Content-Type: multipart/mixed; boundary="-------------0702140429491" This is a multi-part message in MIME format. ---------------0702140429491 Content-Type: text/plain; name="07-36.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-36.keywords" limiting absorption principle Mourre estimate ---------------0702140429491 Content-Type: application/postscript; name="new-proof-lap.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="new-proof-lap.ps" %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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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 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 67 /C put dup 70 /F put dup 73 /I put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 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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 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 65 /A put dup 76 /L put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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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. 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 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 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 6 /plusminus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % 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 (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 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 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % 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 (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 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 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 67 /C put dup 73 /I put dup 82 /R put dup 91 /bracketleft put dup 93 /bracketright put dup 101 /e put dup 105 /i put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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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. 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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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 71 /G put dup 72 /H put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 21EC3643079EECE0C9AB54B4772E5DCA82D0D4ACC7F42FB493AA04A3BF4A1BD6 06ECE186315DBE9CFDCB1A0303E8D3E83027CD3AFA8F0BD466A8E8CA0E7164CF 55B332FAD43482748DD4A1CB3F40CB1F5E67192B8216A0D8FE30F9F05BF016F5 B5CC130A4B0796EE065495422FBA55BEE9BFD99D04464D987AC4D237C208FA86 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 /minus put dup 6 /plusminus put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 29 /greatermuch put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 55 /mapsto put dup 68 /D put dup 72 /H put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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 15 /epsilon1 put dup 21 /lambda put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 1 /Delta put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % 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 (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 67 /C put dup 70 /F put dup 71 /G put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-34 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b(b)s(ounded.)38 b(Then:)662 3196 y(2Im\()p Fm(C)7 b(u;)15 b Fn(\()p Fm(H)28 b Fl(\000)19 b Fm(z)t Fn(\))p Fm(u)p Fn(\))27 b(=)e(\()p Fm(u;)15 b Fn([)p Fm(H)r(;)g Fn(i)p Fm(C)7 b Fn(])p Fm(u)p Fn(\))22 b Fl(\000)e Fn(2Im)p Fm(z)t Fn(\()p Fm(u;)15 b(C)7 b(u)p Fn(\))p Fm(;)77 b(u)25 b Fl(2)g(D)s Fn(\()p Fm(C)7 b Fn(\))p Fm(;)-3118 b Fn(\(1.3\))0 3400 y(where)30 b(the)g(comm)m(utator)i(is)f(understo)s(o)s(d)d(as)j(a) g(quadratic)f(form)g(on)h Fl(D)s Fn(\()p Fm(C)7 b Fn(\).)141 3513 y(Let)31 b(us)f(no)m(w)g(describ)s(e)g(the)g(results)h(of)f(this)g (pap)s(er.)141 3626 y(Let)39 b Fm(H)r(;)15 b(A)39 b Fn(b)s(e)e(t)m(w)m (o)j(selfadjoin)m(t)f(op)s(erators)g(on)f(a)h(Hilb)s(ert)f(space)h Fl(H)g Fn(suc)m(h)f(that)h Fm(H)45 b Fl(2)38 b Fm(C)3320 3593 y Fi(1)3359 3626 y Fn(\()p Fm(A)p Fn(\).)66 b(\(The)0 3739 y(de\014nition)30 b(of)h(the)f(classes)i Fm(C)1026 3706 y Fj(k)1068 3739 y Fn(\()p Fm(A)p Fn(\))f(will)g(b)s(e)e(recalled) j(in)e(Sect.)41 b(2\).)141 3852 y(W)-8 b(e)23 b(assume)e(that)h 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Fm(A)p Fn(\))33 b Fk(and)h(that)g(\(1.4\))f(holds.)44 b(Then:)1216 4857 y Fn(sup)1195 4939 y Fj(z)s Fh(2)p Fj(J)1323 4921 y Fg(\006)1390 4857 y Fl(kh)p Fm(A)p Fl(i)1573 4820 y Fh(\000)p Fj(s)1666 4857 y Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))1976 4820 y Fh(\000)p Fi(1)2071 4857 y Fl(h)p Fm(A)p Fl(i)2209 4820 y Fh(\000)p Fj(s)2302 4857 y Fl(k)25 b Fm(<)g Fl(1)p Fm(;)0 5130 y Fk(for)33 b(al)5 b(l)33 b(close)-5 b(d)34 b(intervals)g Fm(J)g Fl(\032)25 b Fm(I)40 b Fk(and)33 b Fm(s)25 b(>)1521 5094 y Fi(1)p 1521 5109 36 4 v 1521 5161 a(2)1566 5130 y Fk(.)1867 5517 y Fn(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 141 162 a Fn(Thm.)45 b(1)32 b(is)g(w)m(ell)h(kno) m(wn)f(\(see)h([M1)q(],)g([PSS)o(]\))f(under)f(similar)h(h)m(yp)s (otheses.)46 b(It)32 b(w)m(as)g(later)h(impro)m(v)m(ed)g(in)0 275 y([Y)-8 b(a)q(],)31 b([T],)g([JP],)f(the)h(b)s(est)f(results)g(b)s (eing)g(the)h(ones)f(in)g([ABG)q(].)141 388 y(W)-8 b(e)27 b(will)f(also)h(giv)m(e)g(a)f(pro)s(of)f(b)m(y)g(energy)h(estimates)h (of)f(the)g(follo)m(wing)h(t)m(w)m(o)g(resolv)m(en)m(t)g(estimates,)i (where:)1533 544 y Fm(P)1591 558 y Fh(\006)1650 544 y Fn(\()p Fm(A)p Fn(\))d(=)f(1)-20 b(l)1961 569 y Fi(I)-12 b(R)2026 545 y Fg(\006)2083 544 y Fn(\()p Fm(A)p Fn(\))p Fm(:)0 699 y Fn(\(It)32 b(is)f(customary)g(to)h(in)m(terpret)g(the)f (pro)5 b(jections)31 b Fm(P)1843 713 y Fh(\006)1903 699 y Fn(\()p Fm(A)p Fn(\))h(as)f(pro)5 b(jections)32 b(on)f Fk(outgoing)j(/)f(inc)-5 b(oming)34 b(sub-)0 812 y(sp)-5 b(ac)g(es)p Fn(\).)0 959 y Fe(Theorem)35 b(2)46 b Fk(Assume)32 b(that)i Fm(H)e Fl(2)25 b Fm(C)1333 926 y Fi(3)1372 959 y Fn(\()p Fm(A)p Fn(\))33 b Fk(and)h(that)g(\(1.4\))f(holds.)44 b(Then:)1202 1115 y Fn(sup)1181 1197 y Fj(z)s Fh(2)p Fj(J)1309 1178 y Fg(\006)1376 1115 y Fl(kh)p Fm(A)p Fl(i)1559 1077 y Fh(\000)p Fi(1)1654 1115 y Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))1965 1077 y Fh(\000)p Fi(1)2060 1115 y Fm(P)2118 1129 y Fh(\006)2177 1115 y Fn(\()p Fm(A)p Fn(\))p Fl(k)27 b Fm(<)e Fl(1)p Fm(:)0 1324 y Fk(for)33 b(al)5 b(l)33 b(close)-5 b(d)34 b(intervals)g Fm(J)g Fl(\032)25 b Fm(I)7 b Fk(.)0 1488 y Fe(Theorem)35 b(3)46 b Fk(Assume)32 b(that)i Fm(H)e Fl(2)25 b Fm(C)1333 1455 y Fi(4)1372 1488 y Fn(\()p Fm(A)p Fn(\))33 b Fk(and)h(that)g(\(1.4\))f(holds.)44 b(Then:)1191 1643 y Fn(sup)1169 1726 y Fj(z)s Fh(2)p Fj(J)1297 1707 y Fg(\006)1364 1643 y Fl(k)p Fm(P)1467 1657 y Fh(\007)1527 1643 y Fn(\()p Fm(A)p Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))1976 1606 y Fh(\000)p Fi(1)2071 1643 y Fm(P)2129 1657 y Fh(\006)2189 1643 y Fn(\()p Fm(A)p Fn(\))p Fl(k)26 b Fm(<)f Fl(1)p Fm(;)0 1852 y Fk(for)33 b(al)5 b(l)33 b(close)-5 b(d)34 b(intervals)g Fm(J)g Fl(\032)25 b Fm(I)7 b Fk(.)0 2016 y Fn(Thms.)41 b(2,)32 b(3)g(w)m(ere)f(pro)m(v)m(ed)g(b)s(efore)g(in)g([M2)q(].)43 b(The)30 b(pro)s(ofs)g(of)h(these)h(theorems)f(will)h(b)s(e)e(giv)m(en) i(in)f(Sects.)43 b(3,)0 2129 y(4,)31 b(5)f(resp)s(ectiv)m(ely)-8 b(.)43 b(W)-8 b(e)31 b(will)g(only)f(pro)m(v)m(e)h(the)f Fm(J)1687 2096 y Fi(+)1776 2129 y Fn(case,)i(the)e Fm(J)2209 2096 y Fh(\000)2298 2129 y Fn(case)i(b)s(eing)d(similar.)41 b(Some)31 b(preparatory)0 2242 y(estimates)h(will)f(b)s(e)e(giv)m(en)j (in)e(Sect.)41 b(2.)0 2521 y Fo(2)135 b(F)-11 b(unctional)44 b(calculus)h(and)g(comm)l(utator)h(expansions)0 2727 y Fd(2.1)112 b(The)38 b(classes)h Fc(C)918 2691 y Fj(k)960 2727 y Fr(\()p Fc(A)p Fr(\))0 2899 y Fn(W)-8 b(e)32 b(recall)f(in)f (this)h(subsection)f(some)h(de\014nitions)f(from)g([ABG)q(].)141 3012 y(Let)e Fm(A)f Fn(b)s(e)f(a)i(selfadjoin)m(t)g(op)s(erator)f(on)g Fl(H)q Fn(.)40 b(One)26 b(sa)m(ys)i(that)g(a)f(b)s(ounded)e(op)s (erator)j Fm(B)j Fn(is)c(of)h(class)f Fm(C)3598 2979 y Fj(k)3641 3012 y Fn(\()p Fm(A)p Fn(\))0 3125 y(for)j(some)h Fm(k)d Fl(2)d Fn(I)-15 b(N)30 b(if)h(for)f(all)h Fm(u)25 b Fl(2)g(H)31 b Fn(the)g(function)1500 3280 y(I)-15 b(R)24 b Fl(3)h Fm(t)g Fl(7!)g Fn(e)1909 3243 y Fi(i)p Fj(tA)2012 3280 y Fm(B)5 b Fn(e)2126 3243 y Fi(i)p Fj(tA)2228 3280 y Fm(u)0 3436 y Fn(is)36 b(of)g(class)h Fm(C)497 3403 y Fj(k)539 3436 y Fn(.)57 b(It)36 b Fm(B)j Fl(2)34 b Fm(C)1000 3403 y Fi(1)1039 3436 y Fn(\()p Fm(A)p Fn(\),)k(then)e(the)g (comm)m(utator)h([)p Fm(B)5 b(;)15 b(A)p Fn(])37 b(considered)e(as)i(a) f(quadratic)g(form)g(on)0 3549 y Fl(D)s Fn(\()p Fm(A)p Fn(\))30 b(extends)g(as)h(a)f(b)s(ounded)e(op)s(erator)i(on)g Fl(H)h Fn(and)e Fm(B)h Fl(2)25 b Fm(C)2163 3516 y Fj(k)2205 3549 y Fn(\()p Fm(A)p Fn(\))31 b(implies)f([)p Fm(A;)15 b(B)5 b Fn(])26 b Fl(2)f Fm(C)3098 3516 y Fj(k)r Fh(\000)p Fi(1)3230 3549 y Fn(\()p Fm(A)p Fn(\))31 b(for)e Fm(k)g Fl(\025)c Fn(2.)0 3662 y(If)30 b Fm(B)g Fl(2)25 b Fm(C)348 3629 y Fj(k)390 3662 y Fn(\()p Fm(A)p Fn(\),)31 b(w)m(e)g(will)g(use)f (the)g(standard)g(notation:)990 3817 y(ad)1086 3780 y Fj(l)1086 3840 y(A)1143 3817 y Fm(B)g Fn(:=)25 b([)p Fm(A;)15 b Fn(ad)1593 3779 y Fj(l)q Fh(\000)p Fi(1)1593 3845 y Fj(A)1709 3817 y Fm(B)5 b Fn(])p Fm(;)75 b(ad)2003 3780 y Fi(0)2003 3840 y Fj(A)2061 3817 y Fm(B)30 b Fn(:=)25 b Fm(B)5 b(;)45 b Fn(for)30 b Fm(l)d Fl(\024)e Fm(k)s(:)0 3973 y Fn(If)j Fm(H)36 b Fn(is)29 b(a)h(selfadjoin)m(t)g(op)s(erator,)f (one)h(sa)m(ys)f(that)h Fm(H)i Fl(2)25 b Fm(C)2011 3940 y Fj(k)2053 3973 y Fn(\()p Fm(A)p Fn(\))30 b(if)f(for)f(some)i(\(and)e (hence)i(for)e(all\))i Fm(z)g Fl(2)p 3610 3973 4 56 v 25 w Fn(C)o Fl(n)p Fn(I)-15 b(R)0 4086 y(the)31 b(op)s(erator)f(\()p Fm(H)e Fl(\000)20 b Fm(z)t Fn(\))834 4053 y Fh(\000)p Fi(1)959 4086 y Fn(is)30 b(in)g Fm(C)1228 4053 y Fj(k)1271 4086 y Fn(\()p Fm(A)p Fn(\).)141 4199 y(The)g(follo)m(wing)i(facts)f (are)g(w)m(ell)g(kno)m(wn:)0 4346 y Fe(Lemma)k(2.1)46 b Fk(L)-5 b(et)30 b Fm(H)r(;)15 b(A)30 b Fk(b)-5 b(e)29 b(two)i(selfadjoint)g(op)-5 b(er)g(ators)33 b(such)d(that)h Fm(H)h Fl(2)25 b Fm(C)2676 4313 y Fi(1)2715 4346 y Fn(\()p Fm(A)p Fn(\))p Fk(.)42 b(Then)30 b(for)g(al)5 b(l)30 b Fm(z)g Fl(2)p 3610 4346 V 25 w Fn(C)o Fl(n)p Fn(I)-15 b(R)0 4459 y Fk(and)26 b(for)g(al)5 b(l)25 b Fm(\037)g Fl(2)g Fm(C)669 4426 y Fh(1)662 4482 y Fi(0)743 4459 y Fn(\(I)-15 b(R\))p Fk(,)27 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Fm(A)p Fl(i)2069 1024 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)2373 1062 y Fm(\037)2430 1076 y Fi(1)2470 1062 y Fn(\()p Fm(H)g Fn(\))p Fm(;)0 1256 y Fn(with)1253 1369 y Fl(k)p Fm(R)1367 1383 y Fi(3)1407 1369 y Fl(k)26 b(\024)e Fm(C)7 b Fl(k)p Fn(ad)1786 1383 y Fj(A)1843 1369 y Fm(H)1919 1383 y Fj(\034)1962 1369 y Fl(kk)p Fn(ad)2149 1383 y Fj(A)2206 1369 y Fm(\037)2263 1383 y Fi(2)2302 1369 y Fn(\()p Fm(H)g Fn(\))p Fl(k)p Fm(;)0 1531 y Fn(whic)m(h)30 b(completes)i(the)e(pro)s(of)g(of)h Fk(i\))p Fn(.)40 b(The)30 b(pro)s(of)g(of)g Fk(ii\))g Fn(is)h(similar,)g(using)f(again)h(\(2.10\).)43 b Fa(2)141 1691 y Fn(W)-8 b(e)32 b(no)m(w)e(pro)m(v)m(e)h(t)m(w)m(o)h(similar)f (comm)m(utator)h(expansions)e(for)g(a)h(di\013eren)m(t)f Fm(F)13 b Fn(.)141 1804 y(Let)31 b Fm(g)e Fl(2)c Fm(C)534 1771 y Fh(1)608 1804 y Fn(\(I)-15 b(R\))31 b(with)f(0)25 b Fl(\024)g Fm(g)k Fl(\024)c Fn(1,)31 b Fm(g)s Fn(\()p Fm(\025)p Fn(\))c Fl(\021)e Fn(0)30 b(for)h Fm(\025)25 b Fl(\025)g Fn(2,)31 b Fm(g)s Fn(\()p Fm(\025)p Fn(\))26 b Fl(\021)f Fn(1)31 b(for)f Fm(\025)c Fl(\024)f Fn(1.)141 1917 y(Set:)1405 2054 y Fm(F)13 b Fn(\()p Fm(\025)p Fn(\))26 b(:=)f Fl(\000)1832 1939 y Fb(Z)1915 1965 y Fi(+)p Fh(1)1878 2128 y Fj(\025)2060 2054 y Fm(g)2106 2016 y Fi(2)2146 2054 y Fn(\()p Fm(s)p Fn(\))p Fm(ds:)-2374 b Fn(\(2.12\))0 2261 y(Note)32 b(that)f Fm(g)e Fl(2)24 b Fm(S)635 2228 y Fi(0)675 2261 y Fn(,)30 b Fm(F)39 b Fl(2)25 b Fm(S)974 2228 y Fi(1)1013 2261 y Fn(.)141 2374 y(Props.)40 b(2.5,)32 b(2.6)f(will)g(b)s(e)f(used)f(in)h(the)h(pro)s(of)f(of)g(Thms.)40 b(2,)31 b(3)f(resp)s(ectiv)m(ely)-8 b(.)0 2554 y Fe(Prop)s(osition)36 b(2.5)46 b Fk(Ther)-5 b(e)41 b(exists)f(a)g(c)-5 b(onstant)41 b Fm(C)46 b Fk(such)40 b(that)h(for)f(al)5 b(l)40 b(selfadjoint)h(op)-5 b(er)g(ators)43 b Fm(H)r(;)15 b(A)40 b Fk(such)0 2667 y(that)34 b Fm(H)e Fl(2)25 b Fm(C)452 2634 y Fi(3)491 2667 y Fn(\()p Fm(A)p Fn(\))33 b Fk(one)g(has:)464 3019 y Fm(i)p Fn(\))791 2858 y Fm(\037)848 2872 y Fi(1)887 2858 y Fn(\()p 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y Fh(\000)p Fj(s)2746 3179 y Fm(\037)2803 3193 y Fi(1)2843 3179 y Fn(\()p Fm(H)g Fn(\))20 b(+)g(h)p Fm(:)p Fn(c)p Fm(:)q(;)0 3366 y Fk(for)33 b(al)5 b(l)33 b Fn(0)26 b Fl(\024)f Fm(s)g(<)g Fn(1)33 b Fk(wher)-5 b(e:)833 3561 y Fl(k)p Fm(R)947 3575 y Fi(1)987 3561 y Fl(k)21 b Fn(+)f Fl(k)p Fm(R)1258 3575 y Fi(2)1298 3561 y Fl(k)25 b(\024)g Fm(C)7 b Fn(\()1733 3480 y Fb(X)1571 3664 y Fi(2)p Fh(\024)p Fj(k)r Fi(+)p Fj(l)q Fh(\024)p Fi(3)p Fj(;)15 b(l)q Fh(\024)p Fi(1)2029 3561 y Fl(k)p Fn(ad)2171 3523 y Fj(k)2171 3583 y(A)2228 3561 y Fm(H)2304 3575 y Fj(\034)2346 3561 y Fl(kk)p Fn(ad)2533 3523 y Fj(l)2533 3583 y(A)2590 3561 y Fm(\037)2647 3575 y Fi(2)2687 3561 y Fn(\()p Fm(H)7 b Fn(\))p Fl(k)p Fn(\))p Fm(:)698 3855 y(ii)p Fn(\))149 b Fm(\037)1001 3869 y Fi(1)1040 3855 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1434 3822 y Fi(2)1434 3878 y(2)1475 3855 y Fn(\()p Fm(H)g Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1869 3869 y Fi(1)1911 3855 y Fn(\()p Fm(H)g Fn(\))790 4015 y(=)83 b Fm(\037)1001 4029 y Fi(1)1040 4015 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)1239 3982 y Fi(2)1280 4015 y Fn(\()p Fm(A)p Fn(\))p Fm(\037)1475 4029 y Fi(1)1515 4015 y Fn(\()p Fm(H)g Fn(\))21 b(+)f Fm(\037)1837 4029 y Fi(1)1876 4015 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)2167 3982 y Fh(\000)p Fj(s)2260 4015 y Fm(R)2329 4029 y Fi(3)2369 4015 y Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)2610 4029 y Fi(1)2650 4015 y Fn(\()p Fm(H)g Fn(\))21 b(+)f(h)p Fm(:)p Fn(c)p Fm(:)q(;)0 4170 y Fk(for)33 b(al)5 b(l)33 b Fn(0)26 b Fl(\024)f Fm(s)g(<)g Fn(1)33 b Fk(wher)-5 b(e:)1434 4282 y Fl(k)p Fm(R)1548 4296 y Fi(3)1588 4282 y Fl(k)26 b(\024)f Fm(C)7 b Fl(k)p Fn(ad)1968 4296 y Fj(A)2025 4282 y Fm(\037)2082 4296 y Fi(2)2121 4282 y Fn(\()p Fm(H)g Fn(\))p Fl(k)p Fm(:)0 4462 y Fn(The)30 b(next)g(prop)s(osition)g(is)h(similar,)g(with)f(sligh)m(tly)i(b)s (etter)e(remainder)g(terms.)0 4642 y Fe(Prop)s(osition)36 b(2.6)46 b Fk(Ther)-5 b(e)41 b(exists)f(a)g(c)-5 b(onstant)41 b Fm(C)46 b Fk(such)40 b(that)h(for)f(al)5 b(l)40 b(selfadjoint)h(op)-5 b(er)g(ators)43 b Fm(H)r(;)15 b(A)40 b Fk(such)0 4755 y(that)34 b Fm(H)e Fl(2)25 b Fm(C)452 4722 y Fi(4)491 4755 y Fn(\()p Fm(A)p Fn(\))33 b Fk(one)g(has:)460 5107 y Fm(i)p Fn(\))787 4947 y Fm(\037)844 4961 y Fi(1)883 4947 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)1137 4961 y Fj(\034)1181 4947 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()p Fm(A)p Fn(\)])p Fm(\037)1537 4961 y Fi(1)1578 4947 y Fn(\()p Fm(H)7 b Fn(\))633 5107 y(=)83 b Fm(\037)844 5121 y Fi(1)883 5107 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1277 5121 y Fi(2)1319 5107 y Fn(\()p Fm(H)g Fn(\)[)p Fm(H)1573 5121 y Fj(\034)1616 5107 y Fm(;)15 b Fn(i)p Fm(A)p Fn(])p Fm(\037)1831 5121 y Fi(2)1872 5107 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)2266 5121 y Fi(1)2307 5107 y Fn(\()p Fm(H)g Fn(\))633 5267 y(+)83 b Fm(\037)844 5281 y Fi(1)883 5267 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)1174 5234 y Fh(\000)p Fi(1)1270 5267 y Fm(R)1339 5281 y Fi(1)1379 5267 y Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1620 5281 y Fi(1)1661 5267 y Fn(\()p Fm(H)g Fn(\))20 b(+)g Fm(\037)1982 5281 y Fi(1)2022 5267 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)2313 5234 y Fh(\000)p Fi(1)2408 5267 y Fm(R)2477 5281 y Fi(2)2517 5267 y Fl(h)p Fm(A)p Fl(i)2655 5234 y Fh(\000)p Fi(1)2750 5267 y Fm(\037)2807 5281 y Fi(1)2846 5267 y Fn(\()p Fm(H)g Fn(\))21 b(+)f(h)p Fm(:)p Fn(c)p Fm(:)q(;)1867 5517 y Fn(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 0 162 a Fk(wher)-5 b(e:)833 275 y Fl(k)p Fm(R)947 289 y Fi(1)987 275 y Fl(k)21 b Fn(+)f Fl(k)p Fm(R)1258 289 y Fi(2)1298 275 y Fl(k)25 b(\024)g Fm(C)7 b Fn(\()1733 194 y Fb(X)1571 379 y Fi(2)p Fh(\024)p Fj(k)r Fi(+)p Fj(l)q Fh(\024)p Fi(4)p Fj(;)15 b(l)q Fh(\024)p Fi(2)2029 275 y Fl(k)p Fn(ad)2171 238 y Fj(k)2171 298 y(A)2228 275 y Fm(H)2304 289 y Fj(\034)2346 275 y Fl(kk)p Fn(ad)2533 238 y Fj(l)2533 298 y(A)2590 275 y Fm(\037)2647 289 y Fi(2)2687 275 y Fn(\()p Fm(H)7 b Fn(\))p Fl(k)p Fn(\))p Fm(:)696 541 y(ii)p Fn(\))149 b Fm(\037)999 555 y Fi(1)1039 541 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1433 508 y Fi(2)1433 564 y(2)1474 541 y Fn(\()p Fm(H)g Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1868 555 y Fi(1)1910 541 y Fn(\()p Fm(H)g Fn(\))789 702 y(=)82 b Fm(\037)999 716 y Fi(1)1039 702 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)1238 669 y Fi(2)1278 702 y Fn(\()p Fm(A)p Fn(\))p Fm(\037)1473 716 y Fi(1)1514 702 y Fn(\()p Fm(H)g Fn(\))20 b(+)g Fm(\037)1835 716 y Fi(1)1875 702 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)2166 669 y Fh(\000)p Fi(1)2262 702 y Fm(R)2331 716 y Fi(3)2370 702 y Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)2611 716 y Fi(1)2652 702 y Fn(\()p Fm(H)g Fn(\))21 b(+)f(h)p Fm(:)p Fn(c)p Fm(:;)0 861 y Fk(wher)-5 b(e:)1331 1026 y Fl(k)p Fm(R)1445 1040 y Fi(3)1485 1026 y Fl(k)26 b(\024)f Fm(C)7 b Fn(\()1801 920 y Fi(2)1759 945 y Fb(X)1763 1130 y Fj(l)q Fi(=1)1894 1026 y Fl(k)p Fn(ad)2035 988 y Fj(l)2035 1048 y(A)2092 1026 y Fm(\037)2149 1040 y Fi(2)2189 1026 y Fn(\()p Fm(H)g 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Fm(A;)15 b(H)1986 2261 y Fj(\034)2031 2247 y Fn(])20 b(+)g Fl(h)p Fm(A)p Fl(i)2305 2209 y Fh(\000)p Fi(1)2400 2247 y Fm(R)2469 2261 y Fi(1)2509 2247 y Fl(h)p Fm(A)p Fl(i)2647 2209 y Fh(\000)p Fi(1)2742 2247 y Fm(;)0 2407 y Fn(for)1393 2572 y Fl(k)p Fm(R)1507 2586 y Fi(1)1546 2572 y Fl(k)26 b(\024)f Fm(C)7 b Fn(\()1867 2466 y Fi(4)1824 2491 y Fb(X)1820 2676 y Fj(k)r Fi(=2)1964 2572 y Fl(k)p Fn(ad)2105 2535 y Fj(k)2105 2595 y(A)2162 2572 y Fm(H)2238 2586 y Fj(\034)2281 2572 y Fl(k)p Fn(\))p Fm(:)0 2802 y Fn(Next)381 2914 y Fm(g)427 2877 y Fi(2)467 2914 y Fn(\()p Fm(A)p Fn(\)[)p Fm(A;)15 b(H)814 2928 y Fj(\034)858 2914 y Fn(])26 b(=)f Fm(g)s Fn(\()p Fm(A)p Fn(\)[)p Fm(A;)15 b(H)1398 2928 y Fj(\034)1443 2914 y Fn(])p Fm(g)s Fn(\()p Fm(A)p Fn(\))22 b(+)e Fm(g)s Fn(\()p Fm(A)p Fn(\)ad)2046 2877 y Fi(2)2046 2937 y Fj(A)2103 2914 y Fm(H)2179 2928 y Fj(\034)2222 2914 y Fm(g)2268 2877 y Fh(0)2292 2914 y Fn(\()p Fm(A)p Fn(\))h(+)f Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(R)2795 2928 y Fi(2)2836 2914 y Fn(\()p Fm(g)s(;)15 b(A;)g Fn(ad)3163 2928 y Fj(A)3220 2914 y Fm(H)3296 2928 y Fj(\034)3338 2914 y Fn(\))p Fm(;)0 3074 y Fn(and)30 b(the)g(last)i(t)m(w)m(o)f(terms)g(can)f(b)s(e)g(written)g(as)1024 3317 y Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(R)1277 3331 y Fi(2)1318 3317 y Fl(h)p Fm(A)p Fl(i)1456 3280 y Fh(\000)p Fi(1)1551 3317 y Fm(;)46 b Fn(for)30 b Fl(k)p Fm(R)1875 3331 y Fi(2)1915 3317 y Fl(k)c(\024)f Fm(C)7 b Fn(\()2235 3211 y Fi(3)2193 3236 y Fb(X)2189 3421 y Fj(k)r Fi(=2)2332 3317 y Fl(k)p Fn(ad)2474 3280 y Fj(k)2474 3340 y(A)2531 3317 y Fm(H)2607 3331 y Fj(\034)2649 3317 y Fl(k)p Fn(\))p Fm(:)0 3578 y Fn(W)-8 b(e)32 b(obtain:)469 3769 y([)p Fm(H)570 3783 y Fj(\034)613 3769 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()p Fm(A)p Fn(\)])27 b(=)e Fm(g)s Fn(\()p Fm(A)p Fn(\)[)p Fm(H)1320 3783 y Fj(\034)1364 3769 y Fm(;)15 b Fn(i)p Fm(A)p Fn(])p Fm(g)s Fn(\()p Fm(A)p Fn(\))23 b(+)d Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(R)2073 3783 y Fi(1)2114 3769 y Fl(h)p Fm(A)p Fl(i)2252 3731 y Fh(\000)p Fi(1)2367 3769 y Fn(+)g Fl(h)p Fm(A)p Fl(i)2596 3731 y Fh(\000)p Fi(1)2691 3769 y Fm(R)2760 3783 y Fi(2)2800 3769 y Fl(h)p Fm(A)p Fl(i)2938 3731 y Fh(\000)p Fi(1)3053 3769 y Fn(+)g(h)p Fm(:)p Fn(c)p Fm(:)q(;)-3311 b Fn(\(2.22\))0 3959 y(where)1237 4124 y Fl(k)p Fm(R)1351 4138 y Fi(1)1391 4124 y Fl(k)21 b Fn(+)f Fl(k)p Fm(R)1662 4138 y Fi(2)1702 4124 y Fl(k)25 b(\024)g Fm(C)7 b Fn(\()2022 4019 y Fi(4)1979 4044 y Fb(X)1975 4228 y Fj(k)r Fi(=2)2119 4124 y Fl(k)p Fn(ad)2260 4087 y Fj(k)2260 4147 y(A)2317 4124 y Fm(H)2393 4138 y Fj(\034)2436 4124 y Fl(k)p Fn(\))p Fm(:)-2541 b Fn(\(2.23\))0 4355 y(T)-8 b(o)31 b(handle)f(the)g(cuto\013s)h Fm(\037)931 4369 y Fi(1)971 4355 y Fn(,)f Fm(\037)1083 4369 y Fi(2)1122 4355 y Fn(,)h(w)m(e)g(write:)277 4545 y Fm(\037)334 4559 y Fi(1)373 4545 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))27 b(=)e Fm(\037)890 4559 y Fi(1)930 4545 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1324 4559 y Fi(2)1365 4545 y Fn(\()p Fm(H)g Fn(\))21 b(+)f Fm(\037)1687 4559 y Fi(1)1726 4545 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)1925 4508 y Fh(0)1950 4545 y Fn(\()p Fm(A)p Fn(\)ad)2185 4559 y Fj(A)2242 4545 y Fm(\037)2299 4559 y Fi(2)2338 4545 y Fn(\()p Fm(H)g Fn(\))21 b(+)f Fm(\037)2660 4559 y Fi(1)2699 4545 y Fn(\()p Fm(H)7 b Fn(\))p Fm(R)2921 4559 y Fi(2)2961 4545 y Fn(\()p Fm(g)s(;)15 b(A;)g(\037)3247 4559 y Fi(2)3289 4545 y Fn(\()p Fm(H)7 b Fn(\)\))p Fm(;)0 4736 y Fn(and)30 b(since)g Fm(g)445 4703 y Fh(0)495 4736 y Fl(2)25 b Fm(C)653 4703 y Fh(1)646 4759 y Fi(0)727 4736 y Fn(\(I)-15 b(R\),)31 b(the)f(last)h(t)m(w)m(o)h(terms)e(can)h(b)s(e)f(written)g(as)926 5002 y Fm(\037)983 5016 y Fi(1)1022 5002 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)1313 4964 y Fh(\000)p Fi(1)1409 5002 y Fm(R)1478 5016 y Fi(3)1518 5002 y Fm(;)46 b Fn(for)30 b Fl(k)p Fm(R)1842 5016 y Fi(3)1882 5002 y Fl(k)25 b(\024)g Fm(C)7 b Fn(\()2202 4896 y Fi(2)2159 4921 y Fb(X)2155 5105 y Fj(k)r Fi(=1)2299 5002 y Fl(k)p Fn(ad)2440 4964 y Fj(k)2440 5024 y(A)2497 5002 y Fm(\037)2554 5016 y Fi(2)2594 5002 y Fn(\()p Fm(H)g Fn(\))p Fl(k)p Fn(\))p Fm(:)0 5268 y Fn(This)30 b(yields)g Fk(i\))g Fn(and)g Fk(ii\))g Fn(b)m(y)h(the)f(same)h(argumen)m(ts)g(as)f(b)s(efore.)41 b Fa(2)1867 5517 y Fn(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 0 162 a Fo(3)135 b(Pro)t(of)45 b(of)g(Thm)f(1)0 365 y Fn(Let)31 b Fm(H)h Fl(2)25 b Fm(C)429 332 y Fi(1)468 365 y Fn(\()p Fm(A)p Fn(\))31 b(b)s(e)f(a)h(selfadjoin)m (t)g(op)s(erator)g(and)e(let)j Fm(J)i Fl(\032)25 b Fn(I)-15 b(R)30 b(b)s(e)g(a)g(compact)i(in)m(terv)-5 b(al.)42 b(W)-8 b(e)32 b(recall)f(that:)1189 561 y Fm(J)1248 524 y Fi(+)1333 561 y Fn(:=)25 b Fl(f)p Fm(z)30 b Fl(2)p 1683 561 4 56 v 25 w Fn(C)p Fl(j)60 b Fn(Re)q Fm(z)29 b Fl(2)c Fm(J)o(;)76 b Fn(Im)p Fm(z)29 b(>)c Fn(0)p Fl(g)0 757 y Fn(Let)34 b Fm(\034)5 b(;)15 b(\037)308 771 y Fi(1)378 757 y Fl(2)30 b Fm(C)541 724 y Fh(1)534 780 y Fi(0)615 757 y Fn(\(I)-15 b(R\))34 b(b)s(e)e(t)m(w)m(o)j(cuto\013)f(functions)f (suc)m(h)g(that)h Fm(\037)2223 771 y Fi(1)2263 757 y Fn(\()p Fm(x)p Fn(\))c(=)g Fm(\034)10 b Fn(\()p Fm(x)p Fn(\))31 b Fl(\021)f Fn(1)k(on)g Fm(J)42 b Fn(and)33 b Fm(\034)10 b Fn(\()p Fm(x)p Fn(\))31 b(=)f(1)k(on)0 870 y(supp)14 b Fm(\037)260 884 y Fi(1)299 870 y Fn(.)40 b(W)-8 b(e)32 b(set:)1604 983 y Fm(H)1680 997 y Fj(\034)1748 983 y Fn(:=)25 b Fm(\034)10 b Fn(\()p Fm(H)d Fn(\))p Fm(H)r(:)0 1145 y Fn(\(This)33 b(useful)f(idea)h(of)h(replacing)f Fm(H)40 b Fn(b)m(y)33 b(its)h(lo)s(cal)g(v)m(ersion)f Fm(H)2180 1159 y Fj(\034)2256 1145 y Fn(is)g(tak)m(en)h(from)f([S])g (and)f(w)m(as)i(also)g(used)e(in)0 1258 y([GJ].\))0 1439 y Fe(Lemma)j(3.1)46 b Fk(L)-5 b(et)33 b Fn(0)25 b Fm(<)g(s)g Fl(\024)g Fn(1)p Fk(.)42 b(Consider)34 b(the)f(fol)5 b(lowing)34 b(thr)-5 b(e)g(e)34 b(statements:)1031 1635 y Fn(\()p Fm(L)p Fn(1\))102 b(sup)1289 1717 y Fj(z)s Fh(2)p Fj(J)1417 1698 y Ff(+)1483 1635 y Fl(kh)p Fm(A)p Fl(i)1666 1597 y Fh(\000)p Fj(s)1759 1635 y Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))2069 1597 y Fh(\000)p Fi(1)2164 1635 y Fl(h)p Fm(A)p Fl(i)2302 1597 y Fh(\000)p Fj(s)2395 1635 y Fl(k)25 b Fm(<)g Fn(+)p Fl(1)p Fm(;)0 1897 y Fk(\(L2\))33 b(ther)-5 b(e)34 b(exists)f Fm(C)f(>)24 b Fn(0)33 b Fk(such)g(that)h (for)f(al)5 b(l)33 b Fm(z)d Fl(2)25 b Fm(J)1841 1864 y Fi(+)1900 1897 y Fk(,)32 b Fm(u)26 b Fl(2)f Fn(\()p Fm(H)i Fn(+)20 b(i\))2413 1864 y Fh(\000)p Fi(1)2508 1897 y Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)2754 1864 y Fj(s)2792 1897 y Fn(\))32 b Fk(one)h(has:)1279 2093 y Fl(kh)p Fm(A)p Fl(i)1462 2055 y Fh(\000)p Fj(s)1555 2093 y Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2029 2055 y Fj(s)2067 2093 y Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(:)0 2288 y Fk(\(L3\))33 b(ther)-5 b(e)34 b(exists)f Fm(C)f(>)24 b Fn(0)33 b Fk(such)g(that)h (for)f(al)5 b(l)33 b Fm(z)d Fl(2)25 b Fm(J)1841 2255 y Fi(+)1900 2288 y Fk(,)32 b Fm(u)26 b Fl(2)f(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)2370 2255 y Fj(s)2407 2288 y Fn(\))33 b Fk(one)g(has:)1011 2484 y Fl(kh)p Fm(A)p Fl(i)1194 2447 y Fh(\000)p Fj(s)1287 2484 y Fm(\037)1344 2498 y Fi(1)1384 2484 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2011 2447 y Fj(s)2049 2484 y Fn(\()p Fm(H)2160 2498 y Fj(\034)2223 2484 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)2452 2498 y Fi(1)2492 2484 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)p Fm(:)0 2680 y Fk(Then)1438 2793 y Fn(\()p Fm(L)p Fn(3\))47 b Fl(\))e Fn(\()p Fm(L)p Fn(2\))27 b Fl(\))45 b Fn(\()p Fm(L)p Fn(1\))p Fm(:)0 2974 y Fe(Pro)s(of.)90 b Fn(Note)44 b(\014rst)e(that)h (using)f(Lemma)h(2.1,)k(w)m(e)c(see)g(that)g(the)g(estimates)h Fk(\(L2\))g Fn(and)e Fk(\(L3\))h Fn(ha)m(v)m(e)h(a)0 3087 y(meaning,)f(since)e(the)f(op)s(erators)h(\()p Fm(H)34 b Fl(\000)26 b Fm(z)t Fn(\)\()p Fm(H)35 b Fn(+)26 b(i\))1839 3054 y Fh(\000)p Fi(1)1974 3087 y Fn(and)40 b(\()p Fm(H)2272 3101 y Fj(\034)2342 3087 y Fl(\000)26 b Fm(z)t Fn(\))p Fm(\037)2577 3101 y Fi(1)2617 3087 y Fn(\()p Fm(H)7 b Fn(\))41 b(preserv)m(e)f Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)3421 3054 y Fj(s)3459 3087 y Fn(\))h(for)f(all)0 3199 y(0)26 b Fl(\024)f Fm(s)g Fl(\024)f Fn(1.)141 3312 y(Let)45 b(us)f(pro)m(v)m(e)h(that)g(\()p Fm(L)p Fn(2\))79 b Fl(\))f Fn(\()p Fm(L)p Fn(1\).)83 b(Let)45 b Fm(f)58 b Fl(2)48 b(H)d Fn(and)f Fm(u)k Fn(=)h(\()p Fm(H)36 b Fl(\000)30 b Fm(z)t Fn(\))2880 3279 y Fh(\000)p Fi(1)2975 3312 y Fl(h)p Fm(A)p Fl(i)3113 3279 y Fh(\000)p Fj(s)3205 3312 y Fm(f)10 b Fn(.)82 b(Then)43 b Fm(u)49 b Fl(2)0 3425 y Fn(\()p Fm(H)28 b Fn(+)19 b(i\))289 3392 y Fh(\000)p Fi(1)384 3425 y Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)630 3392 y Fj(s)668 3425 y Fn(\))31 b(since:)534 3621 y Fm(u)25 b Fn(=)g(\()p Fm(H)j Fn(+)20 b(i\))997 3584 y Fh(\000)p Fi(1)1092 3621 y Fl(h)p Fm(A)p Fl(i)1230 3584 y Fh(\000)p Fj(s)1322 3621 y Fm(g)s(;)47 b Fn(for)30 b Fm(g)f Fn(=)c Fm(f)k Fn(+)20 b(\()p Fm(z)25 b Fn(+)20 b(i\))p Fl(h)p Fm(A)p Fl(i)2303 3584 y Fj(s)2341 3621 y Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))2652 3584 y Fh(\000)p Fi(1)2746 3621 y Fl(h)p Fm(A)p Fl(i)2884 3584 y Fh(\000)p Fj(s)2977 3621 y Fm(f)35 b Fl(2)24 b(H)q Fm(:)0 3817 y Fn(Applying)30 b Fk(\(L2\))h Fn(to)g Fm(u)g Fn(w)m(e)f(obtain)h Fk(\(L1\))p Fn(.)141 3930 y(Let)g(us)f(no)m(w)g(pro)m(v)m(e)h(that)g(\()p Fm(L)p Fn(3\))47 b Fl(\))e Fn(\()p Fm(L)p Fn(2\).)d(Set)f(~)-56 b Fm(\037)1869 3944 y Fi(1)1934 3930 y Fn(=)25 b(1)c Fl(\000)e Fm(\037)2243 3944 y Fi(1)2283 3930 y Fn(.)40 b(Then:)951 4126 y Fl(kh)p Fm(A)p Fl(i)1134 4088 y Fh(\000)p Fj(s)1227 4126 y Fm(u)p Fl(k)25 b(\024)g(kh)p Fm(A)p Fl(i)1628 4088 y Fh(\000)p Fj(s)1721 4126 y Fm(\037)1778 4140 y Fi(1)1818 4126 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)21 b Fn(+)f Fl(kh)p Fm(A)p Fl(i)2363 4088 y Fh(\000)p Fj(s)2467 4126 y Fn(~)-56 b Fm(\037)2513 4140 y Fi(1)2552 4126 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)p Fm(;)-2827 b Fn(\(3.24\))511 4322 y Fl(kh)p Fm(A)p Fl(i)694 4284 y Fh(\000)p Fj(s)798 4322 y Fn(~)-55 b Fm(\037)845 4336 y Fi(1)884 4322 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b Fn(=)f Fl(kh)p Fm(A)p Fl(i)1439 4284 y Fh(\000)p Fj(s)1543 4322 y Fn(~)-56 b Fm(\037)1589 4336 y Fi(1)1629 4322 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))2093 4284 y Fh(\000)p Fi(1)2188 4322 y Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(k)p Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)0 4484 y Fn(uniformly)29 b(for)i Fm(z)e Fl(2)c Fm(J)770 4451 y Fi(+)859 4484 y Fn(since)42 b(~)-56 b Fm(\037)1139 4498 y Fi(1)1204 4484 y Fl(\021)25 b Fn(0)30 b(on)h Fm(J)9 b Fn(.)41 b(Next)352 4680 y Fl(h)p Fm(A)p Fl(i)490 4643 y Fj(s)527 4680 y Fn(\()p Fm(H)638 4694 y Fj(\034)702 4680 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)931 4694 y Fi(1)971 4680 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)26 b Fn(=)f Fl(h)p Fm(A)p Fl(i)1436 4643 y Fj(s)1473 4680 y Fn(\()p Fm(H)j Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1841 4694 y Fi(1)1881 4680 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)26 b Fn(=)e Fl(h)p Fm(A)p Fl(i)2345 4643 y Fj(s)2383 4680 y Fm(\037)2440 4694 y Fi(1)2479 4680 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)2770 4643 y Fh(\000)p Fj(s)2864 4680 y Fl(h)p Fm(A)p Fl(i)3002 4643 y Fj(s)3039 4680 y Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u;)0 4876 y Fn(since)31 b Fm(\034)k Fl(\021)25 b Fn(1)31 b(on)f(supp)14 b Fm(\037)856 4890 y Fi(1)895 4876 y Fn(.)40 b(By)31 b(Lemma)g(2.1,)g Fm(\037)1656 4890 y Fi(1)1696 4876 y Fn(\()p Fm(H)7 b Fn(\))31 b(is)f(b)s(ounded)e (on)j Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)2715 4843 y Fj(s)2752 4876 y Fn(\))g(hence:)1008 5072 y Fl(kh)p Fm(A)p Fl(i)1191 5034 y Fj(s)1229 5072 y Fn(\()p Fm(H)1340 5086 y Fj(\034)1404 5072 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1633 5086 y Fi(1)1673 5072 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2300 5034 y Fj(s)2337 5072 y Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)-2770 b Fn(\(3.25\))0 5268 y(uniformly)29 b(for)i Fm(z)e Fl(2)c Fm(J)770 5235 y Fi(+)829 5268 y Fn(.)41 b(Com)m(bining)30 b(\(3.24\))j(and)c(\(3.25\),)k(w)m(e)e(see)g (that)g(\()p Fm(L)p Fn(3\))46 b Fl(\))g Fn(\()p Fm(L)p Fn(2\).)1867 5517 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 141 162 a Fa(2)141 325 y Fk(Pr)-5 b(o)g(of)34 b(of)f(Thm.)43 b(1.)141 438 y Fn(W)-8 b(e)35 b(will)e(pro)m(v)m(e)h(the)g(estimate)h(\()p Fm(L)p Fn(3\))f(in)f (Lemma)h(3.1.)50 b(Let)34 b Fm(I)j Fl(\032)30 b Fn(I)-15 b(R)33 b(b)s(e)g(a)g(b)s(ounded)e(op)s(en)i(in)m(terv)-5 b(al)35 b(on)0 551 y(whic)m(h)22 b(the)h(Mourre)g(estimate)i(\(1.1\))f (holds,)g(and)e(let)i Fm(J)34 b Fl(\032)25 b Fm(I)30 b Fn(b)s(e)22 b(a)h(closed)h(in)m(terv)-5 b(al.)39 b(W)-8 b(e)24 b(c)m(ho)s(ose)g Fm(\034)35 b Fl(2)25 b Fm(C)3550 518 y Fh(1)3543 574 y Fi(0)3624 551 y Fn(\(I)-15 b(R\))0 664 y(suc)m(h)33 b(that)g Fm(\034)40 b Fl(\021)29 b Fn(1)34 b(near)f Fm(J)9 b Fn(.)48 b(By)34 b([GJ,)g(Lemma)f(2.5])i(w)m(e)e(kno)m (w)g(that)h(the)f(Mourre)g(estimate)i(for)d Fm(H)3496 678 y Fj(\034)3572 664 y Fn(holds)0 777 y(on)e Fm(I)7 b Fn(,)31 b(i.e.:)1290 890 y Fm(\037)p Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)1601 904 y Fj(\034)1645 890 y Fm(;)15 b Fn(i)p Fm(A)p Fn(])p Fm(\037)p Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)f Fm(c)2174 904 y Fi(0)2214 890 y Fm(\037)2271 852 y Fi(2)2311 890 y Fn(\()p Fm(H)7 b Fn(\))p Fm(;)-2489 b Fn(\(3.26\))0 1056 y(for)29 b(some)g Fm(c)403 1070 y Fi(0)468 1056 y Fm(>)c Fn(0,)30 b(if)e(supp)14 b Fm(\037)25 b Fl(\032)g Fm(I)7 b Fn(.)40 b(Let)30 b(us)e(pic)m(k)h Fm(\037)1762 1070 y Fi(1)1801 1056 y Fm(;)15 b(\037)1898 1070 y Fi(2)1967 1056 y Fn(as)29 b(in)f(Prop.)40 b(2.3)30 b(with)e(supp)14 b Fm(\037)3050 1070 y Fj(i)3103 1056 y Fl(\032)25 b Fm(J)9 b Fn(.)41 b(F)-8 b(rom)29 b(Prop.)0 1169 y(2.3)i Fk(i\))g Fn(and)f(\(3.26\),)i(w)m(e)f(get:)569 1371 y Fm(\037)626 1385 y Fi(1)665 1371 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)919 1385 y Fj(\034)963 1371 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()p Fm(A)p Fn(\)])p Fm(\037)1319 1385 y Fi(1)1360 1371 y Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)83 b Fm(c)1732 1385 y Fi(0)1772 1371 y Fm(\037)1829 1385 y Fi(1)1868 1371 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)2262 1338 y Fi(2)2262 1394 y(2)2303 1371 y Fn(\()p Fm(H)g Fn(\))p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)2697 1385 y Fi(1)2739 1371 y Fn(\()p Fm(H)g Fn(\))1539 1535 y(+)83 b Fm(\037)1750 1549 y Fi(1)1789 1535 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)2080 1502 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)2385 1535 y Fm(R)2454 1549 y Fi(1)2493 1535 y Fl(h)p Fm(A)p Fl(i)2631 1502 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)2935 1535 y Fm(\037)2992 1549 y Fi(1)3032 1535 y Fn(\()p Fm(H)g Fn(\))p Fm(;)0 1736 y Fn(and)30 b(using)g(also)h(Prop.)40 b(2.3)32 b Fk(ii\))p Fn(,)e(w)m(e)h(obtain:)265 1941 y Fm(\037)322 1955 y Fi(1)361 1941 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)615 1955 y Fj(\034)659 1941 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()p Fm(A)p Fn(\)])p Fm(\037)1015 1955 y Fi(1)1056 1941 y Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)f Fn(c)1371 1955 y Fi(0)1411 1941 y Fm(\037)1468 1955 y Fi(1)1507 1941 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)1706 1903 y Fi(2)1747 1941 y Fn(\()p Fm(A)p Fn(\))p Fm(\037)1942 1955 y Fi(1)1982 1941 y Fn(\()p Fm(H)g Fn(\))21 b(+)f Fm(\037)2304 1955 y Fi(1)2343 1941 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)p Fm(A)p Fl(i)2634 1903 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)2939 1941 y Fm(R)3008 1955 y Fi(2)3047 1941 y Fl(h)p Fm(A)p Fl(i)3185 1903 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)3489 1941 y Fm(;)-3514 b Fn(\(3.27\))0 2145 y(where:)639 2258 y Fl(k)p Fm(R)753 2272 y Fi(2)793 2258 y Fl(k)25 b(\024)g Fm(C)7 b Fn(\()p Fl(k)p Fn(ad)1208 2220 y Fi(2)1208 2280 y Fj(A)1265 2258 y Fm(H)1341 2272 y Fj(\034)1383 2258 y Fl(k)21 b Fn(+)f Fl(k)p Fn(ad)1681 2272 y Fj(A)1738 2258 y Fm(H)1814 2272 y Fj(\034)1857 2258 y Fl(kk)p Fn(ad)2044 2272 y Fj(A)2101 2258 y Fm(\037)2158 2272 y Fi(2)2197 2258 y Fn(\()p Fm(H)7 b Fn(\))p Fl(k)22 b Fn(+)d Fm(c)2546 2272 y Fi(0)2586 2258 y Fl(k)p Fn(ad)2728 2272 y Fj(A)2785 2258 y Fm(\037)2842 2272 y Fi(2)2881 2258 y Fn(\()p Fm(H)7 b Fn(\))p Fl(k)p Fn(\))p Fm(:)-3139 b Fn(\(3.28\))141 2425 y(W)-8 b(e)31 b(replace)f(no)m(w)g Fm(A)g Fn(b)m(y)1026 2389 y Fj(A)p 1026 2404 54 4 v 1026 2456 a(R)1119 2425 y Fn(for)f Fm(R)d Fl(\025)f Fn(1.)41 b(Noting)31 b(that)f(b)m(y)f(Prop.)40 b(2.3)31 b(the)f(constan)m(t)g Fm(C)36 b Fn(in)30 b(the)f(r.h.s.)40 b(of)0 2537 y(\(3.28\))33 b(is)d(indep)s(enden)m(t)f(on)h Fm(A)h Fn(and)f(that)h Fm(c)1507 2551 y Fi(0)1577 2537 y Fn(is)f(replaced)h(b)m(y)f Fm(c)2192 2551 y Fi(0)2232 2537 y Fm(R)2302 2505 y Fh(\000)p Fi(1)2396 2537 y Fn(,)h(w)m(e)f (obtain:)271 2783 y Fm(\037)328 2797 y Fi(1)367 2783 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)621 2797 y Fj(\034)665 2783 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()847 2722 y Fm(A)p 846 2762 70 4 v 846 2846 a(R)927 2783 y Fn(\)])p Fm(\037)1044 2797 y Fi(1)1084 2783 y Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)1368 2722 y Fm(c)1407 2736 y Fi(0)p 1368 2762 79 4 v 1373 2846 a Fm(R)1457 2783 y(\037)1514 2797 y Fi(1)1554 2783 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)1753 2722 y Fm(A)p 1752 2762 70 4 v 1752 2846 a(R)1832 2783 y Fl(i)1867 2746 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))2100 2783 y Fm(\037)2157 2797 y Fi(1)2196 2783 y Fn(\()p Fm(H)g Fn(\))21 b Fl(\000)2490 2722 y Fm(C)p 2471 2762 110 4 v 2471 2846 a(R)2541 2820 y Fi(2)2590 2783 y Fm(\037)2647 2797 y Fi(1)2687 2783 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)2886 2722 y Fm(A)p 2885 2762 70 4 v 2885 2846 a(R)2965 2783 y Fl(i)3000 2746 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))3233 2783 y Fm(\037)3290 2797 y Fi(1)3330 2783 y Fn(\()p Fm(H)g Fn(\))p Fm(:)-3508 b Fn(\(3.29\))0 3009 y(Fixing)31 b Fm(R)26 b Fl(\035)f Fn(1)31 b(w)m(e)g(obtain:)819 3250 y Fm(\037)876 3264 y Fi(1)916 3250 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)1170 3264 y Fj(\034)1213 3250 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()1395 3188 y Fm(A)p 1394 3228 V 1394 3312 a(R)1475 3250 y Fn(\)])p Fm(\037)1592 3264 y Fi(1)1632 3250 y Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)1935 3188 y Fm(c)1974 3202 y Fi(0)p 1917 3228 116 4 v 1917 3312 a Fn(2)p Fm(R)2042 3250 y(\037)2099 3264 y Fi(1)2138 3250 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)2337 3188 y Fm(A)p 2336 3228 70 4 v 2336 3312 a(R)2417 3250 y Fl(i)2452 3212 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))2685 3250 y Fm(\037)2742 3264 y Fi(1)2781 3250 y Fn(\()p Fm(H)g Fn(\))p Fm(:)-2959 b Fn(\(3.30\))0 3493 y(Since)33 b Fm(R)h Fn(is)f(\014xed)f(for)h(the)g(rest)g(of)h(the)f(pro)s(of)f(w)m (e)i(can)f(denote)2258 3457 y Fj(A)p 2258 3472 54 4 v 2258 3524 a(R)2354 3493 y Fn(again)h(b)m(y)f Fm(A)p Fn(.)49 b(W)-8 b(e)34 b(apply)f(no)m(w)g(iden)m(tit)m(y)0 3606 y(\(1.3\))f(to)f Fm(C)h Fn(=)25 b Fm(F)13 b Fn(\()p Fm(A)p Fn(\),)32 b Fm(H)g Fn(=)25 b Fm(H)1067 3620 y Fj(\034)1109 3606 y Fn(.)41 b(Since)30 b Fm(F)39 b Fl(\024)25 b Fn(0)30 b(and)g(Im)p Fm(z)f(>)c Fn(0)31 b(w)m(e)g(get)g(for)g Fm(u)25 b Fl(2)g(H)q Fn(:)682 3810 y Fl(kh)p Fm(A)p Fl(i)865 3772 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)1170 3810 y Fm(\037)1227 3824 y Fi(1)1266 3810 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1516 3772 y Fi(2)1582 3810 y Fl(\024)25 b Fm(C)7 b Fl(j)p Fn(\()p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)2076 3824 y Fi(1)2116 3810 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u;)15 b Fn(\()p Fm(H)2472 3824 y Fj(\034)2537 3810 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)2766 3824 y Fi(1)2806 3810 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fn(\))p Fl(j)p Fm(:)-3096 b Fn(\(3.31\))0 4014 y(Using)31 b(that)g Fm(F)43 b Fn(is)30 b(a)h(b)s(ounded)d(function,)j(w)m(e)f(get)i(for)e Fm(u)25 b Fl(2)g(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)2307 3981 y Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)2556 4014 y Fn(\):)363 4218 y Fl(kh)p Fm(A)p Fl(i)546 4181 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)850 4218 y Fm(\037)907 4232 y Fi(1)946 4218 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1196 4181 y Fi(2)1262 4218 y Fl(\024)25 b Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)1613 4181 y Fh(\000)p Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)1917 4218 y Fm(\037)1974 4232 y Fi(1)2014 4218 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(kkh)p Fm(A)p Fl(i)2447 4181 y Fi(\(1+)p Fj(\017)p Fi(\))p Fj(=)p Fi(2)2697 4218 y Fn(\()p Fm(H)2808 4232 y Fj(\034)2872 4218 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)3101 4232 y Fi(1)3141 4218 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)p Fm(:)0 4423 y Fn(This)38 b(implies)h(that)h(the)f(estimate)i Fk(\(L3\))f Fn(of)f(Lemma)g(3.1)h(holds)e(for)h Fm(z)44 b Fl(2)39 b Fm(J)2733 4390 y Fi(+)2831 4423 y Fn(and)g Fm(s)g Fn(=)g(\(1)27 b(+)f Fm(\017)p Fn(\))p Fm(=)p Fn(2.)67 b(By)0 4535 y(Lemma)30 b(3.1,)i(this)e(pro)m(v)m(es)i(Thm.)39 b(1.)i Fa(2)0 4822 y Fo(4)135 b(Pro)t(of)45 b(of)g(Thm.)59 b(2)0 5025 y Fn(Let)31 b Fm(H)h Fl(2)25 b Fm(C)429 4992 y Fi(1)468 5025 y Fn(\()p Fm(A)p Fn(\))31 b(and)f Fm(J)o(;)15 b(\034)5 b(;)15 b(\037)1045 5039 y Fi(1)1115 5025 y Fn(as)31 b(in)f(Sect.)41 b(3.)1844 5517 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 0 162 a Fe(Lemma)35 b(4.1)46 b Fk(Consider)34 b(the)f(fol)5 b(lowing)33 b(thr)-5 b(e)g(e)34 b(statements:)999 367 y Fn(\()p Fm(M)10 b Fn(1\))102 b(sup)1293 449 y Fj(z)s Fh(2)p Fj(J)1421 430 y Ff(+)1487 367 y Fl(k)p Fm(P)1590 381 y Fh(\000)1650 367 y Fn(\()p Fm(A)p Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))2099 329 y Fh(\000)p Fi(1)2194 367 y Fl(h)p Fm(A)p Fl(i)2332 329 y Fh(\000)p Fi(1)2427 367 y Fl(k)26 b Fm(<)f Fn(+)p Fl(1)p Fm(;)0 637 y Fk(\(M2\))33 b(ther)-5 b(e)33 b(exists)g Fm(C)f(>)25 b Fn(0)33 b Fk(such)g(that)h(for)f(al)5 b(l)33 b Fm(z)c Fl(2)c Fm(J)1865 604 y Fi(+)1924 637 y Fk(,)33 b Fm(u)25 b Fl(2)g Fn(\()p Fm(H)j Fn(+)19 b(i\))2437 604 y Fh(\000)p Fi(1)2532 637 y Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\))34 b Fk(one)f(has:)1285 841 y Fl(k)p Fm(P)1388 855 y Fh(\000)1448 841 y Fn(\()p Fm(A)p Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)29 b Fl(\000)19 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(:)0 1045 y Fk(\(M3\))33 b(ther)-5 b(e)33 b(exists)g Fm(C)f(>)25 b Fn(0)33 b Fk(such)g(that)h(for)f(al)5 b(l)33 b Fm(z)c Fl(2)c Fm(J)1865 1012 y Fi(+)1924 1045 y Fk(,)33 b Fm(u)25 b Fl(2)g(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\))34 b Fk(one)f(has:)1017 1249 y Fl(k)p Fm(P)1120 1263 y Fh(\000)1180 1249 y Fn(\()p Fm(A)p Fn(\))p Fm(\037)1375 1263 y Fi(1)1415 1249 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)2153 1263 y Fj(\034)2217 1249 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)2446 1263 y Fi(1)2486 1249 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)p Fm(:)0 1454 y Fk(Then)1384 1567 y Fn(\()p Fm(M)j Fn(3\))47 b Fl(\))e Fn(\()p Fm(M)10 b Fn(2\))27 b Fl(\))45 b Fn(\()p Fm(M)10 b Fn(1\))p Fm(:)0 1754 y Fe(Pro)s(of.)64 b Fn(Again)36 b(b)m(y)g(Lemma)f(2.1)i(w)m(e)e(see)h(that)g(the)g (estimates)g Fk(\(M2\))g Fn(and)f Fk(\(M3\))g Fn(ha)m(v)m(e)i(a)e (meaning.)56 b(W)-8 b(e)0 1867 y(use)29 b(the)g(notation)h(in)f(the)g (pro)s(of)f(of)i(Lemma)f(3.1.)41 b(The)29 b(pro)s(of)f(that)i(\()p Fm(M)10 b Fn(2\))26 b Fl(\))43 b Fn(\()p Fm(M)10 b Fn(1\))30 b(is)f(as)g(in)g(Lemma)g(3.1.)0 1980 y(T)-8 b(o)31 b(pro)m(v)m(e)g (that)g(\()p Fm(M)10 b Fn(3\))47 b Fl(\))e Fn(\()p Fm(M)10 b Fn(2\))32 b(w)m(e)f(write:)486 2181 y Fl(k)p Fm(P)589 2195 y Fh(\000)649 2181 y Fn(\()p Fm(A)p Fn(\))p Fm(u)p Fl(k)26 b(\024)f(k)p Fm(P)1109 2195 y Fh(\000)1169 2181 y Fn(\()p Fm(A)p Fn(\))p Fm(\037)1364 2195 y Fi(1)1404 2181 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)22 b Fn(+)e Fl(k)p Fm(P)1870 2195 y Fh(\000)1929 2181 y Fn(\()p Fm(A)p Fn(\))11 b(~)-56 b Fm(\037)2124 2195 y Fi(1)2165 2181 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)p Fm(;)486 2342 y Fl(k)p Fm(P)589 2356 y Fh(\000)649 2342 y Fn(\()p Fm(A)p Fn(\))k(~)-56 b Fm(\037)844 2356 y Fi(1)884 2342 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b Fn(=)f Fl(k)p Fm(P)1359 2356 y Fh(\000)1419 2342 y Fn(\()p Fm(A)p Fn(\))11 b(~)-56 b Fm(\037)1614 2356 y Fi(1)1654 2342 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))2118 2309 y Fh(\000)p Fi(1)2213 2342 y Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(k)p Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)0 2548 y Fn(uniformly)29 b(for)i Fm(z)e Fl(2)c Fm(J)770 2515 y Fi(+)829 2548 y Fn(,)31 b(since)41 b(~)-56 b Fm(\037)1164 2562 y Fi(1)1229 2548 y Fl(\021)25 b Fn(0)31 b(on)f Fm(J)9 b Fn(.)41 b(By)30 b(\(3.25\))j(for)d Fm(s)25 b 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Fj(C)p 2508 5039 89 4 v 2508 5093 a(R)2561 5074 y Ff(2)2606 5060 y Fm(\037)2663 5074 y Fi(1)2703 5060 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)2902 5027 y Fi(2)2942 5060 y Fn(\()2987 5024 y Fj(A)p 2987 5039 54 4 v 2987 5091 a(R)3051 5060 y Fn(\))p Fm(\037)3143 5074 y Fi(1)3183 5060 y Fn(\()p Fm(H)g Fn(\))1572 5222 y Fl(\000)1669 5186 y Fj(C)p 1653 5201 89 4 v 1653 5255 a(R)1706 5237 y Ff(2)1751 5222 y Fm(\037)1808 5236 y Fi(1)1847 5222 y Fn(\()p Fm(H)g Fn(\))p Fl(h)2045 5186 y Fj(A)p 2045 5201 54 4 v 2045 5254 a(R)2110 5222 y Fl(i)2145 5189 y Fh(\000)p Fi(2)p Fj(s)2272 5222 y Fm(\037)2329 5236 y Fi(1)2368 5222 y Fn(\()p Fm(H)g Fn(\))p Fm(:)0 5140 y Fn(\(4.34\))1844 5517 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 0 162 a Fn(Fixing)31 b Fm(R)26 b Fl(\035)f Fn(1)31 b(large)g(enough,)g(w)m(e)g(obtain:)379 403 y Fm(\037)436 417 y Fi(1)475 403 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)729 417 y Fj(\034)773 403 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()955 342 y Fm(A)p 954 382 70 4 v 954 466 a(R)1035 403 y Fn(\)])p Fm(\037)1152 417 y Fi(1)1192 403 y Fn(\()p Fm(H)7 b Fn(\))25 b Fl(\025)1495 342 y Fm(c)1534 356 y Fi(0)p 1476 382 116 4 v 1476 466 a Fn(2)p Fm(R)1602 403 y(\037)1659 417 y Fi(1)1698 403 y Fn(\()p Fm(H)7 b Fn(\))p Fm(g)1897 366 y Fi(2)1938 403 y Fn(\()1984 342 y Fm(A)p 1983 382 70 4 v 1983 466 a(R)2063 403 y Fn(\))p Fm(\037)2155 417 y Fi(1)2194 403 y Fn(\()p Fm(H)g Fn(\))21 b Fl(\000)2488 342 y Fm(C)p 2469 382 110 4 v 2469 466 a(R)2539 439 y Fi(2)2588 403 y Fm(\037)2645 417 y Fi(1)2684 403 y Fn(\()p Fm(H)7 b Fn(\))p Fl(h)2883 342 y Fm(A)p 2882 382 70 4 v 2882 466 a(R)2963 403 y Fl(i)2998 366 y Fh(\000)p Fi(2)p Fj(s)3125 403 y Fm(\037)3182 417 y Fi(1)3222 403 y Fn(\()p Fm(H)g Fn(\))p Fm(:)-3400 b Fn(\(4.35\))0 646 y(W)-8 b(e)35 b(denote)g(again)713 611 y Fj(A)p 713 626 54 4 v 713 678 a(R)810 646 y Fn(b)m(y)f Fm(A)g Fn(\(note)h(that)g Fm(P)1542 660 y Fh(\000)1601 646 y Fn(\()p Fm(A)p Fn(\))d(=)f Fm(P)1931 660 y Fh(\000)1991 646 y Fn(\()2036 611 y Fj(A)p 2036 626 V 2036 678 a(R)2100 646 y Fn(\)\),)k(and)f(apply)f(iden)m(tit)m(y)j(\(1.3\))g(to)f Fm(C)i Fn(=)31 b Fm(F)13 b Fn(\()p Fm(A)p Fn(\),)0 759 y Fm(H)32 b Fn(=)25 b Fm(H)280 773 y Fj(\034)353 759 y Fn(and)30 b(get:)115 964 y Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)401 978 y Fi(1)442 964 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)692 926 y Fi(2)758 964 y Fl(\024)25 b Fm(C)7 b Fl(j)p Fn(\()p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)1252 978 y Fi(1)1292 964 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u;)15 b Fn(\()p Fm(H)1648 978 y Fj(\034)1713 964 y Fl(\000)k Fm(z)t Fn(\))p Fm(\037)1941 978 y Fi(1)1981 964 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fn(\))p Fl(j)22 b Fn(+)e Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2614 926 y Fh(\000)p Fj(s)2706 964 y Fm(\037)2763 978 y Fi(1)2803 964 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)3053 926 y Fi(2)3093 964 y Fm(;)76 b(u)25 b Fl(2)g(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\))p Fm(:)0 1168 y Fn(Next)31 b(w)m(e)g(note)g(that:)918 1346 y Fl(j)p Fn(\()p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)1244 1360 y Fi(1)1284 1346 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u;)15 b Fn(\()p Fm(H)1640 1360 y Fj(\034)1705 1346 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1934 1360 y Fi(1)1974 1346 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fn(\))p Fl(j)764 1507 y Fn(=)83 b Fl(j)p Fn(\()p Fl(h)p Fm(A)p Fl(i)1116 1474 y Fh(\000)p Fi(1)1211 1507 y Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)1477 1521 y Fi(1)1518 1507 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u;)15 b Fl(h)p Fm(A)p Fl(i)p Fn(\()p Fm(H)2012 1521 y Fj(\034)2077 1507 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)2306 1521 y Fi(1)2346 1507 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fn(\))p Fl(j)764 1667 y(\024)83 b Fm(\017)p Fl(kh)p Fm(A)p Fl(i)1138 1634 y Fh(\000)p Fi(1)1233 1667 y Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)1499 1681 y Fi(1)1539 1667 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1789 1634 y Fi(2)1850 1667 y Fn(+)20 b Fm(\017)1978 1634 y Fh(\000)p Fi(1)2072 1667 y Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)2366 1681 y Fj(\034)2431 1667 y Fl(\000)g Fm(z)t Fn(\))p Fm(\037)2660 1681 y Fi(1)2700 1667 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)2950 1634 y Fi(2)2990 1667 y Fm(;)0 1863 y Fn(for)30 b(all)h Fm(\017)26 b(>)f Fn(0.)41 b(Since)1265 1976 y Fl(h)p Fm(\025)p Fl(i)1388 1939 y Fh(\000)p Fi(1)1483 1976 y Fl(j)p Fm(F)13 b Fl(j)p Fn(\()p Fm(\025)p Fn(\))27 b Fl(\024)e Fm(C)7 b(g)s Fn(\()p Fm(\025)p Fn(\))21 b(+)e Fm(C)7 b Fl(h)p Fm(\025)p Fl(i)2397 1939 y Fh(\000)p Fj(s)2489 1976 y Fm(;)0 2143 y Fn(for)30 b(all)h(0)26 b Fm(<)f(s)g(<)g Fn(1,)31 b(w)m(e)g(get:)971 2339 y Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1257 2353 y Fi(1)1298 2339 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1548 2306 y Fi(2)817 2500 y Fl(\024)83 b Fm(C)7 b(\017)p Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1366 2514 y Fi(1)1407 2500 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)1657 2467 y Fi(2)1717 2500 y Fn(+)20 b Fm(C)7 b(\017)p Fl(kh)p Fm(A)p Fl(i)2100 2467 y Fh(\000)p Fj(s)2193 2500 y Fm(\037)2250 2514 y Fi(1)2289 2500 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)2539 2467 y Fi(2)817 2660 y Fn(+)83 b Fm(C)7 b(\017)1080 2627 y Fh(\000)p Fi(1)1174 2660 y Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)1468 2674 y Fj(\034)1532 2660 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1761 2674 y Fi(1)1801 2660 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)2051 2627 y Fi(2)2112 2660 y Fn(+)20 b Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2458 2627 y Fh(\000)p Fj(s)2550 2660 y Fm(\037)2607 2674 y Fi(1)2647 2660 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)2897 2627 y Fi(2)2937 2660 y Fm(:)0 2856 y Fn(Cho)s(osing)30 b Fm(\017)g Fn(small)h(enough)f(this)g(giv)m(es:)590 3061 y Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)876 3075 y Fi(1)917 3061 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1167 3023 y Fi(2)1233 3061 y Fl(\024)25 b Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)1695 3075 y Fj(\034)1759 3061 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1988 3075 y Fi(1)2028 3061 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)2278 3023 y Fi(2)2339 3061 y Fn(+)20 b Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2685 3023 y Fh(\000)p Fj(s)2777 3061 y Fm(\037)2834 3075 y Fi(1)2874 3061 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)3124 3023 y Fi(2)3164 3061 y Fm(:)0 3265 y Fn(By)31 b(Thm.)39 b(1,)31 b(w)m(e)g(kno)m(w)f(that:)580 3469 y Fl(kh)p Fm(A)p Fl(i)763 3432 y Fh(\000)p Fj(s)856 3469 y Fm(\037)913 3483 y Fi(1)952 3469 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)27 b(\024)e Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)1691 3483 y Fj(\034)1755 3469 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1984 3483 y Fi(1)2023 3469 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)p Fm(;)47 b Fn(uniformly)30 b(for)g Fm(z)f Fl(2)c Fm(J)3115 3432 y Fi(+)3174 3469 y Fm(;)0 3673 y Fn(if)93 3638 y Fi(1)p 93 3653 36 4 v 93 3705 a(2)164 3673 y Fm(<)g(s)p Fn(.)40 b(This)30 b(\014nally)g(giv)m (es:)1013 3878 y Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1299 3892 y Fi(1)1340 3878 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1590 3840 y Fi(2)1656 3878 y Fl(\024)25 b Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)p Fn(\()p Fm(H)2118 3892 y Fj(\034)2182 3878 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)2411 3892 y Fi(1)2451 3878 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)2701 3840 y Fi(2)2741 3878 y Fm(:)0 4082 y Fn(Since)31 b Fm(g)s Fn(\()p Fm(\025)p Fn(\))d Fl(\025)e Fm(P)590 4096 y Fh(\000)650 4082 y Fn(\()p Fm(\025)p Fn(\),)32 b(w)m(e)g(obtain)f(the)h(estimate)h Fk(\(M3\))p Fn(,)f(whic)m(h)e(b)m (y)h(Lemma)h(4.1)g(completes)h(the)e(pro)s(of)f(of)0 4195 y(Thm.)40 b(2.)h Fa(2)0 4481 y Fo(5)135 b(Pro)t(of)45 b(of)g(Thm.)59 b(3)0 4684 y Fn(Let)31 b Fm(H)h Fl(2)25 b Fm(C)429 4651 y Fi(2)468 4684 y Fn(\()p Fm(A)p Fn(\))31 b(and)f Fm(J)o(;)15 b(\034)5 b(;)15 b(\037)1045 4698 y Fi(1)1115 4684 y Fn(as)31 b(in)f(Sect.)41 b(3.)0 4872 y Fe(Lemma)35 b(5.1)46 b Fk(Consider)34 b(the)f(fol)5 b(lowing)33 b(thr)-5 b(e)g(e)34 b(statements:)995 5076 y Fn(\()p Fm(N)10 b Fn(1\))102 b(sup)1274 5158 y Fj(z)s Fh(2)p Fj(J)1402 5139 y Ff(+)1468 5076 y Fl(k)p Fm(P)1571 5090 y Fh(\000)1630 5076 y Fn(\()p Fm(A)p Fn(\)\()p Fm(H)29 b Fl(\000)20 b Fm(z)t Fn(\))2080 5038 y Fh(\000)p Fi(1)2175 5076 y Fm(P)2233 5090 y Fi(+)2292 5076 y Fn(\()p Fm(A)p Fn(\))p Fl(k)27 b Fm(<)e Fn(+)p Fl(1)p Fm(;)1844 5517 y Fn(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 0 162 a Fk(\(N2\))33 b(ther)-5 b(e)33 b(exists)g Fm(C)q(;)15 b(b)26 b(>)f Fn(0)33 b Fk(such)g(that)h(for)f(al)5 b(l)33 b Fm(z)d Fl(2)25 b Fm(J)1926 129 y Fi(+)1985 162 y Fk(,)32 b Fm(u)26 b Fl(2)e Fn(\()p Fm(H)k Fn(+)20 b(i\))2498 129 y Fh(\000)p Fi(1)2593 162 y Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\))34 b Fk(one)e(has:)762 366 y Fl(k)p Fm(P)865 380 y Fh(\000)925 366 y Fn(\()p Fm(A)p Fn(\))p Fm(u)p Fl(k)27 b(\024)e Fm(C)7 b Fl(k)p Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)i Fn(+)d Fm(C)7 b Fl(k)p Fn(1)-20 b(l)2087 384 y Fi(])p Fh(\0001)p Fj(;b)p Fi(])2306 366 y Fn(\()p Fm(A)p Fn(\))p Fl(h)p Fm(A)p Fl(i)p Fn(\()p Fm(H)29 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(:)0 569 y Fk(\(N3\))33 b(ther)-5 b(e)33 b(exists)g Fm(C)q(;)15 b(b)26 b(>)f Fn(0)33 b Fk(such)g(that)h(for)f(al)5 b(l)33 b Fm(z)d Fl(2)25 b Fm(J)1926 536 y Fi(+)1985 569 y Fk(,)32 b Fm(u)26 b Fl(2)e(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p 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b(hence)g(b)m(y)h(\()p Fm(N)10 b Fn(2\):)888 1999 y Fl(k)p Fm(P)991 2013 y Fh(\000)1051 1999 y Fn(\()p Fm(A)p Fn(\))p Fm(u)p Fl(k)27 b(\024)e Fm(C)7 b Fl(k)p Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)26 b Fn(=)f Fm(C)7 b Fl(k)p Fm(P)2230 2013 y Fi(+)2290 1999 y Fn(\()p Fm(A)p Fn(\))p Fm(f)j Fl(k)26 b(\024)e Fm(C)7 b Fl(k)p Fm(f)j Fl(k)p Fm(;)141 2203 y Fn(uniformly)30 b(for)g Fm(z)f Fl(2)c Fm(J)911 2170 y Fi(+)970 2203 y Fn(,)31 b(whic)m(h)f(implies)g(\()p Fm(N)10 b Fn(1\).)141 2316 y(Let)31 b(us)f(no)m(w)g(pro)m(v)m(e)h(that) g(\()p Fm(N)10 b Fn(3\))47 b Fl(\))e Fn(\()p Fm(N)10 b Fn(2\).)42 b(Let)31 b Fm(u)25 b Fl(2)g Fn(\()p Fm(H)i Fn(+)20 b(i\))2313 2283 y Fh(\000)p Fi(1)2408 2316 y Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\).)42 b(As)31 b(b)s(efore)f(w)m(e)g(ha)m(v)m(e)486 2519 y Fl(k)p Fm(P)589 2533 y Fh(\000)649 2519 y Fn(\()p Fm(A)p Fn(\))11 b(~)-56 b Fm(\037)844 2533 y Fi(1)884 2519 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b Fn(=)f Fl(k)p Fm(P)1359 2533 y Fh(\000)1419 2519 y Fn(\()p Fm(A)p Fn(\)\()p Fm(H)j Fl(\000)20 b Fm(z)t Fn(\))1868 2482 y Fh(\000)p Fi(1)1974 2519 y Fn(~)-56 b Fm(\037)2020 2533 y Fi(1)2060 2519 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)26 b(\024)f Fm(C)7 b Fl(k)p Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)0 2723 y Fn(uniformly)29 b(for)i Fm(z)e Fl(2)c Fm(J)770 2690 y Fi(+)829 2723 y Fn(,)31 b(since)41 b(~)-56 b Fm(\037)1164 2737 y Fi(1)1229 2723 y Fl(\021)25 b Fn(0)31 b(on)f Fm(J)9 b Fn(.)41 b(Next)31 b(w)m(e)g(ha)m(v)m(e:)771 2926 y Fl(k)p Fn(\()p Fm(H)927 2940 y Fj(\034)991 2926 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1220 2940 y Fi(1)1260 2926 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b Fn(=)f Fl(k)p Fm(\037)1734 2940 y Fi(1)1774 2926 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)27 b(\024)d Fm(C)7 b Fl(k)p Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)0 3130 y Fn(and)30 3333 y Fl(k)p Fn(1)-20 b(l)126 3352 y Fi(])p Fh(\0001)p Fj(;b)p Fi(])345 3333 y Fn(\()p Fm(A)p Fn(\))p Fl(h)p Fm(A)p Fl(i)p Fn(\()p Fm(H)732 3347 y Fj(\034)797 3333 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1026 3347 y Fi(1)1066 3333 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)26 b Fn(=)f Fl(k)p Fn(1)-20 b(l)1534 3352 y Fi(])p Fh(\0001)p Fj(;b)p Fi(])1753 3333 y Fn(\()p Fm(A)p Fn(\))p Fl(h)p Fm(A)p Fl(i)p Fm(\037)2086 3347 y Fi(1)2127 3333 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)26 b(\024)f(k)p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)3121 3347 y Fi(1)3162 3333 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)0 3537 y Fn(for)30 b Fm(F)38 b Fl(2)25 b Fm(C)393 3504 y Fh(1)468 3537 y Fn(\(I)-15 b(R)o(\),)31 b Fm(F)39 b Fl(\025)25 b Fn(0,)31 b(supp)13 b Fm(F)39 b Fl(\032)p Fn(])20 b Fl(\000)g(1)p Fm(;)15 b Fn(2)p Fm(b)p Fn(],)31 b Fm(F)13 b Fn(\()p Fm(\025)p Fn(\))26 b(=)f Fl(h)p Fm(\025)p Fl(i)31 b Fn(in)f Fm(\025)c Fl(\024)f Fm(b)p Fn(.)141 3649 y(By)31 b(Lemma)f(2.3,)i(w)m(e)f(ha)m(v)m(e)g(as)g (an)f(iden)m(tit)m(y)i(on)e Fl(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\):)873 3853 y Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)1139 3867 y Fi(1)1179 3853 y Fn(\()p Fm(H)7 b Fn(\))26 b(=)f Fm(\037)1511 3867 y Fi(1)1551 3853 y Fn(\()p Fm(H)7 b Fn(\))p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))21 b(+)f(ad)2121 3867 y Fj(A)2178 3853 y Fm(\037)2235 3867 y Fi(1)2275 3853 y Fn(\()p Fm(H)7 b Fn(\))p Fm(F)2499 3815 y Fh(0)2523 3853 y Fn(\()p Fm(A)p Fn(\))21 b(+)f Fm(R)2842 3867 y Fi(2)2881 3853 y Fm(;)0 4056 y Fn(where)30 b Fm(R)332 4070 y Fi(2)402 4056 y Fn(is)g(b)s(ounded.)39 b(Therefore:)459 4248 y Fl(k)p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)770 4262 y Fi(1)810 4248 y Fn(\()p Fm(H)7 b Fn(\)\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)27 b(\024)82 b Fm(C)7 b Fl(k)p Fm(F)13 b Fn(\()p Fm(A)p Fn(\)\()p Fm(H)29 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)h Fn(+)f Fm(C)7 b Fl(k)p Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)1398 4408 y(\024)82 b Fm(C)7 b Fl(k)p Fn(\()p Fm(H)28 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)h Fn(+)f Fm(C)7 b Fl(k)p Fn(1)-20 b(l)2356 4427 y Fi(])p Fh(\0001)p Fj(;)p Fi(2)p Fj(b)p Fi(])2609 4408 y Fn(\()p Fm(A)p Fn(\))p Fl(h)p Fm(A)p Fl(i)p Fn(\()p Fm(H)30 b Fl(\000)20 b Fm(z)t Fn(\))p Fm(u)p Fl(k)p Fm(;)0 4610 y Fn(since)31 b Fm(F)13 b Fn(\()p Fm(\025)p Fn(\))26 b Fl(\024)f Fm(C)7 b Fn(1)-20 b(l)661 4629 y Fi(])p Fh(\0001)p Fj(;)p Fi(2)p Fj(b)p Fi(])915 4610 y Fn(\()p Fm(\025)p Fn(\))p Fl(h)p Fm(\025)p Fl(i)p Fn(.)42 b(This)29 b(completes)j(the)f (pro)s(of)e(that)i(\()p Fm(N)10 b Fn(3\))47 b Fl(\))e Fn(\()p Fm(N)10 b Fn(2\).)42 b Fa(2)141 4773 y Fk(Pr)-5 b(o)g(of)34 b(of)f(Thm.)43 b(3.)141 4886 y Fn(W)-8 b(e)35 b(will)f(pro)m(v)m(e)g(the)g(estimate)i(\()p Fm(N)10 b Fn(3\))34 b(in)g(Lemma)f(5.1.)52 b(Arguing)34 b(as)g(in)f(the)h(pro)s (of)f(of)g(Thm.)50 b(2,)35 b(using)0 4999 y(no)m(w)30 b(the)h(sharp)s(er)e(estimates)j(of)e(Prop.)40 b(2.6,)32 b(w)m(e)f(obtain)g(for)f Fm(R)c Fl(\035)f Fn(1)31 b(large)g(enough:)463 5239 y Fm(\037)520 5253 y Fi(1)559 5239 y Fn(\()p Fm(H)7 b Fn(\)[)p Fm(H)813 5253 y Fj(\034)857 5239 y Fm(;)15 b Fn(i)p Fm(F)e Fn(\()1039 5177 y Fm(A)p 1038 5218 70 4 v 1038 5301 a(R)1119 5239 y Fn(\)])p Fm(\037)1236 5253 y Fi(1)1275 5239 y Fn(\()p Fm(H)7 b Fn(\))26 b Fl(\025)1578 5177 y Fm(c)1617 5191 y Fi(0)p 1560 5218 116 4 v 1560 5301 a Fn(2)p Fm(R)1685 5239 y Fl(k)p Fm(g)1776 5201 y Fi(2)1817 5239 y Fn(\()1863 5177 y Fm(A)p 1862 5218 70 4 v 1862 5301 a(R)1942 5239 y Fn(\))p Fm(\037)2034 5253 y Fi(1)2074 5239 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)2324 5201 y Fi(2)2384 5239 y Fl(\000)2504 5177 y Fm(C)p 2485 5218 110 4 v 2485 5301 a(R)2555 5275 y Fi(2)2604 5239 y Fl(kh)2695 5177 y Fm(A)p 2694 5218 70 4 v 2694 5301 a(R)2775 5239 y Fl(i)2810 5201 y Fh(\000)p Fi(1)2905 5239 y Fm(\037)2962 5253 y Fi(1)3001 5239 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)3251 5201 y Fi(2)3291 5239 y Fm(:)1844 5517 y Fn(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 162 a Fn(Again)40 b(w)m(e)f(denote)727 126 y Fj(A)p 727 141 54 4 v 727 194 a(R)829 162 y Fn(b)m(y)g Fm(A)g Fn(\(note)g(that)h(this)e(c)m(hange)i(only)f(amoun)m(ts)g(to)h (c)m(hanging)f(constan)m(ts)h Fm(C)q(;)15 b(b)40 b Fn(in)0 275 y(\()p Fm(N)10 b Fn(3\)\).)42 b(By)31 b(iden)m(tit)m(y)h(\(1.3\))g (this)e(giv)m(es:)113 479 y Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)399 493 y Fi(1)441 479 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)691 442 y Fi(2)756 479 y Fl(\024)25 b Fm(C)7 b Fl(j)p Fn(\()p Fm(F)13 b Fn(\()p Fm(A)p Fn(\))p Fm(\037)1250 493 y Fi(1)1291 479 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u;)15 b Fn(\()p Fm(H)1647 493 y Fj(\034)1711 479 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1940 493 y Fi(1)1980 479 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fn(\))p Fl(j)22 b Fn(+)d Fm(C)7 b Fl(kh)p Fm(A)p Fl(i)2612 442 y Fh(\000)p Fi(1)2708 479 y Fm(\037)2765 493 y Fi(1)2804 479 y Fn(\()p Fm(H)g Fn(\))p Fm(u)p Fl(k)3054 442 y Fi(2)3095 479 y Fm(;)76 b(u)25 b Fl(2)g(D)s Fn(\()p Fl(h)p Fm(A)p Fl(i)p Fn(\))p Fm(;)0 684 y 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Fm(H)2653 2964 y Fj(\034)2718 2950 y Fl(\000)f Fm(z)t Fn(\))p Fm(\037)2946 2964 y Fi(1)2986 2950 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u:)0 3194 y Fn(By)31 b(Thms.)39 b(1)31 b(and)f(2,)h(w)m(e)f(kno)m(w)h(that:)429 3399 y(sup)408 3481 y Fj(z)s Fh(2)p Fj(J)536 3462 y Ff(+)601 3399 y Fl(kh)p Fm(A)p Fl(i)784 3361 y Fh(\000)p Fi(1)880 3399 y Fn(\()p Fm(H)d Fl(\000)20 b Fm(z)t Fn(\))1191 3361 y Fh(\000)p Fi(1)1286 3399 y Fl(h)p Fm(A)p Fl(i)1424 3361 y Fh(\000)p Fi(1)1519 3399 y Fl(k)25 b(\024)g Fm(C)q(;)67 b Fn(sup)1823 3481 y Fj(z)s Fh(2)p Fj(J)1951 3462 y Ff(+)2016 3399 y Fl(kh)p Fm(A)p Fl(i)2199 3361 y Fh(\000)p Fi(1)2295 3399 y Fn(\()p Fm(H)27 b Fl(\000)20 b Fm(z)t Fn(\))2605 3361 y Fh(\000)p Fi(1)2700 3399 y Fn(1)-20 b(l)2750 3417 y Fi([0)p Fj(;)p Fi(+)p Fh(1)p Fi([)2974 3399 y Fn(\()p Fm(A)p Fn(\))p Fl(k)27 b(\024)e Fm(C)q(;)0 3666 y Fn(\(the)36 b(second)g(estimate)i(follo)m(ws)f(b)m(y)f(taking)g(adjoin)m(ts)h(in)e (the)h(estimate)i(in)d(Thm.)56 b(2)37 b(for)e Fm(z)k Fl(2)34 b Fm(J)3426 3633 y Fh(\000)3485 3666 y Fn(\).)57 b(This)0 3779 y(yields:)416 3983 y Fl(kh)p Fm(A)p Fl(i)599 3946 y Fh(\000)p Fi(1)694 3983 y Fm(\037)751 3997 y Fi(1)791 3983 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1041 3946 y Fi(2)1107 3983 y Fl(\024)24 b Fm(C)7 b Fl(k)p Fn(\()p Fm(H)1430 3997 y Fj(\034)1494 3983 y Fl(\000)20 b Fm(z)t Fn(\))p Fm(\037)1723 3997 y Fi(1)1763 3983 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)2013 3946 y Fi(2)2073 3983 y Fn(+)20 b Fl(kh)p Fm(A)p Fl(i)p Fm(P)2405 3997 y Fh(\000)2466 3983 y Fn(\()p Fm(A)p Fn(\)\()p Fm(H)2715 3997 y Fj(\034)2779 3983 y Fl(\000)g Fm(z)t Fn(\))p Fm(\037)3008 3997 y Fi(1)3048 3983 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)3298 3946 y Fi(2)3339 3983 y Fm(;)0 4187 y Fn(uniformly)29 b(for)i Fm(z)e Fl(2)c Fm(J)770 4154 y Fi(+)829 4187 y Fn(.)41 b(Summing)29 b(up)g(w)m(e)i(ha)m(v)m(e:)772 4384 y Fl(k)p Fm(g)s Fn(\()p Fm(A)p Fn(\))p Fm(\037)1058 4398 y Fi(1)1099 4384 y Fn(\()p Fm(H)7 b Fn(\))p Fm(u)p Fl(k)1349 4351 y Fi(2)618 4544 y Fl(\024)83 b Fm(C)7 b Fl(k)p 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Fn(3\))31 b(holds)f(whic)m(h)g(completes)i(the)f(pro)s (of)e(of)i(Thm.)39 b(3.)i Fa(2)1844 5517 y Fn(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 162 a Fo(References)0 365 y Fn([ABG])47 b(Amrein,)38 b(W.,)i(Boutet)e(de)f(Mon)m(v)m(el,)j(A.,)f (Georgescu,)h(V.:)54 b Fm(C)2492 379 y Fi(0)2532 365 y Fk(-Gr)-5 b(oups,)41 b(Commutator)g(Metho)-5 b(ds)232 478 y(and)34 b(Sp)-5 b(e)g(ctr)g(al)35 b(The)-5 b(ory)34 b(of)f Fm(N)10 b Fk(-Bo)-5 b(dy)33 b(Hamiltonians)p Fn(,)g(Birkh\177) -45 b(auser,)30 b(Basel-Boston-Berlin,)k(\(1996\).)0 666 y([B])118 b(Burq,)34 b(N.:)47 b(Semiclassical)35 b(estimates)g(for)e(the)g(resolv)m(en)m(t)i(in)e(non)f(trapping)h (geometries,)j(I.M.R.N.)232 779 y Fe(5)31 b Fn(\(2002\),)i(221-241.)0 966 y([D])113 b(Da)m(vies,)33 b(E.)d(B.:)41 b(The)30 b(functional)h(calculus.)g(J.)g(London)e(Math.)i(So)s(c.)g Fe(52)p Fn(,)g(2)g(\(1995\),)i(166-176.)0 1154 y([DG])47 b(Derezinski)23 b(J.,)g(G)m(\023)-43 b(erard,)25 b(C.:)58 b(Scattering)23 b(theory)f(of)f(classical)j(and)d(quan)m(tum)h Fm(N)10 b Fl(\000)p Fn(particle)22 b(systems,)232 1267 y(Springer)29 b(V)-8 b(erlag,)33 b(\(1997\).)0 1454 y([GJ])64 b(Gol)m(\023)-43 b(enia,)30 b(S.)c(Jec)m(k)m(o,)j(T.:)39 b(A)26 b(new)f(lo)s(ok)i(at)g(Mourre's)f(comm)m(utator)i(theory)-8 b(,)28 b(\(2006\))g(preprin)m(t)d(mp-arc)232 1567 y(06-138.)0 1755 y([H])114 b(H\177)-45 b(ormander,)60 b(L.:)87 b(On)53 b(the)h(existence)h(and)f(the)g(regularit)m(y)h(of)f(solutions)g(of)g (linear)g(pseudo-)232 1868 y(di\013eren)m(tial)32 b(equations,)f (l'Enseignemen)m(t)h(Math)m(\023)-43 b(ematique,)33 b Fe(18)p Fn(,)e(1)g(\(1971\).)0 2056 y([HSS])45 b(Hunzik)m(er,)30 b(W.,)g(Sigal,)g(I.)f(M.,)h(So\013er,)f(A.:)41 b(Minimal)29 b(escap)s(e)g(v)m(elo)s(cities.)j(C.P)-8 b(.D.E.)30 b Fe(24)g Fn(11,)g(\(1999\),)232 2168 y(2279-2295.)0 2356 y([J])135 b(Jec)m(k)m(o,)44 b(T.:)58 b(F)-8 b(rom)40 b(classical)i(to)e(semiclassical)i(non-trapping)c(b)s(eha)m(vior,)k (C.R.A.S.)e Fe(338)g Fn(\(2004\),)232 2469 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