p Ft(\020)18 b Fp(\025)c Ft(\016)1276 2206 y Fo(0)1287 2226 y Fp(j)p Ft(\020)t Fp(j)p Ft(:)228 2339 y Fu(Th)o(us,)i(instead)g(of)k(\(4.48\)) q(,)c(w)o(e)f(\014nd,)i(for)f(some)f Ft(\016)1150 2321 y Fo(0)o(0)1185 2339 y Ft(>)f Fu(0,)379 2423 y(2)p Ft(sq)448 2430 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)529 2423 y Fu(\(2)p Ft(\024)600 2402 y Fo(0)612 2423 y Ft(v)r(b)c Fp(\000)h Fu(2)p Ft(h)h Fp(\000)e Ft(W)d Fu(\))393 2506 y Fp(\024)13 b(\000)p Ft(\016)508 2485 y Fo(0)o(0)529 2506 y Ft(q)551 2513 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)649 2506 y Ft(f)678 2485 y Fs(2)698 2506 y Ft(q)720 2513 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)830 2506 y Fp(\000)e Fu(2)p Ft(s)935 2465 y Fl(\010)964 2506 y Fu(\()p Ft(h)g Fp(\000)p 1072 2465 26 2 v 11 w Ft(\020)t Fu(\))p Ft(q)1138 2513 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1230 2506 y Fu(+)g Ft(q)1301 2513 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)1382 2506 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1534 2465 y Fl(\011)390 2588 y Fp(\000)g Ft(iC)t(s)527 2548 y Fl(\010)555 2588 y Fu(\()p Ft(h)g Fp(\000)p 663 2548 V 11 w Ft(\020)t Fu(\))p Ft(q)729 2595 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)821 2588 y Fp(\000)g Ft(q)893 2595 y Fs(2)p Fm(s)p Fo(\000)p Fs(1)974 2588 y Fu(\()p Ft(h)g Fp(\000)g Ft(\020)t Fu(\))1126 2548 y Fl(\011)1163 2588 y Ft(:)956 2658 y Fq(33)p eop %%Page: 34 34 34 33 bop 228 213 a Fu(The)16 b(same)g(thing)h(is)f(done)h(in)f (\(4.49\))q(.)22 b(The)17 b(rest)f(of)h(the)f(pro)q(of)i(is)e(no)o(w)h (similar)228 271 y(to)f(Case)h(1.)p 1687 329 2 33 v 1689 298 30 2 v 1689 329 V 1718 329 2 33 v 278 418 a(W)l(e)f(no)o(w)h(remo)o (v)o(e)d(the)j(extra)f Ft(b)p Fu('s)h(app)q(earing)g(in)g(the)f (statemen)o(t)f(of)i(Lemma)228 476 y(4.9)e(compared)f(to)h(\(4.5d\).)21 b(Notice)14 b(that)h(this)g(follo)o(ws)g(readily)e(from)h(the)h(pseu-) 228 535 y(do)q(di\013eren)o(tial)c(calculus)h(since)f(\()p Ft(\024)863 517 y Fo(0)877 535 y Fp(\000)s Ft(b)p Fu(\))959 517 y Fo(\000)p Fm(s)1016 535 y Fu(is)h(b)q(ounded)h(on)g(supp\()p Ft(F)1471 542 y Fo(\000)1500 535 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1591 542 y Fo(\000)1620 535 y Fu(\()p Ft(a)1665 542 y Fs(0)1684 535 y Fu(\)\))228 593 y(\(this)j(w)o(as)h(in)f(fact)g (also)h(used)f(in)g(the)g(pro)q(of)h(of)g(Lemma)d(4.9\).)228 683 y Fy(Lemm)o(a)i(4.10.)24 b Fh(L)n(et)c Ft(s)h Fp(\025)e Fu(0)j Fh(b)n(e)f(arbitr)n(ary.)31 b(L)n(et)20 b Ft(A)g Fu(=)g Ft(A)1337 690 y Fm(\020)1377 683 y Fu(=)g(Op)1492 665 y Fs(w)1511 683 y Fu(\()p Ft(a)p Fu(\))g Fh(b)n(e)i(the)228 747 y(quantization)g(of)f(the)g(symb)n(ol)f Ft(a)g Fu(=)930 734 y(~)919 747 y Ft(F)951 754 y Fo(\000)980 747 y Fu(\()p Ft(b)p Fu(\))p Ft(F)1071 754 y Fo(\000)1100 747 y Fu(\()p Ft(a)1145 754 y Fs(0)1164 747 y Fu(\))p Fh(.)33 b(Then)21 b(for)f(al)r(l)j Ft(\017)c(>)h Fu(0)i Fh(the)228 805 y(op)n(er)n(ator)611 869 y Fp(h)p Ft(x)p Fp(i)677 849 y Fo(\000)p Fm(\017)722 869 y Ft(f)5 b(w)787 849 y Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)886 869 y Ft(AR)p Fu(\()p Ft(\020)t Fu(\))p Ft(w)1059 849 y Fo(\000)p Fm(s)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1185 869 y Ft(f)g Fp(h)p Ft(x)p Fp(i)1280 849 y Fo(\000)p Fm(\017)1325 869 y Ft(;)228 943 y Fh(is)17 b(b)n(ounde)n(d)h(uniformly)f(in)h Ft(\020)g Fp(2)c Fu(\000)864 950 y Fm(\022)884 943 y Fh(.)278 1033 y Fu(W)l(e)i(can)g(no)o(w)h (\014nish)f(the)g(pro)q(of)h(of)j(\(4.5d\))q(:)228 1123 y Fh(Pr)n(o)n(of)c(of)27 b Fu(\(4.5d\))q Fh(.)20 b Fu(The)f(\014nal)f (step)g(in)g(pro)o(ving)h(\(4.5d\))g(consists)f(of)h(replacing)228 1181 y(the)c(w)o(eigh)o(ts)h Ft(f)21 b Fu(and)16 b Ft(w)h Fu(in)f(Lemma)d(4.10)k(b)o(y)e(their)g(limits)f(as)i Fp(j)p Ft(\020)t Fp(j)g Fu(go)q(es)h(to)f(zero.)228 1239 y(That)h(is)f(the)g(con)o(ten)o(t)f(of)i(Lemma)d(4.6.)p 1687 1239 V 1689 1208 30 2 v 1689 1239 V 1718 1239 2 33 v 657 1353 a(5.)28 b Fv(Pr)o(oof)18 b(of)g(main)g(resul)m(ts)228 1440 y Fu(5.1.)28 b Fy(Pro)r(of)16 b(of)g(Theorem)e(1.1.)23 b Fu(W)l(e)14 b(will)f(only)g(consider)h(the)g(case)f Ft(\020)18 b Fp(2)c Fu(\000)1653 1447 y Fm(\022)1673 1440 y Fu(|)228 1499 y(the)i(other)g(case,)p 556 1458 26 2 v 16 w Ft(\020)i Fp(2)c Fu(\000)672 1506 y Fm(\022)692 1499 y Fu(,)i(can)g(b)q(e)h(pro)o(v)o(ed)e(analogously)l(.)278 1557 y(Let)h(us)h(write)e(for)i Ft(\020)h Fp(2)c Fn(C)23 b Fp(n)11 b Fn(R)551 1641 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))k(=)e(\()p Ft(H)j Fp(\000)11 b Ft(\020)t Fu(\))886 1620 y Fo(\000)p Fs(1)933 1641 y Ft(;)21 b(R)1005 1648 y Fs(1)1025 1641 y Fu(\()p Ft(\020)t Fu(\))14 b(=)g(\()p Ft(H)1213 1648 y Fs(1)1244 1641 y Fp(\000)d Ft(\020)t Fu(\))1338 1620 y Fo(\000)p Fs(1)1385 1641 y Ft(;)231 b Fu(\(5.1\))228 1730 y(with)16 b Ft(H)379 1737 y Fs(1)413 1730 y Fu(=)e Ft(p)489 1712 y Fs(2)520 1730 y Fu(+)d Ft(V)597 1737 y Fs(1)631 1730 y Fu(=)j Ft(H)h Fp(\000)c Ft(V)816 1737 y Fs(2)836 1730 y Fu(.)21 b(W)l(e)16 b(shall)g(pro)q (ceed)g(p)q(erturbativ)o(ely)f(using)680 1814 y Ft(R)p Fu(\()p Ft(\020)t Fu(\)\()p Ft(I)g Fu(+)c Ft(V)913 1821 y Fs(2)933 1814 y Ft(R)970 1821 y Fs(1)990 1814 y Fu(\()p Ft(\020)t Fu(\)\))i(=)h Ft(R)1174 1821 y Fs(1)1194 1814 y Fu(\()p Ft(\020)t Fu(\))p Ft(:)351 b Fu(\(5.2\))228 1900 y(First)20 b(w)o(e)g(sho)o(w)h(that)g Ft(R)699 1907 y Fs(1)718 1900 y Fu(\()p Ft(\020)t Fu(\))g(is)f(uniformly)e(H\177)-24 b(older)20 b(con)o(tin)o(uous)g(in)g(\000)1583 1907 y Fm(\022)1603 1900 y Fu(.)34 b(F)l(or)228 1958 y(that)22 b(w)o(e)f(in)o(terp)q(olate)g(\(4.6a\))h(for)g Ft(m)h Fu(=)g(1)f(and)g Ft(m)h Fu(=)g(2.)38 b(W)l(e)21 b(consider)g(the)228 2017 y(family)14 b(of)i(b)q(ounded)i(op)q(erators)332 2100 y Ft(B)s Fu(\()p Ft(z)r Fu(\))13 b(=)h Ft(k)527 2080 y Fo(\000)p Fm(z)q Fo(\000)p Fm(\017)616 2100 y Fp(f)p Ft(R)678 2107 y Fs(1)697 2100 y Fu(\()p Ft(\020)737 2107 y Fs(1)757 2100 y Fu(\))d Fp(\000)g Ft(R)874 2107 y Fs(1)894 2100 y Fu(\()p Ft(\020)934 2107 y Fs(2)954 2100 y Fu(\))p Fp(g)p Ft(k)1025 2080 y Fo(\000)p Fm(z)q Fo(\000)p Fm(\017)1114 2100 y Fu(;)22 b Fp(<)p Fu(\()p Ft(z)r Fu(\))13 b Fp(2)h Fu([1)p Ft(=)p Fu(2)p Ft(;)8 b Fu(3)p Ft(=)p Fu(2])p Ft(:)105 b Fu(\(5.3\))228 2198 y(F)l(or)14 b Fp(<)p Fu(\()p Ft(z)r Fu(\))g(=)g(3)p Ft(=)p Fu(2)h(\(using)g Ft(R)747 2205 y Fs(1)767 2198 y Fu(\()p Ft(\020)807 2205 y Fs(1)827 2198 y Fu(\))7 b Fp(\000)g Ft(R)936 2205 y Fs(1)956 2198 y Fu(\()p Ft(\020)996 2205 y Fs(2)1016 2198 y Fu(\))14 b(=)1101 2158 y Fl(R)1134 2171 y Fs(1)1124 2215 y(0)1173 2178 y Fm(d)p 1167 2187 31 2 v 1167 2215 a(dt)1203 2198 y Ft(R)1240 2205 y Fs(1)1260 2198 y Fu(\()p Ft(\020)1300 2205 y Fs(2)1327 2198 y Fu(+)7 b Ft(t)p Fu(\()p Ft(\020)1430 2205 y Fs(1)1457 2198 y Fp(\000)g Ft(\020)1524 2205 y Fs(2)1544 2198 y Fu(\)\))h Ft(dt)p Fu(\))15 b(w)o(e)228 2257 y(ha)o(v)o(e)j(the)h(b)q(ound)i Fp(k)p Ft(B)s Fu(\()p Ft(z)r Fu(\))p Fp(k)d(\024)g Ft(C)t Fp(j)p Ft(\020)886 2264 y Fs(1)919 2257 y Fp(\000)13 b Ft(\020)992 2264 y Fs(2)1012 2257 y Fp(j)p Fu(,)19 b(and)h(for)f Fp(<)p Fu(\()p Ft(z)r Fu(\))g(=)g(1)p Ft(=)p Fu(2)h(the)f(b)q(ound)228 2315 y Fp(k)p Ft(B)s Fu(\()p Ft(z)r Fu(\))p Fp(k)13 b(\024)g Ft(C)t Fu(,)j(yielding)678 2403 y Fp(k)p Ft(B)s Fu(\()p Ft(z)r Fu(\))p Fp(k)d(\024)g Ft(C)t Fp(j)p Ft(\020)970 2410 y Fs(1)1001 2403 y Fp(\000)e Ft(\020)1072 2410 y Fs(2)1092 2403 y Fp(j)1106 2382 y Fo(<)p Fs(\()p Fm(z)q Fs(\))p Fo(\000)p Fs(1)p Fm(=)p Fs(2)1259 2403 y Ft(:)349 b Fu(\(5.4\))228 2492 y(F)l(or)21 b Ft(s)h Fp(\024)f Fu(3)p Ft(=)p Fu(2\(1)16 b(+)e Ft(\026=)p Fu(2\))22 b(w)o(e)f(c)o(ho)q(ose)g Ft(z)j Fu(=)e Fp(<)p Fu(\()p Ft(z)r Fu(\))f(=)h Ft(s)p Fu(\(1)14 b(+)h Ft(\026=)p Fu(2)q(\))1479 2473 y Fo(\000)p Fs(1)1547 2492 y Fu(in)20 b(\(5.4\))q(;)228 2550 y(otherwise)11 b(w)o(e)f(tak)o(e)g Ft(z)16 b Fu(=)e(3)p Ft(=)p Fu(2.)21 b(This)11 b(pro)o(v)o(es)f(the)h (H\177)-24 b(older)10 b(con)o(tin)o(uit)o(y)f(statemen)o(t)228 2608 y(of)19 b(Theorem)e(1.1)i(in)f(the)g(case)g Ft(V)853 2615 y Fs(2)891 2608 y Fu(=)g(0.)28 b(In)18 b(particular)g Ft(R)1340 2587 y Fs(+)1340 2620 y(1)1388 2608 y Fu(=)f Ft(R)1480 2615 y Fs(1)1500 2608 y Fu(\(0)c(+)g Ft(i)p Fu(0\))k(=)956 2658 y Fq(34)p eop %%Page: 35 35 35 34 bop 228 216 a Fu(lim)296 223 y Fm(\020)r Fo(!)p Fs(0)p Fm(;\020)r Fo(2)p Fs(\000)441 229 y Fi(\022)467 216 y Ft(R)504 223 y Fs(1)524 216 y Fu(\()p Ft(\020)t Fu(\))21 b(and)f Ft(R)743 195 y Fo(\000)743 228 y Fs(1)794 216 y Fu(=)h Ft(R)890 223 y Fs(1)910 216 y Fu(\(0)14 b Fp(\000)f Ft(i)p Fu(0\))21 b(=)g(lim)1227 223 y Fm(\020)r Fo(!)p Fs(0)p Fm(;\020)r Fo(2)p Fs(\000)1372 229 y Fi(\022)1398 216 y Ft(R)1435 223 y Fs(1)1447 175 y Fl(\000)p 1470 175 26 2 v 41 x Ft(\020)1495 175 y Fl(\001)1538 216 y Fu(are)f(w)o(ell-)228 275 y(de\014ned)c(\(in)g(w)o(eigh)o(ted)f (spaces\).)278 333 y(T)l(o)i(sho)o(w)h(\(1.4\))f(\(in)f(the)h(general)f (case\))h(it)f(su\016ces)g(to)h(sho)o(w)h(that)f Fp(h)p Ft(x)p Fp(i)1610 315 y Fm(s)1628 333 y Fu(\()p Ft(I)e Fu(+)228 391 y Ft(V)256 398 y Fs(2)276 391 y Ft(R)313 370 y Fs(+)313 403 y(1)343 391 y Fu(\))p Fp(h)p Ft(x)p Fp(i)428 373 y Fo(\000)p Fm(s)489 391 y Fu(is)g(in)o(v)o(ertible)e(as)j (an)f(op)q(erator)i(on)e Ft(L)1172 373 y Fs(2)1192 391 y Fu(\()p Fn(R)1250 373 y Fm(d)1268 391 y Fu(\).)20 b(\(Here)14 b(w)o(e)h(use)g(\(5.2\))q(,)228 449 y(as)21 b(w)o(ell)f(as)h(the)f (standard)i(limiting)c(absorption)k(principle)d(for)i(p)q(ositiv)o(e)f (en-)228 507 y(ergies)f(and)i(absence)f(of)g(p)q(ositiv)o(e)f(eigen)o (v)m(alues,)g(cf.)32 b([Mou81)q(],)20 b([T)l(am89)o(])g(and)228 565 y([DeG)o(\023)-23 b(e97,)16 b(Section)g(6.5].\))278 623 y(Notice)21 b(that)i Fp(h)p Ft(x)p Fp(i)615 605 y Fm(s)633 623 y Ft(V)661 630 y Fs(2)682 623 y Ft(R)719 603 y Fs(+)719 635 y(1)749 623 y Fp(h)p Ft(x)p Fp(i)815 605 y Fo(\000)p Fm(s)883 623 y Fu(is)f(compact)g(\(b)q(eing)g(the)g (norm-limit)d(of)k(a)228 681 y(compact)e(op)q(erator-v)m(alued)i (function\).)39 b(Whence)22 b(b)o(y)f(F)l(redholm)g(theory)l(,)i(it)228 740 y(su\016ces)16 b(to)g(sho)o(w)h(that)g(the)f(equation)830 833 y Ft(\036)d Fu(=)h Fp(\000)p Ft(V)991 840 y Fs(2)1011 833 y Ft(R)1048 812 y Fs(+)1048 845 y(1)1078 833 y Ft(\036;)509 b Fu(\(5.5\))228 933 y(has)11 b(no)h(nonzero)f(solution)g Ft(\036)j Fp(2)g(h)p Ft(x)p Fp(i)883 915 y Fo(\000)p Fm(s)929 933 y Ft(L)962 915 y Fs(2)982 933 y Fu(\()p Fn(R)1040 915 y Fm(d)1057 933 y Fu(\).)19 b(Let)11 b Ft( )16 b Fu(=)e Ft(R)1328 912 y Fs(+)1328 945 y(1)1358 933 y Ft(\036)f Fu(\()p Fp(2)h(h)p Ft(x)p Fp(i)1532 915 y Fm(s)1551 933 y Ft(L)1584 915 y Fs(2)1604 933 y Fu(\()p Fn(R)1662 915 y Fm(d)1679 933 y Fu(\)\).)228 991 y(Then)i(w)o(e)g(ha)o (v)o(e)f(in)h(the)g(sense)h(of)f(distributions)720 1084 y Ft(H)t( )f Fu(=)f(0)g(and)h(V)1031 1091 y Fs(2)1050 1084 y Ft( )g Fu(=)f Fp(\000)p Ft(\036:)391 b Fu(\(5.6\))278 1177 y(W)l(e)16 b(can)g(calculate)522 1271 y(0)e(=)g Fp(=h)p Ft( )r(;)8 b(V)751 1278 y Fs(2)771 1271 y Ft( )r Fp(i)13 b Fu(=)h Fp(\000=h)p Ft( )r(;)8 b(\036)p Fp(i)560 1351 y Fu(=)14 b Fp(\000=h)p Ft(R)743 1330 y Fs(+)743 1363 y(1)773 1351 y Ft(\036;)8 b(\036)p Fp(i)14 b Fu(=)f(\(2)p Ft(i)p Fu(\))1016 1330 y Fo(\000)p Fs(1)1064 1351 y Fp(h)p Ft(\036;)8 b Fu(\()p Ft(R)1190 1330 y Fs(+)1190 1363 y(1)1231 1351 y Fp(\000)j Ft(R)1318 1330 y Fo(\000)1318 1363 y Fs(1)1348 1351 y Fu(\))p Ft(\036)p Fp(i)p Ft(:)228 1466 y Fu(Since)360 1440 y Fm(R)387 1425 y Fk(+)387 1451 y(1)412 1440 y Fo(\000)p Fm(R)466 1425 y Fg(\000)466 1451 y Fk(1)p 360 1455 133 2 v 412 1484 a Fs(2)p Fm(i)511 1466 y Fp(\025)j Fu(0)j(w)o(e)e(get)i(that)748 1563 y Ft( )f Fu(=)e Ft(R)885 1543 y Fs(+)885 1576 y(1)915 1563 y Ft(\036)f Fu(=)h Ft(R)1046 1543 y Fo(\000)1046 1576 y Fs(1)1076 1563 y Ft(\036:)503 b Fu(\(5.7\))228 1659 y Fy(Lemm)o(a)16 b(5.1.)24 b Ft( )15 b Fp(2)f Ft(L)665 1641 y Fs(2)685 1659 y Fu(\()p Fn(R)743 1641 y Fm(d)760 1659 y Fu(\))p Fh(.)228 1757 y(Pr)n(o)n(of.)19 b Fu(A)e(priori)g Ft(\036)g Fp(2)f(h)p Ft(x)p Fp(i)726 1739 y Fo(\000)p Fm(s)773 1757 y Ft(L)806 1739 y Fs(2)843 1757 y Fu(and)j Ft( )f Fp(2)e(h)p Ft(x)p Fp(i)1105 1739 y Fm(s)1121 1727 y Fg(0)1135 1757 y Ft(L)1168 1739 y Fs(2)1188 1757 y Fu(;)24 b Ft(s)1249 1739 y Fo(0)1277 1757 y Ft(>)1337 1737 y Fs(1)p 1337 1745 18 2 v 1337 1774 a(2)1371 1757 y Fu(+)1426 1735 y Fm(\026)p 1426 1745 22 2 v 1428 1774 a Fs(4)1453 1757 y Fu(.)h(F)l(rom)17 b(\(5.6\))228 1822 y(w)o(e)f(learn)h(that)g(if)g Ft( )f Fp(2)f(h)p Ft(x)p Fp(i)737 1804 y Fm(s)753 1792 y Fg(0)767 1822 y Ft(L)800 1804 y Fs(2)837 1822 y Fu(for)i(some)f(real)h Ft(s)1154 1804 y Fo(0)1182 1822 y Fu(then)g Ft(\036)e Fp(2)g(h)p Ft(x)p Fp(i)1452 1804 y Fm(s)1468 1792 y Fg(0)1480 1804 y Fo(\000)p Fs(1)p Fo(\000)p Fm(\026=)p Fs(2)p Fo(\000)p Fm(\016)1656 1822 y Ft(L)1689 1804 y Fs(2)1709 1822 y Fu(.)228 1880 y(In)e(particular)g(w)o(e)g(ha)o(v)o(e)g Ft(\036)g Fp(2)h(h)p Ft(x)p Fp(i)842 1862 y Fo(\000)q Fs(~)-19 b Fm(s)889 1880 y Ft(L)922 1862 y Fs(2)942 1880 y Fu(;)23 b(~)-26 b Ft(s)14 b(<)1071 1860 y Fs(1)p 1071 1868 18 2 v 1071 1897 a(2)1099 1880 y Fu(+)1147 1858 y Fm(\026)p 1147 1868 22 2 v 1149 1897 a Fs(4)1179 1880 y Fu(+)5 b Ft(\016)r Fu(.)20 b(The)13 b(idea)g(of)h(the)f(pro)q(of)228 1938 y(is)k(to)h(sho)o(w)g(b)o(y)f(a)h(b)q(o)q(otstrap)h(argumen)o(t)d (that)i(w)o(e)f(ma)o(y)f(tak)o(e)h Ft(s)1431 1920 y Fo(0)1460 1938 y Fu(arbitrary)l(.)25 b(A)228 1996 y(b)q(o)q(otstrap)d(argumen)o (t)d(for)h(a)g(similar)e(problem)g(for)i(the)g(free)f(Laplacian)i(w)o (as)228 2054 y(giv)o(en)c(b)o(y)h(Agmon)f(in)h(his)g(pro)q(of)h(of)g ([Ag75,)f(Theorem)f(3.3].)27 b(Our)18 b(analysis)g(is)228 2112 y(based)f(on)f(Theorem)f(4.1.)278 2170 y(W)l(e)i(pic)o(k)g(a)h (real-v)m(alued)f(function)h Ft(F)984 2177 y Fs(+)1031 2170 y Fu(as)g(in)g(Theorem)e(4.1)i(\(ii\))f(suc)o(h)h(that)228 2229 y Ft(F)260 2236 y Fs(+)289 2229 y Fu(\()p Ft(x)p Fu(\))c(=)g(1)j(for)g Fp(j)p Ft(x)p Fp(j)c Ft(>)h Fu(2)p Ft(C)717 2236 y Fs(0)737 2229 y Fu(.)22 b(Let)17 b Ft(F)893 2236 y Fo(\000)936 2229 y Fu(=)d(1)e Fp(\000)f Ft(F)1106 2236 y Fs(+)1135 2229 y Fu(.)22 b(Pic)o(k)15 b(real-v)m(alued)h (functions)239 2278 y(~)228 2291 y Ft(F)260 2298 y Fo(\000)307 2291 y Fu(and)416 2278 y(~)405 2291 y Ft(F)437 2298 y Fs(+)484 2291 y Fu(as)j(in)f(Theorem)g(4.1)h(\(iii\))e(suc)o(h)h(that) 1223 2278 y(~)1212 2291 y Ft(F)1244 2298 y Fo(\000)1286 2291 y Fu(+)1347 2278 y(~)1336 2291 y Ft(F)1368 2298 y Fs(+)1415 2291 y Fu(=)g(1.)28 b(Then)19 b(w)o(e)228 2349 y(decomp)q(ose)g(with)h(the)g(sym)o(b)q(ols)f Ft(a)896 2356 y Fs(0)936 2349 y Fu(and)h Ft(b)g Fu(b)q(eing)g(de\014ned)g(as)h (in)f(\(4.1\))g(with)228 2407 y Ft(E)d Fu(=)c(0)k(in)f(the)g (expression)g(\(4.2\))g(for)h Ft(f)543 2504 y( )f Fu(=)d(Op)699 2484 y Fs(w)719 2504 y Fu(\()p Ft(F)770 2511 y Fs(+)799 2504 y Fu(\()p Ft(a)844 2511 y Fs(0)863 2504 y Fu(\)\))p Ft( )g Fu(+)e(Op)1052 2484 y Fs(w)1071 2504 y Fu(\()p Ft(F)1122 2511 y Fo(\000)1151 2504 y Fu(\()p Ft(a)1196 2511 y Fs(0)1216 2504 y Fu(\))1246 2492 y(~)1235 2504 y Ft(F)1267 2511 y Fo(\000)1296 2504 y Fu(\()p Ft(b)p Fu(\)\))p Ft( )588 2588 y Fu(+)g(Op)694 2568 y Fs(w)713 2588 y Fu(\()p Ft(F)764 2595 y Fo(\000)793 2588 y Fu(\()p Ft(a)838 2595 y Fs(0)858 2588 y Fu(\))888 2576 y(~)877 2588 y Ft(F)909 2595 y Fs(+)938 2588 y Fu(\()p Ft(b)p Fu(\)\))p Ft( )r(:)558 b Fu(\(5.8\))956 2658 y Fq(35)p eop %%Page: 36 36 36 35 bop 228 213 a Fu(By)14 b(\(4.5b\))h(and)g(\(4.5d\))g(the)f (\014rst)g(t)o(w)o(o)h(terms)e(on)i(the)f(righ)o(t)g(hand)h(side)f(of)k (\(5.8\))228 273 y(b)q(elong)f(to)f Fp(h)p Ft(x)p Fp(i)509 255 y Fm(s)525 243 y Fg(0)539 273 y Ft(L)572 255 y Fs(2)608 273 y Fu(where)g(\(assuming)g(here)g Ft(\036)e Fp(2)g(h)p Ft(x)p Fp(i)1242 255 y Fo(\000)p Fm(s)1288 273 y Ft(L)1321 255 y Fs(2)1341 273 y Fu(\))462 381 y Ft(s)485 360 y Fo(0)511 381 y Fu(=)f(\(1)f(+)671 359 y Fm(\026)p 671 369 22 2 v 673 398 a Fs(2)697 381 y Fu(\)\()p Fp(\000)p Ft(t)e Fu(+)856 361 y Fs(1)p 856 369 18 2 v 856 398 a(2)890 381 y Fu(+)h Ft(\017)p Fu(\);)21 b Ft(t)14 b Fu(=)1148 347 y Ft(s)p 1102 369 116 2 v 1102 415 a Fu(1)d(+)1191 393 y Fm(\026)p 1191 403 22 2 v 1193 432 a Fs(2)1233 381 y Fp(\000)1288 361 y Fs(1)p 1288 369 18 2 v 1288 398 a(2)1322 381 y Fp(\000)g Ft(\017:)216 b Fu(\(5.9\))228 510 y(W)l(e)16 b(notice)f(that)i(the)f(b)q(ound)h(\(4.5e\))g(is)f (equiv)m(alen)o(t)e(to)393 608 y Fp(k)p Ft(k)445 588 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)582 608 y Fu(Op)639 588 y Fs(w)659 608 y Fu(\()p Ft(F)710 615 y Fo(\000)739 608 y Fu(\()p Ft(a)784 615 y Fs(0)803 608 y Fu(\))833 596 y(~)822 608 y Ft(F)854 615 y Fs(+)883 608 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)998 615 y Fs(1)1017 608 y Fu(\()p Ft(\020)t Fu(\))1080 588 y Fo(\003)1100 608 y Ft(k)1127 588 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1292 608 y Fp(k)f(\024)h Ft(C)q(:)165 b Fu(\(5.10\))228 705 y(T)l(aking)16 b Ft(\020)i Fp(!)c Fu(0)j(in)e(the)h(sector)h(\000)846 712 y Fm(\022)866 705 y Fu(,)e(\(5.10\))i(leads)g(to)492 804 y Fp(k)p Ft(k)544 783 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)681 804 y Fu(Op)738 783 y Fs(w)757 804 y Fu(\()p Ft(F)808 811 y Fo(\000)837 804 y Fu(\()p Ft(a)882 811 y Fs(0)901 804 y Fu(\))931 791 y(~)920 804 y Ft(F)952 811 y Fs(+)982 804 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)1097 783 y Fo(\000)1097 816 y Fs(1)1126 804 y Ft(k)1153 783 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1318 804 y Fp(k)c(\024)h Ft(C)q(;)147 b Fu(\(5.11\))228 898 y(with)16 b(the)g(same)g(con)o(v)o(en)o(tion)f(for)i Ft(a)891 905 y Fs(0)926 898 y Fu(and)g Ft(b)g Fu(as)g(ab)q(o)o(v)o(e.) 22 b(W)l(e)16 b(use)g(the)h(represen-)228 956 y(tation)g Ft( )g Fu(=)f Ft(R)512 936 y Fo(\000)512 969 y Fs(1)542 956 y Ft(\036)h Fu(of)k(\(5.7\))c(and)h(apply)f(\(5.11\))q(,)g(and)h (conclude)e(that)i(also)f(the)228 1017 y(third)i(term)f(on)i(the)g (righ)o(t)f(hand)i(side)e(of)24 b(\(5.8\))c(b)q(elongs)g(to)g Fp(h)p Ft(x)p Fp(i)1470 999 y Fm(s)1486 987 y Fg(0)1500 1017 y Ft(L)1533 999 y Fs(2)1573 1017 y Fu(with)g Ft(s)1711 999 y Fo(0)228 1077 y Fu(giv)o(en)15 b(b)o(y)i(\(5.9\);)f(so)h Ft( )e Fp(2)f(h)p Ft(x)p Fp(i)774 1059 y Fm(s)790 1047 y Fg(0)804 1077 y Ft(L)837 1059 y Fs(2)857 1077 y Fu(.)278 1135 y(F)l(rom)h(this)h(w)o(e)g(learn)f(that)571 1229 y Ft(\036)f Fp(2)g(h)p Ft(x)p Fp(i)727 1209 y Fm(s)743 1197 y Fg(0)755 1209 y Fo(\000)p Fs(1)p Fo(\000)832 1193 y Fi(\026)p 832 1201 19 2 v 834 1222 a Fk(2)856 1209 y Fo(\000)p Fm(\016)902 1229 y Ft(L)935 1209 y Fs(2)969 1229 y Fu(=)g Fp(h)p Ft(x)p Fp(i)1087 1209 y Fo(\000)p Fm(s)p Fo(\000)p Fm(\016)q Fs(+\(2+)p Fm(\026)p Fs(\))p Fm(\017)1313 1229 y Ft(L)1346 1209 y Fs(2)1366 1229 y Fu(;)228 1323 y(so)23 b(b)o(y)g(taking)g Ft(\017)h(<)h Fu(\(2)16 b(+)g Ft(\026)p Fu(\))794 1305 y Fo(\000)p Fs(1)841 1323 y Ft(\016)24 b Fu(w)o(e)f(impro)o(v)o(e)d(the)j(deca)o(y) f(of)h Ft(\036)p Fu(.)41 b(Iterating)228 1382 y(this)19 b(argumen)o(t)g(\(gaining)h(at)f(eac)o(h)g(iteration)g(almost)g(a)h (factor)g Fp(h)p Ft(x)p Fp(i)1531 1364 y Fo(\000)p Fm(\016)1578 1382 y Fu(\))f(leads)228 1440 y(to)d Ft(s)310 1422 y Fo(0)336 1440 y Fp(\024)d Fu(0)k(ev)o(en)o(tually)l(.)p 1687 1440 2 33 v 1689 1408 30 2 v 1689 1440 V 1718 1440 2 33 v 278 1542 a(Com)o(bining)j(Theorem)g(2.4)h(and)h(Lemma)d(5.1)j (yields)e Ft( )k Fu(=)f Ft(\036)f Fu(=)h(0,)f(com-)228 1600 y(pleting)17 b(the)i(pro)q(of)g(of)j(\(1.4\))d(in)f(the)g(general) g(case.)27 b(It)18 b(remains)f(to)i(sho)o(w)f(the)228 1658 y(H\177)-24 b(older)16 b(con)o(tin)o(uit)o(y)g(statemen)o(t)f(of)j (Theorem)e(1.1)i(in)f(the)g(general)g(case.)24 b(This)228 1716 y(ma)o(y)12 b(easily)g(b)q(e)i(done)f(using)i(\(5.2\))f(and)g(the) f(kno)o(wn)g(result)g(for)h Ft(R)1445 1723 y Fs(1)1465 1716 y Fu(\()p Ft(\020)t Fu(\);)g(w)o(e)f(omit)228 1775 y(the)j(details.)228 1871 y Fy(Remark)g(5.2.)24 b Fu(There)d(exists)f (another)i(approac)o(h)f(to)h(pro)o(ving)f(\(1.4\))h(based)228 1929 y(on)17 b(a)f(virial)f(t)o(yp)q(e)h(argumen)o(t)f(and)i(Theorem)e (2.4:)21 b(F)l(ormally)l(,)14 b(on)j(one)f(hand)700 2023 y Fp(h)p Ft( )r(;)8 b(i)p Fu([)p Ft(H)846 2030 y Fs(1)865 2023 y Ft(;)g(A)p Fu(])p Ft( )r Fp(i)13 b Fu(=)h Fp(h)p Ft( )r(;)8 b(W)f( )r Fp(i)p Ft(;)228 2117 y Fu(while)15 b(on)i(the)f(other)g(hand)642 2212 y Fp(h)p Ft( )r(;)8 b(i)p Fu([)p Ft(H)788 2219 y Fs(1)807 2212 y Ft(;)g(A)p Fu(])p Ft( )r Fp(i)13 b Fu(=)h Fp(\000)p Fu(2)p Fp(=)p Fu(\()p Fp(h)p Ft(\036;)8 b(A )r Fp(i)p Fu(\))p Ft(:)228 2306 y Fu(This)19 b(leads)g(to)g(the)g(conclusion)g(\(rigorously)l(,)g (after)g(some)f(w)o(ork)h(using)g(again)228 2366 y(\(5.7\)\))e(that)g Ft(p)494 2373 y Fm(j)513 2366 y Ft( )r(;)8 b Fp(h)p Ft(x)p Fp(i)635 2348 y Fo(\000)p Fm(\026=)p Fs(2)721 2366 y Ft( )16 b Fp(2)f Ft(L)850 2348 y Fs(2)870 2366 y Fu(\()p Fn(R)928 2348 y Fm(d)945 2366 y Fu(\).)23 b(Therefore,)16 b(w)o(e)g(can)h(apply)g(Theorem)228 2424 y(2.4)j(to)g(conclude)e(that)i Ft( )r Fu(,)g(and)g(therefore)f Ft(\036)p Fu(,)h(v)m(anish)f(iden)o (tically)l(.)29 b(T)l(o)20 b(mak)o(e)228 2482 y(this)h(w)o(ork)g(one)h (needs)f(a)h(stronger)g(deca)o(y)f(assumption)g(than)h(Theorem)e(1.1) 228 2540 y(\(5\).)h(As)16 b(an)g(adv)m(an)o(tage,)g(b)q(eing)g(indep)q (enden)o(t)f(of)h(Theorem)f(4.1,)g(this)h(metho)q(d)228 2598 y(only)g(requires)f(a)i(few)f(deriv)m(ativ)o(es)f(of)h Ft(V)973 2605 y Fs(1)993 2598 y Fu(,)g(cf.)21 b(Remark)14 b(3.6.)956 2658 y Fq(36)p eop %%Page: 37 37 37 36 bop 228 213 a Fu(5.2.)28 b Fy(Pro)r(of)h(of)h(Theorem)d(1.2.)d Fu(W)l(e)h(will)f(only)i(pro)o(v)o(e)f(\(1.5\),)j(since)c(the)228 271 y(H\177)-24 b(older)17 b(con)o(tin)o(uit)o(y)g(follo)o(ws)h(from)f (\(1.5\))i(using)g(in)o(terp)q(olation)f(as)h(in)f(Subsec-)228 329 y(tion)e(5.1.)278 387 y(Let)h Ft(V)27 b Fu(=)16 b Ft(V)503 394 y Fs(1)535 387 y Fu(+)c Ft(V)613 394 y Fs(2)651 387 y Fu(b)q(e)18 b(the)f(decomp)q(osition)f(of)i Ft(V)29 b Fu(giv)o(en)17 b(in)g(the)g(statemen)o(t)228 445 y(of)h(Theorem)f (1.2.)27 b(Tw)o(o)19 b(successiv)o(e)d(applications)i(of)h(the)f (resolv)o(en)o(t)e(iden)o(tit)o(y)228 503 y(\(5.2\))g(giv)o(e)340 592 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))f(=)e Ft(R)543 599 y Fs(1)563 592 y Fu(\()p Ft(\020)t Fu(\))e Fp(\000)g Ft(R)724 599 y Fs(1)744 592 y Fu(\()p Ft(\020)t Fu(\))p Ft(V)835 599 y Fs(2)855 592 y Ft(R)892 599 y Fs(1)912 592 y Fu(\()p Ft(\020)t Fu(\))g(+)g Ft(R)1072 599 y Fs(1)1092 592 y Fu(\()p Ft(\020)t Fu(\))p Ft(V)1183 599 y Fs(2)1203 592 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(V)1331 599 y Fs(2)1352 592 y Ft(R)1389 599 y Fs(1)1409 592 y Fu(\()p Ft(\020)t Fu(\))p Ft(:)112 b Fu(\(5.12\))228 680 y(W)l(e)16 b(will)f(pro)o(v)o(e)g(b)o(y)h(induction)g(in)g Ft(m)g Fu(that)525 728 y Fl(\015)525 757 y(\015)553 770 y Ft(k)580 749 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)790 770 y Ft(R)827 777 y Fs(1)847 770 y Fu(\()p Ft(\020)t Fu(\))910 749 y Fm(s)929 770 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))1029 749 y Fm(t)1044 770 y Ft(k)1071 749 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1281 728 y Fl(\015)1281 757 y(\015)1323 770 y Fp(\024)d Ft(C)q(;)181 b Fu(\(5.13\))228 861 y(for)19 b Ft(s;)8 b(t)18 b Fp(2)h Fn(N)14 b Fp([)f(f)p Fu(0)p Fp(g)19 b Fu(with)g(1)g Fp(\024)f Ft(s)13 b Fu(+)g Ft(t)18 b Fp(\024)g Ft(m)p Fu(.)30 b(It)18 b(is)h(clear)f(from)g (Theorem)g(1.1)228 919 y(that)f(\(5.13\))g(holds)g(for)g Ft(m)e Fu(=)f(1.)23 b(So)17 b(let)f(us)h(assume)f(that)h(\(5.13\))h (holds)f(for)g(all)228 977 y Ft(m)c Fp(\024)h Ft(m)380 984 y Fs(0)416 977 y Fu(and)i(pro)o(v)o(e)g(that)g(it)g(holds)h(for)f Ft(m)e Fu(=)g Ft(m)1150 984 y Fs(0)1180 977 y Fu(+)d(1.)278 1036 y(The)17 b(case)h Ft(s)d Fu(=)h Ft(m)619 1043 y Fs(0)651 1036 y Fu(+)11 b(1,)18 b Ft(t)e Fu(=)g(0)h(is)h(statemen)o(t)d (\(4.6a\))k(of)e(Theorem)f(4.2.)26 b(So)228 1094 y(supp)q(ose)511 1183 y Fp(k)p Ft(k)563 1163 y Fo(\000)p Fs(\()p Fm(m)635 1168 y Fk(0)652 1163 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)790 1183 y Ft(R)827 1190 y Fs(1)847 1183 y Fu(\()p Ft(\020)t Fu(\))910 1163 y Fm(s)929 1183 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))1029 1163 y Fm(t)1044 1183 y Ft(k)1071 1163 y Fo(\000)p Fs(\()p Fm(m)1143 1168 y Fk(0)1161 1163 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1299 1183 y Fp(k)13 b(\024)h Ft(C)q(;)166 b Fu(\(5.14\))228 1272 y(where)16 b Ft(s)e Fp(\025)f Ft(\033)18 b Fu(and)f Ft(s)11 b Fu(+)g Ft(t)i Fu(=)h Ft(m)808 1279 y Fs(0)838 1272 y Fu(+)d(1.)22 b(Then)16 b(need)g(to)h(b)q(ound)291 1360 y Ft(k)318 1340 y Fo(\000)p Fs(\()p Fm(m)390 1345 y Fk(0)408 1340 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)545 1360 y Ft(R)582 1367 y Fs(1)602 1360 y Fu(\()p Ft(\020)t Fu(\))665 1340 y Fm(\033)q Fo(\000)p Fs(1)734 1360 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))834 1340 y Fm(\034)t Fs(+1)901 1360 y Ft(k)928 1340 y Fo(\000)p Fs(\()p Fm(m)1000 1345 y Fk(0)1018 1340 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1156 1360 y Fu(;)k Ft(\034)f Fu(=)14 b Ft(m)1327 1367 y Fs(0)1357 1360 y Fp(\000)d Ft(\033)i Fu(+)e(1)p Ft(:)63 b Fu(\(5.15\))278 1452 y(Up)q(on)16 b(substitution)h(of)j (\(5.12\))d(the)f(expression)g(\(5.15\))h(b)q(ecomes)449 1544 y Ft(k)476 1524 y Fo(\000)p Fs(\()p Fm(m)548 1529 y Fk(0)565 1524 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)703 1544 y Ft(R)740 1551 y Fs(1)760 1544 y Fu(\()p Ft(\020)t Fu(\))823 1524 y Fm(\033)847 1544 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))947 1524 y Fm(\034)969 1544 y Ft(k)996 1524 y Fo(\000)p Fs(\()p Fm(m)1068 1529 y Fk(0)1085 1524 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)410 1628 y Fp(\000)p Ft(k)476 1607 y Fo(\000)p Fs(\()p Fm(m)548 1612 y Fk(0)565 1607 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)703 1628 y Ft(R)740 1635 y Fs(1)760 1628 y Fu(\()p Ft(\020)t Fu(\))823 1607 y Fm(\033)847 1628 y Ft(V)875 1635 y Fs(2)895 1628 y Ft(R)932 1635 y Fs(1)952 1628 y Fu(\()p Ft(\020)t Fu(\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))1115 1607 y Fm(\034)1137 1628 y Ft(k)1164 1607 y Fo(\000)p Fs(\()p Fm(m)1236 1612 y Fk(0)1253 1607 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)411 1711 y Fu(+)p Ft(k)476 1691 y Fo(\000)p Fs(\()p Fm(m)548 1696 y Fk(0)565 1691 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)703 1711 y Ft(R)740 1718 y Fs(1)760 1711 y Fu(\()p Ft(\020)t Fu(\))823 1691 y Fm(\033)847 1711 y Ft(V)875 1718 y Fs(2)895 1711 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))p Ft(V)1023 1718 y Fs(2)1043 1711 y Ft(R)1080 1718 y Fs(1)1100 1711 y Fu(\()p Ft(\020)t Fu(\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))1263 1691 y Fm(\034)1285 1711 y Ft(k)1312 1691 y Fo(\000)p Fs(\()p Fm(m)1384 1696 y Fk(0)1402 1691 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)411 1784 y Fu(=)p Ft(E)485 1791 y Fs(1)516 1784 y Fu(+)11 b Ft(E)601 1791 y Fs(2)632 1784 y Fu(+)g Ft(E)717 1791 y Fs(3)736 1784 y Ft(:)278 1875 y Fu(By)k(the)h(h)o(yp)q(othesis)h (\(5.14\),)f Ft(E)869 1882 y Fs(1)905 1875 y Fu(is)g(uniformly)e(b)q (ounded.)278 1934 y(T)l(o)i(estimate)f Ft(E)581 1941 y Fs(2)617 1934 y Fu(w)o(e)h(write)f(\(with)h Ft(N)22 b Fu(su\016cien)o(tly)14 b(big\))268 2026 y Ft(E)304 2033 y Fs(2)338 2026 y Fu(=)390 1986 y Fl(\000)413 2026 y Ft(k)440 2005 y Fo(\000)p Fs(\()p Fm(m)512 2010 y Fk(0)529 2005 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)667 2026 y Ft(R)704 2033 y Fs(1)724 2026 y Fu(\()p Ft(\020)t Fu(\))787 2005 y Fm(\033)811 2026 y Fp(h)p Ft(x)p Fp(i)877 2005 y Fo(\000)p Fm(N)938 1986 y Fl(\001)8 b(\000)992 2026 y Fp(h)p Ft(x)p Fp(i)1058 2005 y Fm(N)1092 2026 y Ft(V)1120 2033 y Fs(2)1140 2026 y Ft(R)1177 2033 y Fs(1)1197 2026 y Fu(\()p Ft(\020)t Fu(\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))1360 2005 y Fm(\034)1382 2026 y Ft(k)1409 2005 y Fo(\000)p Fs(\()p Fm(m)1481 2010 y Fk(0)1499 2005 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1637 1986 y Fl(\001)1668 2026 y Ft(:)228 2115 y Fu(W)l(e)15 b(only)g(need)g(to)h (estimate)e(the)h(last)h(factor.)21 b(By)15 b(using)h(the)f(resolv)o (en)o(t)f(equa-)228 2173 y(tion)i(it)g(ma)o(y)f(written)g(as)264 2261 y Ft(B)s(B)341 2268 y Fs(1)371 2261 y Fu(+)c(\()p Ft(\020)k Fp(\000)c Ft(i)p Fu(\))p Ft(B)s(B)638 2268 y Fs(2)657 2261 y Fu(;)22 b Ft(B)16 b Fu(=)e Fp(h)p Ft(x)p Fp(i)864 2241 y Fm(N)898 2261 y Ft(V)926 2268 y Fs(2)946 2261 y Ft(R)983 2268 y Fs(1)1003 2261 y Fu(\()p Ft(i)p Fu(\))p Fp(h)p Ft(x)p Fp(i)1124 2241 y Fm(N)1158 2261 y Ft(;)264 2345 y(B)301 2352 y Fs(1)334 2345 y Fu(=)g Fp(h)p Ft(x)p Fp(i)452 2324 y Fo(\000)p Fm(N)514 2345 y Ft(R)p Fu(\()p Ft(\020)t Fu(\))614 2324 y Fm(\034)636 2345 y Ft(k)663 2324 y Fo(\000)p Fs(\()p Fm(m)735 2329 y Fk(0)753 2324 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)890 2345 y Ft(;)22 b(B)963 2352 y Fs(2)997 2345 y Fu(=)14 b Fp(h)p Ft(x)p Fp(i)1115 2324 y Fo(\000)p Fm(N)1176 2345 y Ft(R)1213 2352 y Fs(1)1233 2345 y Fu(\()p Ft(\020)t Fu(\))p Ft(R)p Fu(\()p Ft(\020)t Fu(\))1396 2324 y Fm(\034)1418 2345 y Ft(k)1445 2324 y Fo(\000)p Fs(\()p Fm(m)1517 2329 y Fk(0)1535 2324 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1673 2345 y Ft(:)228 2433 y Fu(Using)24 b(the)g(relativ)o(e)f (b)q(oundedness)i(of)g Ft(V)1033 2440 y Fs(2)1077 2433 y Fu(and)g(the)f(prop)q(ert)o(y)h(of)f(compact)228 2492 y(supp)q(ort,)18 b(w)o(e)f(see)g(that)h Ft(B)i Fu(is)d(b)q(ounded.)26 b(The)17 b(t)o(w)o(o)h(other)f(factors)h Ft(B)1532 2499 y Fs(1)1569 2492 y Fu(and)g Ft(B)1702 2499 y Fs(2)228 2550 y Fu(are)e(b)q(ounded)h(b)o(y)f(the)g(induction)g(h)o(yp)q (otheses.)278 2608 y(The)g(argumen)o(t)f(for)i(the)f(term)e Ft(E)907 2615 y Fs(3)943 2608 y Fu(is)i(similar;)e(it)i(is)g(omitted.) 956 2658 y Fq(37)p eop %%Page: 38 38 38 37 bop 278 213 a Fu(Th)o(us)14 b(w)o(e)g(ha)o(v)o(e)g(a)h(uniform)e (b)q(ound)i(of)k(\(5.15\),)c(and)g(\(5.13\))g(follo)o(ws.)20 b(Clearly)228 271 y(the)c(b)q(ound)i(\(5.13\))f(with)g Ft(s)d Fu(=)h(0,)h Ft(m)e Fu(=)h Ft(t)h Fu(and)h(\(1.5\))g(coincide,)e (so)i(the)g(pro)q(of)g(is)228 329 y(\014nished.)228 438 y(5.3.)28 b Fy(Pro)r(of)21 b(of)27 b Fu(\(1.6\))p Fy(.)d Fu(Supp)q(ose)c(on)g(the)e(con)o(trary)h(that)g Ft(E)1397 420 y Fo(0)1409 438 y Fu(\(+0\))f(=)h(0.)29 b(Let)228 496 y Ft(R)265 478 y Fs(+)310 496 y Fu(=)14 b Ft(R)p Fu(\(0)f(+)e Ft(i)p Fu(0\))17 b(and)g Ft(R)713 478 y Fo(\000)758 496 y Fu(=)e Ft(R)p Fu(\(0)d Fp(\000)g Ft(i)p Fu(0\),)k(so)i(that)f Ft(R)1248 478 y Fs(+)1293 496 y Fu(=)e Ft(R)1383 478 y Fo(\000)1429 496 y Fu(as)j(an)f(iden)o(tit)o(y) 228 554 y(in)j Fp(B)r Fu(\()p Fp(H)385 561 y Fs(1)404 554 y Ft(;)8 b Fp(H)468 561 y Fs(2)488 554 y Fu(\).)36 b(W)l(e)20 b(also)i(use)f(the)g(notation)h Ft(R)1163 534 y Fs(+)1163 566 y(1)1214 554 y Fu(and)g Ft(R)1351 534 y Fo(\000)1351 566 y Fs(1)1402 554 y Fu(of)f(the)g(pro)q(of)h(of) 228 612 y(Theorem)15 b(1.1)h(in)g(Subsection)g(5.1.)278 670 y(W)l(e)g(compute)453 760 y(\(2)p Ft(i)p Fu(\))532 739 y Fo(\000)p Fs(1)580 760 y Fp(h)p Ft(\036;)8 b Fu(\()p Ft(R)706 739 y Fs(+)706 772 y(1)747 760 y Fp(\000)j Ft(R)834 739 y Fo(\000)834 772 y Fs(1)864 760 y Fu(\))p Ft(\036)p Fp(i)467 840 y Fu(=)29 b(lim)519 874 y Fm(\024)p Fo(!)p Fs(0)592 864 y Fk(+)626 840 y Ft(\024)p Fp(k)p Ft(R)716 847 y Fs(1)736 840 y Fu(\()p Ft(i\024)p Fu(\))p Ft(\036)p Fp(k)873 820 y Fs(2)467 944 y Fu(=)g(lim)519 977 y Fm(\024)p Fo(!)p Fs(0)592 968 y Fk(+)626 944 y Ft(\024)p Fp(k)p Ft(R)p Fu(\()p Ft(i\024)p Fu(\))805 930 y(~)799 944 y Ft(\036)p Fu(\()p Ft(\024)p Fu(\))p Fp(k)919 923 y Fs(2)939 944 y Fu(;)981 930 y(~)974 944 y Ft(\036)p Fu(\()p Ft(\024)p Fu(\))14 b(=)g(\()p Ft(I)g Fu(+)d Ft(V)1267 951 y Fs(2)1287 944 y Ft(R)1324 951 y Fs(1)1344 944 y Fu(\()p Ft(i\024)p Fu(\)\))p Ft(\036;)467 1047 y Fu(=)j(\(2)p Ft(i)p Fu(\))598 1027 y Fo(\000)p Fs(1)645 1047 y Fp(h)670 1034 y Fu(~)664 1047 y Ft(\036)p Fu(\(+0\))p Ft(;)8 b Fu(\()p Ft(R)871 1027 y Fs(+)913 1047 y Fp(\000)i Ft(R)999 1027 y Fo(\000)1029 1047 y Fu(\))1054 1034 y(~)1048 1047 y Ft(\036)p Fu(\(+0\))p Fp(i)15 b Fu(=)f(0)p Ft(;)228 1137 y Fu(from)h(whic)o(h)h(w)o(e)f (conclude)h(that)h(also)868 1226 y Ft(R)905 1206 y Fs(+)905 1239 y(1)949 1226 y Fu(=)d Ft(R)1038 1206 y Fo(\000)1038 1239 y Fs(1)1068 1226 y Ft(:)516 b Fu(\(5.16\))278 1319 y(Using)14 b(the)g(pro)q(of)i(of)f(Theorem)e(1.1)i(w)o(e)f(ma)o(y)f (infer)h(from)g(\(5.16\))h(that)g(for)g(all)228 1377 y Ft(t)e(>)h Fu(0)617 1466 y Ft(k)644 1446 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)781 1466 y Ft(R)818 1446 y Fs(+)818 1478 y(1)848 1466 y Ft(k)875 1446 y Fo(\000)p Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)1053 1466 y Fp(2)g(B)r Fu(\()p Ft(L)1187 1446 y Fs(2)1206 1466 y Fu(\()p Fn(R)1264 1446 y Fm(d)1282 1466 y Fu(\)\))p Ft(:)264 b Fu(\(5.17\))278 1562 y(Let)18 b(us)h(consider)g(v)o(ectors)f Ft(\036)g Fp(2)g Ft(L)924 1544 y Fs(2)944 1562 y Fu(\()p Ft(B)1000 1569 y Fm(\032)1020 1562 y Fu(\))h(where)f Ft(B)1238 1569 y Fm(\032)1276 1562 y Fu(=)g Fp(f)p Ft(x)g Fp(2)g Fn(R)1493 1544 y Fm(d)1510 1562 y Fp(j)g(j)p Ft(x)p Fp(j)f(\024)h Ft(\032)p Fp(g)228 1621 y Fu(with)c Ft(\032)g Fu(large.)21 b(By)13 b(taking)i Ft(t)e Fu(=)h(1)p Ft(=)p Fu(2)7 b(+)g Ft(\017)15 b Fu(in)f(\(5.17\))h(w)o(e)f(learn)g(that)h Ft( )g Fu(:=)e Ft(R)1616 1600 y Fs(+)1616 1633 y(1)1646 1621 y Ft(\036)h Fp(2)228 1679 y Ft(L)261 1661 y Fs(2)281 1679 y Fu(\()p Fn(R)339 1661 y Fm(d)356 1679 y Fu(\).)28 b(Since)18 b Ft(H)587 1686 y Fs(1)607 1679 y Ft( )h Fu(=)f Ft(\036)g Fu(w)o(e)g(see)g(that)h Ft(H)1064 1686 y Fs(1)1084 1679 y Ft( )h Fu(=)d(0)i(outside)g Ft(B)1442 1686 y Fm(\032)1462 1679 y Fu(.)28 b(Using)18 b(this)228 1737 y(and)i(Remarks)f(2.5)h(2\))g (w)o(e)f(conclude)g(that)h(indeed)f Ft( )j Fu(=)d(0)i(outside)e Ft(B)1588 1744 y Fm(\032)1609 1737 y Fu(.)31 b(W)l(e)228 1796 y(conclude)13 b(that)i Ft(R)567 1775 y Fs(+)567 1808 y(1)611 1796 y Fp(2)f(B)r Fu(\()p Ft(L)745 1778 y Fs(2)764 1796 y Fu(\()p Ft(B)820 1803 y Fm(\032)840 1796 y Fu(\)\))g(is)g(the)g(in)o(v)o(erse)f(of)h(the)g(Diric)o(hlet)f (Laplacian)228 1854 y Ft(H)268 1861 y Fs(1)p Fm(;\032)330 1854 y Fu(=)g Fp(\000)p Fu(\001)7 b(+)g Ft(V)541 1861 y Fs(1)575 1854 y Fu(for)14 b(the)g(region)h Ft(B)911 1861 y Fm(\032)931 1854 y Fu(.)20 b(W)l(e)14 b(obtain)h(a)g(con)o (tradiction)e(from)g(this)228 1912 y(b)o(y)18 b(c)o(ho)q(osing)i(a)g (large)f Ft(\032)g Fu(suc)o(h)g(that)h(zero)f(is)g(an)g(eigen)o(v)m (alue)f(of)i Ft(H)1494 1919 y Fs(1)p Fm(;\032)1542 1912 y Fu(.)29 b(Notice)228 1970 y(that)17 b(all)f(branc)o(hes)h(of)g(eigen) o(v)m(alues)f(will)f(cross)i(zero)g(as)g Ft(\032)e Fp(!)g(1)p Fu(,)h(cf.)22 b([ReSi78,)228 2028 y(Theorem)15 b(XI)q(I)q(I.6)g(\(V)l (ol.)21 b(IV,)15 b(p.)21 b(87\)].)228 2138 y(5.4.)28 b Fy(Pro)r(of)18 b(of)h(Theorem)e(1.4.)278 2208 y(Pro)r(of)h(of)h (\(i\).)h Fu(Due)d(to)f(Theorem)f(1.2)i(w)o(e)f(ma)o(y)e(de\014ne)285 2329 y Ft(E)324 2308 y Fs(\()p Fm(m)p Fs(\))384 2329 y Fu(\()p Ft(\025)p Fu(\))h(=)536 2295 y Ft(d)561 2277 y Fm(m)p Fo(\000)p Fs(1)p 521 2317 133 2 v 521 2363 a Ft(d\025)574 2348 y Fm(m)p Fo(\000)p Fs(1)659 2329 y Ft(E)698 2308 y Fo(0)709 2329 y Fu(\()p Ft(\025)p Fu(\))g(=)e(\(2)p Ft(\031)r(i)p Fu(\))950 2308 y Fo(\000)p Fs(1)1016 2295 y Ft(d)1041 2277 y Fm(m)p Fo(\000)p Fs(1)p 1002 2317 V 1002 2363 a Ft(d\025)1055 2348 y Fm(m)p Fo(\000)p Fs(1)1147 2329 y Fu(\()p Ft(R)p Fu(\()p Ft(\025)f Fu(+)f Ft(i)p Fu(0\))g Fp(\000)g Ft(R)p Fu(\()p Ft(\025)h Fp(\000)f Ft(i)p Fu(0\)\))790 2427 y Fp(2)j(B)r Fu(\()p Ft(k)918 2406 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fs(~)-18 b Fm(\017)1127 2427 y Ft(L)1160 2406 y Fs(2)1180 2427 y Ft(;)8 b(k)1229 2406 y Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\)+~)-18 b Fm(\017)1412 2427 y Ft(L)1445 2406 y Fs(2)1465 2427 y Fu(\))p Ft(;)228 2517 y Fu(and)699 2586 y Ft(E)738 2565 y Fs(\()p Fm(m)p Fs(\))799 2586 y Fu(\(+0\))14 b(=)29 b(lim)964 2619 y Fm(\025)p Fo(!)p Fs(0)1038 2610 y Fk(+)1072 2586 y Ft(E)1111 2565 y Fs(\()p Fm(m)p Fs(\))1171 2586 y Fu(\()p Ft(\025)p Fu(\))p Ft(:)956 2658 y Fq(38)p eop %%Page: 39 39 39 38 bop 278 213 a Fu(Using)19 b(similar)e(notation)j(for)g(\()p Ft(f)5 b(E)944 195 y Fo(0)956 213 y Fu(\)\()p Ft(\025)p Fu(\))19 b(w)o(e)g(represen)o(t)f(as)i(an)g(op)q(erator)h(in)228 271 y(this)16 b(space,)g(cf.)21 b([JMP84],)291 380 y Ft(e)314 359 y Fo(\000)p Fm(itH)400 380 y Fu(\()p Ft(f)5 b Fu(1)472 388 y Fs([0)p Fm(;)p Fo(1)p Fs(\))561 380 y Fu(\)\()p Ft(H)t Fu(\))14 b(=)727 312 y Fl(Z)777 325 y Fo(1)755 425 y Fs(0)823 380 y Ft(e)846 359 y Fo(\000)p Fm(it\025)920 380 y Ft(f)5 b Fu(\()p Ft(\025)p Fu(\))p Ft(E)1054 359 y Fo(0)1067 380 y Fu(\()p Ft(\025)p Fu(\))p Ft(d\025)412 b Fu(\(5.18\))305 532 y(=)357 470 y Fm(m)p Fo(\000)p Fs(1)359 485 y Fl(X)362 590 y Fm(n)p Fs(=1)433 532 y Fu(\()p Ft(it)p Fu(\))506 512 y Fo(\000)p Fm(n)556 532 y Fu(\()p Ft(f)5 b(E)643 512 y Fo(0)655 532 y Fu(\))674 512 y Fs(\()p Fm(n)p Fo(\000)p Fs(1\))770 532 y Fu(\(+0\))11 b(+)930 464 y Fl(Z)980 478 y Fo(1)958 577 y Fs(0)1026 532 y Ft(e)1049 512 y Fo(\000)p Fm(it\025)1123 532 y Fu(\()p Ft(it)p Fu(\))1196 512 y Fo(\000)p Fm(m)p Fs(+1)1301 532 y Fu(\()p Ft(f)5 b(E)1388 512 y Fo(0)1400 532 y Fu(\))1419 512 y Fs(\()p Fm(m)p Fo(\000)p Fs(1\))1525 532 y Fu(\()p Ft(\025)p Fu(\))p Ft(d\025:)278 657 y Fu(T)l(ak)o(e)15 b Ft(m)f Fu(=)g(3.)278 715 y Fy(Pro)r(of)25 b(of)h(\(ii\).)37 b Fu(W)l(e)22 b(m)o(ultiply)e(\(5.18\))j(b)o(y)e Ft(F)7 b Fu(\()p Fp(h)p Ft(x)p Fp(i)24 b Ft(<)g Fu(2)p Ft(t)1403 697 y Fm(\024)1425 715 y Fu(\))e(\(where)g Ft(\024)i Fu(=)228 775 y(\(1)9 b Fp(\000)f Ft(\017)p Fu(\)\(1)g(+)h Ft(\026=)p Fu(2\))560 757 y Fo(\000)p Fs(1)608 775 y Fu(\))15 b(from)f(the)g(left)h(and)g Ft(k)1043 757 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fs(~)-18 b Fm(\017)1268 775 y Fu(from)14 b(the)h(righ)o(t.)20 b(Using)228 833 y(again)d(Theorem)e(1.2)h(w)o(e)g(estimate)254 939 y Fp(k)p Ft(F)7 b Fu(\()p Fp(h)p Ft(x)p Fp(i)14 b Ft(<)g Fu(2)p Ft(t)511 919 y Fm(\024)533 939 y Fu(\))560 871 y Fl(Z)610 885 y Fo(1)588 984 y Fs(0)656 939 y Ft(e)679 919 y Fo(\000)p Fm(it\025)753 939 y Fu(\()p Ft(it)p Fu(\))826 919 y Fo(\000)p Fm(m)p Fs(+1)931 939 y Fu(\()p Ft(f)5 b(E)1018 919 y Fo(0)1030 939 y Fu(\))1049 919 y Fs(\()p Fm(m)p Fo(\000)p Fs(1\))1155 939 y Fu(\()p Ft(\025)p Fu(\))p Ft(d\025k)1301 919 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fs(~)-18 b Fm(\017)1513 939 y Fp(k)268 1076 y(\024)14 b Ft(C)356 1083 y Fs(1)384 1008 y Fl(Z)434 1021 y Fo(1)411 1121 y Fs(0)479 1076 y Fp(k)p Fu(\()528 1042 y Ft(t)546 1024 y Fs(1)p Fo(\000)p Fm(\017)p 528 1065 80 2 v 554 1110 a Ft(k)612 1076 y Fu(\))631 1055 y Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\)+~)-18 b Fm(\017)814 1076 y Ft(e)837 1055 y Fo(\000)p Fm(it\025)911 1076 y Fu(\()p Ft(it)p Fu(\))984 1055 y Fo(\000)p Fm(m)p Fs(+1)1089 1076 y Fu(\()p Ft(f)5 b(E)1176 1055 y Fo(0)1188 1076 y Fu(\))1207 1055 y Fs(\()p Fm(m)p Fo(\000)p Fs(1\))1313 1076 y Fu(\()p Ft(\025)p Fu(\))p Ft(k)1406 1055 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fs(~)-18 b Fm(\017)1617 1076 y Fp(k)p Ft(d\025)268 1190 y Fp(\024)14 b Ft(C)356 1197 y Fs(2)375 1190 y Ft(t)398 1156 y Fk(1)p 398 1162 16 2 v 398 1182 a(2)418 1169 y Fs(+~)-18 b Fm(\017)q Fs(\(1)p Fo(\000)p Fm(\017)p Fs(\))p Fo(\000)p Fm(\017)p Fs(\()p Fm(m)p Fo(\000)665 1156 y Fk(1)p 666 1162 V 666 1182 a(2)686 1169 y Fs(\))702 1190 y Ft(:)278 1272 y Fu(Supp)q(ose)682 1321 y(1)p 682 1343 25 2 v 682 1389 a(2)722 1355 y Fp(\000)11 b Ft(\017)p Fu(\()p Ft(m)f Fp(\000)919 1321 y Fu(1)p 919 1343 V 919 1389 a(2)948 1355 y(\))k Ft(<)g Fp(\000)p Fu(\(1)d(+)g Ft(\017)1195 1334 y Fo(0)1207 1355 y Fu(\))1231 1321 y(1)p 1231 1343 V 1231 1389 a(2)1260 1355 y Ft(:)228 1441 y Fu(Then)16 b(for)h(all)e(su\016cien)o(tly)g(small)g(~)-25 b Ft(\017)14 b(>)f Fu(0)570 1539 y(1)p 570 1561 V 570 1607 a(2)610 1572 y(+)f(~)-25 b Ft(\017)p Fu(\(1)11 b Fp(\000)g Ft(\017)p Fu(\))g Fp(\000)g Ft(\017)p Fu(\()p Ft(m)f Fp(\000)1030 1539 y Fu(1)p 1030 1561 V 1030 1607 a(2)1059 1572 y(\))k Fp(\024)g(\000)p Fu(\(1)d(+)g Ft(\017)1307 1552 y Fo(0)1318 1572 y Fu(\))1342 1539 y(1)p 1342 1561 V 1342 1607 a(2)1372 1572 y Ft(:)228 1659 y Fu(This)21 b(is)f(the)g(argumen)o(t)g(for)h(the) f(con)o(tribution)g(from)f(the)i(last)g(term)d(on)j(the)228 1717 y(righ)o(t)16 b(hand)h(side)e(of)20 b(\(5.18\))q(.)278 1775 y(W)l(e)c(deal)g(with)g(the)g(other)g(terms)f(in)h(a)g(similar)e (w)o(a)o(y:)402 1858 y Fp(k)p Ft(F)7 b Fu(\()p Fp(h)p Ft(x)p Fp(i)14 b Ft(<)g Fu(2)p Ft(t)659 1837 y Fm(\024)681 1858 y Fu(\)\()p Ft(it)p Fu(\))773 1837 y Fo(\000)p Fm(n)823 1858 y Fu(\()p Ft(f)5 b(E)910 1837 y Fo(0)922 1858 y Fu(\))941 1837 y Fs(\()p Fm(n)p Fo(\000)p Fs(1\))1037 1858 y Fu(\(+0\))p Ft(k)1164 1837 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fs(~)-18 b Fm(\017)1375 1858 y Fp(k)416 1969 y(\024)14 b Ft(C)504 1976 y Fs(1)523 1969 y Fp(k)p Fu(\()572 1936 y Ft(t)590 1918 y Fs(1)p Fo(\000)p Fm(\017)p 572 1958 80 2 v 598 2004 a Ft(k)656 1969 y Fu(\))675 1949 y Fs(\()p Fm(n)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\)+~)-18 b Fm(\017)848 1969 y Fu(\()p Ft(it)p Fu(\))921 1949 y Fo(\000)p Fm(n)971 1969 y Fu(\()p Ft(f)5 b(E)1058 1949 y Fo(0)1070 1969 y Fu(\))1089 1949 y Fs(\()p Fm(n)p Fo(\000)p Fs(1\))1185 1969 y Fu(\(+0\))p Ft(k)1312 1949 y Fo(\000)p Fs(\()p Fm(m)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\))p Fo(\000)p Fs(~)-18 b Fm(\017)1523 1969 y Fp(k)416 2068 y Fu(=)14 b Ft(C)503 2075 y Fs(2)523 2068 y Ft(t)541 2047 y Fs(\(1)p Fo(\000)p Fm(\017)p Fs(\)\(\()p Fm(n)p Fo(\000)p Fs(1)p Fm(=)p Fs(2\)+~)-18 b Fm(\017)o Fs(\))827 2068 y Ft(t)845 2047 y Fo(\000)p Fm(n)416 2156 y Fu(=)14 b Ft(C)503 2163 y Fs(2)523 2156 y Ft(t)541 2135 y Fs(~)-18 b Fm(\017)o Fs(\(1)p Fo(\000)p Fm(\017)p Fs(\)+\()687 2122 y Fk(1)p 687 2128 16 2 v 687 2149 a(2)708 2135 y Fo(\000)p Fm(n)p Fs(\))p Fm(\017)p Fo(\000)816 2122 y Fk(1)p 817 2128 V 817 2149 a(2)839 2156 y Ft(:)228 2238 y Fu(The)16 b(w)o(orst)h(b)q (ound)g(is)f(for)h Ft(n)d Fu(=)f(1.)22 b(Clearly)15 b(w)o(e)h(get)h (the)f(result.)313 2346 y Fv(Appendix)h Fu(A.)26 b Fv(Algebraic)18 b(verifica)m(tion)f(of)h(Theorem)g(4.2)278 2433 y Fu(W)l(e)j(pro)o(v)o (e)g(the)g(equations)h(\(4.6\))f(b)o(y)g(induction)h(in)f Ft(m)p Fu(.)37 b(F)l(or)21 b Ft(m)i Fu(=)g(1)e(the)228 2492 y(equations)13 b(\(4.6\))h(b)q(ecome)e(\(4.5\))q(.)20 b(So)14 b(assume)f(that)h(\(4.6\))f(are)h(true)f(for)h(all)f Ft(m)g Fp(\024)228 2550 y Ft(m)271 2557 y Fs(0)308 2550 y Fu(for)18 b(some)f Ft(m)551 2557 y Fs(0)588 2550 y Fp(\025)f Fu(1.)27 b(W)l(e)18 b(will)f(pro)o(v)o(e)g(\(4.6a\))q(,)h (\(4.6b\),)g(\(4.6d\),)g(\(4.6f\))q(,)g(and)228 2608 y(\(4.6g\))c(for)g Ft(m)f Fu(=)h Ft(m)589 2615 y Fs(0)614 2608 y Fu(+)6 b(1.)20 b(By)13 b(symmetry)d(\(lo)q(ok)k(at)g(the)g (`adjoin)o(t')e(expressions\))956 2658 y Fq(39)p eop %%Page: 40 40 40 39 bop 228 213 a Fu(\(4.6c\),)17 b(\(4.6e\))g(and)h(\(4.6h\))f(will) f(follo)o(w)h(from)f(the)h(previous)g(pro)q(ofs,)i(so)f(these)228 271 y(cases)i(will)f(not)i(need)f(further)g(elab)q(oration.)34 b(F)l(or)20 b(shortness)i(w)o(e)d(write)h Ft(R)h Fu(=)228 329 y Ft(R)p Fu(\()p Ft(\020)t Fu(\).)278 387 y Fy(Pro)r(of)26 b(of)32 b Fu(\(4.6a\))q(.)42 b(T)l(o)24 b(pro)o(v)o(e)f(\(4.6a\))h(for) f Ft(m)j Fu(=)f Ft(m)1320 394 y Fs(0)1356 387 y Fu(+)15 b(1)24 b(w)o(e)f(c)o(ho)q(ose)h(a)228 445 y(real-v)m(alued)17 b(function)g Ft(F)701 452 y Fs(+)748 445 y Fu(as)h(in)f(Theorem)f(4.1)i (\(ii\))f(suc)o(h)g(that)h Ft(F)1487 452 y Fs(+)1532 445 y Fp(\021)e Fu(1)i(on)g(a)228 507 y(neigh)o(b)q(ourho)q(o)q(d)c(of) f(+)p Fp(1)p Fu(.)20 b(Let)12 b Ft(F)846 514 y Fo(\000)889 507 y Fu(=)i(1)s Fp(\000)s Ft(F)1042 514 y Fs(+)1071 507 y Fu(.)20 b(Pic)o(k)12 b(real-v)m(alued)g(functions)1672 495 y(~)1661 507 y Ft(F)1693 514 y Fo(\000)228 568 y Fu(and)332 555 y(~)321 568 y Ft(F)353 575 y Fs(+)397 568 y Fu(as)j(in)f(Theorem)f(4.1)i(\(iii\))f(suc)o(h)g(that)1108 555 y(~)1097 568 y Ft(F)1129 575 y Fo(\000)1166 568 y Fu(+)1222 555 y(~)1212 568 y Ft(F)1244 575 y Fs(+)1286 568 y Fu(=)g(1.)21 b(Then)15 b(ob)o(viously)228 630 y(1)f(=)g Ft(F)350 637 y Fs(+)379 630 y Fu(\()p Ft(a)424 637 y Fs(0)443 630 y Fu(\))d(+)g Ft(F)554 637 y Fo(\000)583 630 y Fu(\()p Ft(a)628 637 y Fs(0)648 630 y Fu(\))678 618 y(~)667 630 y Ft(F)699 637 y Fs(+)728 630 y Fu(\()p Ft(b)p Fu(\))f(+)h Ft(F)878 637 y Fo(\000)908 630 y Fu(\()p Ft(a)953 637 y Fs(0)972 630 y Fu(\))1002 618 y(~)991 630 y Ft(F)1023 637 y Fo(\000)1052 630 y Fu(\()p Ft(b)p Fu(\).)21 b(W)l(e)16 b(decomp)q(ose)311 715 y Ft(k)338 694 y Fo(\000)p Fs(\()p Fm(m)410 699 y Fk(0)428 694 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)566 715 y Ft(R)603 694 y Fm(m)634 699 y Fk(0)652 694 y Fs(+1)699 715 y Ft(k)726 694 y Fo(\000)p Fs(\()p Fm(m)798 699 y Fk(0)816 694 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)325 798 y Fu(=)377 758 y Fl(\000)400 798 y Ft(k)427 778 y Fo(\000)p Fs(\()p Fm(m)499 783 y Fk(0)517 778 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)654 798 y Ft(Rk)718 778 y Fo(\000)p Fm(s)765 758 y Fl(\001)8 b(\000)819 798 y Ft(k)846 778 y Fm(s)864 798 y Fu(Op)921 778 y Fs(w)941 798 y Fu(\()p Ft(F)992 805 y Fs(+)1021 798 y Fu(\()p Ft(a)1066 805 y Fs(0)1085 798 y Fu(\)\))p Ft(R)1160 778 y Fm(m)1191 783 y Fk(0)1211 798 y Ft(k)1238 778 y Fo(\000)p Fs(\()p Fm(m)1310 783 y Fk(0)1328 778 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1466 758 y Fl(\001)1618 798 y Fu(\(A.1\))323 894 y(+)372 853 y Fl(\000)394 894 y Ft(k)421 873 y Fo(\000)p Fs(\()p Fm(m)493 878 y Fk(0)511 873 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)649 894 y Ft(Rk)713 873 y Fo(\000)p Fm(s)759 853 y Fl(\001)791 838 y(\020)820 894 y Ft(k)847 873 y Fm(s)866 894 y Fu(Op)923 873 y Fs(w)942 894 y Fu(\()p Ft(F)993 901 y Fo(\000)1022 894 y Fu(\()p Ft(a)1067 901 y Fs(0)1087 894 y Fu(\))1117 881 y(~)1106 894 y Ft(F)1138 901 y Fo(\000)1167 894 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)1282 873 y Fm(m)1313 878 y Fk(0)1332 894 y Ft(k)1359 873 y Fo(\000)p Fs(\()p Fm(m)1431 878 y Fk(0)1449 873 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1587 838 y Fl(\021)323 1002 y Fu(+)372 947 y Fl(\020)401 1002 y Ft(k)428 981 y Fo(\000)p Fs(\()p Fm(m)500 986 y Fk(0)518 981 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)656 1002 y Ft(R)p Fu(Op)750 981 y Fs(w)770 1002 y Fu(\()p Ft(F)821 1009 y Fo(\000)850 1002 y Fu(\()p Ft(a)895 1009 y Fs(0)914 1002 y Fu(\))944 989 y(~)933 1002 y Ft(F)965 1009 y Fs(+)994 1002 y Fu(\()p Ft(b)p Fu(\)\))p Ft(k)1099 981 y Fm(s)1117 947 y Fl(\021)1155 962 y(\000)1178 1002 y Ft(k)1205 981 y Fo(\000)p Fm(s)1251 1002 y Ft(R)1288 981 y Fm(m)1319 986 y Fk(0)1339 1002 y Ft(k)1366 981 y Fo(\000)p Fs(\()p Fm(m)1438 986 y Fk(0)1456 981 y Fs(+1)p Fm(=)p Fs(2\))p Fo(\000)p Fm(\017)1594 962 y Fl(\001)1625 1002 y Ft(;)228 1098 y Fu(cf.)31 b(\(5.8\).)h(Cho)q(osing)21 b Ft(s)f Fu(=)g(1)p Ft(=)p Fu(2)14 b(+)f Ft(\017=)p Fu(3,)21 b(the)f(\014rst)f (term)f(in)i(\(A.1\))f(is)g(seen)h(to)228 1156 y(b)q(e)g(uniformly)f(b) q(ounded)i(b)o(y)f(using)h(\(4.5a\))g(and)g(\(4.6b\))g(with)f Ft(m)g Fu(=)h Ft(m)1604 1163 y Fs(0)1644 1156 y Fu(and)228 1214 y Ft(t)15 b Fu(=)h(1)c(+)f Ft(\017)p Fu(2)p Ft(=)p Fu(3)19 b(and)e Ft(\017)f Fp(!)f Ft(\017=)p Fu(3.)25 b(The)17 b(second)h(term)d(is)i(treated)g(similarly)e(using)228 1272 y(\(4.6d\))c(instead)h(of)i(\(4.6b\))q(.)19 b(F)l(or)12 b(the)f(third)f(term)g(w)o(e)h(c)o(ho)q(ose)g Ft(s)j Fu(=)g Ft(m)1482 1279 y Fs(0)1502 1272 y Fu(+)q Ft(\017=)p Fu(3)q Fp(\000)q Fu(1)p Ft(=)p Fu(2)228 1330 y(and)j(apply)f(\(4.5e\))h (and)f(\(4.6a\))h(to)g(get)f(the)g(conclusion.)278 1389 y(In)k(the)h(rest)f(of)h(the)g(pro)q(of)h(w)o(e)e(will)g(not)h (explicitly)d(in)o(tro)q(duce)i(factors)i Ft(k)1704 1371 y Fm(s)228 1447 y Fu(and)17 b Ft(k)350 1429 y Fo(\000)p Fm(s)412 1447 y Fu(as)g(w)o(e)e(did)h(ab)q(o)o(v)o(e.)278 1505 y Fy(Pro)r(of)21 b(of)26 b Fu(\(4.6b\).)i(F)l(or)18 b(this)h(part)f(w)o(e)g(in)o(tro)q(duce)g(functions)h Ft(G)1513 1512 y Fs(+)1543 1505 y Ft(;)1575 1492 y Fu(~)1565 1505 y Ft(G)1603 1512 y Fs(+)1632 1505 y Ft(;)1665 1492 y Fu(~)1654 1505 y Ft(G)1692 1512 y Fo(\000)228 1563 y Fu(analogous)j(to)e(the)f Ft(F)7 b Fu('s)19 b(in)h(the)f(argumen)o(t) g(ab)q(o)o(v)o(e)g(and)i(satisfying)f(the)f(same)228 1621 y(conditions.)27 b(W)l(e)18 b(assume)f(that)i Ft(G)888 1628 y Fs(+)935 1621 y Fp(\021)e Fu(1)i(on)f(a)h(neigh)o(b)q(ourho)q(o) q(d)h(of)f(supp)9 b Ft(F)1680 1628 y Fs(+)1709 1621 y Fu(.)228 1679 y(Let)21 b Ft(G)358 1686 y Fo(\000)410 1679 y Fu(=)h(1)15 b Fp(\000)f Ft(G)600 1686 y Fs(+)630 1679 y Fu(.)36 b(Then)21 b(w)o(e)g(write)f(with)h Ft(B)k Fu(=)d(Op)1313 1661 y Fs(w)1332 1679 y Fu(\()p Ft(F)1383 1686 y Fs(+)1412 1679 y Fu(\()p Ft(a)1457 1686 y Fs(0)1477 1679 y Fu(\)\))f(and)g Ft(\034)28 b Fu(=)228 1737 y Ft(t)10 b Fu(+)h Ft(m)348 1744 y Fs(0)379 1737 y Fu(+)g(1)p Ft(=)p Fu(2)h(+)f Ft(\017)p Fu(:)307 1821 y Ft(k)334 1801 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)471 1821 y Ft(B)s(R)548 1801 y Fm(m)579 1806 y Fk(0)597 1801 y Fs(+1)644 1821 y Ft(k)671 1801 y Fo(\000)p Fm(\034)734 1821 y Fu(=)p Ft(k)799 1801 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)936 1821 y Ft(B)s(R)p Fu(Op)1070 1801 y Fs(w)1089 1821 y Fu(\()p Ft(G)1146 1828 y Fs(+)1176 1821 y Fu(\()p Ft(a)1221 1828 y Fs(0)1240 1821 y Fu(\)\))p Ft(R)1315 1801 y Fm(m)1346 1806 y Fk(0)1367 1821 y Ft(k)1394 1801 y Fo(\000)p Fm(\034)1610 1821 y Fu(\(A.2\))783 1905 y(+)g Ft(k)859 1885 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)996 1905 y Ft(B)s(R)p Fu(Op)1130 1885 y Fs(w)1149 1905 y Fu(\()p Ft(G)1206 1912 y Fo(\000)1236 1905 y Fu(\()p Ft(a)1281 1912 y Fs(0)1301 1905 y Fu(\))1331 1893 y(~)1320 1905 y Ft(G)1358 1912 y Fo(\000)1387 1905 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)1502 1885 y Fm(m)1533 1890 y Fk(0)1553 1905 y Ft(k)1580 1885 y Fo(\000)p Fm(\034)783 1989 y Fu(+)g Ft(k)859 1969 y Fm(t)p Fo(\000)p Fs(1)p Fm(=)p Fs(2)p Fo(\000)p Fm(\017)996 1989 y Ft(B)s(R)p Fu(Op)1130 1969 y Fs(w)1149 1989 y Fu(\()p Ft(G)1206 1996 y Fo(\000)1236 1989 y Fu(\()p Ft(a)1281 1996 y Fs(0)1301 1989 y Fu(\))1331 1977 y(~)1320 1989 y Ft(G)1358 1996 y Fs(+)1387 1989 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)1502 1969 y Fm(m)1533 1974 y Fk(0)1553 1989 y Ft(k)1580 1969 y Fo(\000)p Fm(\034)1629 1989 y Ft(:)228 2072 y Fu(The)j(\014rst)h(term)e(in)h(\(A.2\))h(is)f(easily)g (estimated)f(using)i(\(4.5b\))g(and)g(\(4.6b\))g(with)228 2130 y Ft(m)e Fu(=)h Ft(m)379 2137 y Fs(0)398 2130 y Fu(.)21 b(The)13 b(second)h(term)e(is)h(estimated)f(b)o(y)h(com)o (bining)f(\(4.5b\))i(and)g(\(4.6d\))228 2189 y(\(with)g Ft(m)f Fu(=)h Ft(m)507 2196 y Fs(0)526 2189 y Fu(\).)21 b(Finally)l(,)13 b(the)g(third)h(term)f(is)g(estimated,)g(using)h(the)g (supp)q(ort)228 2247 y(prop)q(erties)i(of)h Ft(G)551 2254 y Fo(\000)581 2247 y Fu(,)e(b)o(y)h(com)o(bining)f(\(4.5g\))i(and) g(\(4.6a\))g(\(with)f Ft(m)d Fu(=)h Ft(m)1571 2254 y Fs(0)1591 2247 y Fu(\).)278 2305 y Fy(Pro)r(of)19 b(of)25 b Fu(\(4.6d\))q(.)f(W)l(e)17 b(shall)g(use)g(a)g(set)g(of)h(functions)f Ft(G)p Fu('s)h(as)f(ab)q(o)o(v)o(e)g(suc)o(h)228 2367 y(that)346 2354 y(~)335 2367 y Ft(G)373 2374 y Fo(\000)419 2367 y Fp(\021)g Fu(1)h(on)g(a)g(neigh)o(b)q(ourho)q(o)q(d)i(of)e(supp) 1140 2354 y(~)1129 2367 y Ft(F)1161 2374 y Fo(\000)1190 2367 y Fu(.)26 b(W)l(e)17 b(consider)h(\(A.2\))f(no)o(w)228 2429 y(with)i Ft(B)j Fu(=)d(Op)515 2411 y Fs(w)535 2429 y Fu(\()p Ft(F)586 2436 y Fo(\000)615 2429 y Fu(\()p Ft(a)660 2436 y Fs(0)679 2429 y Fu(\))709 2417 y(~)698 2429 y Ft(F)730 2436 y Fo(\000)759 2429 y Fu(\()p Ft(b)p Fu(\)\).)31 b(The)19 b(\014rst)h(term)e(is)h(b)q(ounded)h(using)h (\(4.5d\))228 2487 y(and)c(\(4.6b\).)23 b(F)l(or)17 b(the)f(second)h (term)e(w)o(e)i(apply)g(\(4.5d\))g(and)g(\(4.6d\))q(.)22 b(F)l(urther-)228 2550 y(more,)16 b(using)j(the)e(supp)q(ort)j(prop)q (erties)e(of)1062 2537 y(~)1051 2550 y Ft(G)1089 2557 y Fo(\000)1119 2550 y Fu(\()p Ft(b)p Fu(\),)f(w)o(e)h(can)g(apply)g (\(4.5f\))h(and)228 2608 y(\(4.6a\))e(to)f(b)q(ound)i(the)e(third)g (term.)956 2658 y Fq(40)p eop %%Page: 41 41 41 40 bop 278 213 a Fy(Pro)r(of)22 b(of)28 b Fu(\(4.6f\).)k(In)20 b(order)g(to)g(pro)o(v)o(e)f(\(4.6f\))h(w)o(e)f(c)o(ho)q(ose)h(a)g(set) g(of)g(func-)228 271 y(tions)e Ft(G)p Fu('s)f(as)h(ab)q(o)o(v)o(e)g (suc)o(h)f(that)h Ft(G)894 278 y Fo(\000)940 271 y Fp(\021)e Fu(1)i(on)g(a)f(neigh)o(b)q(ourho)q(o)q(d)j(of)e(supp)9 b Ft(F)1687 253 y Fs(2)1680 283 y Fo(\000)1709 271 y Fu(,)228 336 y(dist\(supp)446 324 y(~)434 336 y Ft(G)472 343 y Fo(\000)502 336 y Ft(;)f Fu(supp)644 324 y(~)633 336 y Ft(F)665 343 y Fs(+)694 336 y Fu(\))18 b Ft(>)g Fu(0)h(and)g(dist\(supp)1145 324 y(~)1134 336 y Ft(G)1172 343 y Fs(+)1202 336 y Ft(;)8 b Fu(supp)1344 324 y(~)1333 336 y Ft(F)1365 343 y Fo(\000)1394 336 y Fu(\))18 b Ft(>)f Fu(0.)29 b(No)o(w)19 b(w)o(e)228 399 y(write)c(with)i Ft(B)501 406 y Fs(1)534 399 y Fu(=)d(Op)643 380 y Fs(w)662 399 y Fu(\()p Ft(F)720 380 y Fs(1)713 411 y Fo(\000)742 399 y Fu(\()p Ft(a)787 406 y Fs(0)807 399 y Fu(\))837 386 y(~)826 399 y Ft(F)858 406 y Fo(\000)887 399 y Fu(\()p Ft(b)p Fu(\)\))i(and)h Ft(B)1113 406 y Fs(2)1146 399 y Fu(=)d(Op)1255 380 y Fs(w)1274 399 y Fu(\()p Ft(F)1332 380 y Fs(2)1325 411 y Fo(\000)1354 399 y Fu(\()p Ft(a)1399 406 y Fs(0)1419 399 y Fu(\))1449 386 y(~)1438 399 y Ft(F)1470 406 y Fs(+)1499 399 y Fu(\()p Ft(b)p Fu(\)\))389 486 y Ft(k)416 465 y Fm(t)431 486 y Ft(B)468 493 y Fs(1)488 486 y Ft(R)525 465 y Fm(m)556 470 y Fk(0)574 465 y Fs(+1)621 486 y Ft(B)658 493 y Fs(2)678 486 y Ft(k)705 465 y Fm(t)734 486 y Fu(=)p Ft(k)799 465 y Fm(t)814 486 y Ft(B)851 493 y Fs(1)870 486 y Ft(R)p Fu(Op)964 465 y Fs(w)984 486 y Fu(\()p Ft(G)1041 493 y Fs(+)1071 486 y Fu(\()p Ft(a)1116 493 y Fs(0)1136 486 y Fu(\)\))p Ft(R)1211 465 y Fm(m)1242 470 y Fk(0)1262 486 y Ft(B)1299 493 y Fs(2)1318 486 y Ft(k)1345 465 y Fm(t)1610 486 y Fu(\(A.3\))783 570 y(+)d Ft(k)859 549 y Fm(t)874 570 y Ft(B)911 577 y Fs(1)930 570 y Ft(R)p Fu(Op)1024 549 y Fs(w)1044 570 y Fu(\()p Ft(G)1101 577 y Fo(\000)1131 570 y Fu(\()p Ft(a)1176 577 y Fs(0)1196 570 y Fu(\))1226 557 y(~)1215 570 y Ft(G)1253 577 y Fo(\000)1283 570 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)1398 549 y Fm(m)1429 554 y Fk(0)1448 570 y Ft(B)1485 577 y Fs(2)1505 570 y Ft(k)1532 549 y Fm(t)783 653 y Fu(+)g Ft(k)859 633 y Fm(t)874 653 y Ft(B)911 660 y Fs(1)930 653 y Ft(R)p Fu(Op)1024 633 y Fs(w)1044 653 y Fu(\()p Ft(G)1101 660 y Fo(\000)1131 653 y Fu(\()p Ft(a)1176 660 y Fs(0)1196 653 y Fu(\))1226 641 y(~)1215 653 y Ft(G)1253 660 y Fs(+)1283 653 y Fu(\()p Ft(b)p Fu(\)\))p Ft(R)1398 633 y Fm(m)1429 638 y Fk(0)1448 653 y Ft(B)1485 660 y Fs(2)1505 653 y Ft(k)1532 633 y Fm(t)1547 653 y Ft(:)228 740 y Fu(T)l(o)18 b(b)q(ound)g(the)f(\014rst)h(term)e(in)h(\(A.3\))o(,) h(w)o(e)e(use)i(the)f(supp)q(ort)i(prop)q(ert)o(y)e(of)h Ft(G)1679 747 y Fs(+)1709 740 y Fu(,)228 798 y(\(4.5d\))i(and)g (\(4.6g\))q(.)32 b(The)20 b(second)f(term)g(is)g(b)q(ounded)i(using)f (the)f(separation)228 860 y(of)e(the)g(supp)q(orts)h(of)636 848 y(~)625 860 y Ft(G)663 867 y Fo(\000)710 860 y Fu(and)817 848 y(~)806 860 y Ft(F)838 867 y Fs(+)867 860 y Fu(,)f(\(4.5d\))g(and)g (\(4.6f\))q(.)23 b(Finally)16 b(for)h(the)g(third)228 922 y(term)h(w)o(e)i(com)o(bine)f(\(4.5f\),)i(\(4.6e\))f(and)h(the)f (fact)h(that)f(the)h(supp)q(orts)g(of)1672 910 y(~)1661 922 y Ft(F)1693 929 y Fo(\000)228 985 y Fu(and)334 972 y(~)323 985 y Ft(G)361 992 y Fs(+)407 985 y Fu(are)16 b(separated.)278 1043 y Fy(Pro)r(of)27 b(of)33 b Fu(\(4.6g\))q(.)45 b(W)l(e)24 b(\014nally)f(consider)h(\(4.6g\))h(with)f Ft(m)j Fu(=)g Ft(m)1594 1050 y Fs(0)1630 1043 y Fu(+)16 b(1.)228 1101 y(Here)i(w)o(e)g(c)o(ho)q(ose)i Ft(G)p Fu('s)f(as)h(b)q(efore,)f(but)g(this)g(time)e(satisfying)i(the)g (conditions)228 1159 y(dist\(supp)8 b Ft(G)472 1166 y Fs(+)502 1159 y Ft(;)g Fu(supp)h Ft(F)665 1166 y Fo(\000)694 1159 y Fu(\))14 b Ft(>)g Fu(0,)i(dist\(supp)9 b Ft(G)1078 1166 y Fo(\000)1108 1159 y Ft(;)f Fu(supp)g Ft(F)1270 1166 y Fs(+)1299 1159 y Fu(\))14 b Ft(>)g Fu(0)j(and)228 1221 y(dist\(supp)446 1209 y(~)434 1221 y Ft(G)472 1228 y Fo(\000)502 1221 y Ft(;)535 1209 y Fu(~)524 1221 y Ft(F)556 1228 y Fs(+)585 1221 y Fu(\))29 b Ft(>)f Fu(0.)47 b(In)o(tro)q(ducing)25 b(the)f(corresp)q(onding)i(partition)f(of)228 1279 y(unit)o(y)l(,)c(w)o(e)f(ha)o(v)o(e)h(to)g(b)q(ound)h(the)f(terms) f(in)g(\(A.3\))h(with)g Ft(B)1362 1286 y Fs(1)1403 1279 y Fu(=)h(Op)1520 1261 y Fs(w)1540 1279 y Fu(\()p Ft(F)1591 1286 y Fs(+)1620 1279 y Fu(\()p Ft(a)1665 1286 y Fs(0)1684 1279 y Fu(\)\))228 1342 y(and)g Ft(B)365 1349 y Fs(2)409 1342 y Fu(=)h(Op)527 1324 y Fs(w)547 1342 y Fu(\()p Ft(F)598 1349 y Fo(\000)627 1342 y Fu(\()p Ft(a)672 1349 y Fs(0)691 1342 y Fu(\))721 1329 y(~)710 1342 y Ft(F)742 1349 y Fs(+)771 1342 y Fu(\()p Ft(b)p Fu(\)\).)38 b(Since)22 b(dist)o(\(supp)9 b Ft(G)1279 1349 y Fs(+)1309 1342 y Ft(;)f Fu(supp)h Ft(F)1472 1349 y Fo(\000)1501 1342 y Fu(\))24 b Ft(>)f Fu(0,)h(w)o(e)228 1400 y(can)15 b(use)g(\(4.5b\))g (and)g(\(4.6g\))h(to)f(b)q(ound)h(the)e(\014rst)h(term.)k(F)l(or)c(the) g(second)g(term)228 1462 y(w)o(e)f(use)h(\(4.5b\))q(,)f(dist\(supp)755 1449 y(~)744 1462 y Ft(G)782 1469 y Fo(\000)812 1462 y Ft(;)845 1449 y Fu(~)834 1462 y Ft(F)866 1469 y Fs(+)895 1462 y Fu(\))g Ft(>)f Fu(0)j(and)f(\(4.6f\).)21 b(F)l(or)15 b(the)g(third)f(term)g(w)o(e)228 1520 y(apply)i(the)g(supp)q(ort)h (condition)f(on)h Ft(G)946 1527 y Fo(\000)976 1520 y Fu(,)f(\(4.5g\))q(,)f(and)i(\(4.6e\).)383 1633 y Fv(Appendix)g Fu(B.)26 b Fv(Unif)o(ormity)18 b(of)g(H)1133 1629 y(\177)1131 1633 y(ormander)f(metric)278 1720 y Fu(In)d(this)g(app)q(endix)h(w)o(e) f(v)o(erify)f(that)i(the)f(metric)e Ft(g)1203 1727 y Fm(E)1247 1720 y Fu(=)1309 1701 y Fm(dx)1347 1689 y Fk(2)p 1304 1709 65 2 v 1304 1738 a Fo(h)p Fm(x)p Fo(i)1352 1728 y Fk(2)1381 1720 y Fu(+)1459 1699 y Fm(d\030)1494 1687 y Fk(2)p 1432 1709 108 2 v 1432 1738 a Fm(f)1449 1744 y Fi(E)1475 1738 y Fs(\()p Fm(x)p Fs(\))1523 1728 y Fk(2)1559 1720 y Fu(satis\014es)228 1785 y(the)j(estimates)e(in)i (the)g(de\014nition)g(of)g(a)h(H\177)-24 b(ormander)14 b(metric)f Fh(uniformly)i Fu(in)f(the)228 1843 y(parameter)h Ft(E)i Fp(2)d Fu(\(0)p Ft(;)8 b Fu(1],)16 b(cf.)21 b(Lemma)14 b(4.3.)228 1932 y Fh(Pr)n(o)n(of.)19 b Fu(First)d(w)o(e)f(pro)o(v)o(e)h (that)h(for)f(some)f Ft(C)j(>)13 b Fu(0)k(indep)q(enden)o(t)e(of)i Ft(E)s(;)8 b(x)16 b Fu(and)h Ft(y)714 2021 y(f)5 b Fu(\()p Ft(x)p Fu(\))p 714 2044 96 2 v 715 2089 a Ft(f)g Fu(\()p Ft(y)r Fu(\))827 2055 y Fp(\024)14 b Ft(C)927 1985 y Fl(\022)963 2055 y Fu(1)e(+)1054 2021 y Fp(h)p Ft(y)r Fp(i)p 1053 2044 67 2 v 1053 2089 a(h)p Ft(x)p Fp(i)1124 1985 y Fl(\023)1161 1996 y Fm(\026=)p Fs(2)1228 2055 y Ft(:)370 b Fu(\(B.1\))228 2170 y(Supp)q(ose)17 b Fp(h)p Ft(x)p Fp(i)485 2152 y Fo(\000)p Fm(\026)550 2170 y Fp(\024)d Ft(E)s Fu(.)21 b(Then)372 2273 y Ft(f)5 b Fu(\()p Ft(x)p Fu(\))p 372 2295 96 2 v 373 2341 a Ft(f)g Fu(\()p Ft(y)r Fu(\))486 2307 y Fp(\024)539 2216 y Fl(s)p 588 2216 472 2 v 593 2273 a Ft(\024)621 2253 y Fo(\000)p Fs(2)621 2285 y(0)669 2273 y Ft(E)14 b Fu(+)d(\(1)g Fp(\000)g Ft(\026=)p Fu(2\))968 2259 y Fo(\000)p Fs(1)1016 2273 y Ft(E)p 593 2295 462 2 v 767 2343 a(\024)795 2323 y Fo(\000)p Fs(2)795 2356 y(0)842 2343 y Ft(E)1074 2307 y Fu(=)1126 2216 y Fl(s)p 1176 2216 394 2 v 1181 2273 a Ft(\024)1209 2253 y Fo(\000)p Fs(2)1209 2285 y(0)1267 2273 y Fu(+)g(\(1)g Fp(\000)g Ft(\026=)p Fu(2\))1516 2259 y Fo(\000)p Fs(1)p 1181 2295 384 2 v 1335 2343 a Ft(\024)1363 2323 y Fo(\000)p Fs(2)1363 2356 y(0)1569 2307 y Ft(:)228 2431 y Fu(On)16 b(the)g(other)g(hand)h(if)f Fp(h)p Ft(x)p Fp(i)753 2413 y Fo(\000)p Fm(\026)818 2431 y Fp(\025)e Ft(E)s Fu(,)i(then)384 2535 y Ft(f)5 b Fu(\()p Ft(x)p Fu(\))p 384 2557 96 2 v 385 2603 a Ft(f)g Fu(\()p Ft(y)r Fu(\))498 2569 y Fp(\024)550 2477 y Fl(s)p 600 2477 629 2 v 605 2535 a Ft(\024)633 2515 y Fo(\000)p Fs(2)633 2547 y(0)680 2535 y Fp(h)p Ft(x)p Fp(i)746 2521 y Fo(\000)p Fm(\026)809 2535 y Fu(+)11 b(\(1)g Fp(\000)g Ft(\026=)p Fu(2\))1058 2521 y Fo(\000)p Fs(1)1106 2535 y Fp(h)p Ft(x)p Fp(i)1172 2521 y Fo(\000)p Fm(\026)p 605 2557 619 2 v 733 2603 a Fu(\(1)g Fp(\000)g Ft(\026=)p Fu(2\))933 2589 y Fo(\000)p Fs(1)981 2603 y Fp(h)p Ft(y)r Fp(i)1045 2589 y Fo(\000)p Fm(\026)1242 2569 y Fu(=)j Ft(C)1341 2499 y Fl(\022)1384 2535 y Fp(h)p Ft(y)r Fp(i)p 1382 2557 67 2 v 1382 2603 a(h)p Ft(x)p Fp(i)1454 2499 y Fl(\023)1491 2510 y Fm(\026=)p Fs(2)1557 2569 y Ft(:)956 2658 y Fq(41)p eop %%Page: 42 42 42 41 bop 278 213 a Fu(T)l(o)20 b(pro)o(v)o(e)g(the)g(slo)o(w)g(v)m (ariation,)h(let)f(us)h(assume)e(that)i(1)p Ft(=C)1418 220 y Fs(1)1459 213 y Ft(<)g Fu(1)p Ft(=)p Fu(4.)34 b(The)228 271 y(inequalit)o(y)14 b Ft(g)477 278 y Fs(\()p Fm(x;\030)q Fs(\))553 271 y Fu(\(\()p Ft(y)r(;)8 b(\021)r Fu(\)\))13 b Fp(\024)h Fu(1)p Ft(=C)852 278 y Fs(1)889 271 y Fu(implies)f Fp(j)p Ft(y)r Fp(j)1108 253 y Fs(2)1141 271 y Fp(\024)h(h)p Ft(x)p Fp(i)1260 253 y Fs(2)1280 271 y Ft(=C)1339 278 y Fs(1)1375 271 y Fu(and)j(therefore)796 352 y(2)p 796 374 25 2 v 796 420 a(3)840 386 y Fp(\024)940 352 y(h)p Ft(x)p Fp(i)p 897 374 153 2 v 897 420 a(h)p Ft(x)11 b Fu(+)g Ft(y)r Fp(i)1068 386 y(\024)j Fu(2)p Ft(:)453 b Fu(\(B.2\))228 497 y(No)o(w)16 b(using)h(\(B.1\))f(and)h(\(B.2\))o(,) 361 616 y Ft(g)384 624 y Fs(\()p Fm(x;\030)q Fs(\)+\()p Fm(y)q(;\021)q Fs(\))563 616 y Fu(\()p Ft(v)r Fu(\))c(=)760 582 y Ft(v)786 564 y Fs(2)784 594 y Fm(x)p 697 604 172 2 v 697 650 a Fp(h)p Ft(x)e Fu(+)g Ft(y)r Fp(i)849 636 y Fs(2)885 616 y Fu(+)1016 577 y Ft(v)1042 559 y Fs(2)1040 590 y Fm(\030)p 938 604 201 2 v 938 650 a Ft(f)5 b Fu(\()p Ft(x)12 b Fu(+)f Ft(y)r Fu(\))1120 636 y Fs(2)640 762 y Fp(\024)i Fu(sup)774 706 y Fl(n)816 691 y(\022)900 728 y Fp(h)p Ft(x)p Fp(i)p 857 750 153 2 v 857 796 a(h)p Ft(x)f Fu(+)f Ft(y)r Fp(i)1014 691 y Fl(\023)1051 702 y Fs(2)1079 762 y Ft(;)1101 691 y Fl(\022)1185 728 y Ft(f)5 b Fu(\()p Ft(x)p Fu(\))p 1143 750 181 2 v 1143 796 a Ft(f)g Fu(\()p Ft(x)11 b Fu(+)g Ft(y)r Fu(\))1328 691 y Fl(\023)1365 702 y Fs(2)1393 706 y Fl(o)1426 762 y Ft(g)1449 769 y Fs(\()p Fm(x;\030)q Fs(\))1526 762 y Fu(\()p Ft(v)r Fu(\))640 901 y Fp(\024)i Fu(sup)774 846 y Fl(n)807 901 y Fu(4)p Ft(;)8 b(C)900 831 y Fl(\022)942 867 y Fu(5)p 942 889 25 2 v 942 935 a(2)971 831 y Fl(\023)1008 842 y Fm(\026)1040 846 y Fl(o)1073 901 y Ft(g)1096 909 y Fs(\()p Fm(x;\030)q Fs(\))1172 901 y Fu(\()p Ft(v)r Fu(\))p Ft(:)278 1011 y Fu(The)16 b(dual)g(metric)e(is)i(giv)o(en)f(b)o (y)743 1092 y Ft(g)768 1072 y Fm(\033)805 1092 y Fu(=)f Ft(f)886 1072 y Fs(2)906 1092 y Ft(dx)959 1072 y Fs(2)990 1092 y Fu(+)d Fp(h)p Ft(x)p Fp(i)1105 1072 y Fs(2)1125 1092 y Ft(d\030)1173 1072 y Fs(2)1194 1092 y Ft(:)228 1174 y Fu(Since)k(clearly)762 1235 y Fp(h)p Ft(x)p Fp(i)828 1215 y Fo(\000)p Fs(2)889 1235 y Fp(\024)f(h)p Ft(x)p Fp(i)1008 1215 y Fo(\000)p Fm(\026)1073 1235 y Fp(\024)g Ft(f)1155 1215 y Fs(2)1175 1235 y Ft(;)228 1307 y Fu(w)o(e)i(see)f (that)i(the)f(uncertain)o(t)o(y)f(principle)f(is)i(satis\014ed.)278 1365 y(Finally)l(,)e(w)o(e)i(pro)o(v)o(e)f(that)i(the)f(metric)e(is)i (temp)q(erate.)k(Let)c(us)h(\014rst)g(consider)228 1423 y(the)f(case)g Fp(j)p Ft(x)p Fp(j)d(\025)h(j)p Ft(y)r Fp(j)p Fu(.)20 b(Then)578 1532 y Ft(g)601 1540 y Fs(\()p Fm(x;\030)q Fs(\))678 1532 y Fu(\()p Ft(v)r Fu(\))p 578 1556 164 2 v 578 1602 a Ft(g)601 1610 y Fs(\()p Fm(y)q(;\021)q Fs(\))678 1602 y Fu(\()p Ft(v)r Fu(\))760 1568 y(=)836 1499 y Fm(v)854 1487 y Fk(2)853 1507 y Fi(x)p 822 1511 65 2 v 822 1540 a Fo(h)p Fm(x)p Fo(i)870 1531 y Fk(2)902 1523 y Fu(+)981 1491 y Fm(v)999 1479 y Fk(2)998 1503 y Fi(\030)p 956 1511 86 2 v 956 1540 a Fm(f)t Fs(\()p Fm(x)p Fs(\))1025 1531 y Fk(2)p 817 1556 230 2 v 836 1588 a Fm(v)854 1578 y Fk(2)853 1596 y Fi(x)p 823 1600 64 2 v 823 1629 a Fo(h)p Fm(y)q Fo(i)869 1619 y Fk(2)902 1612 y Fu(+)981 1585 y Fm(v)998 1591 y Fi(\030)p 956 1600 84 2 v 956 1629 a Fm(f)t Fs(\()p Fm(y)q Fs(\))1023 1619 y Fk(2)1065 1568 y Fu(=)1122 1529 y Ft(a)1148 1511 y Fs(2)1168 1529 y Ft(v)1194 1511 y Fs(2)1192 1541 y Fm(x)1224 1529 y Fu(+)11 b Ft(b)1294 1511 y Fs(2)1314 1529 y Ft(v)1340 1511 y Fs(2)1338 1542 y Fm(\030)p 1122 1556 237 2 v 1122 1602 a Ft(c)1143 1588 y Fs(2)1163 1602 y Ft(v)1189 1588 y Fs(2)1187 1614 y Fm(x)1220 1602 y Fu(+)g Ft(d)1294 1588 y Fs(2)1314 1602 y Ft(v)1340 1585 y Fs(2)1338 1616 y Fm(\030)1364 1568 y Ft(;)228 1712 y Fu(where)17 b Ft(c)391 1694 y Fs(2)427 1712 y Fp(\025)f Ft(a)508 1694 y Fs(2)527 1712 y Fu(,)h Ft(b)579 1694 y Fs(2)615 1712 y Fp(\025)f Ft(d)695 1694 y Fs(2)715 1712 y Fu(.)25 b(One)17 b(no)o(w)h(sees)f(that)h(the)g(function)f Ft(s)f Fp(7!)1555 1693 y Fm(a)1574 1681 y Fk(2)1591 1693 y Fs(+)p Fm(b)1633 1681 y Fk(2)1651 1693 y Fm(s)p 1555 1701 113 2 v 1555 1729 a(c)1570 1720 y Fk(2)1587 1729 y Fs(+)p Fm(d)1632 1720 y Fk(2)1650 1729 y Fm(s)1689 1712 y Fu(is)228 1774 y(increasing)h(on)h([0)p Ft(;)8 b Fp(1)p Fu(\).)25 b(Therefore,)17 b(using)h(also)h(\(B.1\))o(,)f(w)o (e)f(infer)g(that)h(for)g(all)228 1832 y Ft(v)d Fp(6)p Fu(=)f(0)338 1915 y Ft(g)361 1923 y Fs(\()p Fm(x;\030)q Fs(\))438 1915 y Fu(\()p Ft(v)r Fu(\))p 338 1939 164 2 v 338 1984 a Ft(g)361 1992 y Fs(\()p Fm(y)q(;\021)q Fs(\))438 1984 y Fu(\()p Ft(v)r Fu(\))520 1950 y Fp(\024)580 1917 y Ft(b)601 1899 y Fs(2)p 578 1939 46 2 v 578 1984 a Ft(d)603 1970 y Fs(2)642 1950 y Fu(=)699 1917 y Ft(f)5 b Fu(\()p Ft(y)r Fu(\))792 1899 y Fs(2)p 698 1939 115 2 v 698 1984 a Ft(f)g Fu(\()p Ft(x)p Fu(\))793 1970 y Fs(2)832 1950 y Fp(\024)13 b Ft(C)923 1930 y Fs(2)951 1880 y Fl(\022)988 1950 y Fu(1)e(+)1077 1917 y Fp(h)p Ft(x)p Fp(i)p 1077 1939 67 2 v 1078 1984 a(h)p Ft(y)r Fp(i)1149 1880 y Fl(\023)1185 1891 y Fm(\026)1222 1950 y Fp(\024)j Ft(C)1314 1930 y Fs(2)1341 1880 y Fl(\022)1378 1950 y Fu(1)e(+)1467 1917 y Fp(h)p Ft(x)p Fp(i)p 1467 1939 V 1468 1984 a(h)p Ft(y)r Fp(i)1539 1880 y Fl(\023)1576 1891 y Fs(2)1604 1950 y Ft(:)228 2066 y Fu(A)k(similar)e(argumen)o(t)h (sho)o(ws)i(that)g(in)f(the)g(case)g Fp(j)p Ft(x)p Fp(j)d Ft(<)h Fp(j)p Ft(y)r Fp(j)745 2143 y Ft(g)768 2151 y Fs(\()p Fm(x;\030)q Fs(\))845 2143 y Fu(\()p Ft(v)r Fu(\))p 745 2167 164 2 v 745 2213 a Ft(g)768 2221 y Fs(\()p Fm(y)q(;\021)q Fs(\))845 2213 y Fu(\()p Ft(v)r Fu(\))927 2179 y Fp(\024)985 2145 y Ft(a)1011 2127 y Fs(2)p 985 2167 46 2 v 987 2213 a Ft(c)1008 2199 y Fs(2)1049 2179 y Fu(=)1107 2145 y Fp(h)p Ft(y)r Fp(i)1171 2127 y Fs(2)p 1106 2167 87 2 v 1106 2213 a Fp(h)p Ft(x)p Fp(i)1172 2199 y Fs(2)1197 2179 y Ft(:)228 2303 y Fu(W)l(e)i(need)g(to)g(\014nd)594 2290 y(~)583 2303 y Ft(C)s(;)8 b(N)19 b(>)14 b Fu(0)j(suc)o(h)f(that) 372 2413 y(max)471 2358 y Fl(n)509 2379 y Fp(h)p Ft(x)p Fp(i)575 2361 y Fs(2)p 509 2402 V 510 2447 a Fp(h)p Ft(y)r Fp(i)574 2433 y Fs(2)600 2413 y Ft(;)628 2379 y Fp(h)p Ft(y)r Fp(i)692 2361 y Fs(2)p 627 2402 V 627 2447 a Fp(h)p Ft(x)p Fp(i)693 2433 y Fs(2)718 2358 y Fl(o)765 2413 y Fp(\024)829 2401 y Fu(~)818 2413 y Ft(C)856 2358 y Fl(\020)886 2413 y Fu(1)11 b(+)g Ft(g)995 2393 y Fm(\033)993 2425 y Fs(\()p Fm(x;\030)q Fs(\))1070 2413 y Fu(\(\()p Ft(y)r(;)d(\021)r Fu(\))i Fp(\000)h Fu(\()p Ft(x;)d(\030)r Fu(\)\))1391 2358 y Fl(\021)1421 2369 y Fm(N)1454 2413 y Ft(:)144 b Fu(\(B.3\))228 2522 y(Clearly)371 2604 y Ft(g)396 2583 y Fm(\033)394 2616 y Fs(\()p Fm(x;\030)q Fs(\))470 2604 y Fu(\(\()p Ft(y)r(;)8 b(\021)r Fu(\))i Fp(\000)h Fu(\()p Ft(x;)d(\030)r Fu(\)\))14 b Fp(\025)g Ft(f)5 b Fu(\()p Ft(x)p Fu(\))953 2583 y Fs(2)972 2604 y Fp(j)p Ft(y)13 b Fp(\000)e Ft(x)p Fp(j)1115 2583 y Fs(2)1148 2604 y Fp(\025)i Ft(c)p Fp(h)p Ft(x)p Fp(i)1287 2583 y Fo(\000)p Fm(\026)1339 2604 y Fu(\()p Fp(j)p Ft(x)p Fp(j)d(\000)h(j)p Ft(y)r Fp(j)p Fu(\))1547 2583 y Fs(2)1566 2604 y Ft(:)956 2658 y Fq(42)p eop %%Page: 43 43 43 42 bop 228 213 a Fu(If)16 b Fp(j)p Ft(x)p Fp(j)d(\025)g Fu(2)p Fp(j)p Ft(y)r Fp(j)p Fu(,)j(then)631 296 y Fp(h)p Ft(x)p Fp(i)697 275 y Fo(\000)p Fm(\026)748 296 y Fu(\()p Fp(j)p Ft(x)p Fp(j)11 b(\000)f(j)p Ft(y)r Fp(j)p Fu(\))956 275 y Fs(2)989 296 y Fp(\025)k Fu(4)1066 275 y Fo(\000)p Fs(1)1113 296 y Fp(h)p Ft(x)p Fp(i)1179 275 y Fo(\000)p Fm(\026)1230 296 y Fp(j)p Ft(x)p Fp(j)1286 275 y Fs(2)1306 296 y Ft(:)228 388 y Fu(Using)k(the)f(trivial)g(b)q(ound)756 365 y Fo(h)p Fm(x)p Fo(i)804 353 y Fk(2)p 756 377 65 2 v 757 406 a Fo(h)p Fm(y)q Fo(i)803 396 y Fk(2)843 388 y Fp(\024)f(h)p Ft(x)p Fp(i)964 370 y Fs(2)1002 388 y Fu(w)o(e)i(conclude)f(that)i(in)e(this)h(case)g(an)o(y)228 457 y Ft(N)h Fp(\025)13 b Fu(2)p Ft(=)p Fu(\(2)g Fp(\000)d Ft(\026)p Fu(\))17 b(su\016ces.)k(If)16 b Fp(j)p Ft(x)p Fp(j)d(\024)g Fu(2)935 439 y Fo(\000)p Fs(1)983 457 y Fp(j)p Ft(y)r Fp(j)p Fu(,)i(then)633 541 y Fp(h)p Ft(x)p Fp(i)699 520 y Fo(\000)p Fm(\026)750 541 y Fu(\()p Fp(j)p Ft(x)p Fp(j)c(\000)f(j)p Ft(y)r Fp(j)p Fu(\))958 520 y Fs(2)991 541 y Fp(\025)k Fu(4)1068 520 y Fo(\000)p Fs(1)1115 541 y Fp(h)p Ft(y)r Fp(i)1179 520 y Fo(\000)p Fm(\026)1230 541 y Fp(j)p Ft(y)r Fp(j)1284 520 y Fs(2)1303 541 y Ft(;)228 623 y Fu(and)h(again)h Ft(N)j Fp(\025)13 b Fu(2)p Ft(=)p Fu(\(2)8 b Fp(\000)g Ft(\026)p Fu(\))16 b(su\016ces.)21 b(If,)14 b(on)h(the)f(other)h(hand,)h(2)1449 605 y Fo(\000)p Fs(1)1496 623 y Fp(j)p Ft(y)r Fp(j)d Ft(<)h Fp(j)p Ft(x)p Fp(j)f Ft(<)228 691 y Fu(2)p Fp(j)p Ft(y)r Fp(j)p Fu(,)i(then)h(max)o Fp(f)567 668 y Fo(h)p Fm(x)p Fo(i)615 656 y Fk(2)p 567 680 V 568 708 a Fo(h)p Fm(y)q Fo(i)614 699 y Fk(2)637 691 y Ft(;)664 668 y Fo(h)p Fm(y)q Fo(i)710 656 y Fk(2)p 663 680 V 663 708 a Fo(h)p Fm(x)p Fo(i)711 699 y Fk(2)733 691 y Fp(g)e(\024)f Fu(4.)278 758 y(W)l(e)18 b(ha)o(v)o(e)g(pro)o(v)o(ed)g(\(B.3\))g(and)h(hence)f (that)h(the)g(metric)d(is)i(temp)q(erate)f(uni-)228 816 y(formly)d(in)i Ft(E)s Fu(.)21 b(That)c(\014nishes)g(the)f(pro)q(of)h (of)g(the)f(\014rst)g(part)h(of)f(Lemma)e(4.3.)278 875 y(The)f(second)g(part)g(of)h(Lemma)c(4.3)k(for)f Ft(m)h Fu(=)f Fp(h)p Ft(x)p Fp(i)h Fu(or)f Ft(m)h Fu(=)f Ft(f)1377 882 y Fm(E)1420 875 y Fu(follo)o(ws)g(readily)228 936 y(from)j(\(B.1\),)h(\(B.2\))g(and)h(\(B.3\))o(.)25 b(The)18 b(statemen)o(t)e(for)i Ft(m)d Fu(=)h Fp(h)p Ft(\030)r Fp(i)j Fu(or)f Ft(m)e Fu(=)g Fp(h)1668 914 y Fm(\030)p 1655 925 43 2 v 1655 953 a(f)1672 959 y Fi(E)1703 936 y Fp(i)228 998 y Fu(ma)o(y)d(b)q(e)i(v)o(eri\014ed)e(along)j(the)e (same)g(line;)f(w)o(e)i(omit)e(the)h(pro)q(of.)22 b(The)15 b(statemen)o(t)228 1057 y(for)k(pro)q(ducts)i(of)f(p)q(o)o(w)o(ers)f (is)g(a)h(general)f(prop)q(ert)o(y)h(for)f(uniformly)f(temp)q(erate)228 1115 y(w)o(eigh)o(t)d(functions.)p 1687 1173 2 33 v 1689 1142 30 2 v 1689 1173 V 1718 1173 2 33 v 830 1284 a Fv(References)228 1362 y Fx([Ag75])55 b(S.)10 b(Agmon,)f Fr(Sp)n(e)n(ctr)n(al)j(pr)n(op)n (erties)f(of)g(Schr\177)-21 b(odinger)13 b(op)n(er)n(ators)e(and)i(sc)n (attering)e(the-)401 1412 y(ory)p Fx(,)i(Ann.)h(Scula)f(Norm.)f(Sup.)i (Pisa)g Fd(2)p Fx(,)f(no.)g(2)h(\(1975\),)e(152{218.)228 1462 y([Bea77])39 b(R.)20 b(Beals,)j Fr(Char)n(acterization)d(of)i (pseudo)n(di\013er)n(ential)g(op)n(er)n(ators)f(and)h(applic)n(a-)401 1512 y(tions)p Fx(,)e(Duk)o(e)g(Math.)f(J.)h Fd(44)p Fx(,)g(no.)f(1)g(\(1977\),)h(45{57;)h Fr(c)n(orr)n(e)n(ction)h Fx(Duk)o(e)e(Math.)401 1562 y(J.)13 b Fd(46)p Fx(,)h(no.)f(1)g (\(1979\),)g(215.)228 1612 y([Bea81])39 b(R.)10 b(Beals,)i Fr(Weighte)n(d)h(distribution)f(sp)n(ac)n(es)h(and)g(pseudo)n(di\013er) n(ential)g(op)n(er)n(ators)p Fx(,)e(J.)401 1661 y(Analyse)j(Math.)f Fd(39)h Fx(\(1981\),)e(131{187.)228 1711 y([BSS85])32 b(D.)12 b(Boll)o(\023)-20 b(e,)13 b(F.)g(Gesztesy)j(and)d(W.)g(Sc)o(h)o (w)o(eiger,)g Fr(Sc)n(attering)i(the)n(ory)f(for)g(long-r)n(ange)401 1761 y(systems)h(at)f(thr)n(eshold)p Fx(,)f(J.)h(Math.)g(Ph)o(ys.)f Fd(26)p Fx(,)g(no.)h(7)f(\(1985\),)g(1661{1674.)228 1811 y([BoCh94])19 b(J.M.)14 b(Bon)o(y)h(and)g(J.Y.)f(Chemin,)f Fr(Esp)n(ac)n(es)j(fonctionnels)g(asso)n(ci)o(\022)-20 b(es)16 b(au)g(c)n(alcul)g(de)401 1861 y(Weyl-H\177)-21 b(ormander)p Fx(,)13 b(Bull.)g(So)q(c.)h(Math.)f(F)m(rance)i Fd(122)p Fx(,)e(no.)g(1)g(\(1994\),)g(77{118.)228 1910 y([CyP)o(e84])20 b(H.L.)c(Cycon)i(and)g(P)m(.A.)f(P)o(erry)m(,)h Fr(L)n(o)n(c)n(al)h(time-de)n(c)n(ay)f(of)g(high)h(ener)n(gy)g(sc)n (attering)401 1960 y(states)14 b(for)h(the)g(Schr\177)-21 b(o)n(dinger)15 b(e)n(quation)p Fx(,)f(Math.)f(Z.)h Fd(188)f Fx(\(1984\),)g(125-142.)228 2010 y([DeG)o(\023)-20 b(e97])19 b(J.)14 b(Derezi)q(\023)-22 b(nski)15 b(and)f(C.)g(G)o(\023)-20 b(erard,)14 b Fr(Sc)n(attering)h(the)n(ory)g(of)g(classic)n(al)f(and)i (quantum)401 2060 y Fc(N)5 b Fr(-p)n(article)17 b(systems)p Fx(,)g(T)m(exts)h(and)f(Monographs)g(in)g(Ph)o(ysics,)h(Springer-V)m (erlag,)401 2110 y(Berlin,)13 b(1997.)228 2159 y([G)o(\023)-20 b(er93])40 b(C.)11 b(G)o(\023)-20 b(erard,)12 b Fr(Asymptotic)h(c)n (ompleteness)g(for)g Fx(3)p Fr(-p)n(article)e(systems)p Fx(,)h(In)o(v)o(en)o(t.)g(Math.)401 2209 y Fd(114)h Fx(\(1993\),)g (333{397.)228 2259 y([GIS96])36 b(C.)18 b(G)o(\023)-20 b(erard,)19 b(H.)f(Isozaki,)h(and)f(E.)g(Skibsted,)i Fc(N)5 b Fr(-b)n(o)n(dy)19 b(r)n(esolvent)g(estimates)p Fx(,)f(J.)401 2309 y(Math.)13 b(So)q(c.)h(Japan)g Fd(48)p Fx(,)f(no.)g(1)h(\(1996\),)e(135{160.)228 2359 y([GiT)m(r01])19 b(D.)e(Gilbarg)f(and)i(N.S.)f(T)m(rudinger,)h Fr(El)r(liptic)f(p)n (artial)h(di\013er)n(ential)g(e)n(quations)h(of)401 2409 y(se)n(c)n(ond)c(or)n(der)p Fx(,)e(Springer-V)m(erlag,)g(Berlin,)h (2001,)e(Reprin)o(t)i(of)f(the)i(1998)e(edition.)228 2458 y([Her91])42 b(I.)17 b(Herbst,)j Fr(Sp)n(e)n(ctr)n(al)e(and)h(sc)n (attering)f(the)n(ory)g(for)g(Schr\177)-21 b(o)n(dinger)19 b(op)n(er)n(ators)f(with)401 2508 y(p)n(otentials)c(indep)n(endent)h (of)f Fb(j)p Fc(x)p Fb(j)p Fx(,)d(Amer.)h(J.)g(Math.)h Fd(113)p Fx(,)f(no.)g(3)h(\(1991\),)e(509{565.)228 2558 y([H\177)-21 b(or94])39 b(L.)17 b(H\177)-21 b(ormander,)19 b Fr(The)g(analysis)g(of)g(line)n(ar)f(p)n(artial)g(di\013er)n(ential)h (op)n(er)n(ators.)f(III)p Fx(,)401 2608 y(Springer-V)m(erlag,)13 b(Berlin,)g(1985.)956 2658 y Fq(43)p eop %%Page: 44 44 44 43 bop 228 213 a Fx([Iso85])55 b(H.)16 b(Isozaki,)g Fr(Di\013er)n(entiability)h(of)g(gener)n(alize)n(d)g(Fourier)g(tr)n (ansforms)f(asso)n(ciate)n(d)401 262 y(with)e(Schr\177)-21 b(o)n(dinger)16 b(op)n(er)n(ators)p Fx(,)d(J.)h(Math.)g(Ky)o(oto)g (Univ.)f Fd(25)p Fx(,)h(no.)f(4)h(\(1985\),)f(789{)401 312 y(806.)228 362 y([IsKi85])32 b(H.)15 b(Isozaki,)g(J.Kitada,)g Fr(Sc)n(attering)h(matric)n(es)g(for)g(two-b)n(o)n(dy)g(Schr\177)-21 b(o)n(dinger)16 b(op)n(er-)401 412 y(ators)p Fx(,)e(Scien)o(ti\014c)i (pap)q(ers)g(of)e(the)i(College)e(of)h(Arts)h(and)f(Sciences,)h(T)m (oky)o(o)e(Univ.)401 462 y Fd(35)f Fx(\(1985\),)g(81{107.)228 511 y([Jen85])45 b(A.)16 b(Jensen,)i Fr(Pr)n(op)n(agation)g(estimates)f (for)f(Schr\177)-21 b(o)n(dinger-typ)n(e)18 b(op)n(er)n(ators)p Fx(,)e(T)m(rans.)401 561 y(Amer.)d(Math.)g(So)q(c.)h Fd(291)p Fx(,)f(no.)g(1)g(\(1985\),)g(129{144.)228 611 y([Jen89])45 b(A.)22 b(Jensen,)j Fr(High)e(ener)n(gy)g(r)n(esolvent)f (estimates)g(for)g(gener)n(alize)n(d)g(many-b)n(o)n(dy)401 661 y(Schr\177)-21 b(o)n(dinger)15 b(op)n(er)n(ators)p Fx(,)e(Publ.)g(RIMS,)g(Ky)o(oto)h(Univ.)f Fd(25)g Fx(\(1989\),)g (155{167.)228 711 y([JenKat79])20 b(A.)f(Jensen)i(and)e(T.)f(Kato,)i Fr(Sp)n(e)n(ctr)n(al)f(pr)n(op)n(erties)g(of)h(Schr\177)-21 b(o)n(dinger)20 b(op)n(er)n(ators)401 761 y(and)f(time-de)n(c)n(ay)f (of)g(the)g(wave)g(functions)p Fx(,)g(Duk)o(e)g(Math.)f(J.)g Fd(46)p Fx(,)g(no.)g(3)g(\(1979\),)401 810 y(583{611.)228 860 y([JenNen01])k(A.)15 b(Jensen)j(and)e(G.)f(Nenciu,)i Fr(A)f(uni\014e)n(d)i(appr)n(o)n(ach)g(to)f(r)n(esolvent)f(exp)n (ansions)401 910 y(at)f(thr)n(esholds)p Fx(,)e(Rev.)g(Math.)g(Ph)o (ysics,)i Fd(13)p Fx(,)e(no.)g(6)g(\(2001\),)g(717{754.)228 960 y([JerKen85])21 b(D.)16 b(Jerison)h(and)f(C.E.)g(Kenig,)g Fr(Unique)i(c)n(ontinuation)g(and)g(absenc)n(e)g(of)f(p)n(osi-)401 1010 y(tive)e(eigenvalues)i(for)e(Schr\177)-21 b(o)n(dinger)16 b(op)n(er)n(ators)p Fx(,)e(Ann.)h(of)f(Math.)h(\(2\))g Fd(121)p Fx(,)f(no.)g(3)401 1059 y(\(1985\),463{494,)c(With)j(an)h(app) q(endix)g(b)o(y)g(E.)f(M.)h(Stein.)228 1109 y([JMP84])20 b(A.)14 b(Jensen,)610 1099 y(\023)606 1109 y(E.)h(Mourre,)h(and)f(P)m (.)f(P)o(erry)m(,)h Fr(Multiple)h(c)n(ommutator)g(estimates)f(and)401 1159 y(r)n(esolvent)c(smo)n(othness)i(in)e(quantum)i(sc)n(attering)e (the)n(ory)p Fx(,)g(Ann.)f(Inst.)h(H.)f(P)o(oincar)o(\023)-20 b(e)401 1209 y(Ph)o(ys.)14 b(Th)o(\023)-20 b(eor.)13 b Fd(41)p Fx(,)h(no.)f(2)g(\(1984\),)g(207{225.)228 1259 y([Kat95])38 b(T.)10 b(Kato,)h Fr(Perturb)n(ation)h(the)n(ory)g(for)g (line)n(ar)f(op)n(er)n(ators)p Fx(,)g(Springer-V)m(erlag,)f(Berlin,)401 1308 y(1995,)i(Reprin)o(t)i(of)f(the)i(1980)e(edition.)228 1358 y([Kit84])47 b(K.)23 b(Kitada,)h Fr(Time-de)n(c)n(ay)f(of)h(the)f (high)h(ener)n(gy)f(p)n(art)g(of)g(the)h(solution)f(for)g(a)401 1408 y(Schr\177)-21 b(o)n(dinger)16 b(e)n(quation)p Fx(,)g(J.)f(F)m (ac.)g(Sci.)g(Univ.)f(T)m(oky)o(o)g(Sect.)i(IA)g(Math.)f Fd(31)p Fx(,)g(no.)f(1)401 1458 y(\(1984\),)e(109{146.)228 1508 y([Mou81])405 1497 y(\023)401 1508 y(E.)e(Mourre,)h Fr(A)o(bsenc)n(e)g(of)g(singular)h(c)n(ontinuous)g(sp)n(e)n(ctrum)f (for)g(c)n(ertain)g(selfadjoint)401 1558 y(op)n(er)n(ators)p Fx(,)i(Comm.)d(Math.)j(Ph)o(ys.)h Fd(78)p Fx(,)f(no.)g(3)h (\(1980/81\),)e(391{408.)228 1607 y([M\034Sk02])19 b(J.S.)14 b(M\034ller)h(and)g(E.)g(Skibsted,)g Fr(Sp)n(e)n(ctr)n(al)h(the)n(ory)f (of)h(time-p)n(erio)n(dic)f(many-b)n(o)n(dy)401 1657 y(systems)p Fx(,)e(MaPh)o(ySto)h(Preprin)o(t)g(\(2002\),)f(no.)g (2002-26.)228 1707 y([Nak94])33 b(S.)22 b(Nak)n(am)o(ura,)g Fr(L)n(ow)g(ener)n(gy)h(asymptotics)g(for)g(Schr\177)-21 b(o)n(dinger)23 b(op)n(er)n(ators)f(with)401 1757 y(slow)r(ly)10 b(de)n(cr)n(e)n(asing)h(p)n(otentials)p Fx(,)f(Comm.)d(Math.)i(Ph)o (ys.)h Fd(161)p Fx(,)g(no.)f(1)g(\(1994\),)h(63{76.)228 1807 y([PSS81])33 b(P)m(.)d(P)o(erry)m(,)36 b(I.M.)30 b(Sigal,)j(and)e(B.)g(Simon,)i Fr(Sp)n(e)n(ctr)n(al)d(analysis)h(of)g Fc(N)5 b Fr(-b)n(o)n(dy)401 1856 y(Schr\177)-21 b(o)n(dinger)15 b(op)n(er)n(ators)p Fx(,)e(Ann.)g(of)h(Math.)f Fd(114)p Fx(,)g(no.)g(3)h(\(1981\),)e(519{567.)228 1906 y([Rau78])32 b(J.)17 b(Rauc)o(h,)i Fr(L)n(o)n(c)n(al)f(de)n(c)n(ay)h(of)f(sc)n (attering)g(solutions)h(to)g(Schr\177)-21 b(o)n(dinger's)18 b(e)n(quation)p Fx(,)401 1956 y(Comm.)10 b(Math.)j(Ph)o(ys.)h Fd(61)g Fx(\(1978\),)e(149{168.)228 2006 y([ReSi78])23 b(M.)16 b(Reed)i(and)f(B.)g(Simon,)e Fr(Metho)n(ds)k(of)e(mo)n(dern)h (mathematic)n(al)g(physics)g(I-IV)p Fx(,)401 2056 y(Academic)13 b(Press,)i(1972-78.)228 2106 y([Sk03])62 b(E.)18 b(Skibsted,)h Fr(L)n(ong-r)n(ange)h(sc)n(attering)e(of)h(thr)n(e)n(e-b)n(o)n(dy)g (quantum)h(systems:)27 b(As-)401 2155 y(ymptotic)15 b(c)n(ompleteness)p Fx(,)e(In)o(v)o(en)o(t.)h(Math.)f Fd(151)h Fx(\(2003\),)e(65-99.)228 2205 y([T)m(am89])23 b(H.)15 b(T)m(am)o(ura,)f Fr(Principle)i(of)h (limiting)e(absorption)i(for)f Fc(N)5 b Fr(-b)n(o)n(dy)17 b(Schr\177)-21 b(o)n(dinger)17 b(op-)401 2255 y(er)n(ators)p Fx(,)12 b(Letters)k(in)d(Math.)h(Ph)o(ys.)g Fd(17)f Fx(\(1989\),)g (31-36.)228 2305 y([Y)m(af82])45 b(D.R.)15 b(Y)m(afaev,)h Fr(The)h(low)g(ener)n(gy)g(sc)n(attering)g(for)g(slow)r(ly)f(de)n(cr)n (e)n(asing)i(p)n(otentials)p Fx(,)401 2355 y(Comm.)10 b(Math.)j(Ph)o(ys.)h Fd(85)p Fx(,)f(no.)h(2)f(\(1982\),)g(177{196.)956 2658 y Fq(44)p eop %%Page: 45 45 45 44 bop 278 213 a Fx(\(S.)17 b(F)m(ournais\))g Fw(Labora)m(toire)k (de)e(Ma)m(th)999 210 y(\023)999 213 y(ema)m(tiques,)j(Universit)1443 210 y(\023)1443 213 y(e)e(P)l(aris-Sud)g(-)228 262 y(B)261 259 y(^)260 262 y(a)m(t)c(425,)f(F-91405)g(Orsa)m(y)i(Cedex,)f(France.) 278 312 y Fr(E-mail)e(addr)n(ess)s Fx(:)k Fa(soeren.fournais@math)o (.u-p)o(sud.f)o(r)278 397 y Fx(\(E.)d(Skibsted\))i Fw(Institut)f(f)o (or)i(Ma)m(tema)m(tiske)h(F)-5 b(a)o(g,)18 b(Aarhus)g(Universitet,)g (Ny)228 447 y(Munkegade,)f(8000)f(Aarhus)h(C,)e(Denmark)278 497 y Fr(E-mail)f(addr)n(ess)s Fx(:)k Fa(skibsted@imf.au.dk)956 2658 y Fq(45)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306200736502--