Content-Type: multipart/mixed; boundary="-------------0308140715857" This is a multi-part message in MIME format. ---------------0308140715857 Content-Type: text/plain; name="03-366.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-366.comments" to appear in J. Funct. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 126 /planckover2pi1 put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 112 /radicalbig put dup 114 /radicalbigg put readonly def /FontBBox{-24 -2960 1454 772}readonly 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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b(the)h(trace)f(p)s(er)g(unit)-24 1592 y(v)m(olume)g([5],)f(while)729 1567 y(^)711 1592 y Fk(T)51 b Fq(is)27 b(the)f(\(disorder)g(a)m(v)m (eraged\))h(trace)g(p)s(er)f(unit)g(v)m(olume)h(in)f(the)g(1-direction) g(com)m(bined)-24 1713 y(with)j(the)f(usual)g(trace)h(in)f(the)g (2-direction.)42 b(Finally)28 b(let)g(us)g(remark)h(that)f(the)g (traces)g(are)g(in)m(v)-5 b(arian)m(t)29 b(w.r.t.)-24 1833 y(the)k(deriv)-5 b(ations:)974 2048 y(^)956 2074 y Fk(T)25 b Fq(\()p Fk(r)1156 2089 y Fn(1)1196 2074 y Fp(A)p Fq(\))55 b(=)g(0)32 b(,)228 b Fk(T)26 b Fq(\()p Fk(r)2030 2089 y Fo(j)2066 2074 y Fp(A)p Fq(\))56 b(=)f(0)32 b(,)130 b Fp(j)34 b Fq(=)28 b(1)p Fp(;)17 b Fq(2)31 b(,)-24 2258 y(as)45 b(long)g(as)g Fp(A)g Fq(has)g(join)m(tly)g(con)m(tin)m (uous)i(in)m(tegral)e(k)m(ernel)h(and)f Fk(r)2538 2273 y Fn(1)2578 2258 y Fp(A)g Fq(is)2824 2233 y(^)2806 2258 y Fk(T)25 b Fq(-traceclass)46 b(\(resp.)81 b Fk(r)3772 2273 y Fo(j)3809 2258 y Fp(A)45 b Fq(is)-24 2379 y Fk(T)26 b Fq(-traceclass\).)44 b(Under)34 b(these)g(h)m(yp)s(othesis,)g(this)g (can)e(directly)i(b)s(e)f(v)m(eri\014ed)i(from)d(the)h(expressions)i (\(8\).)-24 2759 y Fr(4)161 b(In)l(tegralk)l(ernels)51 b(asso)t(ciated)i(to)g(the)g(planar)h(Hamiltonian)-24 2978 y Fq(In)36 b(this)g(section,)i(Hamiltonians)e(without)g(b)s (oundary)g(conditions)g(are)g(considered.)54 b(Hence)37 b Fp(s)32 b Fq(=)h Fk(1)i Fq(and)-24 3098 y Fp(!)d Fq(stands)e(for)e (\()p Fp(!)t(;)17 b Fk(1)p Fq(\).)40 b(F)-8 b(ollo)m(wing)29 b(Da)m(vies)g([13,)h(Sec.)43 b(3.4],)29 b(the)g(functional)g(calculus)h (of)e(the)h(Hamiltonian)-24 3218 y Fp(H)57 3233 y Fo(!)135 3218 y Fq(=)f Fp(H)320 3233 y Fo(L)394 3218 y Fq(+)22 b Fp(V)549 3233 y Fo(!)632 3218 y Fq(will)33 b(b)s(e)g(done)g(via)g (the)g(complex)h(heat)f(k)m(ernel:)1248 3485 y Fp(F)14 b Fq(\()p Fp(H)1444 3500 y Fo(!)1494 3485 y Fq(\))56 b(=)1719 3349 y Fi(Z)1818 3376 y Fj(1)1774 3575 y(\0001)1920 3485 y Fp(dt)2045 3460 y Fq(~)2023 3485 y Fp(F)14 b Fq(\()p Fp(t)p Fq(\))j Fp(e)2273 3444 y Fj(\000)p Fo(H)2386 3452 y Fb(!)2430 3444 y Fn(\(1+)p Fo({t)p Fn(\))2661 3485 y Fq(,)1179 b(\(9\))-24 3713 y(where)1263 3955 y(~)1241 3980 y Fp(F)13 b Fq(\()p Fp(t)p Fq(\))56 b(=)f(2)p Fp(\031)1756 3844 y Fi(Z)1855 3871 y Fj(1)1811 4070 y(\0001)1957 3980 y Fp(dE)23 b(e)2148 3939 y Fo({E)t(t)2273 3980 y Fp(e)2318 3939 y Fo(E)2394 3980 y Fp(F)14 b Fq(\()p Fp(E)6 b Fq(\))32 b(.)-24 4221 y(F)-8 b(or)43 b(compactly)i(supp)s(orted)f(di\013eren)m (tiable)i(functions)e Fp(F)60 b Fk(2)47 b Fp(C)2452 4185 y Fo(k)2445 4246 y(c)2494 4221 y Fq(\()p Fm(R)p Fq(\),)g(one)c(has)h (the)g(standard)g(F)-8 b(ourier)-24 4342 y(estimates)36 b Fk(j)456 4316 y Fq(~)435 4342 y Fp(F)13 b Fq(\()p Fp(t)p Fq(\))p Fk(j)29 b(\024)h Fp(c)828 4357 y Fo(k)871 4342 y Fq(\(1)22 b(+)h Fk(j)p Fp(t)p Fk(j)1170 4305 y Fo(k)r Fj(\000)p Fn(1)1302 4342 y Fq(\))1340 4305 y Fj(\000)p Fn(1)1434 4342 y Fq(.)47 b(Suc)m(h)34 b(an)g(estimate)h(ma)m(y)f(also)g (hold)f(for)g(functions)i(with)f(in\014nite)-24 4462 y(supp)s(ort,)f(but)g(w)m(e)h(do)e(not)h(in)m(tend)h(here)f(to)f(giv)m (e)i(the)f(most)g(general)g(form)m(ulation.)-24 4690 y Fh(Prop)s(osition)38 b(1)49 b Fl(L)-5 b(et)44 b Fp(V)921 4705 y Fo(!)1015 4690 y Fk(2)h Fp(L)1192 4654 y Fj(1)1267 4690 y Fq(\()p Fm(R)1377 4654 y Fn(2)1416 4690 y Fq(\))f Fl(and)f Fp(F)58 b Fk(2)44 b Fp(C)2004 4654 y Fo(k)1997 4715 y(c)2047 4690 y Fq(\()p Fm(R)p Fq(\))f Fl(with)h Fp(k)j(>)d Fq(2)p Fl(.)71 b(Then)43 b Fp(F)14 b Fq(\()p Fp(H)3286 4705 y Fo(!)3336 4690 y Fq(\))44 b Fl(is)f(an)h(inte)-5 b(gr)g(al)-24 4811 y(op)g(er)g(ator)35 b(the)g(inte)-5 b(gr)g(al)34 b(kernel)g(of)h(which)f(satis\014es)g(uniformly)h(in)f Fp(!)k Fl(and)d(for)f(any)h Fp(\016)d(>)27 b Fq(0)1182 5051 y Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(F)14 b Fq(\()p Fp(H)1528 5066 y Fo(!)1578 5051 y Fq(\))p Fk(j)o Fp(~)-48 b(y)19 b Fk(ij)55 b(\024)2281 4984 y Fp(c)2323 4999 y Fo(\016)p 1976 5029 691 4 v 1976 5120 a Fq(1)22 b(+)g Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)19 b Fk(j)2445 5091 y Fo(k)r Fj(\000)p Fn(2)p Fj(\000)p Fo(\016)2711 5051 y Fl(.)1960 5611 y Fq(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop -24 171 a Fh(Pro)s(of:)26 b Fq(As)f(the)h(follo)m (wing)g(estimates)h(are)e(p)s(oin)m(t)m(wise)i(in)f Fp(!)t Fq(,)g(the)f(index)i(will)f(b)s(e)g(suppressed.)43 b(Let)26 b(us)g(b)s(egin)-24 291 y(with)38 b(the)g(in)m(tegral)f(k)m(ernel)i(of) e Fp(e)1188 255 y Fj(\000)p Fo(H)1301 266 y Fb(L)1347 255 y Fo(z)1424 291 y Fq(explicitly)j(using)e(Mehler's)h(form)m(ula)e (for)g(the)g(\(shifted\))i(harmonic)-24 411 y(oscillator)33 b Fp(h)p Fq(\()p Fp(k)s Fq(\))28 b(=)722 372 y Fn(1)p 722 388 36 4 v 722 446 a(2)767 411 y Fq(\()p Fk(\000)p Fp(@)938 375 y Fn(2)933 436 y Fo(x)973 445 y Fe(2)1034 411 y Fq(+)22 b Fp(\015)1188 375 y Fn(2)1228 411 y Fq(\()p Fp(X)1347 426 y Fn(2)1408 411 y Fq(+)1517 372 y Fo(k)p 1516 388 41 4 v 1516 446 a(\015)1566 411 y Fq(\))1604 375 y Fn(2)1644 411 y Fq(\))32 b(in)h(the)g(2-direction)g(\()p Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))28 b Fp(>)g Fq(0\):)336 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Fj(\000)p Fo(q)3306 2922 y Fn(~)3288 2939 y Fo(H)5 b(z)3391 2980 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)33 b Fq(.)234 b(\(13\))-24 3247 y(Using)38 b(this)f(iterativ)m(ely)-8 b(,)39 b(one)e(obtains)g(the)g (Dyson)g(series)h(for)e Fp(z)j Fq(=)34 b Fp(t)h(>)f Fq(0)i(whic)m(h)i (is)f(estimated)h(term)f(b)m(y)-24 3367 y(term)c(using)h(\(12\))90 3745 y Fi(\014)90 3805 y(\014)124 3830 y Fk(h)n Fp(~)-47 b(x)p Fk(j)p Fp(e)291 3788 y Fj(\000)p Fo(H)5 b(t)438 3830 y Fk(j)o Fp(~)-48 b(y)20 b Fk(i)573 3745 y Fi(\014)573 3805 y(\014)661 3830 y Fk(\024)56 b Fp(e)839 3788 y Fj(\000)p Fo(tV)960 3797 y Fe(+)1049 3735 y Fi(X)1054 3945 y Fo(n)p Fj(\025)p Fn(0)1272 3759 y Fp(V)1350 3723 y Fo(n)1329 3784 y Fn(+)p 1219 3807 231 4 v 1219 3898 a Fq(\(2)p Fp(\031)t Fq(\))1403 3869 y Fo(n)1476 3659 y Fi( )1597 3705 y Fo(n)1555 3735 y Fi(Y)1563 3947 y Fo(l)q Fn(=1)1699 3694 y Fi(Z)1754 3919 y Ff(R)1802 3901 y Fe(2)1858 3830 y Fp(d)m(~)-46 b(r)1953 3845 y Fo(l)1995 3694 y Fi(Z)2095 3720 y Fo(q)2127 3732 y Fb(l)p 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Fn(2)3003 656 y Fi(\023)3126 796 y Fq(,)-24 1035 y(so)33 b(that)g(the)g(inequalit)m(y)h Fp(e)973 999 y Fj(\000)p Fo(r)1094 1035 y Fk(\024)28 b Fp(c)1241 1050 y Fo(\014)1288 1035 y Fp(=)p Fq(\(1)22 b(+)g Fp(r)s Fq(\))1629 999 y Fo(\014)1708 1035 y Fq(for)32 b Fp(r)m(;)17 b(\014)33 b(>)28 b Fq(0)k(leads)h(to)667 1300 y Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(F)14 b Fq(\()p Fp(H)1013 1315 y Fo(!)1062 1300 y Fq(\))p Fk(j)o Fp(~)-48 b(y)19 b Fk(ij)55 b(\024)1450 1164 y Fi(Z)1566 1300 y Fp(dt)1791 1232 y(c)1833 1247 y Fo(k)1892 1232 y Fp(c)1934 1247 y Fo(\014)p 1690 1277 393 4 v 1690 1368 a Fq(1)22 b(+)g Fk(j)p Fp(t)p Fk(j)1950 1339 y Fo(k)r Fj(\000)p Fn(1)2109 1300 y Fq(\(1)g(+)g Fp(t)2351 1259 y Fn(2)2390 1300 y Fq(\))2428 1259 y Fo(\014)2821 1232 y Fq(1)p 2502 1277 687 4 v 2502 1371 a(\(1)g(+)2719 1332 y Fn(1)p 2719 1348 36 4 v 2719 1405 a(4)2764 1371 y Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)20 b Fk(j)3065 1342 y Fn(2)3104 1371 y Fq(\))3142 1342 y Fo(\014)3231 1300 y Fq(.)-24 1537 y(Hence)34 b(the)f Fp(t)p Fq(-in)m(tegral)g(is)g(b)s(ounded)g(as)g(long)g(as)f(2)p Fp(\014)h(<)28 b(k)d Fk(\000)e Fq(2)32 b(whic)m(h)i(concludes)h(the)e (pro)s(of.)479 b Fa(2)123 1704 y Fq(As)30 b(an)f(aside)i(b)s(e)f(men)m (tioned)h(that)f(there)g(are)g(v)-5 b(arious)30 b(other)f(w)m(a)m(ys)j (to)d(get)h(estimates)h(on)f(the)g(in)m(tegral)-24 1825 y(k)m(ernel)39 b(of)e(the)h(semigroup)g Fp(e)1073 1788 y Fj(\000)p Fo(tH)1221 1825 y Fq(.)58 b(One)37 b(is)h(a)f(Com)m(b)s (es-Thomas-lik)m(e)j(argumen)m(t)e(whic)m(h)h(will)f(b)s(e)f(used)i(in) -24 1945 y(Section)33 b(5.)42 b(Another)32 b(is)g(to)f(simply)i(apply)f (the)g(diamagnetic)g(inequalit)m(y)i([10)o(,)e(Theo.)g(1.13],)f(whic)m (h)i(reads)-24 2065 y Fk(j)p Fp(e)49 2029 y Fj(\000)p Fo(tH)197 2065 y Fp(\036)p Fq(\()n Fp(~)-47 b(x)p Fq(\))p Fk(j)27 b(\024)h Fp(e)591 2029 y Fo(t)p Fn(\001)p Fo(=)p Fn(2)751 2065 y Fk(j)p Fp(\036)p Fk(j)p Fq(\()n Fp(~)-47 b(x)o Fq(\))30 b(for)f(p)s(ositiv)m(e)i Fp(V)51 b Fq(and)30 b(an)m(y)g Fp(\036)e Fk(2)g Fp(L)2251 2029 y Fn(2)2291 2065 y Fq(\()p Fm(R)2401 2029 y Fn(2)2440 2065 y Fq(\))i(and)f Fp(t)f(>)g Fq(0)h(where)i(\001)d(=)g Fp(@)3487 2029 y Fn(2)3482 2090 y(1)3543 2065 y Fq(+)16 b Fp(@)3691 2029 y Fn(2)3686 2090 y(2)3761 2065 y Fq(is)31 b(the)-24 2186 y(t)m(w)m(o-dimensional)39 b(Laplacian.)58 b(As)38 b(is)g(moreo)m(v)m (er)g(kno)m(wn)h(\(consult)f Fc(e.g.)58 b Fq([7)o(]\))38 b(that)f(the)g(in)m(tegral)h(k)m(ernels)-24 2306 y Fk(h)n Fp(~)-47 b(x)q Fk(j)p Fp(e)144 2270 y Fj(\000)p Fo(tH)291 2306 y Fk(j)o Fp(~)f(y)19 b Fk(i)42 b Fq(are)h(join)m(tly)g(con)m(tin)m (uous,)j(one)d(also)f(deduces)j(the)e(p)s(oin)m(t)m(wise)h(estimate)f (\(16\))f(b)s(ecause)i(the)-24 2427 y(r.h.s.)f(of)27 b(\(16\))f(is)i(precisely)i(the)e(in)m(tegral)g(k)m(ernel)h(of)d Fp(e)1968 2390 y Fo(t)p Fn(\001)p Fo(=)p Fn(2)2128 2427 y Fq(.)41 b(Here)28 b(the)g(ab)s(o)m(v)m(e)g(Dyson)g(series)h(argumen)m (t)f(w)m(as)-24 2547 y(used)36 b(b)s(ecause)g(the)f(same)h(tec)m (hnique)h(will)e(b)s(e)g(used)h(to)e(deriv)m(e)i(estimates)g(on)f(the)g (co)m(v)-5 b(arian)m(t)35 b(deriv)-5 b(ativ)m(es)-24 2667 y(of)32 b(the)h(in)m(tegral)g(k)m(ernels.)-24 2896 y Fh(Prop)s(osition)38 b(2)49 b Fl(L)-5 b(et)39 b Fp(V)916 2911 y Fo(!)1001 2896 y Fk(2)c Fp(L)1168 2859 y Fj(1)1243 2896 y Fq(\()p Fm(R)1353 2859 y Fn(2)1392 2896 y Fq(\))k Fl(and)f Fp(F)48 b Fk(2)35 b Fp(C)1951 2859 y Fo(k)1944 2920 y(c)1994 2896 y Fq(\()p Fm(R)p Fq(\))j Fl(with)h Fp(k)f(>)c Fq(6)p Fl(.)56 b(Then)38 b Fp(D)3069 2911 y Fo(j)3105 2896 y Fp(F)14 b Fq(\()p Fp(H)3301 2911 y Fo(!)3351 2896 y Fq(\))39 b Fl(is)f(an)h(inte)-5 b(gr)g(al)-24 3016 y(op)g(er)g(ator)35 b(satisfying)f(for)h(any)g Fp(\016)c(>)d Fq(0)1132 3257 y Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(D)1363 3272 y Fo(j)1399 3257 y Fp(F)14 b Fq(\()p Fp(H)1595 3272 y Fo(!)1645 3257 y Fq(\))p Fk(j)o Fp(~)-48 b(y)s Fk(ij)54 b(\024)2332 3189 y Fp(c)2374 3204 y Fo(\016)p 2026 3234 691 4 v 2026 3325 a Fq(1)22 b(+)g Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)20 b Fk(j)2496 3296 y Fo(k)r Fj(\000)p Fn(6)p Fj(\000)p Fo(\016)2762 3257 y Fl(.)-24 3495 y(Supp)-5 b(ose)41 b(that)g Fp(@)610 3510 y Fo(j)647 3495 y Fp(V)704 3510 y Fo(!)794 3495 y Fk(2)e Fp(L)965 3458 y Fj(1)1040 3495 y Fq(\()p Fm(R)1150 3458 y Fn(2)1190 3495 y Fq(\))i Fl(and)f Fp(F)53 b Fk(2)40 b Fp(C)1763 3458 y Fo(k)1756 3519 y(c)1805 3495 y Fq(\()p Fm(R)p Fq(\))h Fl(with)g Fp(k)h(>)d Fq(10)p Fl(.)63 b(Then)40 b Fp(D)2952 3510 y Fo(j)2989 3495 y Fp(D)3070 3510 y Fo(i)3098 3495 y Fp(F)14 b Fq(\()p Fp(H)3294 3510 y Fo(!)3344 3495 y Fq(\))41 b Fl(is)g(an)g(inte)-5 b(gr)g(al)-24 3615 y(op)g(er)g(ator)35 b(satisfying)f(for)h(any)g Fp(\016)c(>)d Fq(0)1060 3856 y Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(D)1291 3871 y Fo(j)1327 3856 y Fp(D)1408 3871 y Fo(i)1436 3856 y Fp(F)14 b Fq(\()p Fp(H)1632 3871 y Fo(!)1682 3856 y Fq(\))p Fk(j)o Fp(~)-48 b(y)s Fk(ij)54 b(\024)2386 3788 y Fp(c)2428 3803 y Fo(\016)p 2063 3833 726 4 v 2063 3924 a Fq(1)22 b(+)g Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)20 b Fk(j)2533 3895 y Fo(k)r Fj(\000)p Fn(10)p Fj(\000)p Fo(\016)2834 3856 y Fl(.)-24 4182 y Fh(Pro)s(of:)33 b Fq(Again)f(the)h(index)h Fp(!)i Fq(will)d(b)s(e)g(suppressed.)46 b(One)33 b(has)805 4462 y Fp(D)886 4477 y Fo(j)923 4462 y Fp(e)968 4421 y Fj(\000)p Fo(z)s(H)1181 4462 y Fq(=)56 b Fp(D)1394 4477 y Fo(j)1430 4462 y Fp(e)1475 4421 y Fj(\000)p Fo(z)s(H)1624 4432 y Fb(L)1696 4462 y Fk(\000)23 b Fp(z)1879 4327 y Fi(Z)1979 4353 y Fn(1)1934 4552 y(0)2035 4462 y Fp(dq)31 b(D)2241 4477 y Fo(j)2277 4462 y Fp(e)2322 4421 y Fj(\000)p Fo(z)s Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(H)2649 4432 y Fb(L)2728 4462 y Fp(V)38 b(e)2868 4421 y Fj(\000)p Fo(z)s(q)r(H)3093 4462 y Fq(.)699 b(\(18\))-24 4684 y(Hence)32 b(estimates)g(on)e(the)h(co)m(v)-5 b(arian)m(t)31 b(deriv)-5 b(ativ)m(es)32 b(of)e(the)h(Landau)f(Hamiltonian)h(will)g(b) s(e)f(needed.)44 b(Using)-24 4804 y Fk(j)17 b Fq(coth\()p Fp(z)t Fq(\))p Fk(j)28 b(\024)501 4765 y Fo(e)534 4741 y Fg(\000<)p Fb(e)p Fe(\()p Fb(z)r Fe(\))p 501 4781 239 4 v 533 4838 a Fj(<)p Fo(e)p Fn(\()p Fo(z)s Fn(\))771 4804 y Fq(+)21 b(1)32 b(and)h(the)f(inequalities)j(\(11\))c(\(from)h (no)m(w)h(on)f Fp(c)g Fq(denotes)i(v)-5 b(arying)32 b(constan)m(ts)i (and)-24 4938 y Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))29 b Fk(\025)f Fq(0\))1960 5611 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 228 334 a Fi(\014)228 393 y(\014)261 418 y Fq(\()p Fp({@)381 433 y Fo(x)421 443 y Fb(j)480 418 y Fk(\000)23 b Fp(\015)5 b(\016)679 433 y Fo(j;)p Fn(1)767 418 y Fp(x)822 433 y Fn(2)861 418 y Fq(\))17 b Fk(h)n Fp(~)-47 b(x)p Fk(j)p Fp(e)1083 377 y Fj(\000)p Fo(z)s(H)1232 388 y Fb(L)1282 418 y Fk(j)o Fp(~)f(y)19 b Fk(i)1416 334 y Fi(\014)1416 393 y(\014)1532 418 y Fk(\024)1803 351 y Fp(c)p 1703 395 243 4 v 1703 487 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1983 418 y Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)20 b Fk(j)2328 278 y Fi(\022)2401 418 y Fq(1)i(+)2677 351 y(1)p 2580 395 V 2580 487 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))2833 278 y Fi(\023)2950 418 y Fp(e)2995 377 y Fj(\000)3060 350 y Fe(1)p 3060 362 31 3 v 3060 403 a(4)3101 377 y Fj(j)o Fo(~)-34 b(x)o Fj(\000)p Fo(~)f(y)14 b Fj(j)3284 354 y Fe(2)3330 377 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3477 354 y Fg(\000)p Fe(1)3559 377 y Fn(\))3819 621 y Fq(\(19\))1532 789 y Fk(\024)2041 722 y Fp(c)p 1703 766 719 4 v 1703 871 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1945 843 y Fn(2)1985 786 y Fi(p)p 2085 786 338 4 v 85 x Fk(<)p Fp(e)p Fq(\()p Fp(z)2289 843 y Fj(\000)p Fn(1)2384 871 y Fq(\))2476 789 y(\()p Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1\))44 b Fp(e)3053 748 y Fj(\000)3118 721 y Fe(1)p 3118 733 31 3 v 3118 774 a(8)3158 748 y Fj(j)o Fo(~)-34 b(x)p Fj(\000)p Fo(~)f(y)14 b Fj(j)3342 725 y Fe(2)3388 748 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3535 725 y Fg(\000)p Fe(1)3617 748 y Fn(\))3681 789 y Fq(,)-24 1097 y(where)41 b(in)f(the)g(second)h (step)g Fp(ae)1193 1061 y Fj(\000)p Fn(2)p Fo(a)1320 1037 y Fe(2)1399 1097 y Fk(\024)f Fp(e)1561 1061 y Fj(\000)p Fo(a)1653 1037 y Fe(2)1732 1097 y Fq(for)f Fp(a)h(>)g Fq(0)f(w)m(as)h(used.)66 b(Let)40 b(no)m(w)g Fp(I)3094 1112 y Fn(1)3173 1097 y Fq(denote)h(the)f(in)m(tegral)-24 1217 y(k)m(ernel)35 b(of)d(the)h(second)h(con)m(tribution)f(in)g (\(18\).)43 b(Using)33 b(\(17\))f(and)h(then)g(again)f(\(15\),)g(one)h (gets)24 1497 y Fk(j)p Fp(I)95 1512 y Fn(1)135 1497 y Fk(j)49 b(\024)h Fp(c)561 1430 y Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1)p 408 1474 719 4 v 408 1580 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))650 1551 y Fn(2)690 1494 y Fi(p)p 790 1494 338 4 v 86 x Fk(<)p Fp(e)p Fq(\()p Fp(z)994 1551 y Fj(\000)p Fn(1)1089 1580 y Fq(\))1153 1497 y Fk(j)p Fp(z)t Fk(j)1292 1362 y Fi(Z)1391 1388 y Fn(1)1347 1587 y(0)1431 1497 y Fp(dq)1545 1362 y Fi(Z)1645 1497 y Fp(d)m(~)-46 b(r)1918 1430 y Fq(1)p 1768 1474 348 4 v 1768 1582 a(\(1)22 b Fk(\000)h Fp(q)t Fq(\))2072 1519 y Fe(3)p 2072 1531 31 3 v 2072 1572 a(2)2143 1497 y Fp(e)2188 1448 y Fj(\000)2331 1421 y Fe(1)p 2253 1433 188 3 v 2253 1474 a(8\(1)p Fg(\000)p Fb(q)r Fe(\))2450 1448 y Fj(j)o Fo(~)-34 b(x)o Fj(\000)n Fo(~)h(r)14 b Fj(j)2630 1425 y Fe(2)2676 1448 y Fj(<)p Fo(e)p Fn(\()p Fo(z)2823 1425 y Fg(\000)p Fe(1)2905 1448 y Fn(\))3083 1430 y Fq(1)p 2963 1474 290 4 v 2963 1566 a Fp(q)t Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))3279 1497 y Fp(e)3324 1451 y Fj(\000)3404 1424 y Fe(1)p 3389 1436 61 3 v 3389 1477 a(8)p Fb(q)3460 1451 y Fj(j)n Fo(~)-33 b(r)q Fj(\000)p Fo(~)e(y)14 b Fj(j)3637 1427 y Fe(2)3683 1451 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3830 1427 y Fg(\000)p Fe(1)3913 1451 y Fn(\))3819 1718 y Fq(\(20\))212 1918 y Fk(\024)50 b Fp(c)573 1851 y Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))22 b(+)g(1)p 419 1895 719 4 v 419 2000 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))661 1972 y Fn(3)701 1915 y Fi(p)p 801 1915 338 4 v 85 x Fk(<)p Fp(e)p Fq(\()p Fp(z)1005 1972 y Fj(\000)p Fn(1)1100 2000 y Fq(\))1301 1851 y Fk(j)p Fp(z)t Fk(j)p 1185 1895 V 1185 1986 a(<)p Fp(e)p Fq(\()p Fp(z)1389 1958 y Fj(\000)p Fn(1)1485 1986 y Fq(\))1560 1918 y Fp(e)1605 1877 y Fj(\000)1670 1850 y Fe(1)p 1670 1862 31 3 v 1670 1903 a(8)1711 1877 y Fj(j)o Fo(~)-34 b(x)o Fj(\000)p Fo(~)f(y)14 b Fj(j)1894 1854 y Fe(2)1940 1877 y Fj(<)p Fo(e)p Fn(\()p Fo(z)2087 1854 y Fg(\000)p Fe(1)2169 1877 y Fn(\))2233 1918 y Fq(.)-24 2202 y(Hence,)342 2384 y Fi(\014)342 2444 y(\014)376 2469 y Fk(h)n Fp(~)-47 b(x)p Fk(j)p Fp(D)579 2484 y Fo(j)615 2469 y Fp(e)660 2428 y Fj(\000)p Fo(z)s(H)818 2469 y Fk(j)o Fp(~)f(y)19 b Fk(i)952 2384 y Fi(\014)952 2444 y(\014)1041 2469 y Fk(\024)56 b Fp(c)1396 2402 y Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1)p 1242 2446 719 4 v 1242 2551 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1484 2522 y Fn(2)1524 2466 y Fi(p)p 1624 2466 338 4 v 85 x Fk(<)p Fp(e)p Fq(\()p Fp(z)1828 2522 y Fj(\000)p Fn(1)1923 2551 y Fq(\))2004 2329 y Fi(\022)2078 2469 y Fq(1)g(+)2494 2402 y Fk(j)p Fp(z)t Fk(j)p 2256 2446 580 4 v 2256 2537 a(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))p Fk(<)p Fp(e)p Fq(\()p Fp(z)2702 2509 y Fj(\000)p Fn(1)2798 2537 y Fq(\))2846 2329 y Fi(\023)2953 2469 y Fp(e)2998 2428 y Fj(\000)3063 2401 y Fe(1)p 3063 2413 31 3 v 3063 2454 a(8)3103 2428 y Fj(j)o Fo(~)-34 b(x)p Fj(\000)p Fo(~)f(y)14 b Fj(j)3287 2404 y Fe(2)3333 2428 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3480 2404 y Fg(\000)p Fe(1)3562 2428 y Fn(\))3819 2469 y Fq(\(21\))-24 2754 y(yielding)1060 2841 y Fi(\014)1060 2901 y(\014)1094 2926 y Fk(h)n Fp(~)-47 b(x)p Fk(j)p Fp(D)1297 2941 y Fo(j)1333 2926 y Fp(e)1378 2885 y Fj(\000)p Fn(\(1+)p Fo({t)p Fn(\))p Fo(H)1694 2926 y Fk(j)o Fp(~)f(y)20 b Fk(i)1829 2841 y Fi(\014)1829 2901 y(\014)1917 2926 y Fk(\024)56 b Fp(c)17 b Fq(\(1)k(+)h Fp(t)2350 2885 y Fn(2)2390 2926 y Fq(\))2428 2885 y Fn(2)2467 2926 y Fp(e)2512 2868 y Fj(\000)2585 2835 y Fg(j)o Fb(~)-31 b(x)o Fg(\000)p Fb(~)f(y)12 b Fg(j)2749 2814 y Fe(2)p 2577 2853 214 3 v 2577 2903 a(8\(1+)p Fb(t)2732 2889 y Fe(2)2767 2903 y(\))2838 2926 y Fq(.)954 b(\(22\))-24 3098 y(This)34 b(implies)g(just)f(as)g(in)g(Prop)s(osition)g(1)f(that)g Fp(D)1855 3113 y Fo(j)1892 3098 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\))31 b(satis\014es)k(the)e(stated)g(b)s(ound.)123 3266 y(T)-8 b(o)32 b(pro)m(v)m(e)i(the)f(second)h(statemen)m(t,)g(let)f (us)g(use)h(\(7\))e(whic)m(h)i(implies)153 3641 y Fp(D)234 3656 y Fo(j)270 3641 y Fp(D)351 3656 y Fo(i)379 3641 y Fp(e)424 3600 y Fj(\000)p Fo(z)s(H)665 3641 y Fq(=)83 b Fp(D)905 3656 y Fo(j)942 3641 y Fp(D)1023 3656 y Fo(i)1051 3641 y Fp(e)1096 3600 y Fj(\000)p Fo(z)s(H)1245 3611 y Fb(L)1317 3641 y Fq(+)22 b Fp(\016)1458 3656 y Fo(i;)p Fn(1)1569 3641 y Fp({\015)5 b(z)1739 3506 y Fi(Z)1839 3532 y Fn(1)1795 3731 y(0)1895 3641 y Fp(dq)20 b(D)2090 3656 y Fo(j)2143 3561 y Fi(\000)2189 3641 y Fk(r)2272 3656 y Fn(2)2311 3641 y Fp(e)2356 3600 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(z)s(H)2683 3611 y Fb(L)2734 3561 y Fi(\001)2797 3641 y Fp(V)h(e)2920 3600 y Fj(\000)p Fo(q)r(z)s(H)824 3972 y Fq(+)c Fp({z)1031 3836 y Fi(Z)1130 3863 y Fn(1)1086 4062 y(0)1186 3972 y Fp(dq)32 b(D)1393 3987 y Fo(j)1429 3972 y Fp(e)1474 3931 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(z)s(H)1801 3942 y Fb(L)1852 3972 y Fp(@)1903 3987 y Fo(i)1932 3972 y Fp(V)21 b(e)2055 3931 y Fj(\000)p Fo(q)r(z)s(H)2286 3972 y Fq(+)39 b Fp(z)2467 3836 y Fi(Z)2567 3863 y Fn(1)2522 4062 y(0)2623 3972 y Fp(dq)31 b(D)2829 3987 y Fo(j)2865 3972 y Fp(e)2910 3931 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(z)s(H)3237 3942 y Fb(L)3288 3972 y Fp(V)22 b(D)3448 3987 y Fo(i)3476 3972 y Fp(e)3521 3931 y Fj(\000)p Fo(q)r(z)s(H)3746 3972 y Fq(.)-24 4240 y(As)33 b(in)g(\(19\))f(one)h(sho)m(ws)h(for)e(the)h (\014rst)g(term)618 4442 y Fi(\014)618 4502 y(\014)651 4527 y Fk(h)n Fp(~)-47 b(x)p Fk(j)p Fp(D)854 4542 y Fo(j)890 4527 y Fp(D)971 4542 y Fo(i)1000 4527 y Fp(e)1045 4486 y Fj(\000)p Fo(z)s(H)1194 4497 y Fb(L)1244 4527 y Fk(j)o Fp(~)f(y)19 b Fk(i)1378 4442 y Fi(\014)1378 4502 y(\014)1466 4527 y Fk(\024)56 b Fp(c)1679 4460 y Fq(\()p Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1\))2166 4411 y Fn(2)p 1679 4504 528 4 v 1801 4595 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))2043 4567 y Fn(3)2398 4460 y Fq(1)p 2254 4504 338 4 v 2254 4595 a Fk(<)p Fp(e)p Fq(\()p Fp(z)2458 4567 y Fj(\000)p Fn(1)2553 4595 y Fq(\))2628 4527 y Fp(e)2673 4486 y Fj(\000)2738 4452 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)12 b Fg(j)2903 4432 y Fe(2)p 2738 4471 200 3 v 2807 4512 a(16)2959 4486 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3106 4463 y Fg(\000)p Fe(1)3188 4486 y Fn(\))3280 4527 y Fq(.)-24 4750 y(F)-8 b(or)46 b(the)g(second)i(term,)i(let)c(us)h(comm)m(ute)h Fp(D)1760 4765 y Fo(j)1842 4750 y Fq(and)f Fk(r)2129 4765 y Fn(2)2168 4750 y Fq(.)84 b(The)47 b(in)m(tegral)g(k)m(ernel)h (of)e(the)g(con)m(tribution)-24 4803 y Fi(R)43 4829 y Fn(1)23 4917 y(0)99 4883 y Fp(dq)t Fq([)p Fp(D)305 4898 y Fo(j)341 4883 y Fp(;)17 b Fk(r)468 4898 y Fn(2)507 4883 y Fq(])p Fp(e)579 4847 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(z)s(H)906 4858 y Fb(L)957 4883 y Fp(V)22 b(e)1081 4847 y Fj(\000)p Fo(q)r(z)s(H)1325 4883 y Fq(satis\014es)53 b(a)f(b)s(ound)g(as)g(\(20\).)101 b(Let)52 b Fp(I)2802 4898 y Fn(2)2893 4883 y Fq(b)s(e)g(the)h(in)m(tegral)f(k)m(ernel)i(of) -24 4942 y Fi(R)43 4968 y Fn(1)23 5057 y(0)99 5022 y Fp(dq)t Fk(r)280 5037 y Fn(2)319 5022 y Fp(D)400 5037 y Fo(j)436 5022 y Fp(e)481 4986 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(z)s(H)808 4997 y Fb(L)859 5022 y Fp(V)22 b(e)983 4986 y Fj(\000)p Fo(q)r(z)s(H)1175 5022 y Fq(.)43 b(Using)-24 5234 y Fi(\014)-24 5294 y(\014)10 5319 y Fk(h)n Fp(~)-47 b(x)p Fk(jr)215 5334 y Fn(2)254 5319 y Fp(D)335 5334 y Fo(j)371 5319 y Fp(e)416 5278 y Fj(\000)p Fo(z)s(H)565 5289 y Fb(L)616 5319 y Fk(j)o Fp(~)f(y)19 b Fk(i)750 5234 y Fi(\014)750 5294 y(\014)827 5319 y Fk(\024)45 b Fp(c)1029 5251 y Fq(\()p Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))22 b(+)g(1\))1516 5203 y Fn(2)p 1029 5296 528 4 v 1151 5387 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1393 5358 y Fn(2)1582 5319 y Fk(j)n Fp(~)-47 b(x)s Fk(\000)r Fp(~)f(y)20 b Fk(j)1844 5278 y Fn(2)1883 5319 y Fp(e)1928 5278 y Fj(\000)1993 5244 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)12 b Fg(j)2158 5223 y Fe(2)p 1993 5263 200 3 v 2078 5304 a(4)2214 5278 y Fj(<)p Fo(e)p Fn(\()p Fo(z)2361 5254 y Fg(\000)p Fe(1)2444 5278 y Fn(\))2519 5319 y Fk(\024)2668 5251 y Fp(c)17 b Fq(\()p Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1\))3214 5203 y Fn(2)p 2651 5296 620 4 v 2651 5387 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))2893 5358 y Fn(2)2933 5387 y Fk(<)p Fp(e)p Fq(\()p Fp(z)3137 5358 y Fj(\000)p Fn(1)3233 5387 y Fq(\))3308 5319 y Fp(e)3353 5278 y Fj(\000)3418 5244 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)12 b Fg(j)3583 5223 y Fe(2)p 3418 5263 200 3 v 3487 5304 a(16)3639 5278 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3786 5254 y Fg(\000)p Fe(1)3868 5278 y Fn(\))3933 5319 y Fq(,)1960 5611 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop -24 171 a Fq(and)33 b(p)s(erforming)g(a)f (similar)h(calculation)g(as)g(in)g(\(20\))f(one)h(\014nds)g(that)g(it)f (can)h(b)s(e)g(b)s(ounded)g(b)m(y)1121 462 y Fk(j)p Fp(I)1192 477 y Fn(2)1231 462 y Fk(j)55 b(\024)h Fp(c)1532 395 y Fk(j)p Fp(z)t Fk(j)17 b Fq(\()o Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1\))2141 346 y Fn(2)p 1527 439 659 4 v 1527 530 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1769 502 y Fn(3)1809 530 y Fk(<)p Fp(e)p Fq(\()p Fp(z)2013 502 y Fj(\000)p Fn(1)2108 530 y Fq(\))2146 502 y Fn(2)2223 462 y Fp(e)2268 421 y Fj(\000)2333 387 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)12 b Fg(j)2498 366 y Fe(2)p 2333 406 200 3 v 2402 447 a(16)2554 421 y Fj(<)p Fo(e)p Fn(\()p Fo(z)2701 397 y Fg(\000)p Fe(1)2783 421 y Fn(\))-24 732 y Fq(The)38 b(in)m(tegral)g(k)m(ernel)g(of)f(the)g(third)g(con)m (tribution)h(can)g(b)s(e)f(b)s(ounded)h(as)f(ab)s(o)m(v)m(e,)i(hence)f (let)f(us)h(fo)s(cus)f(on)-24 853 y(the)44 b(in)m(tegral)f(k)m(ernel)i Fp(I)861 868 y Fn(4)943 853 y Fq(of)e(the)g(forth)g(con)m(tribution.)75 b(Using)44 b(\(19\))e(and)h(\(21\),)i(one)e(gets)h(b)m(y)g(a)e(similar) -24 973 y(calculation)34 b(as)e(in)h(\(20\))462 1248 y Fk(j)p Fp(I)533 1263 y Fn(4)572 1248 y Fk(j)55 b(\024)g Fp(c)913 1181 y Fq(\()p Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))23 b(+)f(1\))1401 1145 y Fn(2)p 867 1225 620 4 v 867 1317 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1109 1288 y Fn(4)1149 1317 y Fk(<)p Fp(e)p Fq(\()p Fp(z)1353 1288 y Fj(\000)p Fn(1)1449 1317 y Fq(\))1513 1108 y Fi(\022)1586 1248 y Fq(1)g(+)1980 1181 y Fk(j)p Fp(z)t Fk(j)p 1765 1225 535 4 v 1765 1317 a(<)p Fq(\()p Fp(z)t Fq(\))p Fk(<)p Fp(e)p Fq(\()p Fp(z)2166 1288 y Fj(\000)p Fn(1)2262 1317 y Fq(\))2310 1108 y Fi(\023)2553 1181 y Fk(j)p Fp(z)t Fk(j)p 2437 1225 338 4 v 2437 1317 a(<)p Fp(e)p Fq(\()p Fp(z)2641 1288 y Fj(\000)p Fn(1)2737 1317 y Fq(\))2812 1248 y Fp(e)2857 1207 y Fj(\000)2922 1174 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)12 b Fg(j)3087 1153 y Fe(2)p 2922 1192 200 3 v 2991 1233 a(16)3143 1207 y Fj(<)p Fo(e)p Fn(\()p Fo(z)3290 1184 y Fg(\000)p Fe(1)3372 1207 y Fn(\))3437 1248 y Fq(.)-24 1480 y(Finally)-8 b(,)35 b(the)f(n)m(um)m(b)s(er)i Fp(k)g Fq(determining)g(the)e(deca)m(y)h(of)f(the)g(in)m(tegral)g(k)m (ernel)i(of)d Fp(D)3047 1495 y Fo(j)3084 1480 y Fp(D)3165 1495 y Fo(i)3193 1480 y Fp(F)14 b Fq(\()p Fp(H)3389 1495 y Fo(!)3439 1480 y Fq(\))34 b(dep)s(ends)h(on)-24 1600 y(the)j(leading)g(p)s(o)m(w)m(er)h(in)e Fp(t)h Fq(of)1087 1516 y Fi(\014)1087 1575 y(\014)1120 1600 y Fk(h)n Fp(~)-47 b(x)q Fk(j)p Fp(D)1324 1615 y Fo(j)1360 1600 y Fp(D)1441 1615 y Fo(i)1469 1600 y Fp(e)1514 1564 y Fj(\000)p Fn(\(1+)p Fo({t)p Fn(\))p Fo(H)1830 1600 y Fk(j)o Fp(~)f(y)19 b Fk(i)1964 1516 y Fi(\014)1964 1575 y(\014)1997 1600 y Fq(.)59 b(Comparing)38 b(the)f(ab)s(o)m(v)m(e)i(con)m(tributions)g(one) e(sees)-24 1721 y(that)c(this)g(p)s(o)m(w)m(er)g(is)h(determined)g(b)m (y)f Fp(I)1448 1736 y Fn(4)1488 1721 y Fq(,)g(and,)f(setting)i Fp(z)e Fq(=)c(1)22 b(+)g Fp({t)p Fq(,)32 b(one)h(has)1357 1993 y Fk(j)p Fp(I)1428 2008 y Fn(4)1468 1993 y Fk(j)55 b(\024)g Fp(c)28 b Fq(\(1)22 b(+)g Fp(t)1995 1952 y Fn(2)2035 1993 y Fq(\))2073 1952 y Fn(4)2140 1993 y Fp(e)2185 1936 y Fj(\000)2272 1902 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)12 b Fg(j)2437 1881 y Fe(2)p 2250 1921 245 3 v 2250 1970 a(16\(1+)p Fb(t)2435 1956 y Fe(2)2470 1970 y(\))2541 1993 y Fq(.)-24 2168 y(As)33 b(in)g(Prop)s(osition)g(1,)g (the)g(statemen)m(t)h(of)e(the)h(prop)s(osition)f(follo)m(ws.)1324 b Fa(2)123 2335 y Fq(Let)33 b(us)h(remark)g(that)f(Prop)s(osition)g(2)g (implies)i(in)e(particular)h(that)f(the)g(in)m(tegral)h(k)m(ernel)h(of) e Fp(F)14 b Fq(\()p Fp(H)3806 2350 y Fo(!)3855 2335 y Fq(\))34 b(is)-24 2456 y(t)m(wice)f(di\013eren)m(tiable.)45 b(In)32 b(dimension)g(2,)g(the)g(same)g(argumen)m(t)g(go)s(es)f (through)g(for)g Fp(D)3195 2419 y Fn(2)3192 2480 y(1)3234 2456 y Fp(D)3318 2419 y Fn(2)3315 2480 y(2)3374 2456 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\),)31 b(but)g(not)-24 2576 y(for)h Fp(D)209 2540 y Fn(3)206 2601 y Fo(j)265 2576 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\).)43 b(In)33 b(higher)g (dimension,)h(more)f(regularit)m(y)g(can)g(b)s(e)g(obtained.)-24 2909 y Fr(5)161 b(In)l(tegralk)l(ernels)51 b(of)j(op)t(erators)e(on)i (the)e(half-plane)-24 3128 y Fq(The)33 b(aim)e(of)g(this)h(section)h (is)f(to)f(sho)m(w)h(that)g(Prop)s(osition)f(1)g(and)h(the)g(part)f(of) g(Prop)s(osition)h(2)f(concerning)-24 3248 y(co)m(v)-5 b(arian)m(t)39 b(deriv)-5 b(ativ)m(es)41 b(in)e(the)g(1-direction)f (remain)h(essen)m(tially)i(v)-5 b(alid)39 b(for)f(the)h(op)s(erators)f Fp(H)3574 3263 y Fo(!)r(;s)3715 3248 y Fq(on)h(the)-24 3369 y(half-plane.)k(This)31 b(is)f(done)g(b)m(y)h(pro)m(ving)f (estimates)i(lik)m(e)f(\(12\))e(and)h(\(19\))f(for)g(the)h(k)m(ernel)i (of)d(the)h(semigroup)-24 3489 y(generated)38 b(b)m(y)h(the)e(Landau)h (op)s(erator)1519 3464 y(^)1494 3489 y Fp(H)1575 3504 y Fo(L)1664 3489 y Fq(\(and)g(its)f(co)m(v)-5 b(arian)m(t)38 b(deriv)-5 b(ativ)m(e\))39 b(with)f(Diric)m(hlet)g(b)s(oundary)-24 3609 y(conditions)48 b(at)f Fp(s)52 b Fq(=)f(0.)86 b(Co)m(v)-5 b(ariance)48 b(then)f(implies)i(that)d(these)i(estimates)h(also)e(hold) g(for)f(arbitrary)-24 3730 y Fp(s)37 b(<)g Fk(1)h Fq(and)g(the)g(p)s (erturbativ)m(e)i(argumen)m(ts)f(based)g(on)f(the)g(Dyson)h(series)h (expansion)f(can)f(b)s(e)h(directly)-24 3850 y(transp)s(osed)33 b(to)f(obtain)g(a)f(p)s(o)m(w)m(er)i(la)m(w)f(deca)m(y)i(of)d(the)h(in) m(tegral)h(k)m(ernels)h(of)d(functions)i(of)e(the)i(Hamiltonian)-24 3971 y(on)g(the)g(half-plane.)-24 4246 y Fh(Prop)s(osition)38 b(3)49 b Fl(F)-7 b(or)34 b Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))28 b Fp(>)g Fq(0)34 b Fl(and)h Fp(n)27 b Fq(=)h(0)p Fp(;)17 b Fq(1)p Fp(;)g Fq(2)p Fl(.)43 b(Then)638 4407 y Fi(\014)638 4466 y(\014)638 4526 y(\014)699 4521 y Fk(h)n Fp(~)-47 b(x)16 b Fk(j)p Fp(D)921 4480 y Fo(n)918 4546 y Fn(1)984 4521 y Fp(e)1029 4480 y Fj(\000)p Fo(z)1138 4463 y Fn(^)1120 4480 y Fo(H)1178 4491 y Fb(L)1229 4521 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)1390 4407 y Fi(\014)1390 4466 y(\014)1390 4526 y(\014)1479 4521 y Fk(\024)56 b Fp(c)1680 4454 y Fq(1)22 b(+)g Fk(j)p Fp(z)t Fk(j)1954 4418 y Fo(n)p Fn(+1)p 1680 4498 412 4 v 1688 4596 a Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))1940 4532 y Fb(n)p 1941 4544 39 3 v 1945 4585 a Fe(2)1989 4559 y Fn(+1)2118 4521 y Fq(exp)2284 4381 y Fi(\022)2357 4521 y Fk(\000)2444 4454 y(j)n 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y Fj(\000)p Fo(tH)322 377 y Fb(!)r(;s)419 410 y Fk(j)o Fp(~)f(y)19 b Fk(i)553 325 y Fi(\014)553 385 y(\014)641 410 y Fk(\024)56 b(h)n Fp(~)-47 b(x)p Fk(j)p Fp(e)941 369 y Fo(t)p Fn(\001)1025 377 y Fb(s)1063 410 y Fk(j)o Fp(~)f(y)20 b Fk(i)55 b Fq(=)1441 343 y(1)p 1394 387 143 4 v 1394 478 a(4)p Fp(\031)t(t)1563 410 y(e)1608 369 y Fj(\000j)o Fo(~)-34 b(x)p Fj(\000)p Fo(~)f(y)14 b Fj(j)1847 346 y Fe(2)1881 369 y Fo(=t)1962 329 y Fi(\000)2008 410 y Fq(1)22 b Fk(\000)h Fp(e)2224 369 y Fj(\000)p Fn(2\()p Fo(x)2381 378 y Fe(2)2416 369 y Fn(+)p Fo(s)p Fn(\)\()p Fo(y)2593 378 y Fe(2)2628 369 y Fn(+)p Fo(s)p Fn(\))p Fo(=t)2807 329 y Fi(\001)2897 410 y Fp(\037)p Fq(\()p Fp(x)3051 425 y Fn(2)3119 410 y Fk(\025)28 b(\000)p Fp(s)p Fq(\))17 b Fp(\037)p Fq(\()p Fp(y)3549 425 y Fn(2)3616 410 y Fk(\025)28 b(\000)p Fp(s)p Fq(\))33 b(.)3819 561 y(\(23\))-24 682 y(This)h(also)f(sho)m(ws)h(ho)m (w)f(the)g(in)m(tegral)g(k)m(ernels)i(of)d(functions)i(of)e Fp(H)2451 697 y Fo(!)r(;s)2586 682 y Fq(v)-5 b(anish)33 b(near)g Fp(x)3159 697 y Fn(2)3226 682 y Fq(=)28 b Fk(\000)p Fp(s)33 b Fq(or)f Fp(y)3653 697 y Fn(2)3720 682 y Fq(=)27 b Fk(\000)p Fp(s)p Fq(.)123 860 y(F)-8 b(or)28 b(the)i(pro)s(of)f(of)f (Prop)s(osition)i(3,)g(the)g(semigroup)g(of)2207 834 y(^)2182 860 y Fp(H)2263 875 y Fo(L)2344 860 y Fq(is)g(calculated)g (via)g(F)-8 b(ourier)29 b(transform)h(just)-24 980 y(as)j(in)g(\(10\):) 970 1245 y Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(e)1154 1204 y Fj(\000)p Fo(t)1252 1187 y Fn(^)1234 1204 y Fo(H)1292 1215 y Fb(L)1343 1245 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)55 b Fq(=)1664 1110 y Fi(Z)1791 1178 y Fp(dk)p 1790 1222 108 4 v 1790 1314 a Fq(2)p Fp(\031)1935 1245 y(e)1980 1204 y Fo({k)r Fn(\()p Fo(x)2109 1213 y Fe(1)2144 1204 y Fj(\000)p Fo(y)2234 1213 y Fe(1)2268 1204 y Fn(\))2327 1245 y Fk(h)p Fp(x)2421 1260 y Fn(2)2461 1245 y Fk(j)p Fp(e)2534 1204 y Fj(\000)p Fo(t)2615 1187 y Fn(^)2614 1204 y Fo(h)p Fn(\()p Fo(k)r Fn(\))2753 1245 y Fk(j)p Fp(y)2829 1260 y Fn(2)2867 1245 y Fk(i)32 b Fq(,)854 b(\(24\))-24 1489 y(where)255 1463 y(^)254 1489 y Fp(h)p Fq(\()p Fp(k)s Fq(\))28 b(=)581 1450 y Fn(1)p 581 1466 36 4 v 581 1523 a(2)627 1489 y Fq(\()p Fk(\000)p Fp(@)798 1453 y Fn(2)852 1489 y Fq(+)14 b Fp(\015)998 1453 y Fn(2)1037 1489 y Fq(\()p Fp(X)22 b Fq(+)1278 1450 y Fo(k)p 1277 1466 41 4 v 1277 1523 a(\015)1328 1489 y Fq(\))1366 1453 y Fn(2)1405 1489 y Fq(\))28 b(with)h(Diric)m(hlet)h(b)s(oundary)f (conditions)g(at)f(the)h(origin.)42 b(As)29 b(w)m(e)h(did)-24 1609 y(not)f(succeed)h(in)f(calculating)g(this)g(k)m(ernel)h (explicitly)-8 b(,)32 b(recourse)d(to)f(more)h(abstract)g(analytical)g (argumen)m(ts)-24 1730 y(is)j(necessary)-8 b(.)45 b(F)-8 b(or)29 b(a)i(complex)h(dilation)f(argumen)m(t)g(on)g(the)g(heat)f(k)m (ernel,)j(the)e(follo)m(wing)g(will)h(b)s(e)e(needed:)-24 1958 y Fh(Lemma)39 b(2)49 b Fp(k)26 b Fq(+)c Fp({\024)28 b Fk(2)g Fm(C)f Fk(7!)h Fp(e)1134 1922 y Fj(\000)p Fo(t)1215 1904 y Fn(^)1214 1922 y Fo(h)p Fn(\()p Fo(k)r Fn(+)p Fo({\024)p Fn(\))1506 1958 y Fl(is)35 b(entir)-5 b(e)34 b(for)h(al)5 b(l)34 b Fp(t)28 b(>)g Fq(0)35 b Fl(and)f(the)h(inte)-5 b(gr)g(al)34 b(kernel)g(satis\014es)1120 2084 y Fi(\014)1120 2144 y(\014)1120 2204 y(\014)1153 2199 y Fk(h)p Fp(x)p Fk(j)p Fp(e)1320 2158 y Fj(\000)p Fo(t)1401 2140 y Fn(^)1400 2158 y Fo(h)p Fn(\()p Fo(k)r Fn(+)p Fo({\024)p Fn(\))1657 2199 y Fk(j)p Fp(y)t Fk(i)1776 2084 y Fi(\014)1776 2144 y(\014)1776 2204 y(\014)1863 2199 y Fk(\024)56 b Fp(e)2051 2130 y Fe(1)p 2051 2142 31 3 v 2051 2184 a(2)2104 2158 y Fo(t\024)2170 2134 y Fe(2)2236 2199 y Fk(h)p Fp(x)p Fk(j)p Fp(e)2403 2158 y Fj(\000)p Fo(th)p Fn(\()p Fo(k)r Fn(\))2622 2199 y Fk(j)p Fp(y)t Fk(i)33 b Fl(.)1015 b Fq(\(25\))-24 2475 y Fh(Pro)s(of:)28 b Fq(First)f(let)h(us)g(sho)m(w)h (that)e Fp(X)35 b Fq(is)28 b(relativ)m(ely)i(b)s(ounded)e(w.r.t.)2531 2449 y(^)2530 2475 y Fp(h)p Fq(\(0\))f(with)h(relativ)m(e)h(b)s(ound)f (0.)41 b(There-)-24 2595 y(fore)34 b(let)g Fk(j)p Fp(n)p Fk(i)f Fq(denote)h(the)g(Hermite)h(eigenfunctions)h(of)d Fp(h)p 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b(desired)i(analyticit)m(y) f(prop)s(ert)m(y)-8 b(.)123 3389 y(In)30 b(order)h(to)f(pro)m(v)m(e)i (the)f(estimate,)h(let)f(us)g(cite)g(a)f(norm-con)m(v)m(ergen)m(t)i(v)m (ersion)g(of)e(the)h(T)-8 b(rotter)31 b(pro)s(duct)-24 3509 y(form)m(ula)36 b(from)e([8]:)49 b(Giv)m(en)36 b(t)m(w)m(o)f Fp(m)p Fq(-sectorial)h(op)s(erators)f Fp(A;)17 b(B)40 b Fq(on)35 b(a)f(giv)m(en)i(Hilb)s(ert)g(space)g Fk(H)g Fq(satisfying)-24 3629 y(that)d(\()p Fp(A)22 b Fq(+)g(1\))506 3593 y Fj(\000)p Fn(1)632 3629 y Fq(is)34 b(compact)f(and)f Fp(D)s Fq(\()p Fp(A)p Fq(\))22 b Fk(\\)h Fp(D)s Fq(\()p Fp(B)5 b Fq(\))32 b(is)h(dense)h(in)f Fk(H)q Fq(,)1262 3870 y Fp(e)1307 3829 y Fj(\000)p Fo(t)p Fn(\()p Fo(A)1485 3817 y Fn(_)1467 3829 y(+)p Fo(B)s Fn(\))1666 3870 y Fq(=)79 b(lim)1797 3930 y Fo(n)p Fj(!1)1998 3760 y Fi(\020)2057 3870 y Fp(e)2102 3829 y Fj(\000)2175 3802 y Fb(t)p 2167 3814 39 3 v 2167 3855 a(n)2216 3829 y Fo(A)2273 3870 y Fp(e)2318 3829 y Fj(\000)2390 3802 y Fb(t)p 2383 3814 V 2383 3855 a(n)2431 3829 y Fo(B)2492 3760 y Fi(\021)2551 3782 y Fo(n)2647 3870 y Fq(,)-24 4083 y(where)31 b(the)f(con)m(v)m (ergence)i(is)f(in)e(the)h(norm)g(top)s(ology)f(and)h Fp(A)2261 4068 y Fq(_)2237 4083 y(+)p Fp(B)35 b Fq(is)30 b(the)g(form)f(sum.)43 b(This)31 b(will)f(b)s(e)g(applied)-24 4203 y(for)j Fp(A)28 b Fq(=)331 4177 y(^)330 4203 y Fp(h)q Fq(\()p Fp(k)s Fq(\))22 b Fk(\000)h Fp(\013)q(W)14 b Fq(\()p Fp(k)s Fq(\))31 b(and)i Fp(B)g Fq(=)28 b Fp(\013)q(W)14 b Fq(\()p Fp(k)s Fq(\))22 b Fk(\000)g Fp({\015)5 b(\024)p Fq(\()p Fp(X)31 b Fq(+)2192 4164 y Fo(k)p 2191 4180 41 4 v 2191 4238 a(\015)2241 4203 y Fq(\))i(where)h Fp(W)14 b Fq(\()p Fp(k)s Fq(\))27 b(=)2971 4159 y Fo(\015)3011 4136 y Fe(2)p 2971 4180 75 4 v 2991 4238 a Fn(2)3056 4203 y Fq(\()p Fp(X)j Fq(+)3314 4164 y Fo(k)p 3313 4180 41 4 v 3313 4238 a(\015)3363 4203 y Fq(\))3401 4167 y Fn(2)3463 4203 y Fq(+)22 b(1.)44 b(Indeed,)-24 4324 y Fp(A)39 b Fq(is)f(a)g(strictly)i(p)s(ositiv)m(e)g(selfadjoin)m(t)f(op)s 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2522 4749 V 2522 4790 a(n)2571 4764 y Fo(B)2631 4695 y Fi(\021)2691 4717 y Fo(n)2787 4805 y Fq(.)-24 5006 y(Setting)33 b Fp(x)28 b Fq(=)g Fp(r)543 5021 y Fn(0)615 5006 y Fq(and)33 b Fp(y)d Fq(=)e Fp(r)1031 5021 y Fo(n)1078 5006 y Fq(,)33 b(one)f(can)h(therefore)g(b)s(ound)g(as)g (follo)m(ws:)1935 5611 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 167 320 a Fi(\014)167 379 y(\014)167 439 y(\014)201 434 y Fk(h)p Fp(x)p Fk(j)340 324 y Fi(\020)399 434 y Fp(e)444 393 y Fj(\000)516 366 y Fb(t)p 509 378 39 3 v 509 419 a(n)557 393 y Fo(A)614 434 y Fp(e)659 393 y Fj(\000)731 366 y Fb(t)p 724 378 V 724 419 a(n)772 393 y Fo(B)833 324 y Fi(\021)892 346 y Fo(n)956 434 y Fk(j)p Fp(y)t Fk(i)1075 320 y Fi(\014)1075 379 y(\014)1075 439 y(\014)1190 434 y Fk(\024)1350 299 y Fi(Z)1406 524 y Ff(R)1454 494 y Fb(n)p Fg(\000)p Fe(1)1454 545 y(+)1595 434 y Fp(dr)1690 449 y Fn(1)1746 434 y Fk(\001)17 b(\001)g(\001)e Fp(dr)1974 449 y Fo(n)p Fj(\000)p Fn(1)2169 310 y Fo(n)2127 339 y Fi(Y)2130 549 y Fo(j)t Fn(=1)2282 434 y Fk(h)p Fp(r)2365 449 y Fo(j)t Fj(\000)p Fn(1)2492 434 y Fk(j)p Fp(e)2565 393 y Fj(\000)2637 366 y Fb(t)p 2629 378 V 2629 419 a(n)2678 393 y Fn(\()2706 375 y(^)2705 393 y Fo(h)p Fn(\()p Fo(k)r Fn(\))p Fj(\000)p Fo(\013W)10 b Fn(\()p Fo(k)r Fn(\)\))3141 434 y Fk(j)p Fp(r)3213 449 y Fo(j)3249 434 y Fk(i)17 b Fp(e)3350 393 y Fj(\000)3422 366 y Fb(t)p 3414 378 V 3414 419 a(n)3463 393 y Fo(\013W)10 b Fn(\()p Fo(k)r Fn(\)\()p Fo(r)3737 403 y Fb(j)3770 393 y Fn(\))1191 722 y Fq(=)83 b Fk(h)p Fp(x)p Fk(j)1489 612 y Fi(\020)1548 722 y Fp(e)1593 681 y Fj(\000)1666 654 y Fb(t)p 1658 666 V 1658 707 a(n)1707 681 y Fn(\()1735 664 y(^)1734 681 y Fo(h)p Fn(\()p Fo(k)r Fn(\))p Fj(\000)p Fo(\013W)10 b Fn(\()p Fo(k)r Fn(\)\))2170 722 y Fp(e)2215 681 y Fj(\000)2287 654 y Fb(t)p 2280 666 V 2280 707 a(n)2328 681 y Fo(\013W)g Fn(\()p Fo(k)r Fn(\))2547 612 y Fi(\021)2607 634 y Fo(n)2670 722 y Fk(j)p Fp(y)t Fk(i)1191 925 y Fq(=)83 b Fk(h)p Fp(x)p Fk(j)p Fp(e)1517 884 y Fj(\000)p Fo(t)1598 866 y Fn(^)1597 884 y Fo(h)p Fn(\()p Fo(k)r Fn(\))1736 925 y Fk(j)p Fp(y)t Fk(i)31 b Fq(,)-24 1138 y(where)38 b(the)e(last)h(equalit)m(y)h(follo)m(ws)f(from)f(recomp)s(osing)h(with) f(the)h(T)-8 b(rotter)36 b(form)m(ula.)55 b(T)-8 b(o)36 b(conclude,)j(one)-24 1258 y(just)26 b(notes)g(that)f(the)h(in)m (tegral)g(k)m(ernel)h(of)e Fp(e)1552 1222 y Fj(\000)p Fo(t)1633 1204 y Fn(^)1632 1222 y Fo(h)p Fn(\()p Fo(k)r Fn(\))1796 1258 y Fq(is)g(b)s(ounded)h(b)m(y)h(that)e(of)g Fp(e)2759 1222 y Fj(\000)p Fo(th)p Fn(\()p Fo(k)r Fn(\))3003 1258 y Fq(\(this)h(follo)m(ws)g(easily)-8 b(,)28 b(e.g.,)-24 1378 y(from)34 b(the)g(F)-8 b(eynman-Kac)34 b(path-in)m(tegral)f(in)h (whic)m(h)h(Diric)m(hlet)g(b)s(oundary)f(conditions)g(are)g(incorp)s (orated)-24 1499 y(b)m(y)g(c)m(haracteristic)g(functions\).)2755 b Fa(2)-24 1766 y Fh(Lemma)39 b(3)49 b Fl(F)-7 b(or)34 b Fp(t)28 b(>)g Fq(0)34 b Fl(and)h Fp(n)28 b Fq(=)f(0)p Fp(;)17 b Fq(1)p Fp(;)g Fq(2)p Fl(,)809 1943 y Fi(\014)809 2003 y(\014)809 2063 y(\014)870 2058 y Fp(D)954 2017 y Fo(n)951 2082 y Fn(1)1017 2058 y Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(e)1201 2017 y Fj(\000)p Fo(t)1299 2000 y Fn(^)1281 2017 y Fo(H)1339 2028 y Fb(L)1390 2058 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)1552 1943 y Fi(\014)1552 2003 y(\014)1552 2063 y(\014)1640 2058 y Fk(\024)56 b Fp(c)1842 1990 y Fq(\(1)21 b(+)h Fp(t)p Fq(\))2121 1954 y Fo(n)p Fn(+)2229 1927 y Fe(1)p 2229 1939 31 3 v 2229 1980 a(2)p 1842 2035 433 4 v 1964 2132 a Fp(t)2009 2068 y Fb(n)p 2009 2080 39 3 v 2013 2122 a Fe(2)2057 2096 y Fn(+1)2300 2058 y Fq(exp)2466 1917 y Fi(\022)2539 2058 y Fk(\000)2626 1990 y(j)n Fp(~)-47 b(x)23 b Fk(\000)f Fp(~)-48 b(y)s Fk(j)2911 1954 y Fn(2)p 2626 2035 324 4 v 2678 2126 a Fq(2)2727 2097 y Fo(n)p Fn(+1)2864 2126 y Fp(t)2960 1917 y Fi(\023)3085 2058 y Fl(.)-24 2346 y Fh(Pro)s(of:)33 b Fq(Applying)g Fp(D)818 2310 y Fo(n)815 2371 y Fn(1)898 2346 y Fq(to)f(equation)h(\(24\))f(and)h(m)m(ultiplying)h(it)f(with)g Fp(e)2696 2310 y Fo(\024)p Fn(\()p Fo(x)2804 2319 y Fe(1)2839 2310 y Fj(\000)p Fo(y)2929 2319 y Fe(1)2963 2310 y Fn(\))2994 2346 y Fq(,)g Fp(\024)28 b Fk(2)g Fm(R)k Fq(leads)i(to)351 2611 y Fp(e)396 2570 y Fo(\024)p Fn(\()p Fo(x)504 2579 y Fe(1)539 2570 y Fj(\000)p Fo(y)629 2579 y Fe(1)663 2570 y Fn(\))723 2611 y Fp(D)807 2570 y Fo(n)804 2636 y Fn(1)870 2611 y Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(e)1054 2570 y Fj(\000)p Fo(t)1152 2553 y Fn(^)1134 2570 y Fo(H)1192 2581 y Fb(L)1242 2611 y Fk(j)o Fp(~)-48 b(y)20 b Fk(i)55 b Fq(=)1563 2476 y Fi(Z)1619 2701 y Ff(R)1699 2544 y Fp(dk)p 1697 2588 108 4 v 1697 2680 a Fq(2)p Fp(\031)1843 2611 y Fq(\()p Fk(\000)p Fp(k)25 b Fk(\000)e Fp(\015)5 b(x)2245 2626 y Fn(2)2285 2611 y Fq(\))2323 2570 y Fo(n)2397 2611 y Fp(e)2442 2570 y Fo({)p Fn(\()p Fo(k)r Fj(\000)p Fo({\024)p Fn(\)\()p Fo(x)2744 2579 y Fe(1)2780 2570 y Fj(\000)p Fo(y)2870 2579 y Fe(1)2904 2570 y Fn(\))2936 2611 y Fk(h)p Fp(x)3030 2626 y Fn(2)3069 2611 y Fk(j)p Fp(e)3142 2570 y Fj(\000)p Fo(t)3223 2553 y Fn(^)3222 2570 y Fo(h)p Fn(\()p Fo(k)r Fn(\))3361 2611 y Fk(j)p Fp(y)3437 2626 y Fn(2)3475 2611 y Fk(i)33 b Fq(.)-24 2835 y(Let)j(us)g(c)m(hange)h(v)-5 b(ariables)36 b Fp(k)27 b Fk(\000)e Fp({\024)33 b Fk(7!)f Fp(k)s Fq(,)37 b(then)f(use)h (analyticit)m(y)g(\(Lemma)f(2\))f(and)h(deca)m(y)h(prop)s(erties)f(on) -24 2956 y(the)d(b)s(oundaries)h(of)e(a)g(Cauc)m(h)m(y)j(con)m(tour)e (in)g(order)f(to)g(obtain:)188 3313 y Fp(e)233 3272 y Fo(\024)p Fn(\()p Fo(x)341 3281 y Fe(1)375 3272 y Fj(\000)p Fo(y)465 3281 y Fe(1)500 3272 y Fn(\))559 3313 y Fp(D)643 3272 y Fo(n)640 3338 y Fn(1)706 3313 y Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(e)890 3272 y Fj(\000)p Fo(t)988 3255 y Fn(^)970 3272 y Fo(H)1028 3283 y Fb(L)1079 3313 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)55 b Fq(=)1399 3177 y Fi(Z)1455 3403 y Ff(R)1535 3245 y Fp(dk)p 1533 3290 V 1533 3381 a Fq(2)p Fp(\031)1679 3313 y Fq(\()p Fk(\000)p Fp(k)25 b Fk(\000)e Fp({\024)f Fk(\000)h Fp(\015)5 b(x)2290 3328 y Fn(2)2330 3313 y Fq(\))2368 3272 y Fo(n)2443 3313 y Fp(e)2488 3272 y Fo({k)r Fn(\()p Fo(x)2617 3281 y Fe(1)2651 3272 y Fj(\000)p Fo(y)2741 3281 y Fe(1)2776 3272 y Fn(\))2807 3313 y Fk(h)p Fp(x)2901 3328 y Fn(2)2941 3313 y Fk(j)p Fp(e)3014 3272 y Fj(\000)p Fo(t)3095 3254 y Fn(^)3094 3272 y Fo(h)p Fn(\()p Fo(k)r Fn(+)p Fo({\024)p Fn(\))3351 3313 y Fk(j)p Fp(y)3427 3328 y Fn(2)3466 3313 y Fk(i)32 b Fq(.)255 b(\(26\))-24 3589 y(No)m(w)30 b(the)g(estimate)g(\(25\))f (will)h(b)s(e)g(used,)h(along)e(with)h(the)f(fact)g Fk(h)p Fp(x)p Fk(j)p Fp(e)2504 3553 y Fj(\000)p Fo(th)p Fn(\()p Fo(k)r Fn(\))2723 3589 y Fk(j)p Fp(y)t Fk(i)d Fq(=)i Fk(h)p Fp(x)15 b Fq(+)3184 3550 y Fo(k)p 3183 3566 41 4 v 3183 3624 a(\015)3233 3589 y Fk(j)p Fp(e)3306 3553 y Fj(\000)p Fo(th)p Fn(\(0\))3521 3589 y Fk(j)p Fp(y)j Fq(+)3718 3550 y Fo(k)p 3717 3566 V 3717 3624 a(\015)3767 3589 y Fk(i)29 b Fq(and)-24 3710 y(the)k(follo)m(wing)g(estimate)h(for) e(the)h(Mehler)h(k)m(ernel:)138 4075 y Fk(h)p Fp(x)p Fk(j)p Fp(e)305 4034 y Fj(\000)p Fo(th)p Fn(\(0\))521 4075 y Fk(j)p Fp(y)t Fk(i)82 b Fq(=)882 3941 y Fi(r)p 981 3941 486 4 v 1196 4007 a Fp(\015)p 991 4052 466 4 v 991 4143 a Fq(2)p Fp(\031)21 b Fq(sinh)q(\()p Fp(\015)5 b(t)p Fq(\))1511 4075 y(exp)1677 3934 y Fi(\022)1750 4075 y Fk(\000)1837 4007 y Fp(\015)p 1837 4052 57 4 v 1841 4143 a Fq(4)1920 4075 y(coth)2121 3934 y Fi(\022)2204 4007 y Fp(\015)g(t)p 2204 4052 92 4 v 2225 4143 a Fq(2)2305 3934 y Fi(\023)2395 4075 y Fk(j)p Fp(x)22 b Fk(\000)h Fp(y)t Fk(j)2680 4034 y Fn(2)2741 4075 y Fk(\000)2850 4007 y Fp(\015)p 2850 4052 57 4 v 2854 4143 a Fq(4)2933 4075 y(tanh)3144 3934 y Fi(\022)3228 4007 y Fp(\015)5 b(t)p 3228 4052 92 4 v 3249 4143 a Fq(2)3329 3934 y Fi(\023)3419 4075 y Fk(j)p Fp(x)22 b Fq(+)g Fp(y)t Fk(j)3702 4034 y Fn(2)3740 3934 y Fi(\023)721 4418 y Fk(\024)882 4261 y Fi(r)p 981 4261 163 4 v 1038 4350 a Fq(1)p 991 4395 143 4 v 991 4486 a(2)p Fp(\031)t(t)1216 4418 y Fq(exp)1382 4277 y Fi(\022)1455 4418 y Fk(\000)1560 4350 y Fq(1)p 1542 4395 85 4 v 1542 4486 a(2)p Fp(t)1637 4418 y Fk(j)p Fp(x)g Fk(\000)g Fp(y)t Fk(j)1921 4377 y Fn(2)1982 4418 y Fk(\000)2091 4350 y Fp(\015)p 2091 4395 57 4 v 2095 4486 a Fq(4)2174 4418 y(tanh)2386 4277 y Fi(\022)2469 4350 y Fp(\015)5 b(t)p 2469 4395 92 4 v 2490 4486 a Fq(2)2570 4277 y Fi(\023)2660 4418 y Fk(j)p Fp(x)22 b Fq(+)g Fp(y)t Fk(j)2943 4377 y Fn(2)2982 4277 y Fi(\023)3104 4418 y Fq(.)-24 4688 y(Replacing)34 b(this)f(and)f(substituting)i Fp(k)i Fq(for)c Fp(k)25 b Fq(+)1770 4644 y Fo(\015)p 1770 4665 41 4 v 1772 4722 a Fn(2)1821 4688 y Fq(\()p Fp(x)1914 4703 y Fn(2)1975 4688 y Fq(+)d Fp(y)2121 4703 y Fn(2)2160 4688 y Fq(\),)33 b(one)g(obtains)95 4957 y Fi(\014)95 5017 y(\014)95 5077 y(\014)156 5072 y Fp(e)201 5031 y Fo(\024)p Fn(\()p Fo(x)309 5040 y Fe(1)343 5031 y Fj(\000)p Fo(y)433 5040 y Fe(1)468 5031 y Fn(\))527 5072 y Fp(D)611 5031 y Fo(n)608 5097 y Fn(1)674 5072 y Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(e)858 5031 y Fj(\000)p Fo(t)956 5014 y Fn(^)938 5031 y Fo(H)996 5042 y Fb(L)1047 5072 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)1209 4957 y Fi(\014)1209 5017 y(\014)1209 5077 y(\014)1269 5072 y Fk(\024)1375 4936 y Fi(Z)1502 5005 y Fp(dk)p 1501 5049 108 4 v 1501 5140 a Fq(2)p Fp(\031)1652 4957 y Fi(\014)1652 5017 y(\014)1652 5077 y(\014)1685 5072 y Fp(k)25 b Fq(+)1869 5005 y Fp(\015)p 1869 5049 57 4 v 1873 5140 a Fq(2)1935 5072 y(\()p Fp(x)2028 5087 y Fn(2)2090 5072 y Fk(\000)e Fp(y)2238 5087 y Fn(2)2277 5072 y Fq(\))f(+)g Fp({\024)2522 4957 y Fi(\014)2522 5017 y(\014)2522 5077 y(\014)2555 4984 y Fo(n)2636 5005 y Fp(e)2691 4941 y Fe(1)p 2691 4953 31 3 v 2691 4994 a(2)2731 4968 y Fo(t)12 b(\024)2809 4944 y Fe(2)p 2629 5049 226 4 v 2629 5069 a Fk(p)p 2712 5069 143 4 v 82 x Fq(2)p Fp(\031)t(t)2920 5072 y(e)2965 5025 y Fj(\000)3030 4989 y Fg(j)p Fb(x)3084 5004 y Fe(2)3118 4989 y Fg(\000)p Fb(y)3198 5004 y Fe(2)3232 4989 y Fg(j)3251 4969 y Fe(2)p 3030 5010 256 3 v 3131 5052 a(2)p Fb(t)3296 5025 y Fj(\000)3363 4998 y Fe(1)p 3361 5010 36 3 v 3361 5052 a Fb(\015)3418 5025 y Fn(tanh)q(\()3596 4995 y Fb(\015)s(t)p 3597 5010 60 3 v 3611 5052 a Fe(2)3666 5025 y Fn(\))p Fo(k)3732 5002 y Fe(2)3803 5072 y Fq(.)-24 5362 y(No)m(w)34 b(let)e(us)i(c)m(ho)s (ose)f Fp(\024)28 b Fq(=)971 5315 y Fn(\()p Fo(x)1038 5324 y Fe(1)1073 5315 y Fj(\000)p Fo(y)1163 5324 y Fe(1)1198 5315 y Fn(\))p 971 5339 254 4 v 1085 5397 a Fo(t)1267 5362 y Fq(and)33 b(in)m(tegrate)g(o)m(v)m(er)h Fp(k)s Fq(.)43 b(Then)1935 5611 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 594 386 a Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(e)789 345 y Fj(\000)p Fo(t)887 328 y Fn(^)869 345 y Fo(H)927 356 y Fb(L)978 386 y Fk(j)o Fp(~)f(y)19 b Fk(ij)55 b(\024)1328 216 y Fi( )1579 319 y Fp(\031)t(\015)p 1417 363 441 4 v 1417 460 a(t)17 b Fq(tanh)1680 379 y Fi(\000)1736 416 y Fo(\015)t(t)p 1736 437 66 4 v 1751 494 a Fn(2)1811 379 y Fi(\001)1867 216 y(!)1956 211 y Fe(1)p 1956 223 31 3 v 1956 264 a(2)2017 386 y Fp(e)2062 345 y Fj(\000)2127 312 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)q Fg(j)2281 291 y Fe(2)p 2127 330 189 3 v 2194 371 a(2)p Fb(t)2385 386 y Fk(\024)56 b Fp(c)2597 319 y Fq(\(1)22 b(+)g Fp(t)p Fq(\))2887 255 y Fe(1)p 2887 267 31 3 v 2887 308 a(2)p 2597 363 335 4 v 2747 455 a Fp(t)2970 386 y(e)3015 345 y Fj(\000)3080 312 y Fg(j)o Fb(~)-31 b(x)o Fg(\000)p Fb(~)f(y)r Fg(j)3234 291 y Fe(2)p 3080 330 189 3 v 3147 371 a(2)p Fb(t)3315 386 y Fq(,)-24 665 y(b)s(ecause)34 b(coth)q(\()p Fp(t)p Fq(\))27 b Fp(<)774 625 y Fn(1+)p Fo(t)p 774 642 116 4 v 819 699 a(t)899 665 y Fq(.)44 b(Using)1244 584 y Fi(R)1327 665 y Fp(dk)20 b Fk(j)p Fp(k)k Fk(\000)f Fp(b)p Fk(j)17 b Fp(e)1783 629 y Fj(\000)p Fo(ak)1914 605 y Fe(2)1980 665 y Fk(\024)28 b Fp(a)2136 629 y Fj(\000)p Fn(1)2253 665 y Fq(+)2351 588 y Fi(p)p 2451 588 63 4 v 2461 625 a Fo(\031)p 2461 642 43 4 v 2464 699 a(a)2530 665 y Fk(j)p Fp(b)p Fk(j)33 b Fq(for)f Fp(a)27 b(>)h Fq(0)k(it)h(follo)m(ws)g(that)44 1067 y Fk(j)p Fp(D)153 1082 y Fn(1)193 1067 y Fk(h)n Fp(~)-47 b(x)p Fk(j)p Fp(e)360 1026 y Fj(\000)p Fo(t)458 1009 y Fn(^)440 1026 y Fo(H)498 1037 y Fb(L)548 1067 y Fk(j)o Fp(~)f(y)20 b Fk(ij)82 b(\024)954 897 y Fi( )1247 1000 y Fp(\015)1313 936 y Fe(1)p 1313 948 31 3 v 1313 989 a(2)p 1043 1044 520 4 v 1043 1152 a Fq(\(tanh)1292 1072 y Fi(\000)1348 1109 y Fo(\015)t(t)p 1348 1130 66 4 v 1363 1187 a Fn(2)1424 1072 y Fi(\001)1469 1152 y Fq(\))1517 1088 y Fe(1)p 1517 1100 31 3 v 1517 1142 a(2)1594 1067 y Fq(+)1702 928 y Fk(p)p 1785 928 59 4 v 72 x Fp(\031)t Fk(j)p Fp(x)1927 1015 y Fn(1)1988 1000 y Fk(\000)23 b Fp(y)2136 1015 y Fn(1)2175 1000 y Fk(j)p 1702 1044 501 4 v 1935 1136 a Fp(t)2235 1067 y Fq(+)f Fp(\015)2399 928 y Fk(p)p 2482 928 59 4 v 72 x Fp(\031)t Fk(j)p Fp(x)2624 1015 y Fn(2)2685 1000 y Fk(\000)h Fp(y)2833 1015 y Fn(2)2872 1000 y Fk(j)p 2399 1044 501 4 v 2625 1136 a Fq(2)2910 897 y Fi(!)3253 1000 y Fp(\015)3319 936 y Fe(1)p 3319 948 31 3 v 3319 989 a(2)p 3015 1044 587 4 v 3015 1093 a Fi(\000)3061 1174 y Fp(t)17 b Fq(tanh)3324 1093 y Fi(\000)3380 1130 y Fo(\015)t(t)p 3380 1151 66 4 v 3395 1208 a Fn(2)3456 1093 y Fi(\001)o(\001)3557 1088 y Fe(1)p 3557 1100 31 3 v 3557 1142 a(2)3611 1067 y Fp(e)3656 1026 y Fj(\000)3721 993 y Fg(j)o Fb(~)-31 b(x)p Fg(\000)p Fb(~)f(y)q Fg(j)3875 972 y Fe(2)p 3721 1011 189 3 v 3789 1052 a(2)p Fb(t)793 1406 y Fk(\024)84 b Fp(c)1033 1338 y Fq(\(1)22 b(+)g Fp(t)p Fq(\))1323 1274 y Fe(3)p 1323 1286 31 3 v 1323 1327 a(2)p 1033 1383 335 4 v 1156 1491 a Fp(t)1201 1427 y Fe(3)p 1201 1439 31 3 v 1201 1480 a(2)1406 1406 y Fp(e)1451 1364 y Fj(\000)1516 1331 y Fg(j)o Fb(~)-31 b(x)o Fg(\000)p Fb(~)f(y)r Fg(j)1670 1310 y Fe(2)p 1516 1349 189 3 v 1583 1390 a(4)p Fb(t)-24 1690 y Fq(where)41 b(for)e(the)h(second)h(b)s (ound)e Fp(xe)1325 1654 y Fj(\000)p Fn(2)p Fo(x)1455 1631 y Fe(2)1534 1690 y Fk(\024)h Fp(e)1696 1654 y Fj(\000)p Fo(x)1791 1631 y Fe(2)1869 1690 y Fq(w)m(as)g(used.)66 b(The)40 b(last)g(estimate)h(\()p Fp(n)e Fq(=)g(2\))h(is)g(obtained)-24 1811 y(similarly)34 b(up)s(on)f(using)877 1730 y Fi(R)960 1811 y Fp(dk)s Fq(\()p Fp(k)25 b Fk(\000)e Fp(b)p Fq(\))1358 1774 y Fn(2)1398 1811 y Fp(e)1443 1774 y Fj(\000)p Fo(ak)1574 1751 y Fe(2)1640 1811 y Fq(=)28 b(\()p Fp(b)1823 1774 y Fn(2)1885 1811 y Fq(+)22 b(2)p Fp(a)2083 1774 y Fj(\000)p Fn(1)2177 1811 y Fq(\))2215 1734 y Fi(p)p 2314 1734 63 4 v 2324 1771 a Fo(\031)p 2324 1788 43 4 v 2327 1845 a(a)2377 1811 y Fq(.)1514 b Fa(2)-24 1978 y Fh(Pro)s(of)48 b Fq(of)f(Prop)s(osition)g(3)g(\(This)i(argumen)m(t)f(follo)m(ws)g (closely)h([13,)i(Thm.)e(3.4.8])e(and)g(is)h(hence)h(k)m(ept)-24 2111 y(sk)m(etc)m(h)m(y\).)e(Let)33 b(us)g(set)g Fp(K)7 b Fq(\()p Fp(z)t(;)15 b(~)-47 b(x)q(;)16 b(~)-48 b(y)20 b Fq(\))27 b(=)h Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(D)1630 2074 y Fo(n)1692 2111 y Fp(e)1737 2074 y Fj(\000)p Fo(z)1846 2058 y Fn(^)1828 2074 y Fo(H)1886 2085 y Fb(L)1937 2111 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)p Fq(.)43 b(If)33 b Fp(z)f Fq(=)c Fp(t)22 b Fq(+)g Fp({s)p Fq(,)33 b(one)g(has)-9 2451 y Fk(j)p Fp(K)7 b Fq(\()p Fp(z)t(;)15 b(~)-47 b(x;)16 b(~)-48 b(y)20 b Fq(\))p Fk(j)55 b(\024)661 2336 y Fi(\015)661 2396 y(\015)661 2456 y(\015)716 2451 y Fp(D)800 2410 y Fo(n)863 2451 y Fp(e)908 2410 y Fj(\000)p Fo(z)1017 2393 y Fn(^)999 2410 y Fo(H)1057 2421 y Fb(L)1107 2336 y Fi(\015)1107 2396 y(\015)1107 2456 y(\015)1163 2520 y Fj(1)p Fo(;)p Fn(1)1337 2451 y Fk(\024)1469 2336 y Fi(\015)1469 2396 y(\015)1469 2456 y(\015)1525 2451 y Fp(D)1609 2410 y Fo(n)1672 2451 y Fp(e)1717 2410 y Fj(\000)1785 2383 y Fb(t)p 1782 2395 31 3 v 1782 2436 a Fe(2)1840 2393 y Fn(^)1822 2410 y Fo(H)1880 2421 y Fb(L)1931 2336 y Fi(\015)1931 2396 y(\015)1931 2456 y(\015)1986 2520 y Fj(1)p Fo(;)p Fn(2)2132 2336 y Fi(\015)2132 2396 y(\015)2132 2456 y(\015)2188 2451 y Fp(e)2233 2410 y Fj(\000)2301 2383 y Fb(t)p 2298 2395 V 2298 2436 a Fe(2)2356 2393 y Fn(^)2338 2410 y Fo(H)2396 2421 y Fb(L)2447 2336 y Fi(\015)2447 2396 y(\015)2447 2456 y(\015)2502 2520 y Fn(2)p Fo(;)p Fn(1)2641 2451 y Fq(=)2772 2336 y Fi(\015)2772 2396 y(\015)2772 2456 y(\015)2827 2451 y Fp(D)2911 2410 y Fo(n)2975 2451 y Fp(e)3020 2410 y Fj(\000)3088 2383 y Fb(t)p 3085 2395 V 3085 2436 a Fe(2)3143 2393 y Fn(^)3125 2410 y Fo(H)3183 2421 y Fb(L)3233 2336 y Fi(\015)3233 2396 y(\015)3233 2456 y(\015)3289 2520 y Fj(1)p Fo(;)p Fn(2)3435 2336 y Fi(\015)3435 2396 y(\015)3435 2456 y(\015)3490 2451 y Fp(e)3535 2410 y Fj(\000)3604 2383 y Fb(t)p 3600 2395 V 3600 2436 a Fe(2)3659 2393 y Fn(^)3641 2410 y Fo(H)3699 2421 y Fb(L)3749 2336 y Fi(\015)3749 2396 y(\015)3749 2456 y(\015)3805 2520 y Fj(1)p Fo(;)p Fn(2)3951 2451 y Fq(.)-24 2732 y(Since)34 b Fk(k)p Fp(A)p Fk(k)404 2683 y Fn(2)404 2761 y Fj(1)p Fo(;)p Fn(2)561 2732 y Fk(\024)28 b Fq(sup)812 2755 y Fo(~)-34 b(x)874 2652 y Fi(R)921 2766 y Ff(R)p Fj(\002)p Ff(R)1072 2775 y Fe(+)1143 2732 y Fp(d)o(~)-48 b(y)19 b Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(A)p Fk(j)o Fp(~)f(y)19 b Fk(ij)1646 2696 y Fn(2)1685 2732 y Fq(,)33 b(Lemma)g(3)f(implies)1334 2909 y Fi(\015)1334 2969 y(\015)1334 3029 y(\015)1389 3024 y Fp(D)1473 2983 y Fo(n)1536 3024 y Fp(e)1581 2983 y Fj(\000)1650 2956 y Fb(t)p 1646 2968 V 1646 3009 a Fe(2)1705 2966 y Fn(^)1687 2983 y Fo(H)1745 2994 y Fb(L)1795 2909 y Fi(\015)1795 2969 y(\015)1795 3029 y(\015)1851 3093 y Fj(1)p Fo(;)p Fn(2)2008 3024 y Fk(\024)c Fp(c)2193 2957 y Fq(\(1)21 b(+)h Fp(t)p Fq(\))2472 2920 y Fo(n)p Fn(+)2580 2893 y Fe(1)p 2580 2905 V 2580 2946 a(2)p 2193 3001 433 4 v 2321 3110 a Fp(t)2366 3045 y Fb(n)p Fe(+1)p 2367 3058 116 3 v 2409 3099 a(2)-24 3241 y Fq(so)33 b(that)967 3516 y Fk(j)p Fp(K)7 b Fq(\()p Fp(z)t(;)15 b(~)-47 b(x;)16 b(~)-48 b(y)20 b Fq(\))p Fk(j)55 b(\024)h Fp(c)17 b(f)11 b Fq(\()p Fp(t)p Fq(\))31 b(,)228 b Fp(f)11 b Fq(\()p Fp(z)t Fq(\))56 b(=)2533 3449 y(1)22 b(+)g Fp(z)2751 3413 y Fo(n)p Fn(+1)p 2533 3493 356 4 v 2610 3591 a Fp(z)2669 3527 y Fb(n)p 2670 3539 39 3 v 2674 3580 a Fe(2)2718 3554 y Fn(+1)2931 3516 y Fq(.)123 3728 y(No)m(w)33 b(for)f(0)27 b Fk(\024)i Fp(\015)j(<)873 3689 y Fo(\031)p 873 3705 43 4 v 877 3762 a Fn(2)926 3728 y Fq(,)g(let)h Fp(D)d Fq(=)e Fk(f)p Fp(z)21 b Fk(j)c Fq(0)27 b Fk(\024)h Fq(arg\()p Fp(z)t Fq(\))g Fk(\024)g Fp(\015)38 b Fq(,)32 b Fk(j)p Fp(z)t Fk(j)c(\025)g Fq(1)p Fk(g)k Fq(and)h(set)650 4001 y Fp(g)t Fq(\()p Fp(z)t Fq(\))55 b(=)1138 3934 y(1)p 1023 3978 279 4 v 1023 4069 a Fp(f)11 b Fq(\()p Fp(z)1169 4041 y Fj(\000)p Fn(1)1263 4069 y Fq(\))1328 4001 y Fp(K)c Fq(\()p Fp(z)1505 3960 y Fj(\000)p Fn(1)1600 4001 y Fp(;)15 b(~)-47 b(x;)16 b(~)-48 b(y)20 b Fq(\))44 b(exp)2058 3860 y Fi(\022)2142 3934 y Fq(1)p 2142 3978 49 4 v 2142 4069 a(8)2217 4001 y Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)20 b Fk(j)2518 3960 y Fn(2)2557 4001 y Fp(e)2602 3960 y Fo({)p Fn(\()2662 3933 y Fb(\031)p 2663 3945 38 3 v 2667 3986 a Fe(2)2711 3960 y Fj(\000)p Fo(\015)t Fn(\))2965 3934 y Fp(z)p 2864 3978 252 4 v 2864 4069 a Fq(sin)q(\()p Fp(\015)5 b Fq(\))3125 3860 y Fi(\023)3248 4001 y Fq(.)-24 4233 y(The)49 b(h)m(yp)s(othesis)h(of)e(the)g (Phragmen-Lindel\177)-49 b(of)49 b(Theorem)g(can)f(b)s(e)g(v)m (eri\014ed,)54 b(sho)m(wing)49 b(that)f Fk(j)p Fp(g)t Fq(\()p Fp(z)t Fq(\))p Fk(j)53 b(\024)-24 4363 y Fp(c)34 b Fq(cos\()p Fp(\015)5 b Fq(\))314 4326 y Fj(\000)379 4298 y Fb(n)p Fe(+1)p 379 4311 116 3 v 421 4352 a(2)539 4363 y Fq(for)30 b Fp(z)i Fk(2)c Fp(D)s Fq(.)43 b(Applying)31 b(this)g(also)g(to)p 1930 4308 50 4 v 30 w Fp(z)k Fq(and)30 b(c)m(ho)s(osing)h Fp(\015)i Fq(=)2788 4323 y Fo(\031)p 2788 4340 43 4 v 2792 4397 a Fn(2)2841 4363 y Fq(\(1)18 b Fk(\000)g Fp(\017)p Fq(\))g(+)f Fp(\017)p Fk(j)g Fq(arg)q(\()p Fp(z)t Fq(\))p Fk(j)30 b Fq(for)g(some)-24 4483 y Fp(\017)e(<)g Fq(1)k(allo)m(ws)i(to)e(conclude)i(the)f(\014rst)g(statemen)m(t)h(of)e (the)h(Prop)s(osition.)123 4651 y(In)45 b(order)g(to)f(pro)m(v)m(e)i (con)m(tin)m(uit)m(y)h(in)c Fp(~)-47 b(x)q(;)16 b(~)-48 b(y)47 b Fq(of)d Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(D)2070 4615 y Fo(n)2067 4675 y Fn(1)2117 4651 y Fp(e)2162 4615 y Fj(\000)p Fo(z)2270 4598 y Fn(^)2253 4615 y Fo(H)2311 4626 y Fb(L)2361 4651 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)45 b Fq(for)f Fp(n)49 b Fq(=)f(0)p Fp(;)17 b Fq(1,)47 b(one)e(ma)m(y)h (follo)m(w)f(the)-24 4783 y(same)33 b(strategy)f(as)f(ab)s(o)m(v)m(e)h (to)f(obtain)h(a)f(b)s(ound)g(on)g Fk(h)n Fp(~)-47 b(x)17 b Fk(j)p Fp(D)2146 4747 y Fo(n)2143 4808 y Fn(1)2209 4783 y Fp(D)2290 4798 y Fn(2)2330 4783 y Fp(e)2375 4747 y Fj(\000)p Fo(z)2483 4730 y Fn(^)2466 4747 y Fo(H)2524 4758 y Fb(L)2574 4783 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)p Fq(.)43 b(This)33 b(in)m(v)m(olv)m(es)h(calculating)e(the)-24 4903 y(deriv)-5 b(ativ)m(e)34 b(of)f(the)g(half-sided)g(Mehler)h(k)m (ernel)g(with)f(Duhamel's)h(form)m(ula,)251 5284 y Fp(@)302 5299 y Fo(x)363 5284 y Fk(h)p Fp(x)p Fk(j)p Fp(e)530 5243 y Fj(\000)p Fo(t)611 5225 y Fn(^)610 5243 y Fo(h)p Fn(\()p Fo(k)r Fn(\))749 5284 y Fk(j)p Fp(y)t Fk(i)54 b Fq(=)1053 5148 y Fi(Z)1152 5174 y Fn(1)1108 5374 y(0)1208 5284 y Fp(dq)1323 5148 y Fi(Z)1422 5174 y Fj(1)1378 5374 y Fn(0)1514 5284 y Fp(dr)30 b(@)1690 5299 y Fo(x)1751 5284 y Fk(h)p Fp(x)p Fk(j)p Fp(e)1918 5243 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(t)2177 5225 y Fn(^)2176 5243 y Fo(h)q Fn(\(0\))2312 5284 y Fk(j)p Fp(r)s Fk(i)d Fp(\015)5 b(k)s(r)30 b Fk(h)p Fp(r)s Fk(j)p Fp(e)2796 5243 y Fj(\000)p Fo(q)r(t)2911 5225 y Fn(^)2910 5243 y Fo(h)o Fn(\()p Fo(k)r Fn(\))3048 5284 y Fk(j)p Fp(y)t Fk(i)c Fp(e)3238 5243 y Fj(\000)3303 5215 y Fe(1)p 3303 5227 31 3 v 3303 5269 a(2)3344 5243 y Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(tk)3586 5219 y Fe(2)3658 5284 y Fq(,)1935 5611 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop -24 171 a Fq(whic)m(h)30 b(can)e(b)s(e)g(done)g (exactly)i(as)e(the)g(k)m(ernel)i(of)d Fp(e)1825 135 y Fj(\000)p Fo(t)1906 117 y Fn(^)1905 135 y Fo(h)p Fn(\(0\))2069 171 y Fq(is)h(kno)m(wn)h(explicitly)h(b)m(y)f(the)f(re\015ection)i (principle.)-24 291 y(One)j(then)h(replaces)g(in)e(\(26\),)g(carries)i (out)e(the)h Fp(k)s Fq(-in)m(tegral)g(and)g(uses)853 420 y Fi(Z)908 646 y Ff(R)977 556 y Fp(dr)d(r)1149 514 y Fo(p)1188 556 y Fp(e)1233 506 y Fj(\000)1298 473 y Fe(\()p Fb(r)r Fg(\000)p Fb(d)p Fe(\))1457 452 y(2)p 1298 491 194 3 v 1304 532 a(\(1)p Fg(\000)p Fb(q)r Fe(\))p Fb(t)1505 556 y Fp(e)1550 509 y Fj(\000)1615 482 y Fb(r)1646 461 y Fe(2)p 1615 494 66 3 v 1621 535 a Fb(q)r(t)1750 556 y Fk(\024)56 b Fp(c)17 b(e)1987 514 y Fj(\000)2052 487 y Fb(d)2084 466 y Fe(2)p 2051 499 67 3 v 2073 541 a Fb(t)2149 556 y Fq(\(1)22 b Fk(\000)g Fp(q)t Fq(\))2452 487 y Fe(1)p 2452 499 31 3 v 2452 541 a(2)2497 556 y Fp(q)2554 487 y Fe(1)p 2553 499 V 2553 541 a(2)2598 556 y Fp(t)2643 487 y Fe(1)p 2643 499 V 2643 541 a(2)2688 556 y Fq(\(1)f(+)h Fp(t)2939 484 y Fb(p)p 2940 499 33 3 v 2941 541 a Fe(2)2986 556 y Fq(\))32 b(,)-24 800 y(to)48 b(b)s(ound)g(the)g Fp(r)s Fq(-in)m(tegral.)89 b(Application)49 b(of)e(the)i(inequalit)m(y)g Fp(xe)2549 763 y Fj(\000)p Fn(2)p Fo(x)2679 740 y Fe(2)2772 800 y Fk(\024)55 b Fp(e)2949 763 y Fj(\000)p Fo(x)3044 740 y Fe(2)3130 800 y Fq(allo)m(ws)49 b(to)e(obtain)h(an)-24 920 y(expression)c(whic)m(h)e(is)g(in)m (tegrable)g(in)f Fp(q)k Fq(at)c(0)f(and)h(1.)69 b(This)42 b(yields)h(an)e(estimate)h(similar)g(to)e(but)i(more)-24 1040 y(cum)m(b)s(ersome)31 b(than)d(the)g(ones)h(in)g(Lemma)f(3.)42 b(Since)29 b(only)g(the)f(con)m(tin)m(uit)m(y)j(result)e(is)f(needed)i (here,)g(further)-24 1161 y(details)k(are)e(left)h(out.)3131 b Fa(2)-24 1541 y Fr(6)161 b(Comparing)53 b(in)l(tegral)f(k)l(ernels) -24 1760 y Fq(F)-8 b(or)42 b(a)h(giv)m(en)h(function)f Fp(F)14 b Fq(,)46 b(one)d(can)g(compare)g(the)h(in)m(tegral)f(k)m (ernels)i(of)d Fp(F)14 b Fq(\()p Fp(H)3035 1775 y Fo(!)r(;s)3137 1760 y Fq(\))43 b(and)g Fp(F)14 b Fq(\()p Fp(H)3614 1775 y Fo(!)r(;)p Fj(1)3754 1760 y Fq(\))43 b(and)-24 1880 y(estimate)28 b(the)e(di\013erence)i(in)e(particular)h(for)e(argumen)m (ts)j(whic)m(h)f(are)f(far)g(from)g(the)g(b)s(oundary)h(at)e Fp(x)3670 1895 y Fn(2)3738 1880 y Fq(=)j Fk(\000)p Fp(s)p Fq(.)-24 2000 y(Therefore)45 b(let)f(us)g(construct)h(the)e(semigroup)i (of)e Fp(H)2016 2015 y Fo(!)r(;s)2162 2000 y Fq(b)m(y)h(means)h(of)e (the)h(re\015ection)g(principle.)78 b(The)-24 2121 y(re\015ection)35 b Fp(S)460 2136 y Fo(s)526 2121 y Fq(:)29 b Fp(L)648 2085 y Fn(2)688 2121 y Fq(\()p Fm(R)798 2085 y Fn(2)837 2121 y Fq(\))h Fk(!)f Fp(L)1100 2085 y Fn(2)1139 2121 y Fq(\()p Fm(R)1249 2085 y Fn(2)1289 2121 y Fq(\))k(at)g(the)h(line)g Fp(x)1889 2136 y Fn(2)1958 2121 y Fq(=)29 b Fk(\000)p Fp(s)34 b Fq(is)g(de\014ned)h(b)m(y)g(\()p Fp(S)2891 2136 y Fo(s)2927 2121 y Fp( )t Fq(\)\()p Fp(x)3125 2136 y Fn(1)3165 2121 y Fp(;)17 b(x)3264 2136 y Fn(2)3304 2121 y Fq(\))29 b(=)g Fp( )t Fq(\()p Fp(x)3636 2136 y Fn(1)3676 2121 y Fp(;)17 b Fk(\000)p Fp(x)3852 2136 y Fn(2)3915 2121 y Fk(\000)-24 2241 y Fq(2)p Fp(s)p Fq(\).)68 b(Let)41 b(\005)460 2205 y Fo(s)460 2266 y Fn(2)540 2241 y Fq(b)s(e)g(the)g(indicator)g(function)g(on)f(the)h(half-plane)g Fp(x)2502 2256 y Fn(2)2583 2241 y Fk(\025)h(\000)p Fp(s)p Fq(.)68 b(Note)41 b(that)f Fp(S)3443 2256 y Fo(s)3480 2241 y Fp(H)3561 2256 y Fo(L)3613 2241 y Fp(S)3673 2256 y Fo(s)3750 2241 y Fq(is)i(the)-24 2362 y(Landau)33 b(op)s(erator)f (with)h(rev)m(ersed)i(magnetic)e(\014eld.)45 b(No)m(w)33 b(set)1301 2577 y(~)1275 2602 y Fp(H)1356 2617 y Fo(!)r(;s)1514 2602 y Fq(=)55 b(\005)1718 2561 y Fo(s)1718 2627 y Fn(2)1758 2602 y Fp(H)1839 2617 y Fo(!)1911 2602 y Fq(+)22 b(\(1)g Fk(\000)h Fq(\005)2291 2561 y Fo(s)2291 2627 y Fn(2)2330 2602 y Fq(\))p Fp(S)2428 2617 y Fo(s)2465 2602 y Fp(H)2546 2617 y Fo(!)2596 2602 y Fp(S)2656 2617 y Fo(s)-24 2777 y Fq(with)32 b(core)g Fp(C)479 2741 y Fj(1)472 2801 y Fo(c;s)558 2777 y Fq(\()p Fm(R)668 2741 y Fn(2)708 2777 y Fq(\),)f(giv)m(en)h(b)m(y)g(the)g(functions)g(in)f Fp( )h Fk(2)c Fp(C)2155 2741 y Fj(1)2148 2801 y Fo(c)2229 2777 y Fq(\()p Fm(R)2339 2741 y Fn(2)2379 2777 y Fq(\))j(satisfying)h (the)f(an)m(tisymmetry)j(relation)-24 2914 y Fp(S)36 2929 y Fo(s)73 2914 y Fp( )e Fq(=)27 b Fk(\000)p Fp( )t Fq(.)41 b(These)23 b(functions)g(v)-5 b(anish)23 b(on)e(the)h(b)s (oundary)h Fp(x)2220 2929 y Fn(2)2287 2914 y Fq(=)28 b Fk(\000)p Fp(s)p Fq(.)40 b(By)22 b(construction,)j Fp(S)3360 2929 y Fo(s)3422 2889 y Fq(~)3397 2914 y Fp(H)3478 2929 y Fo(!)r(;s)3581 2914 y Fp(S)3641 2929 y Fo(s)3705 2914 y Fq(=)3834 2889 y(~)3809 2914 y Fp(H)3890 2929 y Fo(!)r(;s)-24 3034 y Fq(and)f(therefore)g Fp(C)634 2998 y Fj(1)627 3059 y Fo(c;s)714 3034 y Fq(\()p Fm(R)824 2998 y Fn(2)863 3034 y Fq(\))f(is)h(left)g(in)m(v)-5 b(arian)m(t.)41 b(Moreo)m(v)m(er,)27 b(if)c Fp( )k Fq(is)d(a)f(smo)s (oth)h(compactly)g(supp)s(orted)h(function)-24 3171 y(in)33 b(the)g(domain)g(of)741 3146 y(~)716 3171 y Fp(H)797 3186 y Fo(!)r(;s)932 3171 y Fq(then)1180 3146 y(~)1154 3171 y Fp(H)1235 3186 y Fo(!)r(;s)1338 3171 y Fp( )e Fq(=)d(\005)1609 3135 y Fo(s)1609 3196 y Fn(2)1649 3171 y Fp(H)1730 3186 y Fo(!)r(;s)1832 3171 y Fq(\(1)22 b Fk(\000)h Fp(S)6 b Fq(\))p Fp( )36 b Fq(so)d(that)f(for)g Fk(<)p Fp(e)p Fq(\()p Fp(z)t Fq(\))c Fp(>)g Fq(0)1248 3412 y Fp(e)1293 3371 y Fj(\000)p Fo(z)s(H)1442 3379 y Fb(!)r(;s)1594 3412 y Fq(=)55 b(\005)1798 3371 y Fo(s)1798 3437 y Fn(2)1854 3412 y Fp(e)1899 3371 y Fj(\000)p Fo(z)2008 3354 y Fn(~)1990 3371 y Fo(H)2048 3379 y Fb(!)r(;s)2162 3412 y Fq(\(1)22 b Fk(\000)g Fp(S)2430 3427 y Fo(s)2467 3412 y Fq(\)\005)2578 3371 y Fo(s)2578 3437 y Fn(2)2650 3412 y Fq(.)1142 b(\(27\))123 3587 y(F)-8 b(urthermore)46 b(let)h Fp(\036)k Fk(2)g Fp(C)1159 3550 y Fj(1)1234 3587 y Fq(\()p Fm(R)p Fq(\))46 b(b)s(e)g(monotonously)i(increasing,)j Fp(\036)p Fq(\()p Fk(\0001)p Fq(\))f(=)h(0,)f Fp(\036)p Fq(\()p Fk(1)p Fq(\))g(=)h(1)46 b(and)-24 3707 y(supp)q(\()p Fp(\036)273 3671 y Fj(0)296 3707 y Fq(\))32 b Fk(\032)g Fq([0)p Fp(;)17 b Fq(1])34 b(and)h(set)h Fp(\036)1110 3722 y Fo(s)1146 3707 y Fq(\()p Fp(x)p Fq(\))c(=)g Fp(\036)p Fq(\()p Fp(s)23 b Fq(+)g Fp(x)p Fq(\).)51 b(The)36 b(follo)m(wing)f (result)h(is)f(similar)h(to)e(the)i(discrete)g(case)-24 3827 y([18,)d(14)o(].)-24 4056 y Fh(Theorem)38 b(1)49 b Fl(L)-5 b(et)39 b Fp(V)785 4071 y Fo(!)871 4056 y Fk(2)c Fp(L)1038 4019 y Fj(1)1113 4056 y Fq(\()p Fm(R)1223 4019 y Fn(2)1263 4056 y Fq(\))j Fl(and)h Fp(F)49 b Fk(2)35 b Fp(C)1823 4019 y Fo(k)1816 4080 y(c)1866 4056 y Fl(,)k Fp(k)f(>)d Fq(6)k Fl(and)f Fp(s)d(<)g Fk(1)p Fl(.)57 b(Then)38 b Fp(F)14 b Fq(\()p Fp(H)3249 4071 y Fo(!)r(;s)3351 4056 y Fq(\))38 b Fl(is)h(an)g(inte)-5 b(gr)g(al)-24 4176 y(op)g(er)g(ator)35 b(which)f(c)-5 b(an)34 b(b)-5 b(e)35 b(de)-5 b(c)g(omp)g(ose)g(d)33 b(as)1323 4417 y Fp(F)14 b Fq(\()p Fp(H)1519 4432 y Fo(!)r(;s)1621 4417 y Fq(\))55 b(=)g Fp(\036)1903 4432 y Fo(s)1957 4417 y Fp(F)14 b Fq(\()p Fp(H)2153 4432 y Fo(!)2202 4417 y Fq(\))22 b(+)g Fp(K)2443 4432 y Fo(!)r(;s)2581 4417 y Fl(,)-24 4591 y(wher)-5 b(e)41 b Fp(K)341 4606 y Fo(!)r(;s)485 4591 y Fl(form)g(a)h(c)-5 b(ovariant)40 b(family)h(of)h(inte)-5 b(gr)g(al)41 b(op)-5 b(er)g(ators)41 b(the)g(kernels)g(of)g(which)f (satisfy)i(for)f(any)-24 4711 y Fp(\016)32 b(>)c Fq(0)899 4952 y Fk(jh)n Fp(~)-47 b(x)o Fk(j)p Fp(K)1131 4967 y Fo(!)r(;s)1234 4952 y Fk(j)o Fp(~)f(y)19 b Fk(ij)55 b(\024)2232 4885 y Fp(c)2274 4900 y Fo(\016)p 1594 4929 1357 4 v 1594 5021 a Fq(1)22 b(+)g Fk(j)p Fp(x)1846 5036 y Fn(2)1907 5021 y Fq(+)g Fp(s)p Fk(j)2079 4992 y Fo(k)r Fj(\000)p Fn(6)p Fj(\000)p Fo(\016)2323 5021 y Fq(+)g Fk(j)p Fp(y)2497 5036 y Fn(2)2557 5021 y Fq(+)g Fp(s)p Fk(j)2729 4992 y Fo(k)r Fj(\000)p Fn(6)p Fj(\000)p Fo(\016)2995 4952 y Fl(.)794 b Fq(\(28\))1935 5611 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop -24 171 a Fh(Pro)s(of:)31 b Fq(Again)f(w)m(e)i (set)f Fp(s)c Fq(=)h(0,)j(drop)f(the)h(indices)h Fp(!)i Fq(and)d Fp(s)f Fq(and)h(denote)g(the)g(half-plane)f(op)s(erator)g(b)m (y)3902 145 y(^)3877 171 y Fp(H)7 b Fq(,)-24 291 y(the)33 b(one)g(on)g(the)g(plane)g(b)m(y)g Fp(H)8 b Fq(.)43 b(F)-8 b(urthermore)33 b(set:)297 606 y(~)272 631 y Fp(H)63 b Fq(=)55 b Fp(H)30 b Fq(+)22 b Fp(P)46 b Fq(,)227 b Fp(P)69 b Fq(=)55 b(\(1)22 b Fk(\000)h Fq(\005)1664 646 y Fn(2)1703 631 y Fq(\))1758 551 y Fi(\000)1804 631 y Fk(\000)p Fq(2)p Fp(\015)f(X)2084 646 y Fn(2)2140 631 y Fp(D)2221 646 y Fn(1)2282 631 y Fq(+)g(2)p Fp(\015)2485 590 y Fn(2)2524 631 y Fp(X)2613 590 y Fn(2)2605 656 y(2)2675 631 y Fk(\000)g Fp(V)44 b Fq(+)22 b Fp(S)6 b(V)21 b(S)3183 551 y Fi(\001)3245 631 y Fq(\(1)h Fk(\000)h Fq(\005)3527 646 y Fn(2)3566 631 y Fq(\))33 b(,)-24 851 y(One)46 b(easily)g(v)m (eri\014es)i(the)e(argumen)m(ts)g(of)f(Section)h(5)f(whic)m(h)i(imply)f (that)f(also)g(the)h(in)m(tegral)g(k)m(ernel)h(of)-24 972 y Fp(e)21 936 y Fj(\000)p Fo(z)130 919 y Fn(~)112 936 y Fo(H)213 972 y Fq(satis\014es)35 b(the)f(estimates)i(of)d(Prop)s (osition)h(3.)46 b(Using)34 b(\(27\))f(and)h(Duhamel's)h(form)m(ula,)f (one)g(gets)g(the)-24 1092 y(follo)m(wing)f(op)s(erator)f(iden)m(tit)m (y)j(on)d Fp(L)1343 1056 y Fn(2)1383 1092 y Fq(\()p Fm(R)22 b Fk(\002)h Fm(R)1687 1107 y Fn(+)1745 1092 y Fq(\):)514 1373 y Fp(e)559 1332 y Fj(\000)p Fo(z)668 1315 y Fn(^)650 1332 y Fo(H)772 1373 y Fq(=)56 b(\005)977 1388 y Fn(2)1016 1373 y Fp(e)1061 1332 y Fj(\000)p Fo(z)s(H)1219 1373 y Fq(\005)1292 1388 y Fn(2)1354 1373 y Fq(+)22 b Fp(z)1518 1237 y Fi(Z)1618 1264 y Fn(1)1573 1463 y(0)1674 1373 y Fp(dq)31 b Fq(\005)1872 1388 y Fn(2)1912 1373 y Fp(e)1957 1332 y Fj(\000)p Fn(\(1)p Fj(\000)p Fo(q)r Fn(\))p Fo(z)2244 1315 y Fn(~)2226 1332 y Fo(H)2293 1373 y Fp(P)14 b(e)2415 1332 y Fj(\000)p Fo(q)r(z)s(H)2607 1373 y Fq(\005)2680 1388 y Fn(2)2741 1373 y Fk(\000)23 b Fq(\005)2914 1388 y Fn(2)2970 1373 y Fp(e)3015 1332 y Fj(\000)p Fo(z)3124 1315 y Fn(~)3106 1332 y Fo(H)3173 1373 y Fp(S)6 b Fq(\005)3312 1388 y Fn(2)3384 1373 y Fq(.)-24 1600 y(Replacing)32 b(this)g(in)m(to)g(\(9\),)f(the)h(\014rst)f(term)h(giv)m(es)h(rise)f (to)f(\005)2200 1615 y Fn(2)2239 1600 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\)\005)2554 1615 y Fn(2)2593 1600 y Fq(,)31 b(whic)m(h)i(can)f(easily)g(b)s(e)g(replaced)g(b)m(y)-24 1720 y Fp(\036)34 1735 y Fn(0)74 1720 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\))p Fp(\036)374 1735 y Fn(0)438 1720 y Fq(up)27 b(to)e(an)i(error)f(satisfying)h(\(28\).)41 b(The)27 b(third)g(term)f(leads)h(to)f(\005)2799 1735 y Fn(2)2839 1720 y Fp(F)14 b Fq(\()2979 1695 y(~)2954 1720 y Fp(H)7 b Fq(\))p Fp(S)f Fq(\005)3219 1735 y Fn(2)3258 1720 y Fq(,)28 b(whic)m(h)g(according)-24 1841 y(to)g(Prop)s(osition)h(1)e (\(holding)i(also)f(for)1430 1815 y(~)1404 1841 y Fp(H)8 b Fq(\))28 b(can)g(directly)i(b)s(e)e(seen)h(to)f(satisfy)h(\(28\),)g (ev)m(en)h(with)e(\005)3572 1856 y Fn(2)3640 1841 y Fq(replaced)-24 1961 y(b)m(y)34 b Fp(\036)170 1976 y Fn(0)209 1961 y Fq(.)123 2129 y(No)m(w)27 b(let)f(us)h(consider)h(the)f(second)h(con)m (tribution)f(to)f Fp(e)2144 2092 y Fj(\000)p Fo(z)2253 2076 y Fn(^)2235 2092 y Fo(H)2329 2129 y Fq(and)g(denote)h(it)g Fp(I)8 b Fq(\()p Fp(z)t Fq(\).)42 b(In)26 b(order)h(to)f(estimate)-24 2259 y(it,)32 b(it)f(will)i(b)s(e)e(used)i(that)e(the)h(k)m(ernel)h(of) e Fp(e)1551 2223 y Fj(\000)p Fo(z)1659 2206 y Fn(~)1642 2223 y Fo(H)1740 2259 y Fq(satis\014es)i(the)f(estimate)h(\(17\))d (follo)m(wing)i(from)f(\(23\).)43 b(Then,)-24 2380 y(using)34 b(the)f(particular)f(form)h(of)f Fp(P)46 b Fq(and)32 b(the)h(estimate)h(\(21\),)e(one)h(\014rst)g(obtains)561 2570 y Fi(\014)561 2630 y(\014)594 2655 y Fk(h)m Fp(~)-46 b(r)19 b Fk(j)p Fp(P)14 b(e)846 2614 y Fj(\000)p Fo(q)r Fn(\(1+)p Fo({t)p Fn(\))p Fo(H)1196 2655 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)1330 2570 y Fi(\014)1330 2630 y(\014)1418 2655 y Fk(\024)56 b Fp(c)1637 2514 y Fi(\022)1730 2587 y Fk(j)p Fp(r)1802 2602 y Fn(2)1841 2587 y Fk(j)p 1721 2632 157 4 v 1721 2723 a Fp(q)1768 2694 y Fn(3)p Fo(=)p Fn(2)1904 2655 y Fq(\(1)22 b(+)g Fp(t)2146 2614 y Fn(2)2186 2655 y Fq(\))2224 2614 y Fn(2)2302 2655 y Fq(+)2426 2587 y Fp(r)2473 2551 y Fn(2)2470 2612 y(2)p 2426 2632 87 4 v 2446 2723 a Fp(q)2561 2655 y Fq(+)2686 2587 y Fk(j)p Fp(V)f Fk(j)p 2686 2632 134 4 v 2729 2723 a Fp(q)2830 2514 y Fi(\023)2936 2655 y Fp(e)2981 2597 y Fj(\000)3071 2564 y Fg(j)o Fb(~)-31 b(r)q Fg(\000)p Fb(~)f(y)12 b Fg(j)3231 2543 y Fe(2)p 3046 2582 244 3 v 3046 2632 a(8)p Fb(q)r Fe(\(1+)p Fb(t)3231 2618 y Fe(2)3266 2632 y(\))3337 2655 y Fq(.)-24 2886 y(Due)33 b(to)f(\(17\),)g(one)h(can)g(b)s(ound)-2 3267 y Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(I)8 b Fq(\()p Fp(z)t Fq(\))p Fk(j)o Fp(~)-48 b(y)20 b Fk(ij)82 b(\024)i Fp(c)816 3131 y Fi(Z)915 3157 y Fn(1)871 3357 y(0)971 3267 y Fp(dq)1086 3131 y Fi(Z)1141 3357 y Fo(r)1173 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y(\))569 3599 y Fk(\024)84 b Fp(c)805 3463 y Fi(Z)904 3489 y Fn(1)860 3689 y(0)960 3599 y Fp(dq)1075 3463 y Fi(Z)1174 3489 y Fj(1)1130 3689 y Fn(0)1232 3599 y Fp(dr)1327 3614 y Fn(2)1527 3531 y Fp(q)1574 3495 y Fn(1)p Fo(=)p Fn(2)p 1404 3576 404 4 v 1404 3667 a Fq(\(1)22 b Fk(\000)g Fp(q)t Fq(\))1697 3638 y Fn(1)p Fo(=)p Fn(2)1850 3458 y Fi(\022)1943 3531 y Fk(j)p Fp(r)2015 3546 y Fn(2)2054 3531 y Fk(j)p 1934 3576 157 4 v 1934 3667 a Fp(q)1981 3638 y Fn(3)p Fo(=)p Fn(2)2117 3599 y Fq(\(1)g(+)g Fp(t)2359 3558 y Fn(2)2399 3599 y Fq(\))2437 3558 y Fn(2)2515 3599 y Fq(+)2639 3531 y Fp(r)2686 3495 y Fn(2)2683 3556 y(2)p 2639 3576 87 4 v 2659 3667 a Fp(q)2774 3599 y Fq(+)2899 3531 y Fk(j)p Fp(V)f Fk(j)p 2899 3576 134 4 v 2942 3667 a Fp(q)3043 3458 y Fi(\023)3149 3599 y Fp(e)3194 3541 y Fj(\000)3259 3505 y Fg(j)p Fb(x)3313 3520 y Fe(2)3347 3505 y(+)p Fb(r)3423 3520 y Fe(2)3458 3505 y Fg(j)3477 3484 y Fe(2)p 3259 3526 252 3 v 3263 3575 a(8)p Fb(q)r Fe(\(1+)p Fb(t)3448 3561 y Fe(2)3483 3575 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Fq(simply)f(b)s(ounded)g(b)m(y)f(1,)g(and)g(the)g(third)g(follo)m(ws) h(from)e(the)h(estimate)h Fk(j)p Fp(x)3103 4339 y Fn(2)3159 4324 y Fk(\000)16 b Fp(r)3296 4339 y Fn(2)3336 4324 y Fk(j)3364 4288 y Fn(2)3420 4324 y Fq(+)g Fk(j)p Fp(r)3584 4339 y Fn(2)3639 4324 y Fk(\000)g Fp(y)3780 4339 y Fn(2)3820 4324 y Fk(j)3848 4288 y Fn(2)3915 4324 y Fk(\025)-24 4444 y Fp(x)31 4408 y Fn(2)31 4469 y(2)95 4444 y Fq(+)23 b Fp(y)246 4408 y Fn(2)242 4469 y(2)308 4444 y Fq(+)g(2)p Fp(r)503 4408 y Fn(2)500 4469 y(2)542 4444 y Fq(,)35 b(follo)m(w)m(ed)g(b)m(y)h(another)e(Gaussian)h(in)m(tegration)g (\(then)g(o)m(v)m(er)g(all)f Fp(r)3051 4459 y Fn(2)3121 4444 y Fk(2)e Fm(R)p Fq(\).)48 b(Just)35 b(as)g(in)g(the)-24 4565 y(pro)s(of)d(of)g(Prop)s(osition)h(2)f(the)h(desired)h(b)s(ound)f (on)g(the)g(con)m(tribution)g(to)f Fp(F)14 b Fq(\()2866 4539 y(^)2841 4565 y Fp(H)8 b Fq(\))32 b(follo)m(ws.)604 b Fa(2)-24 4897 y Fr(7)161 b(T)-13 b(raceclass)53 b(estimates)-24 5116 y Fq(T)-8 b(o)31 b(b)s(egin)f(with,)h Fk(T)26 b Fq(-traceclass)31 b(prop)s(erties)g(on)f(compactly)i(supp)s(orted)f (smo)s(oth)f(functions)h(of)f(the)g(planar)-24 5237 y(Hamiltonians)49 b(are)f(examined.)91 b(Prop)s(osition)48 b(2)g(implies)h(the)f(con)m (tin)m(uit)m(y)i(of)e(the)g(in)m(tegral)g(k)m(ernel)i(of)-24 5357 y Fp(D)57 5372 y Fo(j)94 5357 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\))31 b(so)i(that)g(one)f(obtains)h(the)g(follo)m(wing:) 1935 5611 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop -24 171 a Fh(Corollary)38 b(1)49 b Fl(L)-5 b(et)27 b Fp(@)790 186 y Fo(j)827 171 y Fp(V)884 186 y Fo(!)962 171 y Fk(2)h Fp(L)1122 134 y Fj(1)1197 171 y Fq(\()p Fm(R)1307 134 y Fn(2)1346 171 y Fq(\))p Fl(,)g Fp(j)34 b Fq(=)27 b(1)p Fp(;)17 b Fq(2)p Fl(,)28 b(and)e Fp(F)41 b Fk(2)28 b Fp(C)2275 134 y Fj(1)2268 195 y Fo(c)2350 171 y Fq(\()p Fm(R)p Fq(\))p Fl(.)41 b(Then)26 b Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\))27 b Fk(2)h(A)3258 186 y Fj(1)3359 171 y Fl(and)d Fp(D)3620 186 y Fo(j)3657 171 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\))27 b 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Fk(T)26 b Fq(-traceclass)-24 1995 y(b)m(y)34 b(the)f(ab)s(o)m(v)m(e)g (argumen)m(t,)h(so)e(is)h Fp(D)1318 2010 y Fn(1)1358 1995 y Fp(F)14 b Fq(\()p Fp(H)8 b Fq(\).)2291 b Fa(2)123 2163 y Fq(The)30 b(next)h(result)g(do)s(es)f(not)f(allude)i(to)e(prop)s (erties)h(of)g(the)g(Hamiltonian,)g(but)g(rather)g(giv)m(es)h(a)f (general)-24 2283 y(prop)s(ert)m(y)47 b(of)531 2258 y(^)513 2283 y Fk(T)25 b Fq(-traceclass)47 b(op)s(erators.)83 b(Therefore)47 b(let)g Fk(jr)2308 2298 y Fo(j)2344 2283 y Fk(j)e Fq(b)s(e)i(new)f(op)s(erations)h(on)e Fk(A)h Fq(de\014ned)h(b)m(y)-24 2404 y Fk(h)n Fp(~)-47 b(x)17 b Fk(j)g Fq(\()p Fk(jr)281 2419 y Fo(j)317 2404 y Fk(j)p Fp(A)p Fq(\))461 2419 y Fn(^)-40 b Fo(!)506 2404 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)28 b Fq(=)g Fk(j)p Fp(x)855 2419 y Fo(j)914 2404 y Fk(\000)22 b Fp(y)1061 2419 y Fo(j)1097 2404 y Fk(jh)n Fp(~)-47 b(x)17 b Fk(j)p Fp(A)1342 2419 y Fn(^)-40 b Fo(!)1387 2404 y Fk(j)o Fp(~)-48 b(y)19 b Fk(i)p Fq(.)44 b(Whether)34 b Fk(jr)2107 2419 y Fo(j)2143 2404 y 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Fh(Lemma)39 b(4)49 b Fp(\030)k Fl(is)48 b(a)g Fq(1)p Fl(-c)-5 b(o)g(cycle,)51 b(namely)c(it)i (satis\014es)f(whenever)f Fq(\()p Fp(A;)17 b(B)5 b Fq(\))p Fp(;)17 b Fq(\()p Fp(B)5 b(;)17 b(C)7 b Fq(\)\()p Fp(C)r(;)17 b(A)p Fq(\))51 b Fk(2)i(D)s Fq(\()p Fp(\030)5 b Fq(\))47 b Fl(have)-24 4123 y(jointly)35 b(c)-5 b(ontinuous)35 b(inte)-5 b(gr)g(al)34 b(kernels:)-24 4267 y Fq(\(i\))h Fl(Cyclicity:)45 b Fp(\030)5 b Fq(\()p Fp(A;)17 b(B)5 b Fq(\))27 b(=)h Fk(\000)p Fp(\030)5 b Fq(\()p Fp(B)g(;)17 b(A)p Fq(\))p Fl(.)-24 4411 y Fq(\(ii\))35 b Fl(Close)-5 b(dness)33 b(under)i(the)g(Ho)-5 b(chschild)34 b(op)-5 b(er)g(ator:)44 b Fp(\030)5 b Fq(\()p Fp(AB)g(;)17 b(C)7 b Fq(\))22 b Fk(\000)g Fp(\030)5 b Fq(\()p Fp(A;)17 b(B)5 b(C)i Fq(\))22 b(+)g Fp(\030)5 b Fq(\()p Fp(C)i(A;)17 b(B)5 b Fq(\))27 b(=)g(0)p Fl(.)-24 4640 y Fh(Pro)s(of:)38 b Fq(This)h(follo)m(ws)g(from)f(a)f(short)h(algebraic)h(calculation)f (using)h(the)f(Leibniz)h(rule)g(for)e Fk(r)3584 4655 y Fn(1)3662 4640 y Fq(and)h(the)-24 4760 y Fk(r)59 4775 y Fn(1)99 4760 y Fq(-in)m(v)-5 b(ariance)33 b(of)721 4735 y(^)703 4760 y Fk(T)58 b Fq(holding)33 b(under)g(the)g(stated)h(h) m(yp)s(othesis.)1545 b Fa(2)123 4928 y Fq(By)37 b(general)g(principles) h([12])e(\(see)i(also)e([18,)h(19)o(]\),)h(1-co)s(cycles)f(can)g(b)s(e) g(paired)g(with)g(unitaries.)56 b(The)-24 5048 y(pairing)45 b(in)g(the)h(presen)m(t)g(con)m(text)g(stems)h(from)d(a)h(F)-8 b(redholm)45 b(mo)s(dule)g(so)g(that)g(it)g(leads)h(to)e(an)h(index)-24 5168 y(theorem.)i(Let)33 b(\005)644 5183 y Fn(1)717 5168 y Fq(b)s(e)h(the)f(indicator)h(function)g(on)f(the)h(half-space)f(with) h(p)s(ositiv)m(e)h(\014rst)f(co)s(ordinate,)f(i.e.)-24 5289 y(\005)49 5304 y Fn(1)117 5289 y Fq(=)230 5249 y Fn(1)p 230 5266 36 4 v 230 5323 a(2)275 5289 y Fq(\(\006)23 b(+)f(1\).)43 b(The)34 b(pro)5 b(jection)33 b(from)f Fp(L)1621 5253 y Fn(2)1661 5289 y Fq(\()p Fm(R)1771 5253 y Fn(2)1810 5289 y Fq(\))h(on)m(to)f Fp(L)2166 5253 y Fn(2)2206 5289 y Fq(\()p Fm(R)2316 5304 y Fn(+)2381 5289 y Fk(\002)6 b Fm(R)p Fq(\))32 b(is)h(also)g(denoted)g(b)m(y)h(\005)3477 5304 y Fn(1)3516 5289 y Fq(.)1935 5611 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop -24 171 a Fh(Theorem)38 b(2)49 b Fl(L)-5 b(et)32 b Fk(U)42 b Fl(b)-5 b(e)31 b(a)g(unitary)h(such)f (that)h Fk(U)25 b(\000)14 b Fq(1)28 b Fk(2)g(A)j Fl(is)2354 145 y Fq(^)2336 171 y Fk(T)26 b Fl(-tr)-5 b(ac)g(e)g(class)30 b(and)h(has)g(jointly)g(c)-5 b(ontinuous)-24 291 y(inte)g(gr)g(al)37 b(kernel.)51 b(F)-7 b(urthermor)i(e)36 b(let)i Fk(r)1451 306 y Fn(1)1490 291 y Fk(U)k(2)33 b(A)j Fl(and)h Fk(jr)2112 306 y Fn(1)2134 291 y Fk(j)17 b(U)42 b(2)32 b(A)p Fl(.)51 b(Then)37 b(for)g(\014xe)-5 b(d)36 b Fp(s)c(<)g Fk(1)37 b Fl(and)f Fp(!)f Fk(2)e Fq(\012)p Fl(,)-24 421 y Fq(\005)49 436 y Fn(1)106 421 y Fk(U)173 436 y Fn(^)-40 b Fo(!)218 421 y Fq(\005)291 436 y Fn(1)357 421 y Fl(is)25 b(a)h(F)-7 b(r)i(e)g(dholm)24 b(op)-5 b(er)g(ator)25 b(on)h Fp(L)1511 385 y Fn(2)1551 421 y Fq(\()p Fm(R)1661 436 y Fn(+)1705 421 y Fk(\002)r Fm(R)p 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Fq(\005)877 1402 y Fn(1)939 1387 y Fk(\000)22 b Fq(\005)1111 1402 y Fn(1)1151 1387 y Fp(A)1229 1402 y Fn(^)-40 b Fo(!)1275 1387 y Fq(\005)1348 1402 y Fn(1)1387 1387 y Fp(B)1466 1402 y Fn(^)g Fo(!)1512 1387 y Fq(\005)1585 1402 y Fn(1)1680 1387 y Fq(=)55 b Fk(\000)1898 1320 y Fq(1)p 1898 1364 49 4 v 1898 1456 a(4)1985 1387 y(\005)2058 1402 y Fn(1)2130 1387 y Fq([\006)p Fp(;)17 b(A)2349 1402 y Fn(^)-40 b Fo(!)2395 1387 y Fq(])17 b([\006)p Fp(;)g(B)2659 1402 y Fn(^)-40 b Fo(!)2705 1387 y Fq(])49 b(,)228 b Fp(A;)17 b(B)32 b Fk(2)c(A)k Fq(,)327 b(\(32\))-24 1585 y(follo)m(ws)39 b(that)g(\005)593 1600 y Fn(1)658 1585 y Fk(\000)27 b Fq(\005)835 1600 y Fn(1)874 1585 y Fk(U)941 1600 y Fn(^)-40 b Fo(!)987 1585 y Fq(\005)1060 1600 y Fn(1)1100 1585 y Fk(U)1172 1549 y Fj(\003)1167 1611 y Fn(^)g Fo(!)1212 1585 y Fq(\005)1285 1600 y Fn(1)1363 1585 y Fq(and)39 b(\005)1632 1600 y Fn(1)1697 1585 y Fk(\000)27 b Fq(\005)1874 1600 y Fn(1)1914 1585 y Fk(U)1986 1549 y Fj(\003)1981 1611 y Fn(^)-40 b Fo(!)2026 1585 y Fq(\005)2099 1600 y Fn(1)2139 1585 y Fk(U)2206 1600 y Fn(^)g Fo(!)2252 1585 y Fq(\005)2325 1600 y Fn(1)2403 1585 y Fq(are)38 b(traceclass.)62 b(By)39 b(F)-8 b(edoso)m(v's)40 b(form)m(ula)-24 1706 y(\()p Fc(e.g.)77 b Fq([12)o(,)33 b(6,)f(18]\),)g(\005)824 1721 y Fn(1)864 1706 y Fk(U)931 1721 y Fn(^)-40 b Fo(!)977 1706 y Fq(\005)1050 1721 y Fn(1)1122 1706 y Fq(is)33 b(a)f(F)-8 b(redholm)33 b(op)s(erator)f(on)g Fp(L)2330 1670 y Fn(2)2370 1706 y Fq(\()p Fm(R)2480 1721 y Fn(+)2545 1706 y Fk(\002)6 b Fm(R)p Fq(\))32 b(whose)i(index)g(is)f (giv)m(en)h(b)m(y)636 1947 y(Ind)785 1962 y Fn(^)-41 b Fo(!)885 1947 y Fq(=)55 b Fh(T)-9 b(r)16 b Fq(\(\005)1258 1962 y Fn(1)1320 1947 y Fk(\000)23 b Fq(\005)1493 1962 y Fn(1)1549 1947 y Fk(U)1621 1905 y Fj(\003)1616 1971 y Fn(^)-40 b Fo(!)1661 1947 y Fq(\005)1734 1962 y Fn(1)1791 1947 y Fk(U)1858 1962 y Fn(^)g Fo(!)1903 1947 y Fq(\005)1976 1962 y Fn(1)2016 1947 y Fq(\))22 b Fk(\000)h Fh(T)-9 b(r)16 b Fq(\(\005)2418 1962 y Fn(1)2479 1947 y Fk(\000)23 b Fq(\005)2652 1962 y Fn(1)2708 1947 y Fk(U)2775 1962 y Fn(^)-40 b Fo(!)2821 1947 y Fq(\005)2894 1962 y Fn(1)2950 1947 y Fk(U)3022 1905 y Fj(\003)3017 1971 y Fn(^)g Fo(!)3063 1947 y Fq(\005)3136 1962 y Fn(1)3175 1947 y Fq(\))49 b(.)123 2177 y(By)41 b(h)m(yp)s(othesis,)805 2150 y Fp(~)802 2177 y(\030)46 b Fk(2)c Fm(R)1071 2141 y Fn(2)1152 2177 y Fk(7!)f Fq(\005)1366 2192 y Fn(1)1406 2177 y Fk(U)1470 2190 y Fo(~)1468 2207 y(\030)s Fj(\001)p Fo(!)r(;s)1624 2177 y Fq(\005)1697 2192 y Fn(1)1778 2177 y Fq(is)g(a)f(norm-con)m(tin) m(uous)j(family)e(of)f(F)-8 b(redholm)41 b(op)s(erators)-24 2297 y(so)d(that)f(b)m(y)h(homotop)m(y)g(in)m(v)-5 b(ariance)38 b(and)g(ergo)s(dicit)m(y)g(of)f Fh(P)f Fq(their)i(F)-8 b(redholm)38 b(index)g(is)g Fh(P)p Fq(-almost)f(surely)-24 2417 y(constan)m(t.)68 b(The)41 b(iden)m(tit)m(y)h(\005)1080 2432 y Fn(1)1137 2417 y Fk(U)1199 2432 y Fo(!)r(;s)p Fn(+)p Fo(\030)1384 2441 y Fe(2)1422 2417 y Fq(\005)1495 2432 y Fn(1)1575 2417 y Fq(=)f Fp(U)10 b Fq(\(0)p Fp(;)17 b(\030)1942 2432 y Fn(2)1981 2417 y Fq(\))2019 2381 y Fj(\003)2059 2417 y Fq(\005)2132 2432 y Fn(1)2188 2417 y Fk(U)2250 2433 y Fn(\(0)p Fo(;\030)2363 2442 y Fe(2)2398 2433 y Fn(\))p Fj(\001)p Fo(!)r(;s)2548 2417 y Fq(\005)2621 2432 y Fn(1)2660 2417 y Fp(U)10 b Fq(\(0)p Fp(;)17 b(\030)2910 2432 y Fn(2)2949 2417 y Fq(\))41 b(implies)h(that)e Fp(s)g Fk(2)i Fm(R)e Fk(7!)-24 2538 y Fq(\005)49 2553 y Fn(1)106 2538 y Fk(U)168 2553 y Fo(!)r(;s)271 2538 y Fq(\005)344 2553 y Fn(1)416 2538 y Fq(is)33 b(norm)f(con)m(tin)m(uous)j(and)d (therefore)i(Ind)1999 2553 y Fo(!)r(;s)2134 2538 y Fq(constan)m(t)g(in) f Fp(s)p Fq(.)123 2705 y(The)g(almost)g(sure)h(index)f(Ind)h(is,)f(for) f(an)m(y)h Fp(s)28 b Fk(2)g Fm(R)p Fq(,)k(giv)m(en)i(b)m(y)1019 2970 y(Ind)56 b(=)1349 2834 y Fi(Z)1465 2970 y Fp(d)p Fh(P)p Fq(\()p Fp(!)t Fq(\))26 b(Ind)1904 2985 y Fo(!)r(;s)2062 2970 y Fq(=)55 b Fk(\000)28 b Fp(\021)2346 2985 y Fo(s)2383 2970 y Fq(\()p Fk(U)2493 2929 y Fj(\003)2555 2970 y Fk(\000)23 b Fq(1)p Fp(;)17 b Fk(U)10 b Fq(\))32 b(,)-24 3196 y(where)i Fp(\021)306 3211 y Fo(s)343 3196 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))28 b(=)746 3116 y Fi(R)829 3196 y Fp(d)p Fh(P)p Fq(\()p Fp(!)t Fq(\))17 b Fp(\021)1163 3211 y Fo(!)r(;s)1264 3196 y Fq(\()p Fp(A;)g(B)5 b Fq(\))32 b(with)198 3413 y Fp(\021)251 3428 y Fn(^)-40 b Fo(!)296 3413 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))55 b(=)g Fh(T)-9 b(r)16 b Fq(\(\005)996 3428 y Fn(1)1036 3413 y Fp(B)1115 3428 y Fn(^)-40 b Fo(!)1160 3413 y Fp(A)1238 3428 y Fn(^)g Fo(!)1284 3413 y Fq(\005)1357 3428 y Fn(1)1419 3413 y Fk(\000)22 b Fq(\005)1591 3428 y Fn(1)1631 3413 y Fp(B)1710 3428 y Fn(^)-40 b Fo(!)1755 3413 y Fq(\005)1828 3428 y Fn(1)1868 3413 y Fp(A)1946 3428 y Fn(^)g Fo(!)1992 3413 y Fq(\005)2065 3428 y Fn(1)2104 3413 y Fq(\))22 b Fk(\000)h Fh(T)-9 b(r)16 b Fq(\(\005)2506 3428 y Fn(1)2545 3413 y Fp(A)2623 3428 y Fn(^)-40 b Fo(!)2669 3413 y Fp(B)2748 3428 y Fn(^)g Fo(!)2793 3413 y Fq(\005)2866 3428 y Fn(1)2928 3413 y Fk(\000)23 b Fq(\005)3101 3428 y Fn(1)3140 3413 y Fp(A)3218 3428 y Fn(^)-40 b Fo(!)3264 3413 y Fq(\005)3337 3428 y Fn(1)3376 3413 y Fp(B)3455 3428 y Fn(^)g Fo(!)3501 3413 y Fq(\005)3574 3428 y Fn(1)3613 3413 y Fq(\))50 b(.)-24 3676 y(In)m(tro)s(duce)34 b(next)g(the)f(1-co)s (cycle)g Fp(\020)1266 3691 y Fo(s)279 4036 y Fp(\020)322 4051 y Fo(s)359 4036 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))55 b(=)817 3901 y Fi(Z)934 4036 y Fp(d)p Fh(P)p Fq(\()p Fp(!)t Fq(\))26 b Fp(\020)1272 4051 y Fo(!)r(;s)1374 4036 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))32 b(,)228 b Fp(\020)1981 4051 y Fn(^)-40 b Fo(!)2026 4036 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))55 b(=)2494 3969 y(1)p 2494 4013 V 2494 4105 a(4)2570 4036 y Fh(T)-9 b(r)16 b Fq(\(\006)h([\006)p Fp(;)g(A)3040 4051 y Fo(!)r(;s)3143 4036 y Fq(])f([\006)p Fp(;)h(B)3401 4051 y Fo(!)r(;s)3504 4036 y Fq(]\))50 b(.)-24 4301 y(Then)34 b Fp(\021)284 4316 y Fn(^)-40 b Fo(!)330 4301 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))27 b(=)h Fp(\020)781 4316 y Fn(^)-40 b Fo(!)826 4301 y Fq(\()p Fp(A;)17 b(B)5 b Fq(\))32 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b(is)g(one)g(crucial)-24 5242 y(di\013erence.)62 b(The)39 b(in)m(v)-5 b(ariance)40 b(of)d(the)i(index)g(in)g(the)f (2-direction)h(holds)g(for)e(all)h(unitaries,)j(while)e(in)g(the)-24 5362 y(discrete)31 b(case)e(it)g(w)m(as)g(only)h(true)f(for)f (unitaries)h(in)g(the)g(image)g(of)f(the)i(exp)s(onen)m(tial)g(map)e ([18,)i(Prop.)f(4.10].)1935 5611 y(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop -24 171 a Fq(The)33 b(reason)f(is)g(that)g(the) g(exp)s(onen)m(tial)h(map)e(is)i(an)e(isomorphism)i(in)f(the)g(con)m (tin)m(uous)h(case,)g(namely)g(it)f(is)-24 291 y(Connes')i(Thom)g (isomorphism)g([11)o(].)44 b(F)-8 b(urther)33 b(explanations)g(will)h (b)s(e)f(giv)m(en)g(in)g([19].)123 459 y(Note)h(that)f Fk(U)g(\000)23 b Fq(1)34 b(b)s(eing)1132 433 y(^)1114 459 y Fk(T)25 b Fq(-traceclass)35 b(implies)g(that)f(also)f Fk(U)2480 422 y Fo(k)2546 459 y Fk(\000)24 b Fq(1)33 b(is)2846 433 y(^)2828 459 y Fk(T)26 b Fq(-traceclass)34 b(for)g(an)m(y)g Fp(k)f Fk(2)d Fh(Z)p Fq(.)-24 579 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Fq(\()p Fp(t)p Fq(\))17 b Fp(e)834 2135 y Fj(\000)p Fo(E)t Fn(\(1+)p Fo({t)p Fn(\))1142 2172 y Fq(,)32 b(for)h Fp(k)d Fk(6)p Fq(=)e(0,)607 2376 y(^)588 2401 y Fk(T)e Fq(\(\()p Fk(U)816 2360 y Fj(\003)856 2401 y Fq(\(\001\))c Fk(\000)g Fq(1\))17 b Fk(r)1321 2416 y Fn(1)1360 2401 y Fk(U)10 b Fq(\(\001\)\))56 b(=)f(2)p Fp(\031)t({)1999 2376 y Fq(^)1981 2401 y Fk(T)2077 2321 y Fi(\000)2123 2401 y Fq(\()p Fk(U)10 b Fq(\(\001\))2390 2360 y Fo(k)2455 2401 y Fk(\000)22 b Fq(1\))17 b(\()p Fk(r)2779 2416 y Fn(1)2818 2401 y Fp(H)8 b Fq(\))17 b Fp(G)3039 2360 y Fj(0)3061 2401 y Fq(\()p Fp(H)8 b Fq(\))3226 2321 y Fi(\001)3320 2401 y Fq(,)-24 2618 y(while,)34 b(for)e Fp(k)f Fq(=)d(0,)k(the)h(r.h.s.)44 b(v)-5 b(anishes.)123 2786 y(T)d(o)29 b(conclude,)j(let)d Fp(\036)f Fq(:)g([0)p Fp(;)17 b Fq(1])27 b Fk(!)g Fm(R)i Fq(b)s(e)h(a)f(di\013eren)m(tiable)i (function)f(v)-5 b(anishing)30 b(at)f(the)h(b)s(oundary)f(p)s(oin)m(ts) -24 2916 y(0)j(and)f(1.)43 b(Let)31 b(its)h(F)-8 b(ourier)32 b(co)s(e\016cien)m(ts)h(b)s(e)f(denoted)g(b)m(y)g Fp(a)2185 2931 y Fo(k)2256 2916 y Fq(=)2360 2836 y Fi(R)2426 2862 y Fn(1)2407 2951 y(0)2482 2916 y Fp(dx)17 b(e)2650 2880 y Fj(\000)p Fn(2)p Fo(\031)r({k)r(x)2889 2916 y Fp(\036)p Fq(\()p Fp(x)p Fq(\).)43 b(Then)3401 2841 y Fi(P)3506 2945 y Fo(k)3566 2916 y Fp(a)3617 2931 y Fo(k)3660 2916 y Fp(e)3705 2880 y Fn(2)p Fo(\031)r({k)r(x)3916 2916 y Fq(=)-24 3036 y Fp(\036)p Fq(\()p Fp(x)p Fq(\))33 b(and,)g(in)g (particular,)1005 2962 y Fi(P)1111 3065 y Fo(k)1170 3036 y Fp(a)1221 3051 y Fo(k)1291 3036 y Fq(=)28 b(0.)43 b(Hence)235 3266 y Fi( )314 3341 y(X)321 3554 y Fo(k)r Fj(6)p Fn(=0)474 3436 y Fp(a)525 3451 y Fo(k)568 3266 y Fi(!)698 3411 y Fq(^)680 3436 y Fk(T)26 b Fq(\(\()p Fk(U)908 3395 y Fj(\003)947 3436 y Fq(\(\001\))d Fk(\000)f Fq(1\))17 b Fk(r)1413 3451 y Fn(1)1452 3436 y Fk(U)10 b Fq(\(\001\)\))83 b(=)g(2)p Fp(\031)t({)2144 3341 y Fi(X)2197 3554 y Fo(k)2305 3436 y Fp(a)2356 3451 y Fo(k)2445 3411 y Fq(^)2426 3436 y Fk(T)2523 3355 y Fi(\000)2568 3436 y Fq(\()p Fk(U)10 b Fq(\(\001\))2835 3395 y Fo(k)2900 3436 y Fk(\000)23 b Fq(1\))17 b(\()p Fk(r)3225 3451 y Fn(1)3264 3436 y Fp(H)8 b Fq(\))17 b Fp(G)3485 3395 y Fj(0)3507 3436 y Fq(\()p Fp(H)8 b Fq(\))3672 3355 y Fi(\001)1802 3704 y Fq(=)83 b(2)p Fp(\031)t({)2146 3679 y Fq(^)2128 3704 y Fk(T)25 b Fq(\()p Fp(G)2322 3663 y Fj(0)2345 3704 y Fq(\()p Fp(H)8 b Fq(\))17 b Fp(\036)p Fq(\()p Fp(G)p Fq(\()p Fp(H)8 b Fq(\)\))17 b(\()p Fk(r)3041 3719 y Fn(1)3078 3704 y Fp(H)8 b Fq(\)\))32 b(.)-24 3921 y(Let)h(no)m(w)h Fp(\036)f Fq(con)m(v)m(erge)h(to)f(the)g(indicator)h(function)f(of)f ([0)p Fp(;)17 b Fq(1].)44 b(Then)34 b Fp(a)2611 3936 y Fn(0)2679 3921 y Fk(!)28 b Fq(1)33 b(and)3079 3846 y Fi(P)3184 3950 y Fo(k)r Fj(6)p Fn(=0)3334 3921 y Fp(a)3385 3936 y Fo(k)3456 3921 y Fk(!)28 b(\000)p Fq(1,)33 b(while)-24 4042 y Fp(G)53 4005 y Fj(0)77 4042 y Fq(\()p Fp(H)8 b Fq(\))p Fp(\036)p Fq(\()p Fp(G)p Fq(\()p Fp(H)g Fq(\)\))26 b Fk(!)i Fp(G)849 4005 y Fj(0)872 4042 y Fq(\()p Fp(H)8 b 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b(follo)m(wing)f(theorem)g(is)g(w)m(ell-kno)m(wn)i([20,)d(4,)h (3)o(].)1935 5611 y(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop -24 171 a Fh(Theorem)38 b(4)49 b Fl(L)-5 b(et)32 b Fp(P)41 b Fk(2)28 b(A)999 186 y Fj(1)1105 171 y Fl(b)-5 b(e)32 b(a)f(pr)-5 b(oje)g(ction)31 b(with)h(inte)-5 b(gr)g(al)31 b(kernels)g(satisfying)g Fh(P)p Fl(-a.s.)43 b(for)31 b(some)g Fp(\016)h(>)27 b Fq(0)1314 411 y Fk(jh)n Fp(~)-47 b(x)p Fk(j)p Fp(P)1527 426 y Fo(!)1577 411 y Fk(j)o Fp(~)f(y)20 b Fk(ij)54 b(\024)2196 344 y Fp(c)2238 359 y Fo(\016)p 1937 388 598 4 v 1937 480 a Fq(1)22 b(+)g Fk(j)n Fp(~)-47 b(x)22 b Fk(\000)g Fp(~)-48 b(y)20 b Fk(j)2407 451 y Fn(2+)p Fo(\016)2579 411 y Fl(.)1210 b Fq(\(37\))-24 641 y Fl(Then)33 b(ch)p Fq(\()p Fp(P)s(;)17 b(P)s(;)g(P)d Fq(\))31 b Fl(is)i(wel)5 b(l-de\014ne)-5 b(d)32 b(and)h(e)-5 b(qual)33 b(to)h(an)f(inte)-5 b(ger)33 b(given)g(as)g(the)h(index)e(of)i(a)f(F)-7 b(r)i(e)g(dholm)32 b(op)-5 b(er-)-24 761 y(ator.)123 987 y Fq(The)28 b(imp)s(ortance)g(of) f(this)h(result)h(stems)g(from)e(the)h(fact)f(that)g(the)h(bulk)h(Hall) e(conductivit)m(y)j(of)d(a)g(gas)h(of)-24 1107 y(indep)s(enden)m(t)34 b(electrons)f(describ)s(ed)g(b)m(y)g Fp(H)38 b Fq(at)32 b(zero-temp)s(erature,)g(zero)g(dissipation)h(and)e(with)h(c)m(hemical) -24 1228 y(p)s(oten)m(tial)h Fp(\026)f Fq(is)i(giv)m(en)f(b)m(y)h([2,)e (20,)g(4,)h(21)o(,)g(3,)f(6,)h(1])1358 1503 y Fp(\033)1417 1462 y Fj(?)1413 1527 y Fo(b)1476 1503 y Fq(\()p Fp(\026)p Fq(\))55 b(=)1808 1435 y Fp(q)1855 1399 y Fn(2)p 1808 1480 87 4 v 1823 1571 a Fp(h)1931 1503 y Fq(c)m(h)q(\()p Fp(P)2127 1518 y Fo(\026)2174 1503 y Fp(;)17 b(P)2281 1518 y Fo(\026)2327 1503 y Fp(;)g(P)2434 1518 y Fo(\026)2480 1503 y Fq(\))33 b(,)-24 1707 y(where)e Fp(P)318 1722 y Fo(\026)393 1707 y Fq(=)c Fp(\037)557 1723 y Fn(\()p Fj(\0001)p Fo(;\026)p Fn(])796 1707 y Fq(\()p Fp(H)8 b Fq(\))29 b(is)h(the)h(family)f(of)f(asso)s(ciated)i(F)-8 b(ermi)30 b(pro)5 b(jections.)44 b(This)31 b(fact)e(can)h(b)s(e)g (deduced)-24 1827 y(from)j(Kub)s(o's)g(form)m(ula)f([6,)h(1)o(])g(or)f (the)h(adiabatic)g(Laughlin)g Fc(Gedank)m(enexp)s(erimen)m(t)j Fq([3].)123 1995 y(As)c(discussed)i(in)e(great)f(detail)h(in)f([6,)h (1])f(in)h(the)g(discrete)h(setting,)f(\(37\))f(is)h(a)f(dynamical)i (lo)s(calization)-24 2115 y(condition)44 b(on)e(the)h(sp)s(ectral)h (region)f(in)g(the)g(vicinit)m(y)i(of)d(the)h(F)-8 b(ermi)43 b(lev)m(el.)75 b(F)-8 b(or)42 b(the)h(purp)s(oses)h(of)f(the)-24 2236 y(presen)m(t)38 b(article,)g(ho)m(w)m(ev)m(er,)i(w)m(e)d(restrict) g(ourselv)m(es)i(to)d(the)h(situation)f(where)i(the)f(F)-8 b(ermi)36 b(lev)m(el)i Fp(\026)e Fq(is)h(in)f(a)-24 2356 y(gap)31 b(of)e(the)i(sp)s(ectrum)h(of)e Fp(H)8 b Fq(.)42 b(Then)32 b Fp(P)1440 2371 y Fo(\026)1487 2356 y Fq(,)e(de\014ned)i (with)f(a)g(c)m(haracteristic)h(function,)f(can)g(also)g(b)s(e)f (written)-24 2477 y(as)36 b(a)f(smo)s(oth)h(function)g(of)f Fp(H)43 b Fq(for)35 b(whic)m(h)i(the)f(estimate)g(\(37\))f(holds)h(b)m (y)h(Prop)s(osition)e(1.)52 b(By)36 b(homotop)m(y)-24 2597 y(of)c(a)h(F)-8 b(redholm)33 b(index,)h(one)e(then)i(deduces:)-24 2823 y Fh(Corollary)k(3)49 b Fl(L)-5 b(et)40 b(the)f(interval)g Fq(\001)h Fl(b)-5 b(e)39 b(a)g(gap)g(of)g(the)g(sp)-5 b(e)g(ctrum)39 b(of)g Fp(H)2697 2838 y Fo(!)r(;)p Fj(1)2838 2823 y Fl(.)58 b(Then)38 b Fp(\026)e Fk(2)g Fq(\001)g Fk(7!)g Fp(\033)3693 2787 y Fo(b)3689 2849 y Fj(?)3748 2823 y Fq(\()p Fp(\026)p Fq(\))j Fl(is)-24 2958 y(c)-5 b(onstant)35 b(and)f(e)-5 b(qual)35 b(to)g(an)f(inte)-5 b(ger)35 b(multiple)f(of)1878 2914 y Fo(q)1912 2891 y Fe(2)p 1878 2935 69 4 v 1892 2993 a Fo(h)1956 2958 y Fl(.)123 3184 y Fq(The)41 b(follo)m(wing)g(result,)i(analogous)d(to)g (the)h(discrete)h(case)f([23,)f(18,)g(14],)i(alb)s(eit)f(based)g(on)g (Connes')-24 3304 y(Thom)27 b(isomorphism)h(and)e(its)h(dual)f(in)h (cyclic)g(cohomology)g([11,)f(15)o(])h(instead)g(of)e(the)i(Pimsner-V) -8 b(oiculescu)-24 3425 y(sequence)36 b(and)c(its)h(dual,)g(will)h(b)s (e)e(pro)m(v)m(en)i(in)f([19]:)-24 3651 y Fh(Theorem)38 b(5)49 b Fl(L)-5 b(et)35 b(the)g(interval)g Fq(\001)g Fl(b)-5 b(e)34 b(a)h(gap)g(of)f Fp(H)1939 3666 y Fo(!)r(;)p Fj(1)2079 3651 y Fl(.)45 b(Then)34 b Fp(\033)2467 3615 y Fo(b)2463 3677 y Fj(?)2523 3651 y Fq(\()p Fp(\026)p Fq(\))27 b(=)h(Ind\(\005)17 b Fk(U)3127 3666 y Fn(^)-40 b Fo(!)3173 3651 y Fq(\(\001\)\005\))35 b Fl(for)f Fp(\026)28 b Fk(2)g Fq(\001)p Fl(.)-24 3924 y Fh(Ac)m(kno)m(wledgmen)m(t:)75 b Fq(This)49 b(w)m(ork)f(w)m(as)h(supp)s(orted)f(b)m(y)h(the)f(SFB)g (288)f(\\Di\013eren)m(tialgeometrie)i(und)-24 4044 y(Quan)m(tenph)m (ysik".)-24 4377 y Fr(References)25 4596 y Fq([1])g(M.)38 b(Aizenman,)j(G.)d(Graf,)g Fc(Lo)s(calization)g(b)s(ounds)h(for)e(an)h (electron)h(gas)f Fq(J.)g(Ph)m(ys.)i(A:)f(Math.)f(Gen.)177 4716 y Fh(31)p Fq(,)33 b(6783-6806)d(\(1998\).)25 4919 y([2])49 b(J.)38 b(E.)g(Avron,)h(R.)f(Seiler,)i Fc(Quan)m(tization)e (of)f(the)h(Hall)f(Conductance)j(for)d(Gerneral)h(Multiparticle)177 5039 y(Sc)m(hr\177)-49 b(odinger)34 b(Hamiltonians)p Fq(,)f(Ph)m(ys.)i(Rev.)e(Lett.)g Fh(54)p Fq(,)f(259-262)f(\(1985\).)25 5242 y([3])49 b(J.E.)32 b(Avron,)g(R.)f(Seiler,)h(B.)f(Simon,)h Fc(Charge)g(De\014ciency)-8 b(,)33 b(Charge)f(T)-8 b(ransp)s(ort)31 b(and)g(Comparison)i(of)177 5362 y(Dimensions)p Fq(,)h(Com.)f(Math.)g (Ph)m(ys.)h Fh(159)p Fq(,)f(399-422)e(\(1994\).)1935 5611 y(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 25 171 a Fq([4])49 b(J.)29 b(Bellissard,)i Fc(Ordinary)f(quan)m(tum)g(Hall)f(e\013ect)h(and)f(non-comm)m(utativ)m (e)i(cohomology)p Fq(,)f(in)f Fc(Pro)s(c.)h(of)177 291 y(the)25 b(Bad)g(Sc)m(handau)i(Conference)f(on)f(Lo)s(calization,)i (1986)p Fq(,)f(Eds.)g(W.)f(Ziesc)m(he)i(&)e(P)-8 b(.)25 b(W)-8 b(eller,)28 b(T)-8 b(eubner)177 411 y(T)g(exte)34 b(Ph)m(ys.)g(16,)e(\(T)-8 b(eubner-V)g(erlag,)34 b(Leipzig,)f(1988\).) 25 611 y([5])49 b(J.)30 b(Bellissard,)i Fc(Gap)d(lab)s(elling)h (theorems)h(for)f(Sc)m(hr\177)-49 b(odinger)31 b(op)s(erators)p Fq(,)f(538-630,)f(in)h Fc(F)-8 b(rom)29 b(Num)m(b)s(er)177 731 y(Theory)39 b(to)g(Ph)m(ysics)p Fq(,)j(edited)e(b)m(y)f(M.)g(W)-8 b(aldsc)m(hmidt,)42 b(P)-8 b(.)39 b(Moussa,)i(J.)d(M.)h(Luc)m(k)h(and)e (C.)h(Itzykson,)177 851 y(\(Springer,)33 b(Berlin,)h(1992\).)25 1050 y([6])49 b(J.)24 b(Bellissard,)j(A.)c(v)-5 b(an)24 b(Elst,)i(H.)e(Sc)m(h)m(ulz-Baldes,)k Fc(The)c(Non-Comm)m(utativ)m(e)h (Geometry)g(of)e(the)h(Quan-)177 1171 y(tum)33 b(Hall)f(E\013ect)p Fq(,)i(J.)f(Math.)g(Ph)m(ysics)i Fh(35)p Fq(,)e(5373-5451)d(\(1994\).) 25 1370 y([7])49 b(K.)d(Bro)s(derix,)k(D.)45 b(Hundertmark,)51 b(H.)46 b(Lesc)m(hk)m(e,)52 b Fc(Con)m(tin)m(uit)m(y)c(prop)s(erties)f (of)e(Sc)m(hr\177)-49 b(odinger)47 b(semi-)177 1490 y(groups)33 b(with)g(magnetic)g(\014elds)p Fq(,)h(Rev.)f(Math.)g(Ph)m(ys.)i Fh(12)p Fq(,)d(181-225)f(\(2000\).)25 1690 y([8])49 b(V.)36 b(Cac)m(hia,)i(V.)e(A.)h(Zagrebno)m(v,)g Fc(Op)s(erator-norm)e(con)m(v) m(ergence)k(of)d(the)g(T)-8 b(rotter)37 b(pro)s(duct)f(form)m(ula)177 1810 y(for)c(sectorial)h(generators)p Fq(,)g(Lett.)g(Math.)g(Ph)m(ys.)i Fh(50)p Fq(,)d(203-211)f(\(1999\).)25 2009 y([9])49 b(J.-M.)34 b(Com)m(b)s(es,)i(P)-8 b(.)35 b(Hislop,)g Fc(Landau)f(Hamiltonians)h (with)g(random)f(p)s(oten)m(tials:)48 b(lo)s(calization)34 b(and)177 2130 y(the)f(densit)m(y)h(of)f(states)p Fq(,)g(Comm)m(un.)h (Math.)f(Ph)m(ys.)i Fh(177)p Fq(,)d(603-629)f(\(1996\).)-24 2329 y([10])49 b(H.)35 b(L.)g(Cycon,)h(R.)f(G.)f(F)-8 b(ro)s(ese,)35 b(W.)g(Kirsc)m(h,)i(B.)e(Simon,)h Fc(Sc)m(hr\177)-49 b(odinger)36 b(op)s(erators)e(with)i(application)177 2449 y(to)c(quan)m(tum)i(mec)m(hanics)h(and)e(global)f(geometry)p Fq(,)h(\(Springer-V)-8 b(erlag,)33 b(Berlin,)g(1987\).)-24 2648 y([11])49 b(A.)33 b(Connes,)h Fc(An)f(analogue)f(of)g(the)h(Thom)g (isomorphism)p Fq(,)i(Adv.)e(in)g(Math.)g Fh(39)p Fq(,)f(31-55)g (\(1981\).)-24 2847 y([12])49 b(A.)33 b(Connes,)h Fc(Non-Comm)m(utativ) m(e)g(Geometry)p Fq(,)f(\(Acad.)g(Press,)i(San)d(Diego,)g(1994\).)-24 3047 y([13])49 b(E.)24 b(B.)g(Da)m(vies,)j Fc(Heat)d(k)m(ernels)i(and)e (sp)s(ectral)g(theory)p Fq(,)j(\(Cam)m(bridge)e(Univ.)g(Press,)i(Cam)m (bridge,)g(1989\).)-24 3246 y([14])49 b(P)-8 b(.)41 b(Elbau,)k(G.-M.)40 b(Graf,)j Fc(Equalit)m(y)g(of)e(bulk)h(and)f(edge)h(Hall)f(conductance) i(revisited)p Fq(,)j(Comm)m(un.)177 3366 y(Math.)33 b(Ph)m(ys.)h Fh(229)p Fq(,)f(415{432)e(\(2002\).)-24 3565 y([15])49 b(G.)43 b(Elliott,)j(T.)d(Natsume,)k(R.)c(Nest,)k Fc(Cyclic)e (cohomology)e(for)g(one-parameter)g(smo)s(oth)h(crossed)177 3686 y(pro)s(ducts)p Fq(,)33 b(Acta)g(Math.)g Fh(160)p Fq(,)g(285-305)d(\(1988\).)-24 3885 y([16])49 b(F.)38 b(Germinet,)j(A.)d(Klein,)j Fc(Bo)s(otstrap)d(m)m(ultiscale)i(analysis) g(and)f(lo)s(calization)f(in)h(random)f(media)p Fq(,)177 4005 y(Comm)m(un.)c(Math.)f(Ph)m(ys.)h Fh(222)p Fq(,)f(415{448)e (\(2001\).)-24 4204 y([17])49 b(T.)33 b(Kato,)f Fc(P)m(erturbation)h (Theory)p Fq(,)h(Second)g(Edition,)f(\(Springer,)g(Berlin,)h(1980\).) -24 4404 y([18])49 b(J.)29 b(Kellendonk,)i(T.)e(Ric)m(h)m(ter,)i(H.)d (Sc)m(h)m(ulz-Baldes,)k Fc(Edge)e(curren)m(t)f(c)m(hannels)i(and)e (Chern)g(n)m(um)m(b)s(ers)i(in)177 4524 y(the)i(in)m(teger)g(quan)m (tum)h(Hall)f(e\013ect)p Fq(,)g(Rev.)h(Math.)f(Ph)m(ys.)h Fh(14)p Fq(,)f(87-119)e(\(2002\).)-24 4723 y([19])49 b(J.)33 b(Kellendonk,)h(H.)f(Sc)m(h)m(ulz-Baldes,)i(in)d(preparation.) -24 4922 y([20])49 b(H.)37 b(Kunz,)j Fc(The)e(quan)m(tum)h(Hall)e (e\013ect)h(for)f(electrons)i(in)e(a)g(random)h(p)s(oten)m(tial)p Fq(,)h(Comm)m(un.)g(Math.)177 5043 y(Ph)m(ys.)34 b Fh(112)p Fq(,)f(121-145)e(\(1987\).)-24 5242 y([21])49 b(S.)c(Nak)-5 b(am)m(ura,)49 b(J.)c(Bellissard,)k Fc(Lo)m(w)d(energy)g(bands)g(do)f (not)f(con)m(tribute)j(to)d(the)i(quan)m(tum)g(Hall)177 5362 y(e\013ect)p Fq(,)33 b(Comm)m(un.)i(Math.)d(Ph)m(ys.)j Fh(131)p Fq(,)e(282-305)d(\(1990\).)1935 5611 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop -24 171 a Fq([22])49 b(H.)42 b(Sc)m(h)m(ulz-Baldes,)k(J.)41 b(Bellissard,)46 b Fc(A)41 b(Kinetic)h(Theory)h(for)e(Quan)m(tum)h(T)-8 b(ransp)s(ort)42 b(in)g(Ap)s(erio)s(dic)177 291 y(Media)p Fq(,)33 b(J.)g(Stat.)f(Ph)m (ys.)j Fh(91)p Fq(,)e(991-1027)d(\(1998\).)-24 494 y([23])49 b(H.)38 b(Sc)m(h)m(ulz-Baldes,)43 b(J.)38 b(Kellendonk,)k(T.)d(Ric)m(h) m(ter,)i Fc(Sim)m(ultaneous)f(quan)m(tization)f(of)f(the)h(edge)g(and) 177 615 y(bulk)33 b(Hall)g(conductivit)m(y)p Fq(,)i(J.)e(Ph)m(ys.)h(A:) f(Math.)g(Gen.)g Fh(33)p Fq(,)f(L27-L32)f(\(2000\).)-24 818 y([24])49 b(B.)33 b(Simon,)g Fc(Sc)m(hr\177)-49 b(odinger)34 b(Semigroups)p Fq(,)f(Bull.)h(Amer.)f(Math.)g(So)s(c.)f Fh(7)p Fq(,)h(447-526)e(\(1982\).)1935 5611 y(24)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0308140715857--