Content-Type: multipart/mixed; boundary="-------------0205071023756" This is a multi-part message in MIME format. ---------------0205071023756 Content-Type: text/plain; name="02-215.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-215.keywords" Lattice Spin Systems, Witten Laplacian, Gibbs Measure, Exponential Decay of Correlations ---------------0205071023756 Content-Type: application/postscript; name="BachMoeller-I-preprint-02-4-22.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BachMoeller-I-preprint-02-4-22.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: BachMoeller-I-02-4-22.dvi %%Pages: 58 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Italic Times-Bold %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o BachMoeller-I-preprint-02-4-22.ps %+ BachMoeller-I-02-4-22 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.05.07:1616 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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5347 y Ft(2)1464 5413 y Fq(H)q(;Q)1655 5388 y Fr(=)f Fu(d)1834 5347 y Fs(\003)1834 5413 y Fq(H)q(;Q)1989 5388 y Fu(d)2040 5403 y Fq(H)q(;Q)2228 5388 y Fr(+)e Fu(d)2405 5403 y Fq(H)q(;Q)2559 5388 y Fu(d)2610 5347 y Fs(\003)2610 5413 y Fq(H)q(;Q)2771 5388 y Fu(:)523 b Fx(\(I.3\))p eop %%Page: 3 4 3 3 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2415 b Fx(3)257 573 y(By)34 b(construction,)f Fr(\001)1040 588 y Fq(H)q(;Q)1212 573 y Fx(is)g(a)g(positi)n(v)o(e)e(operator)-5 b(.)55 b(Its)32 b(k)o(ernel)i(is)e(one-dimensional)f(and)257 693 y(spanned)25 b(by)f(the)h Fr(0)p Fx(-form)f Fu(e)1225 657 y Fs(\000)p Fq(\014)s(H)1391 693 y Fx(,)h(as)g(we)g(sho)n(w)e(belo) n(w)-6 b(.)404 814 y(The)28 b(\(untwisted,)h Fu(Q)35 b Fr(=)f(0)p Fx(\))29 b(W)l(itten)f(Laplacian)h Fr(\001)2195 829 y Fq(H)q(;)p Ft(0)2342 814 y Fx(w)o(as)g(originally)e(introduced)h (by)257 934 y(W)l(itten)23 b([22])g(in)g(the)g(conte)o(xt)g(of)g(dif)n (ferential)g(geometry)-6 b(.)29 b(Its)23 b(importance)f(for)i(the)f (study)f(of)257 1054 y(\(I.1\))k(lies)e(in)g(the)h(identity)770 1254 y Fp(E)831 1213 y Fq(T)831 1279 y(\014)892 1254 y Fr(\()p Fu(u)17 b Fr(;)32 b Fu(v)t Fr(\))83 b(:=)g Fp(E)1480 1269 y Fq(\014)1534 1254 y Fr(\()p Fu(u)21 b Fo(\001)h Fu(v)t Fr(\))44 b Fo(\000)h Fp(E)2014 1269 y Fq(\014)2067 1254 y Fr(\()p Fu(u)p Fr(\))22 b Fo(\001)g Fp(E)2331 1269 y Fq(\014)2384 1254 y Fr(\()p Fu(v)t Fr(\))810 b Fx(\(I.4\))1248 1457 y Fr(=)1524 1389 y(1)p 1430 1434 236 4 v 1430 1525 a Fu(\014)1491 1496 y Ft(2)1547 1525 y Fo(Z)1619 1540 y Fq(\014)1692 1376 y Fm(\012)1740 1457 y Fu(e)1785 1416 y Fs(\000)p Fq(\014)s(H)1966 1457 y Fu(du)2095 1372 y Fm(\014)2095 1432 y(\014)2156 1376 y(\000)2201 1457 y Fr(\001)2282 1406 y Ft(\(1\))2282 1484 y Fq(H)q(;)p Ft(0)2401 1376 y Fm(\001)2447 1398 y Fs(\000)p Ft(1)2557 1457 y Fu(e)2602 1416 y Fs(\000)p Fq(\014)s(H)2784 1457 y Fu(dv)2886 1376 y Fm(\013)2955 1457 y Fu(;)257 1716 y Fx(for)38 b(the)g(truncated)g(correlation)f (function)g(of)h(the)g(probability)e(measure)i Fu(d\026)p 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y(This)23 b(w)o(as)g(realized)g(by)g(Helf)n(fer)g (and)g(Sj)8 b(\250)-41 b(ostrand)22 b([15])h(and,)h(more)e(e)o (xplicitly)-6 b(,)21 b(by)i(Sj)8 b(\250)-41 b(ostrand)257 2457 y(in)28 b([18)o(].)40 b(In)27 b(particular)l(,)i(the)e(truncated)g (tw)o(o-point)g(correlation)g(function)g(has)g(an)h(e)o(xplicit)257 2577 y(representation)d(as)g(a)g(matrix)f(element)g(of)h(the)g(resolv)o (ent)e(of)i Fr(\001)2481 2592 y Fq(H)q(;)p Ft(0)2600 2577 y Fx(,)745 2825 y Fp(E)806 2784 y Fq(T)806 2849 y(\014)867 2825 y Fr(\()p Fu(x)960 2840 y Fq(i)1005 2825 y Fr(;)33 b Fu(x)1120 2840 y Fq(j)1157 2825 y Fr(\))53 b(=)1480 2757 y(1)p 1386 2802 V 1386 2893 a Fu(\014)1447 2864 y Ft(2)1503 2893 y Fo(Z)1575 2908 y Fq(\014)1649 2744 y Fm(\012)1696 2825 y Fu(e)1741 2784 y Fs(\000)p Fq(\014)s(H)1923 2825 y Fu(dx)2029 2840 y Fq(i)2079 2740 y Fm(\014)2079 2800 y(\014)2140 2744 y(\000)2185 2825 y Fr(\001)2266 2774 y Ft(\(1\))2266 2852 y Fq(H)q(;)p Ft(0)2385 2744 y Fm(\001)2431 2766 y Fs(\000)p Ft(1)2542 2825 y 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Ft(0)1542 3572 y Fx(.)55 b(In)32 b(this)g(paper)h(we)g(pro)o(v)o(e)e(an)i(upper)f (bound)g(on)g(the)257 3693 y(correlations)19 b(and)f(in)h(a)g(future)g (w)o(ork)g(we)g(will)f(deri)n(v)o(e)g(the)g(asymptotics)f(of)i(the)g (correlations,)257 3813 y(follo)n(wing)k(ideas)i(of)g([2].)404 3944 y(A)34 b(certain)g(class)f(of)h(Hamiltonian)e(functions)h(were)i (studied)d(by)i(means)f(of)h Fr(\001)3265 3893 y Ft(\(1\))3265 3971 y Fq(H)q(;)p Ft(0)3418 3944 y Fx(in)257 4064 y(the)29 b(abo)o(v)o(e)f(mentioned)f(paper)i(by)f(Sj)8 b(\250)-41 b(ostrand)29 b([18)o(].)43 b(He)29 b(deri)n(v)o(ed)e(the)i(asymptotics) e(of)h(the)257 4184 y(truncated)j(correlation)f(function)g(in)g (leading)g(order)h(in)f Fu(\014)2326 4148 y Fs(\000)p Ft(1)2451 4184 y Fx(under)h(the)f(assumption)f(that)257 4305 y(the)35 b(Hamiltonian)e(function)h Fu(H)43 b Fx(is)34 b(uniformly)g(strictly)f(con)l(v)o(e)o(x.)60 b(In)35 b([20],)i(he)f(consider)n(-)257 4425 y(ably)c(re\002ned)g(his)f 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(of)h(the)f(correlations,)g(the)g(e)o(xclusion)e(of)i(other)257 5268 y(critical)36 b(points)f(in)g([2])h(appeared)h(to)e(be)h(purely)g (technical)f(and)h(not)g(essential.)63 b(This)35 b(is)257 5388 y(precisely)c(the)f(point)g(to)g(be)h(established)e(in)i(the)f (present)g(paper)-5 b(.)48 b(Loosely)30 b(speaking,)h(the)p eop %%Page: 4 5 4 4 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2415 b Fx(4)257 574 y(introduction)26 b(of)h(the)g(twist)e Fu(d)p Fr(\000\()p Fu(Q)p Fr(\))i Fx(con)l(v)o(e)o(xi\002es)f(the)h(potential) f(in)h Fr(\001)2728 523 y Ft(\(1\))2728 601 y Fq(H)q(;Q)2894 574 y Fx(such)f(that)h(it)f(has)257 694 y(all)i(properties)g(required)h (in)f([2],)h(and)g(all)f(the)g(methods)f(de)n(v)o(eloped)g(therein)h (apply)g(again.)257 815 y(Apart)23 b(from)g(this)f(\(desired\))h(con)l (v)o(e)o(xi\002cation,)e Fu(d)p Fr(\000\()p Fu(Q)p Fr(\))i Fx(also)f(generates)h(\(undesired\))g(e)o(xpo-)257 935 y(nentially)d(lar)n(ge)h(weight)g(f)o(actors.)29 b(These)21 b(lar)n(ge)g(weights)f(lead)h(to)g(technical)f(complications)257 1055 y(in)28 b(our)f(analysis)g(because)i(we)f(ha)n(v)o(e)f(to)h(sho)n (w)e(that)i(the)o(y)f(do)g(not)g(af)n(fect)i(the)e(main)g(relati)n(v)o (e)257 1186 y(form)e(bound)f(implying)f(the)h(e)o(xpected)h(spectral)g (properties)f(of)h Fr(\001)2606 1135 y Ft(\(1\))2606 1213 y Fq(H)q(;Q)2769 1186 y Fx(\(see)h(Sect.)f(II\).)404 1306 y(The)37 b(twist)f(with)h Fu(d)p Fr(\000\()p Fu(Q)p Fr(\))g Fx(leads)g(us)g(to)g(analyze)h(the)f(stability)f(at)h(lo)n(w)g (temperatures)257 1427 y(of)30 b(the)f(partition)g(function)f Fo(Z)1322 1442 y Fq(\014)1370 1427 y Fx(,)i(under)g(perturbations)e(of) i Fu(H)37 b Fx(in)29 b(a)h(single)e(v)n(ariable,)i(more)257 1547 y(precisely)e(of)g(the)g(type)g Fu(H)41 b Fo(!)33 b Fu(H)f Fo(\000)25 b Fu(q)1619 1562 y Fq(i)1648 1547 y Fx(,)k(where)f Fu(q)2016 1562 y Fq(i)2045 1547 y Fr(\()p Fu(x)p Fr(\))34 b(=)f Fu(q)2362 1562 y Fq(i)2391 1547 y Fr(\()p Fu(x)2484 1562 y Fq(i)2512 1547 y Fr(\))28 b Fx(and)g Fu(q)2793 1562 y Fq(i)2850 1547 y Fx(v)n(anishes)f(around) 257 1668 y Fr(0)p Fx(.)51 b(W)-8 b(e)32 b(pro)o(v)o(e)e(that)i (pertubations)e(of)h(this)g(type)g(only)g(change)h Fo(Z)2593 1683 y Fq(\014)2672 1668 y Fx(by)f(an)g(error)i(which)e(is)257 1788 y(e)o(xponentially)21 b(small)g(in)h Fu(\014)6 b Fx(.)29 b(W)-8 b(e)23 b(are)g(inspired)e(by)h(techniques)g(of)g(Sj)8 b(\250)-41 b(ostrand)22 b([19)o(],)h(cf.)31 b(also)257 1908 y([10,)25 b(16].)404 2029 y(A)20 b(v)n(ariety)f(of)h(Hamiltonian)e (functions,)i(especially)f(those)h(of)g Fu(\036)2638 1993 y Ft(4)2677 2029 y Fx(-type,)h(were)f(analyzed)257 2159 y(by)j(Helf)n(fer)g([11,)g(13,)g(12,)g(14)o(])h(by)e(means)h(of)g (the)g(W)l(itten)g(Laplacian)g Fr(\001)2760 2108 y Ft(\(1\))2760 2186 y Fq(H)q(;)p Ft(0)2878 2159 y Fx(.)30 b(He)24 b(also)e(uses)h(a) 257 2280 y(\223twist\224,)k(which)g(is,)f(ho)n(we)n(v)o(er)l(,)h(dif)n (ferent)f(from)h(ours.)36 b(Ne)n(v)o(ertheless,)26 b(Helf)n(fer')-5 b(s)27 b(approach)257 2400 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Fq(j)1583 2961 y Fr(:)47 b Fp(R)42 b Fo(!)30 b Fp(R)37 b Fx(are)27 b(the)f(on-site)g(ener)n(gy)h(functions)e(at)h(the)257 3082 y(lattice)f(point)g Fu(j)34 b Fo(2)29 b Fr(\003)p Fx(,)c(and)h Fu(w)1284 3097 y Fq(i;j)1409 3082 y Fr(:)45 b Fp(R)1547 3045 y Ft(2)1621 3082 y Fo(!)28 b Fp(R)37 b Fx(represent)25 b(the)g(pair)h(interaction)e(between)h(the)257 3202 y(lattice)h(sites)f Fu(i;)17 b(j)36 b Fo(2)31 b Fr(\003)p Fx(.)j(Furthermore,)27 b Fr(0)i Fo(\024)i Fu(\013)g Fo(\024)f Fu(\013)2105 3217 y Ft(0)2175 3202 y Fu(<)g Fr(1)c Fx(is)f(a)i(coupling)e(constant)g(which)257 3322 y(is)h(assumed)f(to)g(be)h(suf)n(\002ciently)e(small,)h(although)g(we)h (do)f(not)g(consider)h(the)f(limit)f Fu(\013)31 b Fo(!)e Fr(0)p Fx(.)257 3443 y(F)o(or)c(the)g(deri)n(v)n(ation)e(of)i(our)f (main)g(results,)g(we)h(require)h(the)e(follo)n(wing)f(hypotheses.)257 3634 y Fv(Hypothesis)37 b(1.)47 b Fn(F)-10 b(or)36 b(any)h Fu(j)55 b Fo(2)50 b Fr(\003)p Fn(,)40 b(zer)l(o)35 b(is)i(the)f(unique) 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y Fu(;)797 b Fx(\(I.8\))743 4355 y Fo(8)p Fu(t)28 b Fo(6)p Fr(=)g(0)f(:)1464 4275 y Fm(\010)1522 4355 y Fu(f)1581 4319 y Fs(0)1570 4380 y Fq(j)1607 4355 y Fr(\()p Fu(t)p Fr(\))g(=)h(0)1898 4275 y Fm(\011)2008 4355 y Fo(\))2161 4275 y Fm(\010)2219 4355 y Fu(f)2267 4370 y Fq(j)2303 4355 y Fr(\()p Fu(t)p Fr(\))g Fo(\025)g Fu(c)2589 4370 y Fq(f)2635 4275 y Fm(\011)2715 4355 y Fu(;)579 b Fx(\(I.9\))729 4511 y Fo(8)p Fu(t)28 b Fo(2)h Fp(R)38 b Fr(:)83 b Fo(j)p Fu(f)1238 4475 y Fs(00)1227 4535 y Fq(j)1280 4511 y Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)h Fu(f)1572 4475 y Fs(00)1561 4535 y Fq(j)1614 4511 y Fr(\(0\))p Fo(j)52 b(\024)h Fu(C)2019 4526 y Fq(f)2064 4430 y Fm(\000)2110 4511 y Fo(j)p Fu(f)2197 4475 y Fs(0)2186 4535 y Fq(j)2222 4511 y Fr(\()p Fu(t)p Fr(\))p Fo(j)39 b Fr(+)f(min)o Fo(f)p Fr(1)p Fu(;)17 b Fo(j)p Fu(t)p Fo(jg)2961 4430 y Fm(\001)3027 4511 y Fu(;)217 b Fx(\(I.10\))615 4666 y Fo(8j)p Fu(t)p Fo(j)28 b(\025)g Fu(R)968 4681 y Fq(f)1041 4666 y Fr(:)91 b Fo(j)p Fu(f)1246 4630 y Fs(0)1235 4691 y Fq(j)1272 4666 y Fr(\()p Fu(t)p Fr(\))p Fo(j)52 b(\025)h Fu(c)1635 4681 y Fq(f)1719 4666 y Fr(max)1917 4586 y Fm(\010)1975 4666 y Fo(j)p Fu(f)2062 4630 y Fs(0)2051 4692 y Fq(k)2094 4666 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2271 4582 y Fm(\014)2271 4641 y(\014)2329 4666 y Fo(j)p Fu(s)p Fo(j)27 b(\024)h(j)p Fu(t)p Fo(j)22 b Fu(;)42 b(k)30 b Fo(2)e Fr(\003)2988 4586 y Fm(\011)3271 4666 y Fx(\(I.11\))257 4872 y Fn(for)d(all)f Fu(j)33 b Fo(2)28 b Fr(\003)p Fn(.)404 5062 y Fx(Hypothesis)37 b(1)i(roughly)f(coincides)g(with)g(what)h(is)f (required)h(in)g([2,)j(Example)c(I.1],)257 5183 y(e)o(xcept)25 b(for)g(Eq.)f(\(I.10\),)h(which)g(is)f(much)g(weak)o(er)i(than)e(the)h (corresponding)f(assumption)1011 5388 y Fo(j)p Fu(f)1098 5347 y Fs(00)1140 5388 y Fr(\()p Fu(t)p Fr(\))f Fo(\000)f Fu(f)1432 5347 y Fs(00)1474 5388 y Fr(\(0\))p Fo(j)52 b(\024)h Fr(const)2049 5307 y Fm(\000)2094 5388 y Fo(j)p Fu(f)2181 5347 y Fs(0)2204 5388 y Fr(\()p Fu(t)p Fr(\))p Fo(j)22 b Fr(+)g Fu(f)2522 5347 y Fs(0)2545 5388 y Fr(\()p Fu(t)p Fr(\))2656 5347 y Ft(2)2695 5307 y Fm(\001)3271 5388 y Fx(\(I.12\))p eop %%Page: 8 9 8 8 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2415 b Fx(8)257 573 y(in)36 b([2,)j(Example)d(I.1].)65 b(Note)36 b(that)g(Eq.)g(\(I.12\))g(yields)g Fu(f)2322 537 y Fs(00)2364 573 y Fr(\()p Fu(t)2437 588 y Fs(\003)2476 573 y Fr(\))49 b(=)g Fu(f)2747 537 y Fs(00)2789 573 y Fr(\(0\))g Fu(>)f Fr(0)p Fx(,)39 b(for)e(an)o(y)257 693 y(critical)25 b(point)f Fu(t)824 708 y Fs(\003)864 693 y Fx(,)h(which)f(implies)f(that)i Fu(t)j Fr(=)g(0)d Fx(is)f(the)h(only)f(critical)h(point)f(of)h Fu(f)11 b Fx(.)30 b(This)25 b(does)257 814 y(not)j(only)g(suggest)f (that)h(Eq.)g(\(I.12\))g(is)g(merely)g(technical,)h(b)n(ut)e(it)h(also) g(illustrates)f(that)g(the)257 934 y(requirement)33 b(\(I.12\))f(e)o (xcludes)g(one)h(of)g(the)f(main)h(e)o(xamples)e(for)i(an)g (application)f(of)g(our)257 1054 y(method:)44 b(an)32 b(Ising-type)f(or)i Fu(\036)1365 1018 y Ft(4)1404 1054 y Fx(-model)e(in)h(a)g(magnetic)f(\002eld,)j(in)e(which)f(case)i Fu(f)3174 1069 y Fq(j)3211 1054 y Fr(\()p Fu(x)3304 1069 y Fq(j)3340 1054 y Fr(\))41 b(=)257 1175 y(const)24 b(+)e(\()p Fu(x)694 1139 y Ft(2)694 1199 y Fq(j)757 1175 y Fo(\000)h Fr(1\))944 1139 y Ft(2)1006 1175 y Fr(+)g Fu(\025)1162 1190 y Fq(j)1198 1175 y Fu(x)1253 1190 y Fq(j)1290 1175 y Fx(,)j(with)f Fu(\025)1601 1190 y Fq(j)1666 1175 y Fo(2)30 b Fp(R)36 b Fx(being)25 b(the)h(magnetic)f(\002eld)h(at)f(site) g Fu(j)6 b Fx(.)33 b(In)26 b(f)o(act,)257 1295 y(the)f(main)f(goal)h (of)g(the)f(present)h(w)o(ork)f(is)h(to)f(weak)o(en)h(Assumption)e (\(I.12\).)404 1416 y(T)-8 b(o)24 b(formulate)h(our)f(hypothesis)f(on)h Fu(w)1746 1431 y Fq(i;j)1826 1416 y Fx(,)h(we)g(introduce)f(functions) 852 1652 y Fu(h)908 1667 y Fq(j)945 1652 y Fr(\()p Fu(s)p Fr(\))83 b(:=)g(min)n Fo(fj)p Fu(f)1624 1667 y Fq(j)1660 1652 y Fr(\()p Fu(s)p Fr(\))p Fo(j)p Fu(;)17 b Fo(j)p Fu(f)1930 1667 y Fq(j)1966 1652 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2126 1583 y Fk(1)p 2125 1595 31 3 v 2125 1637 a(2)2170 1652 y Fo(g)49 b Fr(+)h(min)o Fo(f)p Fr(1)p Fu(;)17 b Fo(j)p Fu(s)p Fo(jg)j Fu(;)371 b Fx(\(I.13\))257 1872 y(for)25 b(all)g Fu(j)34 b Fo(2)28 b Fr(\003)p Fx(.)257 2075 y Fv(Hypothesis)22 b(2.)36 b Fn(F)-10 b(or)21 b(all)g Fu(i;)c(j)33 b Fo(2)28 b Fr(\003)p Fn(,)22 b(the)f(pair)g(inter)o (action)f(functions)f Fu(!)2744 2090 y Fq(i;j)2852 2075 y Fo(2)28 b Fu(C)3023 2039 y Ft(2)3062 2075 y Fr(\()p Fp(R)15 b Fo(\002)c Fp(R)g Fr(;)17 b Fp(R)10 b Fr(\))257 2195 y Fn(vanish)29 b(on-site)f(and)h(at)f(the)h(origin,)g Fu(w)1652 2210 y Fq(i;i)1723 2195 y Fr(\()p Fu(s;)17 b(t)p Fr(\))36 b Fo(\021)g Fr(0)29 b Fn(and)f Fu(w)2399 2210 y Fq(i;j)2479 2195 y Fr(\(0)p Fu(;)17 b Fr(0\))35 b(=)g(0)p Fn(.)43 b(Furthermor)l(e)o(,)257 2316 y(for)23 b Fu(i;)17 b(j)34 b Fo(2)28 b Fr(\003)p Fn(,)23 b(ther)l(e)g(e)n(xists) g(a)g(symmetric)g(matrix)f Fu(a)p 2014 2332 52 4 v 28 w Fr(=)28 b(\()p Fu(a)2286 2331 y Fq(ij)2346 2316 y Fr(\))2384 2331 y Fq(i;j)t Fs(2)p Ft(\003)2584 2316 y Fn(of)22 b(nonne)l(gative)h (weights)257 2436 y(\(with)i Fu(a)538 2451 y Fq(i;i)638 2436 y Fr(=)i(0)e Fn(for)f(con)l(venience\),)i(suc)o(h)e(that)639 2624 y Fm(X)595 2840 y Fq(\034)8 b Fs(2f)p Fq(i;j)t Fs(g)844 2574 y Fm(\014)844 2634 y(\014)844 2693 y(\014)844 2753 y(\014)887 2651 y Fu(@)d(w)1013 2666 y Fq(i;j)p 887 2695 207 4 v 913 2787 a Fu(@)g(x)1024 2802 y Fq(\034)1104 2718 y Fr(\()p Fu(x)1197 2733 y Fq(i)1225 2718 y Fu(;)17 b(x)1324 2733 y Fq(j)1361 2718 y Fr(\))1399 2574 y Fm(\014)1399 2634 y(\014)1399 2693 y(\014)1399 2753 y(\014)1482 2718 y Fr(+)1654 2624 y Fm(X)1580 2840 y Fq(\033)n(;\034)8 b Fs(2f)p Fq(i;j)t Fs(g)1888 2574 y Fm(\014)1888 2634 y(\014)1888 2693 y(\014)1888 2753 y(\014)1973 2651 y Fu(@)2029 2615 y Ft(2)2070 2651 y Fu(w)2140 2666 y Fq(i;j)p 1931 2695 332 4 v 1931 2787 a Fu(@)d(x)2042 2802 y Fq(\033)2107 2787 y Fu(@)g(x)2218 2802 y Fq(\034)2272 2718 y Fr(\()p Fu(x)2365 2733 y Fq(i)2394 2718 y Fu(;)17 b(x)2493 2733 y Fq(j)2529 2718 y Fr(\))39 b Fo(\000)2775 2651 y Fu(@)2831 2615 y Ft(2)2871 2651 y Fu(w)2941 2666 y Fq(i;j)p 2732 2695 V 2732 2787 a Fu(@)5 b(x)2843 2802 y Fq(\033)2908 2787 y Fu(@)g(x)3019 2802 y Fq(\034)3073 2718 y Fr(\(0)p Fu(;)17 b Fr(0\))3291 2574 y Fm(\014)3291 2634 y(\014)3291 2693 y(\014)3291 2753 y(\014)689 3021 y Fo(\024)53 b Fu(a)870 3036 y Fq(i;j)967 3021 y Fr(\()p Fu(h)1061 3036 y Fq(i)1090 3021 y Fr(\()p Fu(x)1183 3036 y Fq(i)1211 3021 y Fr(\))22 b(+)g Fu(h)1425 3036 y Fq(j)1462 3021 y Fr(\()p Fu(x)1555 3036 y Fq(j)1592 3021 y Fr(\)\))g Fu(;)1554 b Fx(\(I.14\))257 3241 y Fn(and)1297 3303 y Fm(X)1223 3519 y Fq(\033)n(;\034)8 b Fs(2f)p Fq(i;j)t Fs(g)1531 3253 y Fm(\014)1531 3313 y(\014)1531 3373 y(\014)1531 3432 y(\014)1617 3330 y Fu(@)1673 3294 y Ft(2)1713 3330 y Fu(w)1783 3345 y Fq(i;j)p 1574 3375 V 1574 3466 a Fu(@)d(x)1685 3481 y Fq(\033)1750 3466 y Fu(@)g(x)1861 3481 y Fq(\034)1916 3397 y Fr(\(0)p Fu(;)17 b Fr(0\))2134 3253 y Fm(\014)2134 3313 y(\014)2134 3373 y(\014)2134 3432 y(\014)2218 3397 y Fo(\024)54 b Fu(a)2400 3412 y Fq(i;j)2502 3397 y Fu(:)742 b Fx(\(I.15\))257 3676 y Fn(The)31 b(weights)e(ar)l(e)i(assumed)e(to)g (have)i(e)n(xponential)e(decay)h(with)g(r)l(espect)g(to)f(the)h(metric) g Fu(\032)257 3796 y Fn(on)25 b Fr(\003)p Fn(,)1343 3916 y Fr(max)1374 3978 y Fq(i)p Fs(2)p Ft(\003)1541 3822 y Fm(X)1549 4033 y Fq(j)t Fs(2)p Ft(\003)1702 3916 y Fu(e)1747 3875 y Fq(\032)p Ft(\()p Fq(i;j)t Ft(\))1935 3916 y Fu(a)1986 3931 y Fq(i;j)2119 3916 y Fo(\024)53 b Fu(C)2319 3931 y Fq(a)2382 3916 y Fu(;)862 b Fx(\(I.16\))257 4187 y Fn(for)25 b(some)f(univer)o(sal)g(constant)f Fu(C)1455 4202 y Fq(a)1525 4187 y Fu(<)k Fo(1)p Fn(.)404 4390 y Fx(Similar)21 b(to)h(the)g(e)o(xample)f(abo)o(v)o(e,)h(Hypothesis)f(2)h (also)f(accommodates)h(for)g(the)g(impor)n(-)257 4510 y(tant)32 b(e)o(xample)f(of)g(an)h(Ising)f(or)h(spin)f(glass)g(type)h (interaction,)g(for)g(which)g Fu(w)3004 4525 y Fq(i;j)3084 4510 y Fr(\()p Fu(x)3177 4525 y Fq(i)3205 4510 y Fu(;)17 b(x)3304 4525 y Fq(j)3341 4510 y Fr(\))40 b(=)257 4631 y Fu(x)312 4646 y Fq(i)358 4631 y Fu(x)413 4646 y Fq(j)482 4631 y Fx(\(e)o(xcept)33 b(that)f Fu(C)1059 4646 y Fq(a)1133 4631 y Fx(may)g(not)g(be)h(uni)n(v)o(ersal)e(for)i(spin)f(glass)f (models\).)54 b(Further)33 b(note)257 4751 y(that)28 b(in)g(the)g(case)h Fr(\003)k Fo(2)i Fp(Z)1162 4715 y Fq(d)1228 4751 y Fx(and)28 b Fu(\032)h Fx(the)f(euclidean)g(metric,)h (the)f(summability)d(condition)i(is)257 4872 y(easily)32 b(ful\002lled)f(for)h(interactions)e(of)i(\002nite)g(range)g Fu(D)43 b(>)d Fr(0)p Fx(,)33 b(i.e.,)g(in)e(case)h(that)f Fu(w)3208 4887 y Fq(i;j)3329 4872 y Fo(\021)40 b Fr(0)257 4992 y Fx(whene)n(v)o(er)25 b Fu(\032)p Fr(\()p Fu(i;)17 b(j)6 b Fr(\))27 b Fo(\025)h Fu(D)s Fx(.)404 5112 y(Hypotheses)g(1)g (and)h(2)g(insure)g(that)f Fu(H)37 b Fx(is)28 b(gro)n(wing)g(at)h (least)f(linearly)h(in)f Fu(x)i Fx(at)f(in\002nity)-6 b(,)257 5233 y(pro)o(vided)35 b Fu(\013)49 b Fo(\025)g Fr(0)36 b Fx(is)g(suf)n(\002ciently)f(small,)j(cf.)65 b(Lemma)35 b(B.1.)65 b(Thus)35 b Fu(u)30 b Fo(\001)g Fr(exp)q([)p Fo(\000)p Fu(\014)6 b(H)i Fr(])49 b Fo(2)257 5353 y Fu(L)323 5317 y Ft(2)363 5353 y Fr(\()p Fp(R)467 5317 y Ft(\003)526 5353 y Fr(\))p Fx(,)33 b(for)e(an)o(y)g Fu(\014)45 b(>)40 b Fr(0)31 b Fx(and)g(an)o(y)g(polynomially)d(bounded) j(measurable)g(function)f Fu(u)39 b Fr(:)p eop %%Page: 9 10 9 9 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2415 b Fx(9)257 573 y Fp(R)323 537 y Ft(\003)427 573 y Fo(!)44 b Fp(C)20 b Fx(.)63 b(In)34 b(particular)l(,)i(the)e(e)o(xpectation)e (v)n(alue)i Fp(E)2236 588 y Fq(\014)2289 573 y Fr(\()p Fu(u)p Fr(\))f Fx(in)h(Eq.)f(\(I.1\))h(e)o(xists)f(for)h(an)o(y)257 693 y(such)25 b(function)f(\(which)g(represents)h(an)g(observ)n (able\).)404 814 y(The)g(main)f(result)g(of)h(this)f(paper)h(is)f(the)h (follo)n(wing.)257 995 y Fv(Theor)n(em)34 b(I.1)e(\(Exponential)i (Decay\).)46 b Fn(Assume)32 b(Hypothesises)f(1)h(and)g(2.)53 b(Ther)l(e)34 b(e)n(xist)257 1115 y(univer)o(sal)25 b(constants)f Fu(\014)1110 1130 y Ft(0)1180 1115 y Fu(<)29 b Fo(1)d Fn(and)f Fu(\013)1648 1130 y Ft(1)1717 1115 y Fu(>)30 b Fr(0)c Fn(suc)o(h)f(that,)g(for)g(all)g Fr(0)30 b Fu(<)f(")g Fo(\024)i Fr(1)p Fu(=)p Fr(2)p Fn(,)25 b Fr(0)k Fo(\024)i Fu(\024)e(<)257 1235 y Fr(1)p Fn(,)c Fu(\014)33 b(>)28 b(\014)603 1250 y Ft(0)642 1235 y Fu(")688 1199 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)853 1235 y Fn(,)d(and)f Fr(0)j Fo(\024)i Fu(\013)f(<)f(\013)1514 1250 y Ft(0)1582 1235 y Fr(:=)g Fu(")p Fr(\(1)22 b Fo(\000)g Fu(\024)p Fr(\))p Fu(\013)2122 1250 y Ft(1)2162 1235 y Fn(,)j(we)g(have)1019 1389 y Fm(\014)1019 1449 y(\014)1053 1474 y Fp(E)1113 1433 y Fq(T)1113 1498 y(\014)1174 1474 y Fr(\()p Fu(x)1267 1489 y Fq(i)1318 1474 y Fr(;)17 b Fu(x)1417 1489 y Fq(j)1453 1474 y Fr(\))1491 1389 y Fm(\014)1491 1449 y(\014)1577 1474 y Fo(\024)53 b Fr(\(1)22 b(+)g Fu(")p Fr(\))2160 1406 y(1)p 2024 1451 322 4 v 2024 1542 a(2)17 b Fu(\014)22 b(\025)2224 1557 y Ft(min)2372 1474 y Fu(e)2417 1433 y Fs(\000)p Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))2706 1474 y Fu(:)538 b Fx(\(I.17\))404 1718 y(Here)23 b Fu(\025)677 1733 y Ft(min)826 1718 y Fr(:=)28 b(min)15 b Fu(\033)t Fr(\()p Fu(H)1322 1682 y Fs(00)1364 1718 y Fr(\(0\)\))22 b Fx(which,)g(for)g Fu(\013)29 b(<)e(\013)2234 1733 y Ft(0)2296 1718 y Fx(small)21 b(enough,)h(satis\002es)g Fu(\025)3270 1733 y Ft(min)3419 1718 y Fu(>)257 1839 y(c)299 1854 y Fq(f)345 1839 y Fu(=)p Fr(2)p Fx(.)28 b(W)-8 b(e)20 b(remark)g(that)f(the)g(ef)n(fect)h(of)f(boundary)g (conditions)f(\(in)h(a)h(gi)n(v)o(en)e(e)o(xample\))g(on)h(the)257 1959 y(decay)26 b(estimate)f(\(I.17\))h(is)f(encoded)h(in)f Fu(\025)1733 1974 y Ft(min)1880 1959 y Fx(and)h(in)f Fu(C)2223 1974 y Fq(a)2264 1959 y Fx(,)h(and)g(hence,)g(is)f(easy)h(to) f(analyze.)257 2079 y(Control)32 b(with)g(respect)g(to)g(boundary)g (conditions)e(were)j(considered)f(in)g([25)o(,)h(23)o(,)f(3,)h(17)o(],) 257 2200 y(ho)n(we)n(v)o(er)l(,)24 b(not)g(with)g(uniform)g(control)g (at)h(lo)n(w)f(temperature.)404 2320 y(An)38 b(important)f(point)h(in)g (our)h(analysis)f(is)g(the)g(observ)n(ation)f(made)i(in)f(Lemma)g(I.2,) 257 2440 y(belo)n(w)-6 b(,)39 b(that)e(our)g(hypothesis)e(on)i Fu(f)1562 2455 y Fq(j)1636 2440 y Fx(suf)n(\002ces)g(to)g(construct)f (an)i(auxilliary)e(function)g Fu(g)3459 2455 y Fq(j)257 2561 y Fx(which)22 b(obe)o(ys)e(\(I.12\))i(and,)h(yet,)f(dif)n(fers)f (from)h Fu(f)1925 2576 y Fq(j)1983 2561 y Fx(only)f(a)o(w)o(ay)h(from)f (its)g(global)g(minimum)e(at)257 2681 y Fu(x)312 2696 y Fq(j)377 2681 y Fr(=)28 b(0)p Fx(,)20 b(i.e.,)g(in)f(a)h(re)o(gion)f (which)g(should)f(be)i(irrele)n(v)n(ant)e(\(and,)j(indeed,)f(is)f (irrele)n(v)n(ant,)g(as)h(we)257 2802 y(pro)o(v)o(e)k(in)h(a)g (subsequent)e(paper\))j(for)f(the)f(correlation)h(asymptotics)e(at)i (lo)n(w)e(temperature.)257 2983 y Fv(Lemma)j(I.2.)41 b Fn(Assume)24 b(Hypothesis)g(1.)30 b(Ther)l(e)c(e)n(xist)e(univer)o (sal)g(number)o(s)g Fu(q)2955 2998 y Ft(max)3118 2983 y Fu(>)k Fr(0)p Fn(,)1098 3177 y Fr(0)52 b Fu(<)1349 3151 y Fr(^)1327 3177 y Fu(R)1401 3192 y 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Fr(1)p Fu(;)17 b Fo(j)p Fu(t)p Fo(jg)k Fu(;)577 b Fx(\(I.27\))755 1252 y Fo(8)p Fu(t)28 b Fo(2)g Fp(R)39 b Fr(:)83 b Fo(j)p Fu(g)1256 1216 y Fs(00)1252 1276 y Fq(j)1298 1252 y Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)g Fu(g)1581 1216 y Fs(00)1577 1276 y Fq(j)1623 1252 y Fr(\(0\))p Fo(j)49 b Fr(+)h Fo(j)s Fr(^)-52 b Fu(g)2030 1216 y Fs(00)2026 1276 y Fq(j)2071 1252 y Fr(\()p Fu(t)p Fr(\))23 b Fo(\000)j Fr(^)-53 b Fu(g)2355 1216 y Fs(00)2351 1276 y Fq(j)2397 1252 y Fr(\(0\))p Fo(j)52 b(\024)h Fu(C)2802 1267 y Fq(g)2864 1252 y Fo(j)p Fu(g)2943 1216 y Fs(0)2939 1276 y Fq(j)2975 1252 y Fr(\()p Fu(t)p Fr(\))p Fo(j)22 b Fu(;)108 b Fx(\(I.28\))755 1484 y Fo(8)p Fu(t)28 b Fo(2)g Fp(R)39 b Fr(:)104 b Fo(j)p Fu(g)1277 1448 y Fs(0)1273 1509 y Fq(j)1309 1484 y Fr(\()p Fu(t)p Fr(\))p Fo(j)52 b(\025)h Fu(c)1672 1499 y Fq(g)1751 1484 y Fr(max)1949 1374 y Fm(n)2015 1484 y Fo(j)p Fu(g)2094 1448 y Fs(0)2090 1510 y Fq(k)2132 1484 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2310 1370 y Fm(\014)2310 1429 y(\014)2310 1489 y(\014)2368 1484 y Fo(j)p Fu(s)p Fo(j)27 b(\024)h(j)p Fu(t)p Fo(j)22 b Fu(;)41 b(k)31 b Fo(2)d Fr(\003)3027 1374 y Fm(o)3115 1484 y Fu(;)129 b Fx(\(I.29\))755 1704 y Fo(8)p Fu(t)28 b Fo(2)g Fp(R)39 b Fr(:)373 b Fu(c)1509 1719 y Fq(g)1566 1704 y Fo(j)p Fu(g)1645 1668 y Fs(0)1641 1729 y Fq(j)1677 1704 y Fr(\()p Fu(t)p Fr(\))p Fo(j)52 b(\024)h(j)s Fr(^)-52 b Fu(g)2077 1668 y Fs(0)2073 1729 y Fq(j)2109 1704 y Fr(\()p Fu(t)p Fr(\))p Fo(j)53 b(\024)g Fu(C)2501 1719 y Fq(g)2557 1704 y Fo(j)p Fu(g)2636 1668 y Fs(0)2632 1729 y Fq(j)2668 1704 y Fr(\()p Fu(t)p Fr(\))p Fo(j)22 b Fu(;)415 b Fx(\(I.30\))751 1907 y Fo(8)p Fu(\034)39 b(>)28 b Fr(0)f(:)438 b(max)1713 1922 y Fs(j)p Fq(t)p Fs(j\024)p Fq(\034)1893 1826 y Fm(\010)1951 1907 y Fo(j)p Fu(g)2030 1871 y Fs(0)2026 1931 y Fq(j)2062 1907 y Fr(\()p Fu(t)p Fr(\))p Fo(j)2201 1826 y Fm(\011)2311 1907 y Fo(\024)53 b Fu(C)2511 1922 y Fq(g)2568 1907 y Fu(e)2613 1871 y Fq(C)2663 1879 y Fj(g)2699 1871 y Fq(\034)2765 1907 y Fu(;)479 b Fx(\(I.31\))751 2110 y 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y Fq(j)2297 3319 y Fr(\))50 b Fo(\000)2565 3251 y Fu(@)2621 3215 y Ft(2)2661 3251 y Fu(w)2731 3266 y Fq(i;j)p 2522 3296 V 2522 3387 a Fu(@)5 b(x)2633 3402 y Fq(\033)2698 3387 y Fu(@)g(x)2809 3402 y Fq(\034)2864 3319 y Fr(\(0)p Fu(;)17 b Fr(0\))3082 3174 y Fm(\014)3082 3234 y(\014)3082 3294 y(\014)3082 3353 y(\014)3271 3319 y Fx(\(I.33\))465 3661 y Fo(\024)84 b Fu(C)696 3676 y Fq(w)769 3661 y Fu(a)820 3676 y Fq(i;j)900 3551 y Fm(\020)976 3661 y Fr(min)1156 3581 y Fm(\010)1214 3661 y Fo(j)p Fu(g)1293 3620 y Fs(0)1289 3686 y Fq(i)1316 3661 y Fr(\()p Fu(x)1409 3676 y Fq(i)1438 3661 y Fr(\))p Fo(j)17 b Fu(;)38 b Fo(j)p Fu(g)1665 3620 y Fs(0)1661 3686 y Fq(i)1688 3661 y Fr(\()p Fu(x)1781 3676 y Fq(i)1810 3661 y Fr(\))p Fo(j)1876 3620 y Ft(1)p Fq(=)p Ft(2)1986 3581 y Fm(\011)2088 3661 y Fr(+)44 b(min)2387 3581 y Fm(\010)2445 3661 y Fo(j)p Fu(g)2524 3620 y Fs(0)2520 3686 y Fq(j)2556 3661 y Fr(\()p Fu(x)2649 3676 y Fq(j)2686 3661 y Fr(\))p Fo(j)17 b Fu(;)38 b Fo(j)p Fu(g)2913 3620 y Fs(0)2909 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y Ft(\003)2078 3949 y Fx(replaced)g(by)f(a)h(compact)f(manifold\))g(in)257 4069 y([22])g(to)g(gi)n(v)o(e)e(a)i(semiclassical)f(proof)h(of)f(the)h (Morse)g(inequalities)e(\(see)i(also)g([7]\).)404 4190 y(The)h(twisted)f(W)l(itten)g(Laplacian)h(preserv)o(es)g(the)g(space)g Fo(H)2492 4153 y Ft(\()p Fq(n)p Ft(\))2620 4190 y Fx(of)h Fu(n)p Fx(-forms)f(on)f Fp(R)3271 4153 y Ft(\003)3330 4190 y Fx(,)i(so,)257 4320 y(writing)d Fr(\001)651 4269 y Ft(\()p Fq(n)p Ft(\))651 4347 y Fq(H)q(;Q)815 4320 y Fx(for)h(the)f(restriction)g(of)h Fr(\001)1720 4335 y Fq(H)q(;Q)1884 4320 y Fx(to)f Fo(H)2071 4284 y Ft(\()p Fq(n)p Ft(\))2173 4320 y Fx(,)h(we)g(ha)n(v)o(e)g(that)1470 4654 y Fr(\001)1551 4669 y Fq(H)q(;Q)1743 4654 y Fr(=)1877 4522 y Fs(j)p Ft(\003)p Fs(j)1846 4559 y Fm(M)1855 4768 y Fq(n)p Ft(=0)2013 4654 y Fr(\001)2094 4603 y Ft(\()p Fq(n)p Ft(\))2094 4681 y Fq(H)q(;Q)2255 4654 y Fu(:)989 b Fx(\(I.48\))257 4949 y(In)27 b(Appendix)e(B,)i(we)f(discuss)f(domain) g(properties)h(of)g Fu(d)2270 4964 y Fq(H)q(;Q)2408 4949 y Fx(,)h Fu(d)2511 4913 y Fs(\003)2511 4974 y Fq(H)q(;Q)2649 4949 y Fx(,)f Fu(D)2781 4964 y Fq(H)q(;Q)2920 4949 y Fx(,)g(and)g Fr(\001)3222 4964 y Fq(H)q(;Q)3361 4949 y Fx(,)g(as)257 5070 y(well)34 b(as,)i(the)e(question)f(of)h (self-adjointness)e(of)i Fu(D)2109 5085 y Fq(H)q(;Q)2282 5070 y Fx(and)f Fr(\001)2540 5085 y Fq(H)q(;Q)2679 5070 y Fx(.)58 b(One)34 b(of)g(the)g(results)257 5190 y(pro)o(v)o(ed)24 b(in)g(Appendix)g(B)i(is)e(the)h(follo)n(wing.)257 5388 y Fv(Theor)n(em)i(I.4.)41 b Fn(Assume)24 b(Hypothesis)g(1)g(and)h (Hypothesis)e(2.)p eop %%Page: 14 15 14 14 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(14)358 573 y Fn(\(i\))49 b(Ther)l(e)26 b(e)n(xist)e Fu(\013)1022 588 y Ft(0)1090 573 y Fu(>)j Fr(0)e Fn(and)f Fu(\014)1496 588 y Ft(0)1563 573 y Fu(<)k Fo(1)c Fn(suc)o(h)g(that,)g (for)g Fr(0)k Fo(\024)g Fu(\013)g(<)g(\013)2781 588 y Ft(0)2845 573 y Fn(and)d Fu(\014)33 b(>)27 b(\014)3266 588 y Ft(0)3306 573 y Fn(,)962 802 y Fr(Ker)p Fo(f)p Fr(\001)1250 751 y Ft(\(0\))1250 829 y Fq(H)q(;Q)1389 802 y Fo(g)52 b Fr(=)h Fp(C)37 b Fu(e)1748 760 y Fs(\000)p Fq(\014)s(H)1966 802 y Fn(and)63 b Fr(Ker)p Fo(f)p Fr(\001)2467 751 y Ft(\(1\))2467 829 y Fq(H)q(;Q)2606 802 y Fo(g)52 b Fr(=)h Fo(f)p Fr(0)p Fo(g)21 b Fu(:)237 b Fx(\(I.49\))331 1063 y Fn(\(ii\))48 b(If)39 b(all)e(the)i Fu(f)953 1078 y Fq(i)981 1063 y Fn(')l(s)f(and)g Fu(!)1336 1078 y Fq(ij)1397 1063 y Fn(')l(s)g(ar)l(e)h(smooth)e(in)h(\(not)g(necessarily)g(univer)o (sal\))g(neigh-)501 1183 y(bourhoods)26 b(of)g Fr(0)p Fn(,)h(then)g(ther)l(e)g(e)n(xists)g(a)f(univer)o(sal)g(constant)g Fu(\013)2734 1198 y Ft(0)2805 1183 y Fu(>)32 b Fr(0)27 b Fn(suc)o(h)f(that)g(for)501 1303 y Fr(0)i Fo(\024)g Fu(\013)g(<)g(\013)939 1318 y Ft(0)1003 1303 y Fn(and)c(all)h Fu(\014)33 b(>)27 b Fr(0)p Fn(,)1424 1523 y Fr(Ker)17 b(\001)1679 1538 y Fq(H)q(;Q)1870 1523 y Fr(=)53 b Fp(C)37 b Fu(e)2127 1482 y Fs(\000)p Fq(\014)s(H)2336 1523 y Fo(\012)j Fr(\012)22 b Fu(:)699 b Fx(\(I.50\))404 1784 y(W)-8 b(e)35 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Fq(i)658 5147 y Fr(\))30 b(=)g Fu(@)883 5162 y Fq(i)930 5122 y Fm(b)911 5147 y Fu(G)p Fx(.)k(The)26 b(intertwining)e(relations)i(\(I.54\))g(are)g(of)g(k)o(e)o(y)g (importance)f(to)g(our)257 5268 y(analysis,)30 b(as)f(it)g(allo)n(ws)e (us)i(to)g(pass)g(from)g Fu(H)37 b Fx(to)28 b(a)i(ne)n(w)f(Hamiltonian) e(function)i Fu(G)g Fx(which)257 5388 y(agrees)d(with)e Fu(H)32 b Fx(at)25 b(zero,)g(b)n(ut)g(yet)f(has)h(no)f(critical)h (points)e(other)i(than)g(zero.)p eop %%Page: 15 16 15 15 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(15)404 573 y(Note)36 b(that)g Fo(H)905 537 y Ft(\(0\))1049 573 y Fr(=)49 b Fu(L)1240 537 y Ft(2)1280 573 y Fr(\()p Fp(R)1384 537 y Ft(\003)1443 573 y Fr(\))30 b Fo(\012)i(F)1702 537 y Ft(\(0\))1832 573 y Fx(can)37 b(be)g(identi\002ed)f(with)f Fu(L)2830 537 y Ft(2)2870 573 y Fr(\()p Fp(R)2974 537 y Ft(\003)3033 573 y Fr(\))p Fx(,)k(and)e Fo(H)3401 537 y Ft(\(1\))257 707 y Fx(with)f Fu(L)537 671 y Ft(2)576 707 y Fr(\()p Fp(R)680 671 y Ft(\003)739 707 y Fr(\))31 b Fo(\012)g Fp(C)982 671 y Ft(\003)1041 707 y Fx(.)64 b(W)-8 b(e)37 b(will)e(use)h(the)g(same)g(notation,)h Fr(\001)2527 656 y Ft(\(0\))2527 734 y Fq(H)q(;Q)2702 707 y Fx(and)f Fr(\001)2963 656 y Ft(\(1\))2963 734 y Fq(H)q(;Q)3102 707 y Fx(,)j(for)d(both)257 827 y(realizations.)404 948 y(W)-8 b(e)33 b(\002rst)h(compute)e(the)i(restriction)e(of)i(the)f (twisted)f(W)l(itten)h(Laplacian)g(to)g Fr(0)p Fx(-forms,)257 1078 y Fr(\001)338 1027 y Ft(\(0\))338 1105 y Fq(H)q(;Q)477 1078 y Fx(,)25 b(using)f(\(I.36\))h(which)f(implies)f(that)i Fu(d)1829 1042 y Fs(\003)1829 1103 y Fq(H)q(;Q)1994 1078 y Fr(=)j(0)d Fx(on)f Fo(H)2381 1042 y Ft(\(0\))2476 1078 y Fx(.)30 b(W)-8 b(e)26 b(obtain)492 1313 y Fr(\001)573 1262 y Ft(\(0\))573 1340 y Fq(H)q(;Q)794 1313 y Fr(=)83 b Fu(d)1004 1271 y Fs(\003)1004 1337 y Fq(H)q(;Q)1159 1313 y Fu(d)1210 1328 y Fq(H)q(;Q)1401 1313 y Fr(=)1529 1218 y Fm(X)1563 1428 y Fq(i;j)1690 1313 y Fu(Z)1764 1271 y Fs(\003)1757 1337 y Fq(i)1803 1313 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)2061 1328 y Fq(i)2105 1313 y Fr(\002)2181 1328 y Fq(j)2234 1313 y Fu(Z)2301 1328 y Fq(j)2337 1313 y Fr(\()p Fu(H)8 b Fr(\))38 b Fo(\012)h Fu(a)2707 1328 y Fq(i)2752 1313 y Fu(a)2803 1271 y Fs(\003)2803 1337 y Fq(j)3271 1313 y Fx(\(I.57\))794 1589 y Fr(=)953 1494 y Fm(X)1013 1704 y Fq(i)1114 1589 y Fr(\002)1190 1604 y Fq(i)1234 1589 y Fu(Z)1308 1547 y Fs(\003)1301 1613 y Fq(i)1348 1589 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)1585 1604 y Fq(i)1612 1589 y Fr(\()p Fu(G)p Fr(\))g(\002)1858 1604 y Fq(i)1924 1589 y Fo(\012)40 b Fi(1)52 b Fo(\021)2279 1494 y Fm(X)2339 1704 y Fq(i)2440 1589 y Fr(\002)2516 1604 y Fq(i)2560 1589 y Fu(Z)2634 1547 y Fs(\003)2627 1613 y Fq(i)2674 1589 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)2911 1604 y Fq(i)2938 1589 y Fr(\()p Fu(G)p Fr(\))g(\002)3184 1604 y Fq(i)3234 1589 y Fu(;)257 1885 y Fx(where)28 b(in)f(the)g(last)f (line,)i(we)f(ha)n(v)o(e)g(gi)n(v)o(en)f(its)g(representation)h(on)g Fu(L)2656 1849 y Ft(2)2695 1885 y Fr(\()p Fp(R)2799 1849 y Ft(\003)2858 1885 y Fr(\))p Fx(.)38 b(In)28 b(the)f(special)257 2006 y(case)f Fu(Q)i Fr(=)f(0)p Fx(,)e(we)g(ha)n(v)o(e)f Fr(\002)1187 2021 y Fq(i)1243 2006 y Fo(\021)k Fr(1)p Fx(,)d(for)g(all)f Fu(i)k Fo(2)g Fr(\003)p Fx(,)d(and)g(hence)345 2219 y Fr(\001)426 2169 y Ft(\(0\))426 2246 y Fq(H)q(;)p Ft(0)597 2219 y Fr(=)725 2125 y Fm(X)785 2335 y Fq(i)902 2219 y Fu(Z)976 2178 y Fs(\003)969 2244 y Fq(i)1016 2219 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1265 2234 y Fq(i)1291 2219 y Fr(\()p Fu(H)8 b Fr(\))52 b(=)h Fo(\000)p Fu(\014)1775 2178 y Fs(\000)p Ft(2)1869 2219 y Fr(\001)45 b(+)f Fo(jr)p Fu(H)8 b Fr(\()p Fu(x)p Fr(\))p Fo(j)2474 2178 y Ft(2)2557 2219 y Fo(\000)45 b Fu(\014)2740 2178 y Fs(\000)p Ft(1)2834 2219 y Fr(\001)p Fu(H)8 b Fr(\()p Fu(x)p Fr(\))22 b Fu(;)87 b Fx(\(I.58\))257 2514 y(where)33 b Fr(\001)41 b(=)772 2439 y Fm(P)877 2543 y Fq(i)p Fs(2)p Ft(\003)1018 2514 y Fu(@)1074 2477 y Ft(2)1069 2538 y Fq(i)1114 2514 y Fx(.)52 b(F)o(or)32 b Fu(Q)42 b Fr(=)e(0)p Fx(,)34 b(the)e(twisted)f(W) l(itten)g(Laplacian)h(restricted)g(to)f Fr(0)p Fx(-)257 2634 y(forms)20 b(thus)f(becomes)h(a)g(standard)f(semiclassical)g(Schr) 8 b(\250)-41 b(odinger)20 b(operator)l(,)h(and)f(the)g(critical)257 2754 y(points)29 b(of)h Fu(H)37 b Fx(become)30 b(global)e(minima)h(of)h (the)f(potential)g(\(if)g(we)h(disre)o(gard)f Fr(\001)p Fu(H)8 b Fx(,)31 b(which)257 2875 y(enters)f(at)g(lo)n(wer)g(order)g (in)f Fu(\014)6 b Fx(\).)46 b(Comparing)30 b(to)f(\(I.57\),)i(we)g(see) f(that)f(the)h(introduction)e(of)257 2995 y Fu(Q)d Fx(remo)o(v)o(es)d (the)i(nonzero)f(local)h(minima,)f(which)g(are)i(irrele)n(v)n(ant)d(in) i(the)f(lo)n(w-temperature)257 3115 y(limit.)39 b(The)28 b(price)g(we)g(pay)g(is)g(the)g(presence)g(of)g(the)g(e)o(xponential)f (weights)g Fr(\002)3025 3130 y Fq(i)3053 3115 y Fx(.)40 b(In)28 b([2])g(the)257 3236 y(assumptions)k(were)i(such)f(that)g(the)h (lo)n(w-temperature)e(study)h(of)g(the)h(W)l(itten)e(Laplacian,)257 3356 y(with)26 b Fu(Q)31 b Fr(=)g(0)p Fx(,)c(became)g(essentially)f(a)g (one-well)h(semiclassical)e(problem.)35 b(Here)28 b(we)f(ha)n(v)o(e)257 3477 y(relax)o(ed)35 b(the)f(assumptions)e(and)i(introduced)g(the)g(e)o (xtra)g(twist)g(by)g Fu(d)p Fr(\000\()p Fu(Q)p Fr(\))g Fx(to)g(restore)g(the)257 3597 y(essential)24 b(features)i(of)e(a)i (one-well)e(problem.)404 3717 y(The)k(full)h(twisted)e(W)l(itten)i (Laplacian)f(is)h(of)f(the)h(form,)h(cf.)43 b(\(I.35\),)30 b(\(I.47\),)f(\(I.53\))g(and)257 3838 y(\(I.54\),)286 4050 y Fr(\001)367 4065 y Fq(H)q(;Q)589 4050 y Fr(=)748 3955 y Fm(X)781 4165 y Fq(i;j)908 4050 y Fr(\002)984 4065 y Fq(i)1029 4050 y Fu(Z)1096 4065 y Fq(i)1124 4050 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1380 4009 y Fs(\003)1373 4075 y Fq(j)1418 4050 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)1676 4065 y Fq(j)1733 4050 y Fo(\012)23 b Fu(a)1884 4009 y Fs(\003)1884 4075 y Fq(i)1940 4050 y Fu(a)1991 4065 y Fq(j)2078 4050 y Fr(+)50 b Fu(Z)2278 4009 y Fs(\003)2271 4075 y Fq(j)2317 4050 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)2575 4065 y Fq(j)2627 4050 y Fr(\002)2703 4065 y Fq(i)2748 4050 y Fu(Z)2815 4065 y Fq(i)2843 4050 y Fr(\()p Fu(H)8 b Fr(\))21 b Fo(\012)i Fu(a)3180 4065 y Fq(j)3233 4050 y Fu(a)3284 4009 y Fs(\003)3284 4075 y Fq(i)589 4359 y Fr(=)748 4265 y Fm(X)803 4475 y Fq(j)908 4249 y Fm(n)974 4359 y Fr(\002)1050 4374 y Fq(j)1103 4359 y Fu(Z)1170 4374 y Fq(j)1207 4359 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1463 4318 y Fs(\003)1456 4384 y Fq(j)1501 4359 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)1759 4374 y Fq(j)1817 4359 y Fo(\012)22 b Fu(a)1967 4318 y Fs(\003)1967 4384 y Fq(j)2023 4359 y Fu(a)2074 4374 y Fq(j)2161 4359 y Fr(+)50 b(\002)2363 4374 y Fq(j)2416 4359 y Fu(Z)2490 4318 y Fs(\003)2483 4384 y Fq(j)2529 4359 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)2766 4374 y Fq(j)2802 4359 y Fr(\()p Fu(G)p Fr(\))g(\002)3048 4374 y Fq(j)3106 4359 y Fo(\012)22 b Fu(a)3256 4374 y Fq(j)3310 4359 y Fu(a)3361 4318 y Fs(\003)3361 4384 y Fq(j)3400 4249 y Fm(o)1220 4659 y Fr(+)1340 4564 y Fm(X)1356 4777 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1501 4578 y Fm(\002)1542 4659 y Fr(\002)1618 4674 y Fq(i)1663 4659 y Fu(Z)1730 4674 y Fq(i)1758 4659 y Fr(\()p Fu(H)8 b Fr(\))21 b Fu(;)39 b(Z)2084 4618 y Fs(\003)2077 4684 y Fq(j)2123 4659 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)2381 4674 y Fq(j)2417 4578 y Fm(\003)2497 4659 y Fo(\012)39 b Fu(a)2664 4618 y Fs(\003)2664 4684 y Fq(i)2721 4659 y Fu(a)2772 4674 y Fq(j)2830 4659 y Fu(;)589 4971 y Fr(=)748 4876 y Fm(X)803 5086 y Fq(j)908 4860 y Fm(n)974 4971 y Fr(\002)1050 4986 y Fq(j)1103 4971 y Fu(Z)1170 4986 y Fq(j)1207 4971 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1463 4930 y Fs(\003)1456 4996 y Fq(j)1501 4971 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)1759 4986 y Fq(j)1817 4971 y Fo(\012)22 b Fu(a)1967 4930 y Fs(\003)1967 4996 y Fq(j)2023 4971 y Fu(a)2074 4986 y Fq(j)2161 4971 y Fr(+)50 b(\002)2363 4986 y Fq(j)2416 4971 y Fu(Z)2490 4930 y Fs(\003)2483 4996 y Fq(j)2529 4971 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)2766 4986 y Fq(j)2802 4971 y Fr(\()p Fu(G)p Fr(\))g(\002)3048 4986 y Fq(j)3106 4971 y Fo(\012)22 b Fu(a)3256 4986 y Fq(j)3310 4971 y Fu(a)3361 4930 y Fs(\003)3361 4996 y Fq(j)3400 4860 y Fm(o)1220 5298 y Fr(+)1339 5230 y(2)p 1333 5275 61 4 v 1333 5366 a Fu(\014)1421 5203 y Fm(X)1437 5415 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1581 5298 y Fr(\002)1657 5313 y Fq(i)1702 5298 y Fr(\002)1778 5313 y Fq(j)1831 5298 y Fu(H)1920 5256 y Fs(00)1912 5322 y Fq(ij)1972 5298 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\012)g Fu(a)2309 5256 y Fs(\003)2309 5322 y Fq(i)2366 5298 y Fu(a)2417 5313 y Fq(j)2476 5298 y Fu(;)768 b Fx(\(I.59\))p eop %%Page: 16 17 16 16 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(16)257 573 y(where)31 b(we)f(use)g Fr([)p Fu(Z)934 588 y Fq(i)962 573 y Fr(\()p Fu(H)8 b Fr(\))p Fu(;)17 b Fr(\002)1247 588 y Fq(j)1283 573 y Fr(])37 b(=)g(0)30 b Fx(and)g Fr([)p Fu(Z)1807 588 y Fq(i)1835 573 y Fr(\()p Fu(H)8 b Fr(\))p Fu(;)17 b(Z)2118 537 y Fs(\003)2111 597 y Fq(j)2156 573 y Fr(\()p Fu(H)8 b Fr(\)])37 b(=)g(2)p Fu(\014)2608 537 y Fs(\000)p Ft(1)2702 573 y Fu(H)2791 537 y Fs(00)2783 597 y Fq(i;j)2863 573 y Fx(,)31 b(for)f Fu(i)38 b Fo(6)p Fr(=)f Fu(j)6 b Fx(,)31 b(and)257 693 y(we)25 b(denote)g Fu(H)778 657 y Fs(00)770 718 y Fq(ij)858 693 y Fr(:=)j Fu(@)1040 708 y Fq(i)1068 693 y Fu(@)1119 708 y Fq(j)1156 693 y Fu(H)8 b Fx(.)404 814 y(Inserting)35 b(\(I.38\))i(into)f(\(I.59\),)j(we)e(arri)n(v)o(e)f(at)g(the)h(twisted) e(W)l(itten)h(Laplacian)g(on)g Fr(1)p Fx(-)257 934 y(forms,)347 1171 y Fr(\001)428 1120 y Ft(\(1\))428 1198 y Fq(H)q(;Q)649 1171 y Fr(=)808 1076 y Fm(X)864 1286 y Fq(j)969 1060 y Fm(n)1035 1171 y Fr(\002)1111 1186 y Fq(j)1164 1171 y Fu(Z)1231 1186 y Fq(j)1268 1171 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1524 1129 y Fs(\003)1517 1195 y Fq(j)1562 1171 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)1820 1186 y Fq(j)1905 1171 y Fr(+)2049 1076 y Fm(X)2031 1292 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2228 1171 y Fr(\002)2304 1186 y Fq(k)2363 1171 y Fu(Z)2437 1129 y Fs(\003)2430 1195 y Fq(k)2477 1171 y Fr(\()p Fu(G)p Fr(\))g Fu(Z)2714 1186 y Fq(k)2755 1171 y Fr(\()p Fu(G)p Fr(\))g(\002)3001 1186 y Fq(k)3043 1060 y Fm(o)3149 1171 y Fo(\012)39 b Fu(E)3337 1186 y Fq(j)t(j)808 1507 y Fr(+)928 1439 y(2)p 922 1484 61 4 v 922 1575 a Fu(\014)1009 1412 y Fm(X)1018 1624 y Fq(j)t Fs(6)p Ft(=)p Fq(k)1170 1507 y Fr(\002)1246 1522 y Fq(j)1299 1507 y Fr(\002)1375 1522 y Fq(k)1434 1507 y Fu(H)1523 1466 y Fs(00)1515 1531 y Fq(j)t(k)1628 1507 y Fo(\012)g Fu(E)1816 1522 y Fq(j)t(k)1914 1507 y Fu(:)1330 b Fx(\(I.60\))257 1884 y Fl(I.4)119 b(Corr)n(elation)30 b(Identity)g(and)h(Exponential)g(W)-8 b(eights)257 2071 y Fx(A)19 b(remarkable)g(consequence)g(of)g(Theorem)g(I.4)g(is)f(that)h (it)f(allo)n(ws)g(us)h(to)f(write)h(the)g(truncated)257 2192 y(correlation)28 b(\(I.2\))g(of)g(tw)o(o)g(observ)n(ables)f Fu(u;)17 b(v)36 b Fo(2)e Fu(C)2063 2155 y Ft(1)2103 2192 y Fr(\()p Fp(R)2207 2155 y Ft(\003)2266 2192 y Fu(;)17 b Fp(R)t Fr(\))34 b Fx(as)28 b(a)h(matrix)e(element)g(of)h(the)257 2325 y(resolv)o(ent)c(of)h Fr(\001)835 2275 y Ft(\(1\))835 2352 y Fq(H)q(;Q)974 2325 y Fx(,)f(as)h(follo)n(ws.)257 2538 y Fv(Theor)n(em)32 b(I.5.)44 b Fn(Assume)29 b(\(I.49\))h(and)f (let)h Fu(u;)17 b(v)40 b Fo(2)e Fu(C)2121 2502 y Ft(1)2160 2538 y Fr(\()p Fp(R)2264 2502 y Ft(\003)2323 2538 y Fu(;)17 b Fp(R)5 b Fr(\))36 b Fn(suc)o(h)29 b(that)g Fo(jr)p Fu(u)p Fo(j)g Fn(and)g Fo(jr)p Fu(v)t Fo(j)257 2659 y Fn(ar)l(e)c(polynomially)f(bounded)g(in)g Fu(x)p Fn(.)31 b(Then)300 2879 y Fp(E)360 2838 y Fq(T)360 2903 y(\014)421 2879 y Fr(\()p Fu(u)17 b Fr(;)33 b Fu(v)t Fr(\))52 b(=)g Fu(\014)922 2838 y Fs(\000)p Ft(2)1033 2879 y Fo(Z)1113 2837 y Fs(\000)p Ft(1)1105 2906 y Fq(\014)3271 2879 y Fx(\(I.61\))466 3097 y Fo(\001)494 2957 y Fm(\034)585 3002 y(X)637 3215 y Fq(k)729 3097 y Fr(\()p Fu(@)818 3112 y Fq(k)861 3097 y Fu(u)p Fr(\))17 b Fu(e)1017 3056 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1292 3068 y Fj(k)1329 3056 y Ft(\))1383 3097 y Fo(\012)39 b Fu(a)1550 3056 y Fs(\003)1550 3122 y Fq(k)1593 3097 y Fr(\012)1680 2953 y Fm(\014)1680 3012 y(\014)1680 3072 y(\014)1680 3132 y(\014)1730 2986 y(\020)1789 3097 y Fr(\001)1870 3046 y Ft(\(1\))1870 3124 y Fq(H)q(;Q)2009 2986 y Fm(\021)2068 3009 y Fs(\000)p Ft(1)2163 2986 y Fm(\020)2239 3002 y(X)2296 3215 y Fq(`)2383 3097 y Fr(\()p Fu(@)2472 3112 y Fq(`)2505 3097 y Fu(v)t Fr(\))17 b Fu(e)2656 3056 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)2931 3068 y Fj(`)2960 3056 y Ft(\))3014 3097 y Fo(\012)39 b Fu(a)3181 3056 y Fs(\003)3181 3122 y Fq(`)3221 3097 y Fr(\012)3291 2986 y Fm(\021)3351 2957 y(\035)3426 3097 y Fu(;)257 3401 y Fn(wher)l(e)1005 3546 y Fo(Z)1077 3561 y Fq(\014)1177 3546 y Fr(:=)1332 3411 y Fm(Z)1388 3636 y Fh(R)1436 3617 y Fk(\003)1502 3546 y Fu(e)1547 3505 y Fs(\000)p Ft(2)p Fq(\014)s(H)5 b Ft(\()p Fq(x)p Ft(\))1859 3546 y Fu(dx)52 b Fr(=)2146 3462 y Fm(\015)2146 3521 y(\015)2201 3546 y Fu(e)2246 3505 y Fs(\000)p Fq(\014)s(H)2434 3546 y Fo(\012)22 b Fr(\012)2603 3462 y Fm(\015)2603 3521 y(\015)2659 3488 y Ft(2)2721 3546 y Fu(:)523 b Fx(\(I.62\))257 3796 y Fn(Pr)l(oof:)30 b Fx(W)-8 b(e)25 b(\002rst)g(observ)o(e)f(that) 307 4060 y Fp(E)368 4075 y Fq(\014)421 4060 y Fr(\()p Fu(u)p Fr(\))51 b(=)g Fo(Z)811 4019 y Fs(\000)p Ft(1)803 4088 y Fq(\014)941 3924 y Fm(Z)997 4150 y Fh(R)1045 4131 y Fk(\003)1111 4060 y Fu(u)p Fr(\()p Fu(x)p Fr(\))17 b Fu(e)1360 4019 y Fs(\000)p Ft(2)p Fq(\014)s(H)5 b Ft(\()p Fq(x)p Ft(\))1671 4060 y Fu(dx)52 b Fr(=)g Fo(Z)2037 4019 y Fs(\000)p Ft(1)2029 4088 y Fq(\014)2150 3979 y Fm(\012)2197 4060 y Fu(e)2242 4019 y Fs(\000)p Fq(\014)s(H)2427 4060 y Fo(\012)19 b Fr(\012)2610 3975 y Fm(\014)2610 4035 y(\014)2660 4060 y Fu(u)e(e)2778 4019 y Fs(\000)p Fq(\014)s(H)2961 4060 y Fo(\012)j Fr(\012)3128 3979 y Fm(\013)3194 4060 y Fu(;)50 b Fx(\(I.63\))257 4326 y(and)25 b(hence)360 4546 y Fp(E)420 4505 y Fq(T)420 4570 y(\014)481 4546 y Fr(\()p Fu(u)17 b Fr(;)32 b Fu(v)t Fr(\))83 b(=)g Fp(E)1042 4561 y Fq(\014)1096 4546 y Fr(\()p Fu(u)21 b Fo(\001)h Fu(v)t Fr(\))44 b Fo(\000)h Fp(E)1576 4561 y Fq(\014)1629 4546 y Fr(\()p Fu(u)p Fr(\))17 b Fp(E)1838 4561 y Fq(\014)1891 4546 y Fr(\()p Fu(v)t Fr(\))823 4701 y(=)83 b Fp(E)1042 4716 y Fq(\014)1096 4620 y Fm(\000)1141 4701 y Fr(\()p Fu(u)38 b Fo(\000)i Fp(E)1450 4716 y Fq(\014)1503 4701 y Fr(\()p Fu(u)p Fr(\)\))22 b Fo(\001)f Fr(\()p Fu(v)43 b Fo(\000)c Fp(E)2049 4716 y Fq(\014)2102 4701 y Fr(\()p Fu(v)t Fr(\)\))2267 4620 y Fm(\001)3271 4701 y Fx(\(I.64\))823 4856 y Fr(=)83 b Fo(Z)1062 4815 y Fs(\000)p Ft(1)1054 4884 y Fq(\014)1173 4776 y Fm(\012)1220 4856 y Fr(\()p Fu(u)38 b Fo(\000)h Fp(E)1529 4871 y Fq(\014)1582 4856 y Fr(\()p Fu(u)p Fr(\)\))17 b Fu(e)1814 4815 y Fs(\000)p Fq(\014)s(H)2000 4856 y Fo(\012)23 b Fr(\012)2187 4772 y Fm(\014)2187 4832 y(\014)2237 4856 y Fu(P)2300 4871 y Ft(0)2356 4856 y Fr(\()p Fu(v)42 b Fo(\000)e Fp(E)2660 4871 y Fq(\014)2713 4856 y Fr(\()p Fu(v)t Fr(\)\))17 b Fu(e)2940 4815 y Fs(\000)p Fq(\014)s(H)3127 4856 y Fo(\012)22 b Fr(\012)3296 4776 y Fm(\013)3366 4856 y Fu(;)257 5076 y Fx(where)k(we)f(use)f(that)h Fu(v)j Fx(is)d(real-v)n (alued)f(and)1001 5296 y Fu(P)1064 5311 y Ft(0)1155 5296 y Fr(:=)1311 5186 y Fm(\020)1370 5296 y Fi(1)50 b Fo(\000)h(Z)1684 5255 y Fs(\000)p Ft(1)1676 5324 y Fq(\014)1795 5296 y Fo(j)p Fu(e)1868 5255 y Fs(\000)p Fq(\014)s(H)2033 5296 y Fo(ih)p Fu(e)2156 5255 y Fs(\000)p Fq(\014)s(H)2321 5296 y Fo(j)2349 5186 y Fm(\021)2458 5296 y Fo(\012)f Fu(P)2648 5311 y Ft(\012)2725 5296 y Fu(;)519 b Fx(\(I.65\))p eop %%Page: 17 18 17 17 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(17)257 573 y(denotes)34 b(the)g(orthogonal)f(projection)g(onto)g (the)h(orthogonal)f(complement)g(to)g Fp(C)55 b Fo(\001)29 b Fu(e)3330 537 y Fs(\000)p Fq(\014)s(H)257 693 y Fx(times)i(the)g (projection)f(onto)h(the)g(F)o(ock)g(v)n(acuum,)h Fu(P)2118 708 y Ft(\012)2212 693 y Fr(:=)40 b Fo(j)p Fr(\012)p Fo(ih)p Fr(\012)p Fo(j)p Fx(.)50 b(W)-8 b(e)32 b(used)e(abo)o(v)o(e)h (that)257 814 y Fr(\()p Fu(u)22 b Fo(\000)h Fp(E)533 829 y Fq(\014)586 814 y Fr(\()p Fu(u)p Fr(\)\))17 b(exp\()p Fo(\000)p Fu(\014)6 b(H)i Fr(\))22 b Fo(\012)g Fr(\012)29 b Fo(2)f Fr(Ran)16 b Fu(P)1792 829 y Ft(0)1831 814 y Fx(.)404 934 y(Ne)o(xt,)24 b(Eq.)h(\(I.49\))g(implies)e(that)i Fr(Ran)p Fo(f)p Fu(D)1855 949 y Fq(H)q(;Q)2010 934 y Fu(P)2073 949 y Ft(0)2112 934 y Fo(g)44 b(?)i Fr(Ker)p Fo(f)p Fr(\001)2617 949 y Fq(H)q(;Q)2755 934 y Fo(g)p Fx(,)25 b(which)g(allo)n(ws)f(us)257 1054 y(to)h(write)627 1228 y Fu(P)690 1243 y Ft(0)813 1228 y Fr(=)82 b Fu(P)1034 1243 y Ft(0)1074 1228 y Fu(D)1155 1243 y Fq(H)q(;Q)1310 1228 y Fr(\001)1391 1187 y Fs(\000)p Ft(1)1391 1255 y Fq(H)q(;Q)1546 1228 y Fu(D)1627 1243 y Fq(H)q(;Q)1782 1228 y Fu(P)1845 1243 y Ft(0)3271 1228 y Fx(\(I.66\))813 1403 y Fr(=)g Fu(P)1034 1418 y Ft(0)1090 1403 y Fu(d)1141 1361 y Fs(\003)1141 1427 y Fq(H)q(;Q)1296 1403 y Fr(\001)1377 1361 y Fs(\000)p Ft(1)1377 1430 y Fq(H)q(;Q)1532 1403 y Fu(d)1583 1418 y Fq(H)q(;Q)1738 1403 y Fu(P)1801 1418 y Ft(0)1893 1403 y Fr(=)52 b Fu(P)2084 1418 y Ft(0)2140 1403 y Fu(d)2191 1361 y Fs(\003)2191 1427 y Fq(H)q(;Q)2346 1322 y Fm(\000)2392 1403 y 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y(correlation.)2704 b Fg(\003)404 2798 y Fx(As)24 b(an)h(immediate)f(consequence)h(of)g(\(I.61\),)f(we)h (ha)n(v)o(e)g(the)g(identity)307 2996 y Fp(E)368 2955 y Fq(T)368 3021 y(\014)429 2996 y Fr(\()p Fu(x)522 3011 y Fq(i)567 2996 y Fr(;)33 b Fu(x)682 3011 y Fq(j)719 2996 y Fr(\))52 b(=)g Fu(\014)998 2955 y Fs(\000)p Ft(2)1109 2996 y Fo(Z)1189 2955 y Fs(\000)p Ft(1)1181 3024 y Fq(\014)1300 2886 y Fm(D)1361 2996 y Fu(e)1406 2955 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1681 2965 y Fj(i)1707 2955 y Ft(\))1776 2996 y Fo(\012)37 b Fu(e)1935 3011 y Fq(i)1980 2882 y Fm(\014)1980 2941 y(\014)1980 3001 y(\014)2030 2916 y(\000)2075 2996 y Fr(\001)2156 2945 y Ft(\(1\))2156 3023 y Fq(H)q(;Q)2295 2916 y Fm(\001)2341 2938 y Fs(\000)p Ft(1)2452 2916 y Fm(\000)2497 2996 y Fu(e)2542 2955 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)2817 2965 y Fj(j)2850 2955 y Ft(\))2919 2996 y Fo(\012)37 b Fu(e)3078 3011 y Fq(j)3115 2916 y Fm(\001)3160 2886 y(E)3271 2996 y Fx(\(I.69\))257 3205 y(for)32 b(the)g(tw)o(o-point)f(correlation)g(function,)i(where)f Fu(e)2152 3220 y Fq(i)2221 3205 y Fo(\021)41 b Fu(a)2390 3169 y Fs(\003)2390 3230 y Fq(i)2430 3205 y Fr(\012)32 b Fx(denotes)g(the)f Fu(i)3055 3169 y Fq(th)3158 3205 y Fx(standard)257 3325 y(basis)24 b(v)o(ector)h(in)f Fp(C)922 3289 y Ft(\003)981 3325 y Fx(.)31 b(The)24 b(presence)i(of)e (the)h(modi\002ers)f Fu(q)2288 3340 y Fq(i)2341 3325 y Fx(and)h Fu(q)2553 3340 y Fq(j)2614 3325 y Fx(in)g(the)f(e)o (xponents)f(leads)257 3446 y(us)33 b(to)f(study)g(the)g(stability)f(of) i(the)f(partition)g(function)g(under)g(perturbations)g(in)g(a)h(single) 257 3566 y(spin)24 b(v)n(ariable.)31 b(W)-8 b(e)25 b(ha)n(v)o(e)f(the)h (follo)n(wing)e(theorem,)h(which)g(is)h(pro)o(v)o(ed)e(in)i(Section)g (III)257 3730 y Fv(Theor)n(em)41 b(I.6.)48 b Fn(Assume)38 b(Hypotheses)h(1)f(and)g(2.)73 b(Ther)l(e)39 b(e)n(xists)f(a)h(univer)o (sal)f(constant)257 3850 y Fu(c)29 b(>)f Fr(0)d Fn(suc)o(h)g(that,)f (for)h(all)f Fu(o)k Fo(2)h Fr(\003)p Fn(,)c 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Fk(2)3108 4148 y Fs(g)3170 4189 y Fu(:)74 b Fx(\(I.70\))404 4425 y(This)20 b(result)h(states)g (that)g(perturbing)g(the)h(Hamiltonian)d(function)i(in)g(one)h(spin-v)n (ariable)257 4545 y(only)35 b(changes)g(the)g(partition)f(function)h (by)g(an)g(e)o(xponentially)f(small)g(error)i(in)f(the)g(lo)n(w-)257 4666 y(temperature)27 b(re)o(gime,)f(pro)o(vided)g(the)h(perturbation)e (is)i(supported)e(a)o(w)o(ay)i(from)g(the)f(global)257 4786 y(minimum.)404 4907 y(Exponential)21 b(decay)i(estimates)e(for)i (the)f(measure)h Fr(exp)q([)p Fo(\000)p Fr(2)p Fu(\014)6 b(H)i Fr(])17 b Fu(d)2741 4870 y Ft(\003)2793 4907 y Fu(x)p Fx(,)23 b(as)g(a)g(single)e(spin)257 5027 y(v)n(ariable)f(is)f (a)o(w)o(ay)h(from)g(the)f(global)g(minimum)f(are)j(complemented)d(by)i (the)g(decay)g(proper)n(-)257 5147 y(ties)k(of)f Fp(E)587 5111 y Fq(T)587 5173 y(\014)648 5147 y Fr(\()p Fu(x)741 5162 y Fq(i)786 5147 y Fr(;)33 b Fu(x)901 5162 y Fq(j)938 5147 y Fr(\))24 b Fx(in)f(\(I.69\),)h(as)g(the)f(distance)h(between)f Fu(i)i Fx(and)e Fu(j)30 b Fx(in)23 b Fr(\003)h Fx(tends)f(to)g(be)h (lar)n(ge.)257 5268 y(F)o(or)k(the)g(analysis)f(of)h(the)g(latter)l(,)h (we)f(introduce)f(e)o(xponential)g(weights)g Fr(exp)q([)p Fu(\024\032)p Fr(\()p Fu(i;)17 b(j)6 b Fr(\)])28 b Fx(and)257 5388 y(observ)o(e)d(the)f(follo)n(wing)f(generalization)h(of)h (\(I.69\).)p eop %%Page: 18 19 18 18 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(18)257 573 y Fv(Lemma)26 b(I.7.)41 b Fn(Let)25 b Fu(\024)j(>)f Fr(0)e Fn(and)f Fu(i;)17 b(j)34 b Fo(2)28 b Fr(\003)p Fn(.)i(Then)381 742 y Fu(e)426 701 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))666 742 y Fp(E)726 701 y Fq(T)726 767 y(\014)787 742 y Fr(\()p Fu(x)880 757 y Fq(i)925 742 y Fr(;)k Fu(x)1041 757 y Fq(j)1078 742 y Fr(\))2155 b Fx(\(I.71\))464 927 y Fr(=)83 b Fu(\014)684 886 y Fs(\000)p Ft(2)794 927 y Fo(Z)874 886 y Fs(\000)p Ft(1)866 955 y Fq(\014)986 817 y Fm(D)1046 927 y Fu(e)1091 886 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1366 896 y Fj(i)1393 886 y 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Fs(\003)2191 1835 y Fq(k)2234 1811 y Fu(a)2285 1826 y Fq(`)2340 1811 y Fu(;)904 b Fx(\(I.72\))257 2089 y Fn(and)25 b Fu(\032)j Fr(:)f(\003)22 b Fo(\002)h Fr(\003)k Fo(!)h Fp(R)1043 2048 y Ft(+)1043 2114 y(0)1133 2089 y Fn(is)c(the)h(metric)f(on)h Fr(\003)p Fn(.)404 2249 y Fx(Before)37 b(we)g(turn)f(to)f(the)i(proof)f(of)g(Lemma)g(I.7,) j(we)d(remark)h(that)f(the)g(e)o(xponential)257 2370 y(decay)d(of)f(the)g(tw)o(o-point)f(correlation)g(function)h(is)f (analyzed)h(in)g(detail)g(in)f(Subsect.)h(I.6.)257 2490 y(Loosely)24 b(speaking,)g(the)g(decay)h(rate)g(of)g(the)f(tw)o (o-point)f(correlation)i(function)e(is)i(gi)n(v)o(en)e(by)257 2610 y(the)34 b(limes)g(superior)g(of)g(all)g Fu(\024)45 b(>)g Fr(0)p Fx(,)37 b(for)d(which)g(\(I.61\))h(stays)e(bounded,)j(as)e Fu(\032)p Fr(\()p Fu(i;)17 b(j)6 b Fr(\))35 b Fx(gets)257 2731 y(lar)n(ge.)404 2861 y(Another)18 b(remark)h(to)g(be)g(made)f(is)h (that)f Fr(Ker)f(\001)2003 2810 y Ft(\(0\))2003 2888 y Fq(H)q(;Q)2141 2861 y Fr(\()p Fu(\024;)g(j)6 b Fr(\))28 b(=)f Fp(C)20 b Fo(\001)p Fu(e)2633 2825 y Fs(\000)p Fq(\014)s(H)2823 2861 y Fx(and)f Fr(Ker)e(\001)3241 2810 y Ft(\(1\))3241 2888 y Fq(H)q(;Q)3380 2861 y Fr(\()p Fu(\024;)g(j)6 b Fr(\))27 b(=)257 2982 y Fo(f)p Fr(0)p Fo(g)p Fx(,)44 b(for)c(all)g Fr(0)55 b Fo(\024)i Fu(\024)f Fo(\024)g Fr(1)40 b Fx(and)g Fu(j)62 b Fo(2)57 b Fr(\003)p Fx(,)43 b(due)d(to)g(the)g(f)o(act)g(that)g Fr(\001)2811 2997 y Fq(H)q(;Q)2950 2982 y Fr(\()p Fu(\024;)17 b(j)6 b Fr(\))39 b Fx(deri)n(v)o(es)257 3102 y(from)f Fr(\001)570 3117 y Fq(H)q(;Q)761 3102 y Fo(\021)53 b Fr(\001)972 3117 y Fq(H)q(;Q)1111 3102 y Fr(\(0)p Fu(;)17 b(j)6 b Fr(\))37 b Fx(by)h(conjugation)e(with)i(the)f(bounded)h(and)g(in)l(v)o (ertible)e(op-)257 3222 y(erator)26 b Fr(exp)q([)p Fu(d)p Fr(\000\()p Fu(M)935 3237 y Fq(j)971 3222 y Fr(\)])p Fx(,)f(see)g(belo)n(w)-6 b(.)257 3343 y Fn(Pr)l(oof:)39 b Fx(De\002ne)30 b(a)f(diagonal)g Fr(\003)c Fo(\002)h Fr(\003)p Fx(-matrix)i Fu(M)1974 3358 y Fq(j)2040 3343 y Fx(by)h Fr(\()p Fu(M)2301 3358 y Fq(j)2338 3343 y Fr(\))2376 3358 y Fq(k)r(`)2483 3343 y Fr(:=)36 b Fu(\024)17 b(\032)p Fr(\()p Fu(k)s(;)g(j)6 b Fr(\))17 b Fu(\016)3025 3358 y Fq(k)r(`)3096 3343 y Fx(.)44 b(Observ)o(e)257 3463 y(that)30 b Fo(\006)p Fu(d)p Fr(\000\()p Fu(M)758 3478 y Fq(j)795 3463 y Fr(\))d Fu(a)911 3427 y Fs(\003)911 3488 y Fq(i)950 3463 y Fr(\012)39 b(=)f Fo(\006)p Fu(\024)17 b(\032)p Fr(\()p Fu(i;)g(j)6 b Fr(\))27 b Fu(a)1650 3427 y Fs(\003)1650 3488 y Fq(i)1690 3463 y Fr(\012)k Fx(and)f(that)g Fu(d)p Fr(\000\()p Fu(M)2389 3478 y Fq(j)2426 3463 y Fr(\))g Fx(lea)n(v)o(es)h Fo(H)2856 3427 y Ft(\()p Fq(n)p Ft(\))2988 3463 y Fx(in)l(v)n(ariant,)g(cf.)257 3584 y(\(I.41\).)g(Hence)296 3753 y Fu(\014)357 3712 y Ft(2)413 3753 y Fo(Z)485 3768 y Fq(\014)560 3753 y Fu(e)605 3712 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))845 3753 y Fp(E)905 3712 y Fq(T)905 3777 y(\014)966 3753 y Fr(\()p Fu(x)1059 3768 y Fq(i)1104 3753 y Fr(;)j Fu(x)1220 3768 y Fq(j)1257 3753 y Fr(\))1976 b Fx(\(I.73\))379 3938 y Fr(=)538 3827 y Fm(D)599 3938 y Fu(e)644 3897 y Fq(d)p Ft(\000\()p Fq(M)819 3907 y Fj(j)853 3897 y Ft(\))901 3857 y Fm(\000)946 3938 y Fu(e)991 3897 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1266 3907 y Fj(i)1293 3897 y Ft(\))1363 3938 y Fo(\012)39 b Fu(a)1530 3897 y Fs(\003)1530 3963 y Fq(i)1570 3938 y Fr(\012)1640 3857 y Fm(\001)1714 3823 y(\014)1714 3883 y(\014)1714 3943 y(\014)1772 3938 y Fr(\001)1853 3897 y Fs(\000)p Ft(1)1853 3965 y Fq(H)q(;Q)2014 3938 y Fu(e)2059 3897 y Fs(\000)p Fq(d)p Ft(\000\()p Fq(M)2289 3907 y Fj(j)2322 3897 y Ft(\))2370 3857 y Fm(\000)2416 3938 y Fu(e)2461 3897 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)2736 3907 y Fj(j)2768 3897 y Ft(\))2839 3938 y Fo(\012)39 b Fu(a)3006 3897 y Fs(\003)3006 3963 y Fq(j)3046 3938 y Fr(\012)3116 3857 y Fm(\001)3162 3827 y(E)379 4151 y Fr(=)538 4040 y Fm(D)599 4070 y(\000)645 4151 y Fu(e)690 4109 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)965 4119 y Fj(i)991 4109 y Ft(\))1062 4151 y Fo(\012)39 b Fu(a)1229 4109 y Fs(\003)1229 4175 y Fq(i)1268 4151 y Fr(\012)1338 4070 y Fm(\001)1412 4036 y(\014)1412 4096 y(\014)1412 4156 y(\014)1470 4070 y(\000)1516 4151 y Fu(e)1561 4109 y Fq(d)p Ft(\000\()p Fq(M)1736 4117 y Fj(o)1771 4109 y Ft(\))1820 4151 y Fr(\001)1901 4166 y Fq(H)q(;Q)2056 4151 y Fu(e)2101 4109 y Fs(\000)p Fq(d)p Ft(\000\()p Fq(M)2331 4117 y Fj(o)2366 4109 y Ft(\))2398 4070 y Fm(\001)2444 4092 y Fs(\000)p Ft(1)2555 4070 y Fm(\000)2600 4151 y Fu(e)2645 4109 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)2920 4119 y Fj(j)2953 4109 y Ft(\))3023 4151 y Fo(\012)39 b Fu(a)3190 4109 y Fs(\003)3190 4175 y Fq(j)3230 4151 y Fr(\012)3300 4070 y Fm(\001)3346 4040 y(E)3429 4151 y Fu(:)257 4340 y Fx(Since)959 4461 y Fu(e)1004 4420 y Fq(d)p Ft(\000\()p Fq(M)1179 4430 y Fj(j)1212 4420 y Ft(\))1261 4461 y Fu(a)1312 4420 y Fs(\003)1312 4485 y Fq(k)1371 4461 y Fu(a)1422 4476 y Fq(`)1472 4461 y Fu(e)1517 4420 y Fs(\000)p Fq(d)p Ft(\000\()p Fq(M)1747 4432 y Fj(l)1772 4420 y Ft(\))1856 4461 y Fr(=)52 b Fu(e)2029 4420 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(k)r(;j)t Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(`;j)t Ft(\)])2550 4461 y Fu(a)2601 4420 y Fs(\003)2601 4485 y Fq(k)2660 4461 y Fu(a)2711 4476 y Fq(`)2766 4461 y Fu(;)478 b Fx(\(I.74\))257 4613 y(we)25 b(obtain)f(Eq.)h(\(I.71\))g (from)f(the)h(computation)385 4782 y Fu(e)430 4741 y Fq(d)p Ft(\000\()p Fq(M)605 4751 y Fj(j)638 4741 y Ft(\))686 4782 y Fr(\001)767 4797 y Fq(H)q(;Q)922 4782 y Fu(e)967 4741 y Fs(\000)p Fq(d)p Ft(\000\()p Fq(M)1197 4751 y Fj(j)1231 4741 y Ft(\))468 4982 y Fr(=)626 4887 y Fm(X)679 5099 y Fq(k)787 4871 y Fm(n)853 4982 y Fr(\002)929 4997 y Fq(k)989 4982 y Fu(Z)1056 4997 y Fq(k)1098 4982 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1354 4940 y Fs(\003)1347 5006 y Fq(k)1392 4982 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)1650 4997 y Fq(k)1714 4982 y Fo(\012)23 b Fu(a)1865 4940 y Fs(\003)1865 5006 y Fq(k)1907 4982 y Fu(a)1958 4997 y Fq(k)2051 4982 y Fr(+)50 b(\002)2253 4997 y Fq(k)2312 4982 y Fu(Z)2386 4940 y Fs(\003)2379 5006 y Fq(k)2425 4982 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)2662 4997 y Fq(k)2704 4982 y Fr(\()p Fu(G)p Fr(\))g(\002)2950 4997 y Fq(k)3014 4982 y Fo(\012)23 b Fu(a)3165 4997 y Fq(k)3208 4982 y Fu(a)3259 4940 y Fs(\003)3259 5006 y Fq(k)3302 4871 y Fm(o)1099 5298 y Fr(+)1218 5230 y(2)p 1212 5275 V 1212 5366 a Fu(\014)1300 5203 y Fm(X)1310 5415 y Fq(k)r Fs(6)p Ft(=)p Fq(`)1460 5298 y Fu(e)1505 5256 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(k)r(;j)t Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(`;j)t Ft(\)])2014 5298 y Fr(\002)2090 5313 y Fq(k)2149 5298 y Fr(\002)2225 5313 y Fq(`)2275 5298 y Fu(H)2364 5256 y Fs(00)2356 5322 y Fq(k)r(`)2427 5298 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\012)g Fu(a)2764 5256 y Fs(\003)2764 5322 y Fq(k)2824 5298 y Fu(a)2875 5313 y Fq(`)2930 5298 y Fu(:)314 b Fx(\(I.75\))p eop %%Page: 19 20 19 19 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(19)3418 573 y Fg(\003)257 868 y Fl(I.5)119 b(Comparison)30 b(Operator)257 1056 y Fx(T)-8 b(o)28 b(moti)n(v)n(ate)e(our)i(ne)o(xt)f (de\002nitions,)g(we)i(observ)o(e)e(that,)h(by)g(setting)f Fu(Q)33 b Fr(=)h(0)28 b Fx(in)f(\(I.60\),)i(we)257 1176 y(obtain)24 b(the)h(W)l(itten)f(Laplacian)h(that)f(has)h(been)g (analyzed)g(in)f([2],)1205 1437 y Fr(\001)1286 1386 y Ft(\(1\))1286 1464 y Fq(H)q(;)p Ft(0)1457 1437 y Fr(=)53 b(\001)1667 1386 y Ft(\(0\))1667 1464 y Fq(H)q(;)p Ft(0)1807 1437 y Fo(\012)23 b Fi(1)50 b Fr(+)2155 1370 y(2)p 2149 1414 61 4 v 2149 1505 a Fu(\014)2236 1437 y(H)2325 1396 y Fs(00)2367 1437 y Fr(\()p Fu(x)p Fr(\))22 b Fu(:)724 b Fx(\(I.76\))257 1701 y(In)30 b([2],)h(the)f(semiclassical)f (con\002ning)g(properties)g(of)h(the)g(one-well)f(problem)g(were)i (used)257 1831 y(to)25 b(sho)n(w)f(that)g Fr(\001)848 1780 y Ft(\(1\))848 1858 y Fq(H)q(;)p Ft(0)991 1831 y Fx(is)h(close)f(to)h(the)f(comparison)g(operator)1224 2111 y Fu(A)1297 2060 y Ft(\(1\))1297 2138 y Fq(H)q(;)p Ft(0)1468 2111 y Fr(:=)52 b(\001)1704 2060 y Ft(\(0\))1704 2138 y Fq(H)q(;)p Ft(0)1845 2111 y Fo(\012)23 b Fi(1)49 b Fr(+)2192 2043 y(2)p 2186 2088 V 2186 2179 a Fu(\014)2273 2111 y(H)2362 2070 y Fs(00)2404 2111 y Fr(\(0\))742 b Fx(\(I.77\))257 2407 y(in)27 b(the)g(sense)g(that)g(their)g(dif)n (ference)g Fu(W)1675 2356 y Ft(\(1\))1661 2434 y Fq(H)q(;)p Ft(0)1811 2407 y Fr(:=)32 b(\001)2027 2356 y Ft(\(1\))2027 2434 y Fq(H)q(;)p Ft(0)2170 2407 y Fo(\000)24 b Fu(A)2344 2356 y Ft(\(1\))2344 2434 y Fq(H)q(;)p Ft(0)2490 2407 y Fx(is)i(a)i(small)e(perturbation)g(of)257 2556 y Fu(A)330 2505 y Ft(\(1\))330 2583 y Fq(H)q(;)p Ft(0)449 2556 y Fx(,)34 b(as)e(a)g(quadratic)g(form.)52 b(More)32 b(speci\002cally)-6 b(,)34 b(it)d(w)o(as)h(sho)n(wn)f(in)h([2])g(that)g(if)g Fu(\013)42 b Fo(\025)f Fr(0)257 2676 y Fx(is)27 b(suf)n(\002ciently)g (small)f(and)i Fu(\014)38 b(<)32 b Fo(1)27 b Fx(is)g(suf)n(\002ciently) g(lar)n(ge,)h(then)f(for)h(an)o(y)f(v)o(ector)g Fu( )k Fx(in)c(the)257 2807 y(form)e(domain)f(of)h Fu(A)981 2756 y Ft(\(1\))981 2834 y Fq(H)q(;)p Ft(0)1099 2807 y Fx(,)1120 3010 y Fm(\012)1167 3090 y Fu( )1251 3006 y Fm(\014)1251 3065 y(\014)1311 3090 y Fu(W)1417 3039 y Ft(\(1\))1403 3117 y Fq(H)q(;)p Ft(0)1549 3090 y Fu( )1616 3010 y Fm(\013)1716 3090 y Fo(\024)1903 3023 y Fu(C)p 1856 3067 171 4 v 1856 3159 a(\014)1917 3130 y Ft(1)p Fq(=)p Ft(2)2059 3010 y Fm(\012)2106 3090 y Fu( )2189 3006 y Fm(\014)2189 3065 y(\014)2250 3090 y Fu(A)2323 3039 y Ft(\(1\))2323 3117 y Fq(H)q(;)p Ft(0)2469 3090 y Fu( )2536 3010 y Fm(\013)2606 3090 y Fu(;)638 b Fx(\(I.78\))257 3354 y(for)27 b(some)e(uni)n(v)o(ersal)f(constant)h Fu(C)37 b(<)30 b Fo(1)p Fx(.)k(The)26 b(deri)n(v)n(ation)e(of)i(F)o(orm)g (Bound)f(\(I.78\))i(in)e([2],)257 3474 y(ho)n(we)n(v)o(er)l(,)j(uses)f (Property)h(\(I.12\),)h(which)f(we)g(are)h(lacking)e(in)h(the)f (present)h(conte)o(xt,)g(in)f(an)257 3595 y(essential)d(w)o(ay)-6 b(.)404 3715 y(T)e(o)23 b(treat)g(Hamiltonian)f(functions)g(that)h (merely)g(obe)o(y)g(the)g(weak)o(er)h(Hypotheses)e(1)h(and)257 3836 y(2,)i(we)g(suppose)f(we)h(ha)n(v)o(e)f(constructed)h Fu(q)1722 3851 y Fq(j)1783 3836 y Fx(and)32 b Fr(^)-56 b Fu(q)1995 3851 y Fq(j)2057 3836 y Fx(according)24 b(to)h(Lemma)f(I.2) h(and)g(formed)257 3969 y Fr(\002)333 3984 y Fq(j)397 3969 y Fr(=)j(exp)q([)p Fu(\014)6 b(q)781 3984 y Fq(j)817 3969 y Fr(])25 b Fx(and)1049 3944 y Fm(b)1038 3969 y Fr(\002)1114 3984 y Fq(j)1178 3969 y Fr(=)j(exp)q([)p Fu(\014)13 b Fr(^)-56 b Fu(q)1562 3984 y Fq(j)1598 3969 y Fr(])p Fx(,)25 b(respecti)n(v)o(ely)-6 b(,)23 b(for)i(all)g Fu(j)33 b Fo(2)28 b Fr(\003)p Fx(.)j(W)-8 b(e)25 b(then)g(de\002ne)356 4236 y Fu(A)429 4251 y Fq(H)q(;Q)620 4236 y Fr(:=)775 4142 y Fm(X)831 4352 y Fq(j)936 4126 y Fm(n)1028 4211 y Fr(\024)1002 4236 y Fu(A)1075 4251 y Fq(j)1134 4236 y Fo(\012)e Fu(a)1285 4195 y Fs(\003)1285 4261 y Fq(j)1341 4236 y Fu(a)1392 4251 y Fq(j)1473 4236 y Fr(+)44 b Fu(A)1666 4251 y Fq(j)1725 4236 y Fo(\012)22 b Fu(a)1875 4251 y Fq(j)1929 4236 y Fu(a)1980 4195 y Fs(\003)1980 4261 y Fq(j)2019 4126 y Fm(o)2108 4236 y Fr(+)2249 4169 y(2)p 2244 4213 61 4 v 2244 4305 a Fu(\014)2347 4142 y Fm(X)2381 4352 y Fq(i;j)2508 4236 y Fu(H)2597 4195 y Fs(00)2589 4261 y Fq(ij)2649 4236 y Fr(\(0\))39 b Fo(\012)g Fu(a)2980 4195 y Fs(\003)2980 4261 y Fq(i)3036 4236 y Fu(a)3087 4251 y Fq(j)3146 4236 y Fu(;)98 b Fx(\(I.79\))257 4550 y(where)872 4745 y Fr(\024)846 4770 y Fu(A)919 4785 y Fq(j)1039 4770 y Fr(:=)83 b(\002)1301 4785 y Fq(j)1354 4770 y Fu(Z)1421 4785 y Fq(j)1457 4770 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1713 4729 y Fs(\003)1706 4795 y Fq(j)1752 4770 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)2010 4785 y Fq(j)2089 4770 y Fo(\000)45 b Fr(2)p Fu(\014)2321 4729 y Fs(\000)p Ft(1)2445 4745 y Fr(\024)2431 4770 y(\002)2507 4729 y Ft(2)2507 4795 y Fq(j)2585 4745 y Fr(^)2563 4770 y Fu(G)2640 4729 y Fs(00)2640 4795 y Fq(j)t(j)2709 4770 y Fr(\()p Fu(x)p Fr(\))23 b Fu(;)381 b Fx(\(I.80\))846 4963 y Fu(A)919 4978 y Fq(j)1039 4963 y Fr(:=)83 b(\002)1301 4978 y Fq(j)1354 4963 y Fu(Z)1428 4922 y Fs(\003)1421 4988 y Fq(j)1467 4963 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)1704 4978 y Fq(j)1740 4963 y Fr(\()p Fu(G)p Fr(\))g(\002)1986 4978 y Fq(j)2044 4963 y Fu(:)1200 b Fx(\(I.81\))257 5196 y(W)-8 b(e)31 b(remark)f(that)f(Eq.)h(\(I.79\))g(coincides)f(with)g(\(I.77\),)i(for)f Fu(Q)37 b Fr(=)g(0)p Fx(.)46 b(\(The)30 b(notation)3315 5171 y Fr(\024)3289 5196 y Fu(A)3362 5211 y Fq(j)3429 5196 y Fx(is)257 5317 y(moti)n(v)n(ated)g(by)i(the)g(estimate)f (\(I.92\).\))54 b(Furthermore,)34 b(for)e Fr(0)41 b Fo(\024)h Fu(\024)f Fo(\024)h Fr(1)32 b Fx(and)g Fu(o)42 b Fo(2)f Fr(\003)p Fx(,)34 b(we)p eop %%Page: 20 21 20 20 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(20)257 573 y(de\002ne)382 738 y Fu(W)474 753 y Fq(H)q(;Q)612 738 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))83 b(:=)g Fu(W)1295 753 y Ft(diag)1430 738 y Fr(\()p Fu(x)p Fr(\))50 b(+)g Fu(W)1829 753 y Ft(o\013)5 b Fs(\000)p Ft(d)2009 738 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))940 b Fx(\(I.82\))1017 884 y Fr(:=)83 b Fu(W)1295 899 y Ft(diag)1430 884 y Fr(\()p Fu(x)p Fr(\))50 b(+)g(\001)p Fu(W)1910 899 y Ft(o\013)5 b Fs(\000)p Ft(d)2090 884 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))49 b(+)h Fu(W)2679 899 y Ft(o\013)5 b Fs(\000)p Ft(d)2859 884 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))k Fu(;)576 1076 y(W)668 1091 y Ft(diag)803 1076 y Fr(\()p Fu(x)p Fr(\))83 b(:=)1219 1009 y(2)p 1213 1053 61 4 v 1213 1145 a Fu(\014)1300 982 y Fm(X)1356 1192 y Fq(j)1461 966 y Fm(\020)1534 1051 y Fr(\024)1520 1076 y(\002)1596 1035 y Ft(2)1596 1101 y Fq(j)1674 1051 y Fr(^)1652 1076 y Fu(G)1729 1035 y Fs(00)1729 1101 y Fq(j)t(j)1798 1076 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)g Fu(H)2173 1035 y Fs(00)2165 1101 y Fq(j)t(j)2234 1076 y Fr(\(0\))2359 966 y Fm(\021)2457 1076 y Fo(\012)g Fu(a)2624 1035 y Fs(\003)2624 1101 y Fq(j)2664 1076 y Fu(a)2715 1091 y Fq(j)2774 1076 y Fu(;)470 b Fx(\(I.83\))340 1379 y Fu(W)432 1394 y Ft(o\013)5 b Fs(\000)p Ft(d)612 1379 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))83 b(:=)1219 1312 y(2)p 1213 1356 V 1213 1448 a Fu(\014)1300 1285 y Fm(X)1317 1497 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1461 1269 y Fm(\020)1520 1379 y Fu(e)1565 1338 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])2046 1379 y Fr(\002)2122 1394 y Fq(i)2150 1379 y Fr(\002)2226 1394 y Fq(j)2279 1379 y Fu(H)2368 1338 y Fs(00)2360 1404 y Fq(ij)2421 1379 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)g Fu(H)2796 1338 y Fs(00)2788 1404 y Fq(ij)2848 1379 y Fr(\(0\))2973 1269 y Fm(\021)3071 1379 y Fo(\012)g Fu(a)3238 1338 y Fs(\003)3238 1404 y Fq(i)3278 1379 y Fu(a)3329 1394 y Fq(j)3271 1609 y Fx(\(I.84\))1031 1780 y Fr(=)1213 1712 y(2)17 b Fu(\013)p 1213 1757 129 4 v 1247 1848 a(\014)1368 1685 y Fm(X)1384 1897 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1528 1669 y Fm(\020)1588 1780 y Fu(e)1633 1739 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])2114 1780 y Fr(\002)2190 1795 y Fq(i)2218 1780 y Fr(\002)2294 1795 y Fq(j)2347 1780 y Fu(w)2420 1739 y Fs(00)2417 1804 y 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Fu(@)945 2805 y Fq(j)982 2790 y Fu(w)1052 2805 y Fq(ij)1112 2790 y Fx(,)e(and)g(we)g(observ)o(e)f(the)h(decomposition)d(identity)1120 2956 y Fr(\001)1201 2971 y Fq(H)q(;Q)1340 2956 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))52 b(=)g Fu(A)1816 2971 y Fq(H)q(;Q)2005 2956 y Fr(+)d Fu(W)2222 2971 y Fq(H)q(;Q)2361 2956 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))22 b Fu(:)638 b Fx(\(I.87\))257 3121 y(The)38 b(ar)n(gument)g(\223)p Fu(x)p Fx(\224)h(in)e (\(I.82\)\226\(I.84\))h(indicates)f(that)g(these)h(operators)g(act)g (as)g(matrix-)257 3242 y(v)n(alued)18 b(multiplication)e(operators)j (and)g(contain)f(no)h(dif)n(ferential)f(operator)-5 b(.)28 b(W)-8 b(e)19 b(frequently)257 3362 y(omit)32 b(to)g(display)f Fu(x)p Fx(.)55 b(Note)32 b(also)g(that)g(we)h(ha)n(v)o(e)g(the)f (identites)f Fu(W)2622 3377 y Ft(o\013)5 b Fs(\000)p Ft(d)2802 3362 y Fr(\(0)p Fu(;)17 b(o)p Fr(;)g(0\))41 b(=)h(0)32 b Fx(and)257 3482 y Fr(\001)p Fu(W)430 3497 y Ft(o\013)5 b Fs(\000)p Ft(d)611 3482 y Fr(\(0)p Fu(;)17 b(o)p Fr(;)g Fu(x)p Fr(\))27 b(=)g Fu(W)1148 3497 y Ft(o\013)5 b Fs(\000)p Ft(d)1328 3482 y Fr(\(0)p Fu(;)17 b(o)p Fr(;)g Fu(x)p Fr(\))p Fx(.)404 3603 y(For)22 b Fu(n)28 b Fr(=)g(0)p Fu(;)17 b Fr(1)p Fx(,)k(we)h(compute)f(the)h(restrictions)f(of)h Fu(A)2232 3618 y Fq(H)q(;Q)2392 3603 y Fx(onto)g Fo(H)2677 3567 y Ft(\()p Fq(n)p Ft(\))2779 3603 y Fx(,)g(which)f(we)i(denote)257 3740 y(by)i Fu(A)455 3689 y Ft(\()p Fq(n)p Ft(\))455 3767 y Fq(H)q(;Q)593 3740 y Fx(,)g(respecti)n(v)o(ely)-6 b(.)29 b(From)c(\(I.36\)\226\(I.38\),)g(we)g(immediately)e(obtain)332 3938 y Fu(A)405 3887 y Ft(\(0\))405 3965 y Fq(H)q(;Q)626 3938 y Fr(=)785 3843 y Fm(X)841 4053 y Fq(j)946 3938 y Fu(A)1019 3953 y Fq(j)1108 3938 y Fr(=)1236 3843 y Fm(X)1292 4053 y Fq(j)1397 3938 y Fr(\002)1473 3953 y Fq(j)1526 3938 y Fu(Z)1600 3897 y Fs(\003)1593 3962 y Fq(j)1639 3938 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)1876 3953 y Fq(j)1912 3938 y Fr(\()p Fu(G)p Fr(\))g(\002)2158 3953 y Fq(j)2216 3938 y Fu(;)1028 b Fx(\(I.88\))332 4254 y Fu(A)405 4203 y 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a Fu(\014)2657 4475 y Fm(X)2674 4687 y Fq(i)p Fs(6)p Ft(=)p Fq(j)2818 4569 y Fu(H)2907 4528 y Fs(00)2899 4594 y Fq(ij)2959 4569 y Fr(\(0\))39 b Fo(\012)g Fu(E)3311 4584 y Fq(ij)3394 4569 y Fu(:)257 4839 y Fx(W)-8 b(e)26 b(further)f(observ)o(e)f(that)730 5004 y Fu(d)p Fr(\000)842 4923 y Fm(\000)888 5004 y Fu(A)961 4953 y Ft(\(1\))961 5031 y Fq(H)q(;Q)1099 4923 y Fm(\001)1228 5004 y Fr(=)83 b Fu(A)1460 5019 y Fq(H)q(;Q)1643 5004 y Fr(+)1763 4910 y Fm(X)1818 5120 y Fq(j)1923 5004 y Fu(A)1996 5019 y Fq(j)2055 5004 y Fo(\012)2155 4894 y Fm(n)2256 4910 y(X)2238 5125 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2435 5004 y Fu(a)2486 4963 y Fs(\003)2486 5029 y Fq(k)2529 5004 y Fu(a)2580 5019 y Fq(k)2661 5004 y Fo(\000)40 b Fu(a)2829 5019 y Fq(j)2865 5004 y Fu(a)2916 4963 y Fs(\003)2916 5029 y Fq(j)2956 4894 y Fm(o)1228 5300 y Fr(=)83 b Fu(A)1460 5315 y Fq(H)q(;Q)1643 5300 y Fr(+)1763 5189 y Fm(\020)1839 5205 y(X)1895 5415 y Fq(j)1999 5300 y Fu(A)2072 5315 y Fq(j)2109 5189 y Fm(\021)2191 5300 y Fo(\012)2290 5219 y Fm(\000)2336 5300 y Fu(N)2414 5315 y Fs(F)2497 5300 y Fo(\000)23 b Fi(1)2653 5219 y Fm(\001)2721 5300 y Fu(;)523 b Fx(\(I.90\))p eop %%Page: 21 22 21 21 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(21)257 573 y(where)1308 693 y Fu(N)1386 708 y Fs(F)1500 693 y Fr(:=)52 b Fu(d)p Fr(\000\()p Fi(1)p Fr(\))g(=)2080 599 y Fm(X)2133 811 y Fq(k)2240 693 y Fu(a)2291 652 y Fs(\003)2291 718 y Fq(k)2351 693 y Fu(a)2402 708 y Fq(k)3271 693 y Fx(\(I.91\))257 951 y(is)28 b(the)g(fermion)f(number)g(\(=)i (form)e(de)o(gree\))h(operator)-5 b(.)40 b(Note)28 b(that)f Fu(N)2710 966 y Fs(F)2799 951 y Fx(is)h(bounded,)f(more)257 1071 y(precisely)e Fo(k)p Fu(N)770 1086 y Fs(F)831 1071 y Fo(k)j Fr(=)f Fo(j)p Fr(\003)p Fo(j)p Fx(,)d(thanks)g(to)g(the)h(P)o (auli)f(principle.)404 1205 y(W)-8 b(e)37 b(will)e(write)h Fu(W)1109 1154 y Ft(\()p Fq(k)r Ft(\))1095 1232 y Fq(H)q(;Q)1234 1205 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))36 b Fx(for)h(the)f(restriction) f(of)i Fu(W)2456 1220 y Fq(H)q(;Q)2594 1205 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))36 b Fx(to)g Fu(k)s Fx(-forms,)j(and)257 1326 y(lik)o(e)n(wise)24 b(for)h(the)g(deri)n(v)o(ed)e(quantities)h (presented)g(in)h(the)f(formula)h(\(I.82\).)257 1621 y Fl(I.6)119 b(Exponential)31 b(Decay)f(of)f(Corr)n(elations)257 1809 y Fx(In)i(this)f(subsection)f(we)i(pro)o(v)o(e)f(our)g(main)g (result,)h(Theorem)g(I.1.)48 b(Our)31 b(proof)f(relies)h(on)f(a)257 1929 y(form)35 b(bound)f(to)g(be)h(presented)f(belo)n(w)-6 b(,)36 b(cf.)f(\(I.94\).)60 b(This)34 b(form)h(bound)e(is)i(contained)f (in)257 2049 y(Theorem)25 b(II.1,)g(which)f(is)h(the)f(technical)h (centerpiece)g(of)g(the)g(present)f(w)o(ork.)257 2241 y Fv(Pr)n(oof)f(of)f(Theor)n(em)i(I.1:)58 b Fx(W)-8 b(e)23 b(be)o(gin)e(with)g(the)h(remark)h(that)e(from)h(\(I.80\),)h(\(I.54\),) f(and)g(the)257 2361 y(f)o(acts)j(that)654 2336 y Fm(b)644 2361 y Fr(\002)720 2376 y Fq(j)770 2336 y Fr(\024)756 2361 y(\002)832 2376 y Fq(j)896 2361 y Fr(=)j(\002)1076 2376 y Fq(j)1137 2361 y Fx(and)1316 2336 y Fm(b)1306 2361 y Fr(\002)1382 2376 y Fq(j)1446 2361 y Fo(\025)g Fr(1)p Fx(,)d(we)g(get)925 2569 y Fr(\024)899 2594 y Fu(A)972 2609 y Fq(j)1093 2594 y Fr(=)1266 2569 y(\024)1252 2594 y(\002)1328 2609 y Fq(j)1381 2594 y Fo(f)p Fu(Z)1498 2609 y Fq(j)1534 2594 y Fr(\()1595 2569 y(^)1572 2594 y Fu(G)p Fr(\))1714 2569 y Fm(b)1704 2594 y Fr(\002)1780 2553 y Ft(2)1780 2618 y Fq(j)1836 2594 y Fu(Z)1903 2609 y Fq(j)1939 2594 y Fr(\()1999 2569 y(^)1977 2594 y Fu(G)p Fr(\))2092 2553 y Fs(\003)2181 2594 y Fo(\000)50 b Fr(2)p Fu(\014)2418 2553 y Fs(\000)p Ft(1)2551 2569 y Fr(^)2529 2594 y Fu(G)2606 2609 y Fq(j)t(j)2675 2594 y Fo(g)2755 2569 y Fr(\024)2742 2594 y(\002)2818 2609 y Fq(j)1092 2759 y Fo(\025)1266 2734 y Fr(\024)1252 2759 y(\002)1328 2774 y Fq(j)1381 2759 y Fo(f)p Fu(Z)1498 2774 y Fq(j)1534 2759 y Fr(\()1595 2734 y(^)1572 2759 y Fu(G)p Fr(\))17 b Fu(Z)1771 2774 y Fq(j)1807 2759 y Fr(\()1867 2734 y(^)1845 2759 y Fu(G)p Fr(\))1960 2718 y Fs(\003)2049 2759 y Fo(\000)51 b Fr(2)p Fu(\014)2287 2718 y Fs(\000)p Ft(1)2419 2734 y Fr(^)2397 2759 y Fu(G)2474 2774 y Fq(j)t(j)2543 2759 y Fo(g)2623 2734 y Fr(\024)2610 2759 y(\002)2686 2774 y Fq(j)1093 2914 y Fr(=)1266 2889 y(\024)1252 2914 y(\002)1328 2929 y Fq(j)1381 2914 y Fu(Z)1448 2929 y Fq(j)1485 2914 y Fr(\()1545 2889 y(^)1523 2914 y Fu(G)p Fr(\))1638 2873 y Fs(\003)1694 2914 y Fu(Z)1761 2929 y Fq(j)1797 2914 y Fr(\()1857 2889 y(^)1835 2914 y Fu(G)p Fr(\))1980 2889 y(\024)1967 2914 y(\002)2043 2929 y Fq(j)2101 2914 y Fu(:)1143 b Fx(\(I.92\))257 3134 y(In)25 b(particular)l(,)g(this)f (implies)f(that)1480 3109 y Fr(\024)1454 3134 y Fu(A)1527 3149 y Fq(j)1592 3134 y Fo(\025)28 b Fr(0)p Fx(,)c(and)h(hence)g(we)h (\002nd)e(that)1213 3402 y Fu(A)1286 3351 y Ft(\(1\))1286 3429 y Fq(H)q(;Q)1477 3402 y Fo(\025)1623 3334 y Fr(2)p 1617 3379 61 4 v 1617 3470 a Fu(\014)1704 3402 y(H)1793 3361 y Fs(00)1835 3402 y Fr(\(0\))52 b Fo(\025)2158 3334 y Fr(2)p 2152 3379 V 2152 3470 a Fu(\014)2239 3402 y(\025)2296 3417 y Ft(min)2435 3402 y Fi(1)22 b Fu(;)731 b Fx(\(I.93\))257 3672 y(where)38 b Fu(\025)595 3687 y Ft(min)767 3672 y Fr(:=)50 b(min)15 b Fu(\033)t Fr(\()p Fu(H)1285 3636 y Fs(00)1327 3672 y Fr(\(0\)\))37 b Fx(is)f(the)h(lo)n(west)f(eigen)l (v)n(alue)g(of)h(the)g(Hessian)g(of)g Fu(H)44 b Fx(at)257 3793 y Fu(x)55 b Fr(=)g(0)p Fx(.)74 b(Observ)o(e)39 b(that,)j(since)d Fu(f)1536 3756 y Fs(00)1525 3817 y Fq(j)1578 3793 y Fr(\(0\))55 b Fo(\025)g Fu(c)1932 3808 y Fq(f)2032 3793 y Fu(>)f Fr(0)39 b Fx(and)h Fo(j)p Fu(@)2513 3808 y Fq(\033)2560 3793 y Fu(@)2611 3808 y Fq(\034)2654 3793 y Fu(w)2727 3756 y Fs(00)2724 3817 y Fq(ij)2784 3793 y Fr(\(0)p Fu(;)17 b Fr(0\))p Fo(j)54 b(\024)h Fu(a)3267 3808 y Fq(ij)3328 3793 y Fx(,)43 b(by)257 3913 y(Hypotheses)37 b(1)h(and)f(2,)k(the)d (eigen)l(v)n(alue)f Fu(\025)1825 3928 y Ft(min)1998 3913 y Fo(\025)52 b Fu(c)2169 3928 y Fq(f)2214 3913 y Fu(=)p Fr(2)g Fu(>)f Fr(0)38 b Fx(is)f(strictly)g(positi)n(v)o(e)e(\(for)257 4033 y Fu(\013)29 b(<)e(\013)513 4048 y Ft(0)578 4033 y Fx(small\).)404 4154 y(Let)k Fu(\017)607 4118 y Fs(0)670 4154 y Fu(>)h Fx(\(to)f(be)g(chosen)g(belo)n(w\).)50 b(Theorem)31 b(II.1)g(implies)f(that)h(there)g(e)o(xists)f(a)i(uni-)257 4274 y(v)o(ersal)g(constant)f Fu(C)48 b(<)41 b Fo(1)p Fx(,)34 b(such)e(that,)h(for)g(all)e Fr(0)41 b Fo(\024)h Fu(\024)f(<)h Fr(1)p Fx(,)33 b Fu(\014)47 b(>)41 b(C)7 b(")2860 4238 y Fs(0)2883 4227 y(\000)p Ft(1)p Fq(=)p Ft(2)3048 4274 y Fx(,)34 b Fr(0)41 b Fo(\024)g Fu(\013)h(<)257 4394 y(C)334 4358 y Fs(\000)p Ft(1)429 4394 y Fu(")475 4358 y Fs(0)498 4394 y Fr(\(1)21 b Fo(\000)i Fu(\024)p Fr(\))i Fx(and)g(all)f Fu(j)34 b Fo(2)28 b Fr(\003)p Fx(,)d(the)f(follo)n(wing)f(estimate)h(holds)g(true)727 4543 y Fm(\014)727 4603 y(\014)760 4547 y(\012)807 4628 y Fu( )891 4543 y Fm(\014)891 4603 y(\014)949 4628 y Fu(W)1055 4577 y Ft(\(1\))1041 4655 y Fq(H)q(;Q)1180 4628 y Fr(\()p Fu(\024;)17 b(j)6 b Fr(\))p Fu(\036)1460 4547 y Fm(\013)1506 4543 y(\014)1506 4603 y(\014)1592 4628 y Fo(\024)53 b Fu(")1768 4587 y Fs(0)1813 4543 y Fm(\015)1813 4603 y(\015)1868 4628 y Fr(\()p Fu(A)1979 4577 y Ft(\(1\))1979 4655 y Fq(H)q(;Q)2118 4628 y Fr(\))2156 4587 y Ft(1)p Fq(=)p Ft(2)2266 4628 y Fu( )2333 4543 y Fm(\015)2333 4603 y(\015)2410 4543 y(\015)2410 4603 y(\015)2466 4628 y Fr(\()p Fu(A)2577 4577 y Ft(\(1\))2577 4655 y Fq(H)q(;Q)2715 4628 y Fr(\))2753 4587 y Ft(1)p Fq(=)p Ft(2)2863 4628 y Fu(\036)2921 4543 y Fm(\015)2921 4603 y(\015)2998 4628 y Fu(:)246 b Fx(\(I.94\))257 4848 y(Write)313 4987 y Fm(\000)359 5068 y Fr(\001)440 5017 y Ft(\(1\))440 5095 y Fq(H)q(;Q)579 5068 y Fr(\()p Fu(\024;)17 b(j)6 b Fr(\))801 4987 y Fm(\001)846 5010 y Fs(\000)p Ft(1)3271 5068 y Fx(\(I.95\))396 5272 y Fr(=)555 5192 y Fm(\000)601 5272 y Fu(A)674 5221 y Ft(\(1\))674 5299 y Fq(H)q(;Q)813 5192 y Fm(\001)858 5214 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1040 5162 y Fm(\020)1099 5272 y Fi(1)50 b Fr(+)1331 5192 y Fm(\000)1376 5272 y Fu(A)1449 5221 y Ft(\(1\))1449 5299 y Fq(H)q(;Q)1588 5192 y Fm(\001)1634 5214 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1815 5272 y Fu(W)1921 5221 y Ft(\(1\))1907 5299 y Fq(H)q(;Q)2046 5272 y Fr(\()p Fu(\024;)17 b(j)6 b Fr(\))2285 5192 y Fm(\000)2330 5272 y Fu(A)2403 5221 y Ft(\(1\))2403 5299 y Fq(H)q(;Q)2541 5192 y Fm(\001)2587 5214 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2752 5162 y Fm(\021)2811 5184 y Fs(\000)p Ft(1)2922 5192 y Fm(\000)2968 5272 y Fu(A)3041 5221 y Ft(\(1\))3041 5299 y Fq(H)q(;Q)3179 5192 y Fm(\001)3225 5214 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)3412 5272 y Fu(:)p eop %%Page: 22 23 22 22 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(22)257 573 y(The)24 b(formbound)e(\(I.94\),)i(with)f Fu(\017)1415 537 y Fs(0)1466 573 y Fu(<)28 b Fr(1)p Fx(,)c(allo)n(ws)e (us)h(to)g(e)o(xpand)g(the)h(resolv)o(ent)e(on)h(the)h(right-)257 693 y(hand)c(side)g(of)h(\(I.95\))f(in)g(a)h(Neumann)f(series.)29 b(This)19 b(observ)n(ation)g(together)h(with)f(Eqs.)h(\(I.93\))257 814 y(imply)k(that)g(for)h(small)f(enough)g Fu(\017)1428 777 y Fs(0)1477 814 y Fx(\(depending)g(on)g Fu(\017)p Fx(\))i(we)f(ha)n(v)o(e)1056 995 y Fm(\015)1056 1055 y(\015)1112 999 y(\000)1157 1080 y Fr(\001)1238 1029 y Ft(\(1\))1238 1107 y Fq(H)q(;Q)1377 1080 y Fr(\()p Fu(\024;)17 b(j)6 b Fr(\))1599 999 y Fm(\001)1644 1021 y Fs(\000)p Ft(1)1739 995 y Fm(\015)1739 1055 y(\015)1847 1080 y Fo(\024)2282 1012 y Fu(\014)p 1987 1057 651 4 v 1987 1148 a Fr(2)17 b(\(1)k Fo(\000)i Fu("=)p Fr(3\))17 b Fu(\025)2517 1163 y Ft(min)2669 1080 y Fu(;)575 b Fx(\(I.96\))257 1350 y(under)25 b(the)g(conditions)e(on)h Fu(\013)i Fx(and)f Fu(\014)30 b Fx(mentioned)24 b(abo)o(v)o(e)g(and)g(for)h(all)g Fu(j)34 b Fo(2)28 b Fr(\003)p Fx(.)404 1470 y(Lemma)c(I.7)h(and)g(the)f (Cauchy-Schw)o(arz)i(inequality)e(yield)355 1669 y Fm(\014)355 1728 y(\014)389 1753 y Fp(E)449 1712 y Fq(T)449 1778 y(\014)510 1753 y Fr(\()p Fu(x)603 1768 y Fq(i)632 1753 y Fr(;)38 b Fu(x)752 1768 y Fq(j)789 1753 y Fr(\))827 1669 y Fm(\014)827 1728 y(\014)913 1753 y Fo(\024)53 b Fu(e)1088 1712 y Fs(\000)p Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))1382 1686 y Fo(k)p Fu(e)1477 1650 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1752 1660 y Fj(i)1778 1650 y Ft(\))1809 1686 y Fo(k)17 b(k)p Fu(e)1971 1650 y Fs(\000)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)2246 1660 y Fj(j)2278 1650 y Ft(\))2310 1686 y Fo(k)p 1382 1730 978 4 v 1753 1822 a Fu(\014)1814 1793 y Ft(2)1869 1822 y Fo(Z)1941 1837 y Fq(\014)2386 1669 y Fm(\015)2386 1728 y(\015)2442 1673 y(\000)2487 1753 y Fr(\001)2568 1702 y Ft(\(1\))2568 1780 y Fq(H)q(;Q)2707 1753 y Fr(\()p Fu(\024;)17 b(j)6 b Fr(\))2929 1673 y Fm(\001)2974 1695 y Fs(\000)p Ft(1)3069 1669 y Fm(\015)3069 1728 y(\015)3146 1753 y Fu(:)98 b Fx(\(I.97\))404 2027 y(Note)22 b(that)g(the)g (Hypothesis)f(1)i(and)f(2)h(remain)f(true)g(if)h(we)g(replace)g(one)f (of)h(the)f Fu(f)3192 2042 y Fq(j)3229 2027 y Fx(')-5 b(s)22 b(with)257 2147 y(the)j(corresponding)f Fu(g)1046 2162 y Fq(j)1082 2147 y Fx(.)31 b(W)-8 b(e)25 b(proceed)g(to)g(apply)f (Theorem)h(I.6,)483 2283 y Fm(Z)538 2508 y Fh(R)586 2489 y Fk(\003)653 2418 y Fu(e)698 2377 y Fs(\000)p Ft(2)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1008 2387 y Fj(i)1034 2377 y Ft(\))1066 2418 y Fu(d)1117 2377 y Ft(\003)1170 2418 y Fu(x)84 b Fr(=)1469 2283 y Fm(Z)1524 2508 y Fs(j)p Fq(x)1584 2518 y Fj(i)1610 2508 y Fs(j)p Fq(<)1699 2492 y Ft(^)1685 2508 y Fq(R)1738 2517 y Fk(0)1793 2418 y Fu(e)1838 2377 y Fs(\000)p Ft(2)p Fq(\014)s(H)2038 2418 y Fu(d)2089 2377 y Ft(\003)2142 2418 y Fu(x)50 b Fr(+)2373 2283 y Fm(Z)2428 2508 y Fs(j)p Fq(x)2488 2518 y Fj(i)2514 2508 y Fs(j)p Fq(>)2604 2492 y Ft(^)2589 2508 y Fq(R)2642 2517 y Fk(0)2697 2418 y Fu(e)2742 2377 y Fs(\000)p Ft(2)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)3052 2387 y Fj(i)3079 2377 y Ft(\))3110 2418 y Fu(d)3161 2377 y Ft(\003)3214 2418 y Fu(x)1308 2694 y Fo(\024)1469 2558 y Fm(Z)1524 2784 y Fh(R)1572 2765 y Fk(\003)1638 2694 y Fu(e)1683 2653 y Fs(\000)p Ft(2)p Fq(\014)s(H)1884 2694 y Fu(d)1935 2653 y Ft(\003)1988 2694 y Fu(x)50 b Fr(+)g Fu(e)2264 2653 y Fs(\000)p Fq(c\014)2413 2558 y Fm(Z)2468 2784 y Fh(R)2516 2765 y Fk(\003)2583 2694 y Fu(e)2628 2653 y Fs(\000)p Ft(2)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)2938 2663 y Fj(i)2964 2653 y Ft(\))2996 2694 y Fu(d)3047 2653 y Ft(\003)3100 2694 y Fu(x)22 b(:)67 b Fx(\(I.98\))257 2960 y(This)24 b(implies,)g(for)h(all)f Fu(i)k Fo(2)g Fr(\003)p Fx(,)1070 3089 y Fm(Z)1126 3315 y Fh(R)1174 3296 y Fk(\003)1240 3225 y Fu(e)1285 3184 y Fs(\000)p Ft(2)p Fq(\014)s Ft(\()p Fq(H)5 b Fs(\000)p Fq(q)1595 3194 y Fj(i)1622 3184 y Ft(\))1653 3225 y Fu(d)1704 3184 y Ft(\003)1757 3225 y Fu(x)53 b Fo(\024)g Fr(\(1)22 b Fo(\000)h Fu(e)2249 3184 y Fs(\000)p Fq(c\014)2381 3225 y Fr(\))2419 3184 y Fs(\000)p Ft(1)2514 3225 y Fo(Z)2586 3240 y Fq(\014)2655 3225 y Fu(;)589 b Fx(\(I.99\))257 3492 y(pro)o(vided)29 b Fu(\014)695 3507 y Ft(0)770 3492 y Fu(<)37 b Fo(1)29 b Fx(is)g(suf)n(\002ciently)f(lar)n(ge.)45 b(Inserting)29 b(\(I.96\))h(and)f(\(I.99\))h(into)e(\(I.97\),)j(we)257 3612 y(arri)n(v)o(e)25 b(at)f(the)h(desired)g(bound,)1173 3781 y Fm(\014)1173 3841 y(\014)1206 3866 y Fp(E)1267 3825 y Fq(T)1267 3890 y(\014)1328 3866 y Fr(\()p Fu(x)1421 3881 y Fq(i)1449 3866 y Fr(;)39 b Fu(x)1570 3881 y Fq(j)1607 3866 y Fr(\))1645 3781 y Fm(\014)1645 3841 y(\014)1731 3866 y Fo(\024)1924 3798 y Fr(1)22 b(+)g Fu(")p 1871 3843 322 4 v 1871 3934 a Fr(2)17 b Fu(\014)k(\025)2070 3949 y Ft(min)2218 3866 y Fu(e)2263 3825 y Fs(\000)p Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))2553 3866 y Fu(:)641 b Fx(\(I.100\))3418 4129 y Fg(\003)404 4249 y Fx(W)-8 b(e)25 b(note)f(that)h(the)f(formbound)g(\(I.94\))h(implies)e(the)i(lo) n(wer)f(bound)1188 4521 y Fr(min)15 b Fu(\033)1426 4440 y Fm(\000)1472 4521 y Fr(\001)1553 4470 y Ft(\(1\))1553 4548 y Fq(H)q(;Q)1691 4440 y Fm(\001)1790 4521 y Fo(\025)1930 4453 y Fr(2)i(\(1)k Fo(\000)h Fu(")2249 4417 y Fs(0)2272 4453 y Fr(\))p 1930 4498 381 4 v 2090 4589 a Fu(\014)2337 4521 y(\025)2394 4536 y Ft(min)2538 4521 y Fu(:)656 b Fx(\(I.101\))p eop %%Page: 23 24 23 23 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(23)257 573 y Fw(II)143 b(F)l(orm)35 b(Bound)g(on)g(the)g(P)m (erturbation)257 796 y Fx(This)i(section)g(is)f(the)i(technical)f (heart)g(of)g(our)h(paper)-5 b(.)68 b(Here)38 b(we)f(pro)o(v)o(e)f(a)i (form)f(bound)257 927 y(sho)n(wing)22 b(that)i Fr(\001)871 876 y Ft(\(1\))871 954 y Fq(H)q(;Q)1037 927 y Fr(=)k Fu(A)1214 876 y Ft(\(1\))1214 954 y Fq(H)q(;Q)1371 927 y Fr(+)18 b Fu(W)1571 876 y Ft(\(1\))1557 954 y Fq(H)q(;Q)1719 927 y Fx(decomposes)23 b(the)h(twisted)f(W)l(itten)g(Laplacian)h(on)257 1076 y Fo(H)342 1040 y Ft(\(1\))470 1076 y Fx(into)32 b(its)h(main)f(part)h Fu(A)1279 1025 y Ft(\(1\))1279 1103 y Fq(H)q(;Q)1418 1076 y Fx(,)i(see)e(\(I.89\),)i(and)e(a)h(small)e (perturbation)g Fu(W)3052 1025 y Ft(\(1\))3038 1103 y Fq(H)q(;Q)3177 1076 y Fx(.)55 b(Being)257 1225 y(slightly)25 b(more)h(general,)g(we)h(sho)n(w)e(that)g Fu(W)1802 1240 y Fq(H)q(;Q)1967 1225 y Fx(is)h(a)g(small)f(compared)h(to)g Fu(d)p Fr(\000\()p Fu(A)3118 1174 y Ft(\(1\))3118 1252 y Fq(H)q(;Q)3256 1225 y Fr(\))p Fx(,)h(dis-)257 1359 y(played)i(in)f(\(I.90\),)h(with)f(relati)n(v)o(e)f(bound)h(of)h(order) g Fu(\014)2155 1323 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2319 1359 y Fx(,)h(which)e(becomes)g(small)f(in)h(the)257 1479 y(lo)n(w-temperature)c(limit)f Fu(\014)34 b Fo(!)27 b Fr(0)p Fx(.)404 1600 y(The)e(main)f(result)g(of)h(this)f(section)g (is)257 1787 y Fv(Theor)n(em)38 b(II.1)d(\(Main)h(F)n(orm)f(Bound\).)50 b Fn(Ther)l(e)36 b(e)n(xist)g(univer)o(sal)e(constants)h Fu(C)55 b(<)48 b Fo(1)p Fn(,)257 1907 y Fu(\013)319 1922 y Ft(0)393 1907 y Fu(>)34 b Fr(0)p Fn(,)29 b(and)f Fu(\014)839 1922 y Ft(0)912 1907 y Fu(<)34 b Fo(1)28 b Fn(suc)o(h)g(that,)g(for)g (all)f Fr(0)34 b Fo(\024)h Fu(\013)g Fo(\024)f Fu(\013)2312 1922 y Ft(0)2352 1907 y Fn(,)29 b(all)f Fu(\014)39 b(>)34 b(\014)2799 1922 y Ft(0)2839 1907 y Fn(,)29 b(all)f Fr(0)34 b Fo(\024)g Fu(\024)h(<)f Fr(1)p Fn(,)257 2028 y(and)25 b(all)f Fu(o)j Fo(2)h Fr(\003)p Fn(,)d(the)g(estimate)423 2176 y Fm(\015)423 2236 y(\015)478 2261 y Fu(d)p Fr(\000\()p Fu(A)701 2210 y Ft(\(1\))701 2288 y Fq(H)q(;Q)839 2261 y Fr(\))877 2220 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1070 2261 y Fu(W)1162 2276 y Fq(H)q(;Q)1300 2261 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))28 b Fu(d)p Fr(\000\()p Fu(A)1774 2210 y Ft(\(1\))1774 2288 y Fq(H)q(;Q)1912 2261 y Fr(\))1950 2220 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2115 2176 y Fm(\015)2115 2236 y(\015)2222 2261 y Fo(\024)54 b Fu(C)2446 2150 y Fm(\020)2576 2193 y Fr(1)p 2516 2238 171 4 v 2516 2329 a Fu(\014)2577 2300 y Ft(1)p Fq(=)p Ft(2)2718 2261 y Fr(+)2872 2193 y Fu(\013)17 b(\024)p 2826 2238 227 4 v 2826 2329 a Fr(1)22 b Fo(\000)h Fu(\024)3063 2150 y Fm(\021)3288 2261 y Fx(\(II.1\))257 2542 y Fn(holds)f(true)h(on)803 2461 y Fm(\000)848 2542 y Fo(H)933 2506 y Ft(\(0\))1028 2461 y Fm(\001)1073 2483 y Fs(?)1132 2542 y Fn(,)h(and)e Fu(W)1445 2557 y Fq(H)q(;Q)1584 2542 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))27 b(=)h(0)22 b Fn(on)h Fo(H)2217 2506 y Ft(\(0\))2311 2542 y Fn(.)31 b(Especially)-5 b(,)22 b(for)g Fu(\024)28 b Fr(=)f(0)c Fn(and)g(on)257 2671 y Fo(H)342 2635 y Ft(\(1\))437 2671 y Fn(,)i(we)g(have)1289 2827 y Fo(\006)p Fu(W)1472 2777 y Ft(\(1\))1458 2854 y Fq(H)q(;Q)1597 2827 y Fr(\(0)p Fu(;)17 b(o)p Fr(\))52 b Fo(\024)2052 2760 y Fu(C)p 2005 2805 171 4 v 2005 2896 a(\014)2066 2867 y Ft(1)p Fq(=)p Ft(2)2202 2827 y Fu(A)2275 2777 y Ft(\(1\))2275 2854 y Fq(H)q(;Q)2436 2827 y Fu(;)825 b Fx(\(II.2\))257 3037 y Fn(in)25 b(the)f(sense)h(of)g(quadr)o(atic)d (forms.)257 3224 y(Pr)l(oof:)33 b Fx(W)-8 b(e)27 b(break)g(up)f(the)h (proof)f(of)g(Theorem)h(II.1)f(into)g(a)h(series)f(of)h(lemmata)e(pro)o (v)o(ed)h(in)257 3345 y(Subsects.)31 b(II.1\226II.5,)i(belo)n(w)-6 b(.)49 b(Each)32 b(of)f(these)g(lemmata)g(contain)g(a)g(bootstrap)g(ar) n(gument)257 3465 y(close)25 b(in)f(spirit)g(to)h(the)f(strate)o(gy)g (applied)g(in)h([2,)f(Sect.)i(III].)404 3586 y(W)-8 b(e)24 b(recall)f(from)h(Eq.)f(\(I.90\))h(the)f(de\002nition)f Fu(N)2055 3601 y Fs(F)2144 3586 y Fr(=)2248 3511 y Fm(P)2353 3615 y Fq(k)2412 3586 y Fi(1)c Fo(\012)g Fu(a)2632 3549 y Fs(\003)2632 3611 y Fq(k)2674 3586 y Fu(a)2725 3601 y Fq(k)2796 3586 y Fr(=)2900 3511 y Fm(L)3010 3537 y Fs(j)p Ft(\003)p Fs(j)3010 3615 y Fq(n)p Ft(=0)3164 3586 y Fu(n)f Fo(\001)g Fi(1)3340 3608 y Fs(H)3400 3589 y Fk(\()p Fj(n)p Fk(\))257 3706 y Fx(of)25 b(the)g(fermion)f(number)g (operator)h(on)g Fo(H)q Fx(.)404 3826 y(W)-8 b(e)30 b(may)h(assume)e (without)g(loss)h(of)g(generality)g(that)g(all)g(operators)g(are)i (restricted)e(to)257 3880 y Fm(\000)303 3961 y Fo(H)388 3925 y Ft(\(0\))482 3880 y Fm(\001)528 3902 y Fs(?)587 3961 y Fx(.)h(Recall)25 b(the)g(decomposition,)e(see)i (\(I.82\)\226\(I.86\),)451 4161 y Fu(W)543 4176 y Fq(H)q(;Q)682 4161 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))52 b(=)g Fu(W)1276 4176 y Ft(diag)1411 4161 y Fr(\()p Fu(x)p Fr(\))22 b(+)g(\001)p Fu(W)1835 4176 y Ft(o\013)5 b Fs(\000)p Ft(d)2016 4161 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))22 b(+)g Fu(W)2550 4176 y Ft(o\013)5 b Fs(\000)p Ft(d)2730 4161 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))k Fu(:)194 b Fx(\(II.3\))257 4362 y(W)-8 b(e)20 b(estimate)e Fu(W)850 4377 y Ft(diag)984 4362 y Fr(\()p Fu(x)p Fr(\)+\001)p Fu(W)1364 4377 y Ft(o\013)5 b Fs(\000)p Ft(d)1545 4362 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))h Fx(and)h Fu(W)2140 4377 y Ft(o\013)5 b Fs(\000)p Ft(d)2320 4362 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))h Fx(separately)-6 b(.)28 b(In)19 b(Lemma)g(II.3)257 4483 y(in)25 b(Subsect.)g(II.1)g(we)g (pro)o(v)o(e)e(that)936 4644 y Fm(\015)936 4704 y(\015)991 4728 y Fu(N)1079 4678 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1069 4755 y Fs(F)1267 4728 y Fu(W)1359 4743 y Ft(o\013)5 b Fs(\000)p Ft(d)1539 4728 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))k Fu(N)1964 4678 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1954 4755 y Fs(F)2130 4644 y Fm(\015)2130 4704 y(\015)2238 4728 y Fo(\024)2403 4661 y Fr(3)c Fu(C)2539 4676 y Fq(a)2596 4661 y Fu(\013)h(\024)p 2378 4706 380 4 v 2378 4797 a Fr(\(1)j Fo(\000)i Fu(\024)p Fr(\))17 b Fu(\014)2789 4728 y(;)472 b Fx(\(II.4\))257 4978 y(and)25 b(the)g(main)f(part)h(of)g (the)f(proof)h(is)f(actually)h(to)f(sho)n(w)g(that)257 5110 y Fm(\015)257 5169 y(\015)257 5229 y(\015)313 5143 y(\000)358 5224 y Fu(M)10 b Fr([)p Fu(b)p 489 5240 42 4 v 2 w Fr(])16 b(+)g Fu(\014)728 5183 y Fs(\000)p Ft(1)821 5224 y Fu(N)899 5239 y Fs(F)961 5143 y Fm(\001)1006 5166 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1171 5143 y Fm(\010)1229 5224 y Fu(W)1321 5239 y Ft(diag)1472 5224 y Fr(+)g(\001)p Fu(W)1737 5239 y Ft(o\013)5 b Fs(\000)p Ft(d)1917 5224 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))2140 5143 y Fm(\011\000)2243 5224 y Fu(M)10 b Fr([)p Fu(b)p 2374 5240 V 2 w Fr(])16 b(+)g Fu(\014)2613 5183 y Fs(\000)p Ft(1)2706 5224 y Fu(N)2784 5239 y Fs(F)2846 5143 y Fm(\001)2891 5166 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)3056 5110 y Fm(\015)3056 5169 y(\015)3056 5229 y(\015)3156 5224 y Fo(\024)3334 5157 y Fu(C)p 3288 5201 171 4 v 3288 5293 a(\014)3349 5264 y Ft(1)p Fq(=)p Ft(2)3468 5224 y Fu(;)3288 5388 y Fx(\(II.5\))p eop %%Page: 24 25 24 24 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(24)257 573 y(for)30 b(some)f(uni)n(v)o(ersal)e(constant)i Fu(C)43 b(<)36 b Fo(1)p Fx(.)44 b(Here,)31 b Fu(M)10 b Fr([)p Fu(b)p 2169 589 42 4 v 1 w Fr(])30 b Fx(belongs)e(to)h(the)g (follo)n(wing)e(chain)257 693 y(of)e(\(possibly)e(unbounded\))h (operators,)680 910 y Fu(J)9 b Fr([)p Fu(a)p 770 926 52 4 v Fr(])53 b(:=)1056 815 y Fm(X)1112 1025 y Fq(j)1217 910 y Fu(J)1271 925 y Fq(j)1307 910 y Fr([)p Fu(a)p 1334 926 V Fr(])23 b Fo(\012)f Fu(a)1585 869 y Fs(\003)1585 934 y Fq(j)1625 910 y Fu(a)1676 925 y Fq(j)1735 910 y Fu(;)166 b(K)7 b Fr([)p Fu(b)p 2045 926 42 4 v 1 w Fr(])52 b(:=)2322 815 y Fm(X)2378 1025 y Fq(j)2482 910 y Fu(K)2565 925 y Fq(j)2602 910 y Fr([)p Fu(b)p 2629 926 V 1 w Fr(])22 b Fo(\012)g Fu(a)2870 869 y Fs(\003)2870 934 y Fq(j)2910 910 y Fu(a)2961 925 y Fq(j)3020 910 y Fu(;)241 b Fx(\(II.6\))683 1186 y Fu(L)p Fr([)p Fu(b)p 776 1202 V 1 w Fr(])53 b(:=)1054 1091 y Fm(X)1109 1301 y Fq(j)1214 1186 y Fu(L)1280 1201 y Fq(j)1317 1186 y Fr([)p Fu(b)p 1344 1202 V Fr(])23 b Fo(\012)f Fu(a)1585 1145 y Fs(\003)1585 1210 y Fq(j)1625 1186 y Fu(a)1676 1201 y Fq(j)1735 1186 y Fu(;)166 b(M)10 b Fr([)p Fu(b)p 2059 1202 V 1 w Fr(])53 b(:=)2337 1091 y Fm(X)2392 1301 y Fq(j)2497 1186 y Fu(M)2591 1201 y Fq(j)2628 1186 y Fr([)p Fu(b)p 2655 1202 V 1 w Fr(])22 b Fo(\012)g Fu(a)2896 1145 y Fs(\003)2896 1210 y Fq(j)2936 1186 y Fu(a)2987 1201 y Fq(j)3046 1186 y Fu(;)215 b Fx(\(II.7\))257 1489 y(where)33 b Fu(J)587 1504 y Fq(j)623 1489 y Fr([)p Fu(a)p 650 1505 52 4 v Fr(])p Fx(,)h Fu(K)870 1504 y Fq(j)906 1489 y Fr([)p Fu(b)p 933 1505 42 4 v 1 w Fr(])p Fx(,)g Fu(L)1127 1504 y Fq(j)1163 1489 y Fr([)p Fu(b)p 1190 1505 V 1 w Fr(])e Fx(and)g Fu(M)1561 1504 y Fq(j)1598 1489 y Fr([)p Fu(b)p 1625 1505 V Fr(])g Fx(are)h(multiplication)c (operators,)k(multiplying)c(by)257 1609 y(the)c(follo)n(wing)e (functions)1034 1826 y Fu(J)1088 1841 y Fq(j)1125 1826 y Fr([)p Fu(a)p 1152 1842 52 4 v Fr(])83 b(:=)1512 1801 y(\024)1499 1826 y(\002)1575 1785 y Ft(2)1575 1851 y Fq(j)1631 1826 y Fo(j)s Fr(^)-52 b Fu(g)1710 1785 y Fs(0)1706 1851 y Fq(j)1742 1826 y Fo(j)44 b Fr(+)1952 1731 y Fm(X)1934 1947 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2131 1826 y Fu(a)2182 1841 y Fq(j)t(k)2274 1826 y Fr(\002)2350 1785 y Ft(2)2350 1851 y Fq(k)2409 1826 y Fo(j)p Fu(g)2488 1785 y Fs(0)2484 1851 y Fq(k)2526 1826 y Fo(j)22 b Fu(;)685 b Fx(\(II.8\))1015 2112 y Fu(K)1098 2127 y Fq(j)1134 2112 y Fr([)p Fu(b)p 1161 2128 42 4 v 1 w Fr(])83 b(:=)1512 2086 y(\024)1499 2112 y(\002)1575 2070 y Ft(2)1575 2136 y Fq(j)1631 2112 y Fo(j)1677 2086 y Fm(b)1659 2112 y Fu(G)1736 2070 y Fs(0)1736 2136 y Fq(j)1772 2112 y Fo(j)44 b Fr(+)1982 2017 y Fm(X)1964 2233 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2161 2112 y Fu(b)2202 2127 y Fq(j)t(k)2294 2112 y Fr(\002)2370 2070 y Ft(2)2370 2136 y Fq(k)2430 2112 y Fo(j)p Fu(G)2535 2070 y Fs(0)2535 2136 y Fq(k)2577 2112 y Fo(j)22 b Fu(;)634 b Fx(\(II.9\))1031 2397 y Fu(L)1097 2412 y Fq(j)1134 2397 y Fr([)p Fu(b)p 1161 2413 V 1 w Fr(])83 b(:=)1512 2372 y(\024)1499 2397 y(\002)1575 2356 y Ft(2)1575 2422 y Fq(j)1631 2397 y Fo(j)1677 2372 y Fm(b)1659 2397 y Fu(G)1736 2356 y Fs(0)1736 2422 y Fq(j)1772 2397 y Fo(j)1800 2356 y Ft(2)1883 2397 y Fr(+)2022 2303 y Fm(X)2004 2518 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2201 2397 y Fu(b)2242 2412 y Fq(j)t(k)2334 2397 y Fr(\002)2410 2356 y Ft(2)2410 2422 y Fq(k)2469 2397 y Fo(j)p Fu(G)2574 2356 y Fs(0)2574 2422 y Fq(k)2616 2397 y Fo(j)2644 2356 y Ft(2)2706 2397 y Fu(;)505 b Fx(\(II.10\))1003 2683 y Fu(M)1097 2698 y Fq(j)1134 2683 y Fr([)p Fu(b)p 1161 2699 V 1 w Fr(])83 b(:=)1512 2658 y(\024)1499 2683 y(\002)1575 2642 y Ft(2)1575 2707 y Fq(j)1642 2658 y Fm(b)1631 2683 y Fu(V)1688 2698 y Fq(j)1768 2683 y Fr(+)1907 2588 y Fm(X)1889 2804 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2086 2683 y Fu(b)2127 2698 y Fq(j)t(k)2219 2683 y Fr(\002)2295 2642 y Ft(2)2295 2707 y Fq(k)2354 2683 y Fu(V)2411 2698 y Fq(k)2476 2683 y Fu(;)735 b Fx(\(II.11\))257 2989 y(and)319 3187 y Fm(b)307 3212 y Fu(V)364 3227 y Fq(j)401 3212 y Fr(\()p Fu(x)p Fr(\))51 b(:=)756 3187 y Fm(b)737 3212 y Fu(G)814 3171 y Fs(0)814 3237 y Fq(j)851 3212 y Fr(\()p Fu(x)p Fr(\))982 3171 y Ft(2)1055 3212 y Fo(\000)1182 3145 y Fr(1)p 1176 3189 61 4 v 1176 3280 a Fu(\014)1265 3187 y Fm(b)1246 3212 y Fu(G)1323 3171 y Fs(00)1323 3237 y Fq(j)t(j)1392 3212 y Fr(\()p Fu(x)p Fr(\))119 b Fx(and)f Fu(V)1961 3227 y Fq(k)2003 3212 y Fr(\()p Fu(x)p Fr(\))51 b(:=)h Fu(G)2417 3171 y Fs(0)2417 3237 y Fq(k)2459 3212 y Fr(\()p Fu(x)p Fr(\))2590 3171 y Ft(2)2663 3212 y Fo(\000)2790 3145 y Fr(1)p 2784 3189 V 2784 3280 a Fu(\014)2855 3212 y(G)2932 3171 y Fs(00)2932 3237 y Fq(k)r(k)3013 3212 y Fr(\()p Fu(x)p Fr(\))17 b Fu(:)50 b Fx(\(II.12\))257 3475 y(Furthermore,)31 b(we)e(denote)g(the)g (weights)f(by)h Fu(a)p 1870 3491 52 4 v 36 w Fr(:=)36 b(\()p Fu(a)2185 3490 y Fq(ij)2245 3475 y Fr(\))2283 3490 y Fq(i;j)t Fs(2)p Ft(\003)2459 3475 y Fx(,)31 b Fu(a)p 2515 3491 V Fr(\()p Fu(\024)p Fr(\))36 b(:=)2872 3394 y Fm(\000)2918 3475 y Fu(a)2969 3490 y Fq(ij)3030 3475 y Fr(\()p Fu(\024)p Fr(\))3162 3394 y Fm(\001)3207 3514 y Fq(i;j)t Fs(2)p Ft(\003)3419 3475 y Fr(=)257 3540 y Fm(\000)303 3621 y Fu(e)348 3585 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))560 3621 y Fu(a)611 3636 y Fq(ij)672 3540 y Fm(\001)717 3660 y Fq(i;j)t Fs(2)p Ft(\003)894 3621 y Fx(,)24 b(and)g(introduce)g(ne)n(w)f(weights)h Fu(b)p 2034 3637 42 4 v 28 w Fr(:=)j(\()p Fu(b)2312 3636 y Fq(ij)2373 3621 y Fr(\))2411 3636 y Fq(i;j)t Fs(2)p Ft(\003)2587 3621 y Fx(,)e(with)e(the)h(property)g(that)257 3767 y Fu(b)298 3782 y Fq(ij)387 3767 y Fo(\025)k Fu(e)537 3731 y Fq(\032)p Ft(\()p Fq(i;j)t Ft(\))709 3767 y Fu(a)760 3782 y Fq(ij)820 3767 y Fx(,)d(in)g(Subsect.)f(II.3.)404 3888 y(T)-8 b(o)21 b(e)o(xplain)f(the)h(general)h(strate)o(gy)e(of)i (the)f(estimates)f(we)h(note)g(that)g Fo(j)p Fu(g)2854 3851 y Fs(0)2850 3912 y Fq(j)2886 3888 y Fo(j)9 b Fr(+)g Fo(j)p Fu(g)3087 3851 y Fs(0)3083 3913 y Fq(k)3125 3888 y Fo(j)21 b Fx(controls)257 4008 y(the)27 b(deri)n(v)n(ati)n(v)o(es)d (of)j Fu(w)1041 4023 y Fq(j)t(k)1115 4008 y Fx(,)g(by)g(Hypothesis)e (2.)36 b(This)26 b(enables)g(us)h(to)f(sho)n(w)g(in)g(Theorem)g(II.6) 257 4128 y(in)f(Subsect.)g(II.3)g(that)f(for)h(some)f(uni)n(v)o(ersal)f (constant)h Fu(C)35 b(<)27 b Fo(1)p Fx(,)504 4254 y Fm(\015)504 4313 y(\015)504 4373 y(\015)560 4368 y Fu(J)9 b Fr([)p Fu(a)p 650 4384 52 4 v Fr(\()p Fu(\024)p Fr(\)])860 4327 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1047 4288 y Fm(\010)1105 4368 y Fu(W)1197 4383 y Ft(diag)1371 4368 y Fr(+)38 b(\001)p Fu(W)1658 4383 y Ft(o\013)5 b Fs(\000)p Ft(d)1839 4368 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))2062 4288 y Fm(\011)2142 4368 y Fu(J)9 b Fr([)p Fu(a)p 2232 4384 V Fr(\()p Fu(\024)p Fr(\)])2442 4327 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2607 4254 y Fm(\015)2607 4313 y(\015)2607 4373 y(\015)2715 4368 y Fo(\024)2855 4301 y Fu(C)p 2855 4345 77 4 v 2863 4437 a(\014)2964 4368 y(:)247 b Fx(\(II.13\))404 4619 y(Ne)o(xt,)39 b(we)e(remark)h(that)e(the)h(dif)n(ference)h(between)f Fu(g)2338 4583 y Fs(0)2334 4645 y Fq(k)2413 4619 y Fx(and)g Fu(G)2671 4583 y Fs(0)2671 4645 y Fq(k)2751 4619 y Fx(also)g(contains)f (only)257 4740 y(deri)n(v)n(ati)n(v)o(es)18 b(of)h Fu(w)878 4755 y Fq(j)t(k)973 4740 y Fx(which)h(we)g(control)f(with)g Fo(j)p Fu(g)1951 4704 y Fs(0)1947 4764 y Fq(j)1983 4740 y Fo(j)t Fr(+)t Fo(j)p Fu(g)2174 4704 y Fs(0)2170 4766 y Fq(k)2211 4740 y Fo(j)p Fx(.)29 b(Bootstrapping)19 b(the)g(correspond-)257 4860 y(ing)30 b(estimate,)h(we)g(deri)n(v)o(e)e (in)h(Theorem)g(II.8)h(in)f(Subsect.)g(II.3)g(that,)i(for)e(some)g(uni) n(v)o(ersal)257 4981 y(constant)24 b Fu(C)35 b(<)27 b Fo(1)p Fx(,)1297 5101 y Fu(J)9 b Fr([)p Fu(a)p 1387 5117 52 4 v Fr(\()p Fu(\024)p Fr(\)])53 b Fo(\024)g Fu(J)9 b Fr([)p Fu(b)p 1870 5117 42 4 v 1 w Fr(])52 b Fo(\024)i Fu(C)29 b(K)7 b Fr([)p Fu(b)p 2338 5117 V Fr(])23 b Fu(:)782 b Fx(\(II.14\))257 5268 y(F)o(or)20 b(the)f(bootstrap)f(ar)n(gument)h (to)g(w)o(ork)h(out,)g(we)f(need)h(to)f(pass)g(from)g(the)g(weights)g Fu(a)3199 5283 y Fq(ij)3260 5268 y Fr(\()p Fu(\024)p Fr(\))27 b(=)257 5388 y Fu(e)302 5352 y Fq(\032)p Ft(\()p Fq(i;j)t Ft(\))474 5388 y Fu(a)525 5403 y Fq(ij)620 5388 y Fx(to)33 b(bigger)h(weights)f Fu(b)1405 5403 y Fq(ij)1510 5388 y Fo(\025)45 b Fu(e)1677 5352 y Fq(\032)p Ft(\()p Fq(i;j)t Ft(\))1849 5388 y Fu(a)1900 5403 y Fq(ij)1961 5388 y Fx(.)58 b(These)34 b(ne)n(w)f(weights)g(are)i(then)f(chosen)p eop %%Page: 25 26 25 25 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(25)257 573 y(as)39 b(to)e(ful\002ll)754 498 y Fm(P)859 602 y Fq(k)918 573 y Fu(b)959 588 y Fq(ik)1027 573 y Fu(a)1078 588 y Fq(k)r(j)1205 573 y Fo(\024)53 b Fu(C)1412 537 y Fs(0)1435 573 y Fu(b)1476 588 y Fq(ij)1537 573 y Fx(,)42 b(where)c Fu(C)1962 537 y Fs(0)2038 573 y Fu(<)52 b Fo(1)38 b Fx(is)f(a)i(uni)n(v)o(ersal)d(constant.)70 b(This)257 693 y(construction)24 b(and)h(the)f(bootstrap)g(ar)n(gument) g(require)h Fu(\013)k Fo(\025)f Fr(0)d Fx(to)f(be)h(small.)404 814 y(Similar)d(ar)n(guments)h(are)h(then)e(used)h(in)g(Theorem)g(II.9) g(in)g(Subsect.)g(II.4)g(to)g(pro)o(v)o(e)f(that,)257 934 y(for)j(some)g(uni)n(v)o(ersal)e(constant)h Fu(C)34 b(<)28 b Fo(1)p Fx(,)1308 1188 y Fu(K)7 b Fr([)p Fu(b)p 1425 1204 42 4 v 1 w Fr(])53 b Fo(\024)1687 1120 y Fr(1)p 1687 1165 49 4 v 1689 1256 a Fu(")1762 1188 y(L)p Fr([)p Fu(b)p 1855 1204 42 4 v 1 w Fr(])d(+)g Fu(C)23 b(")17 b(N)2334 1203 y Fs(F)2417 1188 y Fu(;)794 b Fx(\(II.15\))257 1432 y(pro)o(vided)27 b Fu(\013)34 b Fo(\025)g Fr(0)28 b Fx(and)f Fu(")33 b(>)h Fr(0)27 b Fx(are)i(suf)n(\002ciently)e(small,) g(and)h(\002nally)g(in)f(Theorem)h(II.10)g(in)257 1552 y(Subsect.)d(II.5)g(to)g(pro)o(v)o(e)e(that,)i(for)g(some)f(uni)n(v)o (ersal)f(constant)h Fu(C)35 b(<)27 b Fo(1)p Fx(,)1164 1817 y Fu(L)p Fr([)p Fu(b)p 1257 1833 V 1 w Fr(])53 b Fo(\024)1509 1707 y Fm(\020)1568 1817 y Fr(1)22 b(+)1747 1750 y Fu(C)p 1747 1794 77 4 v 1755 1885 a(\014)1834 1707 y Fm(\021)1910 1817 y Fu(M)10 b Fr([)p Fu(b)p 2041 1833 42 4 v 2 w Fr(])50 b(+)2296 1750 y Fu(C)p 2296 1794 77 4 v 2304 1885 a(\014)2400 1817 y(N)2478 1832 y Fs(F)2561 1817 y Fu(;)650 b Fx(\(II.16\))257 2082 y(pro)o(vided)24 b(that)g Fu(\014)34 b(<)27 b Fo(1)e Fx(is)f(suf)n(\002ciently)g(lar)n (ge.)404 2202 y(Inserting)d(\(II.14\)\226\(II.16\))h(into)f(\(II.13\),) i(we)f(conclude)g(that,)g(for)g(some)f(uni)n(v)o(ersal)f(con-)257 2323 y(stants)k Fu(C)r(;)17 b(C)702 2287 y Fs(0)752 2323 y Fu(<)28 b Fo(1)c Fx(and)h(all)g Fu(\036;)17 b( )28 b Fx(in)c(the)h(form)g(domain)e(of)i Fu(M)10 b Fr([)p Fu(b)p 2497 2339 42 4 v 2 w Fr(])p Fx(,)580 2456 y Fm(\014)580 2515 y(\014)580 2575 y(\014)614 2489 y(\012)661 2570 y Fu(\036)746 2485 y Fm(\014)746 2545 y(\014)804 2489 y(\010)862 2570 y Fr(\001)p Fu(W)1035 2585 y Ft(o\013)5 b Fs(\000)p Ft(d)1216 2570 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))k(+)h Fu(W)1650 2585 y Ft(diag)1785 2489 y Fm(\011)1860 2570 y Fu( )1927 2489 y Fm(\013)1974 2456 y(\014)1974 2515 y(\014)1974 2575 y(\014)3238 2570 y Fx(\(II.17\))663 2806 y Fo(\024)824 2692 y Fm(\015)824 2751 y(\015)824 2811 y(\015)879 2696 y(n)979 2739 y Fu(C)p 956 2783 123 4 v 956 2875 a(")17 b(\014)1105 2696 y Fm(\020)1165 2806 y Fr(1)k(+)1343 2739 y Fu(C)p 1343 2783 77 4 v 1351 2875 a(\014)1430 2696 y Fm(\021)1506 2806 y Fu(M)10 b Fr([)p Fu(b)p 1637 2822 42 4 v 2 w Fr(])50 b(+)1935 2739 y Fu(C)p 1892 2783 163 4 v 1892 2875 a(")17 b(\014)2016 2846 y Ft(2)2081 2806 y Fu(N)2159 2821 y Fs(F)2270 2806 y Fr(+)2406 2739 y Fu(C)24 b(")p 2406 2783 140 4 v 2445 2875 a(\014)2572 2806 y(N)2650 2821 y Fs(F)2711 2696 y Fm(o)2777 2718 y Ft(1)p Fq(=)p Ft(2)2904 2806 y Fu(\036)2962 2692 y Fm(\015)2962 2751 y(\015)2962 2811 y(\015)942 3065 y Fo(\001)970 2951 y Fm(\015)970 3010 y(\015)970 3070 y(\015)1025 2955 y(n)1124 2998 y Fu(C)p 1101 3042 123 4 v 1101 3134 a(")17 b(\014)1251 2955 y Fm(\020)1310 3065 y Fr(1)22 b(+)1489 2998 y Fu(C)p 1489 3042 77 4 v 1497 3134 a(\014)1576 2955 y Fm(\021)1652 3065 y Fu(M)10 b Fr([)p Fu(b)p 1783 3081 42 4 v 2 w Fr(])50 b(+)2081 2998 y Fu(C)p 2038 3042 163 4 v 2038 3134 a(")17 b(\014)2162 3105 y Ft(2)2227 3065 y Fu(N)2305 3080 y Fs(F)2416 3065 y Fr(+)2552 2998 y Fu(C)23 b(")p 2552 3042 140 4 v 2591 3134 a(\014)2718 3065 y(N)2796 3080 y Fs(F)2857 2955 y Fm(o)2923 2977 y Ft(1)p Fq(=)p Ft(2)3050 3065 y Fu( )3117 2951 y Fm(\015)3117 3010 y(\015)3117 3070 y(\015)663 3327 y Fo(\024)869 3259 y Fu(C)946 3223 y Fs(0)p 834 3304 171 4 v 834 3395 a Fu(\014)895 3366 y Ft(1)p Fq(=)p Ft(2)1014 3212 y Fm(\015)1014 3272 y(\015)1014 3332 y(\015)1070 3216 y(\020)1129 3327 y Fu(M)10 b Fr([)p Fu(b)p 1260 3343 42 4 v 2 w Fr(])50 b(+)1521 3259 y(1)p 1515 3304 61 4 v 1515 3395 a Fu(\014)1602 3327 y(N)1680 3342 y Fs(F)1742 3216 y Fm(\021)1801 3238 y Ft(1)p Fq(=)p Ft(2)1928 3327 y Fu(\036)1986 3212 y Fm(\015)1986 3272 y(\015)1986 3332 y(\015)2057 3212 y(\015)2057 3272 y(\015)2057 3332 y(\015)2113 3216 y(\020)2172 3327 y Fu(M)10 b Fr([)p Fu(b)p 2303 3343 42 4 v 2 w Fr(])50 b(+)2564 3259 y(1)p 2558 3304 61 4 v 2558 3395 a Fu(\014)2645 3327 y(N)2723 3342 y Fs(F)2785 3216 y Fm(\021)2844 3238 y Ft(1)p Fq(=)p Ft(2)2971 3327 y Fu( )3038 3212 y Fm(\015)3038 3272 y(\015)3038 3332 y(\015)3115 3327 y Fu(;)257 3608 y Fx(by)27 b(choosing)f Fu(")31 b Fr(:=)h Fu(\014)1044 3572 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1208 3608 y Fx(,)c(pro)o(vided)e (that)g Fu(\013)32 b Fo(\025)h Fr(0)27 b Fx(is)f(suf)n(\002ciently)g (small,)g(and)h Fu(\014)38 b(<)31 b Fo(1)c Fx(is)257 3729 y(suf)n(\002ciently)d(lar)n(ge.)31 b(Additionally)23 b(taking)h(into)g(account)g(\(II.4\),)i(we)f(hence)g(obtain)372 3884 y Fm(\014)372 3944 y(\014)372 4003 y(\014)405 3918 y(\012)452 3999 y Fu(\036)538 3914 y Fm(\014)538 3974 y(\014)596 3999 y Fu(W)688 4014 y Fq(H)q(;Q)826 3999 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))g Fu( )1133 3918 y Fm(\013)1180 3884 y(\014)1180 3944 y(\014)1180 4003 y(\014)1296 3999 y Fo(\024)1492 3931 y Fu(C)1569 3895 y Fs(00)p 1466 3976 171 4 v 1466 4067 a Fu(\014)1527 4038 y Ft(1)p Fq(=)p Ft(2)1647 3884 y Fm(\015)1647 3944 y(\015)1647 4003 y(\015)1702 3888 y(\020)1762 3999 y Fu(M)10 b Fr([)p Fu(b)p 1893 4014 42 4 v 1 w Fr(])50 b(+)2138 3888 y Fm(h)2201 3931 y Fr(1)p 2195 3976 61 4 v 2195 4067 a Fu(\014)2287 3999 y Fr(+)2572 3931 y Fu(\013)18 b(\024)p 2395 3976 490 4 v 2395 4067 a(\014)2456 4038 y Ft(1)p Fq(=)p Ft(2)2583 4067 y Fr(\(1)j Fo(\000)i Fu(\024)p Fr(\))2895 3888 y Fm(i)2959 3999 y Fu(N)3037 4014 y Fs(F)3098 3888 y Fm(\021)3157 3910 y Ft(1)p Fq(=)p Ft(2)3267 3999 y Fu(\036)3325 3884 y Fm(\015)3325 3944 y(\015)3325 4003 y(\015)1220 4148 y(\015)1220 4208 y(\015)1220 4268 y(\015)1275 4152 y(\020)1335 4263 y Fu(M)10 b Fr([)p Fu(b)p 1466 4279 42 4 v 1 w Fr(])50 b(+)1711 4152 y Fm(h)1774 4196 y Fr(1)p 1768 4240 61 4 v 1768 4331 a Fu(\014)1861 4263 y Fr(+)2146 4196 y Fu(\013)17 b(\024)p 1969 4240 490 4 v 1969 4331 a(\014)2030 4303 y Ft(1)p Fq(=)p Ft(2)2156 4331 y Fr(\(1)22 b Fo(\000)g Fu(\024)p Fr(\))2468 4152 y Fm(i)2532 4263 y Fu(N)2610 4278 y Fs(F)2671 4152 y Fm(\021)2731 4175 y Ft(1)p Fq(=)p Ft(2)2841 4263 y Fu( )2908 4148 y Fm(\015)2908 4208 y(\015)2908 4268 y(\015)2985 4263 y Fu(;)226 b Fx(\(II.18\))257 4537 y(with)24 b Fu(C)536 4501 y Fs(00)606 4537 y Fu(<)k Fo(1)d Fx(being)f(a)h(uni)n(v)o(ersal)e(constant.)404 4658 y(The)k(ne)o(xt)g (step)g(is)g(to)g(use)h(the)f(positi)n(vity)e(of)j Fo(\000)p Fu(@)2127 4622 y Ft(2)2122 4682 y Fq(j)2167 4658 y Fx(,)h(as)e(a)h (quadratic)g(form.)38 b(This)27 b(gi)n(v)o(es,)257 4778 y(cf.)32 b(\(II.12\),)888 5035 y Fr(\002)964 4994 y Ft(2)964 5060 y Fq(k)1023 5035 y Fu(V)1080 5050 y Fq(k)1206 5035 y Fo(\024)83 b Fr(\002)1442 5050 y Fq(k)1485 4925 y Fm(\020)1566 5035 y Fo(\000)1702 4968 y Fr(1)p 1676 5012 101 4 v 1676 5104 a Fu(\014)1737 5075 y Ft(2)1786 5035 y Fu(@)1842 4994 y Ft(2)1837 5060 y Fq(k)1927 5035 y Fr(+)44 b(\()p Fu(G)2162 4994 y Fs(0)2162 5060 y Fq(k)2204 5035 y Fr(\))2242 4994 y Ft(2)2326 5035 y Fo(\000)2464 4968 y Fr(1)p 2458 5012 61 4 v 2458 5104 a Fu(\014)2528 5035 y(G)2605 4994 y Fs(00)2605 5060 y Fq(k)r(k)2686 4925 y Fm(\021)2746 5035 y Fr(\002)2822 5050 y Fq(k)1207 5231 y Fr(=)83 b(\002)1442 5246 y Fq(k)1501 5231 y Fu(Z)1575 5190 y Fs(\003)1568 5256 y Fq(k)1615 5231 y Fr(\()p Fu(G)p Fr(\))17 b Fu(Z)1852 5246 y Fq(k)1894 5231 y Fr(\()p Fu(G)p Fr(\))g(\002)2140 5246 y Fq(k)2204 5231 y Fu(;)1007 b Fx(\(II.19\))p eop %%Page: 26 27 26 26 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(26)257 573 y(and)25 b(similarly)926 809 y Fr(\024)912 834 y(\002)988 793 y Ft(2)988 859 y Fq(j)1056 809 y Fm(b)1044 834 y Fu(V)1101 849 y Fq(j)1221 834 y Fo(\024)1395 809 y Fr(\024)1381 834 y(\002)1457 849 y Fq(j)1494 723 y Fm(\020)1575 834 y Fo(\000)1710 767 y Fr(1)p 1685 811 101 4 v 1685 902 a Fu(\014)1746 874 y Ft(2)1795 834 y Fu(@)1846 849 y Fq(j)1927 834 y Fr(+)44 b(\()2104 809 y Fm(b)2085 834 y Fu(G)2162 793 y Fs(0)2162 859 y Fq(j)2199 834 y Fr(\))2237 793 y Ft(2)2320 834 y Fo(\000)2458 767 y Fr(1)p 2452 811 61 4 v 2452 902 a Fu(\014)2541 809 y Fm(b)2523 834 y Fu(G)2600 793 y Fs(00)2600 859 y Fq(j)t(j)2668 723 y Fm(\021)2741 809 y Fr(\024)2728 834 y(\002)2804 849 y Fq(j)1221 1052 y Fr(=)1395 1027 y(\024)1381 1052 y(\002)1457 1067 y Fq(j)1510 1052 y Fu(Z)1584 1011 y Fs(\003)1577 1077 y Fq(j)1624 1052 y Fr(\()1680 1027 y Fm(b)1662 1052 y Fu(G)o Fr(\))17 b Fu(Z)1860 1067 y Fq(j)1896 1052 y Fr(\()1953 1027 y Fm(b)1934 1052 y Fu(G)p Fr(\))2079 1027 y(\024)2066 1052 y(\002)2142 1067 y Fq(j)2200 1052 y Fu(:)1011 b Fx(\(II.20\))257 1285 y(The)21 b(distinction)d (between)1240 1260 y Fr(\024)1226 1285 y(\002)1302 1300 y Fq(j)1359 1285 y Fx(and)i Fr(\002)1599 1300 y Fq(j)1656 1285 y Fx(becomes)g(important)e(no)n(w)-6 b(.)28 b(Namely)-6 b(,)20 b(inserting)f(the)257 1419 y(commutator)k Fr([)p Fu(Z)864 1382 y Fs(\003)857 1443 y Fq(j)904 1419 y Fr(\()961 1393 y Fm(b)942 1419 y Fu(G)p Fr(\))p Fu(;)17 b(Z)1168 1434 y Fq(j)1204 1419 y Fr(\()1261 1393 y Fm(b)1242 1419 y Fu(G)p Fr(\)])27 b(=)h Fo(\000)p Fr(2)p Fu(\014)1702 1382 y Fs(\000)p Ft(1)1815 1393 y Fm(b)1796 1419 y Fu(G)1873 1382 y Fs(00)1873 1443 y Fq(j)t(j)1942 1419 y Fx(,)c(using)g(the)g (second)g(intertwining)f(relation)257 1562 y Fu(Z)324 1577 y Fq(j)361 1562 y Fr(\()418 1536 y Fm(b)399 1562 y Fu(G)p Fr(\))524 1536 y Fm(b)514 1562 y Fr(\002)590 1577 y Fq(j)654 1562 y Fr(=)767 1536 y Fm(b)757 1562 y Fr(\002)833 1577 y Fq(j)869 1562 y Fu(Z)936 1577 y Fq(j)973 1562 y Fr(\()p Fu(H)8 b Fr(\))22 b Fx(in)h(\(I.54\),)g(and)h (taking)e(into)g(account)h(that)2667 1536 y Fm(b)2657 1562 y Fr(\002)2733 1525 y Ft(2)2733 1586 y Fq(j)2800 1562 y Fo(\025)28 b Fr(1)p Fx(,)3012 1536 y Fm(b)3002 1562 y Fr(\002)3078 1577 y Fq(j)3128 1536 y Fr(\024)3114 1562 y(\002)3190 1577 y Fq(j)3254 1562 y Fr(=)g(\002)3434 1577 y Fq(j)3470 1562 y Fx(,)257 1682 y(we)d(ha)n(v)o(e)g(\(cf.)31 b(\(I.92\)\))840 1877 y Fr(\024)827 1902 y(\002)903 1861 y Ft(2)903 1927 y Fq(j)970 1877 y Fm(b)959 1902 y Fu(V)1016 1917 y Fq(j)1135 1902 y Fo(\024)1309 1877 y Fr(\024)1296 1902 y(\002)1372 1917 y Fq(j)1425 1902 y Fu(Z)1499 1861 y Fs(\003)1492 1927 y Fq(j)1538 1902 y Fr(\()1595 1877 y Fm(b)1576 1902 y Fu(G)p Fr(\))17 b Fu(Z)1775 1917 y Fq(j)1811 1902 y Fr(\()1868 1877 y Fm(b)1849 1902 y Fu(G)p Fr(\))1994 1877 y(\024)1981 1902 y(\002)2057 1917 y Fq(j)1136 2104 y Fr(=)1309 2079 y(\024)1296 2104 y(\002)1372 2119 y Fq(j)1425 2104 y Fu(Z)1492 2119 y Fq(j)1528 2104 y Fr(\()1585 2079 y Fm(b)1566 2104 y Fu(G)p Fr(\))g Fu(Z)1772 2063 y Fs(\003)1765 2129 y Fq(j)1811 2104 y Fr(\()1868 2079 y Fm(b)1849 2104 y Fu(G)p Fr(\))1994 2079 y(\024)1981 2104 y(\002)2057 2119 y Fq(j)2142 2104 y Fo(\000)2286 2037 y Fr(2)p 2280 2081 V 2280 2173 a Fu(\014)2380 2079 y Fr(\024)2367 2104 y(\002)2443 2063 y Ft(2)2443 2129 y Fq(j)2518 2079 y Fm(b)2499 2104 y Fu(G)2576 2063 y Fs(00)2576 2129 y Fq(j)t(j)1135 2357 y Fo(\024)1309 2332 y Fr(\024)1296 2357 y(\002)1372 2372 y Fq(j)1425 2357 y Fu(Z)1492 2372 y Fq(j)1528 2357 y Fr(\()1585 2332 y Fm(b)1566 2357 y Fu(G)p Fr(\))1708 2332 y Fm(b)1698 2357 y Fr(\002)1774 2372 y Fq(j)1837 2332 y Fm(b)1826 2357 y Fr(\002)1902 2372 y Fq(j)1955 2357 y Fu(Z)2029 2316 y Fs(\003)2022 2381 y Fq(j)2069 2357 y Fr(\()2126 2332 y Fm(b)2107 2357 y Fu(G)p Fr(\))2252 2332 y(\024)2239 2357 y(\002)2315 2372 y Fq(j)2400 2357 y Fo(\000)2544 2289 y Fr(2)p 2538 2334 V 2538 2425 a Fu(\014)2638 2332 y Fr(\024)2625 2357 y(\002)2701 2316 y Ft(2)2701 2381 y Fq(j)2776 2332 y Fm(b)2757 2357 y Fu(G)2834 2316 y Fs(00)2834 2381 y Fq(j)t(j)1136 2609 y Fr(=)1309 2584 y(\024)1296 2609 y(\002)1372 2624 y Fq(j)1435 2584 y Fm(b)1425 2609 y Fr(\002)1501 2624 y Fq(j)1554 2609 y Fu(Z)1621 2624 y Fq(j)1657 2609 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1913 2568 y Fs(\003)1906 2634 y Fq(j)1951 2609 y Fr(\()p Fu(H)8 b Fr(\))2143 2584 y Fm(b)2133 2609 y Fr(\002)2209 2624 y Fq(j)2275 2584 y Fr(\024)2261 2609 y(\002)2337 2624 y Fq(j)2424 2609 y Fo(\000)2567 2542 y Fr(2)p 2561 2586 V 2561 2678 a Fu(\014)2662 2584 y Fr(\024)2648 2609 y(\002)2724 2568 y Ft(2)2724 2634 y Fq(j)2799 2584 y Fm(b)2780 2609 y Fu(G)2857 2568 y Fs(00)2857 2634 y Fq(j)t(j)1136 2862 y Fr(=)84 b(\002)1372 2877 y Fq(j)1425 2862 y Fu(Z)1492 2877 y Fq(j)1528 2862 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1784 2821 y Fs(\003)1777 2886 y Fq(j)1822 2862 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)2080 2877 y Fq(j)2166 2862 y Fo(\000)2309 2794 y Fr(2)p 2303 2839 V 2303 2930 a Fu(\014)2404 2837 y Fr(\024)2390 2862 y(\002)2466 2821 y Ft(2)2466 2886 y Fq(j)2541 2837 y Fm(b)2522 2862 y Fu(G)2599 2821 y Fs(00)2599 2886 y Fq(j)t(j)2690 2862 y Fu(:)521 b Fx(\(II.21\))257 3132 y(Inserting)24 b(\(II.19\))i(and)e(\(II.21\))h(into)f(the)h(de\002nition)f(\(II.11\))h (of)g Fu(M)2587 3147 y Fq(j)2624 3132 y Fr([)p Fu(b)p 2651 3148 42 4 v Fr(])p Fx(,)g(we)g(obtain)f(that)295 3373 y Fu(M)389 3388 y Fq(j)426 3373 y Fr([)p Fu(b)p 453 3389 V Fr(])84 b(=)778 3347 y(\024)765 3373 y(\002)841 3331 y Ft(2)841 3397 y Fq(j)908 3347 y Fm(b)897 3373 y Fu(V)954 3388 y Fq(j)1034 3373 y Fr(+)1173 3278 y Fm(X)1154 3494 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1352 3373 y Fu(b)1393 3388 y Fq(j)t(k)1485 3373 y Fr(\002)1561 3331 y Ft(2)1561 3397 y Fq(k)1620 3373 y Fu(V)1677 3388 y Fq(k)604 3685 y Fo(\024)g Fr(\002)841 3700 y Fq(j)894 3685 y Fu(Z)961 3700 y Fq(j)997 3685 y Fr(\()p Fu(H)8 b Fr(\))17 b Fu(Z)1253 3644 y Fs(\003)1246 3710 y Fq(j)1291 3685 y Fr(\()p Fu(H)8 b Fr(\))17 b(\002)1549 3700 y Fq(j)1635 3685 y Fo(\000)1778 3618 y Fr(2)p 1772 3662 61 4 v 1772 3754 a Fu(\014)1873 3660 y Fr(\024)1859 3685 y(\002)1935 3644 y Ft(2)1935 3710 y Fq(j)2010 3660 y Fm(b)1991 3685 y Fu(G)2068 3644 y Fs(00)2068 3710 y Fq(j)t(j)2187 3685 y Fr(+)2331 3591 y Fm(X)2312 3806 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2510 3685 y Fu(b)2551 3700 y Fq(j)t(k)2643 3685 y Fr(\002)2719 3700 y Fq(k)2778 3685 y Fu(Z)2852 3644 y Fs(\003)2845 3710 y Fq(k)2891 3685 y Fr(\()p Fu(G)p Fr(\))g Fu(Z)3128 3700 y Fq(k)3170 3685 y Fr(\()p Fu(G)p Fr(\))g(\002)3416 3700 y Fq(k)604 3954 y Fo(\024)84 b Fr(\(1)22 b(+)g Fu(C)1042 3969 y Fq(b)1076 3954 y Fr(\))1131 3873 y Fm(\000)1202 3929 y Fr(\024)1176 3954 y Fu(A)1249 3969 y Fq(j)1330 3954 y Fr(+)44 b Fu(A)1523 3969 y Fq(j)1560 3873 y Fm(\001)1627 3954 y Fu(;)1584 b Fx(\(II.22\))257 4211 y(where)20 b Fu(C)590 4226 y Fq(b)651 4211 y Fr(:=)28 b(max)964 4226 y Fq(j)1017 4101 y Fm(n)1100 4137 y(P)1205 4240 y Fq(k)1264 4211 y Fu(b)1305 4226 y Fq(j)t(k)1381 4101 y Fm(o)1475 4211 y Fu(<)f Fo(1)19 b Fx(is)f(a)h(uni)n(v)o(ersal)f (constant,)h(as)g(pro)o(v)o(ed)e(in)i(Lemma)f(II.7)257 4352 y(belo)n(w)-6 b(.)30 b(F)o(or)24 b(all)h Fu(\015)33 b Fo(\025)28 b Fr(1)p Fx(,)c(Eqs.)h(\(II.22\))g(and)g(\(I.79\))g(yield) 893 4596 y Fu(M)10 b Fr([)p Fu(b)p 1024 4612 42 4 v 1 w Fr(])83 b Fo(\024)h Fr(\(1)21 b(+)i Fu(C)1614 4611 y Fq(b)1648 4596 y Fr(\))17 b Fu(\015)1775 4486 y Fm(\020)1851 4502 y(X)1907 4712 y Fq(j)2011 4516 y Fm(\010)2095 4571 y Fr(\024)2070 4596 y Fu(A)2143 4611 y Fq(j)2218 4596 y Fr(+)39 b Fu(A)2406 4611 y Fq(j)2442 4516 y Fm(\011)2522 4596 y Fo(\012)23 b Fu(a)2673 4555 y Fs(\003)2673 4621 y Fq(j)2713 4596 y Fu(a)2764 4611 y Fq(j)2800 4486 y Fm(\021)1176 4878 y Fo(\024)84 b Fr(\(1)21 b(+)i Fu(C)1614 4893 y Fq(b)1648 4878 y Fr(\))1703 4797 y Fm(\000)1748 4878 y Fu(\015)e(d)p Fr(\000\()p Fu(A)2043 4827 y Ft(\(1\))2043 4905 y Fq(H)q(;Q)2182 4878 y Fr(\))44 b Fo(\000)2396 4811 y Fu(c)2438 4826 y Fq(f)2499 4811 y Fu(\015)p 2396 4855 160 4 v 2445 4946 a(\014)2582 4878 y(N)2660 4893 y Fs(F)2721 4797 y Fm(\001)2789 4878 y Fu(:)422 b Fx(\(II.23\))257 5142 y(Here)38 b(we)g(used)f(the)g(follo)n(wing)e(lo)n(wer)i(bound,)i (cf.)68 b(\(I.8\),)41 b(\(I.15\))c(and)g(\(I.16\):)56 b(F)o(or)37 b(an)o(y)p eop %%Page: 27 28 27 27 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(27)257 573 y Fu( )32 b Fo(2)c(F)10 b Fx(,)363 704 y Fm(D)424 815 y Fu( )519 700 y Fm(\014)519 760 y(\014)519 820 y(\014)593 720 y(X)627 930 y Fq(i;j)754 815 y Fu(H)843 774 y Fs(00)835 839 y Fq(ij)895 815 y Fr(\(0\))22 b Fo(\012)h Fu(a)1193 774 y Fs(\003)1193 839 y Fq(i)1232 815 y Fu(a)1283 830 y Fq(j)1348 815 y Fu( )1415 704 y Fm(E)1559 815 y Fo(\025)83 b Fu(c)1761 830 y Fq(f)1823 720 y Fm(X)1879 930 y Fq(j)1984 815 y Fo(k)p Fu(a)2085 830 y Fq(j)2138 815 y Fu( )t Fo(k)2255 774 y Ft(2)2344 815 y Fo(\000)50 b Fu(\013)2550 720 y Fm(X)2584 930 y Fq(i;j)2711 815 y Fu(a)2762 830 y Fq(ij)2839 815 y Fo(k)p Fu(a)2940 830 y Fq(i)2985 815 y Fu( )t Fo(k)17 b(k)p Fu(a)3220 830 y Fq(j)3273 815 y Fu( )t Fo(k)1559 1091 y(\025)1719 1010 y Fm(\000)1765 1091 y Fu(c)1807 1106 y Fq(f)1902 1091 y Fo(\000)50 b Fu(C)2099 1106 y Fq(a)2141 1091 y Fu(\013)2204 1010 y Fm(\001)2266 996 y(X)2321 1206 y Fq(j)2426 1091 y Fo(k)p Fu(a)2527 1106 y Fq(j)2580 1091 y Fu( )t Fo(k)2697 1050 y Ft(2)1559 1372 y Fo(\025)1729 1305 y Fu(c)1771 1320 y Fq(f)p 1729 1349 88 4 v 1748 1441 a Fr(2)1826 1292 y Fm(\012)1873 1372 y Fu( )1940 1288 y Fm(\014)1940 1348 y(\014)2001 1372 y Fu(N)2079 1387 y Fs(F)2157 1372 y Fu( )2224 1292 y Fm(\013)2293 1372 y Fu(;)918 b Fx(\(II.24\))257 1623 y(pro)o(vided)24 b Fr(0)j Fo(\024)i Fu(\013)f Fo(\024)g Fu(c)1054 1638 y Fq(f)1099 1623 y Fu(=)p Fr(\(2)p Fu(C)1305 1638 y Fq(a)1346 1623 y Fr(\))p Fx(.)j(This)24 b(establishes)g (\(II.23\),)h(for)g(all)f Fu(\015)33 b Fo(\025)28 b Fr(1)p Fx(.)404 1744 y(Finally)-6 b(,)23 b(we)i(insert)g(\(II.23\))g(into)f (\(II.18\))h(and)g(obtain)560 1873 y Fm(\014)560 1933 y(\014)560 1993 y(\014)593 1907 y(\012)640 1988 y Fu(\036)726 1903 y Fm(\014)726 1963 y(\014)784 1988 y Fu(W)876 2003 y Fq(H)q(;Q)1014 1988 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))g Fu( )1321 1907 y Fm(\013)1367 1873 y(\014)1367 1933 y(\014)1367 1993 y(\014)3238 1988 y Fx(\(II.25\))643 2239 y Fo(\024)860 2172 y Fu(C)p 813 2216 171 4 v 813 2308 a(\014)874 2279 y Ft(1)p Fq(=)p Ft(2)994 2125 y Fm(\015)994 2184 y(\015)994 2244 y(\015)1049 2129 y(\020)1109 2239 y Fu(\015)k(d)p Fr(\000\()p Fu(A)1404 2188 y Ft(\(1\))1404 2266 y Fq(H)q(;Q)1543 2239 y Fr(\))49 b(+)1772 2172 y(1)p 1766 2216 61 4 v 1766 2308 a Fu(\014)1837 2129 y Fm(h)1884 2239 y Fr(1)21 b(+)2062 2172 y Fu(\014)2123 2136 y Ft(1)p Fq(=)p Ft(2)2250 2172 y Fu(\013)c(\024)p 2062 2216 323 4 v 2072 2308 a Fr(\(1)22 b Fo(\000)h Fu(\024)p Fr(\))2417 2239 y Fo(\000)g Fu(c)2559 2254 y Fq(f)2621 2239 y Fu(\015)2677 2129 y Fm(i)2740 2239 y Fu(N)2818 2254 y Fs(F)2880 2129 y Fm(\021)2939 2151 y Ft(1)p Fq(=)p Ft(2)3049 2239 y Fu(\036)3107 2125 y Fm(\015)3107 2184 y(\015)3107 2244 y(\015)922 2519 y Fo(\001)967 2404 y Fm(\015)967 2464 y(\015)967 2524 y(\015)1021 2408 y(\020)1081 2519 y Fu(\015)e(d)p Fr(\000\()p Fu(A)1376 2468 y Ft(\(1\))1376 2546 y Fq(H)q(;Q)1514 2519 y Fr(\))50 b(+)1744 2452 y(1)p 1738 2496 61 4 v 1738 2587 a Fu(\014)1808 2408 y Fm(h)1855 2519 y Fr(1)22 b(+)2034 2452 y Fu(\014)2095 2415 y Ft(1)p Fq(=)p Ft(2)2222 2452 y Fu(\013)17 b(\024)p 2034 2496 323 4 v 2044 2587 a Fr(\(1)22 b Fo(\000)h Fu(\024)p Fr(\))2389 2519 y Fo(\000)g Fu(c)2531 2534 y Fq(f)2593 2519 y Fu(\015)2649 2408 y Fm(i)2712 2519 y Fu(N)2790 2534 y Fs(F)2852 2408 y Fm(\021)2911 2431 y Ft(1)p Fq(=)p Ft(2)3021 2519 y Fu( )3088 2404 y Fm(\015)3088 2464 y(\015)3088 2524 y(\015)3166 2519 y Fu(;)257 2788 y Fx(for)i(some)g(uni)n(v)o(ersal)e(constant)h Fu(C)34 b(<)28 b Fo(1)p Fx(.)i(W)-8 b(e)26 b(choose)1375 3052 y Fu(\015)58 b Fr(:=)1669 2985 y(1)p 1649 3029 88 4 v 1649 3121 a Fu(c)1691 3136 y Fq(f)1747 2942 y Fm(\020)1806 3052 y Fr(1)22 b(+)1985 2985 y Fu(\014)2046 2949 y Ft(1)p Fq(=)p Ft(2)2172 2985 y Fu(\013)c(\024)p 1985 3029 323 4 v 2033 3121 a Fr(1)k Fo(\000)h Fu(\024)2318 2942 y Fm(\021)3238 3052 y Fx(\(II.26\))257 3324 y(and)k(observ)o(e)f(that)g Fu(\015)36 b Fo(\025)31 b Fr(1)p Fx(,)26 b(since)h Fu(c)1508 3339 y Fq(f)1584 3324 y Fo(\024)k Fr(1)p Fx(.)36 b(Moreo)o(v)o(er)l(,) 25 b(this)h(choice)h(of)f Fu(\015)32 b Fx(insures)25 b(that)h(the)257 3445 y(terms)f(in)f(brack)o(ets)h(in)f(\(II.25\))h(v)n (anish.)30 b(Therefore,)594 3591 y Fm(\014)594 3651 y(\014)594 3711 y(\014)627 3625 y(\012)674 3706 y Fu(\036)759 3621 y Fm(\014)759 3681 y(\014)818 3706 y Fu(W)910 3721 y Fq(H)q(;Q)1048 3706 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))g Fu( )1355 3625 y Fm(\013)1401 3591 y(\014)1401 3651 y(\014)1401 3711 y(\014)1515 3706 y Fo(\024)1693 3638 y Fu(C)24 b(\015)p 1683 3683 171 4 v 1683 3774 a(\014)1744 3745 y Ft(1)p Fq(=)p Ft(2)1891 3621 y Fm(\015)1891 3681 y(\015)1946 3706 y Fu(d)p Fr(\000\()p Fu(A)2169 3655 y Ft(\(1\))2169 3733 y Fq(H)q(;Q)2307 3706 y Fr(\))2345 3665 y Ft(1)p Fq(=)p Ft(2)2472 3706 y Fu(\036)2530 3621 y Fm(\015)2530 3681 y(\015)2613 3621 y(\015)2613 3681 y(\015)2668 3706 y Fu(d)p Fr(\000\()p Fu(A)2891 3655 y Ft(\(1\))2891 3733 y Fq(H)q(;Q)3029 3706 y Fr(\))3067 3665 y Ft(1)p Fq(=)p Ft(2)3194 3706 y Fu( )3261 3621 y Fm(\015)3261 3681 y(\015)677 3980 y Fr(=)881 3913 y Fu(C)958 3876 y Fs(0)p 846 3957 V 846 4048 a Fu(\014)907 4020 y Ft(1)p Fq(=)p Ft(2)1026 3869 y Fm(\020)1086 3980 y Fr(1)e(+)1265 3913 y Fu(\014)1326 3876 y Ft(1)p Fq(=)p Ft(2)1452 3913 y Fu(\013)17 b(\024)p 1265 3957 323 4 v 1313 4048 a Fr(1)22 b Fo(\000)g Fu(\024)1597 3869 y Fm(\021)1684 3895 y(\015)1684 3955 y(\015)1740 3980 y Fu(d)p Fr(\000\()p Fu(A)1963 3929 y Ft(\(1\))1963 4007 y Fq(H)q(;Q)2101 3980 y Fr(\))2139 3939 y Ft(1)p Fq(=)p Ft(2)2265 3980 y Fu(\036)2323 3895 y Fm(\015)2323 3955 y(\015)2406 3895 y(\015)2406 3955 y(\015)2462 3980 y Fu(d)p Fr(\000\()p Fu(A)2685 3929 y Ft(\(1\))2685 4007 y Fq(H)q(;Q)2823 3980 y Fr(\))2861 3939 y Ft(1)p Fq(=)p Ft(2)2987 3980 y Fu( )3054 3895 y Fm(\015)3054 3955 y(\015)3132 3980 y Fu(;)79 b Fx(\(II.27\))257 4244 y(\002nishing)24 b(the)h(proof.)2399 b Fg(\003)257 4539 y Fl(II.1)119 b(Estimate)28 b(on)i Ff(W)1308 4557 y Fe(o\013)6 b FE(\000)p Fe(d)1519 4539 y Fd(\()p Ff(\024;)20 b(o)p Fd(\))29 b Fl(in)i(T)-11 b(erms)28 b(of)i Ff(J)11 b Fd([)p Fl(a)p 2535 4560 60 5 v -1 w Fd(\()p Ff(\024)p Fd(\)])257 4726 y Fx(W)-8 b(e)26 b(recall)f(from)f(\(I.84\))h(the)g(de\002nition)f(of)h Fu(W)1880 4741 y Ft(o\013)5 b Fs(\000)p Ft(d)2060 4726 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))p Fx(,)307 4991 y Fu(W)399 5006 y Ft(o\013)5 b Fs(\000)p Ft(d)579 4991 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))48 b(=)1083 4923 y(2)17 b Fu(\013)p 1083 4968 129 4 v 1116 5059 a(\014)1237 4896 y Fm(X)1254 5108 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1398 4880 y Fm(\020)1457 4991 y Fu(e)1502 4950 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])1983 4991 y Fr(\002)2059 5006 y Fq(i)2087 4991 y Fr(\002)2163 5006 y Fq(j)2216 4991 y Fu(w)2289 4950 y Fs(00)2286 5015 y Fq(ij)2347 4991 y Fr(\()p Fu(x)p Fr(\))10 b Fo(\000)g Fu(w)2648 4950 y Fs(00)2645 5015 y Fq(ij)2706 4991 y Fr(\(0\))2831 4880 y Fm(\021)2895 4991 y Fo(\012)5 b Fu(a)3028 4950 y Fs(\003)3028 5015 y Fq(i)3068 4991 y Fu(a)3119 5006 y Fq(j)3161 4991 y Fu(;)50 b Fx(\(II.28\))p eop %%Page: 28 29 28 28 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(28)257 573 y(and)25 b(the)f(de\002nition)g(of)g Fr(\002)1162 588 y Fq(j)1226 573 y Fr(:=)k Fu(e)1402 537 y Fq(\014)s(q)1477 547 y Fj(j)1513 573 y Fx(.)j(W)-8 b(e)25 b(decompose)f Fu(W)2289 588 y Ft(o\013)5 b Fs(\000)p Ft(d)2469 573 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))27 b(=)h(\001)p Fu(W)2996 588 y Ft(o\013)5 b Fs(\000)p Ft(d)3176 573 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))j(+)257 693 y Fu(W)349 708 y Ft(o\013)5 b Fs(\000)p Ft(d)529 693 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))p Fx(,)24 b(where)257 859 y Fr(\001)p Fu(W)430 874 y Ft(o\013)5 b Fs(\000)p Ft(d)611 859 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))82 b(:=)h Fu(W)1293 874 y Ft(o\013)5 b Fs(\000)p Ft(d)1474 859 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g Fu(x)p Fr(\))k Fo(\000)i Fu(W)2009 874 y Ft(o\013)5 b Fs(\000)p Ft(d)2189 859 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))746 b Fx(\(II.29\))1029 1048 y Fr(=)1211 981 y(2)17 b Fu(\013)p 1211 1025 129 4 v 1245 1117 a(\014)1366 954 y Fm(X)1382 1166 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1527 1048 y Fu(e)1572 1007 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])2053 938 y Fm(\020)2112 1048 y Fr(\002)2188 1063 y Fq(i)2216 1048 y Fr(\002)2292 1063 y Fq(j)2345 1048 y Fu(w)2418 1007 y Fs(00)2415 1073 y Fq(ij)2475 1048 y Fr(\()p Fu(x)p Fr(\))45 b Fo(\000)g Fu(w)2846 1007 y Fs(00)2843 1073 y Fq(ij)2903 1048 y Fr(\(0\))3028 938 y Fm(\021)3137 1048 y Fo(\012)50 b Fu(a)3315 1007 y Fs(\003)3315 1073 y Fq(i)3355 1048 y Fu(a)3406 1063 y Fq(j)3465 1048 y Fu(;)345 1354 y(W)437 1369 y Ft(o\013)5 b Fs(\000)p Ft(d)617 1354 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))96 b(=)1211 1286 y(2)17 b Fu(\013)p 1211 1331 V 1245 1422 a(\014)1366 1259 y Fm(X)1382 1471 y Fq(i)p Fs(6)p Ft(=)p Fq(j)1527 1243 y Fm(\020)1586 1354 y Fu(e)1631 1312 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])2156 1354 y Fo(\000)45 b Fr(1)2327 1243 y Fm(\021)2403 1354 y Fu(w)2476 1312 y Fs(00)2473 1378 y Fq(ij)2533 1354 y Fr(\(0\))50 b Fo(\012)g Fu(a)2886 1312 y Fs(\003)2886 1378 y Fq(i)2926 1354 y Fu(a)2977 1369 y Fq(j)3036 1354 y Fu(:)188 b Fx(\(II.30\))257 1615 y(Belo)n(w)-6 b(,)24 b(we)h(pro)o(v)o(e)f(the)h(follo)n(wing)e (estimate.)257 1772 y Fv(Lemma)j(II.2.)40 b Fn(F)-10 b(or)24 b(some)h(univer)o(sal)f(constant)f Fu(C)35 b(<)27 b Fo(1)e Fn(and)f(all)g Fu(\024)k Fo(2)g Fr([0)p Fu(;)17 b Fr(1])p Fn(,)847 1871 y Fm(\015)847 1931 y(\015)847 1990 y(\015)902 1985 y Fu(J)9 b Fr([)p Fu(a)p 992 2001 52 4 v 1 w Fr(\()p Fu(\024)p Fr(\)])1203 1944 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1384 1985 y Fr(\001)p Fu(W)1557 2000 y Ft(o\013)c Fs(\000)p Ft(d)1738 1985 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))g Fu(J)9 b Fr([)p Fu(a)p 2068 2001 V -1 w Fr(\()p Fu(\024)p Fr(\)])2277 1944 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2442 1871 y Fm(\015)2442 1931 y(\015)2442 1990 y(\015)2550 1985 y Fo(\024)2690 1918 y Fu(C)24 b(\013)p 2690 1962 157 4 v 2738 2054 a(\014)2879 1985 y(:)332 b Fx(\(II.31\))257 2211 y Fn(wher)l(e)42 b Fu(J)9 b Fr([)p Fu(a)p Fr(])42 b Fn(is)e(de\002ned)h(in)f(Eqs.)h(\(II.6\))g(and)g (\(II.8\),)k(and)40 b Fu(a)p 2424 2227 52 4 v 1 w Fr(\()p Fu(\024)p Fr(\))57 b(:=)2826 2130 y Fm(\000)2871 2211 y Fu(a)2922 2226 y Fq(j)t(k)2997 2211 y Fr(\()p Fu(\024)p Fr(\))3129 2130 y Fm(\001)3175 2250 y Fq(j;k)r Fs(2)p Ft(\003)3419 2211 y Fr(=)257 2277 y Fm(\000)303 2357 y Fu(e)348 2321 y Fq(\024\032)p Ft(\()p Fq(j;k)r Ft(\))587 2357 y Fu(a)638 2372 y Fq(j)t(k)713 2277 y Fm(\001)759 2396 y Fq(j;k)r Fs(2)p Ft(\003)946 2357 y Fn(.)257 2535 y(Pr)l(oof:)f Fx(W)-8 b(e)38 b(use)g(a)g(v)n(ariant)f(of)h(the)g (Cauchy-Schw)o(arz)h(inequality)-6 b(.)68 b(F)o(or)37 b Fu(x)53 b Fo(2)f Fp(R)3254 2499 y Ft(\003)3313 2535 y Fx(,)41 b(we)257 2656 y(introduce)25 b(the)f(characteristic)h (functions)f Fu(\037)1818 2671 y Fq(j)1883 2656 y Fr(:=)j Fi(1)p Fr([)p Fo(j)p Fu(x)2179 2671 y Fq(j)2216 2656 y Fo(j)h Fu(<)2397 2630 y Fr(^)2375 2656 y Fu(R)2449 2671 y Ft(0)2489 2656 y Fr(])d Fx(and)36 b Fr(\026)-60 b Fu(\037)2771 2671 y Fq(j)2835 2656 y Fr(:=)28 b Fi(1)p Fr([)p Fo(j)p Fu(x)3132 2671 y Fq(j)3169 2656 y Fo(j)f(\025)3351 2630 y Fr(^)3330 2656 y Fu(R)3404 2671 y Ft(0)3443 2656 y Fr(])p Fx(,)257 2776 y(so)e(that)1579 2896 y Fr(1)52 b(=)h Fu(\037)1870 2911 y Fq(j)1929 2896 y Fr(+)33 b(\026)-60 b Fu(\037)2088 2911 y Fq(j)2146 2896 y Fu(:)1065 b Fx(\(II.32\))257 3062 y(Note)31 b(that)f(if)h Fo(j)p Fu(x)837 3077 y Fq(i)865 3062 y Fo(j)38 b(\024)1069 3037 y Fr(^)1048 3062 y Fu(R)1122 3077 y Ft(0)1192 3062 y Fx(then)30 b Fr(\002)1470 3077 y Fq(i)1537 3062 y Fo(\021)39 b Fr(1)31 b Fx(and)f Fo(j)p Fu(g)1986 3026 y Fs(0)1982 3087 y Fq(i)2010 3062 y Fr(\()p Fu(x)2103 3077 y Fq(i)2132 3062 y Fr(\))p Fo(j)38 b(\024)h Fu(C)2422 3077 y Fq(g)2462 3062 y Fu(e)2507 3026 y Fq(C)2557 3034 y Fj(g)2609 3009 y Ft(^)2594 3026 y Fq(R)2647 3035 y Fk(0)2685 3062 y Fx(,)33 b(by)d(\(I.31\).)49 b(Further)n(-)257 3193 y(more,)24 b Fo(j)p Fu(g)590 3157 y Fs(0)586 3217 y Fq(j)622 3193 y Fr(\()p Fu(x)715 3208 y Fq(j)751 3193 y Fr(\))p Fo(j)k(\025)g Fu(c)992 3208 y Fq(g)1049 3193 y Fr(min)n Fo(f)p Fr(1)p Fu(;)1375 3168 y Fr(^)1354 3193 y Fu(R)1428 3208 y Ft(0)1467 3193 y Fo(g)p Fx(,)c(for)f Fo(j)p Fu(x)1788 3208 y Fq(j)1825 3193 y Fo(j)k(\025)2006 3168 y Fr(^)1985 3193 y Fu(R)2059 3208 y Ft(0)2099 3193 y Fx(,)c(by)g(\(I.32\).)31 b(Hence,)24 b(for)f(an)o(y)g Fu(i;)17 b(j)34 b Fo(2)28 b Fr(\003)p Fx(,)536 3388 y Fu(\037)597 3403 y Fq(i)653 3388 y Fr(\026)-60 b Fu(\037)703 3403 y Fq(j)756 3307 y Fm(\000)802 3388 y Fo(j)p Fu(g)881 3346 y Fs(0)877 3412 y Fq(i)904 3388 y Fr(\()p Fu(x)997 3403 y Fq(i)1026 3388 y Fr(\))p Fo(j)22 b Fr(+)g Fo(j)p Fu(g)1291 3346 y Fs(0)1287 3412 y Fq(j)1323 3388 y Fr(\()p Fu(x)1416 3403 y Fq(j)1452 3388 y Fr(\))p Fo(j)g Fr(+)g(2)1687 3307 y Fm(\001)1816 3388 y Fo(\024)83 b Fu(\037)2037 3403 y Fq(i)2094 3388 y Fr(\026)-61 b Fu(\037)2143 3403 y Fq(j)2196 3307 y Fm(\000)2242 3388 y Fu(C)2312 3403 y Fq(g)2352 3388 y Fu(e)2397 3346 y Fq(C)2447 3354 y Fj(g)2499 3330 y Ft(^)2484 3346 y Fq(R)2537 3355 y Fk(0)2598 3388 y Fr(+)22 b Fo(j)p Fu(g)2775 3346 y Fs(0)2771 3412 y Fq(j)2807 3388 y Fr(\()p Fu(x)2900 3403 y Fq(j)2936 3388 y Fr(\))p Fo(j)g Fr(+)g(2)3171 3307 y Fm(\001)3238 3388 y Fx(\(II.33\))1816 3533 y Fo(\024)83 b Fu(C)2046 3548 y Ft(1)2102 3533 y Fu(\037)2163 3548 y Fq(i)2219 3533 y Fr(\026)-60 b Fu(\037)2269 3548 y Fq(j)2322 3533 y Fo(j)p Fu(g)2401 3492 y Fs(0)2397 3557 y Fq(j)2433 3533 y Fr(\()p Fu(x)2526 3548 y Fq(j)2563 3533 y Fr(\))p Fo(j)22 b Fu(;)257 3698 y Fx(and)346 3864 y Fr(\026)-61 b Fu(\037)395 3879 y Fq(i)452 3864 y Fr(\026)g Fu(\037)501 3879 y Fq(j)554 3783 y Fm(\000)600 3864 y Fo(j)p Fu(g)679 3822 y Fs(0)675 3888 y Fq(i)702 3864 y Fr(\()p Fu(x)795 3879 y Fq(i)824 3864 y Fr(\))p Fo(j)890 3822 y Ft(1)p Fq(=)p Ft(2)1022 3864 y Fr(+)22 b Fo(j)p Fu(g)1199 3822 y Fs(0)1195 3888 y Fq(j)1231 3864 y Fr(\()p Fu(x)1324 3879 y Fq(j)1361 3864 y Fr(\))p Fo(j)1427 3822 y Ft(1)p Fq(=)p Ft(2)1558 3864 y Fr(+)g(2)1705 3783 y Fm(\001)1803 3864 y Fo(\024)53 b Fu(C)2003 3879 y Ft(2)2071 3864 y Fr(\026)-61 b Fu(\037)2120 3879 y Fq(i)2177 3864 y Fr(\026)g Fu(\037)2226 3879 y Fq(j)2279 3864 y Fo(j)p Fu(g)2358 3822 y Fs(0)2354 3888 y Fq(i)2382 3864 y Fr(\()p Fu(x)2475 3879 y Fq(i)2503 3864 y Fr(\))p Fo(j)2569 3822 y Ft(1)p Fq(=)p Ft(2)2696 3864 y Fo(j)p Fu(g)2775 3822 y Fs(0)2771 3888 y Fq(j)2807 3864 y Fr(\()p Fu(x)2900 3879 y Fq(j)2936 3864 y Fr(\))p Fo(j)3002 3822 y Ft(1)p Fq(=)p Ft(2)3134 3864 y Fu(;)77 b Fx(\(II.34\))257 4055 y(where)20 b Fu(C)590 4070 y Ft(1)657 4055 y Fr(:=)27 b(1+\()p Fu(C)1020 4070 y Fq(g)1060 4055 y Fu(e)1105 4019 y Fq(C)1155 4027 y Fj(g)1206 4002 y Ft(^)1191 4019 y Fq(R)1244 4028 y Fk(0)1283 4055 y Fr(+2\))p Fu(c)1488 4019 y Fs(\000)p Ft(1)1488 4080 y Fq(g)1599 4055 y Fr(min)n Fo(f)p Fr(1)p Fu(;)1925 4030 y Fr(^)1904 4055 y Fu(R)1978 4070 y Ft(0)2017 4055 y Fo(g)2067 4019 y Fs(\000)p Ft(1)2180 4055 y Fx(and)19 b Fu(C)2413 4070 y Ft(2)2480 4055 y Fr(:=)28 b(2)p Fu(c)2702 4004 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2702 4065 y Fq(g)2883 4055 y Fr(min)o Fo(f)p Fr(1)p Fu(;)3209 4030 y Fr(^)3189 4055 y Fu(R)3263 4070 y Ft(0)3302 4055 y Fo(g)3352 4019 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)3516 4055 y Fr(+)257 4195 y(2)p Fu(c)348 4159 y Fs(\000)p Ft(1)348 4220 y Fq(g)459 4195 y Fr(min)o Fo(f)p Fr(1)p Fu(;)785 4170 y Fr(^)765 4195 y Fu(R)839 4210 y Ft(0)878 4195 y Fo(g)928 4159 y Fs(\000)p Ft(1)1022 4195 y Fx(.)j(Additionally)22 b(using)i(that)g Fr(\002)2097 4210 y Fq(j)2134 4195 y Fu(\037)2195 4210 y Fq(j)2259 4195 y Fr(=)k Fu(\037)2424 4210 y Fq(j)2485 4195 y Fx(and)d Fr(\002)2730 4210 y Fq(j)2794 4195 y Fo(\025)j Fr(1)p Fx(,)d(we)g(obtain)401 4286 y Fm(\014)401 4346 y(\014)434 4371 y Fr(\002)510 4386 y Fq(i)538 4371 y Fr(\002)614 4386 y Fq(j)667 4371 y Fu(w)740 4330 y Fs(00)737 4396 y Fq(ij)797 4371 y Fr(\()p Fu(x)p Fr(\))40 b Fo(\000)f Fu(w)1157 4330 y Fs(00)1154 4396 y Fq(ij)1214 4371 y Fr(\(0\))1339 4286 y Fm(\014)1339 4346 y(\014)3238 4371 y Fx(\(II.35\))484 4574 y Fo(\024)83 b Fu(\037)705 4589 y Fq(i)750 4574 y Fu(\037)811 4589 y Fq(j)865 4489 y Fm(\014)865 4549 y(\014)898 4574 y Fu(w)971 4532 y Fs(00)968 4598 y Fq(ij)1028 4574 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)g Fu(w)1387 4532 y Fs(00)1384 4598 y Fq(ij)1445 4574 y Fr(\(0\))1570 4489 y Fm(\014)1570 4549 y(\014)1652 4574 y Fr(+)62 b(\026)-61 b Fu(\037)1839 4589 y Fq(i)1895 4574 y Fr(\026)h Fu(\037)1945 4589 y Fq(j)1998 4574 y Fr(\002)2074 4589 y Fq(i)2119 4574 y Fr(\002)2195 4589 y Fq(j)2231 4493 y Fm(\000)2277 4574 y Fo(j)p Fu(w)2378 4532 y Fs(00)2375 4598 y Fq(ij)2435 4574 y Fr(\()p Fu(x)p Fr(\))p Fo(j)39 b Fr(+)f Fo(j)p Fu(w)2848 4532 y Fs(00)2845 4598 y Fq(ij)2905 4574 y Fr(\(0\))p Fo(j)3058 4493 y Fm(\001)672 4753 y Fr(+)28 b Fu(\037)837 4768 y Fq(i)893 4753 y Fr(\026)-60 b Fu(\037)943 4768 y Fq(j)996 4753 y Fr(\002)1072 4768 y Fq(j)1108 4672 y Fm(\000)1154 4753 y Fo(j)p Fu(w)1255 4711 y Fs(00)1252 4777 y Fq(ij)1312 4753 y Fr(\()p Fu(x)p Fr(\))p Fo(j)39 b Fr(+)f Fo(j)p Fu(w)1725 4711 y Fs(00)1722 4777 y Fq(ij)1782 4753 y Fr(\(0\))p Fo(j)1935 4672 y Fm(\001)2030 4753 y Fr(+)61 b(\026)-60 b Fu(\037)2217 4768 y Fq(i)2261 4753 y Fu(\037)2322 4768 y Fq(j)2376 4753 y Fr(\002)2452 4768 y Fq(i)2480 4672 y Fm(\000)2525 4753 y Fo(j)p Fu(w)2626 4711 y Fs(00)2623 4777 y Fq(ij)2683 4753 y Fr(\()p Fu(x)p Fr(\))p Fo(j)39 b Fr(+)g Fo(j)p Fu(w)3097 4711 y Fs(00)3094 4777 y Fq(ij)3154 4753 y Fr(\(0\))p Fo(j)3307 4672 y Fm(\001)484 4962 y Fo(\024)83 b Fu(C)714 4977 y Fq(w)787 4962 y Fu(a)838 4977 y Fq(ij)916 4962 y Fu(\037)977 4977 y Fq(i)1022 4962 y Fu(\037)1083 4977 y Fq(j)1136 4881 y Fm(\000)1182 4962 y Fo(j)p Fu(g)1261 4920 y Fs(0)1257 4986 y Fq(i)1284 4962 y Fo(j)22 b Fr(+)g Fo(j)p Fu(g)1511 4920 y Fs(0)1507 4986 y Fq(j)1543 4962 y Fo(j)1571 4881 y Fm(\001)1666 4962 y Fr(+)50 b Fu(C)1862 4977 y Fq(w)1935 4962 y Fu(C)2005 4977 y Ft(2)2061 4962 y Fu(a)2112 4977 y Fq(ij)2200 4962 y Fr(\026)-60 b Fu(\037)2250 4977 y Fq(i)2306 4962 y Fr(\026)g Fu(\037)2356 4977 y Fq(j)2409 4962 y Fr(\002)2485 4977 y Fq(i)2530 4962 y Fr(\002)2606 4977 y Fq(j)2659 4962 y Fo(j)p Fu(g)2738 4920 y Fs(0)2734 4986 y Fq(i)2761 4962 y Fo(j)2789 4920 y Ft(1)p Fq(=)p Ft(2)2916 4962 y Fo(j)p Fu(g)2995 4920 y Fs(0)2991 4986 y Fq(j)3026 4962 y Fo(j)3054 4920 y Ft(1)p Fq(=)p Ft(2)672 5141 y Fr(+)28 b Fu(C)846 5156 y Fq(w)919 5141 y Fu(C)989 5156 y Ft(1)1044 5141 y Fu(a)1095 5156 y Fq(ij)1173 5060 y Fm(\000)1219 5141 y Fu(\037)1280 5156 y Fq(i)1336 5141 y Fr(\026)-61 b Fu(\037)1385 5156 y Fq(j)1439 5141 y Fr(\002)1515 5156 y Fq(j)1568 5141 y Fo(j)p Fu(g)1647 5099 y Fs(0)1643 5165 y Fq(j)1679 5141 y Fo(j)43 b Fr(+)56 b(\026)-60 b Fu(\037)1932 5156 y Fq(i)1977 5141 y Fu(\037)2038 5156 y Fq(j)2091 5141 y Fr(\002)2167 5156 y Fq(i)2211 5141 y Fo(j)p Fu(g)2290 5099 y Fs(0)2286 5165 y Fq(i)2314 5141 y Fo(j)2342 5060 y Fm(\001)484 5373 y Fo(\024)83 b Fu(C)714 5388 y Ft(3)770 5373 y Fu(a)821 5388 y Fq(ij)899 5263 y Fm(\020)958 5373 y Fr(\002)1034 5388 y Fq(i)1079 5373 y Fo(j)p Fu(g)1158 5332 y Fs(0)1154 5398 y Fq(i)1181 5373 y Fo(j)44 b Fr(+)g(\002)1449 5388 y Fq(j)1502 5373 y Fo(j)p Fu(g)1581 5332 y Fs(0)1577 5398 y Fq(j)1613 5373 y Fo(j)g Fr(+)g(\002)1881 5388 y Fq(i)1926 5373 y Fr(\002)2002 5388 y Fq(j)2055 5373 y Fo(j)p Fu(g)2134 5332 y Fs(0)2130 5398 y Fq(i)2158 5373 y Fo(j)2186 5332 y Ft(1)p Fq(=)p Ft(2)2295 5373 y Fo(j)p Fu(g)2374 5332 y Fs(0)2370 5398 y Fq(j)2406 5373 y Fo(j)2434 5332 y Ft(1)p Fq(=)p Ft(2)2544 5263 y Fm(\021)2626 5373 y Fu(;)p eop %%Page: 29 30 29 29 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(29)257 573 y(for)33 b(some)e(uni)n(v)o(ersal)f(constant)h Fu(C)1475 588 y Ft(3)1555 573 y Fu(<)41 b Fo(1)p Fx(.)52 b(Ne)o(xt,)33 b(the)e(triangle)h(inequality)e(implies)h(that)257 693 y Fu(\032)p Fr(\()p Fu(i;)17 b(o)p Fr(\))24 b Fo(\000)f Fu(\032)p Fr(\()p Fu(j;)17 b(o)p Fr(\))31 b Fo(\024)g Fu(\032)p Fr(\()p Fu(i;)17 b(j)6 b Fr(\))p Fx(,)27 b(and)f(hence)h Fu(e)1802 657 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])2299 693 y Fu(a)2350 708 y Fq(ij)2441 693 y Fo(\024)k Fu(a)2600 708 y Fq(ij)2661 693 y Fr(\()p Fu(\024)p Fr(\))p Fx(,)c(uniformly)e(in) g(the)257 814 y(reference)i(point)d Fu(o)j Fo(2)h Fr(\003)p Fx(.)j(Thus,)24 b(Eq.)g(\(II.35\))h(implies)f(that)684 990 y Fu(e)729 949 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])1227 905 y Fm(\014)1227 965 y(\014)1260 990 y Fr(\002)1336 1005 y Fq(i)1364 990 y Fr(\002)1440 1005 y Fq(j)1493 990 y Fu(w)1566 949 y Fs(00)1563 1015 y Fq(ij)1624 990 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)g Fu(w)1983 949 y Fs(00)1980 1015 y Fq(ij)2040 990 y Fr(\(0\))2165 905 y Fm(\014)2165 965 y(\014)3238 990 y Fx(\(II.36\))767 1223 y Fo(\024)84 b Fu(C)998 1238 y Ft(3)1054 1223 y Fu(a)1105 1238 y Fq(ij)1166 1223 y Fr(\()p Fu(\024)p Fr(\))1315 1112 y Fm(\020)1374 1223 y Fr(\002)1450 1238 y Fq(i)1494 1223 y Fo(j)p Fu(g)1573 1181 y Fs(0)1569 1247 y Fq(i)1597 1223 y Fo(j)44 b Fr(+)g(\002)1865 1238 y Fq(j)1918 1223 y Fo(j)p Fu(g)1997 1181 y Fs(0)1993 1247 y Fq(j)2029 1223 y Fo(j)g Fr(+)g(\002)2297 1238 y Fq(i)2342 1223 y Fr(\002)2418 1238 y Fq(j)2471 1223 y Fo(j)p Fu(g)2550 1181 y Fs(0)2546 1247 y Fq(i)2573 1223 y Fo(j)2601 1181 y Ft(1)p Fq(=)p Ft(2)2711 1223 y Fo(j)p Fu(g)2790 1181 y Fs(0)2786 1247 y Fq(j)2822 1223 y Fo(j)2850 1181 y Ft(1)p Fq(=)p Ft(2)2959 1112 y Fm(\021)3041 1223 y Fu(:)257 1421 y Fx(No)n(w)-6 b(,)24 b(let)g Fu(\036;)17 b( )32 b Fo(2)c(H)e Fx(and)e(observ)o(e)h(that)409 1514 y Fm(X)425 1727 y Fq(i)p Fs(6)p Ft(=)p Fq(j)570 1609 y Fu(a)621 1624 y Fq(ij)681 1609 y Fr(\()p Fu(\024)p Fr(\))17 b(\002)906 1624 y Fq(j)959 1609 y Fo(j)p Fu(g)1038 1568 y Fs(0)1034 1634 y Fq(j)1070 1609 y Fo(j)g(k)p Fu(a)1216 1624 y Fq(i)1243 1609 y Fu(\036)p Fr(\()p Fu(x)p Fr(\))p Fo(k)g(k)p Fu(a)1600 1624 y Fq(j)1636 1609 y Fu( )t Fr(\()p Fu(x)p Fr(\))p Fo(k)492 1951 y(\024)653 1810 y Fm(\022)743 1856 y(X)759 2068 y Fq(i)p Fs(6)p Ft(=)p Fq(j)903 1951 y Fu(a)954 1966 y Fq(ij)1015 1951 y Fr(\()p Fu(\024)p Fr(\))g(\002)1240 1910 y Ft(2)1240 1975 y Fq(j)1295 1951 y Fo(j)p Fu(g)1374 1910 y Fs(0)1370 1975 y Fq(j)1406 1951 y Fo(j)g(k)p Fu(a)1552 1966 y Fq(i)1580 1951 y Fu(\036)p Fr(\()p Fu(x)p Fr(\))p Fo(k)1819 1910 y Ft(2)1858 1810 y Fm(\023)1931 1833 y Ft(1)p Fq(=)p Ft(2)2069 1810 y Fm(\022)2159 1856 y(X)2175 2068 y Fq(i)p Fs(6)p Ft(=)p Fq(j)2320 1951 y Fu(a)2371 1966 y Fq(ij)2431 1951 y Fr(\()p Fu(\024)p Fr(\))g Fo(j)p Fu(g)2659 1910 y Fs(0)2655 1975 y Fq(j)2691 1951 y Fo(j)g(k)p Fu(a)2837 1966 y Fq(j)2873 1951 y Fu( )t Fr(\()p Fu(x)p Fr(\))p Fo(k)3121 1910 y Ft(2)3160 1810 y Fm(\023)3233 1833 y Ft(1)p Fq(=)p Ft(2)492 2292 y Fo(\024)653 2152 y Fm(\022)743 2198 y(X)798 2408 y Fq(j)903 2182 y Fm(n)969 2292 y Fu(C)1039 2307 y Fq(a)1098 2292 y Fo(j)p Fu(g)1177 2251 y Fs(0)1173 2317 y Fq(j)1208 2292 y Fo(j)50 b Fr(+)1423 2198 y Fm(X)1412 2414 y Fq(i)p Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1594 2292 y Fu(a)1645 2307 y Fq(ij)1706 2292 y Fr(\()p Fu(\024)p Fr(\))17 b(\002)1931 2251 y Ft(2)1931 2317 y Fq(i)1987 2292 y Fo(j)p Fu(g)2066 2251 y Fs(0)2062 2317 y Fq(i)2089 2292 y Fo(j)2117 2182 y Fm(o)2200 2292 y Fo(k)p Fu(a)2301 2307 y Fq(j)2337 2292 y Fu(\036)p Fr(\()p Fu(x)p Fr(\))p Fo(k)2576 2251 y Ft(2)2616 2152 y Fm(\023)2689 2174 y Ft(1)p Fq(=)p Ft(2)771 2641 y Fo(\001)799 2501 y Fm(\022)888 2547 y(X)944 2757 y Fq(j)1049 2531 y Fm(n)1115 2641 y Fu(C)1185 2656 y Fq(a)1243 2641 y Fo(j)p Fu(g)1322 2600 y Fs(0)1318 2666 y Fq(j)1354 2641 y Fo(j)50 b Fr(+)1569 2547 y Fm(X)1557 2762 y Fq(i)p Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1740 2641 y Fu(a)1791 2656 y Fq(ij)1852 2641 y Fr(\()p Fu(\024)p Fr(\))17 b(\002)2077 2600 y Ft(2)2077 2666 y Fq(i)2132 2641 y Fo(j)p Fu(g)2211 2600 y Fs(0)2207 2666 y Fq(i)2235 2641 y Fo(j)2263 2531 y Fm(o)2346 2641 y Fo(k)p Fu(a)2447 2656 y Fq(j)2483 2641 y Fu( )t Fr(\()p Fu(x)p Fr(\))p Fo(k)2731 2600 y Ft(2)2771 2501 y Fm(\023)2844 2523 y Ft(1)p Fq(=)p Ft(2)3238 2641 y Fx(\(II.37\))257 2921 y(and)667 3003 y Fm(X)684 3215 y Fq(i)p Fs(6)p Ft(=)p Fq(j)828 3098 y Fu(a)879 3113 y Fq(ij)940 3098 y Fr(\()p Fu(\024)p Fr(\))g(\002)1165 3113 y Fq(i)1209 3098 y Fr(\002)1285 3113 y Fq(j)1338 3098 y Fo(j)p Fu(g)1417 3057 y Fs(0)1413 3122 y Fq(i)1440 3098 y Fo(j)1468 3057 y Ft(1)p Fq(=)p Ft(2)1578 3098 y Fo(j)p Fu(g)1657 3057 y Fs(0)1653 3122 y Fq(j)1689 3098 y Fo(j)1717 3057 y Ft(1)p Fq(=)p Ft(2)1843 3098 y Fo(k)p Fu(a)1944 3113 y Fq(i)1972 3098 y Fu(\036)p Fr(\()p Fu(x)p Fr(\))p Fo(k)g(k)p Fu(a)2329 3113 y Fq(j)2365 3098 y Fu( )t Fr(\()p Fu(x)p Fr(\))p Fo(k)53 b(\024)495 b Fx(\(II.38\))833 3299 y Fm(\022)923 3345 y(X)979 3555 y Fq(j)1084 3329 y Fm(n)1178 3345 y(X)1167 3560 y Fq(i)p Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1350 3439 y Fu(a)1401 3454 y Fq(ij)1461 3439 y Fr(\()p Fu(\024)p Fr(\))17 b(\002)1686 3398 y Ft(2)1686 3464 y Fq(i)1742 3439 y Fo(j)p Fu(g)1821 3398 y Fs(0)1817 3464 y Fq(i)1844 3439 y Fo(j)1872 3329 y Fm(o)1955 3439 y Fo(k)p Fu(a)2056 3454 y Fq(j)2093 3439 y Fu(\036)p Fr(\()p Fu(x)p Fr(\))p Fo(k)2332 3398 y Ft(2)2371 3299 y Fm(\023)2444 3321 y Ft(1)p Fq(=)p Ft(2)1306 3648 y Fm(\022)1396 3694 y(X)1452 3904 y Fq(j)1556 3678 y Fm(n)1650 3694 y(X)1639 3909 y Fq(i)p Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1822 3788 y Fu(a)1873 3803 y Fq(ij)1934 3788 y Fr(\()p Fu(\024)p Fr(\))g(\002)2159 3747 y Ft(2)2159 3813 y Fq(i)2214 3788 y Fo(j)p Fu(g)2293 3747 y Fs(0)2289 3813 y Fq(i)2317 3788 y Fo(j)2345 3678 y Fm(o)2428 3788 y Fo(k)p Fu(a)2529 3803 y Fq(j)2565 3788 y Fu( )t Fr(\()p Fu(x)p Fr(\))p Fo(k)2813 3747 y Ft(2)2853 3648 y Fm(\023)2926 3670 y Ft(1)p Fq(=)p Ft(2)3058 3788 y Fu(;)257 4088 y Fx(where)26 b Fu(a)577 4103 y Fq(ij)637 4088 y Fr(\()p Fu(\024)p Fr(\))i(:=)g Fu(e)973 4052 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))1201 4088 y Fu(a)1252 4103 y Fq(ij)1313 4088 y Fx(.)j(Eqs.)24 b (\(II.35\)\226\(II.38\))h(and)2340 4063 y Fr(\024)2326 4088 y(\002)2402 4103 y Fq(j)2466 4088 y Fo(\025)j Fr(1)d Fx(imply)e(that)406 4231 y Fm(\014)406 4291 y(\014)439 4235 y(\012)486 4316 y Fu(\036)561 4231 y Fm(\014)561 4291 y(\014)621 4316 y Fr(\001)p Fu(W)794 4331 y Ft(o\013)5 b Fs(\000)p Ft(d)975 4316 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))g Fu( )1282 4235 y Fm(\013)1328 4231 y(\014)1328 4291 y(\014)1414 4316 y Fo(\024)1554 4248 y Fr(3)p Fu(C)1673 4263 y Ft(3)1728 4248 y Fu(\013)p 1554 4293 238 4 v 1642 4384 a(\014)1818 4231 y Fm(\015)1818 4291 y(\015)1873 4316 y Fu(J)9 b Fr([)p Fu(a)p 1963 4332 52 4 v Fr(\()p Fu(\024)p Fr(\)])2173 4275 y Ft(1)p Fq(=)p Ft(2)2300 4316 y Fu(\036)2358 4231 y Fm(\015)2358 4291 y(\015)2435 4231 y(\015)2435 4291 y(\015)2491 4316 y Fu(J)g Fr([)p Fu(a)p 2581 4332 V Fr(\()p Fu(\024)p Fr(\)])2791 4275 y Ft(1)p Fq(=)p Ft(2)2918 4316 y Fu( )2985 4231 y Fm(\015)2985 4291 y(\015)3062 4316 y Fu(;)149 b Fx(\(II.39\))257 4536 y(and)25 b(hence)g(\(II.31\).)2453 b Fg(\003)404 4656 y Fx(Ne)o(xt)20 b(we)h(turn)g(to)g(estimating)e Fu(W)1563 4671 y Ft(o\013)5 b Fs(\000)p Ft(d)1743 4656 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))j Fx(which)h(is)f(a)i(constant)e(matrix.) 28 b(Here)22 b(we)257 4777 y(use)i(the)g(f)o(act)g(that)f Fu(W)997 4792 y Ft(o\013)5 b Fs(\000)p Ft(d)1178 4777 y Fr(\(0)p Fu(;)17 b(o)p Fr(;)g(0\))26 b(=)h(0)d Fx(to)g(bound)f Fu(W)2156 4792 y Ft(o\013)5 b Fs(\000)p Ft(d)2336 4777 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))23 b Fx(by)g(a)i(term)e(of)h (order)g Fu(\024)p Fx(.)257 4943 y Fv(Lemma)i(II.3.)40 b Fn(F)-10 b(or)24 b(all)g Fr(0)k Fo(\024)g Fu(\024)g(<)f Fr(1)p Fn(,)936 5071 y Fm(\015)936 5131 y(\015)991 5156 y Fu(N)1079 5105 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1069 5183 y Fs(F)1267 5156 y Fu(W)1359 5171 y Ft(o\013)5 b Fs(\000)p Ft(d)1539 5156 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))k Fu(N)1964 5105 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1954 5183 y Fs(F)2130 5071 y Fm(\015)2130 5131 y(\015)2238 5156 y Fo(\024)2403 5088 y Fr(3)c Fu(C)2539 5103 y Fq(a)2596 5088 y Fu(\013)h(\024)p 2378 5133 380 4 v 2378 5224 a Fr(\(1)j Fo(\000)i Fu(\024)p Fr(\))17 b Fu(\014)2789 5156 y(;)422 b Fx(\(II.40\))257 5388 y Fn(wher)l(e)26 b Fu(N)600 5403 y Fs(F)689 5388 y Fr(=)793 5313 y Fm(P)898 5417 y Fq(j)951 5388 y Fi(1)c Fo(\012)h Fu(a)1180 5352 y Fs(\003)1180 5413 y Fq(j)1219 5388 y Fu(a)1270 5403 y Fq(j)1307 5388 y Fn(.)p eop %%Page: 30 31 30 30 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(30)257 573 y Fn(Pr)l(oof:)29 b Fx(W)-8 b(e)23 b(pick)g Fu(\036;)17 b( )31 b Fo(2)d(F)33 b Fx(and)23 b(apply)f(the)h (Cauchy-Schw)o(arz)h(inequality)-6 b(.)29 b(Additionally)257 693 y(using)g(the)h(triangle)f(inequality)-6 b(,)30 b(implying)d Fu(e)1872 657 y Fq(\024)p Ft([)p Fq(\032)p Ft(\()p Fq(i;o)p Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(j;o)p Ft(\)])2390 693 y Fo(\024)38 b Fu(e)2550 657 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))2762 693 y Fx(,)31 b(and)f(taking)f(\(I.15\))257 814 y(and)c(the)g(symmetry)e Fu(a)1047 829 y Fq(ij)1136 814 y Fr(=)k Fu(a)1290 829 y Fq(j)t(i)1376 814 y Fx(into)d(account,)g (we)h(obtain)316 956 y Fm(\014)316 1016 y(\014)349 1040 y Fo(h)p Fu(\036)p Fo(j)i Fu(W)593 1055 y Ft(o\013)5 b Fs(\000)p Ft(d)773 1040 y Fr(\()p Fu(\024;)17 b(o)p Fr(;)g(0\))g Fu( )t Fo(i)1212 956 y Fm(\014)1212 1016 y(\014)1324 1040 y Fo(\024)1492 973 y Fr(2)g Fu(\013)p 1492 1018 129 4 v 1525 1109 a(\014)1646 946 y Fm(X)1680 1156 y Fq(i;j)1807 960 y Fm(\000)1852 1040 y Fu(e)1897 999 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))2148 1040 y Fo(\000)39 b Fr(1)2313 960 y Fm(\001)2375 1040 y Fu(a)2426 1055 y Fq(ij)2515 1040 y Fo(k)p Fu(a)2616 1055 y Fq(i)2661 1040 y Fu(\036)p Fo(k)21 b(k)p Fu(a)2891 1055 y Fq(j)2945 1040 y Fu( )t Fo(k)176 b Fx(\(II.41\))399 1350 y Fo(\024)569 1283 y Fr(2)17 b Fu(\013)p 569 1327 V 603 1418 a(\014)724 1350 y Fr(max)803 1410 y Fq(i)922 1239 y Fm(n)1005 1255 y(X)1061 1465 y Fq(j)1166 1269 y Fm(\000)1211 1350 y Fu(e)1256 1309 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))1507 1350 y Fo(\000)39 b Fr(1)1672 1269 y Fm(\001)1734 1350 y Fu(a)1785 1365 y Fq(ij)1846 1239 y Fm(o)17 b(n)2012 1255 y(X)2065 1468 y Fq(k)2173 1350 y Fo(k)p Fu(a)2274 1365 y Fq(k)2333 1350 y Fu(\036)p Fo(k)2441 1309 y Ft(2)2480 1239 y Fm(o)2546 1262 y Ft(1)p Fq(=)p Ft(2)2673 1239 y Fm(n)2756 1255 y(X)2809 1468 y Fq(k)2917 1350 y Fo(k)p Fu(a)3018 1365 y Fq(k)3077 1350 y Fu( )t Fo(k)3194 1309 y Ft(2)3233 1239 y Fm(o)3300 1262 y Ft(1)p Fq(=)p Ft(2)3410 1350 y Fu(;)257 1624 y Fx(and)25 b(hence)g(\(II.40\))h(follo)n(ws)d(if) i(we)g(sho)n(w)e(that,)i(for)g(all)f Fu(i)k Fo(2)g Fr(\003)p Fx(,)1205 1750 y Fm(X)1261 1960 y Fq(j)1366 1764 y Fm(\000)1411 1844 y Fu(e)1456 1803 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))1707 1844 y Fo(\000)39 b Fr(1)1872 1764 y Fm(\001)1934 1844 y Fu(a)1985 1859 y Fq(ij)2099 1844 y Fo(\024)2239 1777 y Fr(3)17 b Fu(C)2375 1792 y Fq(a)2432 1777 y Fu(\024)p 2239 1821 250 4 v 2250 1913 a Fr(1)22 b Fo(\000)h Fu(\024)2520 1844 y(:)691 b Fx(\(II.42\))257 2118 y(T)-8 b(o)25 b(this)f(end,)h(we)g (observ)o(e)f(that)g Fu(e)1441 2082 y Fq(r)1501 2118 y Fo(\000)f Fr(1)28 b Fo(\024)g Fu(r)19 b(e)1891 2082 y Fq(r)1929 2118 y Fx(,)25 b(which)f(implies)329 2217 y Fm(X)385 2427 y Fq(j)490 2231 y Fm(\000)535 2311 y Fu(e)580 2270 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))831 2311 y Fo(\000)39 b Fr(1)996 2231 y Fm(\001)1058 2311 y Fu(a)1109 2326 y Fq(ij)1253 2311 y Fo(\024)1414 2217 y Fm(X)1470 2427 y Fq(j)1574 2311 y Fu(\024)17 b(\032)p Fr(\()p Fu(i;)g(j)6 b Fr(\))17 b Fu(e)1958 2270 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))2186 2311 y Fu(a)2237 2326 y Fq(ij)3238 2311 y Fx(\(II.43\))1254 2593 y Fr(=)1509 2526 y Fu(\024)p 1424 2570 227 4 v 1424 2661 a Fr(1)22 b Fo(\000)g Fu(\024)1677 2498 y Fm(X)1733 2708 y Fq(j)1837 2512 y Fm(\000)1883 2593 y Fr(\(1)g Fo(\000)h Fu(\024)p Fr(\))17 b Fu(\032)p Fr(\()p Fu(i;)g(j)6 b Fr(\))17 b Fu(e)2514 2552 y Fs(\000)p Ft(\(1)p Fs(\000)p Fq(\024)p Ft(\))p Fq(\032)p Ft(\()p Fq(i;j)t Ft(\))2925 2512 y Fm(\001)f(\000)3033 2593 y Fu(e)3078 2552 y Fq(\032)p Ft(\()p Fq(i;j)t Ft(\))3266 2593 y Fu(a)3317 2608 y Fq(ij)3378 2512 y Fm(\001)1253 2900 y Fo(\024)1445 2833 y Fu(C)1515 2848 y Fq(a)1573 2833 y Fu(\024)p 1424 2877 V 1424 2968 a Fr(1)22 b Fo(\000)g Fu(\024)1694 2900 y Fr(max)1721 2961 y Fq(\032)p Fs(\025)p Ft(0)1892 2819 y Fm(\010)1950 2900 y Fu(\032)17 b(e)2062 2859 y Fs(\000)p Fq(\032)2157 2819 y Fm(\011)2296 2900 y Fo(\024)2463 2833 y Fr(3)g Fu(C)2599 2848 y Fq(a)2657 2833 y Fu(\024)p 2463 2877 250 4 v 2474 2968 a Fr(1)23 b Fo(\000)f Fu(\024)2745 2900 y(;)257 3119 y Fx(using)i(\(I.16\).)31 b(This)24 b(gi)n(v)o(es)f(\(II.42\).)1925 b Fg(\003)257 3408 y Fl(II.2)119 b(Estimate)28 b(on)i Fd(\001)p Ff(W)1406 3426 y Fe(o\013)6 b FE(\000)p Fe(d)1616 3408 y Fd(\()p Ff(\024;)20 b(o)p Fd(\))26 b(+)h Ff(W)2139 3426 y Fe(diag)2326 3408 y Fl(in)j(T)-11 b(erms)29 b(of)h Ff(J)11 b Fd([)p Fl(a)p 3045 3429 60 5 v -1 w Fd(\()p Ff(\024)p Fd(\)])257 3596 y Fx(W)-8 b(e)33 b(be)o(gin)f(by)g(deri)n(ving)g(tw)o(o)g(simple)f (estimates)h(which)g(are)h(of)g(k)o(e)o(y)f(importance)g(to)g(our)257 3716 y(approach.)257 3885 y Fv(Lemma)23 b(II.4.)36 b Fn(F)-10 b(or)21 b(some)h(univer)o(sal)f(constant)g Fu(C)2030 3900 y Ft(\002)2116 3885 y Fu(<)28 b Fo(1)p Fn(,)22 b(all)f Fu(j;)c(k)31 b Fo(2)d Fr(\003)p Fn(,)22 b(and)g(all)f Fu(x)28 b Fo(2)g Fp(R)3411 3849 y Ft(\003)3470 3885 y Fn(,)1161 4039 y Fr(\024)1148 4064 y(\002)1224 4023 y Ft(2)1224 4089 y Fq(j)1280 4064 y Fo(j)p Fu(g)1359 4023 y Fs(0)1355 4089 y Fq(k)1397 4064 y Fo(j)82 b(\024)1681 4039 y Fr(\024)1668 4064 y(\002)1744 4023 y Ft(2)1744 4089 y Fq(k)1803 4064 y Fo(j)p Fu(g)1882 4023 y Fs(0)1878 4089 y Fq(k)1920 4064 y Fo(j)49 b Fr(+)h Fu(C)2193 4079 y Ft(\002)2282 4039 y Fr(\024)2269 4064 y(\002)2345 4023 y Ft(2)2345 4089 y Fq(j)2401 4064 y Fo(j)p Fu(g)2480 4023 y Fs(0)2476 4089 y Fq(j)2511 4064 y Fo(j)22 b Fu(;)650 b Fx(\(II.44\))1148 4223 y Fr(\002)1224 4182 y Ft(2)1224 4247 y Fq(j)1280 4223 y Fo(j)p Fu(g)1359 4182 y Fs(0)1355 4247 y Fq(k)1397 4223 y Fo(j)82 b(\024)1681 4198 y Fr(\024)1668 4223 y(\002)1744 4182 y Ft(2)1744 4247 y Fq(k)1803 4223 y Fo(j)p Fu(g)1882 4182 y Fs(0)1878 4247 y Fq(k)1920 4223 y Fo(j)49 b Fr(+)h Fu(C)2193 4238 y Ft(\002)2269 4223 y Fr(\002)2345 4182 y Ft(2)2345 4247 y Fq(j)2401 4223 y Fo(j)p Fu(g)2480 4182 y Fs(0)2476 4247 y Fq(j)2511 4223 y Fo(j)22 b Fu(:)650 b Fx(\(II.45\))257 4402 y Fn(Pr)l(oof:)39 b Fx(First,)30 b(Eqs.)f(\(II.44\))h(and)f(\(II.45\))h(hold)f(true)g (tri)n(vially)-6 b(,)28 b(for)i Fu(x)2696 4417 y Fq(k)2775 4402 y Fo(2)37 b Fu(I)2921 4417 y Fs(1)2995 4402 y Fx(,)31 b(because)f(on)257 4523 y(that)25 b(set)581 4497 y Fr(\024)567 4523 y(\002)643 4538 y Fq(k)714 4523 y Fo(\021)j Fu(e)864 4487 y Fq(\014)s(q)939 4495 y Fk(max)1087 4523 y Fo(\025)g Fr(\002)1268 4538 y Fq(j)1332 4523 y Fo(\025)1451 4497 y Fr(\024)1437 4523 y(\002)1513 4538 y Fq(j)1549 4523 y Fx(.)404 4643 y(Con)l(v)o(ersely)-6 b(,)26 b(if)g Fu(x)1040 4658 y Fq(k)1115 4643 y Fo(62)31 b Fu(I)1255 4658 y Fs(1)1357 4643 y Fx(then)26 b(we)h(distinguish)d(the)i(cases)h Fu(x)2602 4658 y Fq(j)2670 4643 y Fo(2)32 b Fu(I)2811 4658 y Ft(0)2877 4643 y Fx(and)27 b Fu(x)3103 4658 y Fq(j)3171 4643 y Fo(62)k Fu(I)3311 4658 y Ft(0)3351 4643 y Fx(.)36 b(In)257 4763 y(the)25 b(former)g(case,)935 4738 y Fr(\024)921 4763 y(\002)997 4778 y Fq(j)1061 4763 y Fr(=)j(\002)1241 4778 y Fq(j)1305 4763 y Fo(\021)g Fr(1)g Fo(\024)1605 4738 y Fr(\024)1592 4763 y(\002)1668 4778 y Fq(k)1710 4763 y Fx(,)d(and)g(there)g(is)f(nothing)g(to)g(pro)o (v)o(e.)404 4884 y(It)31 b(remains)g(to)h(demonstrate)e(Eqs.)h (\(II.44\))h(and)g(\(II.45\))g(for)g Fu(x)2634 4899 y Fq(k)2717 4884 y Fo(62)41 b Fu(I)2867 4899 y Fs(1)2973 4884 y Fx(and)32 b Fu(x)3204 4899 y Fq(j)3281 4884 y Fo(62)41 b Fu(I)3431 4899 y Ft(0)3470 4884 y Fx(,)257 5014 y(i.e.,)28 b(for)g Fo(j)p Fu(x)659 5029 y Fq(k)702 5014 y Fo(j)k(\024)894 4989 y Fr(\024)873 5014 y Fu(R)947 5029 y Ft(1)1014 5014 y Fx(and)27 b Fo(j)p Fu(x)1268 5029 y Fq(j)1305 5014 y Fo(j)33 b(\025)1497 4989 y Fr(^)1476 5014 y Fu(R)1550 5029 y Ft(0)1589 5014 y Fx(.)40 b(F)o(or)27 b(this,)g(we)h(remark)g(that)f(Eqs.)g(\(I.31\))h(and)f(\(I.32\))257 5135 y(imply)1135 5327 y Fo(j)p Fu(g)1214 5286 y Fs(0)1210 5351 y Fq(k)1252 5327 y Fr(\()p Fu(x)1345 5342 y Fq(k)1388 5327 y Fr(\))p Fo(j)52 b(\024)1763 5259 y Fu(C)1833 5274 y Fq(g)1890 5259 y Fu(e)1935 5223 y Fq(C)1985 5231 y Fj(g)2036 5206 y Ft(\024)2021 5223 y Fq(R)2074 5232 y Fk(1)p 1646 5304 584 4 v 1646 5410 a Fu(c)1688 5425 y Fq(g)1762 5410 y Fr(min)n Fo(f)p Fr(1)p Fu(;)2088 5385 y Fr(^)2067 5410 y Fu(R)2141 5425 y Ft(0)2180 5410 y Fo(g)2262 5327 y(j)p Fu(g)2341 5286 y Fs(0)2337 5351 y Fq(j)2373 5327 y Fr(\()p Fu(x)2466 5342 y Fq(j)2503 5327 y Fr(\))p Fo(j)22 b Fu(;)620 b Fx(\(II.46\))p eop %%Page: 31 32 31 31 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(31)257 573 y(which)25 b(completes)f(the)g(proof)h(upon)f(the)h (choice)g Fu(C)2068 588 y Ft(\002)2154 573 y Fr(:=)j Fu(C)2355 588 y Fq(g)2395 573 y Fu(c)2437 537 y Fs(\000)p Ft(1)2437 597 y Fq(g)2531 573 y Fu(e)2576 537 y Fq(C)2626 545 y Fj(g)2678 520 y Ft(\024)2663 537 y Fq(R)2716 546 y Fk(1)2771 573 y Fr(min)o Fo(f)p Fr(1)p Fu(;)3097 548 y Fr(^)3077 573 y Fu(R)3151 588 y Ft(0)3190 573 y Fo(g)3240 537 y Fs(\000)p Ft(1)3334 573 y Fx(.)59 b Fg(\003)404 693 y Fx(Lemma)24 b(II.4)h(is)f(used)h(to)f(estimate)816 934 y Fu(W)908 949 y Ft(diag)1043 934 y Fr(\()p Fu(x)p Fr(\))53 b(=)1371 867 y(2)p 1365 911 61 4 v 1365 1002 a Fu(\014)1452 839 y Fm(X)1508 1049 y Fq(j)1613 823 y Fm(\020)1686 909 y Fr(\024)1672 934 y(\002)1748 893 y Ft(2)1748 959 y Fq(j)1826 909 y Fr(^)1804 934 y Fu(G)1881 893 y Fs(00)1881 959 y Fq(j)t(j)1950 934 y Fr(\()p Fu(x)p Fr(\))44 b Fo(\000)h Fu(H)2336 893 y Fs(00)2328 959 y Fq(j)t(j)2397 934 y Fr(\(0\))2522 823 y Fm(\021)2631 934 y Fo(\012)50 b Fu(a)2809 893 y Fs(\003)2809 959 y Fq(j)2849 934 y Fu(a)2900 949 y Fq(j)3238 934 y Fx(\(II.47\))257 1255 y(in)26 b(terms)f(of)g Fu(J)9 b Fr([)p Fu(a)p 806 1271 52 4 v 1 w Fr(\()p Fu(\024)p Fr(\)])p Fx(,)25 b(where)h Fu(J)9 b Fr([)p Fu(a)p Fr(])27 b Fx(is)e(de\002ned)h(in)f(Eqs.)g (\(II.6\))h(and)f(\(II.8\).)34 b(Recall)26 b(further)n(-)257 1375 y(more)33 b(that)f Fu(a)p 677 1391 V Fr(\()p Fu(\024)p Fr(\))43 b(:=)1048 1294 y Fm(\000)1094 1375 y Fu(a)1145 1390 y Fq(j)t(k)1220 1375 y Fr(\()p Fu(\024)p Fr(\))1352 1294 y Fm(\001)1398 1414 y Fq(j;k)r Fs(2)p Ft(\003)1626 1375 y Fr(=)1745 1294 y Fm(\000)1790 1375 y Fu(e)1835 1339 y Fq(\024\032)p Ft(\()p Fq(j;k)r Ft(\))2075 1375 y Fu(a)2126 1390 y Fq(j)t(k)2201 1294 y Fm(\001)2246 1414 y Fq(j;k)r Fs(2)p Ft(\003)2433 1375 y Fx(,)35 b(and)d(that)2865 1350 y Fr(\024)2852 1375 y(\002)2928 1390 y Fq(j)3006 1375 y Fr(:=)43 b(exp)q([)p Fu(\014)13 b Fr(\024)-56 b Fu(q)3432 1390 y Fq(j)3468 1375 y Fr(])257 1532 y Fx(and)445 1507 y Fm(b)426 1532 y Fu(G)28 b Fr(=)f Fu(H)j Fo(\000)845 1458 y Fm(P)950 1562 y Fq(k)1016 1532 y Fr(^)-56 b Fu(q)1052 1547 y Fq(k)1095 1532 y Fx(.)31 b(W)-8 b(e)25 b(ha)n(v)o(e)f(the)h (follo)n(wing)e(estimates.)257 1736 y Fv(Lemma)39 b(II.5.)48 b Fn(F)-10 b(or)36 b(some)i(univer)o(sal)f(constant)f Fu(C)59 b(<)51 b Fo(1)p Fn(,)41 b(all)c Fr(0)51 b Fo(\024)h Fu(\013)h Fo(\024)f Fr(1)p Fn(,)41 b(and)c(all)257 1856 y Fu(x)28 b Fo(2)h Fp(R)501 1820 y Ft(\003)560 1856 y Fn(,)961 2051 y Fr(\024)948 2076 y(\002)1024 2035 y Ft(2)1024 2101 y Fq(j)1063 2076 y Fr(\()p Fu(x)1156 2091 y Fq(j)1193 2076 y Fr(\))1253 2051 y(^)1231 2076 y Fu(G)1308 2035 y Fs(00)1308 2101 y Fq(j)t(j)1377 2076 y Fr(\()p Fu(x)p Fr(\))44 b Fo(\000)h Fu(H)1763 2035 y Fs(00)1755 2101 y Fq(j)t(j)1824 2076 y Fr(\(0\))82 b Fo(\024)i Fu(C)35 b(J)2351 2091 y Fq(j)2387 2076 y Fr([)p Fu(a)p 2414 2092 V Fr(\()p Fu(\024)p Fr(\)]\()p Fu(x)p Fr(\))23 b Fu(;)433 b Fx(\(II.48\))948 2405 y Fr(\002)1024 2364 y Ft(2)1024 2430 y Fq(j)1063 2405 y Fr(\()p Fu(x)1156 2420 y Fq(j)1193 2405 y Fr(\))p Fu(G)1308 2364 y Fs(00)1308 2430 y Fq(j)t(j)1377 2405 y Fr(\()p Fu(x)p Fr(\))44 b Fo(\000)h Fu(H)1763 2364 y Fs(00)1755 2430 y Fq(j)t(j)1824 2405 y Fr(\(0\))82 b Fo(\024)i Fu(C)35 b(J)2351 2420 y Fq(j)2387 2405 y Fr([)p Fu(a)p 2414 2421 V Fr(\()p Fu(\024)p Fr(\)]\()p Fu(x)p Fr(\))23 b Fu(:)433 b Fx(\(II.49\))257 2625 y Fn(Pr)l(oof:)44 b Fx(W)-8 b(e)32 b(only)f(gi)n(v)o(e)g(the)g(proof)h (of)g(Eq.)f(\(II.48\),)j(the)e(proof)f(of)h(Eq.)g(\(II.48\))g(is)f (similar)-5 b(.)257 2746 y(Furthermore)29 b(we)g(note)f(that)g Fu(a)1358 2761 y Fq(j)t(k)1468 2746 y Fo(\024)35 b Fu(a)1631 2761 y Fq(j)t(k)1706 2746 y Fr(\()p Fu(\024)p Fr(\))29 b Fx(and)g(it)f(hence)h(suf)n(\002ces)f(to)g(pro)o(v)o(e)g(Eq.)g (\(II.48\))257 2866 y(for)d Fu(a)449 2881 y Fq(j)t(k)525 2866 y Fx(.)404 2997 y(First,)31 b(we)g(estimate)f(in)g(the)h(re)o (gion)f Fo(fj)p Fu(x)1831 3012 y Fq(j)1867 2997 y Fo(j)38 b Fu(<)2069 2971 y Fr(^)2047 2997 y Fu(R)2121 3012 y Ft(0)2161 2997 y Fo(g)p Fx(,)32 b(where)f(we)g(ha)n(v)o(e)2918 2971 y Fr(\024)2904 2997 y(\002)2980 3012 y Fq(j)3055 2997 y Fo(\021)39 b Fr(1)p Fx(.)48 b(This,)257 3117 y(Eq.)25 b(\(I.28\),)g(Eq.)f(\(I.30\),)h(and)g Fu(H)1384 3081 y Fs(00)1376 3142 y Fq(j)t(j)1445 3117 y Fr(\(0\))i(=)g Fu(G)1777 3081 y Fs(00)1777 3142 y Fq(j)t(j)1846 3117 y Fr(\(0\))e Fx(gi)n(v)o(e)257 3272 y Fm(\014)257 3332 y(\014)14 b Fr(\024)291 3357 y(\002)367 3316 y Ft(2)367 3381 y Fq(j)445 3332 y Fr(^)422 3357 y Fu(G)499 3316 y Fs(00)499 3381 y Fq(j)t(j)568 3357 y Fr(\()p Fu(x)p Fr(\))45 b Fo(\000)g Fu(H)955 3316 y Fs(00)947 3381 y Fq(j)t(j)1015 3357 y Fr(\(0\))1140 3272 y Fm(\014)1140 3332 y(\014)1226 3357 y Fr(=)1354 3272 y Fm(\014)1354 3332 y(\014)23 b Fr(^)1387 3357 y Fu(G)1464 3316 y Fs(00)1464 3381 y Fq(j)t(j)1533 3357 y Fr(\()p Fu(x)p Fr(\))45 b Fo(\000)1853 3332 y Fr(^)1831 3357 y Fu(G)1908 3316 y Fs(00)1908 3381 y Fq(j)t(j)1976 3357 y Fr(\(0\))2101 3272 y Fm(\014)2101 3332 y(\014)3267 3357 y Fx(\(II.50\))340 3512 y Fo(\024)501 3428 y Fm(\014)501 3487 y(\014)538 3512 y Fr(^)-53 b Fu(g)585 3471 y Fs(00)581 3537 y Fq(j)627 3512 y Fr(\()p Fu(x)720 3527 y Fq(j)757 3512 y Fr(\))44 b Fo(\000)k Fr(^)-52 b Fu(g)1012 3471 y Fs(00)1008 3537 y Fq(j)1053 3512 y Fr(\(0\))1178 3428 y Fm(\014)1178 3487 y(\014)1261 3512 y Fr(+)501 3698 y Fu(\013)599 3603 y Fm(X)580 3819 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))777 3587 y Fm(n)844 3613 y(\014)844 3673 y(\014)877 3698 y Fu(@)933 3656 y Ft(2)928 3722 y Fq(j)973 3698 y Fu(w)1043 3713 y Fq(k)r(j)1118 3698 y Fr(\()p Fu(x)1211 3713 y Fq(k)1254 3698 y Fu(;)17 b(x)1353 3713 y Fq(j)1390 3698 y Fr(\))39 b Fo(\000)g Fu(@)1639 3656 y Ft(2)1634 3722 y Fq(j)1679 3698 y Fu(w)1749 3713 y Fq(k)r(j)1824 3698 y Fr(\(0)p Fu(;)17 b Fr(0\))2042 3613 y Fm(\014)2042 3673 y(\014)2113 3698 y Fr(+)2228 3613 y Fm(\014)2228 3673 y(\014)2261 3698 y Fu(@)2317 3656 y Ft(2)2312 3722 y Fq(j)2357 3698 y Fu(w)2427 3713 y Fq(j)t(k)2502 3698 y Fr(\()p Fu(x)2595 3713 y Fq(j)2632 3698 y Fu(;)g(x)2731 3713 y Fq(k)2774 3698 y Fr(\))38 b Fo(\000)i Fu(@)3023 3656 y Ft(2)3018 3722 y Fq(j)3063 3698 y Fu(w)3133 3713 y Fq(j)t(k)3208 3698 y Fr(\(0)p Fu(;)17 b Fr(0\))3426 3613 y Fm(\014)3426 3673 y(\014)3458 3587 y(o)340 3983 y Fo(\024)84 b Fu(C)571 3998 y Fq(g)627 3983 y Fo(j)s Fr(^)-52 b Fu(g)706 3942 y Fs(0)702 4008 y Fq(j)738 3983 y Fo(j)44 b Fr(+)g(2)p Fu(C)1049 3998 y Fq(w)1122 3983 y Fu(\013)1220 3889 y Fm(X)1201 4104 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1399 3983 y Fu(a)1450 3998 y Fq(k)r(j)1542 3903 y Fm(\000)1587 3983 y Fo(j)p Fu(g)1666 3942 y Fs(0)1662 4008 y Fq(k)1704 3983 y Fo(j)22 b Fr(+)g Fo(j)p Fu(g)1931 3942 y Fs(0)1927 4008 y Fq(j)1963 3983 y Fo(j)1991 3903 y Fm(\001)340 4269 y Fo(\024)501 4188 y Fm(\000)547 4269 y Fu(C)617 4284 y Fq(g)679 4269 y Fr(+)g(2)p Fu(C)896 4284 y Fq(w)952 4269 y Fu(C)1022 4284 y Fq(a)1063 4269 y Fu(\013)1126 4188 y Fm(\001)1188 4269 y Fo(j)s Fr(^)-52 b Fu(g)1267 4228 y Fs(0)1263 4293 y Fq(j)1299 4269 y Fo(j)44 b Fr(+)g(2)p Fu(C)1610 4284 y Fq(w)1683 4269 y Fu(c)1725 4228 y Fs(\000)p Ft(1)1725 4293 y Fq(g)1836 4269 y Fu(\013)1933 4174 y Fm(X)1915 4390 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2112 4269 y Fu(a)2163 4284 y Fq(k)r(j)2255 4269 y Fr(\002)2331 4228 y Ft(2)2331 4293 y Fq(k)2390 4269 y Fo(j)s Fr(^)-52 b Fu(g)2469 4228 y Fs(0)2465 4293 y Fq(k)2507 4269 y Fo(j)22 b Fu(;)257 4619 y Fx(yielding)i(Eq.)g(\(II.48\))h(in)g(the)g (re)o(gion)f Fo(fj)p Fu(x)1712 4634 y Fq(j)1748 4619 y Fo(j)j Fu(<)1928 4594 y Fr(^)1907 4619 y Fu(R)1981 4634 y Ft(0)2021 4619 y Fo(g)p Fx(.)p eop %%Page: 32 33 32 32 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(32)404 573 y(Con)l(v)o(ersely)-6 b(,)23 b(on)i Fo(fj)p Fu(x)1153 588 y Fq(j)1189 573 y Fo(j)j(\025)1371 548 y Fr(^)1350 573 y Fu(R)1424 588 y Ft(0)1464 573 y Fo(g)p Fx(,)c(we)h(ha)n(v)o(e)526 718 y Fm(\014)526 778 y(\014)14 b Fr(\024)560 803 y(\002)636 762 y Ft(2)636 828 y Fq(j)714 778 y Fr(^)692 803 y Fu(G)769 762 y Fs(00)769 828 y Fq(j)t(j)837 803 y Fr(\()p Fu(x)p Fr(\))45 b Fo(\000)g Fu(H)1224 762 y Fs(00)1216 828 y Fq(j)t(j)1284 803 y Fr(\(0\))1409 718 y Fm(\014)1409 778 y(\014)609 995 y Fo(\024)784 969 y Fr(\024)770 995 y(\002)846 953 y Ft(2)846 1019 y Fq(j)885 910 y Fm(\014)885 970 y(\014)937 969 y(b)918 995 y Fu(G)995 953 y Fs(00)995 1019 y Fq(j)t(j)1064 995 y Fr(\()p Fu(x)p Fr(\))23 b Fo(\000)1336 969 y Fm(b)1317 995 y Fu(G)1394 953 y Fs(00)1394 1019 y Fq(j)t(j)1463 995 y Fr(\(0\))1588 910 y Fm(\014)1588 970 y(\014)1671 995 y Fr(+)1797 914 y Fm(\000)1856 969 y Fr(\024)1842 995 y(\002)1918 953 y Ft(2)1918 1019 y Fq(j)1980 995 y Fr(+)f(1)2127 914 y Fm(\001)2189 910 y(\014)2189 970 y(\014)2222 995 y Fu(G)2299 953 y Fs(00)2299 1019 y Fq(j)t(j)2368 995 y Fr(\(0\))2493 910 y Fm(\014)2493 970 y(\014)609 1177 y Fo(\024)784 1152 y Fr(\024)770 1177 y(\002)846 1136 y Ft(2)846 1201 y Fq(j)885 1092 y Fm(\014)885 1152 y(\014)922 1177 y Fr(^)-53 b Fu(g)969 1136 y Fs(00)965 1201 y Fq(j)1011 1177 y Fr(\()p Fu(x)1104 1192 y Fq(j)1141 1177 y Fr(\))22 b Fo(\000)k Fr(^)-52 b Fu(g)1352 1136 y Fs(00)1348 1201 y Fq(j)1394 1177 y Fr(\(0\))1519 1092 y Fm(\014)1519 1152 y(\014)1601 1177 y Fr(+)1727 1096 y Fm(\000)1786 1152 y Fr(\024)1773 1177 y(\002)1849 1136 y Ft(2)1849 1201 y Fq(j)1910 1177 y Fr(+)22 b(1)2057 1096 y Fm(\001)2119 1092 y(\014)2119 1152 y(\014)2152 1177 y Fu(g)2203 1136 y Fs(00)2199 1201 y Fq(j)2245 1177 y Fr(\(0\))2370 1092 y Fm(\014)2370 1152 y(\014)3238 1177 y Fx(\(II.51\))1265 1362 y Fr(+)g(2)p Fu(C)1482 1377 y Fq(w)1554 1362 y Fu(\013)1652 1267 y Fm(X)1634 1483 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))1831 1362 y Fu(a)1882 1377 y Fq(k)r(j)1974 1251 y Fm(n)2054 1337 y Fr(\024)2040 1362 y(\002)2116 1321 y Ft(2)2116 1387 y Fq(j)2172 1281 y Fm(\000)2218 1362 y Fo(j)p Fu(g)2297 1321 y Fs(0)2293 1387 y Fq(k)2335 1362 y Fo(j)g Fr(+)g Fo(j)p Fu(g)2562 1321 y Fs(0)2558 1387 y Fq(j)2593 1362 y Fo(j)2621 1281 y Fm(\001)2711 1362 y Fr(+)2845 1337 y(\024)2831 1362 y(\002)2907 1321 y Ft(2)2907 1387 y Fq(j)2991 1362 y Fr(+)44 b(1)3160 1251 y Fm(o)609 1681 y Fo(\024)770 1600 y Fm(\000)816 1681 y Fr(1)22 b(+)g Fu(C)1055 1696 y Fq(g)1117 1681 y Fr(+)g(4)p Fu(C)1334 1696 y Fq(w)1390 1681 y Fu(C)1460 1696 y Fq(a)1501 1681 y Fu(\013)1564 1600 y Fm(\001)1640 1656 y Fr(\024)1626 1681 y(\002)1702 1640 y Ft(2)1702 1706 y Fq(j)1758 1571 y Fm(h)1805 1681 y Fo(j)s Fr(^)-52 b Fu(g)1884 1640 y Fs(0)1880 1706 y Fq(j)1916 1681 y Fr(\()p Fu(x)2009 1696 y Fq(j)2046 1681 y Fr(\))p Fo(j)44 b Fr(+)g(2)2325 1600 y Fm(\000)2370 1681 y Fr(1)22 b(+)g Fo(j)p Fu(g)2618 1640 y Fs(00)2614 1706 y Fq(j)2660 1681 y Fr(\(0\))p Fo(j)2813 1600 y Fm(\001)2858 1571 y(i)1265 1884 y Fr(+)g(2)p Fu(C)1482 1899 y Fq(w)1554 1884 y Fu(C)1624 1899 y Ft(\002)1700 1884 y Fu(c)1742 1843 y Fs(\000)p Ft(1)1742 1908 y Fq(g)1853 1884 y Fu(\013)1950 1789 y Fm(X)1932 2005 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2129 1884 y Fu(a)2180 1899 y Fq(k)r(j)2272 1884 y Fr(\002)2348 1843 y Ft(2)2348 1908 y Fq(k)2407 1884 y Fo(j)p Fu(g)2486 1843 y Fs(0)2482 1908 y Fq(k)2524 1884 y Fo(j)g Fu(;)257 2227 y Fx(where)h(we)f(use)g(\(I.30\))g(again)f (and)1478 2202 y Fr(\024)1465 2227 y(\002)1541 2191 y Ft(2)1541 2252 y Fq(j)1580 2227 y Fo(j)p Fu(g)1659 2191 y Fs(0)1655 2253 y Fq(k)1697 2227 y Fo(j)27 b(\024)1871 2202 y Fr(\024)1857 2227 y(\002)1933 2191 y Ft(2)1933 2253 y Fq(k)1976 2227 y Fo(j)p Fu(g)2055 2191 y Fs(0)2051 2253 y Fq(k)2093 2227 y Fo(j)11 b Fr(+)g Fu(C)2289 2242 y Ft(\002)2361 2202 y Fr(\024)2348 2227 y(\002)2424 2191 y Ft(2)2424 2252 y Fq(j)2463 2227 y Fo(j)s Fr(^)-52 b Fu(g)2542 2191 y Fs(0)2538 2252 y Fq(j)2574 2227 y Fo(j)p Fx(,)22 b(which)f(we)h(established)257 2367 y(in)31 b(Lemma)f(II.4.)49 b(Observing)30 b(that)h(on)f Fo(fj)p Fu(x)1806 2382 y Fq(j)1843 2367 y Fo(j)38 b(\025)2047 2342 y Fr(^)2026 2367 y Fu(R)2100 2382 y Ft(0)2139 2367 y Fo(g)p Fx(,)33 b(we)e(ha)n(v)o(e)f Fo(j)p Fu(g)2688 2331 y Fs(0)2684 2392 y Fq(j)2720 2367 y Fo(j)39 b(\025)g Fu(c)2945 2382 y Fq(g)3002 2367 y Fr(min)o Fo(f)p Fr(1)p Fu(;)3328 2342 y Fr(^)3308 2367 y Fu(R)3382 2382 y Ft(0)3421 2367 y Fo(g)p Fx(,)257 2488 y(and)25 b(additionally)e(using)h Fo(j)p Fu(g)1247 2452 y Fs(00)1243 2512 y Fq(j)1288 2488 y Fr(\(0\))p Fo(j)j(\024)h Fu(C)1643 2503 y Fq(f)1689 2488 y Fx(,)c(we)h(further)g(obtain)1034 2718 y Fo(j)p Fu(g)1113 2677 y Fs(0)1109 2743 y Fq(j)1145 2718 y Fr(\()p Fu(x)1238 2733 y Fq(j)1275 2718 y Fr(\))p Fo(j)44 b Fr(+)g(2)1554 2637 y Fm(\000)1599 2718 y Fr(1)22 b(+)g Fo(j)p Fu(g)1847 2677 y Fs(00)1843 2743 y Fq(j)1889 2718 y Fr(\(0\))p Fo(j)2042 2637 y Fm(\001)2139 2718 y Fo(\024)53 b Fu(C)24 b Fo(j)s Fr(^)-52 b Fu(g)2442 2677 y Fs(0)2438 2743 y Fq(j)2474 2718 y Fr(\()p Fu(x)2567 2733 y Fq(j)2604 2718 y Fr(\))p Fo(j)21 b Fu(;)520 b Fx(\(II.52\))257 2958 y(for)28 b(the)e(uni)n(v)o(ersal)g(constant)g Fu(C)39 b Fr(:=)31 b(1)24 b(+)f(2)p Fu(c)1803 2922 y Fs(\000)p Ft(2)1803 2982 y Fq(g)1897 2958 y Fr(\(1)h(+)f Fu(C)2177 2973 y Fq(f)2223 2958 y Fr(\))17 b(min)n Fo(f)p Fr(1)p Fu(;)2603 2933 y Fr(^)2583 2958 y Fu(R)2657 2973 y Ft(0)2696 2958 y Fo(g)2746 2922 y Fs(\000)p Ft(1)2872 2958 y Fu(<)31 b Fo(1)p Fx(.)37 b(Inserting)257 3098 y(\(II.52\))26 b(into)d(\(II.51\),)i(we)h(arri)n(v)o(e)e(at)h(Eq.)f(\(II.48\))h(in)g (the)f(re)o(gion)g Fo(fj)p Fu(x)2620 3113 y Fq(j)2657 3098 y Fo(j)j(\025)2839 3073 y Fr(^)2817 3098 y Fu(R)2891 3113 y Ft(0)2931 3098 y Fo(g)p Fx(,)e(as)f(well.)83 b Fg(\003)404 3218 y Fx(Combining)20 b(Lemma)h(II.2)h(and)g(Eq.)f (\(II.48\))h(in)f(Lemma)h(II.5,)g(we)g(arri)n(v)o(e)f(at)h(the)f(follo) n(w-)257 3339 y(ing)k(intermediate)f(conclusion,)257 3542 y Fv(Theor)n(em)j(II.6.)40 b Fn(F)-10 b(or)24 b(some)h(univer)o (sal)f(constant)f Fu(C)35 b(<)27 b Fo(1)e Fn(and)f(all)g Fr(0)k Fo(\024)g Fu(\024)g Fo(\024)g Fr(1)p Fn(,)521 3715 y Fm(\015)521 3774 y(\015)576 3799 y Fu(J)9 b Fr([)p Fu(a)p 666 3815 52 4 v 1 w Fr(\()p Fu(\024)p Fr(\)])877 3758 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1064 3718 y Fm(\010)1122 3799 y Fr(\001)p Fu(W)1295 3814 y Ft(o\013)c Fs(\000)p Ft(d)1475 3799 y Fr(\()p Fu(\024;)17 b(o)p Fr(\))22 b(+)g Fu(W)1910 3814 y Ft(diag)2045 3718 y Fm(\011)2125 3799 y Fu(J)9 b Fr([)p Fu(a)p 2215 3815 V Fr(\()p Fu(\024)p Fr(\)])2425 3758 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)2590 3715 y Fm(\015)2590 3774 y(\015)2698 3799 y Fo(\024)2838 3732 y Fu(C)p 2838 3776 77 4 v 2846 3868 a(\014)2947 3799 y(;)264 b Fx(\(II.53\))257 4070 y Fn(wher)l(e)26 b Fu(J)9 b Fr([)p Fu(a)p 612 4085 52 4 v 1 w Fr(])25 b Fn(is)f(de\002ned)h(in)f(Eqs.)g(\(II.6\))h(and)g(\(II.8\).)257 4365 y Fl(II.3)119 b(Estimate)28 b(on)i Ff(J)11 b Fd([)p Fl(a)p 1306 4385 60 5 v -1 w Fd(\()p Ff(\024)p Fd(\)])29 b Fl(in)i(T)-11 b(erms)29 b(of)g Ff(K)9 b Fd([)p Fl(b)p 2338 4385 67 5 v -1 w Fd(])257 4565 y Fx(In)21 b(this)e(subsection)g (we)i(replace)g Fo(j)s Fr(^)-52 b Fu(g)1490 4528 y Fs(0)1486 4589 y Fq(j)1522 4565 y Fo(j)20 b Fx(and)h Fo(j)p Fu(g)1814 4528 y Fs(0)1810 4590 y Fq(k)1852 4565 y Fo(j)f Fx(in)g Fu(J)2052 4580 y Fq(j)2088 4565 y Fr([)p Fu(a)p 2115 4580 52 4 v 1 w Fr(])g Fx(by)g Fo(j)2384 4539 y Fr(^)2362 4565 y Fu(G)2439 4528 y Fs(0)2439 4589 y Fq(j)2475 4565 y Fo(j)h Fx(and)f Fo(j)p Fu(G)2793 4528 y Fs(0)2793 4590 y Fq(k)2835 4565 y Fo(j)p Fx(,)h(respecti)n(v)o(ely)-6 b(.)27 b(It)257 4695 y(turns)20 b(out)g(that)f(it)h(is)g(con)l(v)o (enient)f(to)h(slighty)e(increase)j(the)f(weights)f Fu(a)p 2617 4711 V 1 w Fr(\()p Fu(\024)p Fr(\))27 b(:=)2959 4614 y Fm(\000)3005 4695 y Fu(a)3056 4710 y Fq(j)t(k)3131 4695 y Fr(\()p Fu(\024)p Fr(\))3263 4614 y Fm(\001)3309 4734 y Fq(j;k)r Fs(2)p Ft(\003)257 4830 y Fx(by)e(passing)f(to)g Fu(b)p 808 4846 42 4 v 28 w Fr(:=)k(\()p Fu(b)1087 4845 y Fq(j)t(k)1162 4830 y Fr(\))1200 4845 y Fq(j;k)r Fs(2)p Ft(\003)1412 4830 y Fx(de\002ned)d(by)1038 5129 y Fu(b)1079 5144 y Fq(j)t(k)1207 5129 y Fr(:=)1391 5061 y(1)p 1372 5106 86 4 v 1372 5197 a Fu(\013)1435 5168 y Fs(0)1526 5004 y(1)1489 5034 y Fm(X)1485 5246 y Fq(N)7 b Ft(=1)1655 5018 y Fm(\020)1865 5061 y Fu(\013)1928 5025 y Fs(0)p 1724 5106 368 4 v 1724 5197 a Fr(1)22 b Fo(\000)h Fu(C)1965 5212 y Fq(a)2006 5197 y Fu(\013)2069 5168 y Fs(0)2102 5018 y Fm(\021)2161 5041 y Fq(N)2256 5048 y Fm(\000)2302 5129 y Fu(a)p 2302 5145 52 4 v Fr(\(1\))2478 5088 y Fq(N)2545 5048 y Fm(\001)2591 5168 y Fq(j)t(k)2688 5129 y Fu(;)523 b Fx(\(II.54\))p eop %%Page: 33 34 33 33 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(33)257 573 y(where)31 b Fr(0)37 b Fu(<)h(\013)794 537 y Fs(0)854 573 y Fu(<)g Fr(\(2)p Fu(C)1125 588 y Fq(a)1166 573 y Fr(\))1204 537 y Fs(\000)p Ft(1)1328 573 y Fx(and)30 b Fu(a)p 1502 589 52 4 v 1 w Fr(\(1\))1679 537 y Fq(N)1776 573 y Fx(denotes)f(the)h Fu(N)2350 537 y Fq(th)2451 573 y Fx(po)n(wer)g(of)g Fu(a)p 2841 589 V 1 w Fr(\(1\))f Fx(as)i(a)f(matrix)257 693 y(with)24 b Fu(\024)k Fr(=)g(1)p Fx(,)c(i.e.,)581 818 y Fm(\000)627 898 y Fu(a)p 627 914 V Fr(\()p Fu(\024)p Fr(\))810 857 y Fq(N)878 818 y Fm(\001)923 937 y Fq(j)t(k)1051 898 y Fr(:=)1312 804 y Fm(X)1206 1013 y Fq(n)1249 1022 y Fk(1)1284 1013 y Fq(;:::)n(;n)1425 1024 y Fj(N)5 b Fc(\000)p Fk(1)1577 898 y Fu(e)1622 857 y Fq(\032)p Ft(\()p Fq(j;n)1777 866 y Fk(1)1812 857 y Ft(\))1860 898 y Fu(a)1911 913 y Fq(j)15 b(n)1998 922 y Fk(1)2053 898 y Fo(\001)i(\001)g(\001)e Fu(e)2231 857 y Fq(\032)p Ft(\()p Fq(n)2337 868 y Fj(N)5 b Fc(\000)p Fk(1)2474 857 y Fq(;k)r Ft(\))2580 898 y Fu(a)2631 913 y Fq(n)2674 924 y Fj(N)g Fc(\000)p Fk(1)2822 913 y Fq(k)2887 898 y Fu(;)324 b Fx(\(II.55\))257 1220 y(recalling)39 b(that)f Fu(a)884 1235 y Fq(j)t(k)959 1220 y Fr(\()p Fu(\024)p Fr(\))53 b(=)g Fu(e)1318 1184 y Fq(\024\032)p Ft(\()p Fq(i;j)t Ft(\))1547 1220 y Fu(a)1598 1235 y Fq(j)t(k)1673 1220 y Fx(.)73 b(T)-8 b(o)38 b(pro)o(v)o(e)g(the)g (con)l(v)o(er)n(gence)h(of)g(the)f(series)h(in)257 1340 y(Eq.)26 b(\(II.54\))f(we)h(vie)n(w)f Fu(a)p 1060 1356 V Fr(\(1\))h Fx(as)f(a)h(real,)g(symmetric)e(\(and)i(thus)f(self)o (adjoint\))f(matrix)h(on)g Fp(C)3436 1304 y Ft(\003)257 1461 y Fx(equipped)d(with)g(the)g(norm)g(induced)h(by)f(the)g(standard) g(scalar)h(product)f Fo(h)p Fu(')p Fo(j)p Fu( )t Fo(i)27 b Fr(=)3137 1386 y Fm(P)3242 1490 y Fq(i)p 3287 1406 93 4 v 3287 1461 a Fu(')3351 1476 y Fq(i)3379 1461 y Fu( )3442 1476 y Fq(i)3470 1461 y Fx(.)257 1581 y(The)37 b(operator)f(norm)f(of)i Fu(a)p 1183 1597 52 4 v Fr(\(1\))f Fx(is)g(bounded)f(by)h Fu(C)2083 1596 y Fq(a)2124 1581 y Fx(,)j(and)e(hence)f Fo(k)p Fu(\021)20 b(a)p 2755 1597 V 1 w Fo(k)48 b Fu(<)h Fr(1)p Fx(,)39 b(pro)o(vided)257 1701 y Fr(0)31 b Fu(<)f(\021)k Fr(:=)d Fu(\013)722 1665 y Fs(0)745 1701 y Fr(\(1)23 b Fo(\000)h Fu(\013)1019 1665 y Fs(0)1042 1701 y Fu(C)1112 1716 y Fq(a)1153 1701 y Fr(\))30 b Fu(<)h(C)1405 1665 y Fs(\000)p Ft(1)1398 1726 y Fq(a)1526 1701 y Fx(which)26 b(is)f(equi)n(v)n(alent)g(to)h Fr(0)k Fu(<)h(\013)2678 1665 y Fs(0)2731 1701 y Fu(<)g(C)2915 1665 y Fs(\000)p Ft(1)2908 1726 y Fq(a)3009 1701 y Fu(=)p Fr(2)p Fx(.)k(W)-8 b(e)27 b(note)257 1822 y(that)e(tri)n(vially)1231 2005 y Fu(b)1272 2020 y Fq(j)t(k)1400 2005 y Fo(\025)1540 1938 y Fu(e)1585 1901 y Fq(\032)p Ft(\()p Fq(j;k)r Ft(\))1784 1938 y Fu(a)1835 1953 y Fq(j)t(k)p 1540 1982 370 4 v 1541 2073 a Fr(1)d Fo(\000)h Fu(C)1782 2088 y Fq(a)1823 2073 y Fu(\013)1886 2045 y Fs(0)1972 2005 y Fo(\025)54 b Fu(e)2148 1964 y Fq(\032)p Ft(\()p Fq(j;k)r Ft(\))2346 2005 y Fu(a)2397 2020 y Fq(j)t(k)2494 2005 y Fu(;)717 b Fx(\(II.56\))257 2212 y(by)25 b(ignoring)e(all)i(terms)f(in)h(the)f (series)h(\(II.55\))g(of)g(order)g Fu(N)38 b Fo(\025)29 b Fr(2)p Fx(.)404 2333 y(The)23 b(follo)n(wing)d(Proposition)i(sho)n (ws)f(that)i Fu(b)p 1914 2349 42 4 v 23 w Fx(possesses)f(similar)g (summability)e(proper)n(-)257 2453 y(ties)25 b(as)g Fu(a)p 529 2469 52 4 v Fr(\()p Fu(\024)p Fr(\))g Fx(and)f(has)h(the)g (additional)e(property)i(\(II.58\),)g(which)f Fu(a)p 2571 2469 V Fr(\()p Fu(\024)p Fr(\))h Fx(does)g(not.)257 2644 y Fv(Lemma)j(II.7.)41 b Fn(F)-10 b(or)26 b Fr(0)31 b Fu(<)g(\013)1246 2607 y Fs(0)1300 2644 y Fu(<)g(C)1484 2607 y Fs(\000)p Ft(1)1477 2668 y Fq(a)1578 2644 y Fu(=)p Fr(2)p Fn(,)26 b(the)h(weights)f Fu(b)p 2207 2659 42 4 v 32 w Fr(:=)31 b(\()p Fu(b)2493 2659 y Fq(j)t(k)2568 2644 y Fr(\))2606 2659 y Fq(j;k)r Fs(2)p Ft(\003)2819 2644 y Fn(de\002ned)c(in)f(\(II.54\))257 2764 y(e)n(xist,)36 b(ar)l(e)e(symmetric,)h Fu(b)1173 2779 y Fq(j)t(k)1293 2764 y Fr(=)44 b Fu(b)1454 2779 y Fq(k)r(j)1529 2764 y Fn(,)36 b(vanish)d(on)h(the)f(dia)o(gonal,)h Fu(b)2628 2779 y Fq(j)t(j)2742 2764 y Fr(=)44 b(0)p Fn(,)35 b(and)f(obe)m(y)f (the)257 2884 y(summability)23 b(condition)1065 3142 y Fr(max)1140 3203 y Fq(j)1263 3002 y Fm(\032)1355 3048 y(X)1407 3260 y Fq(k)1515 3142 y Fu(b)1556 3157 y Fq(j)t(k)1632 3002 y Fm(\033)1759 3142 y Fo(\024)2052 3075 y Fu(C)2122 3090 y Fq(a)p 1899 3119 417 4 v 1899 3211 a Fr(1)f Fo(\000)h Fr(2)p Fu(C)2189 3226 y Fq(a)2230 3211 y Fu(\013)2293 3182 y Fs(0)2378 3142 y Fr(=:)53 b Fu(C)2604 3157 y Fq(b)2660 3142 y Fu(:)551 b Fx(\(II.57\))257 3431 y Fn(Mor)l(eo)o(ver)25 b(the)m(y)g(additionally)e(obe)m(y)i(the)f(estimate)1292 3618 y Fu(\013)1355 3581 y Fs(0)p 1151 3662 368 4 v 1151 3753 a Fr(1)e Fo(\000)h Fu(C)1392 3768 y Fq(a)1433 3753 y Fu(\013)1496 3724 y Fs(0)1546 3590 y Fm(X)1586 3800 y Fq(m)1706 3685 y Fu(b)1747 3700 y Fq(j)t(m)1863 3685 y Fu(e)1908 3644 y Fq(\032)p Ft(\()p Fq(j;k)r Ft(\))2107 3685 y Fu(a)2158 3700 y Fq(mk)2316 3685 y Fo(\024)53 b Fu(b)2487 3700 y Fq(j)t(k)2584 3685 y Fu(;)627 b Fx(\(II.58\))257 3972 y Fn(for)25 b(all)f Fu(j;)17 b(k)31 b Fo(2)d Fr(\003)p Fn(.)257 4162 y(Pr)l(oof:)i Fx(W)-8 b(e)25 b(\002rst)g(apply)f(the)h (triangle)f(inequality)g(and)g(\(I.16\))h(to)g(deri)n(v)o(e)498 4421 y Fr(max)573 4482 y Fq(j)697 4281 y Fm(\032)788 4327 y(X)841 4539 y Fq(k)948 4340 y Fm(\000)994 4421 y Fu(a)p 994 4437 52 4 v Fr(\(1\))1170 4380 y Fq(N)1237 4340 y Fm(\001)1283 4460 y Fq(j)t(k)1358 4281 y Fm(\033)3238 4421 y Fx(\(II.59\))582 4726 y Fo(\024)83 b Fr(max)817 4787 y Fq(j)940 4586 y Fm(\032)1166 4632 y(X)1032 4844 y Fq(n)1075 4853 y Fk(1)1109 4844 y Fq(;:::)n(;n)1250 4855 y Fj(N)5 b Fc(\000)p Fk(1)1386 4844 y Fq(;k)1461 4726 y Fu(e)1506 4685 y Fq(\032)p Ft(\()p Fq(j;n)1661 4694 y Fk(1)1695 4685 y Ft(\))1743 4726 y Fu(a)1794 4741 y Fq(j)15 b(n)1881 4750 y Fk(1)1937 4726 y Fo(\001)i(\001)g(\001)d Fu(e)2114 4685 y Fq(\032)p Ft(\()p Fq(n)2220 4696 y Fj(N)5 b Fc(\000)p Fk(1)2357 4685 y Fq(;k)r Ft(\))2463 4726 y Fu(a)2514 4741 y Fq(n)2557 4752 y Fj(N)g Fc(\000)p Fk(1)2705 4741 y Fq(k)2748 4586 y Fm(\033)2903 4726 y Fo(\024)81 b Fu(C)3138 4685 y Fq(N)3131 4751 y(a)3227 4726 y Fu(;)257 5033 y Fx(which)25 b(yields)f(\(II.57\),)447 5310 y Fr(max)522 5371 y Fq(j)645 5170 y Fm(\032)737 5216 y(X)789 5428 y Fq(k)897 5310 y Fu(e)942 5269 y Fq(\032)p Ft(\()p Fq(j;k)r Ft(\))1141 5310 y Fu(b)1182 5325 y Fq(j)t(k)1257 5170 y Fm(\033)1385 5310 y Fo(\024)1543 5243 y Fr(1)p 1525 5288 86 4 v 1525 5379 a Fu(\013)1588 5350 y Fs(0)1679 5186 y(1)1642 5216 y Fm(X)1637 5427 y Fq(N)7 b Ft(=1)1807 5200 y Fm(\020)1962 5243 y Fu(C)2032 5258 y Fq(a)2073 5243 y Fu(\013)2136 5207 y Fs(0)p 1877 5288 368 4 v 1877 5379 a Fr(1)21 b Fo(\000)i Fu(C)2117 5394 y Fq(a)2158 5379 y Fu(\013)2221 5350 y Fs(0)2254 5200 y Fm(\021)2314 5222 y Fq(N)2434 5310 y Fr(=)2725 5243 y Fu(C)2795 5258 y Fq(a)p 2572 5288 417 4 v 2572 5379 a Fr(1)f Fo(\000)h Fr(2)p Fu(C)2862 5394 y Fq(a)2903 5379 y Fu(\013)2966 5350 y Fs(0)3021 5310 y Fu(:)190 b Fx(\(II.60\))p eop %%Page: 34 35 34 34 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(34)404 573 y(Ne)o(xt,)20 b(setting)f Fu(\021)31 b Fr(:=)d Fu(\013)1201 537 y Fs(0)1224 573 y Fr(\(1)t Fo(\000)t Fu(\013)1459 537 y Fs(0)1483 573 y Fu(C)1553 588 y Fq(a)1594 573 y Fr(\))p Fx(,)21 b(we)f(note)g(that)g Fr(0)27 b Fu(<)h(\021)j(<)d(C)2616 537 y Fs(\000)p Ft(1)2609 597 y Fq(a)2710 573 y Fx(.)h(Hence)21 b Fi(1)t Fo(\000)t Fu(\021)t(a)p 3232 589 52 4 v 1 w Fr(\(1\))f Fx(is)257 693 y(in)l(v)o(ertible,)j(and)h(its)f(in)l(v)o(erse)g(can)h(be)g(e)o (xpanded)g(in)f(a)h(norm-con)l(v)o(er)n(gent)f(Neumann)g(series,)532 874 y Fm(\000)577 954 y Fi(1)45 b Fo(\000)f Fu(\021)21 b(a)p 868 970 V Fr(\(1\)\))1082 913 y Fs(\000)p Ft(1)1228 954 y Fr(=)1398 830 y Fs(1)1361 860 y Fm(X)1357 1071 y Fq(N)7 b Ft(=0)1527 874 y Fm(\000)1572 954 y Fu(\021)21 b(a)p 1641 970 V Fr(\(1\))1817 874 y Fm(\001)1862 896 y Fq(N)1982 954 y Fr(=)52 b Fi(1)e Fr(+)2383 830 y Fs(1)2347 860 y Fm(X)2342 1071 y Fq(N)7 b Ft(=1)2512 874 y Fm(\000)2557 954 y Fu(\021)21 b(a)p 2626 970 V Fr(\(1\))2802 874 y Fm(\001)2847 896 y Fq(N)2937 954 y Fu(;)274 b Fx(\(II.61\))257 1225 y(and)25 b(hence)g(we)g(ha)n(v)o(e)1251 1265 y Fm(\000)1297 1346 y Fi(1)44 b Fo(\000)h Fu(\021)20 b(a)p 1587 1362 V 1 w Fr(\(1\)\))1802 1305 y Fs(\000)p Ft(1)1948 1346 y Fr(=)52 b Fi(1)e Fr(+)g Fu(\013)2371 1305 y Fs(0)2411 1346 y Fu(b)p 2411 1362 42 4 v 22 w(:)737 b Fx(\(II.62\))257 1507 y(F)o(or)25 b(the)g(deri)n(v)n(ation)e(of)i(\(II.58\),)g(we)g (multiply)d(\(II.62\))k(by)e Fi(1)e Fo(\000)h Fu(\021)t(a)p 2545 1523 52 4 v Fr(\(1\))h Fx(and)h(obtain)1094 1695 y Fi(1)52 b Fr(=)h Fi(1)38 b Fo(\000)i Fu(\021)20 b(a)p 1610 1711 V Fr(\(1\))39 b(+)f Fu(\013)2002 1654 y Fs(0)2025 1695 y Fu(b)p 2025 1711 42 4 v 39 w Fo(\000)i Fu(\013)2285 1654 y Fs(0)2308 1695 y Fu(\021)20 b(b)p 2376 1711 V 17 w(a)p 2434 1711 52 4 v Fr(\(1\))i Fu(:)579 b Fx(\(II.63\))257 1884 y(Since)26 b(the)e(matrix)g(elements)g(of)h Fu(\021)t(a)p 1474 1900 V 25 w Fx(are)h(nonne)o(gati)n(v)o(e,)21 b(this)j(implies) 1476 2072 y Fu(\021)1528 1992 y Fm(\000)1573 2072 y Fu(b)p 1573 2088 42 4 v 18 w(a)p 1632 2088 52 4 v Fr(\(1\))1808 1992 y Fm(\001)1853 2111 y Fq(j)t(k)1981 2072 y Fo(\024)53 b Fu(b)2152 2087 y Fq(j)t(k)2249 2072 y Fu(;)962 b Fx(\(II.64\))257 2261 y(which)25 b(is)f(equi)n(v)n(alent)f(to)h(\(II.58\).)1982 b Fg(\003)404 2381 y Fx(Ne)o(xt)20 b(we)i(use)f(the)g(ne)n(w)g(weights) g Fu(b)p 1569 2397 42 4 v 28 w Fr(=)27 b(\()p Fu(b)1820 2396 y Fq(j)t(k)1896 2381 y Fr(\))1934 2396 y Fq(j;k)r Fs(2)p Ft(\003)2142 2381 y Fx(to)21 b(implement)f(the)h(change)g(from)h Fo(j)p Fu(g)3430 2345 y Fs(0)3426 2407 y Fq(k)3468 2381 y Fo(j)257 2502 y Fx(to)31 b Fo(j)p Fu(G)471 2465 y Fs(0)471 2528 y Fq(k)513 2502 y Fo(j)p Fx(.)49 b(The)31 b(main)g(result)f(of)h (this)f(subsection)g(is)h(a)g(form)g(bound)f(on)h Fu(J)9 b Fr([)p Fu(a)p 2924 2518 52 4 v Fr(\()p Fu(\024)p Fr(\)])31 b Fx(in)g(terms)257 2622 y(of)d Fu(K)7 b Fr([)p Fu(b)p 485 2638 42 4 v 1 w Fr(])p Fx(,)29 b(where)g Fu(J)9 b Fr([)p Fu(a)p 970 2638 52 4 v Fr(])28 b Fx(and)g Fu(K)7 b Fr([)p Fu(a)p 1365 2638 V 1 w Fr(])28 b Fx(are)h(de\002ned)f(in)g (Eqs.)f(\(II.6\),)j(\(II.8\),)f(and)f(\(II.9\),)h(and)f(we)257 2742 y(recall)e(that)e Fu(a)p 678 2758 V Fr(\()p Fu(\024)p Fr(\))k(:=)1019 2662 y Fm(\000)1065 2742 y Fu(a)1116 2757 y Fq(j)t(k)1191 2742 y Fr(\()p Fu(\024)p Fr(\))1323 2662 y Fm(\001)1369 2781 y Fq(j;k)r Fs(2)p Ft(\003)1583 2742 y Fr(=)1687 2662 y Fm(\000)1733 2742 y Fu(e)1778 2706 y Fq(\024\032)p Ft(\()p Fq(j;k)r Ft(\))2017 2742 y Fu(a)2068 2757 y Fq(j)t(k)2143 2662 y Fm(\001)2189 2781 y Fq(j;k)r Fs(2)p Ft(\003)2375 2742 y Fx(.)257 2927 y Fv(Theor)n(em)33 b(II.8.)44 b Fn(Ther)l(e)31 b(e)n(xists)f(a)h (univer)o(sal)f(constant)f Fr(1)38 b Fo(\024)h Fu(C)46 b(<)38 b Fo(1)31 b Fn(suc)o(h)f(that,)h(for)f(all)257 3048 y Fr(0)e Fo(\024)g Fu(\013)g(<)g(\013)696 3012 y Fs(0)747 3048 y Fo(\024)g Fu(C)929 3012 y Fs(\000)p Ft(1)1048 3048 y Fn(and)c(all)g Fr(0)k Fo(\024)g Fu(\024)g Fo(\024)g Fr(1)p Fn(,)1100 3252 y Fu(J)9 b Fr([)p Fu(a)p 1190 3268 V Fr(\()p Fu(\024)p Fr(\)])53 b Fo(\024)g Fu(J)9 b Fr([)p Fu(b)p 1673 3268 42 4 v 1 w Fr(])53 b Fo(\024)1925 3142 y Fm(\020)1984 3252 y Fr(1)22 b(+)g Fu(C)2252 3185 y(\013)p 2240 3229 86 4 v 2240 3321 a(\013)2303 3292 y Fs(0)2336 3142 y Fm(\021)2418 3252 y Fu(K)7 b Fr([)p Fu(b)p 2535 3268 42 4 v Fr(])23 b Fu(:)585 b Fx(\(II.65\))257 3465 y Fn(Pr)l(oof:)30 b Fx(W)-8 b(e)25 b(\002rst)g(remark)g(that)f (\(II.65\))i(is)e(equi)n(v)n(alent)f(to)798 3678 y Fo(8)p Fu(j)34 b Fo(2)29 b Fr(\003)55 b(:)146 b Fu(J)1372 3693 y Fq(j)1408 3678 y Fr([)p Fu(a)p 1435 3694 52 4 v Fr(\()p Fu(\024)p Fr(\)])53 b Fo(\024)g Fu(J)1882 3693 y Fq(j)1918 3678 y Fr([)p Fu(b)p 1945 3694 42 4 v 1 w Fr(])g Fo(\024)2197 3567 y Fm(\020)2256 3678 y Fr(1)22 b(+)g Fu(C)2524 3610 y(\013)p 2512 3655 86 4 v 2512 3746 a(\013)2575 3717 y Fs(0)2608 3567 y Fm(\021)2690 3678 y Fu(K)2773 3693 y Fq(j)2809 3678 y Fr([)p Fu(b)p 2836 3694 42 4 v 1 w Fr(])g Fu(:)284 b Fx(\(II.66\))257 3909 y(Secondly)-6 b(,)31 b(the)f(\002rst)g(inequality)f(in)g(\(II.66\))i(tri)n(vially)d (follo)n(ws)g(from)i Fu(a)2783 3924 y Fq(j)t(k)2896 3909 y Fo(\024)38 b Fu(e)3056 3872 y Fq(\032)p Ft(\()p Fq(j;k)r Ft(\))3254 3909 y Fu(a)3305 3924 y Fq(j)t(k)3418 3909 y Fo(\024)257 4029 y Fu(b)298 4044 y Fq(j)t(k)374 4029 y Fx(.)49 b(Hence)32 b(it)f(remains)f(to)h(pro)o(v)o(e)f(the)h(second)g (inequality)e(in)i(\(II.66\),)i(and)e(for)g(this)f(it)h(is)257 4149 y(suf)n(\002cient)25 b(to)f(sho)n(w)g(the)g(e)o(xistence)g(of)h(a) h(uni)n(v)o(ersal)d(constant)h Fu(C)2524 4113 y Fs(0)2575 4149 y Fu(<)j Fo(1)e Fx(such)f(that)915 4338 y Fo(8)p Fu(j)34 b Fo(2)28 b Fr(\003)55 b(:)1434 4253 y Fm(\014)1434 4313 y(\014)1467 4338 y Fu(J)1521 4353 y Fq(j)1557 4338 y Fr([)p Fu(b)p 1584 4354 V 1 w Fr(])39 b Fo(\000)g Fu(K)1891 4353 y Fq(j)1927 4338 y Fr([)p Fu(b)p 1954 4354 V 1 w Fr(])2023 4253 y Fm(\014)2023 4313 y(\014)2109 4338 y Fo(\024)53 b Fu(C)2316 4297 y Fs(0)2356 4227 y Fm(\020)2437 4271 y Fu(\013)p 2425 4315 86 4 v 2425 4406 a(\013)2488 4377 y Fs(0)2521 4227 y Fm(\021)2603 4338 y Fu(J)2657 4353 y Fq(j)2693 4338 y Fr([)p Fu(b)p 2720 4354 42 4 v 1 w Fr(])22 b Fu(:)400 b Fx(\(II.67\))257 4551 y(T)-8 b(o)36 b(this)f(end,)j(we)e(proceed)g(as)f(in)h(the)f(proof)h(of)f (Lemma)g(II.5)h(and)g(observ)o(e)f(that)g(\(I.33\))257 4671 y(implies)377 4775 y Fm(\014)377 4835 y(\014)410 4860 y Fu(J)464 4875 y Fq(j)501 4860 y Fr([)p Fu(b)p 528 4875 V Fr(])k Fo(\000)g Fu(K)834 4875 y Fq(j)871 4860 y Fr([)p Fu(b)p 898 4875 V 1 w Fr(])967 4775 y Fm(\014)967 4835 y(\014)3238 4860 y Fx(\(II.68\))460 5032 y Fo(\024)634 5007 y Fr(\024)621 5032 y(\002)697 4991 y Ft(2)697 5057 y Fq(j)753 5032 y Fo(j)799 5007 y Fm(b)781 5032 y Fu(G)858 4991 y Fs(0)858 5057 y Fq(j)916 5032 y Fo(\000)26 b Fr(^)-52 b Fu(g)1067 4991 y Fs(0)1063 5057 y Fq(j)1099 5032 y Fo(j)44 b Fr(+)1291 4938 y Fm(X)1344 5150 y Fq(k)1451 5032 y Fu(b)1492 5047 y Fq(j)t(k)1584 5032 y Fr(\002)1660 4991 y Ft(2)1660 5057 y Fq(k)1720 5032 y Fo(j)p Fu(G)1825 4991 y Fs(0)1825 5057 y Fq(k)1889 5032 y Fo(\000)23 b Fu(g)2040 4991 y Fs(0)2036 5057 y Fq(k)2078 5032 y Fo(j)460 5298 y(\024)84 b Fu(\013)700 5203 y Fm(X)741 5412 y Fq(m)861 5298 y Fu(a)912 5313 y Fq(j)t(m)1027 5217 y Fm(\000)1086 5272 y Fr(\024)1073 5298 y(\002)1149 5256 y Ft(2)1149 5322 y Fq(j)1188 5298 y Fo(j)p Fu(g)1267 5256 y Fs(0)1263 5322 y Fq(j)1299 5298 y Fo(j)39 b Fr(+)1494 5272 y(\024)1480 5298 y(\002)1556 5256 y Ft(2)1556 5322 y Fq(j)1595 5298 y Fo(j)p Fu(g)1674 5256 y Fs(0)1670 5322 y Fq(m)1736 5298 y Fo(j)1764 5217 y Fm(\001)1860 5298 y Fr(+)49 b Fu(\013)2065 5203 y Fm(X)2076 5415 y Fq(k)r(;m)2225 5298 y Fu(b)2266 5313 y Fq(j)t(k)2358 5298 y Fu(a)2409 5313 y Fq(k)r(m)2531 5217 y Fm(\000)2577 5298 y Fr(\002)2653 5256 y Ft(2)2653 5322 y Fq(k)2695 5298 y Fo(j)p Fu(g)2774 5256 y Fs(0)2770 5322 y Fq(k)2812 5298 y Fo(j)39 b Fr(+)f(\002)3069 5256 y Ft(2)3069 5322 y Fq(k)3112 5298 y Fo(j)p Fu(g)3191 5256 y Fs(0)3187 5322 y Fq(m)3253 5298 y Fo(j)3281 5217 y Fm(\001)3348 5298 y Fu(:)p eop %%Page: 35 36 35 35 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(35)257 573 y(No)n(w)24 b(we)i(apply)e(Lemma)g(II.4)h(and)g (\(II.58\),)347 716 y Fm(\014)347 776 y(\014)380 801 y Fu(J)434 816 y Fq(j)471 801 y Fr([)p Fu(b)p 498 817 42 4 v Fr(])39 b Fo(\000)h Fu(K)805 816 y Fq(j)841 801 y Fr([)p Fu(b)p 868 817 V 1 w Fr(])937 716 y Fm(\014)937 776 y(\014)1050 801 y Fo(\024)81 b Fu(\013)1287 706 y Fm(X)1328 915 y Fq(m)1448 801 y Fu(a)1499 816 y Fq(j)t(m)1614 720 y Fm(\000)1660 801 y Fr(\(1)22 b(+)g Fu(C)1937 816 y Ft(\002)1996 801 y Fr(\))2048 775 y(\024)2034 801 y(\002)2110 759 y Ft(2)2110 825 y Fq(j)2149 801 y Fo(j)p Fu(g)2228 759 y Fs(0)2224 825 y Fq(j)2260 801 y Fo(j)38 b Fr(+)2455 775 y(\024)2441 801 y(\002)2517 759 y Ft(2)2517 825 y Fq(m)2583 801 y Fo(j)p Fu(g)2662 759 y Fs(0)2658 825 y Fq(m)2724 801 y Fo(j)2752 720 y Fm(\001)1563 1063 y Fr(+)28 b Fu(\013)1746 968 y Fm(X)1758 1180 y Fq(k)r(;m)1907 1063 y Fu(b)1948 1078 y Fq(j)t(k)2040 1063 y Fu(a)2091 1078 y Fq(k)r(m)2213 982 y Fm(\000)2258 1063 y Fr(\(1)22 b(+)g Fu(C)2535 1078 y Ft(\002)2594 1063 y Fr(\)\002)2708 1022 y Ft(2)2708 1088 y Fq(k)2751 1063 y Fo(j)p Fu(g)2830 1022 y Fs(0)2826 1088 y Fq(k)2868 1063 y Fo(j)38 b Fr(+)3062 1038 y(\024)3049 1063 y(\002)3125 1022 y Ft(2)3125 1088 y Fq(m)3191 1063 y Fo(j)p Fu(g)3270 1022 y Fs(0)3266 1088 y Fq(m)3332 1063 y Fo(j)3360 982 y Fm(\001)430 1341 y Fo(\024)84 b Fr(\(1)22 b(+)g Fu(C)868 1356 y Ft(\002)926 1341 y Fr(\))17 b Fu(C)1051 1356 y Fq(a)1109 1341 y Fu(\013)1202 1316 y Fr(\024)1188 1341 y(\002)1264 1300 y Ft(2)1264 1366 y Fq(j)1304 1341 y Fo(j)p Fu(g)1383 1300 y Fs(0)1379 1366 y Fq(j)1415 1341 y Fo(j)38 b Fr(+)h Fu(\013)1675 1247 y Fm(X)1716 1456 y Fq(m)1835 1341 y Fu(a)1886 1356 y Fq(j)t(m)2016 1316 y Fr(\024)2002 1341 y(\002)2078 1300 y Ft(2)2078 1366 y Fq(m)2145 1341 y Fo(j)p Fu(g)2224 1300 y Fs(0)2220 1366 y Fq(m)2286 1341 y Fo(j)973 1613 y Fr(+)27 b Fu(\013)18 b(C)1226 1628 y Fq(a)1284 1613 y Fr(\(1)k(+)g Fu(C)1561 1628 y Ft(\002)1619 1613 y Fr(\))1674 1519 y Fm(X)1727 1731 y Fq(k)1834 1613 y Fu(b)1875 1628 y Fq(j)t(k)1968 1613 y Fr(\002)2044 1572 y Ft(2)2044 1638 y Fq(k)2086 1613 y Fo(j)p Fu(g)2165 1572 y Fs(0)2161 1638 y Fq(k)2203 1613 y Fo(j)50 b Fr(+)2406 1503 y Fm(\020)2487 1546 y Fu(\013)p 2476 1591 86 4 v 2476 1682 a(\013)2539 1653 y Fs(0)2572 1503 y Fm(\021)2648 1519 y(X)2689 1728 y Fq(m)2808 1613 y Fu(b)2849 1628 y Fq(j)t(m)2979 1588 y Fr(\024)2965 1613 y(\002)3041 1572 y Ft(2)3041 1638 y Fq(m)3108 1613 y Fo(j)p Fu(g)3187 1572 y Fs(0)3183 1638 y Fq(m)3249 1613 y Fo(j)430 1889 y(\024)84 b Fr(\(1)22 b(+)g Fu(C)868 1904 y Ft(\002)926 1889 y Fr(\))17 b Fu(C)1051 1904 y Fq(a)1109 1889 y Fu(\013)1202 1864 y Fr(\024)1188 1889 y(\002)1264 1848 y Ft(2)1264 1913 y Fq(j)1304 1889 y Fo(j)p Fu(g)1383 1848 y Fs(0)1379 1913 y Fq(j)1415 1889 y Fo(j)49 b Fr(+)h Fu(C)1695 1848 y Fs(00)1754 1778 y Fm(\020)1835 1821 y Fu(\013)p 1823 1866 V 1823 1957 a(\013)1886 1928 y Fs(0)1919 1778 y Fm(\021)1995 1794 y(X)2048 2006 y Fq(k)2156 1889 y Fu(b)2197 1904 y Fq(j)t(k)2302 1864 y Fr(\024)2289 1889 y(\002)2365 1848 y Ft(2)2365 1913 y Fq(k)2407 1889 y Fo(j)p Fu(g)2486 1848 y Fs(0)2482 1913 y Fq(k)2524 1889 y Fo(j)22 b Fu(;)637 b Fx(\(II.69\))257 2206 y(for)26 b(some)e(uni)n(v)o(ersal)f(constant)h Fu(C)1454 2169 y Fs(00)1524 2206 y Fu(<)k Fo(1)p Fx(,)c(where)i(we)f (additionally)e(use)i Fu(a)2897 2221 y Fq(j)t(k)3000 2206 y Fo(\024)j Fu(b)3146 2221 y Fq(j)t(k)3222 2206 y Fx(,)3285 2180 y Fr(\024)3271 2206 y(\002)3347 2221 y Fq(k)3418 2206 y Fo(\024)257 2326 y Fr(\002)333 2341 y Fq(k)376 2326 y Fx(,)i Fu(\013)494 2290 y Fs(0)553 2326 y Fo(\024)36 b Fr(1)p Fx(,)30 b(and)f Fo(j)p Fu(g)1022 2290 y Fs(0)1018 2351 y Fq(j)1054 2326 y Fo(j)35 b(\024)i Fu(c)1273 2290 y Fs(\000)p Ft(1)1273 2351 y Fq(g)1367 2326 y Fo(j)s Fr(^)-52 b Fu(g)1446 2290 y Fs(0)1442 2351 y Fq(j)1478 2326 y Fo(j)29 b Fx(to)f(deri)n(v)o(e)h(the)g(last)f (inequality)-6 b(.)42 b(This)29 b(clearly)g(implies)257 2446 y(\(II.67\).)2879 b Fg(\003)257 2741 y Fl(II.4)119 b(Estimate)28 b(on)i Ff(K)9 b Fd([)p Fl(b)p 1339 2761 67 5 v -1 w Fd(])30 b Fl(in)g(T)-11 b(erms)29 b(of)h Ff(L)p Fd([)p Fl(b)p 2190 2761 V -1 w Fd(])257 2943 y Fx(The)g(purpose)e(of)i(this)e(section)h(is)g(to)f(replace)i Fo(j)p Fu(G)2008 2907 y Fs(0)2008 2969 y Fq(k)2051 2943 y Fo(j)f Fx(and)g Fo(j)2328 2918 y Fm(b)2309 2943 y Fu(G)2386 2907 y Fs(0)2386 2969 y Fq(k)2428 2943 y Fo(j)g Fx(in)g Fu(K)7 b Fr([)p Fx(b)p 2709 2960 50 4 v Fr(])30 b Fx(by)f Fu(")p Fr(\()p Fu(G)3106 2907 y Fs(0)3106 2969 y Fq(k)3148 2943 y Fr(\))3186 2907 y Ft(2)3251 2943 y Fr(+)c(1)p Fu(=")257 3074 y Fx(and)40 b Fu(")p Fr(\()543 3049 y Fm(b)525 3074 y Fu(G)602 3038 y Fs(0)602 3100 y Fq(k)644 3074 y Fr(\))682 3038 y Ft(2)754 3074 y Fr(+)33 b(1)p Fu(=")p Fx(,)42 b(respecti)n(v)o(ely)-6 b(,)41 b(where)e Fu(")54 b(>)h Fr(0)39 b Fx(is)g(a)g(small)g(parameter)g(which)g(we)257 3195 y(later)32 b(choose)e(to)h(be)g(of)g(order)h Fu(\014)1424 3159 y Fs(\000)p Ft(1)p Fq(=)p Ft(2)1588 3195 y Fx(.)50 b(This)30 b(replacement)h(requires)g(the)g(introduction)e(of)257 3315 y Fu(L)p Fr([)p Fu(b)p 350 3331 42 4 v 1 w Fr(])p Fx(,)k(de\002ned)f(in)f(Eqs.)g(\(II.7\))h(and)f(\(II.10\).)51 b(The)31 b(main)g(result)g(of)g(this)g(subsection)f(is)g(the)257 3436 y(follo)n(wing)23 b(theorem.)257 3634 y Fv(Theor)n(em)33 b(II.9.)44 b Fn(Ther)l(e)31 b(e)n(xists)f(a)h(univer)o(sal)f(constant)f Fr(1)38 b Fo(\024)h Fu(C)46 b(<)38 b Fo(1)31 b Fn(suc)o(h)f(that,)h (for)f(all)257 3754 y Fr(0)e Fo(\024)g Fu(\013)g(<)g(\013)696 3718 y Fs(0)747 3754 y Fo(\024)g Fu(C)929 3718 y Fs(\000)p Ft(1)1048 3754 y Fn(and)c(all)g Fr(0)k Fu(<)f(")h Fo(\024)g Fu(C)1788 3718 y Fs(\000)p Ft(1)1882 3754 y Fn(,)1308 4001 y Fu(K)7 b Fr([)p Fu(b)p 1425 4017 V 1 w Fr(])53 b Fo(\024)1687 3933 y Fr(1)p 1687 3978 49 4 v 1689 4069 a Fu(")1762 4001 y(L)p Fr([)p Fu(b)p 1855 4017 42 4 v 1 w Fr(])d(+)g Fu(C)23 b(")17 b(N)2334 4016 y Fs(F)2417 4001 y Fu(;)794 b Fx(\(II.70\))257 4245 y Fn(wher)l(e)26 b Fu(N)600 4260 y Fs(F)689 4245 y Fr(=)793 4171 y Fm(P)898 4274 y Fq(k)957 4245 y Fi(1)c Fo(\012)h Fu(a)1186 4209 y Fs(\003)1186 4271 y Fq(k)1229 4245 y Fu(a)1280 4260 y Fq(k)1347 4245 y Fn(is)i(the)f(number)h(oper)o(ator)e(on)h Fo(H)q Fn(.)257 4455 y(Pr)l(oof:)41 b Fx(W)-8 b(e)31 b(start)f(with)g(the)g(observ)n(ation)f(that)h(if)g Fo(j)p Fu(t)p Fo(j)38 b(\025)2308 4429 y Fr(^)2286 4455 y Fu(R)2360 4470 y Ft(0)2431 4455 y Fx(then)30 b Fo(j)p Fu(g)2712 4418 y Fs(0)2708 4481 y Fq(k)2750 4455 y Fr(\()p Fu(t)p Fr(\))p Fo(j)g Fx(and)g Fo(j)s Fr(^)-52 b Fu(g)3172 4418 y Fs(0)3168 4479 y Fq(j)3204 4455 y Fr(\()p Fu(t)p Fr(\))p Fo(j)31 b Fx(are)257 4595 y(bounded)23 b(belo)n(w)f(by)h(a)h(uni)n(v)o (ersal,)d(positi)n(v)o(e)g(constant)i Fu(C)2256 4559 y Fs(\000)p Ft(1)2378 4595 y Fu(>)k Fr(0)p Fx(.)j(Con)l(v)o(ersely)-6 b(,)23 b(if)g Fo(j)p Fu(t)p Fo(j)k Fu(<)3403 4570 y Fr(^)3382 4595 y Fu(R)3456 4610 y Ft(0)257 4715 y Fx(then)467 4690 y Fr(\024)454 4715 y(\002)530 4730 y Fq(j)566 4715 y Fr(\()p Fu(t)p Fr(\))h(=)g(\002\()p Fu(t)p Fr(\))f(=)h(1)p Fx(.)i(Thus)24 b(we)i(ha)n(v)o(e)e(the)h(follo)n(wing)e(estimates,)833 4939 y Fr(\002)909 4898 y Ft(2)909 4964 y Fq(k)952 4939 y Fr(\()p Fu(x)1045 4954 y Fq(k)1088 4939 y Fr(\))84 b(=)f Fi(1)1425 4858 y Fm(\002)1467 4939 y Fo(j)p Fu(x)1550 4954 y Fq(k)1593 4939 y Fo(j)27 b Fu(<)1773 4914 y Fr(^)1752 4939 y Fu(R)1826 4954 y Ft(0)1865 4858 y Fm(\003)1951 4939 y Fr(+)44 b Fi(1)2127 4858 y Fm(\002)2169 4939 y Fo(j)p Fu(x)2252 4954 y Fq(k)2294 4939 y Fo(j)28 b(\025)2476 4914 y Fr(^)2455 4939 y Fu(R)2529 4954 y Ft(0)2569 4858 y Fm(\003)2627 4939 y Fr(\002)2703 4898 y Ft(2)2703 4964 y Fq(k)2745 4939 y Fr(\()p Fu(x)2838 4954 y Fq(k)2881 4939 y Fr(\))1209 5084 y Fo(\024)83 b Fr(1)44 b(+)h Fu(C)23 b Fr(\002)1752 5043 y Ft(2)1752 5109 y Fq(k)1795 5084 y Fr(\()p Fu(x)1888 5099 y Fq(k)1931 5084 y Fr(\))p Fo(j)p Fu(g)2048 5043 y Fs(0)2044 5109 y Fq(k)2086 5084 y Fr(\()p Fu(x)2179 5099 y Fq(k)2222 5084 y Fr(\))p Fo(j)e Fu(;)902 b Fx(\(II.71\))856 5204 y Fr(\024)843 5230 y(\002)919 5188 y Ft(2)919 5254 y Fq(j)958 5230 y Fr(\()p Fu(x)1051 5245 y Fq(j)1088 5230 y Fr(\))84 b(=)f Fi(1)1425 5149 y Fm(\002)1467 5230 y Fo(j)p Fu(x)1550 5245 y Fq(j)1587 5230 y Fo(j)27 b Fu(<)1767 5204 y Fr(^)1745 5230 y Fu(R)1819 5245 y Ft(0)1859 5149 y Fm(\003)1945 5230 y Fr(+)44 b Fi(1)2121 5149 y Fm(\002)2163 5230 y Fo(j)p Fu(x)2246 5245 y Fq(j)2282 5230 y Fo(j)28 b(\025)2464 5204 y Fr(^)2443 5230 y Fu(R)2517 5245 y Ft(0)2556 5149 y Fm(\003)2628 5204 y Fr(\024)2615 5230 y(\002)2691 5188 y Ft(2)2691 5254 y Fq(j)2730 5230 y Fr(\()p Fu(x)2823 5245 y Fq(j)2860 5230 y Fr(\))1209 5388 y Fo(\024)83 b Fr(1)44 b(+)h Fu(C)1690 5363 y Fr(\024)1676 5388 y(\002)1752 5347 y Ft(2)1752 5413 y Fq(j)1791 5388 y Fr(\()p Fu(x)1884 5403 y Fq(j)1921 5388 y Fr(\))p Fo(j)s Fr(^)-52 b Fu(g)2038 5347 y Fs(0)2034 5413 y Fq(j)2070 5388 y Fr(\()p Fu(x)2163 5403 y Fq(j)2200 5388 y Fr(\))p Fo(j)22 b Fu(;)923 b Fx(\(II.72\))p eop %%Page: 36 37 36 36 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(36)257 573 y(and,)25 b(as)g(an)g(immediate)e(consequence,)890 834 y Fr(\002)966 793 y Ft(2)966 859 y Fq(k)1009 834 y Fo(j)p Fu(G)1114 793 y Fs(0)1114 859 y Fq(k)1156 834 y Fo(j)83 b(\024)1460 767 y Fr(1)p 1437 811 95 4 v 1437 902 a(2)p Fu(")1558 834 y Fr(\002)1634 793 y Ft(2)1634 859 y Fq(k)1677 834 y Fo(j)p Fu(G)1782 793 y Fs(0)1782 859 y Fq(k)1824 834 y Fo(j)1852 793 y Ft(2)1941 834 y Fr(+)2078 767 y Fu(")p 2077 811 49 4 v 2077 902 a Fr(2)2135 753 y Fm(\000)2181 834 y Fr(1)44 b(+)g Fu(C)24 b Fr(\002)2564 793 y Ft(2)2564 859 y Fq(k)2606 834 y Fo(j)p Fu(g)2685 793 y Fs(0)2681 859 y Fq(k)2723 834 y Fo(j)2751 753 y Fm(\001)2819 834 y Fu(;)392 b Fx(\(II.73\))913 1042 y Fr(\024)900 1067 y(\002)976 1026 y Ft(2)976 1092 y Fq(j)1015 1067 y Fo(j)1061 1042 y Fm(b)1043 1067 y Fu(G)1120 1026 y Fs(0)1120 1092 y Fq(j)1156 1067 y Fo(j)83 b(\024)1460 1000 y Fr(1)p 1437 1044 95 4 v 1437 1135 a(2)p Fu(")1572 1042 y Fr(\024)1558 1067 y(\002)1634 1026 y Ft(2)1634 1092 y Fq(j)1673 1067 y Fo(j)1720 1042 y Fm(b)1701 1067 y Fu(G)1778 1026 y Fs(0)1778 1092 y Fq(j)1815 1067 y Fo(j)1843 1026 y Ft(2)1932 1067 y Fr(+)2069 1000 y Fu(")p 2067 1044 49 4 v 2067 1135 a Fr(2)2126 986 y Fm(\000)2172 1067 y Fr(1)44 b(+)g Fu(C)2492 1042 y Fr(\024)2478 1067 y(\002)2554 1026 y Ft(2)2554 1092 y Fq(j)2594 1067 y Fo(j)s Fr(^)-52 b Fu(g)2673 1026 y Fs(0)2669 1092 y Fq(j)2705 1067 y Fo(j)2733 986 y Fm(\001)2800 1067 y Fu(:)411 b Fx(\(II.74\))257 1318 y(Inserting)28 b(\(II.73\)\226\(II.74\))h(into)e (the)i(de\002nition)e(\(II.9\))i(of)g Fu(K)2408 1333 y Fq(j)2444 1318 y Fr([)p Fu(b)p 2471 1334 42 4 v 1 w Fr(])f Fx(and)h(observing)e(the)i(de\002-)257 1438 y(nition)24 b(\(II.8\))h(of)g Fu(J)909 1453 y Fq(j)945 1438 y Fr([)p Fu(b)p 972 1454 V 1 w Fr(])p Fx(,)g(we)g(obtain)324 1679 y Fu(K)407 1694 y Fq(j)444 1679 y Fr([)p Fu(b)p 471 1695 V Fr(])84 b(=)797 1654 y(\024)783 1679 y(\002)859 1638 y Ft(2)859 1703 y Fq(j)915 1679 y Fo(j)962 1654 y Fm(b)943 1679 y Fu(G)1020 1638 y Fs(0)1020 1703 y Fq(j)1056 1679 y Fo(j)44 b Fr(+)1248 1584 y Fm(X)1301 1796 y Fq(k)1409 1679 y Fu(b)1450 1694 y Fq(j)t(k)1542 1679 y Fr(\002)1618 1638 y Ft(2)1618 1703 y Fq(k)1677 1679 y Fo(j)p Fu(G)1782 1638 y Fs(0)1782 1703 y Fq(k)1824 1679 y Fo(j)1386 b Fx(\(II.75\))623 1975 y Fo(\024)816 1908 y Fr(1)p 793 1952 95 4 v 793 2043 a(2)p Fu(")914 1975 y(L)980 1990 y Fq(j)1017 1975 y Fr([)p Fu(b)p 1044 1991 42 4 v 1 w Fr(])44 b(+)1289 1908 y Fu(")p 1287 1952 49 4 v 1287 2043 a Fr(2)1346 1864 y Fm(n)1426 1950 y Fr(\024)1412 1975 y(\002)1488 1934 y Ft(2)1488 2000 y Fq(j)1566 1975 y Fr(+)1681 1880 y Fm(X)1734 2093 y Fq(k)1841 1975 y Fu(b)1882 1990 y Fq(j)t(k)1958 1864 y Fm(o)2069 1975 y Fr(+)2199 1908 y Fu(C)23 b(")p 2199 1952 140 4 v 2244 2043 a Fr(2)2348 1864 y Fm(n)2428 1950 y Fr(\024)2414 1975 y(\002)2490 1934 y Ft(2)2490 2000 y Fq(j)2530 1975 y Fo(j)s Fr(^)-52 b Fu(g)2609 1934 y Fs(0)2605 2000 y Fq(j)2640 1975 y Fo(j)39 b Fr(+)2821 1880 y Fm(X)2874 2093 y Fq(k)2982 1975 y Fu(b)3023 1990 y Fq(j)t(k)3099 1975 y Fr(\002)3175 1934 y Ft(2)3175 2000 y Fq(k)3217 1975 y Fo(j)p Fu(g)3296 1934 y Fs(0)3292 2000 y Fq(k)3334 1975 y Fo(j)3362 1864 y Fm(o)623 2278 y Fo(\024)816 2210 y Fr(1)p 793 2255 95 4 v 793 2346 a(2)p Fu(")914 2278 y(L)980 2293 y Fq(j)1017 2278 y Fr([)p Fu(b)p 1044 2294 42 4 v 1 w Fr(])44 b(+)1287 2210 y(\(1)22 b(+)g Fu(C)1564 2225 y Fq(b)1598 2210 y Fr(\))17 b Fu(")p 1287 2255 412 4 v 1468 2346 a Fr(2)1752 2278 y(+)1883 2210 y Fu(C)23 b(")p 1883 2255 140 4 v 1928 2346 a Fr(2)2048 2278 y Fu(J)2102 2293 y Fq(j)2139 2278 y Fr([)p Fu(b)p 2166 2294 42 4 v Fr(])g Fu(:)623 2522 y Fo(\024)816 2454 y Fr(1)p 793 2499 95 4 v 793 2590 a(2)p Fu(")914 2522 y(L)980 2537 y Fq(j)1017 2522 y Fr([)p Fu(b)p 1044 2537 42 4 v 1 w Fr(])44 b(+)1287 2454 y(\(1)22 b(+)g Fu(C)1564 2469 y Fq(b)1598 2454 y Fr(\))17 b Fu(")p 1287 2499 412 4 v 1468 2590 a Fr(2)1752 2522 y(+)1883 2454 y Fu(C)23 b(C)2053 2418 y Fs(0)2093 2454 y Fu(")p 1883 2499 257 4 v 1986 2590 a Fr(2)2165 2411 y Fm(\020)2225 2522 y Fr(1)f(+)2415 2454 y Fu(\013)p 2404 2499 86 4 v 2404 2590 a(\013)2467 2561 y Fs(0)2499 2411 y Fm(\021)2576 2522 y Fu(K)2659 2537 y Fq(j)2695 2522 y Fr([)p Fu(b)p 2722 2537 42 4 v 1 w Fr(])g Fu(;)257 2766 y Fx(where)30 b(the)f(last)g(inequality)f(results)g(from)h(Theorem)g(II.8,)h(which)f (ensures)g(the)g(e)o(xistence)257 2886 y(of)h(a)g(uni)n(v)o(ersal)e (constant)h Fu(C)1274 2850 y Fs(0)1334 2886 y Fu(<)37 b Fo(1)29 b Fx(such)h(that)f Fu(J)2022 2901 y Fq(j)2058 2886 y Fr([)p Fu(b)p 2085 2902 V 1 w Fr(])37 b Fo(\024)g Fr(\(1)26 b(+)f Fu(C)2596 2850 y Fs(0)2619 2886 y Fu(\013)q(=\013)2794 2850 y Fs(0)2817 2886 y Fr(\))17 b Fu(K)2955 2901 y Fq(j)2991 2886 y Fr([)p Fu(b)p 3018 2902 V 1 w Fr(])p Fx(,)31 b(pro)o(vided)257 3006 y Fu(\013)q(=\013)432 2970 y Fs(0)483 3006 y Fx(is)c(suf)n (\002ciently)f(small.)39 b(Thus,)28 b(if)f Fu(\013)34 b Fo(\024)f Fu(\013)1941 2970 y Fs(0)1992 3006 y Fx(and)28 b Fu(")33 b Fo(\024)g Fr(\(2)p Fu(C)7 b(C)2594 2970 y Fs(0)2617 3006 y Fr(\))2655 2970 y Fs(\000)p Ft(1)2777 3006 y Fx(then)27 b(we)h(can)g(solv)o(e)257 3127 y(in)d(Eq.)f (\(II.75\))h(for)g Fu(K)1026 3142 y Fq(j)1063 3127 y Fr([)p Fu(b)p 1090 3143 V 1 w Fr(])g Fx(and)f(obtain)1217 3394 y Fu(K)1300 3409 y Fq(j)1337 3394 y Fr([)p Fu(b)p 1364 3410 V 1 w Fr(])52 b Fo(\024)1625 3327 y Fr(1)p 1625 3371 49 4 v 1627 3463 a Fu(")1701 3394 y(L)1767 3409 y Fq(j)1804 3394 y Fr([)p Fu(b)p 1831 3410 42 4 v Fr(])e(+)g(\(1)22 b(+)g Fu(C)2352 3409 y Fq(b)2386 3394 y Fr(\))17 b Fu(")k(;)703 b Fx(\(II.76\))257 3638 y(pro)o(ving)24 b(the)g(assertion.)2301 b Fg(\003)257 3934 y Fl(II.5)119 b(Estimate)28 b(on)i Ff(L)p Fd([)p Fl(b)p 1310 3954 67 5 v Fd(])f Fl(in)i(T)-11 b(erms)29 b(of)g Ff(M)12 b Fd([)p Fl(b)p 2206 3954 V 1 w Fd(])257 4121 y Fx(In)30 b(this)e(subsection)g(we)i(sho)n(w)e(that)h (subtracting)f Fu(\014)2088 4085 y Fs(\000)p Ft(1)2182 4121 y Fu(G)2259 4085 y Fs(00)2259 4147 y Fq(k)r(k)2370 4121 y Fx(from)h Fr(\()p Fu(G)2708 4085 y Fs(0)2708 4147 y Fq(k)2751 4121 y Fr(\))2789 4085 y Ft(2)2857 4121 y Fx(in)g Fu(L)p Fr([)p Fu(b)p 3057 4137 42 4 v 1 w Fr(])h Fx(does)f(not)257 4242 y(cause)d(a)g(signi\002cant)f(error)-5 b(.)34 b(This)25 b(is)g(our)g(last)g(preparatory)h(step)g(for)g(the)f (deri)n(v)n(ation)f(of)h(the)257 4362 y(main)d(form)h(bound)f (\(II.1\).)30 b(Again,)22 b(we)h(\002rst)g(mak)o(e)f(the)h(form)f(of)h (the)f(desired)h(upper)f(bound)257 4482 y(precise)29 b(by)f(using)f(the)h(operator)g Fu(M)10 b Fr([)p Fu(b)p 1581 4498 V 1 w Fr(])p Fx(,)29 b(de\002ned)g(in)f(Eq.)g(\(II.7\))g(and) g(Eqs.)g(\(II.11\)\226\(II.12\).)257 4603 y(The)d(follo)n(wing)e (theorem)h(states)h(the)f(main)h(result)f(of)h(this)f(subsection.)257 4806 y Fv(Theor)n(em)30 b(II.10.)42 b Fn(Ther)l(e)29 b(e)n(xists)e(a)h(univer)o(sal)f(constant)g Fr(1)33 b Fo(\024)h Fu(C)41 b(<)33 b Fo(1)28 b Fn(suc)o(h)f(that,)h(for)f(all)257 4927 y Fu(\014)34 b Fo(\025)28 b Fu(C)7 b Fn(,)1164 5090 y Fu(L)p Fr([)p Fu(b)p 1257 5106 V 1 w Fr(])53 b Fo(\024)1509 4979 y Fm(\020)1568 5090 y Fr(1)22 b(+)1747 5022 y Fu(C)p 1747 5067 77 4 v 1755 5158 a(\014)1834 4979 y Fm(\021)1910 5090 y Fu(M)10 b Fr([)p Fu(b)p 2041 5106 42 4 v 2 w Fr(])50 b(+)2296 5022 y Fu(C)p 2296 5067 77 4 v 2304 5158 a(\014)2400 5090 y(N)2478 5105 y Fs(F)2561 5090 y Fu(;)650 b Fx(\(II.77\))257 5314 y Fn(wher)l(e)26 b Fu(N)600 5329 y Fs(F)689 5314 y Fr(=)793 5240 y Fm(P)898 5343 y Fq(k)957 5314 y Fi(1)c Fo(\012)h Fu(a)1186 5278 y Fs(\003)1186 5340 y Fq(k)1229 5314 y Fu(a)1280 5329 y Fq(k)1347 5314 y Fn(is)i(the)f(number)h(oper)o (ator)e(on)h Fo(H)q Fn(.)p eop %%Page: 37 38 37 37 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(37)257 573 y Fn(Pr)l(oof:)30 b Fx(First)25 b(we)g(note)f(that)g (the)h(dif)n(ference)g(between)g Fu(M)2287 588 y Fq(j)2324 573 y Fr([)p Fu(b)p 2351 589 42 4 v 1 w Fr(])g Fx(and)f Fu(L)2679 588 y Fq(j)2716 573 y Fr([)p Fu(b)p 2743 589 V 1 w Fr(])h Fx(is)f(bounded)g(by)510 756 y Fm(\014)510 815 y(\014)543 840 y Fu(M)637 855 y Fq(j)674 840 y Fr([)p Fu(b)p 701 856 V 1 w Fr(])e Fo(\000)h Fu(L)958 855 y Fq(j)995 840 y Fr([)p Fu(b)p 1022 856 V Fr(])1090 756 y Fm(\014)1090 815 y(\014)1176 840 y Fo(\024)1322 773 y Fr(1)p 1316 817 61 4 v 1316 909 a Fu(\014)1387 730 y Fm(\020)1460 815 y Fr(\024)1446 840 y(\002)1522 799 y Ft(2)1522 865 y Fq(j)1561 840 y Fo(j)1608 815 y Fm(b)1589 840 y Fu(G)1666 799 y Fs(00)1666 865 y Fq(j)t(j)1735 840 y Fr(\()p Fu(x)p Fr(\))p Fo(j)44 b Fr(+)2058 746 y Fm(X)2111 958 y Fq(k)2219 840 y Fu(b)2260 855 y Fq(j)t(k)2352 840 y Fr(\002)2428 799 y Ft(2)2428 865 y Fq(k)2470 840 y Fo(j)p Fu(G)2575 799 y Fs(00)2575 865 y Fq(k)r(k)2656 840 y Fr(\()p Fu(x)p Fr(\))p Fo(j)2815 730 y Fm(\021)593 1133 y Fo(\024)770 1065 y Fr(1)p 764 1110 V 764 1201 a Fu(\014)834 1022 y Fm(\020)894 1133 y Fo(j)p Fu(H)1011 1091 y Fs(00)1003 1157 y Fq(j)t(j)1071 1133 y Fr(\(0\))p Fo(j)g Fr(+)1388 1038 y Fm(X)1440 1250 y Fq(k)1548 1133 y Fu(b)1589 1148 y Fq(j)t(k)1681 1133 y Fo(j)p Fu(H)1798 1091 y Fs(00)1790 1157 y Fq(k)r(k)1871 1133 y Fr(\(0\))p Fo(j)2024 1022 y Fm(\021)2133 1133 y Fr(+)1029 b Fx(\(II.78\))770 1357 y Fr(1)p 764 1402 V 764 1493 a Fu(\014)834 1314 y Fm(\020)894 1340 y(\014)894 1400 y(\014)13 b Fr(\024)927 1425 y(\002)1003 1384 y Ft(2)1003 1449 y Fq(j)1061 1400 y Fm(b)1042 1425 y Fu(G)1119 1384 y Fs(00)1119 1449 y Fq(j)t(j)1188 1425 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)g Fu(H)1563 1384 y Fs(00)1555 1449 y Fq(j)t(j)1624 1425 y Fr(\(0\))1749 1340 y Fm(\014)1749 1400 y(\014)1826 1425 y Fr(+)1946 1330 y Fm(X)1999 1542 y Fq(k)2107 1425 y Fu(b)2148 1440 y Fq(j)t(k)2240 1340 y Fm(\014)2240 1400 y(\014)2273 1425 y Fr(\002)2349 1384 y Ft(2)2349 1449 y Fq(k)2392 1425 y Fu(G)2469 1384 y Fs(00)2469 1449 y Fq(k)r(k)2550 1425 y Fr(\()p Fu(x)p Fr(\))g Fo(\000)g Fu(H)2925 1384 y Fs(00)2917 1449 y Fq(k)r(k)2998 1425 y Fr(\(0\))3123 1340 y Fm(\014)3123 1400 y(\014)3156 1314 y(\021)3215 1425 y Fu(:)257 1728 y Fx(Due)25 b(to)g(Eqs.)f (\(I.8\))h(and)g(\(I.15\),)g(we)g(ha)n(v)o(e)f(that)826 1962 y Fo(j)p Fu(H)943 1921 y Fs(00)935 1987 y Fq(j)t(j)1003 1962 y Fr(\(0\))p Fo(j)44 b Fr(+)1320 1868 y Fm(X)1373 2080 y Fq(k)1480 1962 y Fu(b)1521 1977 y Fq(j)t(k)1614 1962 y Fo(j)p Fu(H)1731 1921 y Fs(00)1723 1987 y Fq(k)r(k)1803 1962 y Fr(\(0\))p Fo(j)52 b(\024)h Fr(\()p Fu(C)2246 1977 y Fq(f)2313 1962 y Fr(+)22 b Fu(\013)q Fr(\))17 b(\(1)22 b(+)g Fu(C)2806 1977 y Fq(b)2840 1962 y Fr(\))g Fu(:)311 b Fx(\(II.79\))257 2266 y(Ne)o(xt,)24 b(it)h(follo)n(ws)e (from)i(Eqs.)f(\(II.48\)\226\(II.49\))h(and)g(\(I.30\))g(that)552 2425 y Fm(\014)552 2485 y(\014)13 b Fr(\024)585 2510 y(\002)661 2469 y Ft(2)661 2534 y Fq(j)739 2485 y Fr(^)717 2510 y Fu(G)794 2469 y Fs(00)794 2534 y Fq(j)t(j)863 2510 y Fr(\()p Fu(x)p Fr(\))44 b Fo(\000)h Fu(H)1249 2469 y Fs(00)1241 2534 y Fq(j)t(j)1310 2510 y Fr(\(0\))1435 2425 y Fm(\014)1435 2485 y(\014)1520 2510 y Fo(\024)53 b Fu(C)1727 2399 y Fm(\020)1800 2485 y Fr(\024)1787 2510 y(\002)1863 2469 y Ft(2)1863 2534 y Fq(j)1919 2510 y Fo(j)s Fr(^)-52 b Fu(g)1998 2469 y Fs(0)1994 2534 y Fq(j)2030 2510 y Fo(j)43 b Fr(+)i Fu(\013)2301 2415 y Fm(X)2354 2627 y Fq(k)2461 2510 y Fu(a)2512 2525 y Fq(j)t(k)2604 2510 y Fr(\002)2680 2469 y Ft(2)2680 2534 y Fq(k)2739 2510 y Fo(j)p Fu(g)2818 2469 y Fs(0)2814 2534 y Fq(k)2856 2510 y Fo(j)2884 2399 y Fm(\021)3238 2510 y Fx(\(II.80\))257 2820 y(and,)25 b(for)g Fu(k)31 b Fo(2)d Fr(\003)22 b Fo(n)g(f)p Fu(j)6 b Fo(g)p Fx(,)280 2983 y Fm(\014)280 3042 y(\014)313 3067 y Fr(\002)389 3082 y Fq(k)432 3025 y Ft(2)488 3067 y Fu(G)565 3082 y Fq(k)r(k)646 3025 y Fs(00)688 3067 y Fr(\()p Fu(x)p Fr(\))45 b Fo(\000)g Fu(H)1075 3026 y Fs(00)1067 3092 y Fq(k)r(k)1148 3067 y Fr(\(0\))1273 2983 y Fm(\014)1273 3042 y(\014)1388 3067 y Fo(\024)84 b Fu(C)1626 2957 y Fm(\020)1685 3067 y Fr(\002)1761 3026 y Ft(2)1761 3092 y Fq(k)1821 3067 y Fo(j)p Fu(g)1900 3026 y Fs(0)1896 3092 y Fq(k)1938 3067 y Fo(j)44 b Fr(+)g Fu(\013)2209 2973 y Fm(X)2267 3185 y Fq(`)2370 3067 y Fu(a)2421 3082 y Fq(k)r(`)2522 3042 y Fr(\024)2509 3067 y(\002)2585 3026 y Ft(2)2585 3092 y Fq(`)2641 3067 y Fo(j)p Fu(g)2720 3026 y Fs(0)2716 3092 y Fq(`)2748 3067 y Fo(j)2776 2957 y Fm(\021)3238 3067 y Fx(\(II.81\))1388 3343 y Fo(\024)84 b Fu(C)1626 3232 y Fm(\020)1685 3343 y Fu(a)1736 3358 y Fq(j)t(k)1825 3317 y Fr(\024)1812 3343 y(\002)1888 3301 y Ft(2)1888 3367 y Fq(j)1944 3343 y Fo(j)s Fr(^)-52 b Fu(g)2023 3301 y Fs(0)2019 3367 y Fq(j)2054 3343 y Fo(j)44 b Fr(+)h(\002)2323 3301 y Ft(2)2323 3367 y Fq(k)2382 3343 y Fo(j)p Fu(g)2461 3301 y Fs(0)2457 3367 y Fq(k)2499 3343 y Fo(j)f Fr(+)g Fu(\013)2784 3248 y Fm(X)2770 3464 y Fq(`)p Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2958 3343 y Fu(a)3009 3358 y Fq(k)r(`)3097 3343 y Fr(\002)3173 3301 y Ft(2)3173 3367 y Fq(`)3229 3343 y Fo(j)p Fu(g)3308 3301 y Fs(0)3304 3367 y Fq(`)3336 3343 y Fo(j)3364 3232 y Fm(\021)3446 3343 y Fu(;)257 3686 y Fx(for)33 b(some)f(uni)n(v)o(ersal)f(constant)h Fu(C)49 b(<)42 b Fo(1)p Fx(,)34 b(since)2055 3661 y Fr(\024)2042 3686 y(\002)2118 3701 y Fq(`)2193 3686 y Fo(\024)42 b Fr(\002)2388 3701 y Fq(`)2421 3686 y Fx(.)54 b(Inserting)32 b(these)g(estimates)257 3806 y(into)24 b(the)h(last)f(term)h(on)f(the)h (right)f(side)g(of)h(\(II.78\),)g(we)g(obtain)351 3956 y Fm(\014)351 4016 y(\014)398 4015 y Fr(\024)384 4041 y(\002)460 3999 y Ft(2)460 4065 y Fq(j)518 4015 y Fm(b)500 4041 y Fu(G)577 3999 y Fs(00)577 4065 y Fq(j)t(j)645 4041 y Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)h Fu(H)1021 3999 y Fs(00)1013 4065 y Fq(j)t(j)1081 4041 y Fr(\(0\))1206 3956 y Fm(\014)1206 4016 y(\014)1283 4041 y Fr(+)1404 3946 y Fm(X)1456 4158 y Fq(k)1564 4041 y Fu(b)1605 4056 y Fq(j)t(k)1697 3956 y Fm(\014)1697 4016 y(\014)1730 4041 y Fr(\002)1806 3999 y Ft(2)1806 4065 y Fq(k)1849 4041 y Fu(G)1926 3999 y Fs(00)1926 4065 y Fq(k)r(k)2007 4041 y Fr(\()p Fu(x)p Fr(\))f Fo(\000)g Fu(H)2382 3999 y Fs(00)2374 4065 y Fq(k)r(k)2455 4041 y Fr(\(0\))2580 3956 y Fm(\014)2580 4016 y(\014)3238 4041 y Fx(\(II.82\))434 4346 y Fo(\024)84 b Fu(C)672 4205 y Fm(\032)760 4320 y Fr(\024)746 4346 y(\002)822 4305 y Ft(2)822 4370 y Fq(j)862 4346 y Fo(j)s Fr(^)-52 b Fu(g)941 4305 y Fs(0)937 4370 y Fq(j)972 4346 y Fo(j)44 b Fr(+)h Fu(\013)1244 4251 y Fm(X)1296 4463 y Fq(k)1404 4346 y Fu(a)1455 4361 y Fq(j)t(k)1547 4346 y Fr(\002)1623 4305 y Ft(2)1623 4370 y Fq(k)1666 4346 y Fo(j)p Fu(g)1745 4305 y Fs(0)1741 4370 y Fq(k)1783 4346 y Fo(j)1089 4651 y Fr(+)1204 4556 y Fm(X)1256 4768 y Fq(k)1364 4651 y Fu(b)1405 4666 y Fq(j)t(k)1481 4540 y Fm(\020)1540 4651 y Fu(a)1591 4666 y Fq(j)t(k)1680 4626 y Fr(\024)1666 4651 y(\002)1742 4610 y Ft(2)1742 4675 y Fq(j)1782 4651 y Fo(j)s Fr(^)-52 b Fu(g)1861 4610 y Fs(0)1857 4675 y Fq(j)1893 4651 y Fo(j)44 b Fr(+)g(\002)2161 4610 y Ft(2)2161 4675 y Fq(k)2203 4651 y Fo(j)p Fu(g)2282 4610 y Fs(0)2278 4675 y Fq(k)2320 4651 y Fo(j)g Fr(+)g Fu(\013)2605 4556 y Fm(X)2592 4772 y Fq(`)p Ft(\()p Fs(6)p Ft(=)p Fq(j)t Ft(\))2779 4651 y Fu(a)2830 4666 y Fq(k)r(`)2919 4651 y Fr(\002)2995 4610 y Ft(2)2995 4675 y Fq(`)3034 4651 y Fo(j)p Fu(g)3113 4610 y Fs(0)3109 4675 y Fq(`)3141 4651 y Fo(j)3169 4540 y Fm(\021)3228 4510 y(\033)434 4976 y Fo(\024)84 b Fu(C)672 4836 y Fm(\032)746 4976 y Fr(\(1)22 b(+)g Fu(C)1023 4991 y Fq(a)1064 4976 y Fr(\))1133 4951 y(\024)1119 4976 y(\002)1195 4935 y Ft(2)1195 5001 y Fq(j)1234 4976 y Fo(j)s Fr(^)-52 b Fu(g)1313 4935 y Fs(0)1309 5001 y Fq(j)1345 4976 y Fo(j)44 b Fr(+)1537 4882 y Fm(X)1590 5094 y Fq(k)1681 4976 y Fr(\()p Fu(a)1770 4991 y Fq(j)t(k)1868 4976 y Fr(+)22 b Fu(b)2007 4991 y Fq(j)t(k)2082 4976 y Fr(\))17 b(\002)2213 4935 y Ft(2)2213 5001 y Fq(k)2255 4976 y Fo(j)p Fu(g)2334 4935 y Fs(0)2330 5001 y Fq(k)2372 4976 y Fo(j)44 b Fr(+)g Fu(\013)2644 4882 y Fm(X)2672 5094 y Fq(k)r(;`)2804 4976 y Fu(b)2845 4991 y Fq(j)t(k)2937 4976 y Fu(a)2988 4991 y Fq(k)r(`)3077 4976 y Fr(\002)3153 4935 y Ft(2)3153 5001 y Fq(`)3192 4976 y Fo(j)p Fu(g)3271 4935 y Fs(0)3267 5001 y Fq(`)3299 4976 y Fo(j)3327 4836 y Fm(\033)434 5294 y Fo(\024)84 b Fu(C)672 5154 y Fm(\032)746 5294 y Fr(\(1)22 b(+)g Fu(C)1023 5309 y Fq(a)1064 5294 y Fr(\))1133 5269 y(\024)1119 5294 y(\002)1195 5253 y Ft(2)1195 5319 y Fq(j)1234 5294 y Fo(j)s Fr(^)-52 b Fu(g)1313 5253 y Fs(0)1309 5319 y Fq(j)1345 5294 y Fo(j)44 b Fr(+)1537 5200 y Fm(X)1590 5412 y Fq(k)1698 5294 y Fr(3)17 b Fu(b)1805 5309 y Fq(j)t(k)1896 5294 y Fr(\002)1972 5253 y Ft(2)1972 5319 y Fq(k)2015 5294 y Fo(j)p Fu(g)2094 5253 y Fs(0)2090 5319 y Fq(k)2132 5294 y Fo(j)2160 5154 y Fm(\033)2256 5294 y Fu(;)p eop %%Page: 38 39 38 38 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(38)257 573 y(where)32 b(we)f(mak)o(e)g(use)h(of)f(\(II.56\),)i (\(II.58\),)g(and)e(assumed)f Fu(\013)q(=\013)2558 537 y Fs(0)2620 573 y Fu(<)39 b Fr(1)31 b Fx(to)g(be)g(suf)n(\002ciently) 257 693 y(small)26 b(to)h(deri)n(v)o(e)f(the)h(last)f(inequality)-6 b(.)35 b(Observ)o(e)27 b(that)f(the)h(last)f(line)h(in)f(\(II.82\))i (is)e(bounded)257 814 y(by)f(a)g(multiple)e(of)i Fu(J)970 829 y Fq(j)1006 814 y Fr([)p Fu(b)p 1033 830 42 4 v 1 w Fr(])p Fx(.)31 b(Thus,)24 b(putting)f(together)i(\(II.78\),)g (\(II.79\),)g(and)f(\(II.82\),)h(we)h(ha)n(v)o(e)1168 1000 y Fm(\014)1168 1060 y(\014)1201 1085 y Fu(M)1295 1100 y Fq(j)1332 1085 y Fr([)p Fu(b)p 1359 1101 V Fr(])d Fo(\000)f Fu(L)1615 1100 y Fq(j)1652 1085 y Fr([)p Fu(b)p 1679 1101 V 1 w Fr(])1748 1000 y Fm(\014)1748 1060 y(\014)1834 1085 y Fo(\024)1974 1017 y Fu(C)p 1974 1062 77 4 v 1982 1153 a(\014)2110 1085 y Fr(+)2246 1017 y Fu(C)p 2246 1062 V 2254 1153 a(\014)2350 1085 y(J)2404 1100 y Fq(j)2440 1085 y Fr([)p Fu(b)p 2467 1101 42 4 v 1 w Fr(])g Fu(;)653 b Fx(\(II.83\))257 1349 y(for)28 b(some)f(uni)n(v)o(ersal)f(constant)h Fu(C)40 b(<)33 b Fo(1)p Fx(.)39 b(Applications)26 b(of)i(Theorem)f (II.8)h(and)g(of)f(Theo-)257 1469 y(rem)e(II.9,)g(choosing)f Fu(")j(>)h Fr(0)c Fx(to)h(be)g(a)g(small,)e(b)n(ut)i(uni)n(v)o(ersal)e (constant,)h(yield)1121 1689 y Fu(J)1175 1704 y Fq(j)1211 1689 y Fr([)p Fu(b)p 1238 1705 V 1 w Fr(])53 b Fo(\024)g Fu(C)1567 1648 y Fs(0)1606 1689 y Fu(K)1689 1704 y Fq(j)1726 1689 y Fr([)p Fu(b)p 1753 1705 V 1 w Fr(])f Fo(\024)h Fu(C)2081 1648 y Fs(00)2124 1608 y Fm(\000)2169 1689 y Fu(L)2235 1704 y Fq(j)2272 1689 y Fr([)p Fu(b)p 2299 1705 V 1 w Fr(])22 b(+)g(1)2537 1608 y Fm(\001)2605 1689 y Fu(;)606 b Fx(\(II.84\))257 1909 y(for)30 b(some)f(uni)n(v)o(ersal)f (constants)h Fu(C)1512 1873 y Fs(0)1535 1909 y Fu(;)17 b(C)1656 1873 y Fs(00)1735 1909 y Fu(<)36 b Fo(1)p Fx(.)45 b(Thus,)30 b(if)f Fu(\014)43 b(<)36 b Fo(1)29 b Fx(is)g(suf)n (\002ciently)g(lar)n(ge,)257 2029 y(we)c(can)h(solv)o(e)d(in)i(Eq.)f (\(II.82\))h(for)g Fu(L)1547 2044 y Fq(j)1584 2029 y Fr([)p Fu(b)p 1611 2045 V 1 w Fr(])g Fx(and)g(get)1149 2306 y Fu(L)1215 2321 y Fq(j)1252 2306 y Fr([)p Fu(b)p 1279 2322 V 1 w Fr(])52 b Fo(\024)1531 2195 y Fm(\020)1590 2306 y Fr(1)22 b(+)1769 2238 y Fu(C)1846 2202 y Fs(000)p 1769 2283 139 4 v 1808 2374 a Fu(\014)1917 2195 y Fm(\021)1994 2306 y Fu(M)2088 2321 y Fq(j)2124 2306 y Fr([)p Fu(b)p 2151 2322 42 4 v 1 w Fr(])50 b(+)2406 2238 y Fu(C)2483 2202 y Fs(000)p 2406 2283 139 4 v 2445 2374 a Fu(\014)2576 2306 y(;)635 b Fx(\(II.85\))257 2574 y(for)25 b(some)g(uni)n(v)o(ersal) e(constant)h Fu(C)1454 2538 y Fs(000)1543 2574 y Fu(<)j Fo(1)p Fx(.)1647 b Fg(\003)p eop %%Page: 39 40 39 39 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(39)257 573 y Fw(III)143 b(Semiclassical)27 b(Localization)f(of)j (the)g(P)o(artition)e(Func-)568 756 y(tion)257 979 y Fx(This)21 b(section)h(is)f(de)n(v)n(oted)g(to)g(the)h(proof)g(of)g (Theorem)f(I.6.)30 b(The)22 b(method)f(we)h(use)g(here)g(is)g(in-)257 1099 y(spired)j(by)g(a)g(related)g(method)f(which)h(appeared)g(\002rst) g(in)g(a)g(paper)h(by)e(Sj)8 b(\250)-41 b(ostrand)25 b([19].)31 b(The)257 1220 y(method)f(w)o(as)g(used)g(to)g(study)f(the)h (\002rst)h(eigen)l(v)n(alue)e(of)i(a)f(Schr)8 b(\250)-41 b(odinger)30 b(operator)h(di)n(vided)257 1340 y(by)f(the)f(dimension.) 43 b(He)29 b(sho)n(wed)g(that)g(this)f(object)h(is)g(in)h(the)f (semiclassical)f(limit)g(gi)n(v)o(en,)257 1461 y(up)34 b(to)f(an)h(e)o(xponentially)e(small)g(error)l(,)37 b(by)c(using)g(the) g(\002rst)h(eigen)l(v)n(alue)f(of)h(the)f(operator)257 1581 y(localized)27 b(to)f(an)h Fu(l)900 1545 y Fs(1)1001 1581 y Fx(neighbourhood)e(of)i(a)g(global)f(minimum.)34 b(The)26 b(e)o(xponential)f(error)j(is)257 1701 y(uniform)j(in)h (dimension)e(\(i.e.,)j(uni)n(v)o(ersal\).)50 b(This)31 b(method)g(w)o(as)g(later)h(applied)g(in)f([10])h(to)257 1822 y(Laplace)22 b(inte)o(grals)e(\(partition)h(functions\))f(and)h (in)g([16])h(to)f(Laplace)h(inte)o(grals)e(and)h(transfer)257 1942 y(operators.)48 b(The)31 b(results)f(mentioned)f(abo)o(v)o(e)h (are)i(for)f(localization)e(in)h(an)h Fu(l)2954 1906 y Fs(1)3060 1942 y Fx(neighbour)n(-)257 2062 y(hood,)f(whereas)h(our)e (result)g(is)g(for)h(localization)f(in)g(a)h(slab)g(of)f(the)h(form)f Fo(f)p Fu(x)37 b Fr(:)g Fo(j)p Fu(x)3167 2077 y Fq(i)3195 2062 y Fo(j)f(\025)i Fu(\016)t Fo(g)p Fx(,)257 2183 y(where)23 b Fu(i)28 b Fo(2)g Fr(\003)22 b Fx(and)g Fu(\016)31 b(>)d Fr(0)p Fx(.)h(The)22 b(tw)o(o)g(methods)e(apply)i(to)f(both)h (localization)f(problems,)g(b)n(ut)257 2303 y(under)28 b(dif)n(ferent)f(sets)g(of)h(conditions.)37 b(The)28 b(method)e(presented)i(here)g(seems)f(particularly)257 2424 y(well)e(adapted)g(for)g(the)g(model)g(considered)f(in)h(this)f (paper)-5 b(.)32 b(W)-8 b(e)25 b(note)g(that)f(the)h Fu(\013)3088 2439 y Ft(0)3153 2424 y Fx(gi)n(v)o(en)f(by)257 2544 y(Theorem)30 b(I.6)g(is)g(not)g(uniform)f(in)g Fu(\024)p Fx(.)47 b(In)30 b(the)g(related)h(results)e(mentioned)g(abo)o(v)o(e,)h Fu(\013)3287 2559 y Ft(0)3357 2544 y Fx(can)257 2664 y(be)25 b(chosen)g(uniformly)e(in)i(the)f(size)h(of)g(the)g (neighbourhood.)257 2959 y Fl(III.1)118 b(The)30 b(Dilated)h (Interaction)257 3147 y Fx(Throughout)e(this)h(section)f(we)i(impose)e (Hypotheses)g(1)i(and)f(2.)47 b(W)-8 b(e)31 b(pro)o(v)o(e)e(Theorem)h (I.6)257 3267 y(through)24 b(a)h(sequence)g(of)g(lemmata.)257 3471 y Fv(Lemma)h(III.1.)40 b Fn(Let)25 b Fu(")p 1018 3487 46 4 v 27 w Fr(:=)j(\()p Fu(")1306 3486 y Fq(k)1348 3471 y Fr(\))1386 3486 y Fq(k)r Fs(2)p Ft(\003)1552 3471 y Fo(\022)g Fr(\(0)p Fu(;)17 b Fr(1\))24 b Fn(and)g Fu(e)2118 3486 y Ft(0)2186 3471 y Fr(=)j(1)22 b Fo(\000)g Fr(max)2640 3486 y Fq(k)r Fs(2)p Ft(\003)2779 3471 y Fo(f)p Fu(")2875 3486 y Fq(k)2917 3471 y Fo(g)p Fn(.)30 b(Ther)l(e)c(e)n(xists)257 3591 y(a)f(univer)o(sal)f(constant)f Fu(C)35 b(>)28 b Fr(0)p Fn(,)c(independent)g(of)g Fu(")p 2010 3607 V Fn(,)h(suc)o(h)f (that)528 3720 y Fm(\014)528 3780 y(\014)528 3839 y(\014)528 3899 y(\014)578 3770 y(X)607 3982 y Fq(j;k)739 3754 y Fm(n)805 3864 y Fu(w)875 3879 y Fq(j)t(k)950 3784 y Fm(\000)996 3864 y Fu(x)1051 3879 y Fq(j)1088 3864 y Fu(;)17 b(x)1187 3879 y Fq(k)1229 3784 y Fm(\001)1325 3864 y Fo(\000)50 b Fu(w)1522 3879 y Fq(j)t(k)1597 3784 y Fm(\000)1643 3864 y Fr(\(1)21 b Fo(\000)i Fu(")1897 3879 y Fq(j)1933 3864 y Fr(\))p Fu(x)2026 3879 y Fq(j)2063 3864 y Fu(;)17 b Fr(\(1)22 b Fo(\000)g Fu(")2361 3879 y Fq(k)2404 3864 y Fr(\))p Fu(x)2497 3879 y Fq(k)2540 3784 y Fm(\001)2585 3754 y(o)2652 3720 y(\014)2652 3780 y(\014)2652 3839 y(\014)2652 3899 y(\014)3255 3864 y Fx(\(III.1\))611 4182 y Fo(\024)786 4115 y Fu(C)p 782 4159 85 4 v 782 4251 a(e)827 4266 y Ft(0)910 4088 y Fm(X)966 4298 y Fq(j)1070 4042 y Fm(\032)1162 4088 y(X)1214 4300 y Fq(k)1322 4182 y Fu(a)1373 4197 y Fq(j)t(k)1448 4042 y Fm(\022)1522 4182 y Fr(1)49 b(+)1756 4115 y Fu(")1802 4130 y Fq(k)p 1756 4159 89 4 v 1759 4251 a Fu(")1805 4266 y Fq(j)1854 4042 y Fm(\023)q(\033)2019 4072 y(n)2085 4182 y Fu(g)2132 4197 y Fq(j)2168 4182 y Fr(\()p Fu(x)2261 4197 y Fq(j)2298 4182 y Fr(\))h Fo(\000)g Fu(g)2560 4197 y Fq(j)2597 4102 y Fm(\000)2642 4182 y Fr(\(1)22 b Fo(\000)h Fu(")2897 4197 y Fq(j)2933 4182 y Fr(\))p Fu(x)3026 4197 y Fq(j)3063 4102 y Fm(\001)3109 4072 y(o)3197 4182 y Fu(;)257 4510 y Fn(for)i(all)f Fu(x)k Fo(2)g Fp(R)772 4474 y Ft(\003)831 4510 y Fn(.)257 4713 y(Pr)l(oof:)i Fx(W)-8 b(e)25 b(abbre)n(viate)809 4933 y Fu(y)857 4948 y Fq(j)946 4933 y Fr(:=)53 b(\(1)22 b Fo(\000)g Fu(")1356 4948 y Fq(j)1392 4933 y Fr(\))17 b Fu(x)1502 4948 y Fq(j)1610 4933 y Fx(and)70 b Fu(z)1869 4948 y Fq(j)1906 4933 y Fr(\()p Fu(t)p Fr(\))53 b(:=)f Fu(tx)2315 4948 y Fq(j)2402 4933 y Fr(+)e(\(1)22 b Fo(\000)h Fu(t)p Fr(\))p Fu(y)2858 4948 y Fq(j)2916 4933 y Fu(:)312 b Fx(\(III.2\))p eop %%Page: 40 41 40 40 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(40)257 573 y(The)25 b(Fundamental)f(Theorem)h(of)g(Calculus)f (yields,)g(cf.)31 b(also)25 b(the)f(estimate)g(\(B.2\),)262 770 y Fm(\014)262 830 y(\014)295 854 y Fu(w)365 869 y Fq(j)t(k)440 854 y Fr(\()p Fu(x)533 869 y Fq(j)570 854 y Fu(;)17 b(x)669 869 y Fq(k)712 854 y Fr(\))50 b Fo(\000)g Fu(w)997 869 y Fq(j)t(k)1072 854 y Fr(\()p Fu(y)1158 869 y Fq(j)1194 854 y Fu(;)17 b(y)1286 869 y Fq(k)1328 854 y Fr(\))1366 770 y Fm(\014)1366 830 y(\014)1479 854 y Fr(=)1635 710 y Fm(\014)1635 770 y(\014)1635 830 y(\014)1635 889 y(\014)1685 719 y(Z)1785 745 y Ft(1)1740 944 y(0)1868 787 y Fu(d)p 1851 832 86 4 v 1851 923 a(dt)1946 774 y Fm(\010)2005 854 y Fu(w)2075 869 y Fq(j)t(k)2149 854 y Fr(\()p Fu(z)2232 869 y Fq(j)2269 854 y Fr(\()p Fu(t)p Fr(\))p Fu(;)g(z)2469 869 y Fq(k)2512 854 y Fr(\()p Fu(t)p Fr(\)\))2661 774 y Fm(\011)2736 854 y Fu(dt)2839 710 y Fm(\014)2839 770 y(\014)2839 830 y(\014)2839 889 y(\014)346 1134 y Fr(=)506 990 y Fm(\014)506 1050 y(\014)506 1110 y(\014)506 1169 y(\014)555 999 y(Z)655 1025 y Ft(1)611 1224 y(0)711 1024 y Fm(n)778 1134 y Fr(\()p Fu(x)871 1149 y Fq(j)930 1134 y Fo(\000)22 b Fu(y)1077 1149 y Fq(j)1113 1134 y Fr(\))17 b Fu(@)1219 1149 y Fq(j)1256 1134 y Fu(w)1326 1149 y Fq(j)t(k)1401 1054 y Fm(\000)1446 1134 y Fu(z)1491 1149 y Fq(j)1528 1134 y Fr(\()p Fu(t)p Fr(\))p Fu(;)g(z)1728 1149 y Fq(k)1771 1134 y Fr(\()p Fu(t)p Fr(\))1882 1054 y Fm(\001)1977 1134 y Fr(+)50 b(\()p Fu(x)2196 1149 y Fq(k)2261 1134 y Fo(\000)23 b Fu(y)2409 1149 y Fq(k)2451 1134 y Fr(\))17 b Fu(@)2557 1149 y Fq(k)2600 1134 y Fu(w)2670 1149 y Fq(j)t(k)2745 1054 y Fm(\000)2790 1134 y Fu(z)2835 1149 y Fq(j)2872 1134 y Fr(\()p Fu(t)p Fr(\))p Fu(;)g(z)3072 1149 y Fq(k)3115 1134 y Fr(\()p Fu(t)p Fr(\))3226 1054 y Fm(\001)3272 1024 y(o)3355 1134 y Fu(dt)3458 990 y Fm(\014)3458 1050 y(\014)3458 1110 y(\014)3458 1169 y(\014)345 1414 y Fo(\024)506 1279 y Fm(Z)605 1305 y Ft(1)561 1504 y(0)661 1414 y Fu(a)712 1429 y Fq(j)t(k)804 1334 y Fm(\000)850 1414 y Fu(")896 1429 y Fq(j)932 1414 y Fo(j)p Fu(x)1015 1429 y Fq(j)1052 1414 y Fo(j)38 b Fr(+)h Fu(")1279 1429 y Fq(k)1321 1414 y Fo(j)p Fu(x)1404 1429 y Fq(k)1447 1414 y Fo(j)1475 1334 y Fm(\001)27 b(\000)1593 1414 y Fo(j)p Fu(g)1672 1373 y Fs(0)1668 1439 y Fq(j)1704 1414 y Fr(\()p Fu(z)1787 1429 y Fq(j)1824 1414 y Fr(\()p Fu(t)p Fr(\)\))p Fo(j)39 b Fr(+)f Fo(j)p Fu(g)2233 1373 y Fs(0)2229 1439 y Fq(k)2271 1414 y Fr(\()p Fu(z)2354 1429 y Fq(k)2397 1414 y Fr(\()p Fu(t)p Fr(\)\))p Fo(j)2574 1334 y Fm(\001)2636 1414 y Fu(dt)22 b(:)484 b Fx(\(III.3\))257 1698 y(Using)32 b(that)h Fo(j)p Fu(g)790 1661 y Fs(0)786 1722 y Fq(j)822 1698 y Fr(\()p Fu(z)905 1713 y Fq(j)941 1698 y Fr(\()p Fu(t)p Fr(\)\))p Fo(j)43 b Fr(=)f(sgn)q(\()p Fu(x)1514 1713 y Fq(j)1551 1698 y Fr(\))p Fu(g)1640 1661 y Fs(0)1636 1722 y Fq(j)1672 1698 y Fr(\()p Fu(z)1755 1713 y Fq(j)1792 1698 y Fr(\()p Fu(t)p Fr(\)\))33 b Fx(and)g Fu(g)2202 1661 y Fs(0)2198 1722 y Fq(j)2234 1698 y Fr(\()p Fu(z)2317 1713 y Fq(j)2354 1698 y Fr(\()p Fu(t)p Fr(\)\))42 b(=)h Fu(")2710 1656 y Fs(\000)p Ft(1)2710 1723 y Fq(j)2804 1698 y Fu(x)2859 1656 y Fs(\000)p Ft(1)2859 1723 y Fq(j)2976 1658 y(d)p 2964 1675 62 4 v 2964 1732 a(dt)3035 1617 y Fm(\002)3077 1698 y Fu(g)3124 1713 y Fq(j)3160 1698 y Fr(\()p Fu(z)3243 1713 y Fq(j)3280 1698 y Fr(\()p Fu(t)p Fr(\)\))3429 1617 y Fm(\003)3470 1698 y Fx(,)257 1818 y(we)25 b(re)n(write)g(the)g(estimate)f(abo)o(v)o(e)f(as)i(follo)n(ws,) 449 1953 y Fm(\014)449 2013 y(\014)482 2038 y Fu(w)552 2053 y Fq(j)t(k)627 2038 y Fr(\()p Fu(x)720 2053 y Fq(j)757 2038 y Fu(;)17 b(x)856 2053 y Fq(k)899 2038 y Fr(\))50 b Fo(\000)g Fu(w)1184 2053 y Fq(j)t(k)1259 2038 y Fr(\()p Fu(y)1345 2053 y Fq(j)1381 2038 y Fu(;)17 b(y)1473 2053 y Fq(k)1515 2038 y Fr(\))1553 1953 y Fm(\014)1553 2013 y(\014)1666 2038 y Fo(\024)81 b Fu(a)1875 2053 y Fq(j)t(k)1967 1957 y Fm(\000)2012 2038 y Fu(")2058 2053 y Fq(j)2094 2038 y Fo(j)p Fu(x)2177 2053 y Fq(j)2214 2038 y Fo(j)38 b Fr(+)h Fu(")2441 2053 y Fq(k)2483 2038 y Fo(j)p Fu(x)2566 2053 y Fq(k)2609 2038 y Fo(j)2637 1957 y Fm(\001)927 2123 y(Z)1027 2149 y Ft(1)983 2348 y(0)1083 2118 y Fm(\032)1258 2191 y Fr(1)p 1168 2235 230 4 v 1168 2327 a Fu(")1214 2342 y Fq(j)1250 2327 y Fo(j)p Fu(x)1333 2342 y Fq(j)1369 2327 y Fo(j)1451 2191 y Fu(d)p 1434 2235 86 4 v 1434 2327 a(dt)1530 2148 y Fm(h)1577 2258 y Fu(g)1624 2273 y Fq(j)1660 2258 y Fr(\()p Fu(z)1743 2273 y Fq(j)1780 2258 y Fr(\()p Fu(t)p Fr(\)\))1929 2148 y Fm(i)2025 2258 y Fr(+)2258 2191 y(1)p 2161 2235 242 4 v 2161 2327 a Fu(")2207 2342 y Fq(k)2249 2327 y Fo(j)p Fu(x)2332 2342 y Fq(k)2375 2327 y Fo(j)2457 2191 y Fu(d)p 2439 2235 86 4 v 2439 2327 a(dt)2535 2148 y Fm(h)2582 2258 y Fu(g)2629 2273 y Fq(k)2672 2258 y Fr(\()p Fu(z)2755 2273 y Fq(k)2798 2258 y Fr(\()p Fu(t)p Fr(\)\))2947 2148 y Fm(i)2994 2118 y(\033)3085 2258 y Fu(dt)532 2532 y Fr(=)83 b Fu(a)742 2547 y Fq(j)t(k)834 2451 y Fm(\000)880 2532 y Fu(")926 2547 y Fq(j)962 2532 y Fo(j)p Fu(x)1045 2547 y Fq(j)1081 2532 y Fo(j)39 b Fr(+)f Fu(")1308 2547 y Fq(k)1351 2532 y Fo(j)p Fu(x)1434 2547 y Fq(k)1476 2532 y Fo(j)1504 2451 y Fm(\001)1566 2392 y(\032)1651 2465 y Fu(g)1698 2480 y Fq(j)1734 2465 y Fr(\()p Fu(x)1827 2480 y Fq(j)1864 2465 y Fr(\))h Fo(\000)g Fu(g)2104 2480 y Fq(j)2140 2465 y Fr(\()p Fu(y)2226 2480 y Fq(j)2262 2465 y Fr(\))p 1651 2509 650 4 v 1861 2600 a Fu(")1907 2615 y Fq(j)1943 2600 y Fo(j)p Fu(x)2026 2615 y Fq(j)2063 2600 y Fo(j)2360 2532 y Fr(+)2496 2465 y Fu(g)2543 2480 y Fq(k)2585 2465 y Fr(\()p Fu(x)2678 2480 y Fq(k)2721 2465 y Fr(\))g 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y Fo(j)p 1746 3335 242 4 v 1752 3426 a Fu(")1798 3441 y Fq(j)1834 3426 y Fo(j)p Fu(x)1917 3441 y Fq(j)1954 3426 y Fo(j)1997 3217 y Fm(\023)2087 3247 y(\020)2147 3358 y Fu(g)2194 3373 y Fq(j)2230 3358 y Fr(\()p Fu(x)2323 3373 y Fq(j)2360 3358 y Fr(\))g Fo(\000)g Fu(g)2622 3373 y Fq(j)2658 3358 y Fr(\()p Fu(y)2744 3373 y Fq(j)2780 3358 y Fr(\))2818 3247 y Fm(\021)2900 3358 y Fu(:)257 3687 y Fx(No)n(w)25 b(we)h(temporarily)f(\002x)h Fu(j;)17 b(k)32 b Fo(2)e Fr(\003)25 b Fx(and)h(decompose)f(the)g (con\002guration)g(space)h Fp(R)3255 3651 y Ft(\003)3340 3687 y Fx(into)257 3808 y(the)f(tw)o(o)f(disjoint)f(re)o(gions)596 3941 y Fm(n)662 4052 y Fu(x)28 b Fo(2)g Fp(R)905 4011 y Ft(\003)992 3937 y Fm(\014)992 3997 y(\014)992 4057 y(\014)1050 4052 y Fo(j)p Fu(x)1133 4067 y Fq(j)1170 4052 y Fo(j)f(\025)h(j)p Fu(y)1406 4067 y Fq(k)1448 4052 y Fo(j)1476 3941 y Fm(o)1660 4052 y Fx(and)1922 3941 y Fm(n)1989 4052 y Fu(x)g Fo(2)g Fp(R)2232 4011 y Ft(\003)2319 3937 y Fm(\014)2319 3997 y(\014)2319 4057 y(\014)2377 4052 y Fo(j)p Fu(x)2460 4067 y Fq(j)2496 4052 y Fo(j)g Fu(<)f Fo(j)p Fu(y)2731 4067 y Fq(k)2773 4052 y Fo(j)2801 3941 y Fm(o)2889 4052 y Fu(:)339 b Fx(\(III.6\))257 4299 y(F)o(or)25 b Fo(j)p Fu(x)502 4314 y Fq(j)539 4299 y Fo(j)i(\025)h(j)p Fu(y)775 4314 y Fq(k)817 4299 y Fo(j)g Fr(=)f(\(1)22 b Fo(\000)h Fu(")1231 4314 y Fq(k)1273 4299 y Fr(\))p Fo(j)p Fu(x)1394 4314 y Fq(k)1437 4299 y Fo(j)p Fx(,)h(we)h(ha)n(v)o(e)1220 4510 y Fu(")1266 4525 y Fq(k)1308 4510 y Fo(j)p Fu(x)1391 4525 y Fq(k)1434 4510 y Fo(j)p 1220 4554 V 1226 4646 a Fu(")1272 4661 y Fq(j)1308 4646 y Fo(j)p Fu(x)1391 4661 y Fq(j)1428 4646 y Fo(j)1524 4577 y(\024)1837 4510 y Fu(")1883 4525 y Fq(k)p 1664 4554 434 4 v 1664 4646 a Fr(\(1)d Fo(\000)h Fu(")1919 4661 y Fq(k)1961 4646 y Fr(\))17 b Fu(")2062 4661 y Fq(j)2160 4577 y Fo(\024)2348 4510 y Fu(")2394 4525 y Fq(k)p 2300 4554 184 4 v 2300 4646 a Fu(e)2345 4661 y Ft(0)2401 4646 y Fu(")2447 4661 y Fq(j)2516 4577 y Fu(;)712 b Fx(\(III.7\))257 4856 y(where)30 b(we)f(used)f(that)h Fr(1)24 b Fo(\000)i Fu(")1287 4871 y Fq(k)1364 4856 y Fo(\025)36 b Fu(e)1522 4871 y Ft(0)1562 4856 y Fx(.)42 b(Con)l(v)o(ersely)-6 b(,)29 b(if)g Fo(j)p Fu(x)2299 4871 y Fq(j)2335 4856 y Fo(j)35 b Fu(<)g Fo(j)p Fu(y)2585 4871 y Fq(k)2627 4856 y Fo(j)g Fr(=)g(\(1)24 b Fo(\000)i Fu(")3061 4871 y Fq(k)3103 4856 y Fr(\))p Fo(j)p Fu(x)3224 4871 y Fq(k)3267 4856 y Fo(j)j Fx(then)p eop %%Page: 41 42 41 41 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(41)257 573 y(\(I.29\))25 b(implies)288 746 y Fu(")334 761 y Fq(k)376 746 y Fo(j)p Fu(x)459 761 y Fq(k)502 746 y Fo(j)p 288 791 242 4 v 294 882 a Fu(")340 897 y Fq(j)376 882 y Fo(j)p Fu(x)459 897 y Fq(j)496 882 y Fo(j)556 733 y Fm(\000)602 814 y Fu(g)649 829 y Fq(j)685 814 y Fr(\()p Fu(x)778 829 y Fq(j)815 814 y Fr(\))49 b Fo(\000)i Fu(g)1077 829 y Fq(j)1113 814 y Fr(\()p Fu(y)1199 829 y Fq(j)1235 814 y Fr(\))1273 733 y Fm(\001)1402 814 y Fo(\024)83 b Fu(")1608 829 y Fq(k)1667 814 y Fo(j)p Fu(x)1750 829 y Fq(k)1793 814 y Fo(j)33 b Fr(max)2052 703 y Fm(n)2118 814 y Fo(j)p Fu(g)2197 773 y Fs(0)2193 838 y Fq(j)2229 814 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2406 699 y Fm(\014)2406 759 y(\014)2406 819 y(\014)2464 814 y Fr(0)45 b Fo(\024)f Fr(sgn)q(\()p Fu(x)2914 829 y Fq(j)2951 814 y Fr(\))p Fu(s)g Fo(\024)h(j)p Fu(x)3284 829 y Fq(j)3320 814 y Fo(j)3348 703 y Fm(o)1402 1102 y Fr(=)1562 967 y Fm(Z)1662 993 y Fs(j)p Fq(x)1722 1005 y Fj(k)1759 993 y Fs(j)1617 1192 y(j)p Fq(y)1672 1204 y Fj(k)1710 1192 y Fs(j)1800 1102 y Fr(max)1998 992 y Fm(n)2064 1102 y Fo(j)p Fu(g)2143 1061 y Fs(0)2139 1127 y Fq(j)2175 1102 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2352 988 y Fm(\014)2352 1047 y(\014)2352 1107 y(\014)2410 1102 y Fr(0)f Fo(\024)h Fr(sgn)q(\()p Fu(x)2860 1117 y Fq(k)2903 1102 y Fr(\))p Fu(s)f Fo(\024)h(j)p Fu(x)3236 1117 y Fq(k)3278 1102 y Fo(j)3306 992 y Fm(o)3389 1102 y Fu(dt)1402 1401 y Fo(\024)83 b Fu(c)1604 1360 y Fs(\000)p Ft(1)1604 1426 y Fq(g)1715 1266 y Fm(Z)1815 1292 y Fs(j)p Fq(x)1875 1304 y Fj(k)1912 1292 y Fs(j)1770 1491 y(j)p Fq(y)1825 1503 y Fj(k)1863 1491 y Fs(j)1952 1401 y Fr(sgn)q(\()p Fu(x)2187 1416 y Fq(k)2230 1401 y Fr(\))17 b Fu(g)2336 1360 y Fs(0)2332 1426 y Fq(k)2374 1320 y Fm(\000)2436 1401 y Fr(sgn)q(\()p Fu(x)2671 1416 y Fq(k)2714 1401 y Fr(\))p Fu(t)2787 1320 y Fm(\001)2849 1401 y Fu(dt)1402 1626 y Fr(=)84 b Fu(c)1604 1584 y Fs(\000)p Ft(1)1604 1650 y Fq(g)1715 1545 y Fm(\000)1761 1626 y Fu(g)1808 1641 y Fq(k)1850 1626 y Fr(\()p Fu(x)1943 1641 y Fq(k)1986 1626 y Fr(\))50 b Fo(\000)g Fu(g)2248 1641 y Fq(k)2291 1626 y Fr(\()p Fu(y)2377 1641 y Fq(k)2419 1626 y Fr(\))2457 1545 y Fm(\001)2524 1626 y Fu(:)704 b Fx(\(III.8\))257 1815 y(Combining)31 b(the)g(estimates)g (\(III.7\)\226\(III.8\))i(and)f(inserting)e(them)h(into)g(\(III.5\))i (yields)d(the)257 1935 y(lemma.)2864 b Fg(\003)404 2056 y Fx(W)-8 b(e)21 b(no)n(w)g(specify)g Fu(")p 1055 2072 46 4 v 20 w Fx(in)g(terms)g(of)g(the)g(metric)g Fu(\032)g Fx(on)g(the)g(lattice)g Fr(\003)g Fx(and)g(a)h(\002x)o(ed)f(reference) 257 2176 y(point)j Fu(o)k Fo(2)g Fr(\003)p Fx(.)257 2353 y Fv(Lemma)35 b(III.2.)46 b Fn(F)l(ix)34 b Fu(o)45 b Fo(2)h Fr(\003)34 b Fn(and)g Fr(0)45 b Fu(<)h(\034)57 b Fo(\024)45 b Fu(")2017 2368 y Fq(o)2101 2353 y Fu(<)g Fr(1)p Fn(,)37 b(and)d(set)g Fu(")2708 2368 y Fq(k)2796 2353 y Fr(:=)45 b Fu(")2990 2368 y Fq(o)3028 2353 y Fu(\016)3071 2368 y Fq(k)r(;o)3197 2353 y Fr(+)29 b(\(1)g Fo(\000)257 2474 y Fu(\016)300 2489 y Fq(k)r(;o)397 2474 y Fr(\))p Fu(\034)11 b(e)533 2437 y Fs(\000)p Fq(\032)p Ft(\()p Fq(k)r(;o)p Ft(\))776 2474 y Fn(.)62 b(Then)35 b(ther)l(e)g(e)n(xists)g (a)g(univer)o(sal)f(constant)g Fu(C)2528 2437 y Fs(0)2598 2474 y Fu(>)46 b Fr(0)35 b Fn(suc)o(h)g(that,)i(for)d(all)257 2594 y Fu(x)28 b Fo(2)h Fp(R)501 2558 y Ft(\003)560 2594 y Fn(,)837 2684 y Fm(\014)837 2744 y(\014)837 2804 y(\014)837 2863 y(\014)887 2734 y(X)916 2946 y Fq(j;k)1047 2718 y Fm(n)1114 2828 y Fu(w)1184 2843 y Fq(j)t(k)1259 2828 y Fr(\()p Fu(x)1352 2843 y Fq(j)1389 2828 y Fu(;)17 b(x)1488 2843 y Fq(k)1530 2828 y Fr(\))50 b Fo(\000)g Fu(w)1815 2843 y Fq(j)t(k)1890 2828 y Fr(\(\(1)22 b Fo(\000)h Fu(")2183 2843 y Fq(j)2219 2828 y Fr(\))p Fu(x)2312 2843 y Fq(j)2349 2828 y Fu(;)17 b Fr(\(1)k Fo(\000)i Fu(")2647 2843 y Fq(k)2689 2828 y Fr(\))p Fu(x)2782 2843 y Fq(k)2825 2828 y Fr(\))2863 2718 y Fm(o)2929 2684 y(\014)2929 2744 y(\014)2929 2804 y(\014)2929 2863 y(\014)920 3142 y Fo(\024)1241 3075 y Fu(C)1318 3039 y Fs(0)p 1091 3120 401 4 v 1091 3211 a Fr(\(1)e Fo(\000)i Fu(")1345 3226 y Fq(o)1383 3211 y Fr(\))17 b Fu(\034)1534 3048 y Fm(X)1590 3258 y Fq(j)1695 3032 y Fm(n)1761 3142 y Fu(g)1808 3157 y Fq(j)1844 3062 y Fm(\000)1890 3142 y Fu(x)1945 3157 y Fq(j)1982 3062 y Fm(\001)2077 3142 y Fo(\000)51 b Fu(g)2252 3157 y Fq(j)2288 3062 y Fm(\000)2334 3142 y Fr(\(1)21 b Fo(\000)i Fu(")2588 3157 y Fq(j)2624 3142 y Fr(\))p Fu(x)2717 3157 y Fq(j)2754 3062 y Fm(\001)2800 3032 y(o)2888 3142 y Fu(:)340 b Fx(\(III.9\))257 3426 y Fn(Pr)l(oof:)28 b Fx(In)22 b(vie)n(w)f(of)h(Lemma)f(III.1)h(and)f(the)h(f)o(act)g(that) f Fu(e)2182 3441 y Ft(0)2249 3426 y Fo(\025)29 b Fr(1)10 b Fo(\000)g Fu(")2547 3441 y Fq(o)2586 3426 y Fx(,)22 b(it)f(is)g(suf)n(\002cient)g(to)g(sho)n(w)257 3546 y(that)1167 3700 y Fr(max)1242 3761 y Fq(j)1365 3560 y Fm(\032)1457 3606 y(X)1509 3818 y Fq(k)1617 3700 y Fu(a)1668 3715 y Fq(j)t(k)1743 3590 y Fm(\020)1803 3700 y Fr(1)h(+)1982 3633 y Fu(")2028 3648 y Fq(k)p 1982 3677 89 4 v 1985 3769 a Fu(")2031 3784 y Fq(j)2080 3590 y Fm(\021)2140 3560 y(\033)2267 3700 y Fo(\024)2407 3633 y Fu(C)2484 3597 y Fs(00)p 2407 3677 120 4 v 2440 3769 a Fu(\034)2558 3700 y(;)620 b Fx(\(III.10\))257 3950 y(for)34 b(some)f(uni)n(v)o (ersal)f(constant)g Fu(C)1488 3914 y Fs(00)1574 3950 y Fu(<)43 b Fo(1)p Fx(.)57 b(F)o(or)33 b Fu(j)49 b Fo(6)p Fr(=)44 b Fu(o)p Fx(,)35 b(we)f(observ)o(e)f(that)g(the)g(triangle)257 4070 y(inequality)24 b(for)h(the)f(metric)h Fu(\032)g Fx(and)g(Eq.)f(\(I.16\))h(implies)356 4189 y Fm(X)409 4401 y Fq(k)516 4284 y Fu(a)567 4299 y Fq(j)t(k)643 4173 y Fm(\020)702 4284 y Fr(1)d(+)881 4216 y Fu(")927 4231 y Fq(k)p 881 4261 89 4 v 884 4352 a Fu(")930 4367 y Fq(j)979 4173 y Fm(\021)1119 4284 y Fo(\024)81 b Fu(e)1322 4243 y Fq(\032)p Ft(\()p Fq(o;j)t Ft(\))1520 4284 y Fu(a)1571 4299 y Fq(j)t(o)1658 4173 y Fm(\020)1718 4284 y Fr(1)22 b(+)1897 4216 y Fu(")1943 4231 y Fq(o)p 1897 4261 84 4 v 1912 4352 a Fu(\034)1990 4173 y Fm(\021)2100 4284 y Fr(+)2245 4189 y Fm(X)2225 4405 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))2424 4284 y Fu(a)2475 4299 y Fq(j)t(k)2567 4173 y Fm(\020)2627 4284 y Fr(1)38 b(+)h Fu(e)2874 4243 y Fq(\032)p Ft(\()p Fq(o;j)t Ft(\))p Fs(\000)p Fq(\032)p Ft(\()p Fq(o;k)r Ft(\))3294 4173 y Fm(\021)439 4609 y Fo(\024)599 4499 y Fm(\020)659 4609 y Fr(1)22 b(+)838 4542 y Fu(")884 4557 y Fq(o)p 838 4586 V 853 4678 a Fu(\034)932 4499 y Fm(\021)1024 4609 y Fr(max)1099 4670 y Fq(j)1222 4469 y Fm(\032)1314 4515 y(X)1366 4727 y Fq(k)1474 4609 y Fu(e)1519 4568 y Fq(\032)p Ft(\()p Fq(j;k)r Ft(\))1718 4609 y Fu(a)1769 4624 y Fq(j)t(k)1844 4469 y Fm(\033)1999 4609 y Fo(\024)81 b Fu(C)2227 4624 y Fq(a)2285 4499 y Fm(\020)2344 4609 y Fr(1)22 b(+)2523 4542 y Fu(")2569 4557 y Fq(o)p 2523 4586 V 2538 4678 a Fu(\034)2617 4499 y Fm(\021)2757 4609 y Fo(\024)2925 4542 y Fr(2)17 b Fu(C)3061 4557 y Fq(a)p 2925 4586 177 4 v 2986 4678 a Fu(\034)3133 4609 y(:)45 b Fx(\(III.11\))257 4882 y(Con)l(v)o(ersely)-6 b(,)24 b(for)h Fu(j)34 b Fr(=)27 b Fu(o)p Fx(,)e(we)g(estimate)871 5001 y Fm(X)923 5213 y Fq(k)1031 5095 y Fu(a)1082 5110 y Fq(ok)1159 4985 y Fm(\020)1219 5095 y Fr(1)d(+)1398 5028 y Fu(")1444 5043 y Fq(k)p 1398 5072 89 4 v 1400 5164 a Fu(")1446 5179 y Fq(o)1496 4985 y Fm(\021)1608 5095 y Fo(\024)1738 4985 y Fm(\020)1797 5095 y Fr(1)g(+)1992 5028 y Fu(\034)p 1976 5072 84 4 v 1976 5164 a(")2022 5179 y Fq(o)2070 4985 y Fm(\021)2166 5001 y(X)2146 5216 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))2345 5095 y Fu(a)2396 5110 y Fq(ok)2526 5095 y Fo(\024)53 b Fr(2)17 b Fu(C)2792 5110 y Fq(a)2855 5095 y Fu(;)323 b Fx(\(III.12\))257 5388 y(using)24 b(\(I.16\).)31 b(This)24 b(completes)g(the)h(proof)f (since)h Fu(\034)39 b(<)28 b Fr(1)p Fx(.)1133 b Fg(\003)p eop %%Page: 42 43 42 42 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(42)257 573 y Fl(III.2)118 b(The)30 b(Dilated)h(Hamiltonian)f (Function)257 760 y Fx(In)36 b(the)f(follo)n(wing)e(we)j(\002x)g Fu(o)47 b Fo(2)h Fr(\003)35 b Fx(and)g Fu(")1782 775 y Fq(o)1820 760 y Fu(;)17 b(\034)58 b Fo(2)48 b Fr(\(0)p Fu(;)17 b Fr(1\))35 b Fx(and)g(set)g Fu(")2702 775 y Fq(k)2792 760 y Fr(:=)47 b Fu(")2988 775 y Fq(o)3026 760 y Fu(\016)3069 775 y Fq(k)r(;o)3195 760 y Fr(+)30 b(\(1)g Fo(\000)257 881 y Fu(\016)300 896 y Fq(k)r(;o)397 881 y Fr(\))p Fu(\034)11 b(e)533 845 y Fs(\000)p Fq(\032)p Ft(\()p Fq(k)r(;o)p Ft(\))776 881 y Fx(.)55 b(Furthermore,)36 b(for)d Fu(x)43 b Fr(=)g(\()p Fu(x)1874 896 y Fq(k)1917 881 y Fr(\))1955 896 y Fq(k)r Fs(2)p Ft(\003)2137 881 y Fo(2)g Fp(R)2312 845 y Ft(\003)2371 881 y Fx(,)35 b(we)e(de\002ne)h Fu(y)46 b Fr(=)d(\()p Fu(y)3156 896 y Fq(k)3198 881 y Fr(\))3236 896 y Fq(k)r Fs(2)p Ft(\003)3418 881 y Fo(\021)257 1001 y Fu(y)t Fr(\()p Fu(x;)17 b(")p 446 1017 46 4 v -1 w Fr(\))28 b Fo(2)g Fp(R)717 965 y Ft(\003)801 1001 y Fx(by)c Fu(y)973 1016 y Fq(k)1043 1001 y Fr(=)k(\(1)22 b Fo(\000)g Fu(")1401 1016 y Fq(k)1444 1001 y Fr(\))p Fu(x)1537 1016 y Fq(k)1580 1001 y Fx(.)257 1186 y Fv(Pr)n(oposition)33 b(III.3.)45 b Fn(Let)33 b Fu(\030)47 b(>)42 b Fr(0)33 b Fn(and)f(de\002ne)h Fu(")1992 1201 y Fq(o)2072 1186 y Fr(:=)42 b(max)2415 1105 y Fm(\010)2473 1186 y Fr(1)p Fu(=)p Fr(2)p Fu(;)17 b Fr(1)27 b Fo(\000)2867 1160 y Fr(^)2846 1186 y Fu(R)2920 1201 y Ft(0)2981 1160 y Fr(\024)2959 1186 y Fu(R)3034 1144 y Fs(\000)p Ft(1)3033 1210 y(1)3129 1105 y Fm(\011)3187 1186 y Fn(.)54 b(Ther)l(e)257 1306 y(e)n(xists)32 b(a)f(univer)o(sal)g(constant)g Fu(c)40 b(>)h Fr(0)31 b Fn(suc)o(h)g(that)g(if)h Fu(\034)52 b Fr(:=)40 b Fu(c)k Fr(min)o Fo(f)p Fr(1)p Fu(;)17 b(\030)5 b Fo(g)2856 1270 y Ft(2)2926 1306 y Fn(and)31 b Fr(0)40 b Fo(\024)h Fu(\013)h Fo(\024)257 1426 y Fu(\013)319 1441 y Ft(0)387 1426 y Fr(:=)27 b Fu(c)17 b(\034)36 b Fn(then)1183 1547 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))50 b Fo(\000)g Fu(H)8 b Fr(\()p Fu(y)t Fr(\))52 b Fo(\025)h Fu(c)17 b(\030)37 b Fr(min)n Fo(f)p Fr(1)p Fu(;)17 b(\030)5 b Fo(g)21 b Fu(;)636 b Fx(\(III.13\))257 1712 y Fn(for)25 b(all)f Fu(x;)17 b(y)31 b Fo(2)d Fp(R)867 1676 y Ft(\003)951 1712 y Fn(with)c Fu(y)1195 1727 y Fq(k)1265 1712 y Fr(=)k(\(1)21 b Fo(\000)i Fu(")1623 1727 y Fq(k)1665 1712 y Fr(\))p Fu(x)1758 1727 y Fq(k)1801 1712 y Fn(,)i Fu(k)31 b Fo(2)d Fr(\003)d Fn(and)f(suc)o(h)g(that)g Fo(j)p Fu(x)2763 1727 y Fq(o)2801 1712 y Fo(j)k(\025)g Fu(\030)5 b Fn(.)257 1896 y(Pr)l(oof:)30 b Fx(W)-8 b(e)25 b(\002rst)g(observ)o(e)f(that)503 2117 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))49 b Fo(\000)i Fu(H)8 b Fr(\()p Fu(y)t Fr(\))81 b(=)1357 2006 y Fm(n)1424 2117 y Fu(f)1472 2132 y Fq(o)1510 2117 y Fr(\()p Fu(x)1603 2132 y Fq(o)1642 2117 y Fr(\))22 b Fo(\000)g Fu(f)1849 2132 y Fq(o)1888 2117 y Fr(\()p Fu(y)1974 2132 y Fq(o)2011 2117 y Fr(\))2049 2006 y Fm(o)2166 2117 y Fr(+)2310 2022 y Fm(X)2291 2238 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))2490 2036 y Fm(\010)2548 2117 y Fu(f)2596 2132 y Fq(k)2639 2117 y Fr(\()p Fu(x)2732 2132 y Fq(k)2775 2117 y Fr(\))g Fo(\000)h Fu(f)2983 2132 y Fq(k)3026 2117 y Fr(\()p Fu(y)3112 2132 y Fq(k)3154 2117 y Fr(\))3192 2036 y Fm(\011)1357 2403 y Fr(+)28 b Fu(\013)1540 2308 y Fm(X)1549 2520 y Fq(j)t Fs(6)p Ft(=)p Fq(k)1701 2322 y Fm(\010)1759 2403 y Fu(w)1829 2418 y Fq(j)t(k)1903 2403 y Fr(\()p Fu(x)1996 2418 y Fq(j)2033 2403 y Fu(;)17 b(x)2132 2418 y Fq(k)2175 2403 y Fr(\))39 b Fo(\000)g Fu(w)2438 2418 y Fq(j)t(k)2513 2403 y Fr(\()p Fu(y)2599 2418 y Fq(j)2635 2403 y Fu(;)17 b(y)2727 2418 y Fq(k)2769 2403 y Fr(\))2807 2322 y Fm(\011)2887 2403 y Fu(:)291 b Fx(\(III.14\))257 2702 y(According)25 b(to)f(Lemma)g(III.2,)i(there)f (e)o(xists)e(a)i(uni)n(v)o(ersal)e(constant)h Fu(C)2722 2666 y Fs(0)2773 2702 y Fu(<)k Fo(1)c Fx(such)h(that)330 2950 y Fu(\013)409 2855 y Fm(X)418 3067 y Fq(j)t Fs(6)p Ft(=)p Fq(k)570 2869 y Fm(\010)628 2950 y Fu(w)698 2965 y Fq(j)t(k)773 2950 y Fr(\()p Fu(x)866 2965 y Fq(j)902 2950 y Fu(;)17 b(x)1001 2965 y Fq(k)1044 2950 y Fr(\))39 b Fo(\000)g Fu(w)1307 2965 y Fq(j)t(k)1382 2950 y Fr(\()p Fu(y)1468 2965 y Fq(j)1504 2950 y Fu(;)17 b(y)1596 2965 y Fq(k)1638 2950 y Fr(\))1676 2869 y Fm(\011)1786 2950 y Fo(\025)54 b(\000)2114 2882 y Fu(C)2191 2846 y Fs(0)2231 2882 y Fu(\013)p 2004 2927 401 4 v 2004 3018 a Fr(\(1)22 b Fo(\000)g Fu(")2258 3033 y Fq(o)2296 3018 y Fr(\))17 b Fu(\034)2448 2855 y Fm(X)2503 3065 y Fq(j)2608 2869 y Fm(\010)2666 2950 y Fu(g)2713 2965 y Fq(j)2749 2950 y Fr(\()p Fu(x)2842 2965 y Fq(j)2879 2950 y Fr(\))39 b Fo(\000)g Fu(g)3119 2965 y Fq(j)3155 2950 y Fr(\()p Fu(y)3241 2965 y Fq(j)3277 2950 y Fr(\))3315 2869 y Fm(\011)3396 2950 y Fu(:)3205 3166 y Fx(\(III.15\))257 3287 y(T)-8 b(o)32 b(bound)g(the)g(second)f(term)h(on)g(the)g(right)g(side)f(of)h (\(III.14\))h(from)f(belo)n(w)-6 b(,)32 b(we)h(use)f(that,)257 3407 y(for)k Fo(j)p Fu(x)492 3422 y Fq(k)534 3407 y Fo(j)47 b Fu(<)752 3382 y Fr(\024)731 3407 y Fu(R)805 3422 y Ft(1)845 3407 y Fx(,)37 b(the)e(boundedness)f(of)h(the)g(deri)n(v)n (ati)n(v)o(e)e(of)i Fu(q)2475 3422 y Fq(k)2553 3407 y Fx(by)f(a)i(uni)n(v)o(ersal)d(constant)257 3527 y Fu(C)334 3491 y Fs(00)404 3527 y Fu(<)28 b Fo(1)d Fx(implies)498 3725 y Fu(q)541 3740 y Fq(k)583 3725 y Fr(\()p Fu(x)676 3740 y Fq(k)719 3725 y Fr(\))e Fo(\000)f Fu(q)922 3740 y Fq(k)965 3725 y Fr(\()p Fu(y)1051 3740 y Fq(k)1093 3725 y Fr(\))53 b Fo(\025)g(\000j)p Fu(x)1474 3740 y Fq(k)1539 3725 y Fo(\000)23 b Fu(y)1687 3740 y Fq(k)1729 3725 y Fo(j)78 b Fr(sup)1801 3820 y Fs(j)p Fq(s)p Fs(j\024)1943 3803 y Ft(\024)1929 3820 y Fq(R)1982 3829 y Fk(1)2032 3641 y Fm(\014)2032 3700 y(\014)2065 3725 y Fu(q)2112 3684 y Fs(0)2108 3750 y Fq(k)2151 3725 y Fr(\()p Fu(s)p Fr(\))p Fo(j)52 b(\024)h(\000)p Fu(C)2637 3684 y Fs(00)2697 3725 y Fu(")2743 3740 y Fq(k)2823 3700 y Fr(\024)2802 3725 y Fu(R)2876 3740 y Ft(1)2938 3725 y Fu(;)240 b Fx(\(III.16\))257 4020 y(which)25 b(also)f(holds)g(for)h Fo(j)p Fu(x)1176 4035 y Fq(k)1219 4020 y Fo(j)i(\025)1401 3995 y Fr(\024)1380 4020 y Fu(R)1454 4035 y Ft(1)1493 4020 y Fx(,)e(since)g(then)g Fu(q)2013 4035 y Fq(k)2055 4020 y Fr(\()p Fu(x)2148 4035 y Fq(k)2191 4020 y Fr(\))j(=)g Fu(q)2404 4035 y Ft(max)2567 4020 y Fo(\025)h Fu(q)2716 4035 y Fq(k)2759 4020 y Fr(\()p Fu(y)2845 4035 y Fq(k)2887 4020 y Fr(\))p Fx(.)i(Thus)24 b(\(III.16\))257 4140 y(holds)g(true)h(for)g(all)g Fu(x)999 4155 y Fq(k)1067 4140 y Fx(and)f Fu(y)1283 4155 y Fq(k)1353 4140 y Fr(=)k(\(1)22 b Fo(\000)g Fu(")1711 4155 y Fq(k)1754 4140 y Fr(\))p Fu(x)1847 4155 y Fq(k)1915 4140 y Fx(which)i(yields)282 4261 y Fm(X)263 4477 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))462 4275 y Fm(\010)520 4356 y Fu(f)568 4371 y Fq(k)611 4356 y Fr(\()p Fu(x)704 4371 y Fq(k)747 4356 y Fr(\))e Fo(\000)g Fu(f)954 4371 y Fq(k)997 4356 y Fr(\()p Fu(y)1083 4371 y Fq(k)1125 4356 y Fr(\))1163 4275 y Fm(\011)1305 4356 y Fr(=)1484 4261 y Fm(X)1465 4477 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))1664 4275 y Fm(\010)1722 4356 y Fu(g)1769 4371 y Fq(k)1811 4356 y Fr(\()p Fu(x)1904 4371 y Fq(k)1947 4356 y Fr(\))h Fo(\000)f Fu(g)2154 4371 y Fq(k)2197 4356 y Fr(\()p Fu(y)2283 4371 y Fq(k)2325 4356 y Fr(\))2363 4275 y Fm(\011)2443 4356 y Fr(+)2560 4261 y Fm(X)2541 4477 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))2740 4275 y Fm(\010)2798 4356 y Fu(q)2841 4371 y Fq(k)2884 4356 y Fr(\()p Fu(x)2977 4371 y Fq(k)3020 4356 y Fr(\))g Fo(\000)h Fu(q)3223 4371 y Fq(k)3266 4356 y Fr(\()p Fu(y)3352 4371 y Fq(k)3394 4356 y Fr(\))3432 4275 y Fm(\011)1304 4641 y Fo(\025)1484 4547 y Fm(X)1465 4762 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))1664 4561 y Fm(\010)1722 4641 y Fu(g)1769 4656 y Fq(k)1811 4641 y Fr(\()p Fu(x)1904 4656 y Fq(k)1947 4641 y Fr(\))g Fo(\000)f Fu(g)2154 4656 y Fq(k)2197 4641 y Fr(\()p Fu(y)2283 4656 y Fq(k)2325 4641 y Fr(\))2363 4561 y Fm(\011)2471 4641 y Fo(\000)50 b Fu(C)2675 4600 y Fs(00)2755 4616 y Fr(\024)2734 4641 y Fu(R)2808 4656 y Ft(1)2883 4547 y Fm(X)2864 4762 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))3063 4641 y Fu(")3109 4656 y Fq(k)1304 4927 y Fo(\025)1484 4832 y Fm(X)1465 5048 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))1664 4846 y Fm(\010)1722 4927 y Fu(g)1769 4942 y Fq(k)1811 4927 y Fr(\()p Fu(x)1904 4942 y Fq(k)1947 4927 y Fr(\))23 b Fo(\000)f Fu(g)2154 4942 y Fq(k)2197 4927 y Fr(\()p Fu(y)2283 4942 y Fq(k)2325 4927 y Fr(\))2363 4846 y Fm(\011)2471 4927 y Fo(\000)50 b Fu(C)2675 4886 y Fs(00)2755 4902 y Fr(\024)2734 4927 y Fu(R)2808 4942 y Ft(1)2864 4927 y Fu(C)2934 4942 y Fq(\032)2991 4927 y Fu(\034)33 b(;)112 b Fx(\(III.17\))257 5246 y(where)26 b Fu(C)596 5261 y Fq(\032)663 5246 y Fo(\025)769 5172 y Fm(P)874 5275 y Fq(k)r Ft(\()p Fs(6)p Ft(=)p Fq(o)p Ft(\))1077 5246 y Fu(e)1122 5210 y Fs(\000)p Fq(\032)p Ft(\()p Fq(o;k)r Ft(\))1389 5246 y Fx(is)f(the)f(uni)n(v)o(ersal)f (constant)h(introduced)g(in)h(Eq.)f(\(I.6\).)257 5388 y(T)-8 b(o)23 b(estimate)f(the)h(\002rst)g(term)g(on)g(the)f(right)h (side)f(of)h(\(III.14\),)h(we)f(observ)o(e)g(that)f(if)h Fo(j)p Fu(x)3184 5403 y Fq(o)3223 5388 y Fo(j)k Fu(<)3403 5363 y Fr(\024)3382 5388 y Fu(R)3456 5403 y Ft(1)p eop %%Page: 43 44 43 43 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(43)257 573 y(and)31 b Fr(0)39 b Fu(<)g Fr(\(1)27 b Fo(\000)g Fu(")899 588 y Fq(o)937 573 y Fr(\))39 b Fo(\024)1152 548 y Fr(^)1130 573 y Fu(R)1204 588 y Ft(0)1265 548 y Fr(\024)1244 573 y Fu(R)1319 532 y Fs(\000)p Ft(1)1318 597 y(1)1445 573 y Fx(then)30 b Fu(y)1695 588 y Fq(o)1772 573 y Fr(=)39 b(\(1)27 b Fo(\000)g Fu(")2151 588 y Fq(o)2189 573 y Fr(\))p Fu(x)2282 588 y Fq(o)2359 573 y Fo(2)40 b Fu(I)2508 588 y Ft(0)2578 573 y Fx(and)31 b Fu(q)2796 588 y Fq(o)2835 573 y Fr(\()p Fu(y)2921 588 y Fq(o)2959 573 y Fr(\))39 b(=)g(0)p Fx(.)49 b(Since)257 693 y Fu(q)300 708 y Fq(o)367 693 y Fo(\025)28 b Fr(0)c Fx(e)n(v)o(erywhere,)h(we)g (therefore)g(ha)n(v)o(e)387 913 y Fu(f)435 928 y Fq(o)473 913 y Fr(\()p Fu(x)566 928 y Fq(o)605 913 y Fr(\))d Fo(\000)g Fu(f)812 928 y Fq(o)851 913 y Fr(\()p Fu(y)937 928 y Fq(o)974 913 y Fr(\))53 b(=)f Fu(g)1240 928 y Fq(o)1278 913 y Fr(\()p Fu(x)1371 928 y Fq(o)1410 913 y Fr(\))22 b Fo(\000)h Fu(g)1617 928 y Fq(o)1655 913 y Fr(\()p Fu(y)1741 928 y Fq(o)1778 913 y Fr(\))39 b(+)g Fu(q)2013 928 y Fq(o)2051 913 y Fr(\()p Fu(x)2144 928 y Fq(o)2183 913 y Fr(\))52 b Fo(\025)h Fu(g)2450 928 y Fq(o)2488 913 y Fr(\()p Fu(x)2581 928 y Fq(o)2620 913 y Fr(\))22 b Fo(\000)h Fu(g)2827 928 y Fq(o)2865 913 y Fr(\()p Fu(y)2951 928 y Fq(o)2988 913 y Fr(\))f Fu(;)130 b Fx(\(III.18\))257 1133 y(which)36 b(also)g(holds)f(for)h Fo(j)p Fu(x)1221 1148 y Fq(o)1259 1133 y Fo(j)49 b(\025)1483 1108 y Fr(\024)1462 1133 y Fu(R)1536 1148 y Ft(1)1575 1133 y Fx(,)39 b(again)d(because)g (then)g Fu(q)2493 1148 y Fq(o)2531 1133 y Fr(\()p Fu(x)2624 1148 y Fq(o)2663 1133 y Fr(\))48 b(=)h Fu(q)2917 1148 y Ft(max)3101 1133 y Fo(\025)g Fu(q)3270 1148 y Fq(o)3309 1133 y Fr(\()p Fu(y)3395 1148 y Fq(o)3432 1133 y Fr(\))p Fx(.)257 1254 y(Thus,)24 b(also)h(\(III.18\))g(holds)f(true,)h(for)g (all)f Fu(x)1773 1269 y Fq(o)1837 1254 y Fx(and)h Fu(y)2054 1269 y Fq(o)2119 1254 y Fr(=)j(\(1)22 b Fo(\000)g Fu(")2477 1269 y Fq(o)2515 1254 y Fr(\))p Fu(x)2608 1269 y Fq(o)2647 1254 y Fx(.)404 1374 y(Inserting)i(\(III.15\),)h(\(III.17\),)g(and)g (\(III.18\))g(into)f(\(III.14\),)h(we)h(obtain)572 1594 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))50 b Fo(\000)g Fu(H)8 b Fr(\()p Fu(y)t Fr(\))655 1797 y Fo(\025)816 1656 y Fm(\022)889 1797 y Fr(1)44 b Fo(\000)1224 1729 y Fu(C)1301 1693 y Fs(0)1341 1729 y Fu(\013)p 1114 1774 401 4 v 1114 1865 a Fr(\(1)22 b Fo(\000)g Fu(")1368 1880 y Fq(o)1406 1865 y Fr(\))17 b Fu(\034)1524 1656 y Fm(\023)1631 1702 y(X)1687 1912 y Fq(j)1791 1716 y Fm(\010)1849 1797 y Fu(g)1896 1812 y Fq(j)1933 1797 y Fr(\()p Fu(x)2026 1812 y Fq(j)2063 1797 y Fr(\))38 b Fo(\000)h Fu(g)2302 1812 y Fq(j)2339 1797 y Fr(\()p Fu(y)2425 1812 y Fq(j)2461 1797 y Fr(\))2499 1716 y Fm(\011)2607 1797 y Fo(\000)50 b Fu(C)2811 1755 y Fs(00)2891 1771 y Fr(\024)2870 1797 y Fu(R)2944 1812 y Ft(1)3000 1797 y Fu(C)3070 1812 y Fq(\032)3127 1797 y Fu(\034)655 2100 y Fo(\025)826 2032 y Fr(1)p 826 2077 49 4 v 826 2168 a(2)918 2005 y Fm(X)973 2215 y Fq(j)1078 2019 y Fm(\010)1136 2100 y Fu(g)1183 2115 y Fq(j)1220 2100 y Fr(\()p Fu(x)1313 2115 y Fq(j)1349 2100 y Fr(\))39 b Fo(\000)g Fu(g)1589 2115 y Fq(j)1626 2100 y Fr(\()p Fu(y)1712 2115 y Fq(j)1748 2100 y Fr(\))1786 2019 y Fm(\011)1893 2100 y Fo(\000)51 b Fu(C)2098 2058 y Fs(00)2178 2074 y Fr(\024)2157 2100 y Fu(R)2231 2115 y Ft(1)2287 2100 y Fu(C)2357 2115 y Fq(\032)2414 2100 y Fu(\034)655 2403 y Fo(\025)826 2335 y Fr(1)p 826 2380 V 826 2471 a(2)901 2322 y Fm(\010)959 2403 y Fu(g)1006 2418 y Fq(o)1044 2403 y Fr(\()p Fu(x)1137 2418 y Fq(o)1176 2403 y Fr(\))38 b Fo(\000)i Fu(g)1416 2418 y Fq(o)1454 2403 y Fr(\()p Fu(y)1540 2418 y Fq(o)1577 2403 y Fr(\))1615 2322 y Fm(\011)1723 2403 y Fo(\000)51 b Fu(C)1928 2361 y Fs(00)2008 2377 y Fr(\024)1987 2403 y Fu(R)2061 2418 y Ft(1)2117 2403 y Fu(C)2187 2418 y Fq(\032)2244 2403 y Fu(\034)33 b(;)859 b Fx(\(III.19\))257 2658 y(pro)o(vided)24 b Fr(0)j Fo(\024)i Fu(\013)f Fo(\024)g Fr(\(1)22 b Fo(\000)h Fu(")1267 2673 y Fq(o)1304 2658 y Fr(\))17 b Fu(\034)28 b Fr(\(2)p Fu(C)1593 2622 y Fs(0)1616 2658 y Fr(\))1654 2622 y Fs(\000)p Ft(1)1748 2658 y Fx(.)404 2778 y(By)d(assumption,)e (we)i(ha)n(v)o(e)f Fo(j)p Fu(x)1482 2793 y Fq(o)1521 2778 y Fo(j)j(\025)h Fu(\030)h Fx(and)c Fu(")1968 2793 y Fq(o)2034 2778 y Fo(\025)j Fr(1)p Fu(=)p Fr(2)p Fx(.)i(Hence)25 b(\(I.29\))g(and)g(\(I.32\))g(gi)n(v)o(e)650 3053 y Fu(g)697 3068 y Fq(o)735 3053 y Fr(\()p Fu(x)828 3068 y Fq(o)867 3053 y Fr(\))d Fo(\000)g Fu(g)1073 3068 y Fq(o)1111 3053 y Fr(\()p Fu(y)1197 3068 y Fq(o)1235 3053 y Fr(\))83 b Fo(\025)g Fu(g)1563 3068 y Fq(o)1601 3053 y Fr(\()p Fu(x)1694 3068 y Fq(o)1733 3053 y Fr(\))22 b Fo(\000)h Fu(g)1940 3068 y Fq(o)1978 3053 y Fr(\()p Fu(x)2071 3068 y Fq(o)2109 3053 y Fu(=)p Fr(2\))52 b Fo(\025)2437 2986 y Fu(c)2479 3001 y Fq(g)2536 2986 y Fo(j)p Fu(x)2619 3001 y Fq(o)2658 2986 y Fo(j)p 2437 3030 248 4 v 2537 3121 a Fr(2)2695 3053 y Fo(j)p Fu(g)2774 3012 y Fs(0)2770 3078 y Fq(o)2808 3053 y Fr(\()p Fu(x)2901 3068 y Fq(o)2939 3053 y Fu(=)p Fr(2\))p Fo(j)1356 3310 y(\025)1526 3234 y Fu(c)1568 3197 y Ft(2)1568 3258 y Fq(g)1625 3234 y Fu(\030)p 1526 3287 147 4 v 1575 3378 a Fr(2)1716 3310 y(min)n Fo(f)p Fr(1)p Fu(;)17 b(\030)5 b(=)p Fr(2)p Fo(g)51 b(\025)2408 3234 y Fu(c)2450 3197 y Ft(2)2450 3258 y Fq(g)p 2408 3287 83 4 v 2425 3378 a Fr(4)2517 3310 y Fu(\030)37 b Fr(min)o Fo(f)p Fr(1)p Fu(;)17 b(\030)5 b Fo(g)21 b Fu(:)156 b Fx(\(III.20\))257 3581 y(Thus,)24 b(choosing)g Fu(")937 3596 y Fq(o)1003 3581 y Fr(:=)j(max)1331 3500 y Fm(\010)1390 3581 y Fr(1)p Fu(=)p Fr(2)17 b Fu(;)37 b Fr(1)22 b Fo(\000)1810 3555 y Fr(^)1789 3581 y Fu(R)1863 3596 y Ft(0)1924 3555 y Fr(\024)1903 3581 y Fu(R)1978 3539 y Fs(\000)p Ft(1)1977 3605 y(1)2072 3500 y Fm(\011)2130 3581 y Fx(,)354 3879 y Fu(\034)61 b Fr(:=)947 3802 y Fu(c)989 3766 y Ft(2)989 3827 y Fq(g)p 620 3856 738 4 v 620 3956 a Fu(c)662 3927 y Ft(2)662 3980 y Fq(g)746 3956 y Fr(+)44 b(16)17 b Fu(C)1058 3927 y Fs(00)1138 3930 y Fr(\024)1116 3956 y Fu(R)1190 3971 y Ft(1)1247 3956 y Fu(C)1317 3971 y Fq(\032)1400 3879 y Fr(min)o Fo(f)p Fr(1)p Fu(;)g(\030)1754 3838 y Ft(2)1792 3879 y Fo(g)22 b Fu(;)158 b Fx(and)142 b Fu(\013)2397 3894 y Ft(0)2486 3879 y Fr(:=)2649 3811 y(\(1)22 b Fo(\000)g Fu(")2903 3826 y Fq(o)2941 3811 y Fr(\))17 b Fu(\034)p 2649 3856 401 4 v 2774 3947 a Fr(2)p Fu(C)2900 3918 y Fs(0)3081 3879 y Fu(;)97 b Fx(\(III.21\))257 4176 y(we)25 b(ha)n(v)o(e)g Fu(")654 4191 y Fq(o)692 4176 y Fu(;)17 b(\034)39 b Fo(2)28 b Fr(\(0)p Fu(;)17 b Fr(1\))p Fx(,)24 b(and)h(we)g(obtain)1146 4464 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))49 b Fo(\000)i Fu(H)8 b Fr(\()p Fu(y)t Fr(\))51 b Fo(\025)1959 4388 y Fu(c)2001 4352 y Ft(2)2001 4412 y Fq(g)p 1951 4441 98 4 v 1951 4532 a Fr(16)2075 4464 y Fu(\030)38 b Fr(min)n Fo(f)p Fr(1)p Fu(;)17 b(\030)5 b Fo(g)21 b Fu(;)598 b Fx(\(III.22\))257 4722 y(for)25 b(all)e Fr(0)28 b Fo(\024)g Fu(\013)g Fo(\024)g Fu(\013)960 4737 y Ft(0)1024 4722 y Fx(and)c Fu(x;)17 b(y)30 b Fo(2)f Fp(R)1530 4686 y Ft(\003)1613 4722 y Fx(with)23 b Fu(y)1862 4737 y Fq(k)1932 4722 y Fr(=)k(\(1)19 b Fo(\000)g Fu(")2283 4737 y Fq(k)2325 4722 y Fr(\))p Fu(x)2418 4737 y Fq(k)2485 4722 y Fx(and)24 b(such)g(that)f Fo(j)p Fu(x)3116 4737 y Fq(o)3154 4722 y Fo(j)28 b(\025)g Fu(\030)5 b Fx(.)30 b Fg(\003)404 4842 y Fx(W)-8 b(e)25 b(are)g(no)n(w)f(in)h(a)g(position) e(to)h(pro)o(v)o(e)g(the)h(main)f(result)g(of)h(this)f(section.)257 4963 y Fv(Pr)n(oof)36 b(of)f(Theor)n(em)i(I.6:)103 b Fx(Fix)36 b Fu(\030)51 b(>)c Fr(0)p Fx(,)38 b Fu(o)47 b Fo(2)g Fr(\003)p Fx(,)38 b(and)d(de\002ne)h Fu(")2658 4978 y Fq(o)2743 4963 y Fr(:=)47 b(max)3091 4882 y Fm(\010)3149 4963 y Fr(1)p Fu(=)p Fr(2)p Fu(;)17 b Fr(1)29 b Fo(\000)279 5075 y Fr(^)257 5100 y Fu(R)331 5115 y Ft(0)392 5075 y Fr(\024)371 5100 y Fu(R)446 5059 y Fs(\000)p Ft(1)445 5125 y(1)540 5019 y Fm(\011)599 5100 y Fx(,)40 b Fu(\034)61 b Fr(:=)50 b Fu(c)e Fr(min)o Fo(f)p Fr(1)p Fu(;)17 b(\030)5 b Fo(g)1414 5064 y Ft(2)1452 5100 y Fx(,)40 b(and)c Fr(0)50 b Fo(\024)h Fu(\013)g Fo(\024)f Fu(\013)2226 5115 y Ft(0)2316 5100 y Fr(:=)g Fu(c)17 b(\034)2581 5064 y Ft(2)2621 5100 y Fx(,)40 b(where)d Fu(c)50 b(>)g Fr(0)37 b Fx(is)f(the)257 5229 y(uni)n(v)o(ersal)g(constant)h(in)f(Proposition)g(III.3.)69 b(Set)38 b Fu(")2116 5244 y Fq(k)2209 5229 y Fr(:=)51 b Fu(")2409 5244 y Fq(o)2447 5229 y Fu(\016)2490 5244 y Fq(k)r(;o)2618 5229 y Fr(+)31 b(\(1)g Fo(\000)h Fu(\016)2995 5244 y Fq(k)r(;o)3091 5229 y Fr(\))p Fu(\034)11 b(e)3227 5193 y Fs(\000)p Fq(\032)p Ft(\()p Fq(k)r(;o)p Ft(\))3470 5229 y Fx(.)p eop %%Page: 44 45 44 44 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(44)257 573 y(Denoting)23 b Fr(\()p Fi(1)18 b Fo(\000)h Fu(")p 860 589 46 4 v -1 w Fr(\))p Fu(x)28 b Fr(:=)1157 492 y Fm(\000)1203 573 y Fr(\(1)17 b Fo(\000)i Fu(")1449 588 y Fq(k)1491 573 y Fr(\))p Fu(x)1584 588 y Fq(k)1627 492 y Fm(\001)1673 612 y Fq(k)r Fs(2)p Ft(\003)1839 573 y Fo(2)28 b Fp(R)1999 537 y Ft(\003)2082 573 y Fx(and)c Fu(m)p Fr(\()p Fu(\030)5 b Fr(\))27 b(:=)h(2)p Fu(c\030)20 b Fr(min)o Fo(f)p Fr(1)p Fu(;)d(\030)5 b Fo(g)p Fx(,)22 b(we)i(use)257 693 y(Proposition)g(III.3)h(and)g(a)g(change)g(of)g (inte)o(gration)e(v)n(ariables)h Fu(y)31 b Fr(:=)c(\()p Fi(1)22 b Fo(\000)h Fu(")p 2856 709 V Fr(\))p Fu(x)i Fx(to)f(estimate)764 826 y Fm(Z)819 1052 y Fs(f)p Fq(x)p Ft(:)p Fs(j)p Fq(x)974 1060 y Fj(o)1008 1052 y Fs(j\025)p Fq(\030)s Fs(g)1172 962 y Fr(exp)1338 881 y Fm(\002)1401 962 y Fo(\000)f Fr(2)p Fu(\014)6 b(H)i Fr(\()p Fu(x)p Fr(\))1831 881 y Fm(\003)1888 962 y Fu(d)1939 921 y Ft(\003)1992 962 y Fu(x)1158 b Fx(\(III.23\))847 1237 y Fo(\024)83 b Fu(e)1052 1196 y Fs(\000)p Fq(\014)s(m)p Ft(\()p Fq(\030)s Ft(\))1344 1101 y Fm(Z)1400 1327 y Fh(R)1448 1308 y Fk(\003)1514 1237 y Fr(exp)1680 1126 y Fm(h)1749 1237 y Fo(\000)22 b Fr(2)p Fu(\014)6 b(H)2047 1156 y Fm(\000)2092 1237 y Fr(\()p Fi(1)22 b Fo(\000)h Fu(")p 2308 1253 V -1 w Fr(\))p Fu(x)2446 1156 y Fm(\001)2492 1126 y(i)2556 1237 y Fu(d)2607 1196 y Ft(\003)2660 1237 y Fu(x)848 1505 y Fr(=)83 b Fu(e)1052 1464 y Fs(\000)p Fq(\014)s(m)p Ft(\()p Fq(\030)s Ft(\))1328 1364 y Fm(\022)1418 1410 y(Y)1462 1622 y Fq(k)1676 1437 y Fr(1)p 1571 1482 259 4 v 1571 1573 a(1)22 b Fo(\000)h Fu(")1788 1588 y Fq(k)1840 1364 y Fm(\023)1952 1369 y(Z)2008 1595 y Fh(R)2056 1576 y Fk(\003)2122 1505 y Fr(exp)2288 1424 y Fm(\002)2351 1505 y Fo(\000)g Fr(2)p Fu(\014)6 b(H)i Fr(\()p Fu(y)t Fr(\))2778 1424 y Fm(\003)2834 1505 y Fu(d)2885 1464 y Ft(\003)2938 1505 y Fu(y)847 1797 y Fo(\024)83 b Fu(e)1052 1756 y Fs(\000)p Fq(\014)s(m)p Ft(\()p Fq(\030)s Ft(\))1328 1686 y Fm(\020)1500 1730 y Fr(1)p 1397 1774 255 4 v 1397 1865 a(1)22 b Fo(\000)h Fu(")1614 1880 y Fq(o)1662 1686 y Fm(\021)1760 1797 y Fr(exp)1925 1686 y Fm(h)2009 1702 y(X)1989 1918 y Fq(k)r Ft(\()p Fs(6)p Ft(=0\))2189 1797 y Fu(")2235 1812 y Fq(k)2278 1686 y Fm(i)2347 1797 y Fo(Z)2419 1812 y Fq(\014)847 2069 y Fo(\024)83 b Fu(C)24 b(e)1146 2028 y Fq(C)1196 2036 y Fj(\032)1253 2069 y Fu(e)1298 2028 y Fs(\000)p Fq(\014)s(m)p Ft(\()p Fq(\030)s Ft(\))1573 2069 y Fo(Z)1645 2084 y Fq(\014)1715 2069 y Fu(:)257 2289 y Fx(This)g(pro)o(v)o(es)g(Eq.)g(\(I.70\).)2262 b Fg(\003)p eop %%Page: 45 46 45 45 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(45)257 573 y Fw(A)143 b(Construction)33 b(of)43 b Fb(^)-73 b Fa(g)1568 594 y Fu(j)1654 573 y Fw(and)35 b Fa(g)1988 594 y Fu(j)2074 573 y Fw(fr)m(om)g Fa(f)2479 594 y Fu(j)257 796 y Fx(In)27 b(this)e(appendix,)h(we)g(consider)g(a)h (f)o(amily)f Fo(f)p Fu(f)1915 811 y Fq(j)1951 796 y Fo(g)2001 811 y Fq(j)t Fs(2)p Ft(\003)2160 796 y Fx(of)g(single-spin)f (potentials)f(ful\002lling)257 917 y(Hypothesis)38 b(1.)72 b(Recall)40 b(that)e(Hypothesis)g(1)g(requires,)43 b(for)c(an)o(y)f Fu(j)60 b Fo(2)54 b Fr(\003)p Fx(,)42 b(that)c(zero)i(is)257 1037 y(the)d(unique,)i(nonde)o(generate)e(minimum)d(of)j Fu(f)1961 1052 y Fq(j)1998 1037 y Fx(,)j(attained)c(at)h Fu(t)51 b Fr(=)f(0)p Fx(,)39 b(i.e.,)h Fu(f)3084 1052 y Fq(j)3121 1037 y Fr(\(0\))50 b(=)g(0)p Fx(,)257 1158 y Fu(f)316 1121 y Fs(00)305 1182 y Fq(j)359 1158 y Fr(\(0\))g Fu(>)h Fr(0)p Fx(,)41 b(and)c Fu(f)1005 1173 y Fq(j)1042 1158 y Fr(\()p Fu(t)p Fr(\))51 b Fu(>)g Fr(0)p Fx(,)40 b(whene)n(v)o(er)d Fu(t)51 b Fo(6)p Fr(=)g(0)p Fx(.)69 b(Moreo)o(v)o(er)l(,)39 b(there)f(e)o(xist)e(uni)n(v)o(ersal)257 1278 y(constants)24 b Fr(0)k Fu(<)f(c)875 1293 y Fq(f)948 1278 y Fo(\024)h Fr(1)g Fo(\024)g Fu(C)1305 1293 y Fq(f)1378 1278 y Fu(<)f Fo(1)p Fx(,)e Fu(R)1705 1293 y Fq(f)1778 1278 y Fu(<)j Fo(1)p Fx(,)c(such)h(that)1681 1445 y Fu(c)1723 1460 y Fq(f)1821 1445 y Fo(\024)53 b Fu(f)2010 1409 y Fs(00)1999 1470 y Fq(j)2053 1445 y Fr(\(0\))f Fo(\024)h Fu(C)2430 1460 y Fq(f)2497 1445 y Fu(;)758 b Fx(\(A.1\))718 1600 y Fo(8)p Fu(t)28 b Fo(6)p Fr(=)g(0)f(:)1464 1520 y Fm(\010)1522 1600 y Fu(f)1581 1564 y Fs(0)1570 1625 y Fq(j)1607 1600 y Fr(\()p Fu(t)p Fr(\))g(=)h(0)1898 1520 y Fm(\011)2008 1600 y Fo(\))2161 1520 y Fm(\010)2219 1600 y Fu(f)2267 1615 y Fq(j)2303 1600 y Fr(\()p Fu(t)p Fr(\))g Fo(\025)g Fu(c)2589 1615 y Fq(f)2635 1520 y Fm(\011)2715 1600 y Fu(;)540 b Fx(\(A.2\))704 1756 y Fo(8)p Fu(t)29 b Fo(2)f Fp(R)38 b Fr(:)108 b Fo(j)p Fu(f)1238 1720 y Fs(00)1227 1781 y Fq(j)1280 1756 y Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)h Fu(f)1572 1720 y Fs(00)1561 1781 y Fq(j)1614 1756 y Fr(\(0\))p Fo(j)52 b(\024)h Fu(C)2019 1771 y Fq(f)2064 1675 y Fm(\000)2110 1756 y Fo(j)p Fu(f)2197 1720 y Fs(0)2186 1781 y Fq(j)2222 1756 y Fr(\()p Fu(t)p Fr(\))p Fo(j)39 b Fr(+)f(min)o Fo(f)p Fr(1)p Fu(;)17 b Fo(j)p Fu(t)p Fo(jg)2961 1675 y Fm(\001)3027 1756 y Fu(;)228 b Fx(\(A.3\))590 1941 y Fo(8j)p Fu(t)p Fo(j)28 b(\025)g Fu(R)943 1956 y Fq(f)1016 1941 y Fr(:)84 b Fo(j)p Fu(f)1214 1905 y Fs(0)1203 1966 y Fq(j)1239 1941 y Fr(\()p Fu(t)p Fr(\))p Fo(j)52 b(\025)h Fu(c)1602 1956 y Fq(f)1686 1941 y Fr(max)1884 1831 y Fm(n)1951 1941 y Fo(j)p Fu(f)2038 1905 y Fs(0)2027 1967 y Fq(k)2069 1941 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2246 1827 y Fm(\014)2246 1886 y(\014)2246 1946 y(\014)2305 1941 y Fo(j)p Fu(s)p Fo(j)27 b(\024)h(j)p Fu(t)p Fo(j)21 b Fu(;)42 b(k)31 b Fo(2)d Fr(\003)2964 1831 y Fm(o)3052 1941 y Fu(;)203 b Fx(\(A.4\))257 2129 y(for)25 b(all)g Fu(j)34 b Fo(2)28 b Fr(\003)p Fx(.)404 2250 y(Gi)n(v)o(en)42 b Fo(f)p Fu(f)785 2265 y Fq(j)821 2250 y Fo(g)871 2265 y Fq(j)t Fs(2)p Ft(\003)1047 2250 y Fx(ful\002lling)h(Hypothesis)f(1,) 48 b(our)43 b(purpose)g(in)g(this)g(appendix)g(is)g(to)257 2370 y(pro)o(v)o(e)35 b(Lemma)g(I.2,)k(i.e.,)f(to)d(construct)h(f)o (amilies)e Fo(f)p Fu(g)2181 2385 y Fq(j)2218 2370 y Fo(g)2268 2385 y Fq(j)t Fs(2)p Ft(\003)2436 2370 y Fx(and)h Fo(f)s Fr(^)-52 b Fu(g)2712 2385 y Fq(j)2749 2370 y Fo(g)2799 2385 y Fq(j)t Fs(2)p Ft(\003)2931 2370 y Fx(,)38 b(which)e(ful\002ll) 257 2490 y(the)29 b(stronger)f(requirement)g(\(I.12\),)i(besides)e (Hypothesis)f(1,)i(and)g(agree)g(with)f Fo(f)p Fu(f)3176 2505 y Fq(j)3212 2490 y Fo(g)3262 2505 y Fq(j)t Fs(2)p Ft(\003)3423 2490 y Fx(at)257 2611 y(zero)e(and)e(at)h(in\002nity)-6 b(.)404 2731 y(F)o(or)29 b(the)h(sak)o(e)g(of)g(readability)-6 b(,)30 b(we)g(shall)f(only)g(gi)n(v)o(e)f(the)i(details)f(of)h(our)g (construction)257 2851 y(on)e(the)g(positi)n(v)o(e)d(half-axis,)k(and)f (we)g(henceforth)g(assume)f(that)h Fu(t;)17 b(s)33 b Fo(\025)h Fr(0)28 b Fx(without)e(further)257 2972 y(mentioning.)72 b(W)-8 b(e)39 b(di)n(vide)f(our)h(construction)f(into)g(se)n(v)o(eral)g (steps,)k(each)e(casted)f(into)f(a)257 3092 y(lemma.)257 3250 y Fv(Lemma)26 b(A.1.)41 b Fn(Assume)24 b(Hypothesis)g(1)g(and)h (let)f Fu(j)34 b Fo(2)28 b Fr(\003)p Fn(.)j(Then)997 3417 y Fo(8)p Fu(t)d Fo(\024)g Fu(C)1297 3376 y Fs(\000)p Ft(1)1290 3445 y Fq(f)1419 3417 y Fr(:)1542 3333 y Fm(\014)1542 3393 y(\014)1575 3417 y Fu(f)1634 3381 y Fs(0)1623 3442 y Fq(j)1660 3417 y Fr(\()p Fu(t)p Fr(\))39 b Fo(\000)g Fu(tf)2020 3381 y Fs(00)2009 3442 y Fq(j)2062 3417 y Fr(\(0\))2187 3333 y Fm(\014)2187 3393 y(\014)2273 3417 y Fo(\024)53 b Fu(C)2480 3381 y Ft(2)2473 3443 y Fq(f)2536 3417 y Fu(t)2571 3381 y Ft(2)2633 3417 y Fu(;)622 b Fx(\(A.5\))1048 3576 y Fo(8)p Fu(t)28 b Fo(2)h Fp(R)1327 3535 y Ft(+)1327 3601 y(0)1419 3576 y Fr(:)1529 3491 y Fm(\014)1529 3551 y(\014)1563 3576 y Fu(f)1622 3540 y Fs(0)1611 3601 y Fq(j)1647 3576 y Fr(\()p Fu(t)p Fr(\))1758 3491 y Fm(\014)1758 3551 y(\014)1844 3576 y Fo(\024)53 b Fr(3)17 b Fu(C)2110 3591 y Fq(f)2188 3576 y Fr(exp)q([2)g Fu(C)2500 3591 y Fq(f)2561 3576 y Fu(t)p Fr(])23 b Fu(:)609 b Fx(\(A.6\))257 3743 y Fn(Pr)l(oof:)28 b Fx(The)21 b(asserted)h(bounds)e(\(A.5\))h(and) h(\(A.6\))f(follo)n(w)f(from)h(Gronw)o(all-type)g(estimates.)257 3864 y(W)-8 b(e)26 b(\002x)f Fu(j)33 b Fo(2)28 b Fr(\003)d Fx(and)g(de\002ne)1137 4031 y Fu(\026)p Fr(\()p Fu(t)p Fr(\))52 b(:=)63 b(max)1514 4092 y Ft(0)p Fs(\024)p Fq(s)p Fs(\024)p Fq(t)1734 3946 y Fm(\014)1734 4006 y(\014)1767 4031 y Fu(s)1813 3990 y Fs(\000)p Ft(1)1924 4031 y Fu(f)1983 3990 y Fs(0)1972 4055 y Fq(j)2009 4031 y Fr(\()p Fu(s)p Fr(\))50 b Fo(\000)g Fu(f)2367 3990 y Fs(00)2356 4055 y Fq(j)2409 4031 y Fr(\(0\))2534 3946 y Fm(\014)2534 4006 y(\014)2589 4031 y Fu(;)666 b Fx(\(A.7\))257 4247 y(observing)35 b(that)g Fr(lim)1007 4262 y Fq(s)p Fs(!)p Ft(0)1150 4247 y Fo(f)p Fu(s)1246 4211 y Fs(\000)p Ft(1)1340 4247 y Fu(f)1399 4211 y Fs(0)1388 4272 y Fq(j)1424 4247 y Fr(\()p Fu(s)p Fr(\))p Fo(g)48 b Fr(=)g Fu(f)1827 4211 y Fs(00)1816 4272 y Fq(j)1869 4247 y Fr(\(0\))35 b Fx(e)o(xists,)i (since)f Fu(f)2618 4211 y Fs(0)2607 4272 y Fq(j)2644 4247 y Fr(\(0\))47 b(=)h(0)p Fx(.)63 b(Using)35 b(this)257 4367 y(de\002nition)24 b(and)h(\(A.3\),)g(we)g(estimate)504 4578 y Fo(j)p Fu(f)591 4537 y Fs(0)580 4603 y Fq(j)616 4578 y Fr(\()p Fu(s)p Fr(\))39 b Fo(\000)g Fu(sf)998 4537 y Fs(00)987 4603 y Fq(j)1040 4578 y Fr(\(0\))p Fo(j)83 b(\024)1436 4442 y Fm(Z)1536 4469 y Fq(s)1491 4668 y Ft(0)1589 4493 y Fm(\014)1589 4553 y(\014)1622 4578 y Fu(f)1681 4537 y Fs(00)1670 4603 y Fq(j)1724 4578 y Fr(\()p Fu(r)s Fr(\))21 b Fo(\000)i Fu(f)2027 4537 y Fs(00)2016 4603 y Fq(j)2069 4578 y Fr(\(0\))2194 4493 y Fm(\014)2194 4553 y(\014)2243 4578 y Fu(dr)944 b Fx(\(A.8\))1276 4839 y Fo(\024)83 b Fu(C)1506 4854 y Fq(f)1568 4703 y Fm(Z)1668 4730 y Fq(s)1623 4929 y Ft(0)1721 4758 y Fm(\000)1767 4839 y Fo(j)p Fu(f)1854 4798 y Fs(0)1843 4864 y Fq(j)1879 4839 y Fr(\()p Fu(r)s Fr(\))p Fo(j)38 b Fr(+)g(min)o Fo(f)p Fr(1)p Fu(;)17 b(r)s Fo(g)2585 4758 y Fm(\001)2646 4839 y Fu(dr)1276 5100 y Fo(\024)83 b Fu(C)1506 5115 y Fq(f)1568 4964 y Fm(Z)1668 4991 y Fq(s)1623 5190 y Ft(0)1721 5019 y Fm(\000)1767 5100 y Fo(j)p Fu(f)1854 5059 y Fs(0)1843 5125 y Fq(j)1879 5100 y Fr(\()p Fu(r)s Fr(\))22 b Fo(\000)g Fu(r)s(f)2229 5059 y Fs(00)2218 5125 y Fq(j)2271 5100 y Fr(\(0\))p Fo(j)38 b Fr(+)h([1)22 b(+)g Fu(f)2832 5059 y Fs(00)2821 5125 y Fq(j)2874 5100 y Fr(\(0\)])17 b Fu(r)3090 5019 y Fm(\001)3151 5100 y Fu(dr)1276 5370 y Fo(\024)1446 5302 y Fu(C)1516 5317 y Fq(f)1578 5302 y Fu(s)1624 5266 y Ft(2)p 1446 5347 218 4 v 1530 5438 a Fr(2)1690 5289 y Fm(\000)1735 5370 y Fu(\026)p Fr(\()p Fu(s)p Fr(\))22 b(+)g(1)g(+)g Fu(f)2264 5328 y Fs(00)2253 5394 y Fq(j)2306 5370 y Fr(\(0\))2431 5289 y Fm(\001)2499 5370 y Fu(:)p eop %%Page: 46 47 46 46 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(46)257 573 y(Since)26 b Fu(\026)e Fx(is)h(increasing,)f(\(A.8\),)h Fu(f)1460 537 y Fs(00)1449 597 y Fq(j)1502 573 y Fr(\(0\))i Fo(\024)h Fu(C)1829 588 y Fq(f)1874 573 y Fx(,)d(and)g Fu(C)2163 588 y Fq(f)2236 573 y Fo(\025)j Fr(1)d Fx(imply)739 869 y Fu(\026)p Fr(\()p Fu(t)p Fr(\))53 b Fo(\024)1102 801 y Fu(C)1172 816 y Fq(f)1234 801 y Fu(t)p 1102 846 167 4 v 1161 937 a Fr(2)1295 869 y Fu(\026)p Fr(\()p Fu(t)p Fr(\))d(+)1651 791 y Fu(C)1721 806 y Fq(f)1788 791 y Fr(+)22 b Fu(C)1963 755 y Ft(2)1956 817 y Fq(f)p 1651 846 352 4 v 1802 937 a Fr(2)2029 869 y Fu(t)53 b Fo(\024)2257 801 y Fu(C)2327 816 y Fq(f)2389 801 y Fu(t)p 2257 846 167 4 v 2316 937 a Fr(2)2450 869 y Fu(\026)p Fr(\()p Fu(t)p Fr(\))d(+)g Fu(C)2873 827 y Ft(2)2866 893 y Fq(f)2929 869 y Fu(t)22 b(;)269 b Fx(\(A.9\))257 1106 y(which)25 b(yields)1083 1142 y Fm(\014)1083 1202 y(\014)1116 1227 y Fu(f)1175 1186 y Fs(0)1164 1251 y Fq(j)1200 1227 y Fr(\()p Fu(t)p Fr(\))39 b Fo(\000)h Fu(tf)1561 1186 y Fs(00)1550 1251 y Fq(j)1603 1227 y Fr(\(0\))1728 1142 y Fm(\014)1728 1202 y(\014)1813 1227 y Fo(\024)53 b Fu(t)17 b(\026)p Fr(\()p Fu(t)p Fr(\))53 b Fo(\024)g Fr(2)17 b Fu(C)2491 1186 y Ft(2)2484 1251 y Fq(f)2546 1227 y Fu(t)2581 1186 y Ft(2)2643 1227 y Fu(;)562 b Fx(\(A.10\))257 1408 y(for)25 b(all)g Fr(0)i Fo(\024)i Fu(t)e Fo(\024)i Fu(C)950 1367 y Fs(\000)p Ft(1)943 1436 y Fq(f)1044 1408 y Fx(.)404 1528 y(Similarly)-6 b(,)23 b(we)i(de\002ne)1250 1742 y Fu(\027)1298 1757 y Fq(\024)1343 1742 y Fr(\()p Fu(t)p Fr(\))52 b(:=)64 b(max)1662 1803 y Ft(0)p Fs(\024)p Fq(s)p Fs(\024)p Fq(t)1882 1661 y Fm(\010)1940 1742 y Fu(e)1985 1701 y Fs(\000)p Fq(\024s)2134 1742 y Fo(j)p Fu(f)2221 1701 y Fs(0)2210 1766 y Fq(j)2246 1742 y Fr(\()p Fu(s)p Fr(\))p Fo(j)2396 1661 y Fm(\011)2476 1742 y Fu(;)729 b Fx(\(A.11\))257 1993 y(where)26 b Fu(\024)h(>)h Fr(0)d Fx(is)f(chosen)h(later)l(,)f (and)h(observ)o(e)g(that,)f(by)g(\(A.3\),)753 2254 y Fo(j)p Fu(f)840 2213 y Fs(0)829 2279 y Fq(j)865 2254 y Fr(\()p Fu(s)p Fr(\))p Fo(j)83 b(\024)g Fu(C)1328 2269 y Fq(f)1390 2119 y Fm(Z)1489 2145 y Fq(s)1445 2344 y Ft(0)1543 2174 y Fm(\000)1589 2254 y Fo(j)p Fu(f)1676 2213 y Fs(0)1665 2279 y Fq(j)1701 2254 y Fr(\()p Fu(r)s Fr(\))p Fo(j)38 b Fr(+)g Fo(j)p Fu(f)2091 2213 y Fs(00)2080 2279 y Fq(j)2133 2254 y Fr(\(0\))p Fo(j)g Fr(+)h(min)n Fo(f)p Fr(1)p Fu(;)17 b(r)s Fo(g)2841 2174 y Fm(\001)2902 2254 y Fu(dr)235 b Fx(\(A.12\))1098 2515 y Fo(\024)83 b Fu(C)1328 2530 y Fq(f)1390 2380 y Fm(Z)1489 2406 y Fq(s)1445 2605 y Ft(0)1543 2515 y Fu(e)1588 2474 y Fq(\024r)1683 2515 y Fu(\027)1731 2530 y Fq(\024)1776 2515 y Fr(\()p Fu(r)s Fr(\))17 b Fu(dr)52 b Fr(+)d(2)17 b Fu(C)2331 2474 y Ft(2)2324 2540 y Fq(f)2387 2515 y Fu(s)1098 2774 y Fo(\024)1268 2707 y Fu(C)1338 2722 y Fq(f)p 1268 2751 116 4 v 1298 2842 a Fu(\024)1410 2774 y(e)1455 2733 y Fq(\024s)1549 2774 y Fu(\027)1597 2789 y Fq(\024)1642 2774 y Fr(\()p Fu(t)p Fr(\))50 b(+)g(2)17 b Fu(C)2072 2733 y Ft(2)2065 2799 y Fq(f)2127 2774 y Fu(s)22 b(:)257 3012 y Fx(Since)k Fu(\027)552 3027 y Fq(\024)622 3012 y Fx(is)e(increasing,)g(\(A.12\))h(implies,)1181 3145 y Fm(\000)1227 3225 y Fr(1)44 b Fo(\000)h Fu(C)1512 3240 y Fq(f)1574 3225 y Fu(\024)1630 3184 y Fs(\000)p Ft(1)1724 3145 y Fm(\001)1786 3225 y Fu(\027)1834 3240 y Fq(\024)1879 3225 y Fr(\()p Fu(t)p Fr(\))53 b Fo(\024)g Fr(6)17 b Fu(C)2316 3184 y Ft(2)2309 3250 y Fq(f)2372 3225 y Fu(\024)2428 3184 y Fs(\000)p Ft(1)2544 3225 y Fu(;)661 b Fx(\(A.13\))257 3439 y(Choosing)24 b Fu(\024)k Fr(:=)g(2)p Fu(C)998 3454 y Fq(f)1043 3439 y Fx(,)c(we)i(arri)n(v)o(e)e(at)h(\(A.6\).)1596 b Fg(\003)404 3559 y Fx(F)o(or)24 b(the)h(ne)o(xt)f(lemmata,)g(we)h (de\002ne)1053 3785 y Fr(~)-53 b Fu(r)1093 3800 y Ft(0)1160 3785 y Fr(:=)1315 3718 y Fu(c)1357 3733 y Fq(f)p 1301 3762 117 4 v 1301 3854 a Fu(C)1378 3819 y Ft(2)1371 3881 y Fq(f)1545 3785 y Fx(and)123 b Fr(^)-54 b Fu(r)1851 3800 y Ft(1)1918 3785 y Fr(:=)28 b(max)2247 3704 y Fm(\010)2305 3785 y Fu(R)2379 3800 y Fq(f)2441 3785 y Fu(;)49 b Fr(~)-53 b Fu(r)2557 3800 y Ft(0)2596 3704 y Fm(\011)2676 3785 y Fu(:)529 b Fx(\(A.14\))257 4064 y Fv(Lemma)26 b(A.2.)41 b Fn(Under)24 b(the)h(assumption)e(of)h(Hypothesis)g(1,)1029 4278 y Fo(8)k Fu(t)g Fo(\024)33 b Fr(~)-54 b Fu(r)1324 4293 y Ft(0)1391 4278 y Fr(:)1511 4239 y Ft(1)p 1511 4255 36 4 v 1511 4312 a(4)1573 4278 y Fu(c)1615 4293 y Fq(f)1677 4278 y Fu(t)1712 4242 y Ft(2)1804 4278 y Fo(\024)54 b Fu(f)1983 4293 y Fq(j)2019 4278 y Fr(\()p Fu(t)p Fr(\))f Fo(\024)2323 4239 y Ft(3)p 2323 4255 V 2323 4312 a(4)2385 4278 y Fu(C)2455 4293 y Fq(f)2516 4278 y Fu(t)2551 4242 y Ft(2)2613 4278 y Fu(;)592 b Fx(\(A.15\))1029 4423 y Fo(8)28 b Fu(t)g Fo(\024)33 b Fr(~)-54 b Fu(r)1324 4438 y Ft(0)1391 4423 y Fr(:)1551 4384 y Ft(1)p 1551 4400 V 1551 4458 a(2)1613 4423 y Fu(c)1655 4438 y Fq(f)1717 4423 y Fu(t)52 b Fo(\024)i Fu(f)1994 4387 y Fs(0)1983 4448 y Fq(j)2019 4423 y Fr(\()p Fu(t)p Fr(\))f Fo(\024)2323 4384 y Ft(3)p 2323 4400 V 2323 4458 a(2)2385 4423 y Fu(C)2455 4438 y Fq(f)2516 4423 y Fu(t)23 b(;)631 b Fx(\(A.16\))1029 4578 y Fo(8)28 b Fu(t)g Fo(\025)33 b Fr(~)-54 b Fu(r)1324 4593 y Ft(0)1391 4578 y Fr(:)266 b Fu(f)1732 4593 y Fq(j)1768 4578 y Fr(\()p Fu(t)p Fr(\))53 b Fo(\025)2072 4539 y Ft(1)p 2072 4555 V 2072 4612 a(4)2134 4578 y Fu(c)2176 4542 y Ft(3)2176 4604 y Fq(f)2238 4578 y Fu(C)2315 4537 y Fs(\000)p Ft(4)2308 4606 y Fq(f)2431 4578 y Fu(;)774 b Fx(\(A.17\))1029 4736 y Fo(8)28 b Fu(t)g Fo(\025)33 b Fr(^)-54 b Fu(r)1324 4751 y Ft(1)1391 4736 y Fr(:)266 b Fu(f)1743 4700 y Fs(0)1732 4761 y Fq(j)1768 4736 y Fr(\()p Fu(t)p Fr(\))53 b Fo(\025)2072 4697 y Ft(1)p 2072 4713 V 2072 4770 a(2)2134 4736 y Fu(c)2176 4700 y Ft(2)2176 4762 y Fq(f)2238 4736 y Fu(C)2315 4695 y Fs(\000)p Ft(2)2308 4764 y Fq(f)2431 4736 y Fu(;)774 b Fx(\(A.18\))257 4949 y Fn(for)25 b(all)f Fu(j)33 b Fo(2)28 b Fr(\003)p Fn(.)257 5147 y(Pr)l(oof:)47 b Fx(Estimate)32 b(\(A.16\))i(follo)n(ws)d(from)j(Lemma)e(A.1,)k(taking)c(into)h (account)g(that)g Fu(t)44 b Fo(\024)262 5268 y Fr(~)-54 b Fu(r)301 5283 y Ft(0)369 5268 y Fr(=)28 b Fu(c)515 5283 y Fq(f)577 5268 y Fu(C)654 5227 y Fs(\000)p Ft(2)647 5295 y Fq(f)773 5268 y Fx(implies)23 b Fu(t)29 b Fo(\024)f Fu(C)1337 5227 y Fs(\000)p Ft(1)1330 5295 y Fq(f)1431 5268 y Fx(,)d(since)g Fu(c)1753 5283 y Fq(f)1826 5268 y Fo(\024)k Fu(C)2002 5283 y Fq(f)2047 5268 y Fx(,)c(and)g(\(A.15\))g (is)f(obtained)h(from)f(\(A.16\))257 5388 y(by)h(inte)o(gration)e(with) h(respect)h(to)f Fu(t)p Fx(.)p eop %%Page: 47 48 47 47 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(47)404 573 y(Eq.)24 b(\(A.18\))h(follo)n(ws)e(from)i(\(A.16\))g (and)f(\(I.11\),)890 848 y Fu(f)949 806 y Fs(0)938 872 y Fq(j)975 848 y Fr(\()p Fu(t)p Fr(\))52 b Fo(\025)i Fu(c)1311 863 y Fq(f)1400 848 y Fr(max)1389 909 y Ft(0)p Fs(\024)p Fq(s)p Fs(\024)p Fq(t)1592 848 y Fo(f)p Fu(f)1701 806 y Fs(0)1690 872 y Fq(j)1727 848 y Fr(\()p Fu(s)p Fr(\))p Fo(g)e(\025)h Fu(c)2123 863 y Fq(f)2185 848 y Fu(f)2244 806 y Fs(0)2233 872 y Fq(j)2270 848 y Fr(\()t(~)-53 b Fu(r)2352 863 y Ft(0)2391 848 y Fr(\))52 b Fo(\025)2669 770 y Fu(c)2711 734 y Ft(2)2711 796 y Fq(f)p 2621 825 182 4 v 2621 916 a Fr(2)17 b Fu(C)2764 882 y Ft(2)2757 944 y Fq(f)2835 848 y Fu(;)370 b Fx(\(A.19\))257 1122 y(additionally)23 b(using)h(that)29 b Fr(~)-54 b Fu(r)1217 1137 y Ft(0)1284 1122 y Fo(\024)33 b Fr(^)-53 b Fu(r)1434 1137 y Ft(1)1501 1122 y Fo(\024)28 b Fu(t)p Fx(.)404 1243 y(F)o(or)22 b(the)h(deri)n(v)n(ation)e(of)i(\(A.17\),)g(we)g (remark)f(that)h Fu(f)2211 1258 y Fq(j)2270 1243 y Fx(is)f(strict)g (monotonically)f(increas-)257 1363 y(ing)k(on)f Fr([)t(^)-53 b Fu(r)605 1378 y Ft(1)645 1363 y Fu(;)17 b Fo(1)p Fr(\))p Fx(,)24 b(by)g(\(A.18\).)31 b(Ne)o(xt,)24 b(if)h Fu(t)1682 1378 y Fs(\003)1749 1363 y Fo(2)j Fr(\()t(~)-53 b Fu(r)1925 1378 y Ft(0)1964 1363 y Fu(;)21 b Fr(^)-53 b Fu(r)2052 1378 y Ft(1)2091 1363 y Fr(\))25 b Fx(is)g(a)g(critical)f(point)g(of)h Fu(f)3002 1378 y Fq(j)3039 1363 y Fx(,)f(then)1202 1644 y Fu(f)1250 1659 y Fq(j)1287 1644 y Fr(\()p Fu(t)1360 1659 y Fs(\003)1399 1644 y Fr(\))53 b Fo(\025)g Fu(c)1662 1659 y Fq(f)1760 1644 y Fo(\025)1914 1566 y Fr(3)p Fu(c)2005 1530 y Ft(2)2005 1592 y Fq(f)p 1900 1621 166 4 v 1900 1712 a Fr(4)p Fu(C)2026 1678 y Ft(3)2019 1740 y Fq(f)2127 1644 y Fo(\025)h Fu(f)2306 1659 y Fq(j)2342 1644 y Fr(\()t(~)-53 b Fu(r)2424 1659 y Ft(0)2463 1644 y Fr(\))23 b Fu(;)681 b Fx(\(A.20\))257 1918 y(using)24 b(\(I.9\),)h(\(A.15\),)g(and)g Fu(C)1274 1933 y Fq(f)1346 1918 y Fo(\025)k Fr(1)e Fo(\025)h Fu(c)1675 1933 y Fq(f)1721 1918 y Fx(.)i(Thus)1218 2199 y Fr(min)1226 2260 y Fq(t)p Fs(\025)s Ft(~)-38 b Fq(r)1338 2269 y Fk(0)1381 2199 y Fo(f)p Fu(f)1479 2214 y Fq(j)1515 2199 y Fr(\()p Fu(t)p Fr(\))p Fo(g)53 b Fr(=)f Fu(f)1905 2214 y Fq(j)1942 2199 y Fr(\()t(~)-53 b Fu(r)2024 2214 y Ft(0)2063 2199 y Fr(\))52 b Fo(\025)2341 2121 y Fu(c)2383 2085 y Ft(3)2383 2147 y Fq(f)p 2294 2176 182 4 v 2294 2267 a Fr(4)17 b Fu(C)2437 2233 y Ft(4)2430 2295 y Fq(f)2507 2199 y Fu(;)698 b Fx(\(A.21\))257 2472 y(once)25 b(more)g(emplo)o(ying) e(\(A.15\).)1980 b Fg(\003)404 2592 y Fx(F)o(or)23 b(the)g (construction)f(of)i Fu(g)1379 2607 y Fq(j)1439 2592 y Fx(and)j Fr(^)-53 b Fu(g)1653 2607 y Fq(j)1689 2592 y Fx(,)24 b(we)g(introduce)e(the)i(continuously)d(dif)n(ferentiable)257 2713 y(function)406 2985 y Fu(\015)5 b Fr(\()p Fu(t)p Fr(\))53 b(:=)781 2815 y Fm(\()1452 2861 y Ft(1)p 1435 2877 71 4 v 1435 2935 a(16)1532 2900 y Fu(c)1574 2915 y Fq(f)1636 2900 y Fu(t)1671 2864 y Ft(2)2315 2900 y Fx(if)83 b Fu(t)27 b Fo(\024)33 b Fr(~)-53 b Fu(r)2674 2864 y Ft(2)2671 2925 y(0)2729 2900 y Fr(\()t(^)g Fu(r)2811 2915 y Ft(1)2851 2900 y Fr(\))2889 2864 y Fs(\000)p Ft(1)3005 2900 y Fu(;)913 3029 y Ft(1)p 913 3045 36 4 v 913 3102 a(8)975 3068 y Fu(c)1017 3083 y Fq(f)1083 3068 y Fr(~)g Fu(r)1126 3032 y Ft(2)1123 3092 y(0)1182 3068 y Fr(\()t(^)g Fu(r)1264 3083 y Ft(1)1303 3068 y Fr(\))1341 3032 y Fs(\000)p Ft(1)1452 3068 y Fu(t)50 b Fo(\000)1692 3029 y Ft(1)p 1674 3045 71 4 v 1674 3102 a(16)1771 3068 y Fu(c)1813 3083 y Fq(f)1880 3068 y Fr(~)-54 b Fu(r)1922 3032 y Ft(4)1919 3092 y(0)1978 3068 y Fr(\()t(^)h Fu(r)2060 3083 y Ft(1)2099 3068 y Fr(\))2137 3032 y Fs(\000)p Ft(2)2315 3068 y Fx(if)83 b Fu(t)27 b Fo(\025)33 b Fr(~)-53 b Fu(r)2674 3032 y Ft(2)2671 3092 y(0)2729 3068 y Fr(\()t(^)g Fu(r)2811 3083 y Ft(1)2851 3068 y Fr(\))2889 3032 y Fs(\000)p Ft(1)3005 3068 y Fu(:)3232 2985 y Fx(\(A.22\))257 3274 y(Note)25 b(that,)f(by)h(Lemma)f(A.2,)1376 3500 y Fu(\015)5 b Fr(\()p Fu(t)p Fr(\))50 b Fo(\024)1730 3447 y Fq(c)1761 3459 y Fj(f)1812 3447 y Fq(t)1837 3424 y Fk(2)p 1730 3477 143 4 v 1784 3534 a Ft(8)1932 3500 y Fo(\024)2070 3461 y Ft(1)p 2070 3477 36 4 v 2070 3534 a(2)2115 3500 y Fu(f)2163 3515 y Fq(j)2199 3500 y Fr(\()p Fu(t)p Fr(\))23 b Fu(;)872 b Fx(\(A.23\))1390 3660 y Fu(\015)1446 3624 y Fs(0)1469 3660 y Fr(\()p Fu(t)p Fr(\))50 b Fo(\024)1768 3608 y Fq(c)1799 3620 y Fj(f)1850 3608 y Fq(t)p 1768 3638 108 4 v 1804 3695 a Ft(4)1935 3660 y Fo(\024)2072 3621 y Ft(1)p 2072 3638 36 4 v 2072 3695 a(2)2118 3660 y Fu(f)2177 3624 y Fs(0)2166 3685 y Fq(j)2202 3660 y Fr(\()p Fu(t)p Fr(\))22 b Fu(;)870 b Fx(\(A.24\))257 3869 y(for)25 b(all)g Fr(0)i Fo(\024)i Fu(t)e Fo(\024)33 b Fr(~)-53 b Fu(r)917 3884 y Ft(0)956 3869 y Fx(,)999 4112 y Fu(f)1047 4127 y Fq(j)1083 4112 y Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)h Fu(\015)5 b Fr(\()p Fu(t)p Fr(\))53 b Fo(\025)g Fu(f)1714 4127 y Fq(j)1750 4112 y Fr(\()t(~)-53 b Fu(r)1832 4127 y Ft(0)1872 4112 y Fr(\))22 b Fo(\000)g Fu(\015)5 b Fr(\()t(^)-53 b Fu(r)2169 4127 y Ft(1)2209 4112 y Fr(\))52 b Fo(\025)2439 4045 y Fr(1)p 2439 4089 49 4 v 2439 4180 a(8)2515 4112 y Fu(c)2557 4127 y Fq(f)2623 4112 y Fr(~)-53 b Fu(r)2666 4071 y Ft(2)2663 4137 y(0)2727 4112 y Fu(;)478 b Fx(\(A.25\))257 4346 y(for)30 b Fr(~)-54 b Fu(r)442 4361 y Ft(0)510 4346 y Fo(\024)28 b Fu(t)g Fo(\024)k Fr(^)-53 b Fu(r)827 4361 y Ft(1)866 4346 y Fx(,)25 b(and)985 4614 y Fu(\015)1041 4573 y Fs(0)1065 4614 y Fr(\()p Fu(t)p Fr(\))52 b(=)1367 4547 y Fu(c)1409 4562 y Fq(f)1475 4547 y Fr(~)-53 b Fu(r)1518 4511 y Ft(2)1515 4571 y(0)p 1367 4591 191 4 v 1388 4683 a Fr(4)20 b(^)-52 b Fu(r)1498 4698 y Ft(1)1619 4614 y Fo(\024)1760 4547 y Fu(c)1802 4562 y Fq(f)1868 4547 y Fr(~)e Fu(r)1907 4562 y Ft(0)p 1760 4591 188 4 v 1829 4683 a Fr(4)2009 4614 y(=)2195 4537 y Fu(c)2237 4500 y Ft(2)2237 4562 y Fq(f)p 2148 4591 182 4 v 2148 4683 a Fr(8)17 b Fu(C)2291 4648 y Ft(2)2284 4710 y Fq(f)2392 4614 y Fo(\024)53 b Fu(f)2581 4573 y Fs(0)2570 4639 y Fq(j)2607 4614 y Fr(\()p Fu(t)p Fr(\))22 b Fu(;)465 b Fx(\(A.26\))257 4889 y(pro)o(vided)24 b Fu(t)k Fo(\025)k Fr(^)-53 b Fu(r)847 4904 y Ft(1)886 4889 y Fx(.)31 b(From)25 b(\(A.23\)\226\(A.26\))f(we)h(deduce)g(that)1111 5098 y Fo(8)j Fu(t)g Fo(2)g Fp(R)1417 5056 y Ft(+)1510 5098 y Fr(:)95 b(0)52 b Fu(<)h(\015)5 b Fr(\()p Fu(t)p Fr(\))52 b Fu(<)h(f)2258 5113 y Fq(j)2294 5098 y Fr(\()p Fu(t)p Fr(\))23 b Fu(;)777 b Fx(\(A.27\))731 5243 y Fo(8)28 b Fu(t)g Fo(2)g Fp(R)1037 5202 y Ft(+)1124 5243 y Fo(n)22 b Fr(\()t(~)-53 b Fu(r)1278 5258 y Ft(0)1317 5243 y Fu(;)21 b Fr(^)-53 b Fu(r)1405 5258 y Ft(1)1444 5243 y Fr(\))28 b(:)83 b(0)52 b Fu(<)h(\015)1906 5207 y Fs(0)1929 5243 y Fr(\()p Fu(t)p Fr(\))g Fu(<)f(f)2280 5207 y Fs(0)2269 5267 y Fq(j)2306 5243 y Fr(\()p Fu(t)p Fr(\))22 b Fu(:)766 b Fx(\(A.28\))3232 5388 y(\(A.29\))p eop %%Page: 48 49 48 48 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(48)257 573 y(W)-8 b(e)26 b(de\002ne)1086 693 y Fr(^)-53 b Fu(\015)1133 708 y Fq(j)1169 693 y Fr(\()p Fu(t)p Fr(\))53 b(:=)f(min)1513 754 y Fq(s)p Fs(\025)p Fq(t)1668 613 y Fm(\010)1726 693 y Fu(f)1774 708 y Fq(j)1810 693 y Fr(\()p Fu(s)p Fr(\))22 b Fo(\000)h Fu(\015)5 b Fr(\()p Fu(s)p Fr(\))2232 613 y Fm(\011)2334 693 y Fr(+)44 b Fu(\015)5 b Fr(\()p Fu(t)p Fr(\))22 b Fu(;)562 b Fx(\(A.30\))257 905 y(which)29 b(is)g(reminiscent)f(of)i(the)f(Le)o(gendre)g(transform) g(of)g Fu(f)2365 920 y Fq(j)2402 905 y Fx(.)44 b(W)-8 b(e)30 b(no)n(w)f(sho)n(w)f(that)k Fr(^)-52 b Fu(\015)3297 920 y Fq(j)3362 905 y Fx(has)257 1026 y(all)25 b(the)g(desired)f (properties)g(of)29 b Fr(^)-53 b Fu(g)1419 1041 y Fq(j)1481 1026 y Fx(e)o(xcept)24 b(for)h(being)f(twice)h(continuously)e(dif)n (ferentiable.)257 1226 y Fv(Lemma)j(A.3.)41 b Fn(Assume)24 b(Hypothesis)g(1.)30 b(Then)f Fr(^)-53 b Fu(\015)1999 1241 y Fq(j)2036 1226 y Fn(,)24 b(de\002ned)h(as)g(in)f(\(A.30\),)g (ful\002lls)1389 1443 y Fu(\015)5 b Fr(\()p Fu(t)p Fr(\))52 b Fo(\024)57 b Fr(^)-53 b Fu(\015)1789 1458 y Fq(j)1825 1443 y Fr(\()p Fu(t)p Fr(\))53 b Fo(\024)g Fu(f)2167 1458 y Fq(j)2204 1443 y Fr(\()p Fu(t)p Fr(\))22 b Fu(;)868 b Fx(\(A.31\))257 1660 y Fn(for)27 b(all)f Fu(t)32 b Fo(2)h Fp(R)765 1619 y Ft(+)765 1684 y(0)830 1660 y Fn(.)38 b(Mor)l(eo)o(ver)-11 b(,)31 b Fr(^)-53 b Fu(\015)1373 1675 y Fq(j)1437 1660 y Fn(is)26 b(left)h(and)g(right)f(dif)n(fer)l (entiable)f(on)i Fp(R)2834 1624 y Ft(+)2899 1660 y Fn(,)h(i.e)o(.,)f (the)g(limits)257 1780 y Fu(@)308 1795 y Fs(\006)372 1780 y Fr(^)-53 b Fu(\015)419 1795 y Fq(j)455 1780 y Fr(\()p Fu(t)p Fr(\))28 b(:=)f(lim)860 1795 y Fq(h)p Fs(!)p Ft(0)p Fs(\006)1082 1780 y Fu(h)1138 1744 y Fs(\000)p Ft(1)1233 1780 y Fr([)t(^)-53 b Fu(\015)1311 1795 y Fq(j)1347 1780 y Fr(\()p Fu(t)19 b Fr(+)g Fu(h)p Fr(\))g Fo(\000)24 b Fr(^)-53 b Fu(\015)1795 1795 y Fq(j)1831 1780 y Fr(\()p Fu(t)p Fr(\)])25 b Fn(e)n(xist,)e(and)h(its)f(left)h(and)f(right)g (derivatives)257 1901 y(obe)m(y)785 2021 y Fu(\015)841 1980 y Fs(0)864 2021 y Fr(\()p Fu(t)p Fr(\))53 b Fo(\024)g Fu(@)1209 2036 y Fs(\006)1272 2021 y Fr(^)-53 b Fu(\015)1319 2036 y Fq(j)1355 2021 y Fr(\()p Fu(t)p Fr(\))53 b Fo(\024)g Fu(\015)1705 1980 y Fs(0)1728 2021 y Fr(\()p Fu(t)p Fr(\))45 b(+)f(max)p Fo(f)p Fu(f)2295 1980 y Fs(0)2284 2046 y Fq(j)2320 2021 y Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)h Fu(\015)2609 1980 y Fs(0)2632 2021 y Fr(\()p Fu(t)p Fr(\))17 b Fu(;)33 b Fr(0)p Fo(g)22 b Fu(:)264 b Fx(\(A.32\))257 2194 y Fn(for)25 b(all)f Fu(t)k Fo(2)g Fp(R)752 2158 y Ft(+)817 2194 y Fn(.)257 2395 y(Pr)l(oof:)45 b Fx(It)32 b(suf)n(\002ces)h(to)f(pro)o(v)o(e)f(that)h Fu(n)p Fr(\()p Fu(t)p Fr(\))42 b(:=)f(min)2053 2410 y Fq(s)p Fs(\025)p Fq(t)2170 2395 y Fo(f)p Fu(m)p Fr(\()p Fu(s)p Fr(\))p Fo(g)h(\024)g Fu(m)p Fr(\()p Fu(t)p Fr(\))33 b Fx(is)e(left)i(and)f (right)257 2515 y(dif)n(ferentiable)27 b(and)g(that)f Fr(0)32 b Fo(\024)g Fu(@)1407 2530 y Fs(\006)1467 2515 y Fu(n)p Fr(\()p Fu(t)p Fr(\))g Fo(\024)g Fu(m)1862 2479 y Fs(0)1886 2515 y Fr(\()p Fu(t)p Fr(\))p Fx(,)c(where)f Fu(m)32 b Fr(:=)g Fu(f)2620 2530 y Fq(j)2681 2515 y Fo(\000)24 b Fu(\015)5 b Fx(.)37 b(W)-8 b(e)28 b(distinguish)257 2635 y(se)n(v)o(eral)c(re)o(gions)g(of)h Fp(R)1051 2599 y Ft(+)1116 2635 y Fx(.)378 2836 y(1.)48 b(Since)28 b Fu(m)g Fx(is)e(continuous,)g(so)h(is)g Fu(n)p Fx(,)h(and)f Fo(A)2018 2851 y Ft(1)2090 2836 y Fr(:=)32 b Fo(f)p Fu(t)p Fo(j)p Fu(n)p Fr(\()p Fu(t)p Fr(\))g Fu(<)g(m)p Fr(\()p Fu(t)p Fr(\))p Fo(g)c Fx(is)e(open.)38 b(Thus,)501 2956 y(if)h Fu(t)53 b Fo(2)h(A)889 2971 y Ft(1)928 2956 y Fx(,)42 b(then)c Fu(n)h Fx(is)g(constant)e(in)i(a)g(neighborhood)e(of)h Fu(t)h Fx(and,)j(in)d(particular)l(,)501 3077 y(dif)n(ferentiable,)24 b(with)g Fu(@)1338 3092 y Ft(+)1398 3077 y Fu(n)p Fr(\()p Fu(t)p Fr(\))k(=)f Fu(@)1749 3092 y Fs(\000)1809 3077 y Fu(n)p Fr(\()p Fu(t)p Fr(\))h(=)f(0)p Fx(.)378 3279 y(2.)48 b(Consider)27 b(the)g(complement,)e Fo(A)1661 3294 y Ft(2)1732 3279 y Fr(:=)31 b Fo(f)p Fu(t)p Fo(j)p Fu(n)p Fr(\()p Fu(t)p Fr(\))h(=)f Fu(m)p Fr(\()p Fu(t)p Fr(\))p Fo(g)p Fx(.)37 b(Let)27 b Fu(t)k Fo(2)h(A)2999 3294 y Ft(2)3065 3279 y Fx(and)27 b Fu(\016)35 b(>)d Fr(0)p Fx(.)501 3399 y(Then)662 3616 y Fu(n)p Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)h Fu(n)p Fr(\()p Fu(t)f Fo(\000)h Fu(\016)t Fr(\))83 b(=)g Fu(m)p Fr(\()p Fu(t)p Fr(\))50 b Fo(\000)g Fr(min)2085 3536 y Fm(\002)2127 3616 y Fu(m)p Fr(\()p Fu(t)p Fr(\))17 b Fu(;)101 b Fr(min)2409 3679 y Fq(t)p Fs(\000)p Fq(\016)r Fs(\024)p Fq(s)p Fs(\024)p Fq(t)2691 3616 y Fo(f)p Fu(m)p Fr(\()p Fu(s)p Fr(\))p Fo(g)2998 3536 y Fm(\003)3232 3616 y Fx(\(A.33\))1374 3818 y Fr(=)83 b(max)1731 3737 y Fm(\002)1772 3818 y Fr(0)17 b Fu(;)92 b Fr(max)1906 3881 y Fq(t)p Fs(\000)p Fq(\016)r Fs(\024)p Fq(s)p Fs(\024)p Fq(t)2188 3818 y Fo(f)p Fu(m)p Fr(\()p Fu(t)p Fr(\))23 b Fo(\000)f Fu(m)p Fr(\()p Fu(s)p Fr(\))p Fo(g)2813 3737 y Fm(\003)1374 4020 y Fr(=)83 b(max)1731 3939 y Fm(\002)1772 4020 y Fr(0)17 b Fu(;)92 b Fr(max)1906 4082 y Fq(t)p Fs(\000)p Fq(\016)r Fs(\024)p Fq(s)p Fs(\024)p Fq(t)2188 4020 y Fo(f)p Fu(m)2323 3979 y Fs(0)2347 4020 y Fr(\()p Fu(t)p Fr(\))17 b(\()p Fu(t)22 b Fo(\000)g Fu(s)p Fr(\))45 b(+)f Fu(o)p Fr(\()p Fu(t)22 b Fo(\000)h Fu(s)p Fr(\))p Fo(g)3294 3939 y Fm(\003)1374 4222 y Fr(=)83 b(max)1731 4141 y Fm(\002)1772 4222 y Fr(0)17 b Fu(;)41 b(\016)21 b(m)2055 4180 y Fs(0)2079 4222 y Fr(\()p Fu(t)p Fr(\))2190 4141 y Fm(\003)2281 4222 y Fr(+)50 b Fu(o)p Fr(\()p Fu(\016)t Fr(\))22 b Fu(:)501 4438 y Fx(Hence,)j Fu(@)856 4453 y Fs(\000)916 4438 y Fu(n)j Fr(=)f(max)p Fo(f)p Fu(m)1422 4402 y Fs(0)1445 4438 y Fu(;)17 b Fr(0)p Fo(g)25 b Fx(on)f Fo(A)1817 4453 y Ft(2)1856 4438 y Fx(.)328 4641 y(2.1)48 b(Observ)o(e)25 b(that)f Fo(A)1111 4656 y Ft(2)1172 4641 y Fo(\\)f(f)p Fu(t)p Fo(j)p Fu(m)1459 4605 y Fs(0)1482 4641 y Fr(\()p Fu(t)p Fr(\))28 b Fu(<)f Fr(0)p Fo(g)h Fr(=)f Fo(;)p Fx(.)328 4843 y(2.2)48 b(Let)25 b Fu(t)j Fo(2)g(A)896 4858 y Ft(2)957 4843 y Fo(\\)23 b(f)p Fu(t)p Fo(j)p Fu(m)1244 4807 y Fs(0)1267 4843 y Fr(\()p Fu(t)p Fr(\))28 b(=)f(0)p Fo(g)e Fx(and)g Fu(t)1837 4807 y Fs(0)1888 4843 y Fu(>)i(t)p Fx(.)k(Then)545 5115 y Fr(0)52 b Fo(\024)786 5047 y Fu(n)p Fr(\()p Fu(t)917 5011 y Fs(0)941 5047 y Fr(\))22 b Fo(\000)h Fu(n)p Fr(\()p Fu(t)p Fr(\))p 786 5092 484 4 v 921 5183 a Fu(t)956 5154 y Fs(0)1001 5183 y Fo(\000)g Fu(t)1333 5115 y Fr(=)1471 5047 y(min)1634 5062 y Fq(s)p Fs(\025)p Fq(t)1747 5043 y Fc(0)1773 5047 y Fo(f)p Fu(m)p Fr(\()p Fu(s)p Fr(\))p Fo(g)f(\000)h Fu(m)p Fr(\()p Fu(t)p Fr(\))p 1471 5092 928 4 v 1827 5183 a Fu(t)1862 5154 y Fs(0)1908 5183 y Fo(\000)f Fu(t)2461 5115 y Fo(\024)2601 5047 y Fu(m)p Fr(\()p Fu(t)2759 5011 y Fs(0)2783 5047 y Fr(\))g Fo(\000)g Fu(m)p Fr(\()p Fu(t)p Fr(\))p 2601 5092 538 4 v 2762 5183 a Fu(t)2797 5154 y Fs(0)2843 5183 y Fo(\000)g Fu(t)3201 5115 y Fo(!)52 b Fr(0)22 b Fu(;)3232 5268 y Fx(\(A.34\))501 5388 y(as)j Fu(t)644 5352 y Fs(0)695 5388 y Fo(&)j Fu(t)p Fx(,)d(and)f(therefore)i Fu(@)1512 5403 y Ft(+)1571 5388 y Fu(n)p Fr(\()p Fu(t)p Fr(\))i(=)g(0)p Fx(.)p eop %%Page: 49 50 49 49 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(49)253 573 y(2.3.1)48 b(Let)33 b Fu(t)42 b Fo(2)g(A)932 588 y Ft(2)999 573 y Fo(\\)28 b(f)p Fu(t)p Fo(j)p Fu(m)1291 537 y Fs(0)1314 573 y Fr(\()p Fu(t)p Fr(\))42 b Fu(>)g Fr(0)p Fo(g)32 b Fx(and)h(suppose)e(that)h Fu(m)p Fr(\()p Fu(s)p Fr(\))42 b Fu(>)g(m)p Fr(\()p Fu(t)p Fr(\))p Fx(,)35 b(for)d(all)h Fu(s)41 b(>)501 693 y(t)p Fx(.)68 b(Since)37 b Fu(m)972 657 y Fs(0)996 693 y Fr(\()p Fu(t)p Fr(\))50 b Fu(>)g Fr(0)36 b Fx(and)h Fu(m)1634 657 y Fs(0)1708 693 y Fu(>)50 b Fr(0)37 b Fx(at)g(in\002nity)-6 b(,)38 b Fu(m)p Fr(\()p Fu(t)p Fr(\))50 b Fu(<)g(m)2832 708 y Fs(\003)2872 693 y Fx(,)40 b(where)d Fu(m)3302 708 y Fs(\003)3392 693 y Fr(:=)501 814 y(min)o Fo(f)p Fu(m)p Fr(\()p Fu(s)p Fr(\))17 b Fo(j)31 b Fu(m)1082 777 y Fs(0)1106 814 y Fr(\()p Fu(s)p Fr(\))h(=)h(0)p Fu(;)49 b(s)32 b(>)g(t)p Fo(g)p Fx(,)c(if)g(such)f(critical)g(points)f(e)o(xist,)h(and)g Fu(m)3188 829 y Fs(\003)3260 814 y Fr(:=)33 b Fo(1)501 934 y Fx(otherwise.)d(But)25 b(then)g Fu(n)j Fo(\021)g Fu(m)d Fx(in)f(a)i(neighborhood)d(of)i Fu(t)p Fx(,)g(and)g Fu(@)2774 949 y Ft(+)2833 934 y Fu(n)p Fr(\()p Fu(t)p Fr(\))j(=)g Fu(m)3219 898 y Fs(0)3242 934 y Fr(\()p Fu(t)p Fr(\))p Fx(.)253 1131 y(2.3.2)48 b(Let)35 b Fu(t)48 b Fo(2)f(A)945 1146 y Ft(2)1014 1131 y Fo(\\)31 b(f)p Fu(t)p Fo(j)p Fu(m)1309 1095 y Fs(0)1332 1131 y Fr(\()p Fu(t)p Fr(\))47 b Fu(>)g Fr(0)p Fo(g)35 b Fx(and)h(suppose)e(that)h Fu(m)p Fr(\()p Fu(t)2626 1146 y Fs(\003)2666 1131 y Fr(\))47 b(=)g Fu(m)p Fr(\()p Fu(t)p Fr(\))p Fx(,)38 b(for)e(some)501 1251 y Fu(t)536 1266 y Fs(\003)604 1251 y Fu(>)27 b(t)p Fx(.)k(Then)25 b Fu(n)j Fo(\021)g Fu(m)p Fr(\()p Fu(t)p Fr(\))d Fx(on)g Fr([)p Fu(t;)17 b(t)1706 1266 y Fs(\003)1745 1251 y Fr(])p Fx(,)25 b(and)g Fu(@)2042 1266 y Ft(+)2102 1251 y Fu(n)p Fr(\()p Fu(t)p Fr(\))j(=)f(0)p Fx(.)942 b Fg(\003)257 1436 y Fx(Ha)n(ving)19 b(established)f(the)h(left)g(and)g (right)f(dif)n(ferentiability)f(of)23 b Fr(^)-53 b Fu(\015)2488 1451 y Fq(j)2525 1436 y Fx(,)20 b(we)f(deri)n(v)o(e)f(the)h(follo)n (wing)257 1556 y(bounds)24 b(on)k Fr(^)-52 b Fu(\015)746 1571 y Fq(j)782 1556 y Fx(.)257 1741 y Fv(Lemma)26 b(A.4.)41 b Fn(Assume)24 b(Hypothesis)g(1,)g(and)h(set)799 2004 y Fr(^)-53 b Fu(r)839 2019 y Ft(0)931 2004 y Fr(:=)1144 1926 y Fu(c)1186 1890 y Ft(3)1186 1952 y Fq(f)p 1097 1981 182 4 v 1097 2072 a Fr(3)17 b Fu(C)1240 2038 y Ft(5)1233 2100 y Fq(f)1525 2004 y Fn(and)242 b Fr(^)-56 b Fu(q)1953 2019 y Ft(max)2142 2004 y Fr(:=)52 b(2)33 b(exp)2545 1923 y Fm(\002)2586 2004 y Fr(2)17 b Fu(C)2722 2019 y Fq(f)2788 2004 y Fr(^)-54 b Fu(r)2827 2019 y Ft(1)2867 1923 y Fm(\003)2930 2004 y Fu(:)275 b Fx(\(A.35\))257 2267 y Fn(Then)1126 2464 y Fo(8)28 b Fu(t)g Fo(\024)33 b Fr(^)-54 b Fu(r)1421 2479 y Ft(0)1488 2464 y Fr(:)354 b(^)-53 b Fu(\015)1916 2479 y Fq(j)1952 2464 y Fr(\()p Fu(t)p Fr(\))53 b(=)g Fu(f)2293 2479 y Fq(j)2329 2464 y Fr(\()p Fu(t)p Fr(\))22 b Fu(;)743 b Fx(\(A.36\))913 2657 y Fo(8)33 b Fr(^)-54 b Fu(r)1040 2672 y Ft(0)1107 2657 y Fu(<)28 b(t)g(<)k Fr(^)-54 b Fu(r)1421 2672 y Ft(1)1488 2657 y Fr(:)84 b(0)52 b Fo(\024)h Fu(f)1878 2672 y Fq(j)1915 2657 y Fr(\()p Fu(t)p Fr(\))22 b Fo(\000)k Fr(^)-53 b Fu(\015)2198 2672 y Fq(j)2235 2657 y Fr(\()p Fu(t)p Fr(\))52 b Fo(\024)60 b 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Ft(2)1255 4943 y(0)1350 4919 y Fo(\025)1490 4851 y Fr(3)p 1490 4896 V 1490 4987 a(4)1565 4919 y Fu(C)1635 4934 y Fq(f)1697 4919 y Fr(\()t(^)h Fu(r)1779 4934 y Ft(0)1818 4919 y Fr(\))1856 4877 y Ft(2)1948 4919 y Fo(\025)53 b Fu(f)2126 4934 y Fq(j)2163 4919 y Fr(\()t(^)-53 b Fu(r)2245 4934 y Ft(0)2284 4919 y Fr(\))53 b Fo(\025)g Fu(f)2553 4934 y Fq(j)2590 4919 y Fr(\()t(^)-53 b Fu(r)2672 4934 y Ft(0)2711 4919 y Fr(\))22 b Fo(\000)h Fu(\015)5 b Fr(\()t(^)-53 b Fu(r)3009 4934 y Ft(0)3048 4919 y Fr(\))22 b Fu(:)97 b Fx(\(A.42\))257 5147 y(Moreo)o(v)o(er)l(,)21 b Fu(f)738 5162 y Fq(j)783 5147 y Fo(\000)9 b Fu(\015)27 b Fx(is)20 b(strict)h(monotonically)e(increasing)i(on)f Fr(\(0)p Fu(;)h Fr(^)-53 b Fu(r)2578 5162 y Ft(0)2617 5147 y Fr(\))28 b Fo(\022)g Fr(\(0)p Fu(;)21 b Fr(~)-53 b Fu(r)2963 5162 y Ft(0)3002 5147 y Fr(\))p Fx(,)22 b(by)f(\(A.28\).)257 5268 y(Thus,)i(for)f Fu(t)28 b Fo(\024)33 b Fr(^)-54 b Fu(r)855 5283 y Ft(0)895 5268 y Fx(,)23 b(we)g(ha)n(v)o(e)f Fr(min)1451 5283 y Fq(s)p 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810 y Ft(1)2451 795 y Fr(\))2489 754 y Fs(\000)p Ft(1)2605 795 y Fu(;)45 b(t)2712 715 y Fm(\011)2792 795 y Fu(:)413 b Fx(\(A.43\))404 1008 y(F)o(or)35 b(the)h(deri)n(v)n(ation)e(of)i(\(A.40\))f(we)i (distinguish)c(the)i(re)o(gions)g Fr(\(0)p Fu(;)21 b Fr(^)-53 b Fu(r)2872 1023 y Ft(0)2911 1008 y Fr(])p Fx(,)38 b Fr(\()t(^)-53 b Fu(r)3083 1023 y Ft(0)3123 1008 y Fu(;)21 b Fr(^)-53 b Fu(r)3211 1023 y Ft(1)3250 1008 y Fr(\))p Fx(,)38 b(and)257 1128 y Fr([)t(^)-53 b Fu(r)328 1143 y Ft(1)368 1128 y Fu(;)17 b Fo(1)p Fr(\))p Fx(.)257 1248 y(1\))26 b(On)f Fr(\(0)p Fu(;)c Fr(^)-53 b Fu(r)688 1263 y Ft(0)727 1248 y Fr(])30 b Fo(\022)f Fr(\(0)p Fu(;)21 b Fr(~)-53 b Fu(r)1065 1263 y Ft(0)1104 1248 y Fr(])p Fx(,)26 b(the)f(function)g Fu(f)1735 1263 y Fq(j)1797 1248 y Fx(is)h(con)l(v)o(e)o(x,)e(according)i(to)f(Lemma)g(A.1,)g(and)h (it)257 1369 y(agrees)g(with)h Fr(^)-52 b Fu(\015)790 1384 y Fq(j)826 1369 y Fx(.)31 b(Hence)25 b(\(A.40\))g(holds)f(true)g (on)h Fr(\(0)p Fu(;)c Fr(^)-53 b Fu(r)2169 1384 y Ft(0)2208 1369 y Fr(])25 b Fx(\(with)f(constant)i Fr(^)-51 b Fu(c)2894 1384 y Fq(g)2962 1369 y Fr(=)27 b(1)p Fx(\).)257 1489 y(2\))32 b(On)f Fr(\()t(^)-53 b Fu(r)607 1504 y Ft(0)646 1489 y Fu(;)21 b Fr(^)-53 b Fu(r)734 1504 y Ft(1)773 1489 y Fr(\))p Fx(,)33 b(Eq.)d(\(A.40\))i(follo)n(ws)d(from)i(the)g(f)o (act)h(that)e Fu(@)2446 1504 y Fq(\033)2497 1489 y Fr(^)-52 b Fu(\015)2545 1504 y Fq(j)2620 1489 y Fu(>)39 b Fr(0)31 b Fx(is)g(bounded)g(a)o(w)o(ay)257 1609 y(from)25 b(zero,)g(by)g (\(A.39\),)f(and)h(bounded)f(abo)o(v)o(e,)g(by)g(\(A.32\).)257 1730 y(3\))g(On)f Fr([)t(^)-53 b Fu(r)580 1745 y Ft(1)619 1730 y Fu(;)17 b Fo(1)p Fr(\))p Fx(,)23 b(we)h(ha)n(v)o(e)j Fr(^)-53 b Fu(\015)1248 1745 y Fq(j)1312 1730 y Fo(\021)28 b Fu(f)1465 1745 y Fq(j)1502 1730 y Fx(,)23 b(by)g(\(A.38\).)30 b(Then)24 b(Eq.)f(\(A.40\))g(follo)n(ws)f(from)h(the)g(f)o(act)257 1850 y(that)i Fu(f)480 1865 y Fq(j)541 1850 y Fx(obe)o(ys)f(\(A.4\))h (and)g(\(A.32\).)1926 b Fg(\003)404 1971 y Fx(W)-8 b(e)25 b(turn)f(to)h(the)f(construction)f(of)i(a)g(Lipshitz)f(continuous)f (function)h Fu(\015)2909 1986 y Fq(j)2970 1971 y Fx(which)g(has)h(all) 257 2091 y(the)f(properties)g(desired)g(from)g Fu(g)1403 2106 y Fq(j)1439 2091 y Fx(,)g(e)o(xcept)g(for)h(being)e(twice)h (continuously)e(dif)n(ferentiable.)257 2211 y(W)-8 b(e)26 b(\002rst)f(introduce)f(tw)o(o)g(uni)n(v)o(ersal)f(constants,)544 2451 y Fr(\024)-53 b Fu(r)584 2466 y Ft(0)676 2451 y Fr(:=)52 b(2)17 b Fu(R)971 2466 y Fq(f)1069 2451 y Fo(\025)53 b Fr(2)21 b(^)-53 b Fu(r)1309 2466 y Ft(1)1584 2451 y Fx(and)241 b Fr(\024)-54 b Fu(r)2008 2466 y Ft(1)2100 2451 y Fr(:=)57 b(\024)-53 b Fu(r)2300 2466 y Ft(0)2383 2451 y Fr(+)2514 2374 y(4)17 b Fu(C)2657 2337 y Ft(2)2650 2399 y Fq(f)2719 2374 y Fr(^)-56 b Fu(q)2755 2389 y Ft(max)p 2514 2428 378 4 v 2658 2520 a Fu(c)2700 2485 y Ft(2)2700 2547 y Fq(f)2923 2451 y Fu(;)282 b Fx(\(A.44\))257 2697 y(and)25 b(we)g(de\002ne)30 b Fr(\024)-53 b Fu(r)880 2712 y Ft(1)p Fq(;j)999 2697 y Fu(>)32 b 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3657 y Fr(\()p Fu(t)p Fr(\))53 b(:=)924 3423 y Fm(8)924 3513 y(>)924 3543 y(<)924 3722 y(>)924 3752 y(:)1494 3489 y Fr(^)-53 b Fu(\015)1541 3504 y Fq(j)1577 3489 y Fr(\()p Fu(t)p Fr(\))520 b Fx(if)126 b Fr(0)27 b Fo(\024)i Fu(t)e Fo(\024)33 b Fr(\024)-53 b Fu(r)2789 3504 y Ft(0)2850 3489 y Fu(;)1054 3657 y(f)1102 3672 y Fq(j)1139 3657 y Fr(\()t(\024)g Fu(r)1221 3672 y Ft(0)1260 3657 y Fr(\))44 b(+)1472 3618 y Ft(1)p 1472 3634 36 4 v 1472 3691 a(4)1518 3657 y Fu(c)1560 3621 y Ft(2)1560 3683 y Fq(f)1622 3657 y Fu(C)1699 3616 y Fs(\000)p Ft(2)1692 3685 y Fq(f)1793 3576 y Fm(\000)1839 3657 y Fu(t)22 b Fo(\000)27 b Fr(\024)-53 b Fu(r)2040 3672 y Ft(0)2079 3576 y Fm(\001)2208 3657 y Fx(if)87 b Fr(\024)-53 b Fu(r)2396 3672 y Ft(0)2463 3657 y Fo(\024)28 b Fu(t)g Fo(\024)k Fr(\024)-53 b Fu(r)2780 3672 y Ft(1)p Fq(;j)2893 3657 y Fu(;)1325 3824 y(f)1373 3839 y Fq(j)1409 3824 y Fr(\()p Fu(t)p Fr(\))39 b Fo(\000)47 b Fr(^)-57 b Fu(q)1718 3839 y Ft(max)2208 3824 y Fx(if)195 b Fr(\024)-53 b Fu(r)2504 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Fx(\(A.49\))p eop %%Page: 51 52 51 51 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(51)257 573 y(where)32 b Fu(")39 b(>)g Fr(0)31 b Fx(is)f(suf)n(\002ciently)g(small)g(uni)n(v)o(ersal)f(number)-5 b(.)49 b(Furthermore,)32 b(we)g(pick)e Fu(m)40 b Fo(2)257 693 y Fu(C)334 657 y Fs(1)327 718 y Ft(0)409 693 y Fr(\(\()p Fo(\000)p Fr(1)p Fu(;)17 b Fr(0\))p Fu(;)g Fp(R)851 652 y Ft(+)851 718 y(0)916 693 y Fr(\))25 b Fx(with)1181 613 y Fm(R)1228 728 y Fh(R)1297 693 y Fu(m)j Fr(=)f(1)e Fx(and)g(de\002ne)g(a)g(molli\002er)g(by)f Fu(m)2671 708 y Fq(")2708 693 y Fr(\()p Fu(t)p Fr(\))k(:=)g Fu(")3024 657 y Fs(\000)p Ft(2)3118 693 y Fu(m)p Fr(\()p Fu(t=")3371 657 y Ft(2)3410 693 y Fr(\))p Fx(.)404 814 y(No)n(w)h(the)h(functions)i Fr(^)-52 b Fu(g)1225 829 y Fq(j)1291 814 y Fx(and)30 b Fu(g)1512 829 y Fq(j)1578 814 y Fx(are)h(obtained)e(from)34 b Fr(^)-53 b Fu(\015)2378 829 y Fq(j)2444 814 y Fx(and)30 b Fu(\015)2669 829 y Fq(j)2735 814 y Fx(by)g(smearing)f(out)h(in)257 934 y(small)j(neighborhoods)f(of)i(the)f(appropriate)h(re)o(gions)f Fr(\()t(^)-53 b Fu(r)2301 949 y Ft(0)2340 934 y Fu(;)21 b Fr(^)-53 b Fu(r)2428 949 y Ft(1)2467 934 y Fr(\))34 b Fx(and)f Fr(\()t(\024)-53 b Fu(r)2798 949 y Ft(0)2838 934 y Fu(;)21 b Fr(\024)-53 b Fu(r)2926 949 y Ft(1)2965 934 y Fr(\))p Fx(.)57 b(More)34 b(pre-)257 1054 y(cisely)-6 b(,)24 b(we)h(set)914 1274 y Fr(^)-52 b Fu(g)958 1289 y Fq(j)1077 1274 y Fr(:=)1263 1194 y Fm(\000)1308 1274 y Fr(1)22 b Fo(\000)35 b Fr(^)-61 b Fu(\037)1540 1194 y Fm(\001)1606 1274 y Fr(^)-53 b Fu(\015)1653 1289 y Fq(j)1739 1274 y Fr(+)62 b(^)-61 b Fu(\037)1926 1194 y Fm(\000)1975 1274 y Fr(^)-52 b Fu(\015)2023 1289 y Fq(j)2081 1274 y Fo(\003)22 b Fu(m)2238 1289 y Fq(")2275 1194 y Fm(\001)2343 1274 y Fu(;)862 b Fx(\(A.50\))911 1467 y Fu(g)958 1482 y Fq(j)1077 1467 y Fr(:=)1263 1386 y Fm(\000)1308 1467 y Fr(1)22 b Fo(\000)35 b Fr(^)-61 b Fu(\037)22 b Fo(\000)34 b Fr(\024)-60 b Fu(\037)1723 1386 y Fm(\001)1785 1467 y Fu(\015)1836 1482 y Fq(j)1922 1467 y Fr(+)2048 1386 y Fm(\000)2105 1467 y Fr(^)g Fu(\037)22 b Fr(+)33 b(\024)-60 b Fu(\037)2336 1386 y Fm(\001)16 b(\000)2444 1467 y Fu(\015)2495 1482 y Fq(j)2553 1467 y Fo(\003)22 b Fu(m)2710 1482 y Fq(")2747 1386 y Fm(\001)2815 1467 y Fu(:)390 b Fx(\(A.51\))257 1687 y(Ob)o(viously)-6 b(,)28 b Fr(^)-52 b Fu(g)764 1702 y Fq(j)827 1687 y Fx(and)27 b Fu(g)1045 1702 y Fq(j)1108 1687 y Fx(are)g(twice)g(continuously)d (dif)n(ferentiable.)36 b(Since)27 b Fr(supp)18 b Fu(m)3187 1702 y Fq(")3255 1687 y Fo(\022)32 b Fp(R)3430 1651 y Fs(\000)257 1807 y Fx(and)d Fr(^)-53 b Fu(\015)477 1822 y Fq(j)538 1807 y Fx(and)25 b Fu(\015)758 1822 y Fq(j)819 1807 y Fx(are)g(monotonically)e(increasing,)h(we)h(ha)n(v)o(e)g(that)j Fr(^)-53 b Fu(g)2597 1822 y Fq(j)2661 1807 y Fo(\024)32 b Fr(^)-53 b Fu(\015)2817 1822 y Fq(j)2878 1807 y Fx(and)25 b Fu(g)3094 1822 y Fq(j)3158 1807 y Fo(\024)j Fu(\015)3314 1822 y Fq(j)3350 1807 y Fx(.)404 1928 y(Moreo)o(v)o(er)l(,)j Fr(^)-52 b Fu(g)891 1943 y Fq(j)956 1928 y Fx(dif)n(fers)28 b(from)j Fr(^)-52 b Fu(\015)1515 1943 y Fq(j)1579 1928 y Fx(only)28 b(in)g Fr(\()t(^)-53 b Fu(r)1973 1943 y Ft(0)2037 1928 y Fo(\000)26 b Fu(";)20 b Fr(^)-52 b Fu(r)2274 1943 y Ft(1)2337 1928 y Fr(+)25 b Fu(")p Fr(\))p Fx(,)k(and)f(on)g (this)g(interv)n(al)f(we)257 2048 y(estimate)891 2084 y Fm(\014)891 2144 y(\014)927 2168 y Fr(^)-52 b Fu(\015)975 2183 y Fq(j)1050 2168 y Fo(\000)39 b Fr(\()t(^)-53 b Fu(\015)1255 2183 y Fq(j)1313 2168 y Fo(\003)22 b Fu(m)1470 2183 y Fq(")1507 2168 y Fr(\))1545 2084 y Fm(\014)1545 2144 y(\014)1631 2168 y Fo(\024)53 b Fr(3)17 b Fu(C)1897 2183 y Fq(f)1975 2168 y Fr(exp)q([2)g Fu(C)2287 2183 y Fq(f)2348 2168 y Fr(\()t(\024)-53 b Fu(r)2430 2183 y Ft(1)2491 2168 y Fr(+)22 b Fu(")p Fr(\)])28 b Fu(")2774 2127 y Ft(2)2835 2168 y Fu(;)370 b Fx(\(A.52\))257 2343 y(using)24 b(\(A.32\))h(and)g(\(A.6\).)30 b(Hence,)c(on)e Fr(\()t(^)-53 b Fu(r)1734 2358 y Ft(0)1796 2343 y Fo(\000)22 b Fu(";)f Fr(^)-53 b Fu(r)2029 2358 y Ft(1)2090 2343 y Fr(+)22 b 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Fr(\))1413 2667 y Fs(0)1486 2708 y Fr(+)61 b(^)-61 b Fu(\037)1672 2627 y Fm(\000)1718 2708 y Fr(\()t(^)-53 b Fu(\015)1807 2723 y Fq(j)1843 2708 y Fr(\))1881 2667 y Fs(0)1926 2708 y Fo(\003)22 b Fu(m)2083 2723 y Fq(")2120 2627 y Fm(\001)2216 2708 y Fo(\000)50 b(O)s Fr(\()p Fu(")p Fr(\))22 b Fu(;)636 b Fx(\(A.53\))257 2928 y(and)33 b Fr(\()s(^)-52 b Fu(g)519 2943 y Fq(j)556 2928 y Fr(\))594 2892 y Fs(0)650 2928 y Fx(is)33 b(bounded)f(abo)o(v)o (e)g(and)i(belo)n(w)e(on)h Fr(\()t(^)-53 b Fu(r)2061 2943 y Ft(0)2128 2928 y Fo(\000)29 b Fu(";)21 b Fr(^)-53 b Fu(r)2368 2943 y Ft(1)2435 2928 y Fr(+)28 b Fu(")p Fr(\))33 b Fx(by)g(uni)n(v)o(ersal)f(positi)n(v)o(e)257 3048 y(constants.)70 b(A)38 b(similar)f(ar)n(gument)g(applies)h(to)f Fu(g)2044 3063 y Fq(j)2081 3048 y Fx(.)70 b(It)38 b(is)g(hence)g(clear) h(that)f(all)f(bounds)257 3169 y(asserted)24 b(in)f(Lemma)g(I.2)h (hold,)f(including)f(those)h(that)g(in)l(v)n(olv)o(e)f(the)i(second)f (deri)n(v)n(ati)n(v)o(es,)e(if)257 3289 y(we)k(set)1185 3484 y Fr(^)1164 3509 y Fu(R)1238 3524 y Ft(0)1330 3509 y Fr(:=)57 b(^)-53 b Fu(r)1530 3524 y Ft(0)1591 3509 y Fo(\000)23 b Fu(")f(;)1973 3484 y Fr(^)1952 3509 y Fu(R)2026 3524 y Ft(1)2118 3509 y Fr(:=)57 b(^)-53 b Fu(r)2318 3524 y Ft(1)2379 3509 y Fr(+)22 b Fu(")g(;)660 b Fx(\(A.54\))1185 3629 y Fr(\024)1164 3655 y Fu(R)1238 3670 y Ft(0)1330 3655 y Fr(:=)57 b(\024)-53 b Fu(r)1530 3670 y Ft(0)1591 3655 y Fo(\000)23 b Fu(")f(;)1973 3629 y Fr(\024)1952 3655 y Fu(R)2026 3670 y Ft(1)2118 3655 y Fr(:=)57 b(\024)-53 b Fu(r)2318 3670 y Ft(1)2379 3655 y Fr(+)22 b Fu(")g(:)660 b Fx(\(A.55\))257 3875 y(W)-8 b(e)26 b(lea)n(v)o(es)e(the)h(details)f(of)h(their)f(v)o(eri\002cation) g(to)h(the)f(reader)-5 b(.)994 b Fg(\003)257 4214 y Fw(B)144 b(Some)34 b(Basic)h(Pr)m(operties)257 4438 y Fx(W)-8 b(e)26 b(be)o(gin)d(with)h(an)h(estimate)f(which)h(implies)e(that)h (the)h(Gibbs)f(measure)h(is)f(well-de\002ned,)257 4641 y Fv(Lemma)e(B.1.)36 b Fn(Suppose)20 b(Hypothesesis)f(1)i(and)f(2.)29 b(Ther)l(e)22 b(e)n(xist)e(univer)o(sal)g(constants)f Fu(\013)3352 4656 y Ft(0)3419 4641 y Fu(>)257 4761 y Fr(0)25 b Fn(and)f Fu(C)35 b(>)28 b Fr(0)c Fn(suc)o(h)g(that)g(for)g Fr(0)k Fo(\024)g Fu(\013)g(<)g(\013)1752 4776 y Ft(0)1791 4761 y Fn(,)1190 4995 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))52 b Fo(\025)h Fu(C)1669 4953 y Fs(\000)p Ft(1)1780 4900 y Fm(X)1836 5110 y Fq(j)1941 4995 y Fu(g)1988 5010 y Fq(j)2024 4995 y Fr(\()p Fu(x)2117 5010 y Fq(j)2154 4995 y Fr(\))22 b Fo(\000)h Fu(C)7 b Fo(j)p Fr(\003)p Fo(j)21 b Fu(:)725 b Fx(\(B.1\))p eop %%Page: 52 53 52 52 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(52)257 573 y Fn(Pr)l(oof:)30 b Fx(Using)24 b(\(I.29\))h(we)g(get)f (the)h(follo)n(wing)e(estimate,)h(cf.)31 b(also)24 b(\(III.3\))i(and)f (\(III.4\),)521 837 y Fo(j)p Fu(w)619 852 y Fq(ij)678 837 y Fr(\()p Fu(x)771 852 y Fq(i)800 837 y Fu(;)17 b(x)899 852 y Fq(j)936 837 y Fr(\))p Fo(j)82 b(\024)1245 701 y Fm(Z)1344 727 y Ft(1)1300 927 y(0)1400 752 y Fm(\014)1400 812 y(\014)1434 837 y Fu(x)1489 852 y Fq(i)1534 837 y Fu(@)1585 852 y Fq(i)1614 837 y Fu(w)1684 852 y Fq(ij)1744 837 y Fr(\()p Fu(\033)t(x)1896 852 y Fq(i)1924 837 y Fu(;)17 b(\033)t(x)2082 852 y Fq(j)2119 837 y Fr(\))50 b(+)g Fu(x)2388 852 y Fq(j)2441 837 y Fu(@)2492 852 y Fq(j)2529 837 y Fu(w)2599 852 y Fq(ij)2659 837 y Fr(\()p Fu(\033)t(x)2811 852 y Fq(i)2840 837 y Fu(;)17 b(\033)t(x)2998 852 y Fq(j)3035 837 y Fr(\))3073 752 y Fm(\014)3073 812 y(\014)3122 837 y Fu(d\033)1084 1112 y Fo(\024)84 b Fu(a)1296 1127 y Fq(ij)1373 976 y Fm(Z)1473 1003 y Ft(1)1429 1202 y(0)1512 1112 y Fr(\()p Fo(j)p Fu(x)1633 1127 y Fq(i)1661 1112 y Fo(j)39 b Fr(+)f Fo(j)p Fu(x)1925 1127 y Fq(j)1962 1112 y Fo(j)p Fr(\))17 b(\()p Fo(j)p Fu(g)2162 1071 y Fs(0)2158 1137 y Fq(i)2185 1112 y Fo(j)38 b Fr(+)h Fo(j)p Fu(g)2445 1071 y Fs(0)2441 1137 y Fq(j)2477 1112 y Fo(j)p Fr(\))17 b Fu(d\033)1084 1387 y Fo(\024)84 b Fr(4)17 b Fu(c)1353 1346 y Fs(\000)p Ft(1)1353 1412 y Fq(g)1463 1387 y Fu(a)1514 1402 y Fq(ij)1591 1252 y Fm(Z)1691 1278 y Ft(1)1647 1477 y(0)1747 1307 y Fm(\010)1805 1387 y Fo(j)p Fu(x)1888 1402 y Fq(i)1917 1387 y Fo(j)g(j)p Fu(g)2041 1346 y Fs(0)2037 1412 y Fq(i)2063 1387 y Fo(j)39 b Fr(+)f Fo(j)p Fu(x)2327 1402 y Fq(j)2364 1387 y Fo(j)17 b(j)p Fu(g)2488 1346 y Fs(0)2484 1412 y Fq(j)2519 1387 y Fo(j)2547 1307 y Fm(\011)2622 1387 y Fu(d\033)1085 1595 y Fr(=)84 b(4)17 b Fu(c)1353 1554 y Fs(\000)p Ft(1)1353 1620 y Fq(g)1463 1595 y Fu(a)1514 1610 y Fq(ij)1591 1514 y Fm(\000)1637 1595 y Fu(g)1684 1610 y Fq(i)1712 1595 y Fr(\()p Fu(x)1805 1610 y Fq(i)1834 1595 y Fr(\))38 b(+)h Fu(g)2072 1610 y Fq(j)2108 1595 y Fr(\()p Fu(x)2201 1610 y Fq(j)2238 1595 y Fr(\))2276 1514 y Fm(\001)2344 1595 y Fu(:)917 b Fx(\(B.2\))257 1797 y(This)24 b(implies)g(the)g(lemma)g(since)h Fu(g)1497 1812 y Fq(j)1561 1797 y Fr(=)j Fu(f)1713 1812 y Fq(j)1771 1797 y Fo(\000)23 b Fu(q)1914 1812 y Fq(j)1976 1797 y Fx(and)h Fu(q)2187 1812 y Fq(j)2249 1797 y Fx(is)g(bounded.)709 b Fg(\003)404 1918 y Fx(Lemma)28 b(B.1)g(together)g(with)g(the)g (estimate)f Fu(g)2024 1933 y Fq(j)2061 1918 y Fr(\()p Fu(x)2154 1933 y Fq(j)2191 1918 y Fr(\))34 b Fo(\025)g Fu(C)2451 1882 y Fs(\000)p Ft(1)2546 1918 y Fo(j)p Fu(x)2629 1933 y Fq(j)2665 1918 y Fo(j)25 b(\000)g Fu(C)7 b Fx(,)29 b(for)g(some)f(uni-)257 2038 y(v)o(ersal)36 b(constant)e Fu(C)55 b(>)48 b Fr(0)p Fx(,)38 b(implies)c(that)i(the)f(measure)h (\(I.1\))g(is)f(normalizable)g(and)h(that)257 2159 y(polynomially)23 b(bounded)h(measurable)g(functions)g(are)i(inte)o(grable.)404 2279 y(W)-8 b(e)22 b(will)f(in)g(this)g(appendix)g(identify)g Fo(H)h Fx(with)f Fu(L)2088 2243 y Ft(2)2128 2279 y Fr(\()p Fp(R)2232 2243 y Ft(\003)2291 2279 y Fr(;)c Fo(F)10 b Fr(\))p Fx(.)29 b(Let)22 b Fo(D)30 b Fr(:=)e Fu(C)2979 2243 y Fs(1)2972 2304 y Ft(0)3053 2279 y Fr(\()p Fp(R)3157 2243 y Ft(\003)3216 2279 y Fr(;)17 b Fo(F)10 b Fr(\))21 b Fx(be)257 2399 y(the)26 b(space)g(of)g(test-functions.)31 b(W)-8 b(e)26 b(write)g Fo(D)1828 2363 y Fs(0)1876 2399 y Fx(for)g(the)g(dual)f(space)h(of)g(distrib)n(utions)d(\(recall)257 2520 y(that)i Fo(F)35 b Fx(is)24 b(a)i(\002nite)f(dimensional)e(v)o (ector)i(space\).)32 b(Note)25 b(that)f Fu(d)2470 2535 y Fq(H)q(;Q)2608 2520 y Fu(;)17 b(d)2703 2484 y Fs(\003)2703 2545 y Fq(H)q(;Q)2866 2520 y Fx(maps)25 b Fo(H)h Fx(into)e Fo(D)3472 2484 y Fs(0)257 2640 y Fx(and)961 2842 y Fr(dom)o(\()p Fu(d)1234 2857 y Fq(H)q(;Q)1372 2842 y Fr(\))83 b(=)g Fo(f)p Fu( )32 b Fo(2)c(H)56 b Fr(:)f Fu(d)2164 2857 y Fq(H)q(;Q)2319 2842 y Fu( )60 b Fo(2)55 b(H)29 b(g)22 b Fu(;)513 b Fx(\(B.3\))961 2988 y Fr(dom)o(\()p Fu(d)1234 2947 y Fs(\003)1234 3012 y Fq(H)q(;Q)1372 2988 y Fr(\))83 b(=)g Fo(f)p Fu( )32 b Fo(2)c(H)56 b Fr(:)f Fu(d)2164 2947 y Fs(\003)2164 3012 y Fq(H)q(;Q)2319 2988 y Fu( )60 b Fo(2)55 b(H)29 b(g)22 b Fu(:)513 b Fx(\(B.4\))257 3190 y(W)-8 b(e)26 b(proceed)f(to)f(pro)o(v)o(e)g(the)h(follo)n(wing)d (proposition.)257 3378 y Fv(Pr)n(oposition)32 b(B.2.)46 b Fn(The)32 b(Dir)o(ac)f(oper)o(ator)f Fu(D)1896 3393 y Fq(H)q(;Q)2066 3378 y Fn(is)h(essentially)g(self-adjoint)f(on)h Fo(D)k Fn(and)257 3499 y(self-adjoint)23 b(on)i Fr(dom)o(\()p Fu(d)1134 3514 y Fq(H)q(;Q)1272 3499 y Fr(\))d Fo(\\)h Fr(dom)o(\()p Fu(d)1694 3463 y Fs(\003)1694 3524 y Fq(H)q(;Q)1832 3499 y Fr(\))p Fn(.)257 3687 y(Pr)l(oof:)35 b Fx(As)27 b(for)h(the)f(self-adjointness)f(part)h(of)g(the)h(statement)e(we)i (refer)g(the)f(reader)i(to)d(the)257 3807 y(proof)e(of)g([2,)g(Lemma)g (B.1].)31 b(In)24 b(order)g(to)f(handle)h(the)g(presence)g(of)g Fu(Q)h Fx(we)f(need)g(to)g(replace)257 3928 y(the)33 b(observ)n(ation)e Fr(\()p Fu(D)1029 3892 y Ft(2)1026 3952 y(0)p Fq(;)p Ft(0)1148 3928 y Fr(+)d Fi(1)p Fr(\))1346 3892 y Fs(\000)p Ft(1)1483 3928 y Fr(:)43 b Fu(H)1642 3892 y Fq(k)1684 3928 y Fr(\()p Fp(R)1788 3892 y Ft(\003)1847 3928 y Fr(;)17 b Fo(F)10 b Fr(\))41 b Fo(!)h Fu(H)2283 3892 y Fq(k)r Ft(+2)2416 3928 y Fr(\()p Fp(R)2519 3892 y Ft(\003)2579 3928 y Fr(;)17 b Fo(F)10 b Fr(\))p Fx(,)33 b(for)g Fn(any)g Fu(k)s Fx(,)i(by)d(the)257 4048 y(follo)n(wing)23 b(statement)906 4250 y Fr(\()p Fu(D)1028 4209 y Ft(2)1025 4275 y(0)p Fq(;Q)1161 4250 y Fr(+)f Fi(1)p Fr(\))1353 4209 y Fs(\000)p Ft(1)1503 4250 y Fr(:)53 b Fu(H)1672 4209 y Fq(k)1714 4250 y Fr(\()p Fp(R)1818 4209 y Ft(\003)1877 4250 y Fr(;)17 b Fo(F)10 b Fr(\))54 b Fo(!)h Fu(H)2339 4209 y Fq(k)r Ft(+2)2472 4250 y Fr(\()p Fp(R)2575 4209 y Ft(\003)2635 4250 y Fr(;)17 b Fo(F)10 b Fr(\))21 b Fu(;)441 b Fx(\(B.5\))257 4453 y(where)37 b Fu(k)j Fx(is)c(restricted)h (to)f Fu(k)52 b Fr(=)e(0)36 b Fx(and)h Fu(k)52 b Fr(=)e(1)p Fx(.)66 b(Here)37 b Fu(H)2436 4417 y Fq(k)2478 4453 y Fr(\()p Fp(R)2582 4417 y Ft(\003)2641 4453 y Fr(;)17 b Fo(F)10 b Fr(\))36 b Fx(are)h(the)g(standard)257 4573 y(Sobole)n(v)24 b(spaces.)404 4694 y(W)-8 b(e)19 b(proceed)g(to)g(pro)o (v)o(e)e(that)i Fu(D)1476 4709 y Fq(H)q(;Q)1633 4694 y Fx(is)f(self-adjoint)g(on)h Fo(K)29 b Fr(:=)e(dom\()p Fu(d)2819 4709 y Fq(H)q(;Q)2957 4694 y Fr(\))p Fo(\\)p Fr(dom)o(\()p Fu(d)3334 4657 y Fs(\003)3334 4719 y Fq(H)q(;Q)3472 4694 y Fr(\))p Fx(.)257 4814 y(Clearly)g Fu(D)658 4829 y Fq(H)q(;Q)824 4814 y Fx(is)f(symmetric)f(on)i Fo(K)q Fx(,)g(and)g(it)f(remains)g(to)h(sho)n(w)e(that)h(it)h(is)f(closed.)36 b(T)-8 b(o)26 b(this)257 4934 y(end,)37 b(let)d Fo(f)p Fu(u)703 4949 y Fq(n)749 4934 y Fu(;)17 b(v)840 4949 y Fq(n)887 4934 y Fo(g)937 4949 y Fq(n)p Fs(2)p Fh(N)1113 4934 y Fx(be)35 b(a)g(sequence)f(in)g Fr(Graph\()p Fu(D)2223 4949 y Fq(H)q(;Q)2361 4934 y Fo(j)2389 4949 y Fs(K)2447 4934 y Fr(\))p Fx(,)j(satisfying)c Fu(u)3025 4949 y Fq(n)3116 4934 y Fo(!)45 b Fu(u)34 b Fx(and)257 5055 y Fu(v)304 5070 y Fq(n)379 5055 y Fo(!)27 b Fu(v)i Fx(in)c Fo(H)q Fx(.)404 5175 y(W)-8 b(e)25 b(ha)n(v)o(e)f(the)h(follo)n(wing)e (equality)h(on)g Fo(K)q Fx(,)1049 5303 y Fm(\015)1049 5363 y(\015)1104 5388 y Fu(D)1185 5403 y Fq(H)q(;Q)1324 5388 y Fu( )1391 5303 y Fm(\015)1391 5363 y(\015)1446 5330 y Ft(2)1538 5388 y Fr(=)1666 5303 y Fm(\015)1666 5363 y(\015)1722 5388 y Fu(d)1773 5403 y Fq(H)q(;Q)1911 5388 y Fu( )1978 5303 y Fm(\015)1978 5363 y(\015)2033 5330 y Ft(2)2123 5388 y Fr(+)2248 5303 y Fm(\015)2248 5363 y(\015)2304 5388 y Fu(d)2355 5347 y Fs(\003)2355 5413 y Fq(H)q(;Q)2493 5388 y Fu( )2560 5303 y Fm(\015)2560 5363 y(\015)2615 5330 y Ft(2)2677 5388 y Fu(:)584 b Fx(\(B.6\))p eop %%Page: 53 54 53 53 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(53)257 573 y(Since)30 b Fu(D)589 588 y Fq(H)q(;Q)744 573 y Fu(u)800 588 y Fq(n)883 573 y Fr(=)36 b Fu(v)1042 588 y Fq(n)1118 573 y Fx(is)29 b(bounded)g(uniformly)f(in)h Fu(n)p Fx(,)i(we)e(\002nd)h(that)f(so)g(are)h Fu(d)3065 588 y Fq(H)q(;Q)3219 573 y Fu(u)3275 588 y Fq(n)3351 573 y Fx(and)257 693 y Fu(d)308 657 y Fs(\003)308 718 y Fq(H)q(;Q)463 693 y Fu(u)519 708 y Fq(n)566 693 y Fx(.)404 814 y(Clearly)25 b Fu(d)773 829 y Fq(H)q(;Q)928 814 y Fu(u)i Fo(2)h(D)1185 777 y Fs(0)1208 814 y Fx(.)j(Let)24 b Fu( )32 b Fo(2)c(D)s Fx(.)i(W)-8 b(e)26 b(ha)n(v)o(e)661 1021 y Fo(jh)p Fu(d)779 1036 y Fq(H)q(;Q)917 1021 y Fu(u)p Fo(j)g Fu( )t Fo(ij)83 b Fr(=)g Fo(jh)p Fu(u)p Fo(j)26 b Fu(d)1631 980 y Fs(\003)1631 1046 y Fq(H)q(;Q)1769 1021 y Fu( )t Fo(ij)52 b Fr(=)77 b(lim)2084 1081 y Fq(n)p Fs(!1)2284 1021 y Fo(jh)p Fu(u)2407 1036 y Fq(n)2453 1021 y Fo(j)27 b Fu(d)2559 980 y Fs(\003)2559 1046 y Fq(H)q(;Q)2697 1021 y Fu( )t Fo(ij)1244 1199 y Fr(=)107 b(lim)1403 1259 y Fq(n)p Fs(!1)1603 1199 y Fo(jh)p Fu(d)1721 1214 y Fq(H)q(;Q)1859 1199 y Fu(u)1915 1214 y Fq(n)1961 1199 y Fu(;)17 b( )t Fo(ij)52 b(\024)h Fr(sup)2373 1278 y Fq(n)2485 1199 y Fo(k)p Fu(d)2586 1214 y Fq(H)q(;Q)2723 1199 y Fu(u)2779 1214 y Fq(n)2826 1199 y Fo(kk)p Fu( )t Fo(k)22 b Fu(;)196 b Fx(\(B.7\))257 1458 y(which)22 b(implies)e(by)i (\(B.3\))g(that)g Fu(u)27 b Fo(2)h Fr(dom)o(\()p Fu(d)1811 1473 y Fq(H)q(;Q)1949 1458 y Fr(\))p Fx(.)i(Similar)21 b(we)h(\002nd)g(that)g Fu(u)27 b Fo(2)h Fr(dom)o(\()p Fu(d)3294 1422 y Fs(\003)3294 1483 y Fq(H)q(;Q)3432 1458 y Fr(\))p Fx(.)257 1578 y(Since)i Fo(K)g Fx(is)f(contained)g(in)f(the)h (domain)f(of)h(the)g(closure)g(of)g Fu(D)2485 1593 y Fq(H)q(;Q)2624 1578 y Fo(j)2652 1593 y Fs(D)2712 1578 y Fx(,)h(we)g(conclude)f(from)257 1699 y(closedness)37 b(of)h(the)f(latter)g(that)g Fu(D)1511 1714 y Fq(H)q(;Q)1650 1699 y Fu(u)50 b Fr(=)h Fu(v)42 b Fx(and)37 b(hence)h Fr(\()p Fu(u;)17 b(v)t Fr(\))50 b Fo(2)h Fr(Graph\()p Fu(D)3208 1714 y Fq(H)q(;Q)3346 1699 y Fo(j)3374 1714 y Fs(K)3432 1699 y Fr(\))p Fx(.)257 1819 y(This)35 b(completes)g(the)g (proof.)64 b(\(W)-8 b(e)36 b(note)f(that)g(this)g(proof)h(is)f (slightly)f(simpler)g(than)h(the)257 1939 y(corresponding)24 b(proof)h(in)f([2].\))2049 b Fg(\003)404 2060 y Fx(F)o(or)24 b Fu(u)621 2075 y Fq(k)691 2060 y Fo(2)k(H)870 2024 y Ft(\()p Fq(k)r Ft(\))993 2060 y Fx(we)d(de\002ne)g(the)g(linear)g(maps) 351 2329 y Fr(\()p Fu(T)446 2344 y Fq(k)506 2329 y Fu(u)562 2344 y Fq(k)604 2329 y Fr(\)\()p Fu(x)p Fr(\))83 b(:=)g Fu(e)1087 2288 y Fs(\000)p Fq(\014)s Fs(f)p Fq(H)5 b Ft(\()p Fq(x)p Ft(\))p Fs(\000)p Fq(d)p Ft(\000\()p Fq(Q)p Ft(\()p Fq(x)p Ft(\)\))p Fs(g)1774 2193 y Fm(Z)1874 2220 y Ft(1)1830 2419 y(0)1930 2329 y Fu(t)1965 2288 y Fq(k)r Fs(\000)p Ft(1)2115 2329 y Fu(e)2160 2288 y Fq(\014)s Fs(f)p Fq(H)g Ft(\()p Fq(tx)p Ft(\))p Fs(\000)p Fq(d)p Ft(\000\()p Fq(Q)p Ft(\()p Fq(tx)p Ft(\)\))p Fs(g)2827 2329 y Fu(u)2883 2344 y Fq(k)2925 2329 y Fr(\()p Fu(tx)p Fr(\))17 b Fu(dt)22 b(;)45 b Fx(\(B.8\))707 2557 y Fu(S)89 b Fr(:=)1042 2462 y Fm(X)1100 2674 y Fq(`)1202 2557 y Fu(\014)6 b(x)1318 2572 y Fq(`)1390 2557 y Fo(\012)39 b Fu(c)1548 2572 y Fq(`)1604 2557 y Fu(;)1657 b Fx(\(B.9\))654 2785 y Fr(\011)730 2800 y Fq(k)856 2785 y Fr(:=)83 b Fu(S)28 b Fo(\016)22 b Fu(T)1259 2800 y Fq(k)1324 2785 y Fu(:)1887 b Fx(\(B.10\))257 2993 y(W)-8 b(e)26 b(ha)n(v)o(e)e(the)h (follo)n(wing)e(result.)257 3186 y Fv(Lemma)j(B.3.)42 b Fn(Let)26 b Fu(v)1017 3201 y Fq(k)1089 3186 y Fo(2)j(H)1269 3149 y Ft(\()p Fq(k)r Ft(\))1392 3186 y Fn(be)c(continuous)f(in)h(a)h (\(not)f(necessarily)f(univer)o(sal\))h(neigh-)257 3306 y(bourhood)f(of)g Fr(0)p Fn(.)31 b(Then)25 b Fu(T)1158 3321 y Fq(k)1201 3306 y Fu(v)1248 3321 y Fq(k)1318 3306 y Fo(2)j(H)1497 3270 y Ft(\()p Fq(k)r Ft(\))257 3499 y Fn(Pr)l(oof:)46 b Fx(W)-8 b(e)33 b(be)o(gin)e(by)i(analyzing)f(the)g (dif)n(ference)i Fu(H)8 b Fr(\()p Fu(x)p Fr(\))27 b Fo(\000)i Fu(H)8 b Fr(\()p Fu(tx)p Fr(\))p Fx(.)54 b(Assume)32 b(\002rst)h(that)257 3619 y Fr(0)28 b Fu(<)f(t)h(<)g Fr(1)p Fu(=)p Fr(2)p Fx(.)i(From)25 b(\(I.29\))g(we)g(get)f(the)h (estimate)597 3891 y Fu(g)644 3906 y Fq(j)681 3891 y Fr(\()p Fu(s)p Fr(\))49 b Fo(\000)h Fu(g)1026 3906 y Fq(j)1063 3891 y Fr(\()p Fu(s=)p Fr(2\))83 b(=)h(1)p Fu(=)p Fr(2)1690 3756 y Fm(Z)1788 3782 y Ft(1)1744 3981 y(0)1844 3891 y Fu(sg)1941 3850 y Fs(0)1937 3916 y Fq(j)1973 3891 y Fr(\(\()p Fu(\033)t(s)22 b Fr(+)g Fu(s)p Fr(\))p Fu(=)p Fr(2\))17 b Fu(d\033)1365 4167 y Fo(\025)84 b Fr(\(2)p Fu(c)1655 4182 y Fq(g)1694 4167 y Fr(\))1732 4126 y Fs(\000)p Ft(1)1843 4031 y Fm(Z)1943 4057 y Ft(1)1899 4257 y(0)1999 4167 y Fu(sg)2096 4126 y Fs(0)2092 4191 y Fq(j)2128 4167 y Fr(\()p Fu(\033)t(s)p Fr(\))17 b Fu(d\033)56 b Fr(=)c(\(2)p Fu(c)2745 4182 y Fq(g)2785 4167 y Fr(\))2823 4126 y Fs(\000)p Ft(1)2934 4167 y Fu(g)t Fr(\()p Fu(s)p Fr(\))21 b Fu(:)83 b Fx(\(B.11\))257 4420 y(Then,)30 b(by)f(\(B.2\))h(and)f(the)g(boundedness)f(of)h(the)g Fu(q)2057 4435 y Fq(j)2094 4420 y Fx(')-5 b(s,)30 b(there)f(e)o(xists)f (a)i(uni)n(v)o(ersal)d(constant)257 4540 y Fu(C)35 b(<)28 b Fo(1)c Fx(such)h(that)502 4748 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))50 b Fo(\000)g Fu(H)8 b Fr(\()p Fu(tx)p Fr(\))83 b Fo(\025)1397 4653 y Fm(X)1453 4863 y Fq(j)1541 4748 y Fo(f)p Fu(f)1639 4763 y Fq(j)1676 4748 y Fr(\()p Fu(x)1769 4763 y Fq(j)1806 4748 y Fr(\))22 b Fo(\000)g Fu(f)2013 4763 y Fq(j)2050 4748 y Fr(\()p Fu(tx)2178 4763 y Fq(j)2215 4748 y Fr(\))p Fo(g)50 b(\000)g Fu(\013)q(C)2636 4653 y Fm(X)2692 4863 y Fq(j)2797 4748 y Fu(g)2844 4763 y Fq(j)2880 4748 y Fr(\()p Fu(x)2973 4763 y Fq(j)3010 4748 y Fr(\))1237 5024 y Fo(\025)1397 4929 y Fm(X)1453 5139 y Fq(j)1541 5024 y Fo(f)p Fu(g)1638 5039 y Fq(j)1674 5024 y Fr(\()p Fu(x)1767 5039 y Fq(j)1804 5024 y Fr(\))22 b Fo(\000)h Fu(g)2011 5039 y Fq(j)2047 5024 y Fr(\()p Fu(x)2140 5039 y Fq(j)2177 5024 y Fu(=)p Fr(2\))f Fo(\000)g Fu(C)7 b Fo(g)22 b(\000)h Fu(\013)q(C)2839 4929 y Fm(X)2895 5139 y Fq(j)3000 5024 y Fu(g)3047 5039 y Fq(j)3083 5024 y Fr(\()p Fu(x)3176 5039 y Fq(j)3213 5024 y Fr(\))1237 5300 y Fo(\025)1397 5205 y Fm(X)1453 5415 y Fq(j)1558 5219 y Fm(\000)1604 5300 y Fr(\(2)p Fu(c)1733 5315 y Fq(g)1772 5300 y Fr(\))1810 5259 y Fs(\000)p Ft(1)1927 5300 y Fo(\000)f Fu(\013)q(C)2166 5219 y Fm(\001)2212 5300 y Fu(g)2259 5315 y Fq(j)2295 5300 y Fr(\()p Fu(x)2388 5315 y Fq(j)2425 5300 y Fr(\))g Fo(\000)h Fu(C)7 b Fo(j)p Fr(\003)p Fo(j)21 b Fu(:)404 b Fx(\(B.12\))p eop %%Page: 54 55 54 54 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(54)257 573 y(Secondly)-6 b(,)32 b(if)f Fr(1)p Fu(=)p Fr(2)39 b Fu(<)f(t)i(<)f Fr(1)p Fx(,)32 b(we)f(in)l(v)n(ok)o(e)g(Lemma) f(III.1,)j(applied)d(with)g Fu(\017)2948 588 y Fq(k)3030 573 y Fr(=)39 b(1)27 b Fo(\000)g Fu(t)p Fx(,)33 b(to)257 693 y(obtain)24 b(\(for)i Fu(\013)i(<)f(\013)961 708 y Ft(0)1001 693 y Fx(\))307 892 y Fu(H)8 b Fr(\()p Fu(x)p Fr(\))39 b Fo(\000)g Fu(H)8 b Fr(\()p Fu(tx)p Fr(\))49 b Fo(\025)1113 798 y Fm(X)1169 1008 y Fq(j)1257 892 y Fr(\(1)27 b Fo(\000)h Fu(\013)q(C)7 b Fr(\))1654 812 y Fm(\000)1699 892 y Fu(g)1746 907 y Fq(j)1783 892 y Fr(\()p Fu(x)p Fr(\))27 b Fo(\000)h Fu(g)2093 907 y Fq(j)2129 892 y Fr(\()p Fu(tx)p Fr(\))2295 812 y Fm(\001)2380 892 y Fo(\000)39 b Fu(C)7 b Fo(j)p Fr(\003)p Fo(j)49 b(\025)h(\000)p Fu(C)7 b Fo(j)p Fr(\003)p Fo(j)k Fu(;)49 b Fx(\(B.13\))257 1176 y(for)25 b(some)g(uni)n(v)o(ersal)e(constant)h Fu(C)34 b(<)28 b Fo(1)p Fx(.)404 1296 y(No)n(w)-6 b(,)23 b(we)i(estimate,)f (using)g(\(B.12\))h(and)g(\(B.13\),)346 1394 y Fm(Z)402 1620 y Fh(R)450 1601 y Fk(\003)516 1530 y Fo(j)p Fr(\()p Fu(T)639 1545 y Fq(k)681 1530 y Fu(v)728 1545 y Fq(k)771 1530 y Fr(\)\()p Fu(x)p Fr(\))p Fo(j)968 1489 y Ft(2)1024 1530 y Fu(d)1075 1489 y Ft(\003)1128 1530 y Fu(x)83 b Fo(\024)h Fr(sup)1458 1609 y 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y Fu(dt)1545 2966 y Fr(+)1638 2830 y Fm(Z)1737 2856 y Ft(1)p Fq(=)p Ft(2)1693 3056 y(0)1864 2830 y Fm(Z)1919 3056 y Fh(R)1967 3037 y Fk(\003)2033 2966 y Fu(t)2068 2924 y Ft(2)p Fq(k)r Fs(\000)p Ft(2)p Fs(\000j)p Ft(\003)p Fs(j)2380 2966 y Fu(e)2425 2924 y Fs(\000)2490 2876 y Fj(C)2534 2894 y(\014)2575 2876 y Fc(j)p Fj(x)p Fc(j)2648 2891 y Fk(1)p 2490 2909 193 3 v 2574 2951 a Fj(t)2697 2966 y Fo(j)p Fu(v)2772 2981 y Fq(k)2814 2966 y Fr(\()p Fu(x)p Fr(\))p Fo(j)2973 2924 y Ft(2)3029 2966 y Fu(d)3080 2924 y Ft(\003)3133 2966 y Fu(x)17 b(dt)3291 2855 y Fm(o)3379 2966 y Fu(:)257 3207 y Fx(Here)32 b Fo(j)p Fu(x)p Fo(j)593 3222 y Ft(1)671 3207 y Fr(:=)812 3132 y Fm(P)918 3236 y Fq(j)971 3207 y Fo(j)p Fu(x)1054 3222 y Fq(j)1090 3207 y Fo(j)p Fx(.)49 b(As)30 b(for)h(the)g(last)f(term)g(on)h(the)g(right)f (side,)h(we)g(note)g(that)f(in)g(the)257 3328 y(case)f Fr(2)p Fu(k)36 b Fo(\025)e(j)p Fr(\003)p Fo(j)24 b Fr(+)g(2)p Fx(,)k(it)g(can)g(be)g(estimated)f(in)h(the)f(same)h(w)o(ay)g(as)g(the) 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y(identity)i(holds)g (between)h(functions,)925 5360 y Fr(\()p Fu(d)1014 5375 y Fq(H)q(;Q)1169 5360 y Fr(\011)1245 5375 y Fq(k)1304 5360 y Fu(v)1351 5375 y Fq(k)1394 5360 y Fr(\)\()p Fu(x)p Fr(\))50 b(+)f(\(\011)1852 5375 y Fq(k)1911 5360 y Fu(d)1962 5375 y Fq(H)q(;Q)2117 5360 y Fu(v)2164 5375 y Fq(k)2207 5360 y Fr(\)\()p Fu(x)p Fr(\))k(=)f Fu(v)2604 5375 y Fq(k)2647 5360 y Fr(\()p Fu(x)p Fr(\))23 b Fu(:)410 b Fx(\(B.16\))p eop %%Page: 55 56 55 55 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(55)257 573 y Fv(Pr)n(oof)24 b(of)f(Theor)n(em)i(I.4:)60 b Fx(As)23 b(for)h(the)f(\002rst)g(part)h(of)f(the)g(Theorem)g(we)h (refer)g(the)f(reader)i(to)257 693 y(the)f(proof)f(of)h([2)o(,)g(Lemma) f(B.3].)31 b(W)-8 b(e)24 b(note)f(that)g(the)g(pertubation)f(ar)n (gument)h(of)h([2],)g(which)257 814 y(is)29 b(based)f(on)g(the)h(main)f (form)g(bound)g(in)g(Theorem)g(II.1,)i(breaks)f(do)n(wn)e(on)i Fu(k)s Fx(-forms)f(with)257 934 y Fu(k)j Fo(\025)d Fr(2)p Fx(,)d(since)g(we)g(cannot)f(bound)g 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Fw(Refer)m(ences)307 4392 y Fx([1])49 b(S.)29 b(Albe)n(v)o(erio,)f(A.)g(Daletskii,)h(and)f(Y)-13 b(.)29 b(K)m(ondratie)n(v)-6 b(.)45 b(Stochastic)29 b(analysis)e(on)h (prod-)472 4512 y(uct)37 b(manifolds:)54 b(Dirichlet)36 b(operators)h(on)g(dif)n(ferential)f(forms.)75 b Fn(J)n(.)37 b(Funct.)f(Anal.)p Fx(,)472 4633 y(176:280\226316,)23 b(2000.)307 4830 y([2])49 b(V)-13 b(.)38 b(Bach,)j(T)-7 b(.)37 b(Jeck)o(o,)k(and)c(J.)g(Sj)8 b(\250)-41 b(ostrand.)76 b(Correlation)38 b(asymptotics)d(of)j(classical)472 4950 y(lattice)29 b(spin)f(systems)g(with)g(noncon)l(v)o(e)o(x)f(Hamilton)h (function)g(at)h(lo)n(w)g(temperature.)472 5071 y Fn(Ann.)24 b(Henri)h(P)-8 b(oincar)1259 5072 y(\264)1254 5071 y(e)p Fx(,)24 b(1:59\226100,)f(2000.)307 5268 y([3])49 b(T)-7 b(.)29 b(Bodineau)g(and)g(B.)g(Helf)n(fer)-5 b(.)49 b(Correlations,)30 b(spectral)f(gap)g(and)g(log-Sobole)n(v)f(in-)472 5388 y(equalities)c(for)h(unbounded)f(spin)g(systems.)34 b Fn(Stud.)24 b(Adv)-7 b(.)24 b(Math.)p Fx(,)g(16:51\22666,)f(2000.)p eop %%Page: 56 57 56 56 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(56)307 573 y([4])49 b(A.)21 b(Bo)o(vier)l(,)h(M.)f(Eckhof)n(f,)g (V)-13 b(.)21 b(Gayrard,)i(and)e(M.)g(Klein.)27 b(Metastability)19 b(in)i(stochastic)472 693 y(dynamics)26 b(of)g(disordered)g (mean-\002eld)h(models.)40 b Fn(Pr)l(obab)l(.)25 b(Theory)i(Related)f (F)l(ields)p Fx(,)472 814 y(119\(1\):99\226161,)d(2001.)307 1017 y([5])49 b(D.)21 b(Bur)n(ghelea.)27 b(Lectures)20 b(on)h(Witten-Helf)n(fer)n(-Sjstrand)e(theory)-6 b(.)26 b(In)21 b Fn(Pr)l(oceedings)e(of)472 1137 y(the)25 b(Thir)l(d)f (International)f(W)-9 b(orkshop)24 b(on)h(Dif)n(fer)l(ential)e (Geometry)i(and)f(its)g(Applica-)472 1258 y(tions,)32 b(the)f(F)l(ir)o(st)e(German-Romanian)h(Seminar)g(on)h(Geometry)g (\(Sibiu,)h(Romania,)472 1378 y(1997\))24 b(Gen.)h(Math.)f(5)p Fx(,)h(pages)g(85\22699,)f(1997.)307 1582 y([6])49 b(H.)25 b(Cartan.)36 b Fn(Dif)n(fer)l(ential)24 b(forms)p Fx(.)34 b(Hermann,)25 b(P)o(aris,)f(1970.)307 1785 y([7])49 b(H.)37 b(Cycon,)j(R.)d(Froese,)k(W)-9 b(.)37 b(Kirsch,)i(and)e(B.)g(Simon.)74 b Fn(Sc)o(hr)2726 1786 y(\250)2718 1785 y(oding)o(er)35 b(Oper)o(ator)o(s)p Fx(.)472 1905 y(Springer)l(,)25 b(Berlin,)g (Heidelber)n(g,)g(Ne)n(w)f(Y)-11 b(ork,)25 b(1)f(edition,)g(1987.)307 2109 y([8])49 b(R.)28 b(Dobrushin)e(and)i(M.)f(Zahradnik.)43 b(Phase)28 b(diagrams)f(for)h(continuous)d(spin)i(mod-)472 2229 y(els:)55 b(An)37 b(e)o(xtension)e(of)i(the)g(Pirogo)o(v-Sinai)f (theory)-6 b(.)74 b(In)37 b(R.)g(Dobroushin,)i(editor)l(,)472 2350 y Fn(Mathematical)23 b(Pr)l(oblems)h(of)g(Statistical)f(Mec)o (hanics)p Fx(.)h(Reidel,)g(Dordrecht,)h(1986.)307 2553 y([9])49 b(R.)33 b(L.)f(Dobrushin)g(and)g(S.)h(B.)g(Shlosman.)59 b(Phases)33 b(corresponding)e(to)h(minima)f(of)472 2673 y(the)25 b(local)f(ener)n(gy)-6 b(.)36 b Fn(Selecta)25 b(Math.)f(So)o(viet)p Fx(,)g(1:317\226338,)f(1981.)257 2877 y([10])49 b(B.)25 b(Helf)n(fer)-5 b(.)35 b(Around)24 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b(Funct.)f(Anal.)p Fx(,)472 4209 y(155:571\226586,)e(1998.)257 4413 y([14])49 b(B.)33 b(Helf)n(fer)-5 b(.)60 b(Remarks)32 b(on)h(the)f(decay)h(of)f (correlations)g(and)g(Witten')-5 b(s)32 b(Laplacians)472 4533 y(\226)26 b(III.)h(applications)d(to)i(logarithmic)e(Sobole)n(v)h (inequalities.)39 b Fn(Ann.)25 b(de)h(l'Inst.)g(Henri)472 4653 y(P)-8 b(oincar)791 4654 y(\264)786 4653 y(e)24 b(\(to)h(appear\))p Fx(,)f(1998.)257 4857 y([15])49 b(B.)30 b(Helf)n(fer)h(and)e(J.)h(Sj)8 b(\250)-41 b(ostrand.)51 b(On)30 b(the)g(correlations)f(for)h(Kac)h(lik)o(e)e(models)g(in)g(the) 472 4977 y(con)l(v)o(e)o(x)24 b(case.)36 b Fn(J)n(.)25 b(Stat.)e(Phys.)p Fx(,)i(74:349\226369,)e(1994.)257 5181 y([16])49 b(J.)57 b(Moller)-5 b(.)136 b(Lo)n(w-temperature)56 b(localization)g(for)h(continuous)e(spin-systems.)472 5301 y Fn(Pr)l(eprint)p Fx(,)24 b(2001.)p eop %%Page: 57 58 57 57 bop 257 266 a Fy(BM-April)25 b(22,)f(2002)2366 b Fx(57)257 573 y([17])49 b(A.)28 b(Procacci)g(and)g(B.)f(Scoppola.)44 b(On)28 b(decay)g(of)f(correlations)g(for)h(unbounded)e(spin)472 693 y(systems)35 b(with)h(arbitrary)g(boundary)g(conditions.)70 b Fn(J)n(.)36 b(Stat.)g(Phys.)p Fx(,)i(105:453\226482,)472 814 y(2001.)257 1017 y([18])49 b(J.)21 b(Sjoestrand.)27 b(Correlation)21 b(asymptotics)e(and)i(Witten)f(Laplacians.)28 b Fn(St.)20 b(P)-8 b(eter)o(sb)n(ur)l(g)472 1137 y(Math.)27 b(J)n(.)h(\(AMS\))p Fx(,)g(8:123\226148,)f(1997.)44 b(also)28 b(in)f(St.)h(Petersb)n(ur)n(g)g(Math.)f(J.)h(8)g(\(1996\),)472 1258 y(no.)d(1,)f(pp.)h(160\226161.)257 1461 y([19])49 b(J.)21 b(Sj)8 b(\250)-41 b(ostrand.)29 b(Potential)20 b(wells)h(in)h(high)e(dimensions)g(I.)29 b Fn(Ann.)21 b(Inst.)g(Henri)g(P)-8 b(oincar)3432 1462 y(\264)3427 1461 y(e)p Fx(,)472 1582 y(58:1\22641,)24 b(1993.)257 1785 y([20])49 b(J.)30 b(Sj)8 b(\250)-41 b(ostrand.)51 b(Complete)29 b(asymptotics)f(for)i(correlations)f(of)h(Laplace)g(inte) o(grals)f(in)472 1905 y(the)c(semi-classical)f(limit.)34 b Fn(M)1540 1906 y(\264)1535 1905 y(em.)25 b(Soc.)f(Math.)g(F)-5 b(r)-11 b(.)24 b(\(N.S.\))p Fx(,)i(83:i)n(v+104)c(pp,)j(2000.)257 2109 y([21])49 b(M.)24 b(E.)g(T)-8 b(aylor)j(.)34 b Fn(P)-8 b(artial)23 b(dif)n(fer)l(ential)g(equations)g(I:)h(Basic)g(theory)p Fx(.)34 b(Springer)25 b(v)o(erlag,)472 2229 y(Ne)n(w)g(Y)-11 b(ork,)24 b(1996.)257 2433 y([22])49 b(E.)30 b(W)l(itten.)50 b(Supersymmetry)28 b(and)i(Morse)f(theory)-6 b(.)50 b Fn(J)n(.)29 b(Dif)n(f)o(.)g(Geom.)p Fx(,)h(17:661\226692,)472 2553 y(1982.)257 2756 y([23])49 b(N.)19 b(Y)-11 b(oshida.)24 b(The)19 b(equi)n(v)n(alence)f(of)i(the)f(log-Sobole)n(v)f(inequality)g (and)i(a)f(mixing)f(con-)472 2877 y(dition)26 b(for)h(unbounded)e(spin) h(systems)g(on)g(the)h(lattice.)41 b Fn(Ann.)26 b(Inst.)h(Henri)f(P)-8 b(oincar)3457 2878 y(\264)3452 2877 y(e)472 2997 y(Pr)l(obab)l(.)23 b(Statis.)p Fx(,)h(37:223\226243,)e(2001.)257 3200 y([24])49 b(M.)34 b(Zahradnik.)66 b(Contour)34 b(methods)f(and)h(Pirogo)o (v-Sinai)f(theory)h(for)h(continuous)472 3321 y(spin)24 b(models.)35 b Fn(Pr)l(eprint)p Fx(,)23 b(1998.)257 3524 y([25])49 b(B.)27 b(Ze)o(garlinski.)39 b(Dobrushin)25 b(uniqueness)h(theorem)g(and)g(logarithmic)f(Sobole)n(v)h(in-)472 3645 y(equalities.)35 b Fn(J)n(.)24 b(Funct.)h(Anal.)p Fx(,)e(105:77\226111,)g(1992.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0205071023756--