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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 18 /theta put
dup 33 /omega put
dup 66 /B put
dup 67 /C put
dup 70 /F put
dup 86 /V put
dup 87 /W put
dup 97 /a put
dup 100 /d put
dup 117 /u put
dup 120 /x put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueXX 5087385 def
currentdict end
currentfile eexec
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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 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 34 /quotedblright put
dup 39 /quoteright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 65 /A 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 92 /quotedblleft 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
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dup 108 /l put
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dup 110 /n put
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dup 112 /p put
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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
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueXX 5000828 def
currentdict end
currentfile eexec
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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 /Gamma put
dup 10 /Omega put
dup 11 /ff put
dup 12 /fi put
dup 14 /ffi put
dup 16 /dotlessi put
dup 18 /grave put
dup 19 /acute put
dup 21 /breve 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 47 /slash 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 59 /semicolon 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 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q 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 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 123 /endash put
dup 127 /dieresis put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueXX 5000793 def
currentdict end
currentfile eexec
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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 78 /N put
dup 82 /R put
readonly def
/FontBBox{-59 -250 1235 750}readonly def
/UniqueXX 5000766 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53

% 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 (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 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 59 /comma put
dup 105 /i put
dup 106 /j put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueXX 5087382 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 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 53 /five put
dup 54 /six put
dup 56 /eight put
dup 61 /equal put
dup 100 /d put
dup 105 /i put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueXX 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 0 /minus put
dup 3 /asteriskmath put
dup 18 /reflexsubset put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 59 /emptyset put
dup 92 /intersection put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueXX 5000816 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52

% 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 (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 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 11 /alpha put
dup 12 /beta put
dup 16 /zeta put
dup 18 /theta put
dup 20 /kappa put
dup 26 /rho put
dup 27 /sigma put
dup 33 /omega put
dup 34 /epsilon put
dup 59 /comma put
dup 61 /slash put
dup 63 /star put
dup 64 /partialdiff put
dup 66 /B put
dup 68 /D put
dup 69 /E put
dup 77 /M put
dup 82 /R put
dup 84 /T put
dup 97 /a put
dup 100 /d put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 110 /n put
dup 114 /r put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueXX 5087381 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21

% 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 (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 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 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueXX 5000790 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 1 /periodcentered put
dup 3 /asteriskmath put
dup 6 /plusminus put
dup 20 /lessequal put
dup 21 /greaterequal 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
/UniqueXX 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 2 /Theta put
dup 10 /Omega 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 53 /five put
dup 56 /eight put
dup 57 /nine put
dup 61 /equal put
dup 67 /C put
dup 76 /L put
dup 87 /W put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 99 /c put
dup 100 /d 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 115 /s put
dup 116 /t put
dup 117 /u put
dup 120 /x put
dup 126 /tilde put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 def
currentdict end
currentfile eexec
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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. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 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 11 /alpha put
dup 14 /delta put
dup 16 /zeta put
dup 17 /eta put
dup 18 /theta put
dup 20 /kappa put
dup 22 /mu put
dup 23 /nu put
dup 26 /rho put
dup 27 /sigma put
dup 28 /tau put
dup 33 /omega put
dup 34 /epsilon put
dup 39 /phi1 put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 63 /star put
dup 64 /partialdiff put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 72 /H put
dup 75 /K put
dup 77 /M put
dup 78 /N put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 97 /a put
dup 100 /d put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 114 /r 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
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 def
currentdict end
currentfile eexec
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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 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 80 /summationtext put
dup 82 /integraltext put
dup 83 /uniontext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
dup 112 /radicalbig put
dup 113 /radicalBig put
dup 114 /radicalbigg put
dup 115 /radicalBigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55

% 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 (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 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 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 16 /zeta put
dup 17 /eta put
dup 18 /theta put
dup 20 /kappa put
dup 21 /lambda put
dup 22 /mu put
dup 23 /nu put
dup 26 /rho put
dup 27 /sigma put
dup 28 /tau put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 33 /omega put
dup 34 /epsilon put
dup 36 /pi1 put
dup 37 /rho1 put
dup 39 /phi1 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 69 /E put
dup 70 /F put
dup 72 /H put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q 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 96 /lscript put
dup 97 /a 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 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 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueXX 5087386 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: EUSM10
%!PS-AdobeFont-1.1: EUSM10 2.1
%%CreationDate: 1992 Nov 20 17:36:44

% Euler fonts were designed by Hermann Zapf.
% 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 (EUSM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /EUSM10 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 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 74 /J put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
readonly def
/FontBBox{-8 -194 963 738}readonly def
/UniqueXX 5031988 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 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 19 /acute put
dup 33 /exclam put
dup 40 /parenleft put
dup 41 /parenright 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 58 /colon 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 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q 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 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 127 /dieresis put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53

% 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 (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 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 11 /ff put
dup 12 /fi put
dup 18 /grave put
dup 34 /quotedblright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
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 55 /seven put
dup 58 /colon put
dup 59 /semicolon put
dup 65 /A put
dup 66 /B put
dup 70 /F put
dup 72 /H put
dup 73 /I 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 84 /T put
dup 87 /W put
dup 92 /quotedblleft 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 127 /dieresis put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueXX 5000829 def
currentdict end
currentfile eexec
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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 1 /Delta put
dup 2 /Theta put
dup 3 /Lambda put
dup 8 /Phi put
dup 9 /Psi put
dup 10 /Omega 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 35 /numbersign 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 59 /semicolon 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 75 /K 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 88 /X put
dup 89 /Y put
dup 90 /Z put
dup 91 /bracketleft put
dup 92 /quotedblleft 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 123 /endash put
dup 124 /emdash put
dup 126 /tilde put
dup 127 /dieresis put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueXX 5000794 def
currentdict end
currentfile eexec
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6 5 bop 675 475 a FB(In)32 b(particular,)f(the)h(results)h(apply)f(to)f
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Fv(u)3571 438 y Fr(2)3610 475 y FB(\),)375 670 y(and)j
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634 y Fr(2)1327 670 y FB(,)25 b(whic)m(h)f(ha)m(v)m(e)g(application)f
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1201 y Fw(J)p FB(\()p Fv(u)p FB(\))28 b(=)1365 1065 y
Fu(Z)1482 1201 y Fv(a)1533 1216 y Ft(i;j)1613 1201 y
FB(\()p Fv(x)p FB(\))17 b Fv(@)1812 1216 y Ft(i)1840
1201 y Fv(u)g(@)1964 1216 y Ft(j)2001 1201 y Fv(u)38
b FB(+)h Fv(Q)p FB(\()p Fv(x)p FB(\))17 b Fv(\037)2496
1217 y Fr(\()p Fq(\000)p Fr(1)p Ft(;)p Fr(1\))2700 1201
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(zero.)84 b(Suc)m(h)47 b(functional)f Fw(J)h FB(has)375
1691 y(application)33 b(in)g(mo)s(dels)g(for)f(\015uid)h(jets)h(and)e
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(constan)m(t)f Fv(M)1937 2138 y Fr(0)1976 2123 y FB(,)i(dep)s(ending)f
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y Fv(a)426 2333 y Ft(i;j)548 2318 y FB(and)c Fv(Q)p FB(,)j(suc)m(h)e
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2282 y Ft(n)2200 2318 y FB(,)g(there)f(exists)h(a)d(class)i(A)f
(minimizer)375 2513 y Fv(u)g FB(for)f(the)i(functional)f
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2708 y Fs(f)p Fv(x)j FB(s)p Fv(:)p FB(t)p Fv(:)f(x)23
b Fs(\001)f Fv(!)31 b Fs(2)d FB([0)p Fv(;)17 b(M)1227
2723 y Fr(0)1266 2708 y Fs(j)p Fv(!)t Fs(j)p FB(])p Fs(g)p
FB(.)675 2945 y(As)29 b(a)f(limiting)h(case,)i(w)m(e)e(re-obtain)f(the)
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3140 y(a)d(Lipsc)m(hitz)j(Riemannian)e(metric)g Fv(g)j
FB(on)c Fz(R)1991 3103 y Ft(n)2038 3140 y FB(,)h(giv)m(en)h(an)m(y)f
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y Fr(0)628 3530 y Fs(j)p Fv(!)t Fs(j)p FB(])p Fs(g)p
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7 6 bop 1398 699 a Fx(T)-13 b(able)53 b(of)h(Con)l(ten)l(ts)375
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(Ca\013arelli)581 2285 y(and)33 b(de)g(la)f(Lla)m(v)m(e)89
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2426 y(1.3)81 b(\(More)33 b(or)f(less\))i(ph)m(ysical)g(motiv)-5
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Fz(R)1501 2543 y Fr(2)1630 2579 y FB(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g
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(.)h(.)151 b(54)1972 5175 y(vii)p eop
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8 7 bop 375 475 a Fz(Chapter)38 b(5.)112 b(Plane-lik)m(e)39
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616 y FB(5.1)81 b(Existence)35 b(and)e(regularit)m(y)g(of)f
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b(74)581 2787 y(7.1.3)81 b(The)33 b(p)s(erimeter)h(on)e
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9 8 bop 1512 699 a Fx(List)53 b(of)h(Figures)521 1172
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1 9 bop 1669 699 a Fx(Chapter)53 b(1)1318 1047 y(Set)g(up)g(and)h
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1929 y Fu(Z)1303 2154 y Fr(\012)1375 2064 y Fv(a)1426
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3588 y Fv(M)38 b Fs(\025)29 b FB(10)p Fs(j)p Fv(!)t Fs(j)p
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%%Page: 14 23
14 22 bop 3727 201 a FB(14)675 475 y(W)-8 b(e)33 b(no)m(w)g(address)h
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%%Page: 15 24
15 23 bop 3727 201 a FB(15)375 475 y(whic)m(h)34 b(is)g(a)e(con)m
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3604 670 4 66 v 3608 607 59 4 v 3608 670 V 3666 670 4
66 v eop
%%Page: 16 25
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eop
%%Page: 17 26
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4646 a Fp(1)524 4676 y Fo(H\177)-42 b(older)39 b(con)n(tin)n(uit)n(y)f
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4875 y(here)d(rely)g(on)g(the)h(Lipsc)n(hitz)g(regularit)n(y)e(of)h
(the)h(co)r(e\016cien)n(ts.)p eop
%%Page: 18 27
18 26 bop 3727 201 a FB(18)375 475 y Fz(R)22 b Fs(\002)g
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702 y(1)p 1282 746 52 4 v 1282 838 a Fv(a)1343 769 y
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1610 y Fr(supp)p Ft(')1713 1595 y FB(\()p Fv(u)p FB(\))27
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1890 y FB(for)32 b(an)m(y)h(test)h(function)f Fv(')f
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(satisfying)375 2363 y(the)e(assumptions)g(ab)-5 b(ove,)36
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3463 y(idea)28 b(of)g(the)g(pro)s(of)f(is)h(that)g(lo)s(cal)f(minima)i
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i(mo)s(di\014cation)e(of)7 b(\))43 b(De)h(Giorgi's)f(iteration)375
3854 y(metho)s(d.)675 4090 y(F)-8 b(or)38 b(the)i(sak)m(e)h(of)d
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eop
%%Page: 19 28
19 27 bop 3727 201 a FB(19)375 475 y(Notice)33 b(that,)g(a.e.)43
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2264 3553 a FB(\()p Fv(R)h Fs(\000)g Fv(\032)p FB(\))2587
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3663 59 4 v 3608 3726 V 3666 3726 4 66 v 675 4030 a FB(In)33
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%%Page: 20 29
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a(\()p Fv(k)i Fs(\000)e Fv(h)p FB(\))1765 3512 y Fr(2)1831
3337 y Fu(Z)1886 3562 y Ft(A)p Fr(\()p Ft(k)r(;r)r Fr(\))2090
3472 y FB(\()p Fv(u)f Fs(\000)g Fv(h)p FB(\))2399 3431
y Fr(2)2467 3472 y Fs(\025)28 b Fw(L)2643 3431 y Ft(n)2690
3472 y FB(\()p Fv(A)p FB(\()p Fv(k)s(;)17 b(r)s FB(\)\))g
Fv(:)370 b FB(\(3.2\))375 3740 y(F)-8 b(rom)32 b(\(3.1\))g(and)h
(\(3.2\),)f(one)h(gets)g(that)664 3941 y(1)p 650 3986
77 4 v 650 4077 a Fv(C)754 4009 y(U)10 b FB(\()p Fv(k)s(;)17
b(\032)p FB(\))28 b Fs(\024)g Fv(U)10 b FB(\()p Fv(h;)17
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4009 y FB(\()p Fv(A)p FB(\()p Fv(h;)17 b(r)s FB(\)\))2096
3940 y Fg(2)p 2091 3952 39 3 v 2091 3994 a Fj(n)2182
4009 y Fs(\001)2249 3898 y Fu(\020)2461 3941 y FB(1)p
2318 3986 335 4 v 2318 4077 a(\()p Fv(r)25 b Fs(\000)e
Fv(\032)p FB(\))2613 4048 y Fr(2)2684 4009 y FB(+)2942
3941 y(1)p 2792 3986 348 4 v 2792 4077 a(\()p Fv(k)i
Fs(\000)e Fv(h)p FB(\))3100 4048 y Fr(2)3150 3898 y Fu(\021)3474
4009 y FB(\(3.3\))375 4277 y(F)-8 b(rom)32 b(this,)h(using)h(again)e
(\(3.2\))g(and)g(the)h(de\014nition)h(of)e Fv(\013)q
FB(,)g(w)m(e)i(infer)f(that)481 4478 y(1)p 467 4522 77
4 v 467 4614 a Fv(C)571 4545 y Fw(L)642 4504 y Ft(n)689
4545 y FB(\()p Fv(A)p FB(\()p Fv(k)s(;)17 b(\032)p FB(\)\))1062
4504 y Ft(\013)1111 4545 y Fv(U)10 b FB(\()p Fv(k)s(;)17
b(\032)p FB(\))29 b Fs(\024)1555 4478 y Fv(U)10 b FB(\()p
Fv(h;)17 b(r)s FB(\))1854 4442 y Fr(1+)p Ft(\013)2010
4478 y Fw(L)2081 4442 y Ft(n)2128 4478 y FB(\()p Fv(A)p
FB(\()p Fv(h;)g(r)s FB(\)\))2514 4414 y Fg(2)p 2510 4426
39 3 v 2510 4467 a Fj(n)p 1555 4522 1008 4 v 1862 4614
a FB(\()p Fv(k)25 b Fs(\000)e Fv(h)p FB(\))2170 4585
y Fr(2)p Ft(\013)2589 4435 y Fu(\020)2801 4478 y FB(1)p
2658 4522 335 4 v 2658 4614 a(\()p Fv(r)i Fs(\000)e Fv(\032)p
FB(\))2953 4585 y Fr(2)3024 4545 y FB(+)3282 4478 y(1)p
3132 4522 348 4 v 3132 4614 a(\()p Fv(k)i Fs(\000)e Fv(h)p
FB(\))3440 4585 y Fr(2)3490 4435 y Fu(\021)3566 4545
y Fv(:)375 4813 y FB(Set)33 b(no)m(w)g(\010\()p Fv(k)s(;)17
b(r)s FB(\))28 b(=)f Fv(U)10 b FB(\()p Fv(k)s(;)17 b(r)s
FB(\))g Fw(L)1553 4777 y Ft(n)1600 4813 y FB(\()p Fv(A)p
FB(\()p Fv(k)s(;)g(r)s FB(\)\))1970 4777 y Ft(\013)2019
4813 y FB(.)43 b(The)34 b(ab)s(o)m(v)m(e)f(form)m(ula)g(then)g(reads)
862 5082 y(\010\()p Fv(k)s(;)17 b(\032)p FB(\))28 b Fs(\024)1457
5014 y Fv(C)p 1299 5059 393 4 v 1299 5150 a FB(\()p Fv(k)d
Fs(\000)e Fv(h)p FB(\))1607 5121 y Fr(2)p Ft(\013)1718
4971 y Fu(\020)1930 5014 y FB(1)p 1788 5059 335 4 v 1788
5150 a(\()p Fv(r)i Fs(\000)d Fv(\032)p FB(\))2082 5121
y Fr(2)2154 5082 y FB(+)2411 5014 y(1)p 2262 5059 348
4 v 2262 5150 a(\()p Fv(k)j Fs(\000)e Fv(h)p FB(\))2570
5121 y Fr(2)2619 4971 y Fu(\021)2695 5082 y FB(\010\()p
Fv(h;)17 b(r)s FB(\))2988 5041 y Fr(1+)p Ft(\013)3144
5082 y Fv(;)303 b FB(\(3.4\))p eop
%%Page: 21 30
21 29 bop 3727 201 a FB(21)375 475 y(where)32 b(w)m(e)f(also)f(used)i
(that)e Fv(\013)q FB(\()p Fv(\013)18 b FB(+)g(1\))27
b(=)g(2)p Fv(=n)p FB(.)43 b(Let)31 b Fv(d)c Fs(\025)h
Fv(R)j FB(b)s(e)g(a)f(suitable)i(constan)m(t,)f(to)375
670 y(b)s(e)i(c)m(hosen)h(later.)43 b(Set)1072 964 y
Fv(k)1123 979 y Ft(i)1179 964 y FB(=)28 b Fv(d)p FB(\(1)21
b Fs(\000)i FB(2)1591 923 y Fq(\000)p Ft(i)1674 964 y
FB(\))195 b(and)g Fv(r)2303 979 y Ft(i)2359 964 y FB(=)2472
897 y Fv(R)p 2472 942 76 4 v 2485 1033 a FB(2)2557 964
y(\(1)22 b(+)g(2)2813 923 y Fq(\000)p Ft(i)2896 964 y
FB(\))17 b Fv(:)375 1259 y FB(W)-8 b(e)29 b(no)m(w)h(apply)f(\(3.4\),)g
(with)g Fv(\032)f FB(=)g Fv(r)1704 1274 y Ft(i)p Fr(+1)1822
1259 y FB(,)i Fv(r)g FB(=)e Fv(r)2101 1274 y Ft(i)2129
1259 y FB(,)h Fv(k)i FB(=)d Fv(k)2422 1274 y Ft(i)p Fr(+1)2540
1259 y FB(,)h Fv(h)f FB(=)g Fv(k)2835 1274 y Ft(i)2891
1259 y FB(and)h(\010)3147 1274 y Ft(i)3203 1259 y FB(=)f(\010\()p
Fv(k)3466 1274 y Ft(i)3494 1259 y Fv(;)17 b(r)3582 1274
y Ft(i)3610 1259 y FB(\):)375 1454 y(from)43 b(this)h(inductiv)m(e)h
(relation)f(it)f(is)h(standard)f(\(see,)48 b(for)42 b(instance,)48
b(Lemma)c(7.1)e(of)375 1649 y([G2]\))32 b(to)g(deduce)i(that)e(\010)
1350 1664 y Ft(i)1411 1649 y FB(go)s(es)g(to)g(zero)h(for)f
Fv(i)g FB(going)g(to)g(in\014nit)m(y)-8 b(,)34 b(if)e(one)h(c)m(ho)s
(oses)g Fv(d)f FB(of)375 1844 y(the)j(order)g(of)g Fv(R)25
b FB(+)e Fv(R)1189 1808 y Fq(\000)p Fr(1)p Ft(=\013)1364
1844 y FB(\010)1434 1794 y Fr(1)p Ft(=)p Fr(2)1434 1869
y(0)1545 1844 y FB(.)50 b(Th)m(us,)37 b(\010\()p Fv(d;)17
b(R)q(=)p FB(2\))31 b(=)h(0)j(and)g(so)g Fw(L)2921 1808
y Ft(n)2968 1844 y FB(\()p Fv(A)p FB(\()p Fv(d;)17 b(R)q(=)p
FB(2\)\))31 b(=)g(0.)375 2040 y(Therefore,)j(sup)991
2063 y Ft(Q)1047 2077 y Fj(R=)p Fg(2)1179 2040 y Fv(u)28
b Fs(\024)g Fv(C)7 b(d)p FB(.)p 3604 2040 4 66 v 3608
1977 59 4 v 3608 2040 V 3666 2040 4 66 v 675 2351 a(W)-8
b(e)49 b(can)g(no)m(w)g(state)g(the)g(Oscillation)h(Lemma)f(w)m(e)h
(need)g(here:)76 b(w)m(e)50 b(de\014ne)375 2546 y Fv(M)10
b FB(\()p Fv(R)q FB(\))28 b(=)g(sup)909 2570 y Ft(Q)965
2581 y Fj(R)1035 2546 y Fv(u)p FB(,)k Fv(m)p FB(\()p
Fv(R)q FB(\))c(=)g(inf)1636 2561 y Ft(Q)1692 2572 y Fj(R)1763
2546 y Fv(u)p FB(,)k Fv(o)p FB(\()p Fv(R)q FB(\))27 b(=)44
b(osc)2371 2561 y Ft(Q)2427 2572 y Fj(R)2480 2546 y Fv(u)p
FB(.)375 2824 y Fz(Lemma)39 b(3.1.4.)49 b Fy(L)-5 b(et)36
b Fv(k)1302 2839 y Fr(0)1368 2824 y FB(=)28 b(\()p Fv(M)10
b FB(\(2)p Fv(R)q FB(\))23 b(+)f Fv(m)p FB(\(2)p Fv(R)q
FB(\)\))p Fv(=)p FB(2)34 b Fy(and)g(assume)h(that)520
3144 y Fs(\017)49 b Fw(L)690 3108 y Ft(n)737 3144 y FB(\()p
Fv(A)p FB(\()p Fv(k)937 3159 y Fr(0)976 3144 y Fv(;)17
b(R)q FB(\)\))28 b Fs(\024)g Fv(\015)5 b Fw(L)1431 3108
y Ft(n)1478 3144 y FB(\()p Fv(Q)1593 3159 y Ft(R)1651
3144 y FB(\))p Fy(,)35 b(for)f(some)g Fv(\015)f Fs(2)28
b FB(\(0)p Fv(;)17 b FB(1\))p Fy(;)520 3422 y Fs(\017)49
b Fv(o)p FB(\(2)p Fv(R)q FB(\))27 b Fs(\025)h FB(2)1047
3386 y Ft(\027)t Fr(+1)1180 3422 y Fv(R)q Fy(,)35 b(for)g(some)f
Fv(\027)g Fs(2)28 b Fz(N)p Fy(.)375 3742 y(Set)35 b Fv(k)593
3757 y Ft(\027)663 3742 y FB(=)28 b Fv(M)10 b FB(\(2)p
Fv(R)q FB(\))23 b Fs(\000)f FB(2)1242 3705 y Fq(\000)p
Ft(\027)t Fq(\000)p Fr(1)1430 3742 y Fv(o)p FB(\(2)p
Fv(R)q FB(\))p Fy(.)45 b(Then,)1330 4036 y Fw(L)1401
3995 y Ft(n)1448 4036 y FB(\()p Fv(A)p FB(\()p Fv(k)1648
4051 y Ft(\027)1691 4036 y Fv(;)17 b(R)q FB(\)\))27 b
Fs(\024)i Fv(\027)2073 3987 y Fq(\000)2216 3960 y Fj(n)p
2138 3972 196 3 v 2138 4013 a Fg(2\()p Fj(n)p Fi(\000)p
Fg(1\))2347 4036 y Fw(L)2418 3995 y Ft(n)2465 4036 y
FB(\()p Fv(Q)2580 4051 y Ft(R)2638 4036 y FB(\))17 b
Fv(:)375 4348 y Fy(Pr)-5 b(o)g(of.)48 b FB(Fix)31 b Fv(k)904
4363 y Fr(0)971 4348 y Fv(<)c(h)h(<)g(k)s FB(.)42 b(Let)31
b Fv(v)j FB(b)s(e)d(the)f(con)m(tin)m(uous)i(function)f(equal)g(to)f
Fv(u)17 b Fs(\000)i Fv(h)30 b FB(in)h(the)375 4543 y(set)i
Fs(f)p Fv(h)28 b(<)f(u)h(<)f(k)s Fs(g)33 b FB(and)f(constan)m(t)i
(outside.)44 b(Ob)m(viously)-8 b(,)35 b Fv(v)c FB(=)d(0)k(outside)h
Fv(A)p FB(\()p Fv(k)3284 4558 y Fr(0)3324 4543 y Fv(;)17
b(R)q FB(\))32 b(and)1208 4837 y Fw(L)1279 4796 y Ft(n)1327
4837 y FB(\()p Fv(Q)1442 4852 y Ft(R)1522 4837 y Fs(\000)22
b Fv(A)p FB(\()p Fv(k)1783 4852 y Fr(0)1822 4837 y Fv(;)17
b(R)q FB(\)\))28 b Fs(\025)g FB(\(1)22 b Fs(\000)g Fv(\015)5
b FB(\))17 b Fw(L)2540 4796 y Ft(n)2587 4837 y FB(\()p
Fv(Q)2702 4852 y Ft(R)2760 4837 y FB(\))g Fv(:)p eop
%%Page: 22 31
22 30 bop 3727 201 a FB(22)375 475 y(Th)m(us,)34 b(b)m(y)g(Sob)s(olev)f
(Inequalit)m(y)-8 b(,)1303 659 y Fu(\020)1379 634 y(Z)1434
859 y Ft(Q)1490 870 y Fj(R)1561 769 y Fv(v)1660 699 y
Fj(n)p 1621 711 117 3 v 1621 752 a(n)p Fi(\000)p Fg(1)1752
659 y Fu(\021)1812 681 y Fr(1)p Fq(\000)1916 654 y Fg(1)p
1912 666 39 3 v 1912 707 a Fj(n)1992 769 y Fs(\024)2097
634 y Fu(Z)2152 859 y Ft(A)p Fr(\()p Ft(h;R)p Fr(\))p
Fq(\000)p Ft(A)p Fr(\()p Ft(k)r(;R)p Fr(\))2669 769 y
Fs(jr)p Fv(v)t Fs(j)27 b FB(=)1144 1086 y(=)1303 951
y Fu(Z)1358 1176 y Ft(A)p Fr(\()p Ft(h;R)p Fr(\))p Fq(\000)p
Ft(A)p Fr(\()p Ft(k)r(;R)p Fr(\))1875 1086 y Fs(jr)p
Fv(u)p Fs(j)f(\024)i Fv(D)2302 917 y Fu(s)p 2401 917
532 4 v 2401 951 a(Z)2457 1176 y Ft(A)p Fr(\()p Ft(h;R)p
Fr(\))2699 1086 y Fs(jr)p Fv(u)p Fs(j)2894 1058 y Fr(2)2949
1086 y Fv(;)375 1413 y FB(where)34 b(w)m(e)f(set)h Fv(D)c
FB(=)1168 1328 y Fu(p)p 1267 1328 962 4 v 1267 1413 a
Fw(L)1338 1384 y Ft(n)1386 1413 y FB(\()p Fv(A)p FB(\()p
Fv(h;)17 b(R)q FB(\))22 b Fs(\000)g Fv(A)p FB(\()p Fv(k)s(;)17
b(R)q FB(\)\).)43 b(Also,)984 1597 y Fu(\020)1060 1572
y(Z)1116 1798 y Ft(A)p Fr(\()p Ft(k)r(;R)p Fr(\))1356
1708 y Fs(jr)p Fv(v)t Fs(j)1594 1637 y Fj(n)p 1555 1649
117 3 v 1555 1690 a(n)p Fi(\000)p Fg(1)1686 1597 y Fu(\021)1745
1620 y Fr(1)p Fq(\000)1849 1592 y Fg(1)p 1845 1604 39
3 v 1845 1646 a Fj(n)1926 1708 y FB(=)27 b(\()p Fv(k)e
Fs(\000)e Fv(h)p FB(\))17 b Fw(L)2425 1667 y Ft(n)2472
1708 y FB(\()p Fv(A)p FB(\()p Fv(k)s(;)g(R)q FB(\)\))2870
1667 y Fr(1)p Fq(\000)2974 1639 y Fg(1)p 2970 1651 V
2970 1693 a Fj(n)3039 1708 y Fv(:)375 2003 y FB(Therefore,)1075
2307 y(\()p Fv(k)25 b Fs(\000)e Fv(h)p FB(\))17 b Fw(L)1471
2266 y Ft(n)1517 2307 y FB(\()p Fv(A)p FB(\()p Fv(k)s(;)g(R)q
FB(\)\))1915 2266 y Fr(1)p Fq(\000)2019 2239 y Fg(1)p
2015 2251 V 2015 2292 a Fj(n)2095 2307 y Fs(\024)29 b
Fv(D)2301 2138 y Fu(s)p 2401 2138 532 4 v 34 x(Z)2456
2397 y Ft(A)p Fr(\()p Ft(h;R)p Fr(\))2698 2307 y Fs(jr)p
Fv(u)p Fs(j)2893 2278 y Fr(2)2948 2307 y Fv(:)499 b FB(\(3.5\))375
2602 y(F)-8 b(rom)32 b(Prop)s(osition)h(3.1.2,)1047 2761
y Fu(Z)1103 2987 y Ft(A)p Fr(\()p Ft(h;R)p Fr(\))1345
2897 y Fs(jr)p Fv(u)p Fs(j)1540 2855 y Fr(2)1606 2897
y Fs(\024)28 b Fv(C)1788 2786 y Fu(\020)1847 2897 y Fv(R)1922
2855 y Ft(n)p Fq(\000)p Fr(2)2060 2897 y FB(\()p Fv(M)10
b FB(\(2)p Fv(R)q FB(\))22 b Fs(\000)h Fv(h)p FB(\))2618
2855 y Fr(2)2680 2897 y FB(+)f Fv(R)2853 2855 y Ft(n)2900
2786 y Fu(\021)2976 2897 y Fv(:)471 b FB(\(3.6\))375
3191 y(If)33 b Fv(h)27 b Fs(\024)i Fv(k)713 3206 y Ft(\027)755
3191 y FB(,)1051 3386 y Fv(M)10 b FB(\(2)p Fv(R)q FB(\))23
b Fs(\000)g Fv(h)28 b Fs(\025)g Fv(M)10 b FB(\(2)p Fv(R)q
FB(\))22 b Fs(\000)h Fv(k)2144 3401 y Ft(\027)2215 3386
y FB(=)k(2)2367 3345 y Ft(\027)t Fq(\000)p Fr(1)2500
3386 y Fv(o)p FB(\(2)p Fv(R)q FB(\))h Fs(\025)g Fv(R)17
b(;)375 3635 y FB(so,)33 b(b)m(y)g(means)h(of)e(\(3.5\))g(and)h
(\(3.6\),)956 3930 y(\()p Fv(k)25 b Fs(\000)e Fv(h)p
FB(\))17 b Fw(L)1352 3889 y Ft(n)1399 3930 y FB(\()p
Fv(A)p FB(\()p Fv(k)s(;)g(R)q FB(\)\))1797 3889 y Fr(1)p
Fq(\000)1901 3862 y Fg(1)p 1897 3874 39 3 v 1897 3915
a Fj(n)1977 3930 y Fs(\024)28 b Fv(C)7 b(D)19 b(R)2344
3861 y Fj(n)p Fi(\000)p Fg(2)p 2344 3874 117 3 v 2388
3915 a(2)2492 3930 y FB(\()p Fv(M)10 b FB(\(2)p Fv(R)q
FB(\))22 b Fs(\000)h Fv(h)p FB(\))17 b Fv(:)375 4225
y FB(Hence,)34 b(taking)920 4520 y Fv(k)c FB(=)e Fv(k)1156
4535 y Ft(i)1212 4520 y FB(=)f Fv(M)10 b FB(\(2)p Fv(R)q
FB(\))23 b Fs(\000)f FB(2)1790 4479 y Fq(\000)p Ft(i)p
Fq(\000)p Fr(1)1963 4520 y Fv(o)p FB(\(2)p Fv(R)q FB(\))195
b(and)g Fv(h)28 b FB(=)g Fv(k)2996 4535 y Ft(i)p Fq(\000)p
Fr(1)375 4814 y FB(w)m(e)34 b(obtain)1395 5009 y Fw(L)1466
4968 y Ft(n)1513 5009 y FB(\()p Fv(A)p FB(\()p Fv(k)1713
5024 y Ft(i)1741 5009 y Fv(;)17 b(R)q FB(\)\))1946 4940
y Fj(n)p Fi(\000)p Fg(1)p 1946 4953 V 1985 4994 a Fj(n)2104
5009 y Fs(\024)28 b Fv(C)7 b(D)2367 5024 y Ft(i)2395
5009 y Fv(R)2480 4940 y Fj(n)p Fi(\000)p Fg(2)p 2481
4953 V 2524 4994 a(2)2628 5009 y Fv(;)p eop
%%Page: 23 32
23 31 bop 3727 201 a FB(23)375 475 y(where)34 b Fv(D)738
490 y Ft(i)794 475 y FB(=)897 389 y Fu(p)p 997 389 1294
4 v 86 x Fw(L)1068 446 y Ft(n)1115 475 y FB(\()p Fv(A)p
FB(\()p Fv(k)1315 490 y Ft(i)1343 475 y Fv(;)17 b(R)q
FB(\)\))22 b Fs(\000)g Fw(L)1730 446 y Ft(n)1778 475
y FB(\()p Fv(A)p FB(\()p Fv(k)1978 490 y Ft(i)p Fq(\000)p
Fr(1)2096 475 y Fv(;)17 b(R)q FB(\)\))o(.)44 b(Hence,)845
748 y Fv(\027)23 b Fw(L)987 707 y Ft(n)1035 748 y FB(\()p
Fv(A)p FB(\()p Fv(k)1235 763 y Ft(\027)1278 748 y Fv(;)17
b(R)q FB(\)\))1483 674 y Fg(2\()p Fj(n)p Fi(\000)p Fg(1\))p
1482 692 196 3 v 1561 733 a Fj(n)1719 748 y Fs(\024)1877
624 y Ft(\027)1824 654 y Fu(X)1839 864 y Ft(i)p Fr(=1)1985
748 y Fw(L)2056 707 y Ft(n)2103 748 y FB(\()p Fv(A)p
FB(\()p Fv(k)2303 763 y Ft(i)2331 748 y Fv(;)g(R)q FB(\)\))2536
674 y Fg(2\()p Fj(n)p Fi(\000)p Fg(1\))p 2536 692 V 2614
733 a Fj(n)2773 748 y Fs(\024)685 1070 y(\024)898 945
y Ft(\027)845 975 y Fu(X)860 1185 y Ft(i)p Fr(=1)1006
1070 y Fv(C)7 b(D)1167 1029 y Fr(2)1164 1094 y Ft(i)1206
1070 y Fv(R)1281 1029 y Ft(n)p Fq(\000)p Fr(2)1446 1070
y Fs(\024)28 b Fv(C)7 b(R)1703 1029 y Ft(n)p Fq(\000)p
Fr(2)1857 1070 y Fw(L)1928 1029 y Ft(n)1975 1070 y FB(\()p
Fv(A)p FB(\()p Fv(k)2175 1085 y Ft(\027)2218 1070 y Fv(;)17
b(R)q FB(\))22 b Fs(\000)g Fv(A)p FB(\()p Fv(k)2658 1085
y Fr(0)2698 1070 y Fv(;)17 b(R)q FB(\)\))27 b Fs(\024)h
Fv(C)7 b(R)3177 1029 y Fr(2\()p Ft(n)p Fq(\000)p Fr(1\))3421
1070 y Fv(:)p 3604 1343 4 66 v 3608 1281 59 4 v 3608
1343 V 3666 1343 4 66 v 675 1647 a FB(W)-8 b(e)33 b(are)g(no)m(w)h(in)f
(p)s(osition)g(to)g(\014nish)h(the)f(pro)s(of)f(of)h(Theorem)h(3.1.1:)
44 b(follo)m(wing)375 1841 y(the)29 b(notation)g(in)f(Lemma)i(3.1.4)e
(ab)s(o)m(v)m(e,)i(without)g(loss)f(of)f(generalit)m(y)-8
b(,)31 b(w)m(e)f(ma)m(y)f(assume)375 2036 y(that)40 b
Fw(L)665 2000 y Ft(n)713 2036 y FB(\()p Fv(A)p FB(\()p
Fv(k)913 2051 y Fr(0)952 2036 y Fv(;)17 b(R)q FB(\)\))41
b Fs(\024)1317 1997 y Fr(1)p 1317 2013 36 4 v 1317 2070
a(2)1362 2036 y Fw(L)1433 2000 y Ft(n)1480 2036 y FB(\()p
Fv(Q)1595 2051 y Ft(R)1653 2036 y FB(\).)68 b(W)-8 b(e)41
b(ma)m(y)g(also)g(assume)h(that)f Fv(\027)3004 1987 y
Fq(\000)3148 1959 y Fj(n)p 3069 1971 196 3 v 3069 2013
a Fg(2\()p Fj(n)p Fi(\000)p Fg(1\))3319 2036 y FB(is)g(small,)375
2230 y(sa)m(y)34 b(less)f(than)g(some)h Fv(")1242 2245
y Fr(0)1281 2230 y FB(.)43 b(F)-8 b(rom)32 b(Lemma)h(3.1.4,)f(if)h
Fv(o)p FB(\(2)p Fv(R)q FB(\))27 b Fs(\025)h FB(2)2733
2194 y Ft(\027)t Fr(+1)2866 2230 y Fv(R)q FB(,)33 b(then)1443
2504 y Fw(L)1514 2463 y Ft(n)1562 2504 y FB(\()p Fv(A)p
FB(\()p Fv(k)1762 2519 y Ft(\027)1805 2504 y Fv(;)17
b(R)q FB(\)\))27 b Fs(\024)h Fv(")2178 2519 y Fr(0)2234
2504 y Fw(L)2305 2463 y Ft(n)2352 2504 y FB(\()p Fv(Q)2467
2519 y Ft(R)2525 2504 y FB(\))17 b Fv(:)375 2778 y FB(F)-8
b(rom)32 b(this)h(and)g(Prop)s(osition)g(3.1.3,)f(w)m(e)i(get)e(that)
634 3051 y Fv(M)10 b FB(\()p Fv(R)q(=)p FB(2\))23 b Fs(\000)f
Fv(k)1160 3066 y Ft(\027)1286 3051 y Fs(\024)84 b Fv(C)1524
2941 y Fu(h\020)1630 3051 y Fv(M)10 b FB(\(2)p Fv(R)q
FB(\))23 b Fs(\000)f Fv(k)2107 3066 y Ft(\027)2150 2941
y Fu(\021)2232 3051 y Fs(\001)2282 2941 y Fu(\020)2351
2984 y Fw(L)2422 2948 y Ft(n)2470 2984 y FB(\()p Fv(A)p
FB(\()p Fv(k)2670 2999 y Ft(\027)2712 2984 y Fv(;)17
b(R)q FB(\)\))p 2351 3028 556 4 v 2568 3119 a Fv(R)2643
3091 y Ft(n)2917 2941 y Fu(\021)2987 2936 y Fj(\013)p
2987 2948 41 3 v 2992 2989 a Fg(2)3063 3051 y FB(+)22
b Fv(R)3236 2941 y Fu(i)3311 3051 y Fs(\024)1286 3271
y(\024)84 b Fv(C)7 b FB([)p Fv(")1597 3220 y Ft(\013=)p
Fr(2)1597 3295 y(0)1716 3271 y FB(\()p Fv(M)j FB(\(2)p
Fv(R)q FB(\))23 b Fs(\000)g Fv(k)2232 3286 y Ft(\027)2275
3271 y FB(\))f(+)g Fv(R)q FB(])17 b(;)375 3544 y(therefore,)33
b(if)g Fv(")947 3559 y Fr(0)1018 3544 y FB(is)g(small)h(enough,)1184
3818 y Fv(M)10 b FB(\()p Fv(R)q(=)p FB(2\))22 b Fs(\000)h
Fv(k)1710 3833 y Ft(\027)1780 3818 y Fs(\024)1896 3750
y FB(1)p 1896 3795 49 4 v 1896 3886 a(2)1954 3818 y(\()p
Fv(M)10 b FB(\(2)p Fv(R)q FB(\))23 b Fs(\000)f Fv(k)2469
3833 y Ft(\027)2512 3818 y FB(\))g(+)g Fv(C)7 b(R)18
b(:)375 4091 y FB(Subtracting)33 b Fv(m)p FB(\(2)p Fv(R)q
FB(\))g(to)f(b)s(oth)g(sides,)i(w)m(e)g(get)1284 4365
y Fv(o)p FB(\()p Fv(R)q(=)p FB(2\))27 b Fs(\024)1712
4254 y Fu(\020)1772 4365 y FB(1)21 b Fs(\000)2019 4298
y FB(1)p 1952 4342 183 4 v 1952 4433 a(2)2001 4405 y
Ft(\027)t Fr(+1)2144 4254 y Fu(\021)2204 4365 y Fv(o)p
FB(\(2)p Fv(R)q FB(\))g(+)h Fv(C)7 b(R)18 b(:)708 b FB(\(3.7\))375
4639 y(F)-8 b(rom)32 b(these)j(considerations,)f(w)m(e)g(ha)m(v)m(e)g
(that)f(either)h(\(3.7\))e(holds)i(or)e Fv(o)p FB(\(2)p
Fv(R)q FB(\))c Fs(\024)h FB(2)3440 4602 y Ft(\027)t Fr(+1)3573
4639 y Fv(R)q FB(.)375 4833 y(In)k(an)m(y)g(case,)1238
5107 y Fv(o)p FB(\()p Fv(R)q(=)p FB(2\))27 b Fs(\024)1666
4996 y Fu(\020)1726 5107 y FB(1)22 b Fs(\000)1973 5039
y FB(1)p 1906 5084 V 1906 5175 a(2)1955 5146 y Ft(\027)t
Fr(+1)2098 4996 y Fu(\021)2158 5107 y Fv(o)p FB(\(2)p
Fv(R)q FB(\))f(+)h(2)2573 5066 y Ft(\027)2616 5107 y
Fv(C)7 b(R)18 b(:)662 b FB(\(3.8\))p eop
%%Page: 24 33
24 32 bop 3727 201 a FB(24)375 475 y(Iterating)27 b(\(3.8\),)h(a)e
(standard)i(recursiv)m(e)h(argumen)m(t)f(\(see,)h(for)e(instance,)i
(Lemma)f(7.3)e(of)375 669 y([G2])34 b(at)g(page)g(248\))g(pro)m(v)m(es)
i(that)e Fv(u)g FB(is)h(H\177)-49 b(older)35 b(con)m(tin)m(uous.)50
b(This)36 b(\014nishes)g(the)f(pro)s(of)375 864 y(of)d(Theorem)i
(3.1.1.)375 1253 y FA(3.2)135 b(Lipsc)l(hitz)45 b(regularit)l(y)375
1502 y FB(W)-8 b(e)33 b(will)g(no)m(w)g(pro)m(v)m(e)h(the)f(follo)m
(wing)375 1770 y Fz(Theorem)38 b(3.2.1.)49 b Fy(If)41
b Fv(u)e Fs(2)g Fw(M)1567 1734 y Ft(!)1567 1795 y(M)1646
1770 y Fy(,)k(and)d Fv(a)1965 1785 y Ft(i;j)2085 1770
y Fs(2)f FB(C)2260 1734 y Fr(0)p Ft(;)p Fr(1)2355 1770
y FB(\()p Fz(R)2477 1734 y Ft(n)2524 1770 y FB(\))p Fy(,)j(then)f
Fv(u)g Fy(is)f FB(C)3134 1734 y Fr(0)p Ft(;)p Fr(1)3229
1770 y Fy(.)63 b(F)-7 b(urther-)375 1965 y(mor)i(e,)44
b(given)e(any)h Fv(K)49 b Fs(\032\032)43 b Fv(S)1512
1929 y Ft(!)1506 1990 y(M)1585 1965 y Fy(,)i(the)e(Lipschitz)f(norm)g
(of)g Fv(u)h Fy(in)f Fv(K)50 b Fy(is)42 b(b)-5 b(ounde)g(d)42
b(by)h(a)375 2160 y(c)-5 b(onstant)30 b(dep)-5 b(ending)29
b(only)h(on)g Fv(n)p Fy(,)i(the)e(b)-5 b(ounds)30 b(on)g
Fv(a)2330 2175 y Ft(i;j)2441 2160 y Fy(and)g Fv(Q)h Fy(and)f(on)47
b FB(dist)17 b(\()p Fv(K)r(;)g(@)5 b(S)3534 2123 y Ft(!)3528
2185 y(M)3607 2160 y FB(\))p Fy(.)675 2468 y FB(Theorem)37
b(3.2.1)e(can)h(b)s(e)g(seen)h(as)f(an)f(extension)j(of)d(the)h
(results)h(b)m(y)g(Alt,)f(Caf-)375 2663 y(farelli)23
b(and)h(F)-8 b(riedman)23 b(to)g(the)h(functional)g(in)f(\(1.1\))g
(under)h(a)f(more)h(general)g(assumption)375 2858 y(on)32
b(the)h(co)s(e\016cien)m(ts.)46 b(The)34 b(pro)s(of)d(will)j(b)s(e)e
(accomplished)j(in)e(sev)m(eral)h(steps.)675 3094 y(First,)f(w)m(e)g(p)
s(oin)m(t)g(out)f(the)h(PDE-prop)s(erties)h(of)e(the)h(minimizers:)375
3362 y Fz(Lemma)39 b(3.2.2.)49 b Fy(L)-5 b(et)36 b Fv(u)27
b Fs(2)h Fw(M)1526 3326 y Ft(!)1526 3387 y(M)1605 3362
y Fy(.)44 b(Then)35 b Fv(@)1985 3377 y Ft(i)2013 3362
y FB(\()p Fv(a)2102 3377 y Ft(i;j)2183 3362 y Fv(@)2234
3377 y Ft(j)2271 3362 y Fv(u)p FB(\))27 b(=)g(0)35 b
Fy(we)-5 b(akly)34 b(in)h Fs(fj)p Fv(u)p Fs(j)26 b Fv(<)i
FB(1)p Fs(g)p Fy(..)375 3669 y(Pr)-5 b(o)g(of.)48 b FB(T)-8
b(ak)m(e)39 b Fv(')c Fs(2)h FB(C)1192 3633 y Fq(1)1192
3693 y Fr(0)1267 3669 y FB(\()p Fs(fj)p Fv(u)p Fs(j)e
Fv(<)i FB(1)p Fs(g)p FB(\))g(and)h(compute)i(the)e(\014rst)h(v)-5
b(ariation)37 b(of)f(the)i(func-)375 3864 y(tional.)p
3604 3864 4 66 v 3608 3801 59 4 v 3608 3864 V 3666 3864
4 66 v 375 4132 a Fz(Lemma)h(3.2.3.)49 b Fy(L)-5 b(et)40
b Fv(u)c Fs(2)h Fw(M)1548 4096 y Ft(!)1548 4157 y(M)1627
4132 y Fy(.)59 b(Then)39 b Fv(@)2026 4147 y Ft(i)2054
4132 y FB(\()p Fv(a)2143 4147 y Ft(i;j)2224 4132 y Fv(@)2275
4147 y Ft(j)2312 4132 y Fv(u)p FB(\))c Fs(\025)i FB(0)j
Fy(we)-5 b(akly)39 b(in)g Fs(f)p Fv(u)d(<)g FB(1)p Fs(g)p
Fy(.)59 b(and)375 4327 y Fv(@)426 4342 y Ft(i)455 4327
y FB(\()p Fv(a)544 4342 y Ft(i;j)624 4327 y Fv(@)675
4342 y Ft(j)712 4327 y Fv(u)p FB(\))27 b Fs(\024)h FB(0)35
b Fy(we)-5 b(akly)34 b(in)h Fs(f)p Fv(u)27 b(>)g Fs(\000)p
FB(1)p Fs(g)p Fy(.)375 4634 y(Pr)-5 b(o)g(of.)48 b FB(T)-8
b(aking,)52 b(a)47 b(non-negativ)m(e)h Fv(')k Fs(2)g
FB(C)2040 4598 y Fq(1)2040 4658 y Fr(0)2116 4634 y FB(\()p
Fs(f)p Fv(u)f(<)h FB(1)p Fs(g)p FB(\))47 b(and)g(computing)h(the)f
(\014rst)375 4828 y(v)-5 b(ariation,)32 b(w)m(e)i(obtain)749
5112 y(0)28 b Fs(\024)g FB(lim)17 b(sup)964 5198 y Ft(")p
Fq(\000)-12 b(!)p Fr(0)1146 5180 y Fg(+)1279 5044 y FB(1)p
1256 5089 95 4 v 1256 5180 a(2)p Fv(")1361 5001 y Fu(\020)1420
5112 y Fw(J)1470 5127 y Ft(S)1517 5104 y Fj(!)1513 5150
y(M)1585 5112 y FB(\()p Fv(u)22 b Fs(\000)g Fv("')p FB(\))g
Fs(\000)h Fw(J)2120 5127 y Ft(S)2167 5104 y Fj(!)2163
5150 y(M)2234 5112 y FB(\()p Fv(u)p FB(\))2366 5001 y
Fu(\021)2453 5112 y Fs(\024)28 b(\000)2652 4976 y Fu(Z)2708
5202 y Ft(S)2755 5179 y Fj(!)2751 5225 y(M)2839 5112
y Fv(a)2890 5127 y Ft(i;j)2970 5112 y Fv(@)3021 5127
y Ft(i)3050 5112 y Fv('@)3165 5127 y Ft(j)3202 5112 y
Fv(u)17 b(;)p eop
%%Page: 25 34
25 33 bop 3727 201 a FB(25)375 475 y(whic)m(h)49 b(pro)m(v)m(es)g(the)g
(\014rst)f(claim.)89 b(The)48 b(other)g(claim)g(can)g(b)s(e)g(pro)m(v)m
(ed)h(in)f(a)f(similar)375 669 y(w)m(a)m(y)-8 b(.)p 3604
669 4 66 v 3608 607 59 4 v 3608 669 V 3666 669 4 66 v
675 976 a(No)m(w,)33 b(a)g(barrier)f(argumen)m(t:)375
1244 y Fz(Lemma)39 b(3.2.4.)49 b Fy(Fix)38 b Fv(\026)d(>)g
FB(0)p Fy(.)56 b(L)-5 b(et)39 b Fv(\014)h Fs(2)35 b FB(\(0)p
Fv(;)17 b FB(1\))p Fy(.)56 b(Assume)39 b(that)g Fv(v)f
Fs(2)d FB(H)3101 1208 y Fr(1)3141 1244 y FB(\()p Fv(B)3253
1259 y Ft(r)3291 1244 y FB(\))p Fy(,)k Fv(v)g Fs(\025)c
FB(0)p Fy(,)375 1439 y Fv(@)426 1454 y Ft(i)455 1439
y FB(\()p Fv(a)544 1454 y Ft(i;j)624 1439 y Fv(@)675
1454 y Ft(j)712 1439 y Fv(v)t FB(\))27 b Fs(\024)h FB(0)p
Fy(,)35 b(and)1209 1721 y Fv(v)c Fs(\025)d Fv(\026)100
b FB(on)f Fv(@)5 b(B)1883 1736 y Ft(r)1944 1721 y Fs(\\)23
b(f)p Fv(x)2138 1736 y Ft(n)2213 1721 y Fs(\024)28 b
FB(\()p Fs(\000)p FB(1)22 b(+)g Fv(\014)6 b FB(\))p Fv(r)s
Fs(g)17 b Fv(:)375 2004 y Fy(Then,)28 b(ther)-5 b(e)28
b(exists)f(a)g(c)-5 b(onstant)27 b Fv(\022)k Fs(2)d FB(\(0)p
Fv(;)17 b(\014)6 b FB(\))p Fy(,)28 b(dep)-5 b(ending)26
b(only)i(on)f Fv(n)p Fy(,)i Fv(\014)k Fy(and)27 b(the)h(b)-5
b(ounds)375 2199 y(on)34 b Fv(a)565 2214 y Ft(i;j)646
2199 y Fy(,)h(such)f(that:)45 b Fv(v)t FB(\(0)p Fv(;)17
b(:)g(:)g(:)f(;)h FB(\()p Fs(\000)p FB(1)22 b(+)g Fv(\022)s
FB(\))p Fv(r)s FB(\))27 b Fs(\025)h Fv(\026\014)6 b(=)p
FB(2)p Fy(.)375 2505 y(Pr)-5 b(o)g(of.)48 b FB(By)i(scaling,)55
b(it)49 b(is)h(enough)g(to)f(pro)m(v)m(e)i(the)f(claim)g(for)e
Fv(r)59 b FB(=)e(1.)93 b(Set)50 b Fv(c)3476 2469 y Ft(?)3572
2505 y FB(:=)375 2700 y(2)p Fv(n)p Fs(jj)p Fv(a)589 2715
y Ft(i;j)669 2700 y Fs(jj)725 2717 y Fr(W)798 2698 y
Fg(1)p Fj(;)p Fi(1)916 2700 y FB(.)43 b(Let)33 b Fv( )j
FB(b)s(e)c(the)h(solution)g(of)e Fv(@)2090 2715 y Ft(i)2119
2700 y FB(\()p Fv(a)2208 2715 y Ft(i;j)2288 2700 y Fv(@)2339
2715 y Ft(j)2376 2700 y Fv( )t FB(\))d(=)f Fv(c)2654
2664 y Ft(?)2694 2700 y Fv(\026)32 b FB(in)g Fv(B)2972
2715 y Fr(1)3012 2700 y FB(,)g(with)h Fv( )f FB(=)27
b(0)32 b(on)375 2895 y Fv(@)5 b(B)505 2910 y Fr(1)545
2895 y FB(.)43 b(By)28 b(elliptic)i(estimates)g(\(see,)g(for)e
(instance)i([GT],)f(Theorem)g(8.16)f(and)h(Theorem)375
3090 y(8.27\))904 3372 y Fs(j)p Fv( )t FB(\(0)p Fv(;)17
b(:)g(:)g(:)e(;)i FB(0)p Fv(;)g FB(\()p Fs(\000)p FB(1)22
b(+)g Fv(\022)s FB(\)\))p Fs(j)27 b(\024)i FB(osc)2096
3388 y Ft(B)2149 3397 y Fg(1)2184 3388 y Fq(\\)p Ft(B)2284
3400 y Fg(2)p Fj(\022)2350 3388 y Fr(\(0)p Ft(;:::)o(;)p
Fr(0)p Ft(;)p Fq(\000)p Fr(1\))2687 3372 y Fv( )j Fs(\024)c
Fv(c\022)2977 3331 y Ft(\013)3027 3372 y Fv(\026)17 b(;)344
b FB(\(3.9\))375 3655 y(for)37 b(suitable)h(constan)m(ts)h
Fv(c)e FB(and)h Fv(\013)q FB(,)g(dep)s(ending)h(only)f(on)f
Fv(n)h FB(and)f(the)h(b)s(ounds)g(on)g Fv(a)3568 3670
y Ft(i;j)3648 3655 y FB(.)375 3850 y(De\014ne)1133 4132
y Fv(\036)28 b FB(:=)f Fv(\026)p FB([)p Fs(\000)p FB(\()p
Fv(x)1605 4147 y Ft(n)1675 4132 y FB(+)22 b(1\))g(+)g
Fv(\014)6 b FB(])97 b(and)h Fv(\020)35 b FB(:=)27 b Fv(\036)22
b FB(+)g Fv( )f(:)509 b FB(\(3.10\))375 4415 y(W)-8 b(e)35
b(ha)m(v)m(e)g Fv(@)822 4430 y Ft(i)851 4415 y FB(\()p
Fv(a)940 4430 y Ft(i;j)1020 4415 y Fv(@)1071 4430 y Ft(j)1108
4415 y Fv(\020)8 b FB(\))29 b(=)i Fs(\000)p Fv(@)1461
4430 y Ft(i)1490 4415 y Fv(a)1541 4430 y Ft(i;n)1632
4415 y Fv(\026)22 b FB(+)h Fv(c)1854 4379 y Ft(?)1894
4415 y Fv(\026)30 b(>)g FB(0)17 b Fv(:)34 b FB(W)-8 b(e)34
b(no)m(w)h(c)m(hec)m(k)h(that)e Fv(v)g Fs(\025)d Fv(\020)41
b FB(on)34 b Fv(@)5 b(B)3607 4430 y Fr(1)3648 4415 y
FB(.)375 4610 y(Indeed,)34 b(if)f Fv(x)28 b Fs(2)g Fv(@)5
b(B)1116 4625 y Fr(1)1178 4610 y Fs(\\)23 b(f)p Fv(x)1372
4625 y Ft(n)1447 4610 y Fs(\024)28 b(\000)p FB(1)22 b(+)g
Fv(\014)6 b Fs(g)p FB(,)1469 4892 y Fv(\020)i FB(\()p
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y Fs(\\)23 b(f)p Fv(x)1857 5190 y Ft(n)1932 5175 y Fv(>)k
Fs(\000)p FB(1)c(+)f Fv(\014)6 b Fs(g)p FB(,)32 b Fv(\020)8
b FB(\()p Fv(x)p FB(\))27 b Fs(\024)h FB(0)g Fs(\024)g
Fv(v)t FB(\()p Fv(x)p FB(\).)p eop
%%Page: 26 35
26 34 bop 3727 201 a FB(26)675 475 y(Hence,)30 b(b)m(y)f(Maxim)m(um)h
(Principle,)h Fv(v)g Fs(\025)d Fv(\020)35 b FB(in)28
b Fv(B)2459 490 y Fr(1)2499 475 y FB(.)42 b(In)28 b(particular,)h(b)m
(y)g(\(3.9\))f(and)375 669 y(\(3.10\),)k Fv(v)t FB(\(0)p
Fv(;)17 b(:)g(:)g(:)e(;)i FB(0)p Fv(;)g Fs(\000)p FB(1)k(+)g
Fv(\022)s FB(\))28 b Fs(\025)g Fv(\026)p FB(\()p Fv(\014)e
Fs(\000)c Fv(c\022)1963 633 y Ft(\013)2034 669 y Fs(\000)g
Fv(\022)s FB(\))28 b Fs(\025)g Fv(\026\014)6 b(=)p FB(2,)31
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y(resp)s(ect)i(to)e Fv(\014)6 b FB(.)p 3604 863 4 66
v 3608 801 59 4 v 3608 863 V 3666 863 4 66 v 675 1162
a(W)-8 b(e)29 b(w)m(ould)h(lik)m(e)g(to)f(remark)g(that)g(an)g
(alternativ)m(e)h(pro)s(of)e(of)g(the)i(Lemma)f(ab)s(o)m(v)m(e)375
1357 y(can)k(b)s(e)g(accomplished)h(b)m(y)g(using)f(the)g(b)s(oundary)g
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1571 y Ft(!)1526 1632 y(M)1605 1607 y Fy(.)44 b(Supp)-5
b(ose)35 b(that)g Fv(x)2305 1622 y Fr(0)2372 1607 y Fs(2)28
b Fv(S)2532 1571 y Ft(!)2526 1632 y(M)2640 1607 y Fy(veri\014es)1231
1870 y FB(dist)18 b(\()p Fv(x)1499 1885 y Fr(0)1538 1870
y Fv(;)f(@)5 b(S)1704 1828 y Ft(!)1698 1894 y(M)1778
1870 y FB(\))28 b Fv(>)f FB(2)17 b(dist)g(\()p Fv(x)2280
1885 y Fr(0)2320 1870 y Fv(;)g Fs(fj)p Fv(u)p Fs(j)26
b FB(=)h(1)p Fs(g)p FB(\))17 b Fv(:)375 2132 y Fy(Then,)943
2326 y FB(min)q Fs(f)p FB(1)k Fs(\000)i Fv(u)p FB(\()p
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b(+)g Fv(u)p FB(\()p Fv(x)1931 2341 y Fr(0)1970 2326
y FB(\))p Fs(g)27 b(\024)h Fv(C)c FB(dist)17 b(\()p Fv(x)2551
2341 y Fr(0)2591 2326 y Fv(;)g Fs(fj)p Fv(u)p Fs(j)26
b FB(=)i(1)p Fs(g)p FB(\))17 b Fv(;)375 2561 y Fy(wher)-5
b(e)30 b Fv(C)38 b Fy(is)30 b(a)h(suitable)f(c)-5 b(onstant,)31
b(dep)-5 b(ending)29 b(only)i(on)f Fv(n)p Fy(,)i(the)f(b)-5
b(ounds)30 b(on)g Fv(a)3272 2576 y Ft(i;j)3383 2561 y
Fy(and)g Fv(F)14 b Fy(.)375 2860 y(Pr)-5 b(o)g(of.)48
b FB(If)38 b Fv(u)p FB(\()p Fv(x)932 2875 y Fr(0)971
2860 y FB(\))e(=)g Fs(\006)p FB(1,)j(there)f(is)g(nothing)f(to)g(pro)m
(v)m(e.)59 b(Set)38 b Fv(U)2746 2875 y Fq(\006)2841 2860
y FB(=)e(1)25 b Fs(\006)h Fv(u)p FB(,)39 b(and)e Fv(d)3497
2875 y Fr(0)3572 2860 y FB(:=)392 3055 y(dist)17 b(\()p
Fv(x)659 3070 y Fr(0)699 3055 y Fv(;)g Fs(fj)p Fv(u)p
Fs(j)26 b FB(=)h(1)p Fs(g)p FB(\).)675 3288 y(W)-8 b(e)44
b(assume)i(that)e(min)q Fs(f)p Fv(U)1705 3303 y Fr(+)1764
3288 y FB(\()p Fv(x)1857 3303 y Fr(0)1896 3288 y FB(\))p
Fv(;)34 b(U)2061 3303 y Fq(\000)2120 3288 y FB(\()p Fv(x)2213
3303 y Fr(0)2253 3288 y FB(\))p Fs(g)47 b(\025)h Fv(M)10
b(d)2668 3303 y Fr(0)2707 3288 y FB(,)47 b(and)e(deriv)m(e)g(an)f(upp)s
(er)375 3482 y(b)s(ound)39 b(for)f Fv(M)10 b FB(.)63
b(Notice)40 b(that)f Fv(B)1636 3498 y Fr(9)p Ft(d)1707
3507 y Fg(0)1742 3498 y Ft(=)p Fr(10)1852 3482 y FB(\()p
Fv(x)1945 3497 y Fr(0)1985 3482 y FB(\))f Fs(\032)h(fj)p
Fv(u)p Fs(j)e Fv(<)h FB(1)p Fs(g)p FB(,)i(so)f(b)m(y)h(Lemma)g(3.2.2)e
(and)375 3676 y(Harnac)m(k)33 b(inequalit)m(y)1113 3939
y(inf)1005 4001 y Ft(B)1058 4015 y Fg(3)p Fj(d)1120 4030
y Fg(0)1155 4015 y Fj(=)p Fg(4\()p Fj(x)1276 4030 y Fg(0)1311
4015 y(\))1356 3939 y Fv(U)1422 3954 y Fq(\006)1509 3939
y Fs(\025)44 b FB(const)c Fs(\001)115 b FB(sup)1942 4020
y Ft(B)1995 4034 y Fg(3)p Fj(d)2057 4049 y Fg(0)2092
4034 y Fj(=)p Fg(4\()p Fj(x)2213 4049 y Fg(0)2248 4034
y(\))2292 3939 y Fv(U)2358 3954 y Fq(\006)2445 3939 y
Fs(\025)45 b FB(const)17 b Fv(M)10 b(d)2961 3954 y Fr(0)3018
3939 y Fv(:)381 b FB(\(3.11\))375 4211 y(Let)33 b Fv(y)e
Fs(2)d Fv(B)797 4226 y Ft(d)833 4235 y Fg(0)872 4211
y FB(\()p Fv(x)965 4226 y Fr(0)1005 4211 y FB(\))22 b
Fs(\\)g(fj)p Fv(u)p Fs(j)27 b FB(=)g(1)p Fs(g)p FB(.)675
4445 y(Without)34 b(loss)g(of)g(generalit)m(y)-8 b(,)35
b(w)m(e)g(ma)m(y)f(assume)h Fv(y)e FB(=)d(0.)46 b(W)-8
b(e)34 b(no)m(w)h(denote,)f(for)375 4639 y(short,)f Fv(B)g
FB(:=)27 b Fv(B)963 4654 y Ft(d)999 4663 y Fg(0)1071
4639 y FB(and)33 b Fv(U)38 b FB(:=)28 b Fv(U)1562 4654
y Fq(\006)1621 4639 y FB(.)675 4872 y(Let)33 b Fv(v)e
FB(:=)d Fv(v)1106 4887 y Fq(\006)1198 4872 y FB(b)s(e)k(the)h(minim)m
(um)i(of)1759 4999 y Fu(Z)1814 5225 y Ft(B)1892 5135
y Fv(a)1943 5150 y Ft(i;j)2023 5135 y Fv(@)2074 5150
y Ft(i)2103 5135 y Fv(\020)8 b(@)2205 5150 y Ft(j)2241
5135 y Fv(\020)p eop
%%Page: 27 36
27 35 bop 3727 201 a FB(27)375 475 y(among)32 b(the)h(functions)h
Fv(\020)h Fs(2)28 b FB(H)1523 438 y Fr(1)1562 475 y FB(\()p
Fv(B)5 b FB(\))33 b(with)g Fv(\020)i FB(=)27 b Fv(U)2219
490 y Fq(\006)2311 475 y FB(on)33 b Fv(@)5 b(B)g FB(.)44
b(Hence,)1036 769 y(const)1292 634 y Fu(Z)1347 859 y
Ft(B)1424 769 y Fs(jr)p FB(\()p Fv(U)33 b Fs(\000)22
b Fv(v)t FB(\))p Fs(j)1888 728 y Fr(2)1955 769 y Fs(\024)2060
634 y Fu(Z)2115 859 y Ft(B)2193 769 y Fv(a)2244 784 y
Ft(i;j)2324 769 y Fv(@)2375 784 y Ft(i)2404 769 y FB(\()p
Fv(U)32 b Fs(\000)23 b Fv(v)t FB(\))p Fv(@)2780 784 y
Ft(j)2817 769 y FB(\()p Fv(U)32 b Fs(\000)23 b Fv(v)t
FB(\))k(=)860 1028 y(=)1019 893 y Fu(Z)1074 1118 y Ft(B)1152
1028 y Fv(a)1203 1043 y Ft(i;j)1283 1028 y Fv(@)1334
1043 y Ft(i)1363 1028 y Fv(u@)1470 1043 y Ft(j)1506 1028
y Fv(u)22 b Fs(\000)1684 893 y Fu(Z)1739 1118 y Ft(B)1816
1028 y Fv(a)1867 1043 y Ft(i;j)1947 1028 y Fv(@)1998
1043 y Ft(i)2027 1028 y Fv(v)t(@)2129 1043 y Ft(j)2166
1028 y Fv(v)e(:)1166 b FB(\(3.12\))375 1323 y(By)39 b(construction,)i
Fv(V)59 b FB(=)38 b Fv(V)1414 1338 y Fq(\006)1510 1323
y FB(:=)g Fs(\006)p FB(\()p Fv(v)30 b Fs(\000)d FB(1\))37
b(=)h Fv(u)g FB(on)g Fv(@)5 b(B)g FB(:)56 b(hence)40
b(b)m(y)f(\(3.12\))f(and)h(the)375 1518 y(fact)32 b(that)h
Fv(u)27 b Fs(2)h Fw(M)1054 1482 y Ft(!)1054 1543 y(M)1133
1518 y FB(,)909 1677 y Fu(Z)964 1903 y Ft(B)1041 1813
y Fs(jr)p FB(\()p Fv(U)33 b Fs(\000)22 b Fv(v)t FB(\))1477
1772 y Fr(2)1517 1813 y Fs(j)27 b(\024)45 b FB(const)1950
1677 y Fu(Z)2005 1903 y Ft(B)2082 1813 y Fv(Q\037)2220
1828 y Fr(\()p Fq(\000)p Fr(1)p Ft(;)p Fr(1\))2425 1813
y FB(\()p Fv(V)21 b FB(\))h Fs(\000)h Fv(\037)2762 1828
y Fr(\()p Fq(\000)p Fr(1)p Ft(;)p Fr(1\))2966 1813 y
FB(\()p Fv(u)p FB(\))17 b Fv(:)284 b FB(\(3.13\))375
2107 y(By)38 b(Maxim)m(um)h(Principle)g(and)e(the)h(fact)f(that)g(0)f
Fs(\024)g Fv(U)46 b Fs(\024)36 b FB(2,)j(w)m(e)f(deduce)h(that)e
(either)375 2302 y(0)27 b Fv(<)h(v)j(<)d FB(2)k(or)g
Fv(v)37 b FB(is)c(constan)m(t)g(on)g Fv(B)5 b FB(.)675
2539 y(In)25 b(the)h(latter)f(case,)i(since)f Fv(u)i
Fs(2)g Fw(M)1942 2503 y Ft(!)1942 2564 y(M)2021 2539
y FB(,)e Fv(u)f FB(m)m(ust)h(b)s(e)f(constan)m(t)h(in)f
Fv(B)5 b FB(:)40 b(in)25 b(particular)375 2734 y Fv(u)p
FB(\()p Fv(x)524 2749 y Fr(0)563 2734 y FB(\))j(=)f Fv(u)p
FB(\(0\))g(=)h Fs(\006)p FB(1,)33 b(and)f(the)h(claim)h(is)f(pro)m(v)m
(en.)675 2971 y(Th)m(us,)43 b(w)m(e)e(can)f(no)m(w)g(restrict)h
(ourselv)m(es)h(to)d(the)h(case)g(0)g Fv(<)g(v)j(<)d
FB(2)f(in)h Fv(B)5 b FB(,)41 b(i.e.)375 3166 y Fs(\000)p
FB(1)28 b Fv(<)f(V)50 b(<)27 b FB(1.)43 b(F)-8 b(rom)33
b(\(3.13\),)1104 3325 y Fu(Z)1159 3551 y Ft(B)1236 3461
y Fs(jr)p FB(\()p Fv(U)g Fs(\000)22 b Fv(v)t FB(\))p
Fs(j)1700 3420 y Fr(2)1767 3461 y Fs(\024)45 b FB(const)17
b Fw(L)2199 3420 y Ft(n)2246 3461 y FB(\()p Fs(fj)p Fv(u)p
Fs(j)27 b FB(=)g(1)p Fs(g)22 b(\\)g Fv(B)5 b FB(\))17
b Fv(:)480 b FB(\(3.14\))375 3755 y(F)-8 b(rom)39 b(\(3.11\))g(and)h
(Lemma)g(3.2.4,)i(there)e(is)g(a)g(suitable)h(constan)m(t)f
Fv(\024)g Fs(2)h FB(\(0)p Fv(;)17 b FB(1\))38 b(and)i(a)375
3950 y(suitable)34 b(p)s(oin)m(t)f Fv(x)1049 3914 y Ft(?)1117
3950 y Fs(2)c Fv(B)1286 3965 y Ft(\024d)1363 3974 y Fg(0)1435
3950 y FB(so)k(that)g Fv(v)t FB(\()p Fv(x)1911 3914 y
Ft(?)1950 3950 y FB(\))28 b Fs(\025)46 b FB(const)17
b Fv(M)10 b(d)2533 3965 y Fr(0)2573 3950 y FB(.)45 b(Harnac)m(k)33
b(inequalit)m(y)i(then)375 4146 y(implies)1619 4440 y(inf)1598
4502 y Ft(B)1651 4514 y Fj(\024d)1719 4529 y Fg(0)1774
4440 y Fv(v)d Fs(\025)44 b FB(const)18 b Fv(M)10 b(d)2369
4455 y Fr(0)2425 4440 y Fv(:)974 b FB(\(3.15\))375 4735
y(Set)40 b(no)m(w)g Fv(v)811 4699 y Fq(\003)850 4735
y FB(\()p Fv(x)p FB(\))g(:=)1191 4696 y Fr(1)p 1173 4712
71 4 v 1173 4769 a Ft(d)1209 4778 y Fg(0)1254 4735 y
Fv(v)t FB(\()p Fv(d)1394 4750 y Fr(0)1433 4735 y Fv(x)p
FB(\),)i(for)d Fv(x)h Fs(2)f Fv(B)2025 4750 y Fr(1)2065
4735 y FB(.)64 b(Let)40 b(us)g(no)m(w)g(in)m(tro)s(duce)h(the)f(follo)m
(wing)375 4930 y(barrier:)1460 5125 y Fv(w)s FB(\()p
Fv(x)p FB(\))27 b(=)j(\026)-51 b Fv(cM)10 b FB(\()p Fv(e)2024
5084 y Fq(\000)p Ft(\026)p Fq(j)p Ft(x)p Fq(j)2201 5061
y Fg(2)2262 5125 y Fs(\000)23 b Fv(e)2407 5084 y Fq(\000)p
Ft(\026)2508 5125 y FB(\))17 b Fv(;)p eop
%%Page: 28 37
28 36 bop 3727 201 a FB(28)375 475 y(with)35 b(\026)-51
b Fv(c)33 b FB(and)f Fv(\026)h FB(suitable)g(constan)m(ts,)h(prop)s
(erly)f(c)m(hosen)i(to)d(satisfy)1023 741 y Fv(@)1074
756 y Ft(i)1103 741 y FB(\()p Fv(a)1192 756 y Ft(i;j)1272
741 y Fv(@)1323 756 y Ft(j)1360 741 y Fv(w)s FB(\))27
b Fv(>)h FB(0)195 b(in)44 b Fv(B)2045 756 y Fr(1)2107
741 y Fs(\000)22 b Fv(B)2280 756 y Ft(\024)2342 741 y
Fv(;)1023 960 y(w)30 b FB(=)e(0)195 b(on)44 b Fv(@)5
b(B)1748 975 y Fr(1)1983 960 y FB(and)196 b Fv(w)30 b
Fs(\024)e Fv(v)2592 919 y Fq(\003)2826 960 y FB(on)44
b Fv(@)5 b(B)3103 975 y Ft(\024)3166 960 y Fv(;)375 1226
y FB(where)44 b(the)e(last)h(relation)f(can)h(b)s(e)f(ful\014lled)h(in)
g(ligh)m(t)g(of)e(\(3.15\).)72 b(So,)45 b(b)m(y)e(Maxim)m(um)375
1421 y(Principle,)961 1687 y Fv(v)1012 1646 y Fq(\003)1051
1687 y FB(\()p Fv(x)p FB(\))28 b Fs(\025)h Fv(w)s FB(\()p
Fv(x)p FB(\))e Fs(\025)45 b FB(const)17 b Fv(M)10 b FB(\(1)23
b Fs(\000)f(j)p Fv(x)p Fs(j)p FB(\))195 b(in)45 b Fv(B)2765
1702 y Fr(1)2826 1687 y Fs(\000)23 b Fv(B)3000 1702 y
Ft(\024)3062 1687 y Fv(:)375 1953 y FB(Recalling)33 b(\(3.15\),)f(w)m
(e)i(get)1250 2219 y Fv(v)1301 2178 y Fq(\003)1340 2219
y FB(\()p Fv(x)p FB(\))28 b Fs(\025)45 b FB(const)17
b Fv(M)10 b FB(\(1)23 b Fs(\000)f(j)p Fv(x)p Fs(j)p FB(\))195
b(in)45 b Fv(B)2717 2234 y Fr(1)2773 2219 y Fv(:)375
2486 y FB(Rescaling)33 b(bac)m(k:)1197 2752 y Fv(v)t
FB(\()p Fv(x)p FB(\))28 b Fs(\025)44 b FB(const)18 b
Fv(M)10 b(d)1923 2767 y Fr(0)1963 2752 y FB(\(1)21 b
Fs(\000)2181 2684 y(j)p Fv(x)p Fs(j)p 2181 2729 111 4
v 2191 2820 a Fv(d)2242 2835 y Fr(0)2302 2752 y FB(\))195
b(in)44 b Fv(B)2734 2767 y Ft(d)2770 2776 y Fg(0)2826
2752 y Fv(:)573 b FB(\(3.16\))375 3018 y(W)-8 b(e)31
b(can)f(no)m(w)h(follo)m(w)f([A)m(C],)h(p.)43 b(112:)f(Fix)30
b(an)m(y)h Fv(y)g Fs(2)d Fv(B)2390 3034 y Ft(d)2426 3043
y Fg(0)2461 3034 y Ft(=)p Fr(2)2565 3018 y FB(and)j(consider)g(the)g
(follo)m(wing)375 3212 y(transformation)h(of)h Fv(B)1225
3227 y Ft(d)1261 3236 y Fg(0)1332 3212 y FB(suc)m(h)h(that)f
Fv(y)i FB(b)s(ecomes)f(the)f(new)h(origin:)1524 3479
y Fv(T)1581 3494 y Ft(y)1623 3479 y FB(\()p Fv(x)p FB(\))28
b(:=)1923 3411 y Fv(d)1974 3426 y Fr(0)2035 3411 y Fs(\000)22
b(j)p Fv(x)p Fs(j)p 1923 3456 323 4 v 2039 3547 a Fv(d)2090
3562 y Fr(0)2255 3479 y Fv(y)j FB(+)d Fv(x)17 b(;)375
3745 y FB(and)33 b(set,)g(for)f(an)m(y)h(function)g Fv(f)11
b FB(,)33 b Fv(f)1626 3760 y Ft(y)1667 3745 y FB(\()p
Fv(x)p FB(\))28 b(:=)g Fv(f)11 b FB(\()p Fv(T)2111 3760
y Ft(y)2152 3745 y FB(\()p Fv(x)p FB(\)\).)675 3979 y(F)-8
b(or)32 b(an)m(y)h Fv(\020)i Fs(2)28 b Fv(@)5 b(B)1336
3994 y Fr(1)1376 3979 y FB(,)33 b(de\014ne)1177 4245
y Fv(r)1221 4260 y Ft(\020)1288 4245 y FB(:=)28 b(inf)7
b Fs(f)p Fv(r)30 b Fs(2)e FB([)p Fv(d)1833 4260 y Fr(0)1873
4245 y Fv(=)p FB(8)p Fv(;)17 b(d)2066 4260 y Fr(0)2104
4245 y FB(])44 b(s)p Fv(:)p FB(t)p Fv(:)h(U)10 b FB(\()p
Fv(r)s(\020)e FB(\))27 b(=)h(0)p Fs(g)17 b Fv(;)375 4511
y FB(if)32 b(this)i(set)f(is)g(non-empt)m(y)-8 b(.)45
b(F)-8 b(or)31 b(almost)i(all)g(suc)m(h)h Fv(\020)39
b FB(w)m(e)34 b(ha)m(v)m(e)912 4777 y Fv(v)959 4792 y
Ft(y)1001 4777 y FB(\()p Fv(r)1083 4792 y Ft(\020)1122
4777 y Fv(\020)8 b FB(\))83 b(=)1454 4642 y Fu(Z)1554
4668 y Ft(d)1590 4677 y Fg(0)1509 4867 y Ft(r)1541 4879
y Fj(\020)1645 4777 y Fs(r)p FB(\()p Fv(U)1832 4792 y
Ft(y)1896 4777 y Fs(\000)23 b Fv(v)2043 4792 y Ft(y)2084
4777 y FB(\)\()p Fv(\032\020)8 b FB(\))22 b Fs(\001)f
Fv(\020)j(d\032)k Fs(\024)1293 5107 y(\024)1454 4930
y Fu(s)p 1554 4930 1057 4 v 41 x(Z)1653 4997 y Ft(d)1689
5006 y Fg(0)1609 5197 y Ft(r)1641 5209 y Fj(\020)1745
5107 y Fs(jr)p FB(\()p Fv(U)1960 5122 y Ft(y)2023 5107
y Fs(\000)23 b Fv(v)2170 5122 y Ft(y)2211 5107 y FB(\)\()p
Fv(\032\020)8 b FB(\))p Fs(j)2454 5078 y Fr(2)2509 5107
y Fv(d\032)39 b Fs(\001)2699 5021 y Fu(p)p 2799 5021
296 4 v 86 x Fv(d)2850 5122 y Fr(0)2911 5107 y Fs(\000)23
b Fv(r)3055 5122 y Ft(\020)3111 5107 y Fv(:)288 b FB(\(3.17\))p
eop
%%Page: 29 38
29 37 bop 3727 201 a FB(29)375 475 y(By)33 b(\(3.16\),)f(for)g(an)m(y)h
Fv(r)e Fs(2)d FB([0)p Fv(;)17 b(d)1510 490 y Fr(0)1548
475 y FB(],)734 755 y Fv(v)781 770 y Ft(y)822 755 y FB(\()p
Fv(r)s(\020)8 b FB(\))82 b Fs(\025)100 b FB(const)18
b Fv(M)1599 644 y Fu(\020)1659 755 y Fv(d)1710 770 y
Fr(0)1771 755 y Fs(\000)1871 640 y Fu(\014)1871 700 y(\014)1871
760 y(\014)1914 687 y Fv(d)1965 702 y Fr(0)2026 687 y
Fs(\000)23 b Fv(r)p 1914 732 259 4 v 1998 823 a(d)2049
838 y Fr(0)2183 755 y Fv(y)i FB(+)d Fv(r)s(\020)2452
640 y Fu(\014)2452 700 y(\014)2452 760 y(\014)2484 644
y(\021)2571 755 y Fs(\025)1078 1008 y(\025)100 b FB(const)18
b Fv(M)1599 897 y Fu(\020)1659 1008 y Fv(d)1710 1023
y Fr(0)1771 1008 y Fs(\000)1881 940 y Fv(d)1932 955 y
Fr(0)1993 940 y Fs(\000)23 b Fv(r)p 1881 985 V 1965 1076
a(d)2016 1091 y Fr(0)2149 1008 y Fs(j)p Fv(y)t Fs(j)e(\000)i
Fv(r)2425 897 y Fu(\021)2511 1008 y Fs(\025)45 b FB(const)18
b Fv(M)3004 940 y(d)3055 955 y Fr(0)3116 940 y Fs(\000)23
b Fv(r)p 3004 985 V 3109 1076 a FB(2)3289 1008 y Fv(;)375
1288 y FB(since)34 b Fs(j)p Fv(y)t Fs(j)26 b(\024)i Fv(d)904
1303 y Fr(0)943 1288 y Fv(=)p FB(2.)43 b(F)-8 b(rom)32
b(this)i(and)e(\(3.17\))1014 1568 y Fv(M)1118 1527 y
Fr(2)1158 1568 y FB(\()p Fv(d)1247 1583 y Fr(0)1309 1568
y Fs(\000)22 b Fv(r)1452 1583 y Ft(\020)1492 1568 y FB(\))28
b Fs(\024)44 b FB(const)1935 1432 y Fu(Z)2035 1459 y
Ft(d)2071 1468 y Fg(0)1991 1658 y Ft(r)2023 1670 y Fj(\020)2127
1568 y Fs(jr)p FB(\()p Fv(U)2342 1583 y Ft(y)2405 1568
y Fs(\000)23 b Fv(v)2552 1583 y Ft(y)2593 1568 y FB(\)\()p
Fv(\032\020)8 b FB(\))p Fs(j)2836 1527 y Fr(2)2891 1568
y Fv(d\032)17 b(:)390 b FB(\(3.18\))375 1848 y(Also,)33
b(using)g(p)s(olar)f(co)s(ordinates)h(and)g(the)g(de\014nition)h(of)e
Fv(r)2594 1863 y Ft(\020)2633 1848 y FB(,)1056 1993 y
Fu(Z)1111 2218 y Ft(B)1164 2230 y Fj(d)1196 2245 y Fg(0)1235
2218 y Fq(\000)p Ft(B)1343 2232 y Fj(d)1375 2247 y Fg(0)1410
2232 y Fj(=)p Fg(4)1476 2218 y Fr(\()p Ft(y)r Fr(\))1589
2128 y Fv(\037)1650 2144 y Fq(f)p Ft(U)7 b Fr(=0)p Fq(g)1897
2128 y Fs(\024)45 b FB(const)2275 1993 y Fu(Z)2331 2218
y Ft(B)2384 2230 y Fj(d)2416 2245 y Fg(0)2454 2218 y
Fq(\000)p Ft(B)2562 2232 y Fj(d)2594 2247 y Fg(0)2629
2232 y Fj(=)p Fg(8)2716 2128 y Fv(\037)2777 2144 y Fq(f)p
Ft(U)2860 2152 y Fj(y)2898 2144 y Fr(=0)p Fq(g)3055 2128
y Fs(\024)1056 2436 y(\024)f FB(const)1433 2301 y Fu(Z)1489
2526 y Ft(@)t(B)1583 2535 y Fg(1)1638 2301 y Fu(Z)1738
2327 y Ft(d)1774 2336 y Fg(0)1694 2526 y Ft(d)1730 2535
y Fg(0)1765 2526 y Ft(=)p Fr(8)1856 2436 y Fv(\037)1917
2452 y Fq(f)p Ft(U)2000 2460 y Fj(y)2038 2452 y Fr(=0)p
Fq(g)2167 2436 y FB(\()p Fv(\032\020)8 b FB(\))17 b Fv(d\032)g(d\020)33
b FB(=)1056 2731 y(=)44 b(const)1432 2596 y Fu(Z)1487
2821 y Ft(@)t(B)1581 2830 y Fg(1)1637 2596 y Fu(Z)1736
2622 y Ft(d)1772 2631 y Fg(0)1692 2821 y Ft(r)1724 2833
y Fj(\020)1828 2731 y Fv(\037)1889 2747 y Fq(f)p Ft(U)1972
2755 y Fj(y)2010 2747 y Fr(=0)p Fq(g)2139 2731 y FB(\()p
Fv(\032\020)8 b FB(\))17 b Fv(d\032)g(d\020)33 b Fs(\024)1056
3012 y(\024)44 b FB(const)1433 2876 y Fu(Z)1489 3101
y Ft(@)t(B)1583 3110 y Fg(1)1622 3012 y FB(\()p Fv(d)1711
3027 y Fr(0)1772 3012 y Fs(\000)23 b Fv(r)1916 3027 y
Ft(\020)1955 3012 y FB(\))17 b Fv(d\020)23 b(:)375 3292
y FB(F)-8 b(rom)32 b(this,)h(in)m(tegrating)g(b)s(oth)g(sides)h(of)e
(\(3.18\),)g(it)g(follo)m(ws)i(that)712 3572 y(const)18
b Fv(M)1056 3531 y Fr(2)1112 3436 y Fu(Z)1168 3662 y
Ft(B)1221 3674 y Fj(d)1253 3689 y Fg(0)1292 3662 y Fq(\000)p
Ft(B)1400 3676 y Fj(d)1432 3691 y Fg(0)1466 3676 y Fj(=)p
Fg(4)1533 3662 y Fr(\()p Ft(y)r Fr(\))1646 3572 y Fv(\037)1707
3587 y Fq(f)p Ft(U)7 b Fr(=0)p Fq(g)1954 3572 y Fs(\024)2059
3436 y Fu(Z)2115 3662 y Ft(@)t(B)2209 3671 y Fg(1)2264
3436 y Fu(Z)2364 3462 y Ft(d)2400 3471 y Fg(0)2320 3662
y Ft(d)2356 3671 y Fg(0)2391 3662 y Ft(=)p Fr(8)2482
3572 y Fs(jr)p FB(\()p Fv(U)2697 3587 y Ft(y)2760 3572
y Fs(\000)23 b Fv(v)2907 3587 y Ft(y)2949 3572 y FB(\)\()p
Fv(\032\020)8 b FB(\))p Fs(j)3192 3531 y Fr(2)3247 3572
y Fv(d\032)17 b(d\020)34 b Fs(\024)535 3860 y(\024)100
b FB(const)968 3724 y Fu(Z)1024 3950 y Ft(B)1077 3962
y Fj(d)1109 3977 y Fg(0)1168 3860 y Fs(jr)p FB(\()p Fv(U)33
b Fs(\000)22 b Fv(v)t FB(\))p Fs(j)1632 3819 y Fr(2)1688
3860 y Fv(:)375 4140 y FB(Applying)42 b(this)g(result)g(to)f(t)m(w)m(o)
h(p)s(oin)m(ts)g Fv(y)1956 4155 y Fr(1)2036 4140 y FB(and)f
Fv(y)2282 4155 y Fr(2)2362 4140 y FB(in)h Fv(B)2559 4156
y Ft(d)2595 4165 y Fg(0)2630 4156 y Ft(=)p Fr(2)2746
4140 y FB(so)f(that)g Fv(B)3168 4156 y Ft(d)3204 4165
y Fg(0)3239 4156 y Ft(=)p Fr(4)3314 4140 y FB(\()p Fv(y)3400
4155 y Fr(1)3439 4140 y FB(\))g(and)375 4335 y Fv(B)449
4350 y Ft(d)485 4359 y Fg(0)520 4350 y Ft(=)p Fr(4)595
4335 y FB(\()p Fv(y)681 4350 y Fr(2)720 4335 y FB(\))32
b(do)h(not)f(o)m(v)m(erlap,)i(and)f(summing)g(up)g(the)g(t)m(w)m(o)h
(estimates,)g(w)m(e)f(obtain)1128 4615 y Fv(M)1232 4574
y Fr(2)1289 4479 y Fu(Z)1344 4705 y Ft(B)1397 4717 y
Fj(d)1429 4732 y Fg(0)1489 4615 y Fv(\037)1550 4630 y
Fq(f)p Ft(U)7 b Fr(=0)p Fq(g)1798 4615 y Fs(\024)44 b
FB(const)2175 4479 y Fu(Z)2231 4705 y Ft(B)2284 4717
y Fj(d)2316 4732 y Fg(0)2375 4615 y Fs(jr)p FB(\()p Fv(U)33
b Fs(\000)22 b Fv(v)t FB(\))p Fs(j)2839 4574 y Fr(2)2895
4615 y Fv(:)375 4895 y FB(Th)m(us,)34 b(recalling)f(\(3.14\),)f(and)h
(writing)g Fv(U)38 b FB(=)28 b Fv(U)2148 4910 y Fq(\006)2240
4895 y FB(explicitly:)885 5175 y Fv(M)989 5134 y Fr(2)1029
5175 y Fw(L)1100 5134 y Ft(n)1147 5175 y FB(\()p Fs(f)p
Fv(U)1301 5190 y Fq(\006)1388 5175 y FB(=)g(0)p Fs(g)21
b(\\)i Fv(B)1775 5190 y Ft(d)1811 5199 y Fg(0)1850 5175
y FB(\))28 b Fs(\024)45 b FB(const)17 b Fw(L)2348 5134
y Ft(n)2395 5175 y FB(\()p Fs(fj)p Fv(u)p Fs(j)26 b FB(=)i(1)p
Fs(g)22 b(\\)g Fv(B)3008 5190 y Ft(d)3044 5199 y Fg(0)3083
5175 y FB(\))17 b Fv(:)p eop
%%Page: 30 39
30 38 bop 3727 201 a FB(30)375 475 y(Since)34 b Fs(fj)p
Fv(u)p Fs(j)26 b FB(=)i(1)p Fs(g)f(\022)h(f)p Fv(U)1269
490 y Fr(+)1356 475 y FB(=)f(0)p Fs(g)22 b([)h(f)p Fv(U)1785
490 y Fq(\000)1871 475 y FB(=)28 b(0)p Fs(g)p FB(,)k(w)m(e)i(deduce)g
(that)868 760 y Fv(M)972 719 y Fr(2)1012 760 y Fw(L)1083
719 y Ft(n)1130 760 y FB(\()p Fs(fj)p Fv(u)p Fs(j)27
b FB(=)g(1)p Fs(g)22 b(\\)g Fv(B)1743 775 y Ft(d)1779
784 y Fg(0)1819 760 y FB(\))27 b Fs(\024)h FB(2)17 b(const)g
Fw(L)2365 719 y Ft(n)2412 760 y FB(\()p Fs(fj)p Fv(u)p
Fs(j)27 b FB(=)g(1)p Fs(g)22 b(\\)g Fv(B)3025 775 y Ft(d)3061
784 y Fg(0)3101 760 y FB(\))17 b Fv(:)375 1045 y FB(If)36
b Fv(M)47 b FB(w)m(ere)38 b(to)s(o)d(big,)i(recalling)g(\(3.14\),)f(w)m
(e)i(get)e Fs(j)p Fv(u)p Fs(j)d Fv(<)g FB(1)j(in)h Fv(B)2766
1060 y Ft(d)2802 1069 y Fg(0)2841 1045 y FB(,)g(whic)m(h)h(is)e(a)g
(con)m(tra-)375 1240 y(diction.)p 3604 1240 4 66 v 3608
1177 59 4 v 3608 1240 V 3666 1240 4 66 v 375 1509 a Fz(Theorem)i
(3.2.6.)49 b Fy(If)35 b Fv(u)28 b Fs(2)g Fw(M)1539 1473
y Ft(!)1539 1535 y(M)1618 1509 y Fy(,)36 b(then)f Fv(u)27
b Fs(2)i FB(C)2149 1473 y Fr(0)p Ft(;)p Fr(1)2244 1509
y FB(\()p Fv(S)2348 1473 y Ft(!)2342 1535 y(M)2421 1509
y FB(\))p Fy(.)45 b(F)-7 b(urthermor)i(e,)35 b(for)g(any)g(c)-5
b(om-)375 1704 y(p)g(act)35 b(domain)e Fv(D)38 b Fy(wel)5
b(l)34 b(c)-5 b(ontaine)g(d)34 b(in)h Fv(S)1870 1668
y Ft(!)1864 1729 y(M)1978 1704 y Fy(with)1612 1989 y
Fv(D)25 b Fs(\\)d(fj)p Fv(u)p Fs(j)27 b FB(=)g(1)p Fs(g)g(6)p
FB(=)h Fs(;)17 b Fv(;)1004 b FB(\(3.19\))375 2275 y Fy(ther)-5
b(e)34 b(exists)f(a)h(suitable)f(p)-5 b(ositive)34 b(c)-5
b(onstant)33 b Fv(C)2126 2290 y Ft(D)2190 2275 y Fy(,)h(dep)-5
b(ending)32 b(only)i(on)50 b FB(dist)17 b(\()p Fv(D)s(;)g(@)5
b(S)3534 2238 y Ft(!)3528 2300 y(M)3607 2275 y FB(\))p
Fy(,)375 2469 y Fv(n)35 b Fy(and)f(the)h(b)-5 b(ounds)35
b(on)f Fv(a)1331 2484 y Ft(i;j)1446 2469 y Fy(and)h Fv(F)48
b Fy(such)35 b(that)1717 2754 y Fs(jj)p Fv(u)p Fs(jj)1885
2771 y Fr(C)1936 2752 y Fg(0)p Fj(;)p Fg(1)2050 2754
y Fs(\024)28 b Fv(C)2225 2769 y Ft(D)2306 2754 y Fv(:)1093
b FB(\(3.20\))375 3062 y Fy(Pr)-5 b(o)g(of.)48 b FB(First)26
b(of)f(all,)i(w)m(e)g(sho)m(w)g(that)e Fs(r)p Fv(u)g
FB(is)h(b)s(ounded)g(in)g Fv(D)11 b Fs(\\)d Fw(N)q FB(,)28
b(where)e Fw(N)h FB(is)g(a)e(suitable)375 3257 y(neigh)m(b)s(orho)s(o)s
(d)32 b(of)g Fv(@)5 b Fs(fj)p Fv(u)p Fs(j)28 b FB(=)f(1)p
Fs(g)p FB(.)675 3493 y(Indeed:)54 b(if)g(dist)17 b(\()p
Fv(x;)g Fs(fj)p Fv(u)p Fs(j)34 b FB(=)h(1)p Fs(g)p FB(\))g
Fs(\024)2081 3453 y Fr(1)p 2063 3470 71 4 v 2063 3527
a(10)2160 3493 y FB(dist)18 b(\()p Fv(x;)f(@)5 b(S)2594
3457 y Ft(!)2588 3518 y(M)2668 3493 y FB(\),)38 b(w)m(e)g(are)f(under)h
(the)g(h)m(y-)375 3688 y(p)s(otheses)c(of)e(Lemma)h(3.2.5,)f(so)h(that)
g(\(setting)g Fv(B)g FB(:=)27 b Fv(B)2501 3703 y Fr(dist)12
b(\()p Ft(x;)p Fq(fj)p Ft(u)p Fq(j)p Fr(=1)p Fq(g)p Fr(\))p
Ft(=)p Fr(10)3092 3688 y FB(\()p Fv(x)p FB(\))33 b(for)f(short\))667
3973 y(sup)651 4054 y Ft(x)691 4063 y Fg(0)726 4054 y
Fq(2)p Ft(B)846 3973 y FB(min)p Fs(f)p FB(1)22 b Fs(\000)h
Fv(u)p FB(\()p Fv(x)1378 3988 y Fr(0)1417 3973 y FB(\))p
Fv(;)33 b FB(1)22 b(+)g Fv(u)p FB(\()p Fv(x)1833 3988
y Fr(0)1873 3973 y FB(\))p Fs(g)27 b(\024)45 b FB(const)34
b(dist)17 b(\()p Fv(x;)g Fs(fj)p Fv(u)p Fs(j)27 b FB(=)g(1)p
Fs(g)p FB(\))17 b Fv(:)276 b FB(\(3.21\))375 4258 y(Also,)41
b(b)m(y)e(Lemma)g(3.2.2,)h Fv(@)1444 4273 y Ft(i)1473
4258 y FB(\()p Fv(a)1562 4273 y Ft(i;j)1642 4258 y Fv(@)1693
4273 y Ft(j)1730 4258 y FB(\(1)26 b Fs(\006)h Fv(u)p
FB(\)\))37 b(=)h(0)g(in)h Fv(B)5 b FB(;)42 b(therefore,)e(b)m(y)g
(elliptic)g(esti-)375 4453 y(mates,)1135 4647 y Fs(jr)p
FB(\(1)21 b Fs(\006)i Fv(u)p FB(\)\()p Fv(x)p FB(\))p
Fs(j)k(\024)45 b FB(const)2231 4580 y(sup)2377 4604 y
Ft(B)2455 4580 y Fs(j)p FB(1)21 b Fs(\006)i Fv(u)p Fs(j)p
2105 4624 757 4 v 2122 4716 a FB(dist)17 b(\()p Fv(x;)g
Fs(fj)p Fv(u)p Fs(j)26 b FB(=)i(1)p Fs(g)p FB(\))2888
4647 y Fv(:)375 4892 y FB(F)-8 b(rom)32 b(this)h(and)g(\(3.21\),)f(w)m
(e)h(deduce)i(a)d(b)s(ound)h(on)f Fs(r)p Fv(u)g FB(in)h
Fv(D)25 b Fs(\\)d Fw(N)q FB(.)675 5175 y(This)34 b(and)e(standard)h
(elliptic)h(estimates)g(imply)g(that)e Fv(u)c Fs(2)g
FB(C)2976 5139 y Fr(0)p Ft(;)p Fr(1)3070 5175 y FB(\()p
Fv(S)3174 5139 y Ft(!)3168 5200 y(M)3247 5175 y FB(\).)p
eop
%%Page: 31 40
31 39 bop 3727 201 a FB(31)675 475 y(W)-8 b(e)33 b(no)m(w)g(pro)m(v)m
(e)h(\(3.20\):)43 b(T)-8 b(ak)m(e)34 b Fv(D)1949 438
y Fq(0)2004 475 y FB(suc)m(h)g(that)e Fv(D)f Fs(\032\032)d
Fv(D)2813 438 y Fq(0)2864 475 y Fs(\032\032)h Fv(S)3113
438 y Ft(!)3107 500 y(M)3186 475 y FB(,)j(with)1293 744
y(dist)17 b(\()p Fv(D)1589 703 y Fq(0)1612 744 y Fv(;)g(@)5
b(S)1778 703 y Ft(!)1772 769 y(M)1852 744 y FB(\))27
b Fs(\025)2057 677 y FB(9)p 2032 721 98 4 v 2032 813
a(10)2157 744 y(dist)17 b(\()p Fv(D)s(;)g(@)5 b(S)2619
703 y Ft(!)2613 769 y(M)2692 744 y FB(\))17 b Fv(:)375
1014 y FB(With)41 b(no)f(loss)g(of)g(generalit)m(y)-8
b(,)43 b(w)m(e)e(ma)m(y)g(assume)h Fv(D)h FB(and)d Fv(D)2687
978 y Fq(0)2750 1014 y FB(to)g(b)s(e)g(connected.)68
b(Fix)375 1208 y(an)m(y)43 b Fv(x)j Fs(2)f Fv(D)865 1172
y Fq(0)930 1208 y FB(and)e(set)g Fv(r)1336 1223 y Fr(0)1420
1208 y FB(:=)1578 1169 y Fr(1)p 1578 1185 36 4 v 1578
1243 a(5)1640 1208 y FB(dist)17 b(\()p Fv(D)s(;)g(@)5
b(S)2102 1172 y Ft(!)2096 1233 y(M)2176 1208 y FB(\).)73
b(Consider)44 b(a)f(co)m(v)m(ering)h(of)e Fv(D)3420 1172
y Fq(0)3485 1208 y FB(with)375 1403 y(balls)32 b Fv(B)676
1418 y Ft(r)708 1427 y Fg(0)747 1403 y FB(\()p Fv(x)840
1418 y Ft(j)877 1403 y FB(\),)g(cen)m(tered)i(in)e Fv(D)1558
1366 y Fq(0)1581 1403 y FB(,)g(with)g Fv(x)1916 1418
y Fr(0)1984 1403 y FB(:=)27 b Fv(x)33 b FB(and)e Fv(x)2445
1418 y Ft(j)2510 1403 y Fs(2)d Fv(B)2678 1418 y Ft(r)2710
1427 y Fg(0)2745 1418 y Ft(=)p Fr(2)2819 1403 y FB(\()p
Fv(x)2912 1418 y Ft(j)t Fq(\000)p Fr(1)3040 1403 y FB(\).)43
b(Let)32 b Fv(k)i FB(b)s(e)f(the)375 1597 y(n)m(um)m(b)s(er)41
b(of)e(suc)m(h)i(balls:)59 b(ob)m(viously)-8 b(,)43 b
Fv(k)g FB(dep)s(ends)e(only)f(on)g Fv(n)p FB(,)i Fv(D)2875
1561 y Fq(0)2937 1597 y FB(and)e Fv(r)3178 1612 y Fr(0)3217
1597 y FB(,)i(hence)f(w)m(e)375 1791 y(ma)m(y)33 b(think)h
Fv(k)h FB(to)e(dep)s(end)g(only)g(on)g Fv(n)p FB(,)g
Fv(D)i FB(and)49 b(dist)18 b(\()p Fv(D)s(;)f(S)2585 1755
y Ft(!)2579 1817 y(M)2657 1791 y FB(\).)675 2026 y(By)33
b(\(3.19\),)f(there)h(exists)i Fv(k)1710 1989 y Fq(\003)1777
2026 y Fs(\024)28 b Fv(k)36 b FB(so)c(that)1485 2295
y Fv(B)1559 2310 y Ft(r)1591 2319 y Fg(0)1629 2295 y
FB(\()p Fv(x)1722 2310 y Ft(k)1761 2291 y Fi(\003)1801
2295 y FB(\))23 b Fs(\\)f(fj)p Fv(u)p Fs(j)k FB(=)i(1)p
Fs(g)f(6)p FB(=)h Fs(;)17 b Fv(;)375 2565 y FB(while)1282
2759 y Fv(B)1356 2774 y Ft(r)1388 2783 y Fg(0)1427 2759
y FB(\()p Fv(x)1520 2774 y Ft(j)1557 2759 y FB(\))22
b Fs(\\)g(fj)p Fv(u)p Fs(j)27 b FB(=)g(1)p Fs(g)h FB(=)f
Fs(;)17 b Fv(;)77 b Fs(8)p Fv(j)34 b(<)28 b(k)2685 2718
y Fq(\003)2741 2759 y Fv(:)375 2997 y FB(It)33 b(follo)m(ws)g(that)1181
3267 y(dist)18 b(\()p Fv(x)1449 3282 y Ft(k)1488 3263
y Fi(\003)1528 3267 y Fv(;)f Fs(fj)p Fv(u)p Fs(j)26 b
FB(=)h(1)p Fs(g)p FB(\))g Fs(\024)2142 3199 y FB(2)p
2142 3244 49 4 v 2142 3335 a(9)2218 3267 y(dist)17 b(\()p
Fv(x)2485 3282 y Ft(k)2524 3263 y Fi(\003)2564 3267 y
Fv(;)g(@)5 b(S)2730 3225 y Ft(!)2724 3291 y(M)2804 3267
y FB(\))17 b Fv(;)375 3536 y FB(so)33 b(w)m(e)g(can)g(apply)g(Lemma)h
(3.2.5)e(and)g(infer)h(that)1171 3806 y(min)p Fs(f)p
FB(1)22 b Fs(\000)h Fv(u)p FB(\()p Fv(x)1703 3821 y Ft(k)1742
3802 y Fi(\003)1781 3806 y FB(\))p Fv(;)33 b FB(1)22
b(+)g Fv(u)p FB(\()p Fv(x)2197 3821 y Ft(k)2236 3802
y Fi(\003)2276 3806 y FB(\))p Fs(g)27 b(\024)45 b FB(const)17
b Fv(r)2796 3821 y Fr(0)2852 3806 y Fv(:)547 b FB(\(3.22\))375
4075 y(Also,)35 b Fv(@)675 4090 y Ft(i)704 4075 y FB(\()p
Fv(a)793 4090 y Ft(i;j)873 4075 y Fv(@)924 4090 y Ft(j)961
4075 y FB(\(1)23 b Fs(\006)h Fv(u)p FB(\)\))29 b(=)h(0)k(in)h
Fv(B)1712 4090 y Ft(r)1744 4099 y Fg(0)1782 4075 y FB(\()p
Fv(x)1875 4090 y Ft(j)1912 4075 y FB(\))f(for)g Fv(j)i(<)30
b(k)2371 4039 y Fq(\003)2445 4075 y FB(b)s(ecause)35
b(of)f(Lemma)g(3.2.2.)48 b(So,)375 4270 y(w)m(e)34 b(deduce)g(from)e
(Harnac)m(k)i(inequalit)m(y)g(that)1348 4539 y(const)18
b(\(1)k Fs(\006)g Fv(u)p FB(\()p Fv(x)1945 4554 y Ft(j)t
Fr(+1)2072 4539 y FB(\)\))27 b Fs(\025)i FB(1)21 b Fs(\006)i
Fv(u)p FB(\()p Fv(x)2600 4554 y Ft(j)2637 4539 y FB(\))17
b Fv(:)375 4809 y FB(Iterating,)33 b(using)g(\(3.22\))f(and)h
(recalling)g(that)f Fv(x)2189 4824 y Fr(0)2257 4809 y
FB(=)27 b Fv(x)p FB(,)33 b(w)m(e)h(get)1168 5078 y(sup)1156
5161 y Ft(x)p Fq(2)p Ft(D)1303 5142 y Fi(0)1342 5078
y FB(min)q Fs(f)p FB(1)21 b Fs(\000)i Fv(u)p FB(\()p
Fv(x)p FB(\))p Fv(;)33 b FB(1)22 b(+)g Fv(u)p FB(\()p
Fv(x)p FB(\))p Fs(g)27 b(\024)45 b FB(const)17 b Fv(r)2810
5093 y Fr(0)2866 5078 y Fv(:)533 b FB(\(3.23\))p eop
%%Page: 32 41
32 40 bop 3727 201 a FB(32)375 475 y(T)-8 b(ak)m(e)34
b(no)m(w)f Fv(\020)i Fs(2)28 b Fv(D)s FB(,)k(with)h Fs(j)p
Fv(u)p FB(\()p Fv(\020)8 b FB(\))p Fs(j)26 b Fv(<)h FB(1.)675
711 y(Tw)m(o)33 b(cases)h(are)f(p)s(ossible:)45 b(If)1264
1006 y(dist)18 b(\()p Fv(\020)8 b(;)17 b Fs(fj)p Fv(u)p
Fs(j)25 b FB(=)i(1)p Fs(g)p FB(\))h Fs(\025)2142 939
y FB(1)p 2142 983 49 4 v 2142 1074 a(3)2217 1006 y(dist)17
b(\()p Fv(D)s(;)g(@)5 b(D)2697 965 y Fq(0)2721 1006 y
FB(\))17 b Fv(;)375 1301 y FB(then)1069 1496 y Fv(@)1120
1511 y Ft(i)1148 1496 y FB(\()p Fv(a)1237 1511 y Ft(i;j)1317
1496 y Fv(@)1368 1511 y Ft(j)1405 1496 y FB(\(1)22 b
Fs(\006)h Fv(u)p FB(\)\))k(=)h(0)194 b(on)45 b Fv(B)2354
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36 44 bop 3727 201 a FB(36)375 475 y Fy(among)38 b(al)5
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FB(\(\012\))j Fy(so)g(that)h Fv(v)f FB(=)d Fv(u)i Fy(on)375
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642 y FB(~)1138 669 y Fw(J)1188 684 y Fr(\012)1243 669
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b(age)34 b(1.)675 1100 y(Then,)g(for)h(any)f Fv(x)29
b Fs(2)f(f)p Fv(u)f(>)g FB(0)p Fs(g)p Fy(,)1471 1384
y FB(dist)q(\()p Fv(x;)17 b Fs(f)p Fv(u)27 b FB(=)g(0)p
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Fy(and)f Fv(Q)p Fy(.)375 2170 y(Pr)-5 b(o)g(of.)48 b
FB(By)e(scaling,)j(w)m(e)d(ma)m(y)g(assume)h Fv(x)j FB(=)e(0)d(and)62
b(dist)18 b(\()p Fv(x;)f Fs(f)p Fv(u)48 b FB(=)h(0)p
Fs(g)p FB(\))g(=)g(1.)81 b(Set)375 2365 y Fv(")27 b FB(:=)h
Fv(u)p FB(\(0\).)42 b(Since)31 b Fv(B)1155 2380 y Fr(1)1222
2365 y Fs(\022)d(f)p Fv(u)f(>)h FB(0)p Fs(g)p FB(,)i(the)g(\014rst)g(v)
-5 b(ariation)30 b(of)f(the)h(functional)h(sho)m(ws)g(that)375
2560 y Fv(@)426 2575 y Ft(i)455 2560 y FB(\()p Fv(a)544
2575 y Ft(i;j)624 2560 y Fv(@)675 2575 y Ft(j)712 2560
y Fv(u)p FB(\))c(=)h(0)k(on)g Fv(B)1227 2575 y Fr(1)1267
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b(,)1764 2844 y(sup)1762 2925 y Ft(B)1815 2939 y Fg(1)p
Fj(=)p Fg(2)1929 2844 y Fv(u)27 b Fs(\024)h Fv(c)2159
2803 y Fq(\003)2199 2844 y Fv(")17 b(;)375 3128 y FB(for)32
b(a)g(univ)m(ersal)j(constan)m(t)e Fv(c)1453 3092 y Fq(\003)1493
3128 y FB(.)675 3363 y(Let)j Fv(')f FB(b)s(e)h(the)g(radially)f
(symmetric)j(function)e(de\014ned)h(as)f(follo)m(ws:)50
b Fv(')32 b FB(=)h(0)i(in)375 3558 y Fv(B)449 3574 y
Fr(1)p Ft(=)p Fr(4)559 3558 y FB(,)f Fv(')c FB(=)f(2)p
Fv(c)910 3522 y Fq(\003)949 3558 y Fv(")34 b FB(on)f
Fv(@)5 b(B)1295 3574 y Fr(1)p Ft(=)p Fr(2)1440 3558 y
FB(and)34 b(gro)m(wing)g(linearly)g(on)g(the)g(ra)m(ys)h(of)e
Fv(B)3048 3574 y Fr(1)p Ft(=)p Fr(2)3181 3558 y Fs(\000)24
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4037 y Fv(@)1290 4052 y Ft(i)1319 4037 y Fv(u@)1426 4052
y Ft(j)1462 4037 y Fv(u)22 b FB(+)g Fv(Q\037)1776 4053
y Fq(f)p Ft(u>)p Fr(0)p Fq(g)2010 4037 y Fs(\024)2115
3901 y Fu(Z)2170 4127 y Ft(B)2223 4141 y Fg(1)p Fj(=)p
Fg(2)2341 4037 y Fv(a)2392 4052 y Ft(i;j)2472 4037 y
Fv(@)2523 4052 y Ft(i)2552 4037 y Fv(v)t(@)2654 4052
y Ft(j)2691 4037 y Fv(v)k FB(+)c Fv(Q\037)3000 4053 y
Fq(f)p Ft(v)r(>)p Fr(0)p Fq(g)3229 4037 y Fs(\024)881
4323 y(\024)987 4187 y Fu(Z)1042 4413 y Ft(B)1095 4427
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4323 y Fv(@)1395 4338 y Ft(i)1424 4323 y Fv(u@)1531 4338
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4413 y Ft(B)1851 4427 y Fg(1)p Fj(=)p Fg(2)1969 4323
y Fv(a)2020 4338 y Ft(i;j)2101 4323 y Fv(@)2152 4338
y Ft(i)2180 4323 y Fv('@)2295 4338 y Ft(j)2332 4323 y
Fv(')g FB(+)2516 4187 y Fu(Z)2572 4413 y Ft(B)2625 4427
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4427 y Fg(1)p Fj(=)p Fg(4)2947 4323 y Fv(Q\037)3085 4338
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y FB(,)h(the)g(previous)h(inequalit)m(y)g(implies)877
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4891 y Fs(\000)1787 4755 y Fu(Z)1842 4981 y Ft(B)1895
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4906 y Fr(1)p Ft(=)p Fr(4)3102 4891 y Fs(j)17 b Fv(:)p
3604 5175 4 66 v 3608 5112 59 4 v 3608 5175 V 3666 5175
4 66 v eop
%%Page: 37 46
37 45 bop 3727 201 a FB(37)375 475 y Fz(Lemma)39 b(4.1.5.)49
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448 y FB(~)1699 475 y Fw(J)i Fy(b)-5 b(e)42 b(as)h(in)f(L)-5
b(emma)42 b(4.1.4.)68 b(L)-5 b(et)43 b Fv(x)g Fs(2)g
Fv(@)5 b Fs(f)p Fv(u)42 b(>)h FB(0)p Fs(g)p Fy(.)375
669 y(Then,)34 b(for)h(any)f Fv(r)d Fs(\025)d FB(0)p
Fy(,)1765 862 y FB(sup)1748 948 y Ft(B)1801 956 y Fj(r)1835
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Fv(C)7 b(r)19 b(;)375 1093 y Fy(wher)-5 b(e)32 b Fv(C)39
b Fy(is)32 b(a)h(p)-5 b(ositive)31 b(c)-5 b(onstant)33
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Fy(and)i Fv(Q)p Fy(\),)h(pr)-5 b(ovide)g(d)34 b Fv(B)1205
1302 y Ft(r)1243 1287 y FB(\()p Fv(x)p FB(\))28 b Fs(\022)g
FB(\012)p Fy(.)375 1582 y(Pr)-5 b(o)g(of.)48 b FB(By)36
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Fv(r)e FB(=)e(1.)50 b(Since)36 b(0)31 b Fs(2)g Fv(@)5
b Fs(f)p Fv(u)32 b(>)f FB(0)p Fs(g)p FB(,)j(for)375 1776
y(an)m(y)h Fv(")30 b(>)g FB(0)k(there)h(exists)h(a)e(p)s(oin)m(t)h
Fv(x)1743 1791 y Fr(1)1813 1776 y Fs(2)c Fv(B)1984 1791
y Ft(")2021 1776 y FB(,)k(so)f(that)g Fv(u)p FB(\()p
Fv(x)2566 1791 y Fr(1)2605 1776 y FB(\))d Fv(>)f FB(0.)48
b(W)-8 b(e)35 b(no)m(w)g(construct)375 1970 y(a)h(p)s(olygonal,)i
(starting)f(as)f(close)i(as)f(w)m(e)g(wish)h(to)f(zero,)h(along)e(whic)
m(h)i Fv(u)e FB(gro)m(ws)h(up)g(at)375 2164 y(least)c(linearly)-8
b(.)45 b(Set)1249 2417 y Fv(d)1300 2432 y Fr(1)1367 2417
y FB(:=)f(dist)18 b(\()p Fv(x)1782 2432 y Fr(1)1822 2417
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Fv(")17 b(:)375 2669 y FB(By)33 b(Lemma)g(4.1.4,)1730
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y FB(\))27 b Fs(\025)i Fv(c)2131 2881 y Fq(\003)2170
2922 y Fv(d)2221 2937 y Fr(1)2277 2922 y Fv(;)1170 b
FB(\(4.2\))375 3175 y(for)32 b(a)g(suitable)i(constan)m(t)f
Fv(c)1404 3138 y Fq(\003)1444 3175 y FB(.)43 b(Let)33
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3190 y Ft(d)2062 3199 y Fg(1)2102 3175 y FB(\()p Fv(x)2195
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Fq(\003)p 2364 4356 82 4 v 2380 4447 a FB(2)2472 4379
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5056 y Fv(\032)2274 5071 y Fr(1)2330 5056 y Fv(:)1117
b FB(\(4.3\))p eop
%%Page: 38 47
38 46 bop 3727 201 a FB(38)375 475 y(T)-8 b(ak)m(e)40
b(~)-55 b Fv(x)666 490 y Fr(1)735 475 y Fs(2)853 449
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Fg(1)978 475 y FB(\()p Fv(z)1061 490 y Fr(1)1101 475
y FB(\))33 b(attaining)g(the)g(sup)h(in)g(\(4.3\).)44
b(In)33 b(fact,)g(b)m(y)h(Maxim)m(um)h(Principle,)381
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Fv(@)5 b(B)739 684 y Ft(\032)775 693 y Fg(1)815 669 y
FB(\()p Fv(z)898 684 y Fr(1)938 669 y FB(\).)58 b(W)-8
b(e)38 b(w)m(an)m(t)h(no)m(w)f(to)f(sho)m(w)i(that)e(there)i(exist)g
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1783 y Fg(1)1672 1759 y Fr(\()p Ft(z)1732 1768 y Fg(1)1767
1759 y Fr(\))1811 1673 y Fv(u)27 b(<)h FB(\(1)21 b(+)i
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2138 y FB(+)22 b Fv(\032)2035 2153 y Fr(1)2075 2138 y
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2646 y Fs(jj)p Fv(v)t Fs(jj)1353 2661 y Fr(L)1397 2642
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2646 y Fv(;)631 b FB(\(4.5\))375 2916 y(ha)m(ving)28
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4698 y Fg(1)3120 4666 y Fv(w)h Fs(\024)f Fv(\025)p FB(,)i(for)d(a)375
4860 y(univ)m(ersal)c Fv(\025)28 b Fs(2)g FB(\(0)p Fv(;)17
b FB(1\).)43 b(In)33 b(particular,)f(since)41 b(~)-56
b Fv(x)2146 4875 y Fr(1)2214 4860 y Fs(2)2331 4835 y
FB(\026)2308 4860 y Fv(B)2382 4875 y Ft(\032)2418 4884
y Fg(1)2457 4860 y FB(\()p Fv(z)2540 4875 y Fr(1)2580
4860 y FB(\),)1361 5130 y Fv(\025)28 b Fs(\025)g Fv(w)1640
4990 y Fu(\022)1729 5063 y FB(~)-55 b Fv(x)1778 5078
y Fr(1)1840 5063 y Fs(\000)23 b Fv(z)1985 5078 y Fr(1)p
1723 5107 302 4 v 1829 5199 a Fv(\032)1879 5214 y Fr(1)2034
4990 y Fu(\023)2135 5130 y FB(=)2332 5063 y(2)p 2249
5107 216 4 v 2249 5199 a(1)f(+)g Fv(\016)2497 5130 y
Fs(\000)h FB(1)17 b Fv(;)p eop
%%Page: 39 48
39 47 bop 3727 201 a FB(39)375 475 y(whic)m(h)35 b(is)f(a)f(con)m
(tradiction)h(if)f Fv(\016)k FB(is)d(small)g(enough.)47
b(This)34 b(pro)m(v)m(es)h(\(4.4\).)45 b(By)34 b(construc-)375
670 y(tion,)42 b Fv(u)p FB(\()6 b(~)-55 b Fv(x)761 685
y Fr(1)801 670 y FB(\))40 b Fs(\025)i Fv(u)p FB(\()p
Fv(x)1147 685 y Fr(1)1186 670 y FB(\).)67 b(Hence,)43
b(there)e(exists)i(a)d(p)s(oin)m(t)g Fv(x)2585 685 y
Fr(2)2666 670 y Fs(2)2796 645 y FB(\026)2773 670 y Fv(B)2847
685 y Fr(2)p Ft(\032)2918 694 y Fg(1)2957 670 y FB(\()p
Fv(z)3040 685 y Fr(1)3080 670 y FB(\))g(\(and)h(in)f(fact)375
865 y Fv(x)430 880 y Fr(2)498 865 y Fs(2)28 b Fv(@)5
b(B)722 880 y Fr(2)p Ft(\032)793 889 y Fg(1)833 865 y
FB(\()p Fv(z)916 880 y Fr(1)956 865 y FB(\))32 b(b)m(y)h(Maxim)m(um)i
(Principle\))f(suc)m(h)g(that)1094 1160 y Fv(u)p FB(\()p
Fv(x)1243 1175 y Fr(2)1283 1160 y FB(\))27 b Fs(\025)h
Fv(u)p FB(\()6 b(~)-55 b Fv(x)1602 1175 y Fr(1)1642 1160
y FB(\))22 b(+)g Fv(\016)t(c)1889 1118 y Fq(\003)1928
1160 y Fv(\032)1978 1175 y Fr(1)2045 1160 y Fs(\025)29
b Fv(u)p FB(\()p Fv(x)2300 1175 y Fr(1)2339 1160 y FB(\))22
b(+)g Fv(\022)s Fs(j)p Fv(x)2628 1175 y Fr(1)2690 1160
y Fs(\000)h Fv(x)2845 1175 y Fr(2)2884 1160 y Fs(j)17
b Fv(;)375 1454 y FB(for)32 b(univ)m(ersal)i Fv(\016)n(;)45
b(\022)31 b(>)d FB(0,)k(pro)m(vided)i(that)f Fv(B)2024
1469 y Fr(3)p Ft(\032)2095 1478 y Fg(1)2134 1454 y FB(\()p
Fv(x)2227 1469 y Fr(1)2267 1454 y FB(\))27 b Fs(\022)h
Fv(B)2511 1469 y Fr(1)2551 1454 y FB(.)675 1691 y(Iterating,)g(w)m(e)f
(obtain)f(a)g(p)s(olygonal)f(of)h(v)m(ertexes)j Fv(x)2578
1706 y Fr(1)2618 1691 y Fv(;)17 b(x)2717 1706 y Fr(2)2756
1691 y Fv(;)g(:)g(:)g(:)f(;)h(x)3030 1706 y Ft(k)3099
1691 y FB(suc)m(h)27 b(that:)40 b(for)375 1886 y(2)27
b Fs(\024)i Fv(j)k Fs(\024)28 b Fv(k)s FB(,)33 b Fv(x)904
1901 y Ft(j)969 1886 y Fs(2)28 b Fv(@)5 b(B)1193 1901
y Fr(3)p Ft(\032)1264 1911 y Fj(j)s Fi(\000)p Fg(1)1380
1886 y FB(\()p Fv(x)1473 1901 y Ft(j)t Fq(\000)p Fr(1)1600
1886 y FB(\),)33 b Fs(j)p Fv(x)1781 1901 y Ft(j)1840
1886 y Fs(\000)22 b Fv(x)1994 1901 y Ft(j)t Fq(\000)p
Fr(1)2121 1886 y Fs(j)28 b(\025)g Fv(\032)2332 1901 y
Ft(j)2401 1886 y FB(and)1303 2181 y Fv(u)p FB(\()p Fv(x)1452
2196 y Ft(j)1489 2181 y FB(\))83 b Fs(\025)g FB(\(1)22
b(+)g Fv(\016)t FB(\))p Fv(u)p FB(\()p Fv(x)2211 2196
y Ft(j)t Fq(\000)p Fr(1)2337 2181 y FB(\))28 b Fs(\025)994
b FB(\(4.7\))1610 2401 y Fs(\025)83 b Fv(u)p FB(\()p
Fv(x)1919 2416 y Ft(j)t Fq(\000)p Fr(1)2046 2401 y FB(\))22
b(+)g Fv(\022)s Fs(j)p Fv(x)2335 2416 y Ft(j)2394 2401
y Fs(\000)g Fv(x)2548 2416 y Ft(j)t Fq(\000)p Fr(1)2675
2401 y Fs(j)17 b Fv(;)727 b FB(\(4.8\))375 2695 y(pro)m(vided)34
b Fv(B)850 2710 y Fr(3)p Ft(\032)921 2720 y Fj(j)s Fi(\000)p
Fg(1)1037 2695 y FB(\()p Fv(x)1130 2710 y Ft(j)t Fq(\000)p
Fr(1)1257 2695 y FB(\))27 b Fs(\022)h Fv(B)1501 2710
y Fr(1)1541 2695 y FB(.)43 b(In)33 b(particular,)g(iterating)g
(\(4.8\),)1776 2990 y Fv(u)p FB(\()p Fv(x)1925 3005 y
Ft(j)1961 2990 y FB(\))28 b Fs(\025)g Fv(\022)s(`)2221
3005 y Ft(j)375 3285 y FB(where)38 b Fv(`)702 3300 y
Ft(j)776 3285 y FB(is)f(the)h(length)f(of)g(the)g(p)s(olygonal)g
(joining)g Fv(x)2472 3300 y Fr(1)2512 3285 y Fv(;)17
b(:)g(:)g(:)e(;)i(x)2785 3300 y Ft(j)2822 3285 y FB(.)57
b(W)-8 b(e)37 b(w)m(an)m(t)h(to)f(pro)m(v)m(e)375 3480
y(that)31 b(w)m(e)h(can)f(tak)m(e)h Fv(j)37 b FB(big)31
b(enough,)h(and)f(mak)m(e)h Fv(`)2196 3495 y Ft(j)2263
3480 y FB(bigger)f(than)g(some)h(univ)m(ersal)h(quan-)375
3675 y(tit)m(y)-8 b(.)44 b(This)34 b(w)m(ould)f(\014nish)h(the)f(pro)s
(of)f(of)g(the)h(lemma.)675 3962 y(Indeed,)26 b(w)m(e)e(m)m(ust)g(b)s
(e)g(able)f(to)f(iterate)i(the)f(sc)m(heme)i(and)e(\014nd)h(a)e
(suitable)i Fv(j)3425 3925 y Fq(\003)3487 3962 y FB(suc)m(h)375
4157 y(that)31 b Fs(j)p Fv(x)668 4172 y Ft(j)701 4153
y Fi(\003)741 4157 y Fs(j)c(\025)h FB(1)p Fv(=)p FB(20.)43
b(Otherwise,)33 b(if)f Fs(j)p Fv(x)1826 4172 y Ft(j)1862
4157 y Fs(j)27 b Fv(<)h FB(1)p Fv(=)p FB(20)j(for)g(an)m(y)h
Fv(j)6 b FB(,)32 b(then)g Fv(\032)2955 4172 y Ft(j)3019
4157 y Fs(\024)c(j)p Fv(x)3207 4172 y Ft(j)3264 4157
y Fs(\000)21 b Fv(x)3417 4172 y Ft(j)t Fq(\000)p Fr(1)3544
4157 y Fs(j)27 b Fv(<)375 4352 y FB(1)p Fv(=)p FB(10)34
b(for)g(an)m(y)i Fv(j)6 b FB(;)36 b(and)f(so)g Fv(B)1440
4367 y Fr(3)p Ft(\032)1511 4377 y Fj(j)1548 4352 y FB(\()p
Fv(x)1641 4367 y Ft(j)1678 4352 y FB(\))d Fs(\032)g Fv(B)1931
4367 y Fr(1)2005 4352 y FB(for)j(an)m(y)g Fv(j)6 b FB(,)36
b(and)f(iterating)g(\(4.7\))f(w)m(e)i(w)m(ould)375 4547
y(get)d(sup)684 4570 y Ft(B)737 4579 y Fg(1)793 4547
y Fv(u)27 b FB(=)h(+)p Fs(1)p FB(.)675 4783 y(Therefore,)34
b Fv(`)1185 4798 y Ft(j)1218 4780 y Fi(\003)1285 4783
y Fs(\025)28 b(j)p Fv(x)1473 4798 y Ft(j)1506 4780 y
Fi(\003)1546 4783 y Fs(j)21 b(\000)i(j)p Fv(x)1778 4798
y Fr(1)1818 4783 y Fs(j)k(\025)2006 4744 y Fr(1)p 1988
4761 71 4 v 1988 4818 a(20)2091 4783 y Fs(\000)22 b Fv(")p
FB(.)p 3604 4783 4 66 v 3608 4721 59 4 v 3608 4783 V
3666 4783 4 66 v 675 5095 a(W)-8 b(e)33 b(can)g(no)m(w)g(sharp)s(en)g
(the)g(result)h(of)e(Lemma)h(4.1.2:)p eop
%%Page: 40 49
40 48 bop 3727 201 a FB(40)375 475 y Fz(Lemma)39 b(4.1.6.)49
b Fy(L)-5 b(et)38 b Fv(u)30 b Fs(2)i Fw(M)1535 438 y
Ft(!)1535 500 y(M)1614 475 y Fy(.)50 b(Assume)37 b(that)g
Fv(u)p FB(\()6 b(\026)-55 b Fv(x)p FB(\))31 b Fs(2)h
FB(\()p Fs(\000)p FB(1)p Fv(;)17 b FB(1\))p Fy(.)50 b(Then,)36
b(ther)-5 b(e)37 b(exist)375 669 y Fv(r)419 684 y Fr(0)497
669 y Fy(and)g Fv(r)733 684 y Fr(1)811 669 y Fy(\(dep)-5
b(ending)36 b(only)i(on)g Fv(n)p Fy(,)h(the)f(b)-5 b(ounds)38
b(on)g Fv(a)2476 684 y Ft(i;j)2594 669 y Fy(and)g Fv(Q)p
Fy(\))g(such)g(that)h Fv(u)e Fy(takes)375 863 y(b)-5
b(oth)35 b(the)g(values)f FB(1)h Fy(and)f Fs(\000)p FB(1)h
Fy(in)g Fv(B)1663 878 y Ft(r)1695 887 y Fg(0)1733 863
y FB(\()6 b(\026)-55 b Fv(x)q FB(\))p Fy(,)34 b(pr)-5
b(ovide)g(d)34 b(that)h Fv(B)2587 878 y Ft(r)2619 887
y Fg(0)2654 878 y Fr(+)p Ft(r)2741 887 y Fg(1)2779 863
y FB(\()6 b(\026)-55 b Fv(x)q FB(\))27 b Fs(\032)i Fv(S)3110
827 y Ft(!)3104 888 y(M)3183 863 y Fy(.)375 1162 y(Pr)-5
b(o)g(of.)48 b FB(By)32 b(Lemma)g(4.1.2,)f Fv(u)g FB(tak)m(es,)i(sa)m
(y)-8 b(,)32 b(the)g(v)-5 b(alue)32 b Fs(\000)p FB(1)f(in)h
Fv(B)2749 1177 y Ft(r)2781 1186 y Fg(0)2819 1162 y FB(\()6
b(\026)-55 b Fv(x)q FB(\).)43 b(Without)31 b(loss)h(of)375
1356 y(generalit)m(y)-8 b(,)37 b(w)m(e)g(assume)g Fv(r)1378
1371 y Fr(0)1450 1356 y Fs(\025)c FB(3)p Fv(=C)7 b FB(,)36
b(where)g Fv(C)43 b FB(is)35 b(the)h(constan)m(t)h(in)e(Lemma)h(4.1.5.)
52 b(If)375 1551 y Fv(u)32 b FB(did)h(not)f(assume)j(the)e(v)-5
b(alue)33 b(1)f(to)s(o,)g(using)h(Lemmas)h(4.1.3)e(and)g(4.1.5,)g(w)m
(e)i(get)1429 1813 y(2)28 b Fs(\025)80 b FB(sup)1611
1899 y Ft(B)1664 1907 y Fj(r)1693 1922 y Fg(0)1732 1899
y Fr(\()p Ft(x)1799 1908 y Fg(0)1834 1899 y Fr(\))1878
1813 y Fv(u)21 b FB(+)h(1)28 b Fs(\025)g Fv(C)7 b(r)2356
1828 y Fr(0)2423 1813 y Fs(\025)28 b FB(3)17 b Fv(;)375
2075 y FB(whic)m(h)34 b(is)f(a)f(con)m(tradiction.)p
3604 2075 4 66 v 3608 2012 59 4 v 3608 2075 V 3666 2075
4 66 v 675 2374 a(W)-8 b(e)33 b(are)g(no)m(w)g(in)g(p)s(osition)f(to)h
(impro)m(v)m(e)h(the)f(previous)h(densit)m(y)g(estimates:)375
2624 y Fz(Lemma)39 b(4.1.7.)49 b Fy(L)-5 b(et)39 b Fv(u)33
b Fs(2)h Fw(M)1541 2588 y Ft(!)1541 2649 y(M)1620 2624
y Fy(.)54 b(Assume)38 b(that)h Fv(x)34 b Fs(2)g(fj)p
Fv(u)p Fs(j)e Fv(<)h FB(1)p Fs(g)p Fy(.)54 b(Then,)38
b(ther)-5 b(e)38 b(exist)375 2818 y(p)-5 b(ositive)33
b(c)-5 b(onstants)33 b Fv(r)1196 2833 y Fr(0)1235 2818
y Fy(,)g Fv(r)1342 2833 y Fr(1)1415 2818 y Fy(and)g Fv(c)p
Fy(,)g(dep)-5 b(ending)32 b(only)h(on)g Fv(n)h Fy(and)f(on)g(the)g(b)-5
b(ounds)33 b(on)g Fv(a)3595 2833 y Ft(i;j)375 3013 y
Fy(and)h Fv(Q)p Fy(,)h(such)g(that)1640 3207 y Fw(J)1690
3222 y Ft(B)1743 3230 y Fj(r)1778 3222 y Fr(\()p Ft(x)p
Fr(\))1877 3207 y FB(\()p Fv(u)p FB(\))27 b Fs(\025)h
Fv(cr)2230 3166 y Ft(n)p Fq(\000)p Fr(1)2383 3207 y Fv(;)375
3441 y Fy(for)35 b(any)f Fv(r)d Fs(\025)d Fv(r)941 3456
y Fr(0)980 3441 y Fy(,)35 b(pr)-5 b(ovide)g(d)34 b Fv(B)1504
3456 y Ft(r)r Fr(+)p Ft(r)1625 3465 y Fg(1)1663 3441
y FB(\()p Fv(x)p FB(\))28 b Fs(\032)g Fv(S)1993 3405
y Ft(!)1987 3467 y(M)2066 3441 y Fy(.)375 3741 y(Pr)-5
b(o)g(of.)48 b FB(By)33 b(Lemma)h(4.1.6,)e(Giaquin)m(ta)h(and)f
(Giusti's)i(Theorem)g(and)e(Theorem)i(4.1.1)1247 4003
y Fw(L)1318 3962 y Ft(n)1366 3892 y Fu(\020)1425 4003
y Fv(B)1499 4018 y Ft(r)1537 4003 y FB(\()p Fv(x)p FB(\))23
b Fs(\\)f(f)p Fv(u)27 b(>)h FB(9)p Fv(=)p FB(10)p Fs(g)2262
3892 y Fu(\021)2348 4003 y Fs(\025)45 b FB(const)17 b
Fv(r)2756 3962 y Ft(n)375 4265 y FB(and)1187 4459 y Fw(L)1258
4418 y Ft(n)1305 4349 y Fu(\020)1365 4459 y Fv(B)1439
4474 y Ft(r)1477 4459 y FB(\()p Fv(x)p FB(\))22 b Fs(\\)h(f)p
Fv(u)k(<)g Fs(\000)p FB(9)p Fv(=)p FB(10)p Fs(g)2278
4349 y Fu(\021)2365 4459 y Fs(\025)45 b FB(const)17 b
Fv(r)2773 4418 y Ft(n)2836 4459 y Fv(:)375 4694 y FB(Let)33
b Fv(a)f FB(b)s(e)h(the)g(a)m(v)m(erage)h(of)e Fv(u)g
FB(in)h Fv(B)1671 4709 y Ft(r)1709 4694 y FB(\()p Fv(x)p
FB(\).)44 b(If)32 b Fv(a)c Fs(\024)g FB(0,)454 4820 y
Fu(Z)509 5046 y Ft(B)562 5054 y Fj(r)597 5046 y Fr(\()p
Ft(x)p Fr(\))712 4956 y Fs(j)p Fv(u)22 b Fs(\000)g Fv(a)p
Fs(j)83 b(\025)1240 4820 y Fu(Z)1295 5046 y Ft(B)1348
5054 y Fj(r)1383 5046 y Fr(\()p Ft(x)p Fr(\))p Fq(\\f)p
Ft(u>)p Fr(9)p Ft(=)p Fr(10)p Fq(g)1836 4956 y FB(\()p
Fv(u)22 b Fs(\000)g Fv(a)p FB(\))28 b Fs(\025)2308 4888
y FB(9)p 2283 4933 98 4 v 2283 5024 a(10)2391 4956 y
Fw(L)2462 4915 y Ft(n)2509 4845 y Fu(\020)2569 4956 y
Fv(B)2643 4971 y Ft(r)2681 4956 y FB(\()p Fv(x)p FB(\))22
b Fs(\\)h(f)p Fv(u)k(>)g FB(9)p Fv(=)p FB(10)p Fs(g)3405
4845 y Fu(\021)3491 4956 y Fs(\025)1079 5175 y(\025)100
b FB(const)18 b Fv(r)1543 5134 y Ft(n)1606 5175 y Fv(:)p
eop
%%Page: 41 50
41 49 bop 3727 201 a FB(41)375 475 y(Analogously)-8 b(,)33
b(if)g Fv(a)28 b Fs(\025)g FB(0,)790 634 y Fu(Z)846 859
y Ft(B)899 867 y Fj(r)934 859 y Fr(\()p Ft(x)p Fr(\))1049
769 y Fs(j)p Fv(u)21 b Fs(\000)i Fv(a)p Fs(j)28 b(\025)1500
702 y FB(9)p 1476 746 98 4 v 1476 838 a(10)1583 769 y
Fw(L)1654 728 y Ft(n)1701 659 y Fu(\020)1761 769 y Fv(B)1835
784 y Ft(r)1873 769 y FB(\()p Fv(x)p FB(\))22 b Fs(\\)h(f)p
Fv(u)k(<)h Fs(\000)p FB(9)p Fv(=)p FB(10)p Fs(g)2675
659 y Fu(\021)2761 769 y Fs(\025)45 b FB(const)17 b Fv(r)3169
728 y Ft(n)3232 769 y Fv(:)375 1064 y FB(So,)33 b(in)f(an)m(y)i(case,)f
(using)g(P)m(oincar)m(\023)-46 b(e)34 b(and)f(Cauc)m(h)m(y)h
(Inequalities:)661 1359 y(const)17 b Fv(r)947 1318 y
Ft(n)p Fq(\000)p Fr(1)1167 1359 y Fs(\024)1337 1291 y
FB(1)p 1337 1336 49 4 v 1338 1427 a Fv(r)1413 1223 y
Fu(Z)1468 1449 y Ft(B)1521 1457 y Fj(r)1556 1449 y Fr(\()p
Ft(x)p Fr(\))1671 1359 y Fs(j)p Fv(u)22 b Fs(\000)g Fv(a)p
Fs(j)28 b(\024)45 b FB(const)2361 1223 y Fu(Z)2416 1449
y Ft(B)2469 1457 y Fj(r)2504 1449 y Fr(\()p Ft(x)p Fr(\))2619
1359 y Fs(jr)p Fv(u)p Fs(j)27 b(\024)1167 1676 y(\024)100
b FB(const)1583 1507 y Fu(s)p 1683 1507 492 4 v 33 x(Z)1738
1766 y Ft(B)1791 1774 y Fj(r)1826 1766 y Fr(\()p Ft(x)p
Fr(\))1942 1676 y Fs(jr)p Fv(u)p Fs(j)2137 1647 y Fr(2)2192
1526 y Fu(r)p 2291 1526 982 4 v 2291 1676 a Fw(L)2362
1647 y Ft(n)2409 1565 y Fu(\020)2469 1676 y Fv(B)2543
1691 y Ft(r)2581 1676 y FB(\()p Fv(x)p FB(\))22 b Fs(\\)h(fj)p
Fv(u)p Fs(j)j Fv(<)i FB(1)p Fs(g)3214 1565 y Fu(\021)3301
1676 y Fs(\024)1167 1970 y(\024)100 b FB(const)1583 1860
y Fu(\020)1660 1835 y(Z)1715 2060 y Ft(B)1768 2068 y
Fj(r)1803 2060 y Fr(\()p Ft(x)p Fr(\))1918 1970 y Fv(a)1969
1985 y Ft(i;j)2049 1970 y Fv(@)2100 1985 y Ft(i)2129
1970 y Fv(u@)2236 1985 y Ft(j)2273 1970 y Fv(u)21 b FB(+)h
Fv(Q\037)2586 1986 y Fr(\()p Fq(\000)p Fr(1)p Ft(;)p
Fr(1\))2791 1970 y FB(\()p Fv(u)p FB(\))2923 1860 y Fu(\021)2998
1970 y Fv(:)p 3604 2265 4 66 v 3608 2202 59 4 v 3608
2265 V 3666 2265 4 66 v 675 2576 a FB(Let)h(us)h(summarize)h(some)f(of)
f(the)h(densit)m(y)h(estimates)g(obtained)e(in)h(this)g(section:)375
2854 y Fz(Prop)s(osition)38 b(4.1.8.)49 b Fy(L)-5 b(et)39
b Fv(u)c Fs(2)h Fw(M)1754 2818 y Ft(!)1754 2880 y(M)1833
2854 y Fy(.)57 b(Assume)39 b(that)h Fv(x)c Fs(2)f(fj)p
Fv(u)p Fs(j)g Fv(<)g FB(1)p Fs(g)p Fy(.)57 b(Then,)39
b(ther)-5 b(e)375 3050 y(exist)29 b(p)-5 b(ositive)28
b(c)-5 b(onstants)29 b Fv(r)1408 3065 y Fr(0)1447 3050
y Fy(,)h Fv(r)1551 3065 y Fr(1)1590 3050 y Fy(,)g Fv(c)f
Fy(and)g Fv(C)7 b Fy(,)30 b(dep)-5 b(ending)27 b(only)i(on)f
Fv(n)i Fy(and)e(on)h(the)g(b)-5 b(ounds)375 3245 y(on)34
b Fv(a)565 3260 y Ft(i;j)681 3245 y Fy(and)g Fv(Q)p Fy(,)h(such)g
(that:)575 3539 y Fw(J)625 3555 y Ft(B)678 3563 y Fj(r)713
3555 y Fr(\()p Ft(x)p Fr(\))812 3539 y FB(\()p Fv(u)p
FB(\))27 b Fs(\025)h Fv(cr)1165 3498 y Ft(n)p Fq(\000)p
Fr(1)1318 3539 y Fv(;)575 3759 y Fw(L)646 3718 y Ft(n)693
3649 y Fu(\020)752 3759 y Fv(B)826 3774 y Ft(r)864 3759
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4079 y(M)3254 4054 y Fy(,)1689 4349 y Fw(J)1739 4364
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Fr(1)2334 4349 y Fv(;)375 4644 y Fy(for)35 b(any)f Fv(r)d
Fs(\025)d Fv(r)941 4659 y Fr(0)980 4644 y Fy(,)35 b(pr)-5
b(ovide)g(d)34 b Fv(B)1504 4659 y Ft(r)r Fr(+)p Ft(r)1625
4668 y Fg(1)1663 4644 y FB(\()p Fv(y)t FB(\))27 b Fs(\032)h
Fv(S)1989 4607 y Ft(!)1983 4669 y(M)2062 4644 y Fy(.)375
4955 y(Pr)-5 b(o)g(of.)48 b FB(Put)33 b(together)g(Theorem)h(4.1.1,)e
(Lemma)h(4.1.6)f(and)h(Lemma)g(4.1.7.)p 3604 4955 V 3608
4892 59 4 v 3608 4955 V 3666 4955 4 66 v eop
%%Page: 42 51
42 50 bop 3727 201 a FB(42)375 475 y FA(4.2)135 b(Densit)l(y)46
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y Fv(=)f Fs(\030)h FB(\(top)s(ologically)-8 b(,)2144
696 y(~)2126 721 y Fz(R)2210 685 y Ft(n)2296 721 y FB(is)40
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1178 y FB(:=)28 b Fs(f)p Fv(u)f Fs(2)h FB(H)2174 1137
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y(~)2304 1178 y Fz(R)2388 1137 y Ft(n)2435 1178 y FB(\))1022
1442 y(with)33 b Fv(u)p FB(\()p Fv(x)p FB(\))28 b Fs(\025)g
FB(99)p Fv(=)p FB(100)j(for)h(an)m(y)i Fv(x)44 b FB(s)p
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FB(0)k(and)1115 1677 y Fv(u)p FB(\()p Fv(x)p FB(\))27
b Fs(\024)h(\000)p FB(99)p Fv(=)p FB(100)k(for)g(an)m(y)h
Fv(x)45 b FB(s)p Fv(:)p FB(t)p Fv(:)g(!)25 b Fs(\001)d
Fv(x)28 b Fs(\025)g Fv(M)11 b Fs(g)17 b Fv(:)375 1912
y FB(W)-8 b(e)34 b(denote)g(b)m(y)h Fw(M)1094 1876 y
Ft(!)1094 1937 y(M)1173 1912 y FB(\()p Fw(G)p FB(\))e(the)h(set)h(of)e
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1912 y FB(for)f(the)h(functional)g Fw(G)p FB(.)47 b(First)375
2106 y(w)m(e)34 b(recall:)375 2357 y Fz(Theorem)k(4.2.1)g
(\(Ca\013arelli)g(and)h(C\023)-56 b(ordoba\).)49 b Fy(Assume)35
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2357 y FB(\()p Fw(G)p FB(\))p Fy(.)45 b(Then:)461 2647
y(\(i\))j(ther)-5 b(e)38 b(exist)f(p)-5 b(ositive)38
b(c)-5 b(onstants)37 b Fv(c)p Fy(,)i Fv(r)2035 2662 y
Fr(0)2074 2647 y Fy(,)f Fv(r)2186 2662 y Fr(1)2264 2647
y Fy(\(dep)-5 b(ending)36 b(only)i(on)f Fv(n)h Fy(and)f(on)h(the)619
2842 y(b)-5 b(ounds)34 b(on)h Fv(a)1131 2857 y Ft(i;j)1246
2842 y Fy(and)f Fv(F)14 b Fy(\))35 b(such)f(that)1748
3105 y Fw(G)1808 3120 y Ft(B)1861 3128 y Fj(r)1896 3120
y Fr(\()p Ft(x)p Fr(\))1995 3105 y FB(\()p Fv(u)p FB(\))27
b Fs(\024)h Fv(c)17 b(r)2365 3064 y Ft(n)p Fq(\000)p
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Fs(\025)d Fv(r)1185 3383 y Fr(0)1224 3368 y Fy(,)35 b(pr)-5
b(ovide)g(d)34 b Fv(B)1748 3383 y Ft(r)r Fr(+)p Ft(r)1869
3392 y Fg(1)1907 3368 y FB(\()p Fv(x)p FB(\))28 b Fs(\032)g
Fv(S)2237 3332 y Ft(!)2231 3393 y(M)2310 3368 y Fy(;)431
3637 y(\(ii\))48 b(for)36 b(any)h Fv(\022)1010 3652 y
Fr(0)1080 3637 y Fs(2)31 b FB([0)p Fv(;)17 b FB(1\))p
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3637 y FB(])37 b Fy(and)e Fv(\026)2257 3652 y Fr(0)2327
3637 y Fv(>)c FB(0)p Fy(,)36 b(if)h Fw(L)2717 3600 y
Ft(n)2764 3637 y FB(\()p Fv(B)2876 3652 y Ft(K)2944 3637
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Fr(0)717 3831 y Fy(,)41 b(ther)-5 b(e)40 b(exist)g(p)-5
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Ft(?)2139 3831 y Fy(,)i Fv(r)2254 3846 y Fr(0)2293 3831
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4025 y Fy(and)34 b(the)h(b)-5 b(ounds)34 b(on)h Fv(a)1482
4040 y Ft(i;j)1597 4025 y Fy(and)f Fv(F)14 b Fy(\))35
b(such)f(that)1508 4288 y Fw(L)1579 4247 y Ft(n)1626
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Ft(n)2759 4288 y Fv(;)619 4551 y Fy(for)41 b(any)g Fv(r)g
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4575 y Fg(1)1956 4551 y FB(\()p Fv(x)p FB(\))g Fs(\032)f
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Fs(g)p FB(\))h Fs(\025)g Fv(\026)1183 4761 y Fr(0)1222
4746 y Fy(,)1508 4940 y Fw(L)1579 4899 y Ft(n)1626 4829
y Fu(\020)1686 4940 y Fs(f)p Fv(u)f(<)h(\022)s Fs(g)22
b(\\)g Fv(B)2205 4955 y Ft(r)2243 4940 y FB(\()p Fv(x)p
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4899 y Ft(?)2648 4940 y Fv(r)2695 4899 y Ft(n)2759 4940
y Fv(;)619 5175 y Fy(for)35 b(any)f Fv(r)d Fs(\025)d
Fv(r)1185 5190 y Fr(0)1224 5175 y Fy(,)35 b(pr)-5 b(ovide)g(d)34
b Fv(B)1748 5190 y Ft(r)r Fr(+)p Ft(r)1869 5199 y Fg(1)1907
5175 y FB(\()p Fv(x)p FB(\))28 b Fs(\032)g Fv(S)2237
5139 y Ft(!)2231 5200 y(M)2310 5175 y Fy(.)p eop
%%Page: 43 52
43 51 bop 3727 201 a FB(43)675 475 y(This)34 b(statemen)m(t)h(w)m(as)f
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(in)g(the)375 668 y(case)k Fv(a)637 683 y Ft(i;j)753
668 y FB(=)e Fv(\016)908 683 y Ft(i;j)988 668 y FB(.)58
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b Fy(Fix)38 b Fv(\022)1405 1539 y Fr(0)1480 1524 y Fs(2)d
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b Fs(2)h Fw(M)2530 1488 y Ft(!)2530 1550 y(M)2609 1524
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1524 y Fv(;)17 b(\022)3578 1539 y Fr(0)3618 1524 y FB(])p
Fy(.)375 1718 y(Then,)35 b(ther)-5 b(e)36 b(exist)f(p)-5
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1718 y Fy(\(dep)-5 b(ending)34 b(only)h(on)h Fv(\022)3181
1733 y Fr(0)3220 1718 y Fy(,)g Fv(n)g Fy(and)f(on)375
1912 y(the)g(b)-5 b(ounds)34 b(on)h Fv(a)1049 1927 y
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y Fs(f)p Fv(u)27 b(>)h(\022)915 2174 y Fr(0)955 2159
y Fs(g)22 b(\\)g Fv(B)1189 2174 y Ft(r)1227 2159 y FB(\()p
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b Fv(cr)1640 2117 y Ft(n)1886 2159 y FB(and)199 b Fw(L)2313
2117 y Ft(n)2360 2048 y Fu(\020)2420 2159 y Fs(f)p Fv(u)27
b(<)h Fs(\000)p Fv(\022)2779 2174 y Fr(0)2819 2159 y
Fs(g)22 b(\\)g Fv(B)3053 2174 y Ft(r)3091 2159 y FB(\()p
Fv(x)p FB(\))3222 2048 y Fu(\021)3310 2159 y Fs(\025)28
b Fv(cr)3504 2117 y Ft(n)3567 2159 y Fv(;)375 2405 y
Fy(for)35 b(any)f Fv(r)d Fs(\025)d Fv(r)941 2420 y Fr(0)980
2405 y Fy(,)35 b(pr)-5 b(ovide)g(d)34 b Fv(B)1504 2420
y Ft(r)r Fr(+)p Ft(r)1625 2429 y Fg(1)1663 2405 y FB(\()p
Fv(x)p FB(\))28 b Fs(\032)g Fv(S)1993 2369 y Ft(!)1987
2430 y(M)2066 2405 y Fy(.)375 2698 y(Pr)-5 b(o)g(of.)48
b FB(De\014ne)31 b Fv(\022)1028 2662 y Fq(\003)1096 2698
y FB(:=)c(\(1)18 b(+)g Fv(\022)1470 2713 y Fr(0)1510
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Fv(u)f FB(is)h(H\177)-49 b(older)31 b(con)m(tin)m(uous)h(and)375
2892 y(\()p Fv(ii)p FB(\))h(of)f(Theorem)i(4.2.1,)449
3139 y Fw(L)520 3097 y Ft(n)567 3028 y Fu(\020)626 3139
y Fs(f)p Fv(u)27 b(>)h Fs(\000)p Fv(\022)988 3097 y Fq(\003)1028
3139 y Fs(g)22 b(\\)h Fv(B)1263 3154 y Ft(r)1301 3139
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b Fw(L)2378 3097 y Ft(n)2426 3028 y Fu(\020)2485 3139
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y Fs(\025)28 b Fv(cr)3494 3097 y Ft(n)3558 3139 y Fv(;)3474
3296 y FB(\(4.9\))375 3462 y(so)33 b(that)893 3709 y
Fw(L)964 3668 y Ft(n)1011 3598 y Fu(\020)1070 3709 y
Fs(f)p Fv(u)27 b(>)h Fs(\000)p Fv(\022)1432 3668 y Fq(\003)1472
3709 y Fs(g)22 b(\\)h Fv(B)1707 3724 y Ft(r)1745 3709
y FB(\()p Fv(x)p FB(\))1876 3598 y Fu(\021)1958 3709
y Fs(\000)f Fw(L)2128 3668 y Ft(n)2176 3598 y Fu(\020)2235
3709 y Fs(f)p Fv(u)27 b(>)h(\022)2517 3724 y Fr(0)2556
3709 y Fs(g)22 b(\\)h Fv(B)2791 3724 y Ft(r)2829 3709
y FB(\()p Fv(x)p FB(\))2960 3598 y Fu(\021)3048 3709
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y Fu(\020)1070 3942 y Fs(f\000)p Fv(\022)1245 3901 y
Fq(\003)1313 3942 y Fv(<)28 b(u)f Fs(\024)h Fv(\022)1650
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y Fv(F)30 b(;)375 4189 y FB(th)m(us,)k(b)m(y)f(\()p Fv(i)p
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b(\\)g Fv(B)1574 4450 y Ft(r)1612 4435 y FB(\()p Fv(x)p
FB(\))1743 4325 y Fu(\021)1825 4435 y Fs(\000)h Fw(L)1996
4394 y Ft(n)2043 4325 y Fu(\020)2103 4435 y Fs(f)p Fv(u)k(>)g(\022)2384
4450 y Fr(0)2424 4435 y Fs(g)22 b(\\)h Fv(B)2659 4450
y Ft(r)2697 4435 y FB(\()p Fv(x)p FB(\))2828 4325 y Fu(\021)2915
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Fr(1)3263 4435 y Fv(:)375 4682 y FB(F)-8 b(rom)32 b(this)h(and)g
(\(4.9\),)759 4928 y Fw(L)830 4887 y Ft(n)877 4818 y
Fu(\020)936 4928 y Fs(f)p Fv(u)27 b(>)h(\022)1218 4943
y Fr(0)1258 4928 y Fs(g)22 b(\\)g Fv(B)1492 4943 y Ft(r)1530
4928 y FB(\()p Fv(x)p FB(\))1661 4818 y Fu(\021)1749
4928 y Fs(\025)45 b FB(const)17 b Fv(r)2157 4887 y Ft(n)2226
4928 y Fs(\000)39 b FB(const)17 b Fv(r)2628 4887 y Ft(n)p
Fq(\000)p Fr(1)2793 4928 y Fs(\025)45 b FB(const)17 b
Fv(r)3201 4887 y Ft(n)3264 4928 y Fv(;)375 5175 y FB(if)32
b Fv(r)k FB(big)c(enough.)p 3604 5175 4 66 v 3608 5112
59 4 v 3608 5175 V 3666 5175 4 66 v eop
%%Page: 44 53
44 52 bop 3727 201 a FB(44)675 475 y(In)26 b(the)f(sequel,)k(follo)m
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670 y(ory)39 b(\(see,)j(for)d(instance)h([G1]\),)h(w)m(e)f(denote)g
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b FB(in)39 b(an)375 865 y(op)s(en)33 b(set)g(\012)g(as)769
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b(:=)f(sup)q Fs(f)1469 1024 y Fu(Z)1524 1249 y Ft(E)1617
1160 y FB(div)19 b Fv(')44 b FB(s)p Fv(:)p FB(t)p Fv(:)h(')27
b Fs(2)h FB(C)2306 1118 y Fr(1)2306 1184 y(0)2346 1160
y FB(\(\012)p Fv(;)17 b Fz(R)2582 1118 y Ft(n)2629 1160
y FB(\))g Fv(;)44 b Fs(j)p Fv(')p FB(\()p Fv(x)p FB(\))p
Fs(j)27 b(\024)h FB(1)p Fs(g)17 b Fv(:)675 1496 y FB(The)34
b(follo)m(wing)e(result)i(complete)g(the)f(densit)m(y)h(estimates)h(w)m
(e)e(need)h(here:)375 1774 y Fz(Lemma)39 b(4.2.3.)49
b Fy(Fix)44 b Fv(\022)1310 1789 y Fr(0)1395 1774 y Fs(2)h
FB([0)p Fv(;)17 b FB(1\))p Fy(.)73 b(Assume)44 b Fv(u)g
Fs(2)i Fw(M)2499 1738 y Ft(!)2499 1799 y(M)2578 1774
y FB(\()p Fw(G)p FB(\))e Fy(and)g Fv(u)p FB(\()p Fv(x)p
FB(\))g Fs(2)i FB([)p Fs(\000)p Fv(\022)3449 1789 y Fr(0)3489
1774 y Fv(;)17 b(\022)3578 1789 y Fr(0)3618 1774 y FB(])p
Fy(.)375 1969 y(Then,)35 b(ther)-5 b(e)36 b(exist)f(p)-5
b(ositive)35 b(c)-5 b(onstants)36 b Fv(c)p Fy(,)f Fv(r)2064
1984 y Fr(0)2104 1969 y Fy(,)h Fv(r)2214 1984 y Fr(1)2289
1969 y Fy(\(dep)-5 b(ending)34 b(only)h(on)h Fv(\022)3181
1984 y Fr(0)3220 1969 y Fy(,)g Fv(n)g Fy(and)f(on)375
2164 y(the)g(b)-5 b(ounds)34 b(on)h Fv(a)1049 2179 y
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%%Page: 45 54
45 53 bop 3727 201 a FB(45)375 475 y(instance,)34 b([G1],)e(p.)44
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%%Page: 46 55
46 54 bop 3727 201 a FB(46)375 475 y Fy(for)35 b(any)f
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%%Page: 47 56
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1745 y(1)p 2125 1789 501 4 v 2125 1881 a(inf)2244 1896
y Ft(u)p Fq(2)p Fr([)p Ft(\022)2387 1877 y Fj(?)2422
1896 y Ft(;\022)2475 1905 y Fg(0)2509 1896 y Fr(])2549
1881 y Fv(F)2653 1677 y Fu(Z)2708 1902 y Ft(B)2761 1910
y Fj(r)2817 1812 y Fv(F)41 b Fs(\024)1040 2061 y(\024)84
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3123 y Fz(Case)38 b(1:)44 b FB(if)1131 3087 y Fr(1)1203
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3433 y Ft(r)2130 3418 y Fs(\\)23 b(f)p Fv(u)k(>)g(\022)s
Fs(g)p FB(\))17 b Fv(;)1399 3638 y Fw(A)p FB(\()p Fv(r)s
FB(\))82 b(=)1840 3502 y Fu(Z)1895 3728 y Ft(B)1948 3736
y Fj(r)1983 3728 y Fq(\\f)p Ft(u)p Fq(\024)p Ft(\022)r
Fq(g)2252 3638 y Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))g
Fv(:)375 3933 y FB(W)-8 b(e)33 b(w)m(an)m(t)g(to)g(pro)m(v)m(e)g(that)
1384 4227 y Fv(C)1461 4117 y Fu(\020)1521 4227 y FB(\()p
Fw(V)p FB(\()p Fv(r)25 b Fs(\000)d FB(1\)\))1966 4158
y Fj(n)p Fi(\000)p Fg(1)p 1966 4171 117 3 v 2005 4212
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FB(1\))17 b Fv(:)516 b FB(\(4.10\))p 375 4615 1320 4
v 487 4676 a Fp(1)524 4707 y Fo(An)24 b(easy)e(mo)r(di\014cation)h(of)g
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(also)i(to)g(consider)f(b)r(ounded)375 4806 y(measurable)40
b(\(instead)h(of)g(Lipsc)n(hitz\))g(co)r(e\016cien)n(ts.)77
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%%Page: 48 57
48 56 bop 3727 201 a FB(48)675 475 y(Indeed,)31 b(if)d(\(4.10\))f
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b(e)35 b(will)375 1535 y(denote)41 b(b)m(y)g Fv(C)47
b FB(suitable)41 b(constan)m(ts)h(and)e(b)m(y)h Fv(A)f
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1730 y(the)37 b(sequel.)56 b(Let)36 b Fv(\033)i FB(=)c(min)q
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b(+)h Fv(\022)s Fs(g)36 b FB(Applying)h(Sob)s(olev)375
1924 y(Inequalit)m(y)k(to)e Fv(\014)1029 1888 y Fr(2)1107
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g(that)g Fs(r)p Fv(\014)44 b FB(v)-5 b(anishes)375 2119
y(outside)34 b Fs(f)p Fv(u)21 b Fs(\000)i Fv(\033)31
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2400 y Fv(\014)1054 2329 y Fg(2)p Fj(n)p 1030 2341 117
3 v 1030 2382 a(n)p Fi(\000)p Fg(1)1161 2290 y Fu(\021)1230
2284 y Fj(n)p Fi(\000)p Fg(1)p 1230 2297 V 1270 2338
a Fj(n)1389 2400 y Fs(\024)559 2667 y(\024)83 b Fv(C)813
2532 y Fu(Z)868 2757 y Ft(B)921 2765 y Fj(r)956 2757
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b(jr)p Fv(\014)6 b Fs(j)25 b(\024)559 2943 y(\024)83
b Fv(C)7 b(A)869 2832 y Fu(\020)946 2807 y(Z)1001 3032
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2958 y Ft(i)1778 2943 y Fv(u@)1885 2958 y Ft(j)1921 2943
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2958 y Ft(j)2457 2943 y Fv(\033)j Fs(\000)d FB(2)p Fv(a)2738
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2943 y FB(\()p Fv(u)f Fs(\000)h Fv(\033)t FB(\))p Fv(@)3261
2958 y Ft(j)3298 2943 y Fv(\033)t FB(\))3395 2832 y Fu(\021)3476
2943 y FB(+)719 3218 y(+)805 3150 y Fv(C)p 805 3195 77
4 v 807 3286 a(A)909 3082 y Fu(Z)964 3308 y Ft(B)1017
3316 y Fj(r)1052 3308 y Fq(\\f)p Ft(u)p Fq(\000)p Ft(\033)r
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3218 y Fv(:)1539 b FB(\(4.11\))375 3499 y(In)m(tegration)33
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3644 y(Z)998 3870 y Ft(B)1051 3878 y Fj(r)1106 3780 y
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3762 a(n)p Fi(\000)p Fg(1)1308 3669 y Fu(\021)1377 3664
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Fq(\\f)p Ft(u)p Fq(\000)p Ft(\033)r Fq(\024)p Fr(1+)p
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4337 y Ft(i;j)1798 4322 y Fv(@)1854 4281 y Fr(2)1849
4347 y Ft(i;j)1930 4322 y Fv(\033)g FB(\()p Fv(u)g Fs(\000)i
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4322 y Fv(a)2607 4337 y Ft(i;j)2687 4322 y FB(\))p Fv(@)2776
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%%Page: 49 58
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1130 180 4 v 2261 1221 a FB(2)2310 1192 y Fr(2)p Fq(\000)p
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3800 y Fr(2)1981 3841 y FB(+)2089 3773 y Fv(C)p 2089
3818 V 2091 3909 a(A)2176 3841 y Fw(L)2247 3800 y Ft(n)2294
3841 y FB(\()p Fs(f)p Fv(u)k(>)g(\022)s Fs(g)c(\\)f FB(\()p
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%%Page: 50 59
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Fq(\000)p Fr(1)2620 475 y FB(,)942 769 y Fs(jr)p Fv(h)p
Fs(j)g FB(+)g Fs(j)p Fv(D)1369 728 y Fr(2)1408 769 y
Fv(h)p Fs(j)27 b(\024)h Fv(C)7 b FB(\()p Fv(\032)23 b
Fs(\000)f Fv(r)j FB(+)d(1\))2175 688 y Fg(2\()p Fj(d)2261
667 y Fi(0)2284 688 y(\000)p Fg(1\))p 2175 707 212 3
v 2214 752 a(2)p Fi(\000)p Fj(d)2324 738 y Fi(0)2428
769 y Fs(\024)28 b Fv(C)7 b FB(\()p Fv(h)22 b FB(+)g(1\))2911
728 y Ft(d)2947 705 y Fi(0)2970 728 y Fq(\000)p Fr(1)3081
769 y Fv(:)375 1064 y FB(Also,)33 b(in)g(the)g(set)g
Fs(f)p Fv(u)27 b(>)h(\033)t Fs(g)k FB(w)m(e)i(ha)m(v)m(e)g
Fv(\033)d FB(=)d Fv(h)33 b FB(and)f(then)h(the)g(last)g(form)m(ula)g
(implies)876 1359 y(\()p Fs(jr)p Fv(h)p Fs(j)22 b FB(+)g
Fs(j)p Fv(D)1341 1318 y Fr(2)1379 1359 y Fv(h)p Fs(j)p
FB(\))g Fs(\001)g FB(\()p Fv(u)g Fs(\000)g Fv(\033)t
FB(\))83 b Fs(\024)h Fv(C)7 b FB(\()p Fv(h)22 b FB(+)g(1\))2507
1318 y Ft(d)2543 1294 y Fi(0)2566 1318 y Fq(\000)p Fr(1)2682
1359 y Fs(\001)g FB(\()p Fv(u)f Fs(\000)i Fv(h)p FB(\))28
b Fs(\024)1968 1579 y(\024)84 b Fv(C)23 b FB(\()p Fv(u)f
FB(+)g(1\))2523 1538 y Ft(d)2559 1514 y Fi(0)2602 1579
y Fv(;)797 b FB(\(4.14\))375 1874 y(in)33 b(\()p Fv(B)601
1889 y Ft(r)661 1874 y Fs(\000)23 b Fv(B)835 1889 y Ft(r)r
Fq(\000)p Fr(1)963 1874 y FB(\))f Fs(\\)g(f)p Fv(u)27
b(>)h(\033)t Fs(g)p FB(.)43 b(Therefore,)34 b(recalling)f(that)g
Fv(\033)e FB(=)d Fs(\000)p FB(1)33 b(in)f Fv(B)3134 1889
y Ft(r)r Fq(\000)p Fr(1)3263 1874 y FB(,)g(splitting)375
2069 y(the)f(domain)g(of)f(in)m(tegration)g(in)m(to)h(the)g(sets)g
Fs(f)p Fv(u)d Fs(\024)g Fv(\022)s Fs(g)i FB(and)h Fs(f)p
Fv(u)c(>)g(\022)s Fs(g)p FB(,)k(using)g(\(4.14\))f(and)375
2264 y(our)i(h)m(yp)s(otheses)j(on)e Fv(F)14 b FB(,)32
b(w)m(e)i(get)729 2423 y Fu(Z)784 2648 y Ft(B)837 2656
y Fj(r)893 2448 y Fu(\020)953 2558 y Fv(a)1004 2573 y
Ft(i;j)1084 2558 y Fv(@)1140 2517 y Fr(2)1135 2583 y
Ft(i;j)1216 2558 y Fv(\033)20 b FB(\()p Fv(u)i Fs(\000)g
Fv(\033)t FB(\))g(+)h(\()p Fv(@)1813 2573 y Ft(i)1841
2558 y Fv(a)1892 2573 y Ft(i;j)1972 2558 y FB(\))p Fv(@)2061
2573 y Ft(j)2098 2558 y Fv(\033)e FB(\()p Fv(u)h Fs(\000)g
Fv(\033)t FB(\))2486 2448 y Fu(\021)2573 2558 y FB(=)729
2826 y(=)833 2690 y Fu(Z)888 2915 y Fr(\()p Ft(B)968
2923 y Fj(r)1003 2915 y Fq(\000)p Ft(B)1111 2924 y Fj(r)r
Fi(\000)p Fg(1)1224 2915 y Fr(\))p Fq(\\f)p Ft(u>\033)r
Fq(g)1529 2715 y Fu(\020)1588 2826 y Fv(a)1639 2841 y
Ft(i;j)1720 2826 y Fv(@)1776 2784 y Fr(2)1771 2850 y
Ft(i;j)1851 2826 y Fv(\033)f FB(\()p Fv(u)g Fs(\000)i
Fv(\033)t FB(\))f(+)g(\()p Fv(@)2448 2841 y Ft(i)2477
2826 y Fv(a)2528 2841 y Ft(i;j)2608 2826 y FB(\))p Fv(@)2697
2841 y Ft(j)2734 2826 y Fv(\033)f FB(\()p Fv(u)g Fs(\000)i
Fv(\033)t FB(\))3122 2715 y Fu(\021)3209 2826 y Fs(\024)729
3101 y(\024)28 b Fv(C)7 b Fw(L)982 3060 y Ft(n)1029 2990
y Fu(\020)1089 3101 y FB(\()p Fv(B)1201 3116 y Ft(r)1261
3101 y Fs(\000)23 b Fv(B)1435 3116 y Ft(r)r Fq(\000)p
Fr(1)1563 3101 y FB(\))f Fs(\\)h(f)p Fv(u)k(>)g(\022)s
Fs(g)2046 2990 y Fu(\021)2128 3101 y FB(+)22 b Fv(C)2319
2965 y Fu(Z)2375 3191 y Fr(\()p Ft(B)2455 3199 y Fj(r)2490
3191 y Fq(\000)p Ft(B)2598 3200 y Fj(r)r Fi(\000)p Fg(1)2711
3191 y Fr(\))p Fq(\\f)p Ft(u)p Fq(\024)p Ft(\022)r Fq(g)2991
3101 y FB(\()p Fv(u)g FB(+)g(1\))3292 3060 y Ft(d)3328
3036 y Fi(0)3382 3101 y Fs(\024)729 3376 y(\024)28 b
Fv(C)7 b Fw(L)982 3335 y Ft(n)1029 3266 y Fu(\020)1089
3376 y FB(\()p Fv(B)1201 3391 y Ft(r)1261 3376 y Fs(\000)23
b Fv(B)1435 3391 y Ft(r)r Fq(\000)p Fr(1)1563 3376 y
FB(\))f Fs(\\)h(f)p Fv(u)k(>)g(\022)s Fs(g)2046 3266
y Fu(\021)2128 3376 y FB(+)22 b Fv(C)2319 3241 y Fu(Z)2375
3466 y Fr(\()p Ft(B)2455 3474 y Fj(r)2490 3466 y Fq(\000)p
Ft(B)2598 3475 y Fj(r)r Fi(\000)p Fg(1)2711 3466 y Fr(\))p
Fq(\\f)p Ft(u)p Fq(\024)p Ft(\022)r Fq(g)3008 3376 y
Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))g Fv(:)375 3671 y
FB(Collecting)43 b(all)f(these)h(estimates,)j(the)c(pro)s(of)f(of)g
(\(4.10\))h(follo)m(ws.)72 b(This)43 b(\014nishes)h(the)375
3866 y(pro)s(of)32 b(of)g(\(ii\))g(in)h(the)g(\014rst)g(case.)675
4153 y Fz(Case)44 b(2:)53 b FB(if)37 b Fv(d)f FB(=)g(2.)57
b(W)-8 b(e)38 b(use)h(suitable)f(p)s(ositiv)m(e)h(parameters)f(\002)g
(and)f Fv(T)14 b FB(:)53 b(w)m(e)375 4348 y(will)33 b(\014x)g(\002)g
(small)g(enough)g(and)g(then)g(c)m(ho)s(ose)g Fv(T)47
b FB(suitably)33 b(large.)675 4584 y(Let)39 b Fv(k)j
Fs(2)c Fz(N)p FB(.)63 b(W)-8 b(e)39 b(use)h(a)e(barrier)h(function)h
Fv(h)e FB(=)h Fv(h)2647 4599 y Ft(k)2728 4584 y Fs(2)g
FB(C)2903 4548 y Fr(2)2943 4584 y FB(\()p Fv(B)3055 4600
y Fr(\()p Ft(k)r Fr(+1\))p Ft(T)3293 4584 y FB(\))g(so)g(that)375
4779 y Fs(\000)p FB(1)28 b Fs(\024)g Fv(h)g Fs(\024)g
FB(1,)k Fv(h)c FB(=)g(1)k(on)h Fv(@)5 b(B)1466 4795 y
Fr(\()p Ft(k)r Fr(+1\))p Ft(T)1705 4779 y FB(,)1075 5074
y(\()p Fv(h)22 b FB(+)g(1\))g(+)g Fs(jr)p Fv(h)p Fs(j)f
FB(+)h Fs(j)p Fv(D)1922 5033 y Fr(2)1961 5074 y Fv(h)p
Fs(j)28 b(\024)g Fv(C)7 b FB(\()p Fv(h)22 b FB(+)g(1\))27
b Fs(\024)i Fv(C)7 b(e)2811 5033 y Fq(\000)p Fr(\002)p
Ft(T)3426 5074 y FB(\(4.15\))p eop
%%Page: 51 60
51 59 bop 3727 201 a FB(51)375 475 y(in)33 b Fv(B)563
490 y Ft(k)r(T)656 475 y FB(,)g(and)1457 769 y Fs(jr)p
Fv(h)p Fs(j)22 b FB(+)g Fs(j)p Fv(D)1884 728 y Fr(2)1923
769 y Fv(h)p Fs(j)27 b(\024)h Fv(C)7 b FB(\002\()p Fv(h)22
b FB(+)g(1\))833 b(\(4.16\))375 1064 y(in)32 b Fv(B)562
1080 y Fr(\()p Ft(k)r Fr(+1\))p Ft(T)800 1064 y FB(.)44
b(F)-8 b(rom)31 b(the)g(last)h(form)m(ula)g(and)f(our)h(assumptions)h
(on)e Fv(F)14 b FB(,)32 b(w)m(e)g(deduce,)i(if)d(\002)375
1259 y(is)i(suitably)h(small,)1435 1554 y Fs(jr)p Fv(h)p
Fs(j)21 b FB(+)h Fs(j)p Fv(D)1861 1513 y Fr(2)1900 1554
y Fv(h)p Fs(j)28 b(\024)2117 1465 y(p)p 2200 1465 76
4 v 89 x FB(\002)p Fv(F)2339 1569 y Ft(u)2384 1554 y
FB(\()p Fv(x;)17 b(h)p FB(\))811 b(\(4.17\))375 1849
y(in)34 b Fv(B)564 1864 y Fr(\()p Ft(k)r Fr(+1\))p Ft(T)826
1849 y Fs(\\)24 b(f)p Fv(u)29 b Fs(\024)h Fv(\022)s Fs(g)p
FB(.)48 b(W)-8 b(e)35 b(p)s(ostp)s(one)f(the)g(explicit)i(construction)
f(of)e(suc)m(h)j Fv(h)e FB(to)g(the)375 2044 y(end)d(of)e(the)h(pro)s
(of.)42 b(De\014ne)30 b Fv(\022)1469 2008 y Fq(0)1520
2044 y FB(=)e Fv(\022)20 b Fs(\000)d Fv(C)7 b(e)1905
2008 y Fq(\000)p Fr(\002)p Ft(T)2070 2044 y FB(;)31 b(if)e
Fv(T)44 b FB(is)30 b(large)g(enough,)h Fv(\022)3055 2008
y Fq(0)3106 2044 y Fv(>)d Fs(\000)p FB(1.)42 b(De\014ne)375
2239 y(also)1039 2434 y Fv(\033)31 b FB(=)d(min)p Fs(f)p
Fv(u;)17 b(h)p Fs(g)97 b FB(and)h Fv(\014)33 b FB(=)28
b(min)p Fs(f)p Fv(u)22 b Fs(\000)g Fv(\033)n(;)17 b FB(1)22
b(+)g Fv(\022)2894 2393 y Fq(0)2918 2434 y Fs(g)17 b
Fv(:)375 2683 y FB(As)33 b(in)g(\(4.11\){\(4.12\),)e(w)m(e)j(ha)m(v)m
(e)f(that)805 2867 y Fu(\020)881 2842 y(Z)937 3068 y
Ft(B)990 3082 y Fg(\()p Fj(k)q Fg(+1\))p Fj(T)1218 2978
y Fv(\014)1313 2907 y Fg(2)p Fj(n)p 1289 2919 117 3 v
1289 2960 a(n)p Fi(\000)p Fg(1)1420 2867 y Fu(\021)1489
2861 y Fj(n)p Fi(\000)p Fg(1)p 1489 2874 V 1529 2916
a Fj(n)1648 2978 y Fs(\024)645 3263 y(\024)83 b Fv(C)7
b(A)955 3153 y Fu(h)1019 3128 y(Z)1074 3353 y Ft(B)1127
3367 y Fg(\()p Fj(k)q Fg(+1\))p Fj(T)1335 3353 y Fq(\\f)p
Ft(u)p Fq(\000)p Ft(\033)r Fq(\024)p Fr(1+)p Ft(\022)1735
3334 y Fi(0)1758 3353 y Fq(g)1814 3153 y Fu(\020)1874
3263 y Fv(F)14 b FB(\()p Fv(x;)j(\033)t FB(\))22 b Fs(\000)g
Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))22 b(+)g Fv(a)2785
3278 y Ft(i;j)2865 3263 y Fv(@)2921 3222 y Fr(2)2916
3288 y Ft(i;j)2997 3263 y Fv(\033)t FB(\()p Fv(u)f Fs(\000)i
Fv(\033)t FB(\))f(+)805 3528 y(+)p Fv(@)932 3543 y Ft(i)961
3528 y Fv(a)1012 3543 y Ft(i;j)1092 3528 y Fv(@)1143
3543 y Ft(j)1180 3528 y Fv(\033)t FB(\()p Fv(u)g Fs(\000)h
Fv(\033)t FB(\))1552 3418 y Fu(\021i)1680 3528 y FB(+)805
3762 y(+)891 3694 y Fv(C)p 891 3739 77 4 v 893 3830 a(A)995
3626 y Fu(Z)1050 3852 y Ft(B)1103 3866 y Fg(\()p Fj(k)q
Fg(+1\))p Fj(T)1311 3852 y Fq(\\f)p Ft(u)p Fq(\000)p
Ft(\033)r Fq(\024)p Fr(1+)p Ft(\022)1711 3833 y Fi(0)1734
3852 y Fq(g)1773 3762 y FB(\()p Fv(u)f Fs(\000)h Fv(\033)t
FB(\))2086 3721 y Fr(2)2142 3762 y Fv(;)1257 b FB(\(4.18\))375
4056 y(where)35 b Fv(A)g FB(is)g(a)f(free)g(parameter,)h(to)f(b)s(e)h
(c)m(hosen)h(suitably)f(large)f(in)h(what)f(follo)m(ws.)50
b(W)-8 b(e)375 4251 y(no)m(w)33 b(estimate)h(the)g(left)f(hand)g(side)h
(of)e(\(4.18\).)44 b(F)-8 b(rom)32 b(\(4.15\),)g(if)h
Fv(T)46 b FB(is)34 b(big)e(enough,)i(w)m(e)375 4447 y(get)29
b Fv(\022)19 b Fs(\000)c Fv(h)29 b Fs(\025)f FB(\(1)15
b Fs(\000)g Fv(\022)1119 4462 y Fr(0)1160 4447 y FB(\))p
Fv(=)p FB(2)29 b(in)g Fv(B)1509 4462 y Ft(k)r(T)1603
4447 y FB(.)42 b(Th)m(us,)32 b(the)e(left)f(hand)h(side)g(of)f
(\(4.18\))g(is)h(bigger)f(than)375 4642 y Fv(C)7 b Fw(V)p
FB(\()p Fv(k)s(T)14 b FB(\))729 4577 y Fj(n)p Fi(\000)p
Fg(1)p 729 4590 117 3 v 768 4631 a Fj(n)860 4642 y FB(.)675
4878 y(Let)47 b(us)f(no)m(w)h(estimate)h(the)f(righ)m(t)f(hand)h(side)g
(of)f(\(4.18\).)84 b(First)46 b(of)g(all,)k(w)m(e)375
5073 y(consider)40 b(the)f(con)m(tribution)h(in)f Fs(f)p
Fv(u)e Fs(\024)i Fv(\022)s Fs(g)p FB(.)62 b(Let)39 b(us)g(observ)m(e)i
(that,)f(since)g Fs(\000)p FB(1)f Fs(\024)f Fv(\033)43
b Fs(\024)p eop
%%Page: 52 61
52 60 bop 3727 201 a FB(52)375 475 y Fv(u)27 b Fs(\024)h
FB(1,)1249 757 y(\()p Fv(u)22 b FB(+)g(1\))1550 716 y
Fr(2)1611 757 y Fs(\000)h FB(\()p Fv(\033)j FB(+)c(1\))2015
716 y Fr(2)2076 757 y Fs(\000)2186 690 y FB(1)p 2186
734 49 4 v 2186 826 a(2)2244 757 y(\()p Fv(u)g Fs(\000)h
Fv(\033)t FB(\))2557 716 y Fr(2)2624 757 y FB(=)1090
990 y(=)83 b(\()p Fv(u)22 b FB(+)g Fv(\033)t FB(\))17
b(\()p Fv(u)k Fs(\000)h Fv(\033)t FB(\))h(+)f(2\()p Fv(u)f
Fs(\000)i Fv(\033)t FB(\))f Fs(\000)2501 923 y FB(1)p
2501 968 V 2501 1059 a(2)2560 990 y(\()p Fv(u)g Fs(\000)g
Fv(\033)t FB(\))2872 949 y Fr(2)2939 990 y FB(=)1090
1224 y(=)83 b(\()p Fv(u)22 b Fs(\000)g Fv(\033)t FB(\))17
b(\()1626 1156 y(1)p 1626 1201 V 1626 1292 a(2)1685 1224
y Fv(u)k FB(+)1870 1156 y(3)p 1870 1201 V 1870 1292 a(2)1929
1224 y Fv(\033)26 b FB(+)c(2\))27 b Fs(\025)i FB(0)17
b(;)375 1506 y(accordingly)-8 b(,)34 b(in)e Fs(f)p Fv(\033)g(<)27
b(u)h Fs(\024)g Fv(\022)s Fs(g)p FB(,)1112 1789 y Fv(F)14
b FB(\()p Fv(x;)j(u)p FB(\))k Fs(\000)i Fv(F)14 b FB(\()p
Fv(x;)j(\033)t FB(\))27 b(=)1983 1653 y Fu(Z)2082 1680
y Ft(u)2038 1879 y(h)2144 1789 y Fv(F)2207 1804 y Ft(u)2252
1789 y FB(\()p Fv(x;)17 b(\020)8 b FB(\))17 b Fv(d\020)33
b Fs(\025)1112 2050 y(\025)28 b Fv(C)1310 1914 y Fu(Z)1410
1941 y Ft(u)1366 2140 y(h)1455 2050 y FB(\()p Fv(\020)h
FB(+)22 b(1\))17 b Fv(d\020)34 b FB(=)28 b Fv(C)2076
1939 y Fu(h)2123 2050 y FB(\()p Fv(u)21 b FB(+)h(1\))2423
2009 y Fr(2)2484 2050 y Fs(\000)h FB(\()p Fv(h)f FB(+)g(1\))2885
2009 y Fr(2)2924 1939 y Fu(i)2999 2050 y Fs(\025)1112
2270 y(\025)28 b Fv(C)7 b FB(\()p Fv(u)21 b Fs(\000)i
Fv(h)p FB(\))1603 2229 y Fr(2)1659 2270 y Fv(:)1740 b
FB(\(4.19\))375 2552 y(Consequen)m(tly)-8 b(,)34 b(c)m(ho)s(osing)e
Fv(A)f FB(suitably)g(large)g(and)g(recalling)g(\(4.17\),)g(the)g(con)m
(tribution)375 2747 y(of)h(the)h(righ)m(t)g(hand)g(side)g(of)f
(\(4.18\))g(in)h Fs(f)p Fv(u)27 b Fs(\024)h Fv(\022)s
Fs(g)33 b FB(is)g(con)m(trolled)g(b)m(y)603 2894 y Fu(Z)658
3120 y Ft(B)711 3134 y Fg(\()p Fj(k)q Fg(+1\))p Fj(T)919
3120 y Fq(\\f)p Ft(\033)r(<u)p Fq(\024)p Ft(\022)r Fq(g)1285
2919 y Fu(\020)1345 3030 y Fv(F)14 b FB(\()p Fv(x;)j(\033)t
FB(\))22 b Fs(\000)g Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))22
b(+)g Fv(C)2282 2941 y Fs(p)p 2365 2941 76 4 v 89 x FB(\002)p
Fv(F)2504 3045 y Ft(u)2549 3030 y FB(\()p Fv(x;)17 b(\033)t
FB(\)\()p Fv(u)k Fs(\000)i Fv(\033)t FB(\))3095 2919
y Fu(\021)3171 3030 y Fv(:)228 b FB(\(4.20\))375 3312
y(W)-8 b(e)38 b(no)m(w)g(sho)m(w)g(that)g(this)g(quan)m(tit)m(y)h(is)f
(indeed)g(negativ)m(e.)59 b(Since)39 b(w)m(e)f(ruled)h(out)e
Fv(\022)j FB(to)375 3507 y(b)s(e)g(close)g(to)f Fs(\000)p
FB(1,)j(w)m(e)e(ha)m(v)m(e)h(that)e Fv(F)53 b FB(and)40
b Fv(F)2052 3522 y Ft(u)2136 3507 y FB(are)g(monotone)f(in)h(\()p
Fs(\000)p FB(1)p Fv(;)17 b(\022)s FB(\).)64 b(Hence,)43
b(in)375 3702 y Fs(f)p Fv(\033)32 b(<)27 b(u)g Fs(\024)h
Fv(\022)s Fs(g)p FB(,)33 b Fv(F)14 b FB(\()p Fv(x;)j(\033)t
FB(\))22 b Fs(\000)g Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))32
b(is)h(negativ)m(e)h(and,)f(furthermore,)1184 3985 y
Fs(j)p Fv(F)1275 4000 y Ft(u)1320 3985 y FB(\()p Fv(x;)17
b(\033)t FB(\)\()p Fv(u)22 b Fs(\000)g Fv(\033)t FB(\))p
Fs(j)28 b(\024)g(j)p Fv(F)14 b FB(\()p Fv(x;)j(\033)t
FB(\))k Fs(\000)i Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))p
Fs(j)g Fv(:)375 4267 y FB(Since)32 b(w)m(e)g(assumed)g(\002)f(to)f(b)s
(e)h(small,)h(w)m(e)f(ha)m(v)m(e)i(sho)m(wn)f(that)e(the)h(quan)m(tit)m
(y)i(in)d(\(4.20\))g(is)375 4462 y(negativ)m(e.)675 4698
y(Let)36 b(us)h(no)m(w)g(b)s(ound)f(the)g(righ)m(t)h(hand)f(side)h(of)f
(\(4)p Fv(:)p FB(18\))f(in)h Fs(f)p Fv(u)d(>)h(\022)s
Fs(g)p FB(.)54 b(First,)37 b(w)m(e)375 4892 y(notice)c(that)g(this)g
(term)g(has)g(no)f(con)m(tribution)i(in)f Fv(B)2350 4907
y Ft(k)r(T)2443 4892 y FB(:)44 b(indeed,)34 b(from)e(\(4.15\),)946
5175 y Fv(B)1020 5190 y Ft(k)r(T)1136 5175 y Fs(\\)23
b(f)p Fv(\033)31 b(<)d(u)f Fs(\024)h Fv(\033)e FB(+)c(1)g(+)g
Fv(\022)2049 5134 y Fq(0)2073 5175 y Fs(g)27 b(\022)h
Fv(B)2329 5190 y Ft(k)r(T)2445 5175 y Fs(\\)23 b(f)p
Fv(\033)31 b(<)d(u)f Fs(\024)h Fv(\022)s Fs(g)17 b Fv(:)p
eop
%%Page: 53 62
53 61 bop 3727 201 a FB(53)375 475 y(Th)m(us,)34 b(the)f(righ)m(t)g
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Fv(F)14 b FB(\()p Fv(x;)j(u)p FB(\))k(+)h(\()p Fv(\033)k
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a Fj(n)1794 1897 y Fs(\024)29 b Fw(V)p FB(\(\()p Fv(k)c
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b(Notice)30 b(that)2836 2081 y Fu(P)2942 2185 y Fr(1)p
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Ft(j)1973 2544 y FB(\))2021 2475 y Fj(n)p Fi(\000)p Fg(1)p
2021 2488 V 2060 2529 a Fj(n)2179 2544 y Fs(\024)29 b
Fv(\013)2347 2559 y Ft(k)r Fr(+1)2496 2544 y Fv(:)375
2811 y FB(By)36 b(induction)h(o)m(v)m(er)f Fv(k)s FB(,)h(one)e(easily)i
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Fr(+)2088 3623 y Fg(15)p 2087 3635 V 2103 3676 a(8)2158
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b(\(4.22\))p eop
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54 62 bop 3727 201 a FB(54)375 475 y(in)33 b([0)p Fv(;)17
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eop
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eop
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%%Page: 60 69
60 68 bop 3727 201 a FB(60)375 475 y(W)-8 b(e)36 b(will)g(denote)g(b)m
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1101 y(W)d(e)33 b(also)g(set)1176 1396 y Fw(X)1247 1355
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(\(Alt)e(and)i(Ca\013arelli\).)51 b Fy(F)-7 b(or)28 b(any)g
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(order)g(to)f(restrict)i(ourselv)m(es)375 3609 y(to)e(a)g(compact)h
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s(ed)g(in)g(Section)g(5.6,)f(b)m(y)i(an)e(appro)m(ximation)i(argumen)m
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61 69 bop 3727 201 a FB(61)675 475 y(It)38 b(is)h(a)f(standard)h
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(that,)f(indeed,)j Fv(u)37 b Fs(2)375 670 y FB(H)448
634 y Fr(1)448 696 y(lo)r(c)540 670 y FB(\()p Fz(R)662
634 y Ft(n)709 670 y FB(\))24 b(\(see,)k(for)c(instance)i([S]\).)f(F)-8
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(wing)375 948 y Fz(Theorem)38 b(5.1.2)g(\(Giaquin)m(ta)g(and)h
(Giusti\).)49 b Fy(If)f Fv(u)54 b Fs(2)h Fw(M)2768 912
y Ft(!)2768 973 y(M)2847 948 y Fy(,)e(then)c Fv(u)g Fy(is)g
FB(C)3455 912 y Ft(\013)3554 948 y Fy(for)375 1143 y(any)35
b Fv(\013)30 b Fs(2)f FB([0)p Fv(;)17 b FB(1\))p Fy(.)46
b(F)-7 b(urthermor)i(e,)34 b(given)h(any)g Fv(K)h Fs(\032\032)30
b Fv(S)2436 1107 y Ft(!)2430 1168 y(M)2509 1143 y Fy(,)36
b(the)f FB(C)2807 1107 y Ft(\013)2892 1143 y Fy(norm)g(of)h
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b(by)h(a)f(c)-5 b(onstant)37 b(dep)-5 b(ending)35 b(only)i(on)f
Fv(\013)q Fy(,)h Fv(n)p Fy(,)g(the)g(b)-5 b(ounds)36
b(on)g Fv(a)3098 1353 y Ft(i;j)3215 1338 y Fy(and)h Fv(Q)g
Fy(and)375 1533 y(on)51 b FB(dist)17 b(\()p Fv(K)r(;)g(@)5
b(S)994 1497 y Ft(!)988 1558 y(M)1068 1533 y FB(\))p
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f(Section)h(3.1.)41 b(The)28 b(in)m(terested)g(reader)375
2048 y(ma)m(y)33 b(also)g(see)h([GG],)e(or)g(Theorem)i(7.6)e(of)g
([G2],)g(or)g(Chapter)i(5)e(of)g([G].)675 2285 y(Also,)41
b(b)m(y)f(using)g(the)f(Lipsc)m(hitz)i(regularit)m(y)f(of)f(the)g(co)s
(e\016cien)m(ts)i Fv(a)3225 2300 y Ft(i;j)3306 2285 y
FB(,)f(w)m(e)g(will)375 2480 y(see)34 b(that)e(the)h(minimizers)i(are)d
(actually)i(Lipsc)m(hitz)g(\(Theorem)g(3.2.1\).)375 2918
y FA(5.2)135 b(The)45 b(minimal)h(minimizer)375 3167
y FB(F)-8 b(ollo)m(wing)32 b(the)h(ideas)g(dev)m(elop)s(ed)h(in)f
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375 3557 y(de\014ned)34 b(b)m(y)1576 3752 y Fv(u)1632
3711 y Ft(!)1632 3777 y(M)1711 3752 y FB(\()p Fv(x)p
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4296 y Fs(f)p Fv(u)1669 4255 y Ft(!)1669 4321 y(M)1775
4296 y Fv(>)f(\022)s Fs(g)h(\022)g(f)p Fv(u)f(>)h(\022)s
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4596 y Ft(!)1988 4658 y(M)2108 4633 y FB(is)h(still)h(a)e(minimizer:)63
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%%Page: 62 71
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438 y Fr(1)1800 501 y(lo)r(c)1893 475 y FB(\()1947 449
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475 y FB(\))35 b Fy(with)f Fs(r)p Fv(u;)f Fs(r)p Fv(v)e
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Fw(J)1001 1685 y Ft(S)1048 1662 y Fj(!)1044 1708 y(M)1116
1670 y FB(\()p Fv(u)p FB(\))k Fs(\024)i Fw(J)1485 1685
y Ft(S)1532 1662 y Fj(!)1528 1708 y(M)1600 1670 y FB(\(min)p
Fs(f)p Fv(u;)17 b(v)t Fs(g)p FB(\))47 b(and)i Fw(J)2392
1685 y Ft(S)2439 1662 y Fj(!)2435 1708 y(M)2507 1670
y FB(\()p Fv(v)t FB(\))54 b Fs(\024)i Fw(J)2871 1685
y Ft(S)2918 1662 y Fj(!)2914 1708 y(M)2985 1670 y FB(\(max)q
Fs(f)p Fv(u;)17 b(v)t Fs(g)p FB(\),)51 b(b)m(y)375 1865
y(Lemma)33 b(5.2.1)f(w)m(e)i(deduce)723 2160 y Fw(J)773
2175 y Ft(S)820 2152 y Fj(!)816 2198 y(M)888 2160 y FB(\()p
Fv(u)p FB(\))27 b(=)g Fw(J)1200 2175 y Ft(S)1247 2152
y Fj(!)1243 2198 y(M)1315 2160 y FB(\(min)q Fs(f)p Fv(u;)17
b(v)t Fs(g)p FB(\))112 b(and)j Fw(J)2239 2175 y Ft(S)2286
2152 y Fj(!)2282 2198 y(M)2354 2160 y FB(\()p Fv(v)t
FB(\))27 b(=)g Fw(J)2661 2175 y Ft(S)2708 2152 y Fj(!)2704
2198 y(M)2776 2160 y FB(\(max)q Fs(f)p Fv(u;)17 b(v)t
Fs(g)p FB(\))g Fv(:)p 3604 2455 V 3608 2392 59 4 v 3608
2455 V 3666 2455 4 66 v 375 2733 a Fz(Prop)s(osition)38
b(5.2.3.)49 b Fv(u)1347 2697 y Ft(!)1347 2758 y(M)1453
2733 y Fs(2)28 b Fw(M)1645 2697 y Ft(!)1645 2758 y(M)1724
2733 y Fy(.)675 3052 y FB(The)46 b(pro)s(of)e(of)g(Prop)s(osition)h
(5.2.3)g(is)g(no)m(w)h(a)e(standard)i(application)f(of)g(the)375
3248 y(Theorem)34 b(of)e(Ascoli.)375 3636 y FA(5.3)135
b(Birkho\013)45 b(prop)t(ert)l(y)h(for)f(the)g(minimal)h(minimizer)375
3885 y Fz(Prop)s(osition)38 b(5.3.1.)49 b Fy(L)-5 b(et)31
b Fv(\022)g Fs(2)d Fz(R)p Fy(.)43 b(Then,)30 b(the)h(set)g
Fs(f)p Fv(u)2474 3849 y Ft(!)2474 3910 y(M)2580 3885
y Fv(>)c(\022)s Fs(g)k Fy(satis\014es)f(the)g(Birkho\013)375
4080 y(pr)-5 b(op)g(erty)35 b(with)g(r)-5 b(esp)g(e)g(ct)34
b(to)h Fv(!)t Fy(.)44 b(Namely:)520 4400 y Fs(\017)49
b Fy(if)35 b Fv(k)30 b Fs(2)e Fz(Z)957 4364 y Ft(n)1040
4400 y Fy(and)34 b Fv(k)25 b Fs(\001)d Fv(!)31 b Fs(\024)d
FB(0)p Fy(,)35 b(then)f Fs(f)p Fv(u)1988 4364 y Ft(!)1988
4425 y(M)2094 4400 y Fv(>)28 b(\022)s Fs(g)g(\023)g(f)p
Fw(T)2537 4415 y Ft(k)2580 4400 y Fv(u)2636 4364 y Ft(!)2636
4425 y(M)2742 4400 y Fv(>)f(\022)s Fs(g)p Fy(;)520 4678
y Fs(\017)49 b Fy(if)35 b Fv(k)30 b Fs(2)e Fz(Z)957 4642
y Ft(n)1040 4678 y Fy(and)34 b Fv(k)25 b Fs(\001)d Fv(!)31
b Fs(\025)d FB(0)p Fy(,)35 b(then)f Fs(f)p Fv(u)1988
4642 y Ft(!)1988 4703 y(M)2094 4678 y Fv(>)28 b(\022)s
Fs(g)g(\022)g(f)p Fw(T)2537 4693 y Ft(k)2580 4678 y Fv(u)2636
4642 y Ft(!)2636 4703 y(M)2742 4678 y Fv(>)f(\022)s Fs(g)p
Fy(.)p eop
%%Page: 63 72
63 71 bop 3727 201 a FB(63)375 475 y Fy(Pr)-5 b(o)g(of.)48
b FB(Set)g Fv(v)56 b FB(:=)c(min)q Fs(f)p Fv(u)1390 438
y Ft(!)1390 500 y(M)1468 475 y Fv(;)17 b Fw(T)1570 490
y Ft(k)1613 475 y Fv(u)1669 438 y Ft(!)1669 500 y(M)1748
475 y Fs(g)p FB(.)86 b(If)47 b Fv(k)36 b Fs(\001)31 b
Fv(!)56 b Fs(\024)d FB(0,)d(b)m(y)e(Corollary)g(5.2.2,)i
Fv(v)h FB(is)c(a)375 669 y(minimizer)29 b(in)f(the)g(strip)g
Fw(T)1378 684 y Ft(k)1421 669 y Fv(S)1487 633 y Ft(!)1481
695 y(M)1560 669 y FB(.)42 b(By)28 b(construction,)i
Fw(T)2417 684 y Ft(k)2460 669 y Fv(u)2516 633 y Ft(!)2516
695 y(M)2622 669 y FB(is)e(the)g(minimal)g(minimizer)375
864 y(in)33 b Fw(T)547 879 y Ft(k)590 864 y Fv(S)656
828 y Ft(!)650 890 y(M)729 864 y FB(.)43 b(Th)m(us,)35
b(from)d(\(5.2\),)1241 1151 y Fs(f)p Fw(T)1349 1166 y
Ft(k)1392 1151 y Fv(u)1448 1109 y Ft(!)1448 1175 y(M)1554
1151 y Fv(>)c(\022)s Fs(g)g(\022)g(f)p Fv(v)j(>)d(\022)s
Fs(g)f(\022)h(f)p Fv(u)2457 1109 y Ft(!)2457 1175 y(M)2563
1151 y Fv(>)g(\022)s Fs(g)17 b Fv(:)375 1437 y FB(Analogously)-8
b(,)33 b(if)g Fv(k)25 b Fs(\001)d Fv(!)31 b Fs(\025)d
FB(0,)k Fv(v)37 b FB(is)c(a)f(minimizer)i(in)f Fv(S)2366
1401 y Ft(!)2360 1462 y(M)2439 1437 y FB(,)g(therefore)1331
1723 y Fs(f)p Fv(u)1437 1682 y Ft(!)1437 1748 y(M)1543
1723 y Fv(>)28 b(\022)s Fs(g)f(\022)h(f)p Fv(v)k(>)27
b(\022)s Fs(g)h(\022)g(f)p Fw(T)2448 1738 y Ft(k)2491
1723 y Fv(u)2547 1682 y Ft(!)2547 1748 y(M)2625 1723
y Fs(g)17 b Fv(:)p 3604 2009 4 66 v 3608 1947 59 4 v
3608 2009 V 3666 2009 4 66 v 375 2398 a FA(5.4)135 b(The)45
b(minimal)h(minimizer)g(is)f(unconstrained)375 2647 y
Fz(Prop)s(osition)38 b(5.4.1.)49 b Fy(Ther)-5 b(e)42
b(exists)h Fv(M)1948 2662 y Fr(0)2030 2647 y Fv(>)g FB(0)p
Fy(,)i(dep)-5 b(ending)41 b(only)i(on)g Fv(n)g Fy(and)f(on)h(the)375
2842 y(b)-5 b(ounds)37 b(on)g Fv(a)892 2857 y Ft(i;j)1010
2842 y Fy(and)g Fv(Q)p Fy(,)h(such)g(that)g(if)f Fv(M)43
b Fs(\025)34 b Fv(M)2213 2857 y Fr(0)2252 2842 y Fs(j)p
Fv(!)t Fs(j)p Fy(,)j(then)h(the)f(set)h Fs(fj)p Fv(u)3113
2806 y Ft(!)3113 2867 y(M)3191 2842 y Fs(j)32 b Fv(<)h
FB(1)p Fs(g)k Fy(do)-5 b(es)375 3037 y(not)35 b(touch)g(the)g(upp)-5
b(er)35 b(c)-5 b(onstr)g(aint)34 b Fs(f)p Fv(!)25 b Fs(\001)d
Fv(x)28 b FB(=)g Fv(M)10 b Fs(g)p Fy(.)45 b(A)n(lso,)34
b Fs(8)p Fv(\013)29 b Fs(\025)f FB(0)p Fy(,)35 b Fv(u)2967
3001 y Ft(!)2967 3062 y(M)3073 3037 y FB(=)27 b Fv(u)3232
3001 y Ft(!)3232 3062 y(M)7 b Fr(+)p Ft(\013)3411 3037
y Fy(.)375 3345 y(Pr)-5 b(o)g(of.)48 b FB(Let)29 b Fv(S)34
b FB(b)s(e)29 b(a)f(strip)i(of)e(width)h(3)f(around)h(the)g(h)m(yp)s
(erplane)h Fv(L)e FB(:=)g Fs(f)p Fv(x)14 b Fs(\001)g
Fv(!)31 b FB(=)c Fv(M)5 b(=)p FB(2)p Fs(g)p FB(.)375
3540 y(First)31 b(of)f(all,)h(w)m(e)g(will)h(pro)m(v)m(e)g(that)e
(there)h(exists)i(a)d(strip)h Fv(H)38 b FB(of)31 b(width)g(1)f
(parallel)h(to)g(the)375 3735 y(constrain)m(ts)j(and)f(at)f(distance)i
(univ)m(ersally)h(b)s(ounded)e(from)f Fv(L)p FB(,)h(on)g(whic)m(h)1411
4021 y(either)50 b Fv(u)1760 3985 y Ft(!)1760 4046 y(M)1866
4021 y FB(=)28 b Fs(\000)p FB(1)49 b(or)g Fv(u)2337 3985
y Ft(!)2337 4046 y(M)2443 4021 y FB(=)28 b(1)17 b(.)834
b(\(5.3\))375 4307 y(F)-8 b(or)32 b(that,)g(w)m(e)i(consider)g(t)m(w)m
(o)f(cases:)375 4590 y Fz(Case)45 b(1:)56 b FB(there)39
b(exists)i(a)d(p)s(oin)m(t)h Fv(x)1727 4554 y Fq(\003)1805
4590 y Fs(2)f Fv(S)44 b FB(so)39 b(that)g Fv(u)2413 4554
y Ft(!)2413 4616 y(M)2491 4590 y FB(\()p Fv(x)2584 4554
y Fq(\003)2624 4590 y FB(\))f Fs(2)g FB(\()p Fs(\000)p
FB(1)p Fv(;)17 b FB(1\).)62 b(Then,)41 b(from)375 4785
y(Prop)s(osition)26 b(4.4.1)g(there)h(exists)h(a)e(ball)g(\(whose)h
(radius)g(is)g(big)f(if)g Fv(M)37 b FB(is)26 b(big\),)i(con)m(tained)
375 4980 y(in)g Fs(f)p Fv(u)590 4944 y Ft(!)590 5005
y(M)696 4980 y FB(=)f Fs(\000)p FB(1)p Fs(g)h FB(=)f
Fw(C)p Fs(f)p Fv(u)1272 4944 y Ft(!)1272 5005 y(M)1378
4980 y Fv(>)h Fs(\000)p FB(1)p Fs(g)p FB(.)42 b(By)28
b(the)f(results)i(in)f(Section)g(5.3,)g(w)m(e)g(obtain)f(a)g(strip)375
5175 y(of)32 b(width)h(1)g(in)m(tersecting)h(suc)m(h)g(ball)f(and)g
(con)m(tained)g(in)g Fs(f)p Fv(u)2628 5139 y Ft(!)2628
5200 y(M)2734 5175 y FB(=)28 b Fs(\000)p FB(1)p Fs(g)p
FB(.)p eop
%%Page: 64 73
64 72 bop 3727 201 a FB(64)375 475 y Fz(Case)55 b(2:)72
b Fv(u)875 438 y Ft(!)875 500 y(M)954 475 y FB(\()p Fv(S)6
b FB(\))52 b Fs(\022)h FB(\()p Fs(\0001)p Fv(;)17 b Fs(\000)p
FB(1])32 b Fs([)h FB([1)p Fv(;)17 b FB(+)p Fs(1)p FB(\).)86
b(By)47 b(the)h(con)m(tin)m(uit)m(y)h(of)e Fv(u)3275
438 y Ft(!)3275 500 y(M)3353 475 y FB(,)k(either)375
669 y Fv(u)431 633 y Ft(!)431 694 y(M)510 669 y FB(\()p
Fv(S)6 b FB(\))27 b Fs(\022)h FB(\()p Fs(\0001)p Fv(;)17
b Fs(\000)p FB(1])33 b(or)f Fv(u)1404 633 y Ft(!)1404
694 y(M)1482 669 y FB(\()p Fv(S)6 b FB(\))28 b Fs(\022)g
FB([1)p Fv(;)17 b FB(+)p Fs(1)p FB(\).)675 950 y(In)41
b(an)m(y)h(case,)i(\(5.3\))c(is)i(pro)m(v)m(ed.)70 b(With)41
b(no)g(loss)h(of)e(generalit)m(y)-8 b(,)44 b(assume)f(also)375
1144 y Fv(!)436 1159 y Fr(1)504 1144 y Fv(>)29 b FB(0.)46
b(No)m(w)34 b(w)m(e)h(infer)f(that)f(it)g(is)h(not)g(p)s(ossible)h
(that)e Fv(u)2545 1108 y Ft(!)2545 1169 y(M)2623 1030
y Fu(\014)2623 1089 y(\014)2623 1149 y(\014)2657 1213
y Ft(H)2753 1144 y FB(=)c(1:)45 b(Otherwise,)36 b(since)375
1339 y(the)29 b(set)g Fs(f)p Fv(u)793 1303 y Ft(!)793
1364 y(M)899 1339 y FB(=)f(1)p Fs(g)g FB(satis\014es)j(the)e
(Birkho\013)g(prop)s(ert)m(y)g(with)g(resp)s(ect)h(to)e
Fv(!)k FB(\(see)e(Section)375 1534 y(5.3\),)i(w)m(e)i(obtain)e(that)880
1717 y Fu([)863 1929 y Ft(j)t Fq(2)p Fe(N)1023 1812 y
Fw(T)1081 1827 y Fr(\()p Fq(\000)p Ft(j;)p Fr(0)p Ft(;:::)n(;)p
Fr(0\))1412 1812 y Fv(H)j Fs(\022)1649 1717 y Fu([)1633
1929 y Ft(j)t Fq(2)p Fe(N)1793 1812 y Fw(T)1851 1827
y Fr(\()p Fq(\000)p Ft(j;)p Fr(0)p Ft(;:::)n(;)p Fr(0\))2182
1812 y Fs(f)p Fv(u)2288 1771 y Ft(!)2288 1837 y(M)2394
1812 y FB(=)27 b(1)p Fs(g)h(\022)g(f)p Fv(u)2835 1771
y Ft(!)2835 1837 y(M)2941 1812 y FB(=)f(1)p Fs(g)17 b
Fv(;)375 2113 y FB(th)m(us)23 b Fv(u)635 2077 y Ft(!)635
2138 y(M)741 2113 y FB(=)28 b(1)21 b(in)i(the)f(whole)h(p)s(ortion)f
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(constrain)m(t)375 2308 y(and)38 b Fv(H)8 b FB(.)59 b(But)38
b(then)h(w)m(e)g(can)f(consider)h(the)f(in)m(teger)h(translation)f
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Ft(;)p Fr(0)p Ft(;:::)o(;)p Fr(0\))3513 2308 y Fv(u)3569
2272 y Ft(!)3569 2333 y(M)3648 2308 y FB(,)375 2502 y(whic)m(h)24
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2528 y(M)1747 2502 y FB(,)i(con)m(tradicting)e(the)h(fact)e(that)h
Fv(u)2981 2466 y Ft(!)2981 2528 y(M)3082 2502 y FB(is)g(the)g(minimal)
375 2697 y(minimizer.)675 2977 y(Therefore,)32 b Fv(u)1198
2941 y Ft(!)1198 3003 y(M)1277 2863 y Fu(\014)1277 2923
y(\014)1277 2982 y(\014)1310 3046 y Ft(H)1405 2977 y
FB(=)c Fs(\000)p FB(1.)43 b(Then,)32 b(again,)f(since)i(the)e(set)h
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y Fv(>)27 b Fs(\000)p FB(1)p Fs(g)k FB(satis\014es)375
3172 y(the)i(Birkho\013)g(prop)s(ert)m(y)g(with)h(resp)s(ect)f(to)g
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3450 y Fw(T)1458 3466 y Fr(\()p Ft(j;)p Fr(0)p Ft(;:::)o(;)p
Fr(0\))1734 3450 y Fv(H)h Fs(\\)22 b(f)p Fv(u)2039 3409
y Ft(!)2039 3475 y(M)2146 3450 y Fv(>)27 b Fs(\000)p
FB(1)p Fs(g)h FB(=)f Fs(;)17 b Fv(;)375 3729 y FB(so)40
b Fv(u)558 3693 y Ft(!)558 3754 y(M)677 3729 y FB(=)g
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y(strain)m(t)33 b(and)g Fv(H)8 b FB(.)p 3604 3924 4 66
v 3608 3861 59 4 v 3608 3924 V 3666 3924 4 66 v 675 4229
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(in)m(tro)s(duce)f(the)g(follo)m(wing)375 4423 y(notation.)d(If)33
b Fv(a)28 b(>)f FB(0,)32 b(w)m(e)i(de\014ne)1128 4702
y Fv(S)1194 4661 y Ft(!)1188 4727 y(M)s(;a)1348 4702
y FB(:=)28 b Fs(f)p Fv(x)g Fs(2)g Fz(R)1790 4661 y Ft(n)1881
4702 y FB(s)p Fv(:)p FB(t)p Fv(:)45 b(x)22 b Fs(\001)g
Fv(!)31 b Fs(2)d FB([)p Fs(\000)p Fv(a;)17 b(M)33 b FB(+)22
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5134 y Ft(!)1220 5200 y(M)s(;a)1380 5175 y FB(:=)28 b
Fs(f)p Fv(u)f Fs(2)h FB(L)1799 5134 y Fr(1)1799 5200
y(lo)r(c)1891 5175 y FB(\()1946 5150 y(~)1929 5175 y
Fv(S)1995 5134 y Ft(!)1989 5200 y(M)2068 5175 y FB(\))45
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5134 y Fr(2)2686 5175 y FB(\()2741 5150 y(~)2724 5175
y Fv(S)2790 5134 y Ft(!)2784 5200 y(M)2863 5175 y FB(\))p
eop
%%Page: 65 74
65 73 bop 3727 201 a FB(65)1106 475 y(with)33 b Fv(u)p
FB(\()p Fv(x)p FB(\))28 b(=)f(1)32 b(for)g(an)m(y)i Fv(x)44
b FB(s)p Fv(:)p FB(t)p Fv(:)h(!)26 b Fs(\001)c Fv(x)28
b FB(=)f Fs(\000)p Fv(a)33 b FB(and)1153 724 y Fv(u)p
FB(\()p Fv(x)p FB(\))27 b(=)h Fs(\000)p FB(1)k(for)g(an)m(y)i
Fv(x)44 b FB(s)p Fv(:)p FB(t)p Fv(:)h(!)26 b Fs(\001)c
Fv(x)28 b FB(=)f Fv(M)33 b FB(+)22 b Fv(a)p Fs(g)17 b
Fv(:)375 1002 y Fz(Theorem)38 b(5.4.2.)49 b Fy(Ther)-5
b(e)45 b(exists)f(a)h(c)-5 b(onstant)45 b Fv(M)2315 1017
y Fr(0)2355 1002 y Fy(,)i(dep)-5 b(ending)44 b(only)h(on)g
Fv(n)g Fy(and)f(on)375 1197 y(the)36 b(b)-5 b(ounds)35
b(on)g Fv(a)1051 1212 y Ft(i;j)1167 1197 y Fy(and)g Fv(Q)p
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1212 y Fr(0)2295 1197 y Fs(j)p Fv(!)t Fs(j)p Fy(,)35
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1356 y Ft(!)431 1417 y(M)554 1392 y Fy(is)44 b(an)g(unc)-5
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Fw(X)3513 1356 y Ft(!)3513 1417 y(M)s(;a)3645 1392 y
Fy(,)375 1587 y Fw(J)425 1602 y Ft(S)472 1579 y Fj(!)468
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Fv(k)52 b Fs(2)d Fz(Z)3496 2253 y Ft(n)3588 2289 y FB(so)375
2484 y(that)37 b Fv(k)28 b Fs(\001)d Fv(!)39 b(>)d(a)p
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%%Page: 70 79
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%%Page: 71 80
71 79 bop 3727 201 a FB(71)675 475 y(The)38 b(set)g(of)e(minimizers)j
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%%Page: 72 81
72 80 bop 1669 696 a Fx(Chapter)53 b(7)996 1041 y(Bac)l(k)g(to)h(the)f
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%%Page: 73 82
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%%Page: 74 83
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y Fu(\020)1811 1724 y Fw(L)1882 1683 y Ft(n)1929 1724
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1724 y Fv(g)2424 1683 y Fq(\000)p Fr(1)2420 1750 y Ft(i;j)2534
1724 y Fv(:)375 1968 y FB(Hence,)30 b(if)c Fv(g)31 b
FB(is)c(Lipsc)m(hitz,)j(b)m(y)e(Lemma)g(7.1.5,)f(the)h(co)s(e\016cien)m
(ts)h Fv(a)2810 1983 y Ft(i;j)2917 1968 y FB(are)e(also)g(Lipsc)m
(hitz.)375 2162 y(Also,)i(they)g(are)f(uniformly)h(elliptic)g(b)s
(ecause)h(of)d(the)i(nondegeneracy)h(of)d Fv(g)t FB(.)41
b(Moreo)m(v)m(er,)375 2357 y(using)36 b(the)g(de\014nition)g(of)e
(functionals)i(on)g(measures)h(\(see,)g(for)d(instance,)k([B])d(at)g
(page)375 2552 y(293\),)d(and)h(recalling)g(that)1008
2834 y Fv(D)s(\037)1153 2849 y Ft(A)1237 2834 y FB(=)28
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2834 y Fs(j)195 b FB(and)g Fs(j)p Fv(D)s(\037)2389 2849
y Ft(A)2446 2834 y Fs(j)27 b FB(=)h Fw(H)2694 2792 y
Ft(n)p Fq(\000)p Fr(1)2831 2719 y Fu(\014)2831 2779 y(\014)2831
2839 y(\014)2865 2903 y Ft(@)2906 2884 y Fj(?)2942 2903
y Ft(A)3015 2834 y Fv(;)375 3115 y FB(\(see,)34 b(for)e(instance,)i
([G1])e(pages)h(44)f(and)h(52\),)f(w)m(e)h(ha)m(v)m(e)h(that)1387
3397 y Fv(P)14 b FB(\()p Fv(A;)j FB(\012\))27 b(=)1858
3262 y Fu(Z)1913 3487 y Fr(\012)1985 3311 y Fu(p)p 2084
3311 536 4 v 2084 3397 a Fv(a)2135 3412 y Ft(i;j)2216
3397 y Fv(@)2267 3412 y Ft(i)2295 3397 y Fv(\037)2356
3412 y Ft(A)2413 3397 y Fv(@)2464 3412 y Ft(j)2501 3397
y Fv(\037)2562 3412 y Ft(A)2636 3397 y Fv(:)675 3720
y FB(W)-8 b(e)33 b(address)h(no)m(w)f(the)g(follo)m(wing)g(result:)375
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b(minimizers)f(for)h Fw(E)2842 3950 y Ft(")2879 3987
y Fy(.)80 b(Then)45 b(ther)-5 b(e)47 b(is)f(a)375 4181
y(subse)-5 b(quenc)g(e)25 b Fv(")945 4196 y Ft(h)1015
4181 y Fy(c)-5 b(onver)g(ging)24 b(to)i(zer)-5 b(o)26
b(such)f(that)h Fv(u)2248 4196 y Ft(")2281 4208 y Fj(h)2351
4181 y Fy(c)-5 b(onver)g(ges)24 b(in)i FB(L)2948 4145
y Fr(1)2987 4181 y FB(\()3028 4154 y(~)3025 4181 y Fw(S)p
FB(\))g Fy(to)g(a)f(function)375 4376 y Fv(u)431 4391
y Fr(0)499 4376 y Fy(so)j(that)h Fv(u)866 4391 y Fr(0)905
4376 y FB(\()p Fv(x)p FB(\))f(=)f Fs(\006)p FB(1)i Fy(for)f(almost)h
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Fs(g)i Fy(satis\014es)866 4658 y Fv(P)14 b FB(\()p Fv(E)6
b(;)1104 4631 y FB(~)1103 4658 y Fw(S)o FB(\))27 b(=)h(min)1504
4547 y Fu(n)1570 4658 y Fv(P)14 b FB(\()p Fv(F)s(;)1797
4631 y FB(~)1795 4658 y Fw(S)o FB(\))j Fv(;)44 b FB(\()p
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%%Page: 77 86
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490 y Ft(")2904 502 y Fj(h)2998 475 y FB(=)k Fv(\034)11
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b(to)g(the)g(minimal)e(surfac)-5 b(e)35 b Fv(@)5 b(E)h
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1184 y(extended)i(in)e([OS])g(and)g([B],)g(with)g(di\013eren)m(t)h(pro)
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1380 y(see)d([BPV].)g(Here,)i(w)m(e)e(giv)m(e)g(a)f(pro)s(of)f(of)h
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1575 y(one)36 b(in)f([Md],)i(b)m(y)f(using)g(the)g(to)s(ols)f(dev)m
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b(A)36 b(useful)375 1770 y(function)d(will)g(b)s(e)g(the)g(primitiv)m
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(namely)1574 2064 y Fv(\036)p FB(\()p Fv(t)p FB(\))28
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d(the)h(con)m(v)m(erging)h(subsequence)j(in)c(Theorem)375
3036 y(7.2.1)25 b(is)i(giv)m(en)g(b)m(y)g(the)f(follo)m(wing)h
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4048 a(2)1429 3844 y Fu(Z)1486 4051 y Fr(~)1484 4069
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31 3 v 1600 3964 a(2)1644 3980 y Fs(jr)p Fv(u)1811 3995
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FB(+)22 b Fv(")2081 3938 y Fq(\000)2146 3911 y Fg(1)p
2145 3923 V 2145 3964 a(2)2190 3980 y Fv(F)14 b FB(\()p
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%%Page: 78 87
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%%Page: 79 88
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2037 y(=)1539 1901 y Fu(Z)1595 2127 y Fe(R)1676 2037
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FB(=)f Fs(f)p Fv(x)g Fs(2)1103 2298 y FB(~)1100 2325
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2808 y Fs(\000)23 b Fv(v)1779 2823 y Fr(0)1818 2808 y
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2808 y FB(sup)2129 2894 y Ft(t)p Fq(2)p Fr([0)p Ft(;\021)2311
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3384 y Fw(S)p FB(\))17 b Fv(:)950 b FB(\(7.6\))375 3672
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3867 y FB(.)675 4151 y(T)-8 b(ak)m(e)42 b(no)m(w)f Fv(x)h
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FB(1.)68 b(W)-8 b(e)41 b(w)m(an)m(t)g(to)g(pro)m(v)m(e)h(that)f
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4370 y Fg(1)835 4361 y Fq(\025)p Fr(1)946 4346 y Fv(h)p
FB(\()p Fv(x)p FB(\):)48 b(b)m(y)35 b(h)m(yp)s(otheses,)j
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5127 y Ft(")1762 5112 y FB(=)72 b(sup)1865 5198 y Ft(t)p
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Fr(])2117 5112 y Fv(P)14 b FB(\()p Fv(S)2292 5127 y Ft(t)2321
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%%Page: 80 89
80 88 bop 3727 201 a FB(80)375 475 y(As)33 b(ab)s(o)m(v)m(e,)h(w)m(e)f
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736 y Fv(")862 695 y Fq(\000)927 668 y Fg(1)p 927 680
31 3 v 927 721 a(2)971 736 y Fw(E)1032 695 y Ft(")1069
736 y FB(\()p Fv(v)1154 751 y Ft(")1190 736 y FB(\))c
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627 y Ft(\021)1616 635 y Fj(")1537 826 y Fr(0)1654 736
y FB([)p Fv(")1737 668 y Fg(1)p 1737 680 V 1737 721 a(2)1782
736 y FB(\()p Fv(\036)1878 695 y Fq(0)1878 761 y Ft(")1914
736 y FB(\()p Fv(t)p FB(\)\))2063 695 y Fr(2)2125 736
y FB(+)22 b Fv(")2269 695 y Fq(\000)2334 668 y Fg(1)p
2333 680 V 2333 721 a(2)2378 736 y FB(\()2416 659 y Fs(p)p
2499 659 46 4 v 77 x Fv(")f FB(+)h Fv(F)14 b FB(\()p
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1431 1258 y Fv(")1487 1190 y Fg(1)p 1487 1202 31 3 v
1487 1243 a(2)1532 1258 y Fv(\036)1590 1217 y Fq(0)1590
1283 y Ft(")1626 1258 y FB(\()p Fv(t)p FB(\))28 b(=)1868
1134 y Fu(q)p 1968 1134 607 4 v 48 x Fs(p)p 2051 1182
46 4 v 76 x Fv(")22 b FB(+)g Fv(F)14 b FB(\()p Fv(\036)2390
1273 y Ft(")2426 1258 y FB(\()p Fv(t)p FB(\)\))j Fv(;)375
1520 y FB(so)33 b(that)f(\(7.7\))g(b)s(ecomes)937 1781
y Fv(")983 1740 y Fq(\000)1048 1713 y Fg(1)p 1047 1725
31 3 v 1047 1766 a(2)1092 1781 y Fw(E)1153 1740 y Ft(")1189
1781 y FB(\()p Fv(v)1274 1796 y Ft(")1311 1781 y FB(\))83
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y Fr(0)1951 1656 y Fu(q)p 2050 1656 607 4 v 2050 1704
a Fs(p)p 2133 1704 46 4 v 77 x Fv(")22 b FB(+)g Fv(F)14
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Ft(")2769 1781 y FB(\()p Fv(t)p FB(\))g Fv(dt)27 b FB(=)1433
2056 y(=)83 b(2)p Fv(\015)1692 2071 y Ft(")1762 1921
y Fu(Z)1861 1947 y Fr(1)1817 2146 y Fq(\000)p Fr(1)1928
1931 y Fu(q)p 2028 1931 448 4 v 49 x Fs(p)p 2111 1980
46 4 v 76 x Fv(")21 b FB(+)h Fv(F)14 b FB(\()p Fv(s)p
FB(\))j Fv(ds)g(:)375 2317 y FB(F)-8 b(rom)32 b(this)h(and)g(\(7.6\),)f
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2317 4 66 v 3608 2255 59 4 v 3608 2317 V 3666 2317 4
66 v 675 2616 a(W)-8 b(e)47 b(can)g(no)m(w)g(complete)h(the)f(pro)s(of)
e(of)h(Theorem)i(7.2.1.)85 b(It)46 b(is)h(enough)g(to)375
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Fw(S)o FB(\))32 b(for)g(an)m(y)g(op)s(en)h(b)s(ounded)f(set)h(con)m
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3005 y(and)45 b(con)m(tained)g(in)g(\()p Fs(\0001)p Fv(;)17
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%%Page: 81 90
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1706 y Fv(w)2745 1721 y Ft(")2798 1706 y Fv(:)375 1987
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4237 y(1)p 2367 4281 49 4 v 2369 4373 a Fv(")2426 4304
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%%Page: 84 93
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b(1,)31 b(29{68.)309 1895 y([A)m(CF2)50 b(])36 b(Alt,)j(H.)e(W.;)j
(Ca\013arelli,)f(L.)e(A.;)j(F)-8 b(riedman,)38 b(A.)f({)f
Fc(A)i(fr)-5 b(e)g(e)39 b(b)-5 b(oundary)39 b(pr)-5 b(oblem)40
b(for)619 2008 y(quasiline)-5 b(ar)37 b(el)5 b(liptic)36
b(e)-5 b(quations)p Fd(,)35 b(Ann.)50 b(Scuola)34 b(Norm.)51
b(Sup.)e(Pisa)34 b(Cl.)51 b(Sci.)f(\(4\))35 b(11)619
2121 y(\(1984\),)e(no.)41 b(1,)31 b(1{44.)309 2309 y([A)m(CF3)50
b(])37 b(Alt,)i(H.)d(W.;)41 b(Ca\013arelli,)e(L.)e(A.;)j(F)-8
b(riedman,)39 b(A.)e({)f Fc(Jets)j(with)g(two)g(\015uids.)59
b(I.)38 b(One)619 2422 y(fr)-5 b(e)g(e)33 b(b)-5 b(oundary)p
Fd(,)32 b(Indiana)e(Univ.)41 b(Math.)g(J.)30 b(33)i(\(1984\),)h(no.)40
b(2,)31 b(213{247.)435 2609 y([Ba)50 b(])21 b(Bangert,)k(V.)d({)g
Fc(L)-5 b(aminations)26 b(of)f Fd(3)p Fc(-tori)g(by)f(le)-5
b(ast)26 b(ar)-5 b(e)g(a)26 b(surfac)-5 b(es)p Fd(,)24
b(Analysis,)g(et)e(cetera,)619 2722 y(85{114,)33 b(Academic)f(Press,)e
(Boston,)i(MA,)f(1990.)425 2910 y([BZ)49 b(])25 b(Birkho\013,)i(G.;)h
(Zaran)m(tonello,)g(E.)d(H.)h({)f Fc(Jets,)k(wakes,)g(and)g(c)-5
b(avities)p Fd(,)27 b(Academic)f(Press)619 3023 y(Inc.,)k(Publishers,)g
(New)g(Y)-8 b(ork,)31 b(1957.)350 3210 y([BPV)50 b(])27
b(Bellettini,)k(G.;)e(P)m(aolini,)h(M.;)f(V)-8 b(en)m(turini)28
b(S.)f({)h Fc(Some)j(R)-5 b(esults)30 b(on)h(Surfac)-5
b(e)30 b(Me)-5 b(asur)g(es)619 3323 y(in)32 b(Calculus)h(of)g(V)-7
b(ariations)34 b Fd(,)d(Ann.)40 b(Mat.)i(Pura)29 b(Appl.)40
b(\(4\),)32 b(170)g(\(1996\),)h(329{359.)480 3511 y([B)50
b(])32 b(Bouc)m(hitt)m(\022)-43 b(e,)36 b(G.)d({)g Fc(Singular)i(p)-5
b(erturb)g(ations)37 b(of)e(variational)h(pr)-5 b(oblems)37
b(arising)e(fr)-5 b(om)619 3624 y(a)28 b(two-phase)h(tr)-5
b(ansition)30 b(mo)-5 b(del)p Fd(,)28 b(Appl.)38 b(Math.)i(Optim.)e(21)
26 b(\(1990\),)j(no.)39 b(3,)27 b(289{314.)479 3811 y([C)49
b(])30 b(Ca\013arelli,)i(L.)f(A.)f({)h Fc(The)i(obstacle)h(pr)-5
b(oblem)p Fd(,)32 b(preprin)m(t,)e(1998.)414 3999 y([CC)48
b(])28 b(Ca\013arelli,)i(L.)e(A.;)i(C\023)-45 b(ordoba,)28
b(A.)h({)f Fc(Uniform)j(c)-5 b(onver)g(genc)g(e)32 b(of)e(a)h(singular)
h(p)-5 b(erturb)g(a-)619 4112 y(tion)33 b(pr)-5 b(oblem)p
Fd(,)32 b(Comm.)41 b(Pure)29 b(Appl.)41 b(Math.)g(48)31
b(\(1995\),)i(no.)41 b(1,)31 b(1{12.)422 4299 y([CL)49
b(])33 b(Ca\013arelli,)i(L.)d(A.;)j(de)d(la)h(Lla)m(v)m(e,)j(R.)d({)g
Fc(Plane-like)h(minimizers)h(in)g(p)-5 b(erio)g(dic)36
b(me)-5 b(dia)p Fd(,)619 4412 y(preprin)m(t,)30 b(mp)p
1128 4412 28 4 v 32 w(arc)g(00-368.)392 4600 y([DM)50
b(])31 b(Dal)h(Maso,)h(G.)f({)f Fc(Inte)-5 b(gr)g(al)35
b(r)-5 b(epr)g(esentation)36 b(on)e Fb(B)5 b(V)20 b Fd(\(\012\))33
b Fc(of)h Fd(\000)p Fa(\000)p Fc(limit)f(of)h(variational)619
4713 y(inte)-5 b(gr)g(als)p Fd(,)32 b(Man)m(uscripta)f(Math.,)h(30)f
(\(1980\),)i(387{416.)354 4901 y([DEF)50 b(])27 b(Dolcini,)j(A.;)e(Esp)
s(osito,)g(L.;)h(F)-8 b(usco,)29 b(N.)e({)h Fb(C)2221
4868 y Fr(0)p Ft(;\013)2354 4901 y Fc(r)-5 b(e)g(gularity)31
b(of)f Fb(!)s Fc(-minima)p Fd(,)e(Boll.)41 b(Un.)619
5013 y(Mat.)h(Ital.)f(A)31 b(\(7\))g(10)g(\(1996\),)i(no.)41
b(1,)31 b(113{125.)p eop
%%Page: 86 95
86 94 bop 3727 201 a FB(86)483 475 y Fd([E)49 b(])26
b(Elliot,)i(R.)d(J.)h({)f Fc(Visc)-5 b(osity)29 b(solutions)h(and)f
(optimal)h(c)-5 b(ontr)g(ol)p Fd(,)29 b(Pitman)c(researc)m(h)h(notes)
619 588 y(in)k(mathematics)i(series)e(165,)i(Longman,)f(Halo)m(w,)h
(1987.)474 775 y([G)49 b(])33 b(Giaquin)m(ta,)j(M.)d({)h
Fc(Intr)-5 b(o)g(duction)37 b(to)f(r)-5 b(e)g(gularity)37
b(the)-5 b(ory)36 b(for)g(nonline)-5 b(ar)37 b(el)5 b(liptic)35
b(sys-)619 888 y(tems)p Fd(,)c(Birkh\177)-45 b(auser)30
b(V)-8 b(erlag,)32 b(Basel,)g(1993.)402 1076 y([GG)50
b(])35 b(Giaquin)m(ta,)j(M.;)f(Giusti,)g(E.)e({)g Fc(On)i(the)g(r)-5
b(e)g(gularity)38 b(of)f(the)g(minima)h(of)f(variational)619
1189 y(inte)-5 b(gr)g(als)p Fd(,)32 b(Acta)g(Math.)41
b(148)32 b(\(1982\),)h(31{46.)408 1376 y([GT)49 b(])23
b(Gilbarg,)i(D.;)h(T)-8 b(rudinger,)23 b(N.)g(S.)f({)h
Fc(El)5 b(liptic)26 b(p)-5 b(artial)28 b(di\013er)-5
b(ential)26 b(e)-5 b(quations)27 b(of)e(se)-5 b(c)g(ond)619
1489 y(or)g(der.)43 b(Se)-5 b(c)g(ond)34 b(e)-5 b(dition)p
Fd(,)32 b(Springer-V)-8 b(erlag,)32 b(Berlin-New)f(Y)-8
b(ork,)31 b(1983.)417 1677 y([GL)49 b(])30 b(Ginzburg,)f(V.;)i(Landau,)
e(L.)h({)g Fc(On)h(the)i(the)-5 b(ory)33 b(of)f(sup)-5
b(er)g(c)g(onductivity)p Fd(,)33 b(Zh.)3383 1654 y(\022)3375
1677 y(Eksp)s(er.)619 1790 y(T)-8 b(eoret.)42 b(Fiz.)f(20)32
b(\(1950\),)h(1064{1082.)412 1977 y([GP)49 b(])38 b(Ginzburg,)g(V.)g
(L.;)k(Pitaevski)-10 b(\025)-35 b(\020,)41 b(L.)c(P)-8
b(.)38 b({)g Fc(On)h(the)g(the)-5 b(ory)41 b(of)e(sup)-5
b(er\015uidity)p Fd(,)42 b(So)m(viet)619 2090 y(Ph)m(ysics.)f(JETP)29
b(34/7)j(\(1958\),)h(858{861.)428 2278 y([G1)50 b(])40
b(Giusti,)j(E.)d({)g Fc(Minimal)i(surfac)-5 b(es)42 b(and)g(functions)g
(of)f(b)-5 b(ounde)g(d)43 b(variation)p Fd(,)h(Mono-)619
2391 y(graphs)30 b(in)g(Mathematics,)i(80.)42 b(Birkh\177)-45
b(auser)30 b(V)-8 b(erlag,)32 b(Basel-Boston,)h(Mass.,)f(1984.)428
2578 y([G2)50 b(])35 b(Giusti,)i(E.)e({)g Fc(Meto)-5
b(di)37 b(dir)-5 b(etti)38 b(nel)f(c)-5 b(alc)g(olo)39
b(del)5 b(le)37 b(variazioni)p Fd(,)h(Unione)d(Matematica)619
2691 y(Italiana,)d(Bologna,)h(1994.)423 2879 y([Gu)49
b(])40 b(Gurtin,)h(M.)f(E.)f({)h Fc(On)h(a)g(the)-5 b(ory)43
b(of)e(phase)h(tr)-5 b(ansitions)44 b(with)d(interfacial)h(ener)-5
b(gy)p Fd(,)619 2992 y(Arc)m(h.)41 b(Rational)32 b(Mec)m(h.)41
b(Anal.)g(87)32 b(\(1985\),)h(no.)40 b(3,)31 b(187{212.)420
3179 y([HL)49 b(])39 b(Han,)j(Q.;)i(Lin,)d(F.)f({)g Fc(El)5
b(liptic)40 b(p)-5 b(artial)44 b(di\013er)-5 b(ential)42
b(e)-5 b(quations)p Fd(,)42 b(Couran)m(t)e(Lecture)619
3292 y(Notes)33 b(in)e(Mathematics,)k(1.)45 b(New)32
b(Y)-8 b(ork)33 b(Univ)m(ersit)m(y)-8 b(,)34 b(Couran)m(t)e(Institute)g
(of)g(Mathe-)619 3405 y(matical)g(Sciences,)f(New)g(Y)-8
b(ork,)31 b(1997.)477 3593 y([H)49 b(])25 b(Hedlund,)g(G.)g(A.)g(-)g
Fc(Ge)-5 b(o)g(desics)29 b(on)f(a)g(two-dimensional)i(R)n(iemannian)f
(manifold)g(with)619 3706 y(p)-5 b(erio)g(dic)34 b(c)-5
b(o)g(e\016cients)p Fd(,)32 b(Ann.)40 b(of)30 b(Math.)41
b(33)32 b(\(1932\),)h(719{739.)431 3893 y([LL)49 b(])27
b(Landau,)g(L.)f(D.;)j(Lifshitz,)f(E.)f(M.)g({)g Fc(Statistic)-5
b(al)32 b(physics)p Fd(,)c(Oxford)e(Univ)m(ersit)m(y)i(Press,)619
4006 y(Oxford,)h(1938.)292 4194 y([MaMe)51 b(])30 b(Man)m(tegazza,)35
b(C.;)30 b(Menn)m(ucci,)i(A.)e(C.)h(G.)f({)h Fc(Hamilton-Jac)-5
b(obi)34 b(Equations)f(and)h(Dis-)619 4307 y(tanc)-5
b(e)33 b(F)-7 b(unctions)34 b(on)f(R)n(iemannian)g(Manifolds)p
Fd(,)f(preprin)m(t,)e(1999.)378 4495 y([MM)50 b(])38
b(Massari,)i(U.;)i(Miranda,)e(M.)e({)h Fc(Minimal)g(surfac)-5
b(es)40 b(of)g(c)-5 b(o)g(dimension)41 b(one)p Fd(,)g(North-)619
4607 y(Holland)31 b(Mathematics)h(Studies,)e(Amsterdam,)h(1984.)416
4795 y([Ma)50 b(])38 b(Mather,)j(J.)d(N.)g({)h Fc(Invariant)i(subsets)e
(for)i(ar)-5 b(e)g(a)41 b(pr)-5 b(eserving)41 b(home)-5
b(omorphisms)44 b(of)619 4908 y(surfac)-5 b(es)p Fd(,)31
b(Mathematical)i(analysis)e(and)f(applications,)i(P)m(art)f(B,)g(pp.)39
b(531{562,)34 b(Adv.)619 5021 y(in)c(Math.)41 b(Suppl.)e(Stud.,)30
b(7b,)g(Academic)i(Press,)e(New)h(Y)-8 b(ork-London,)30
b(1981.)p eop
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87 95 bop 3727 201 a FB(87)402 475 y Fd([MF)50 b(])26
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b(ction)29 b(minimizing)g(orbits)g(in)g(Hamiltonian)h(systems)p
Fd(,)619 588 y(T)-8 b(ransition)30 b(to)h(c)m(haos)g(in)f(classical)i
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b(Anal.)f(98)31 b(\(1987\),)i(no.)41 b(2,)31 b(123{142.)416
1189 y([Mo)50 b(])36 b(Morse,)j(H.)e(M.)g({)f Fc(A)i(fundamental)i
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b(d)39 b(surfac)-5 b(e)39 b(of)619 1302 y(genus)32 b(gr)-5
b(e)g(ater)34 b(than)g(one)p Fd(,)d(T)-8 b(rans.)40 b(Amer.)h(Math.)g
(So)s(c.)f(26)32 b(\(1924\),)h(no.)40 b(1,)31 b(25{60.)462
1489 y([M)49 b(])27 b(Moser,)h(J.)e({)h Fc(Minimal)j(solutions)g(of)g
(variational)h(pr)-5 b(oblems)31 b(on)f(a)f(torus)p Fd(,)g(Ann.)38
b(Inst.)619 1602 y(H.)31 b(P)m(oinc.)41 b(Anal.)g(Non)31
b(Lin.)40 b(3)31 b(\(1986\),)i(no.)40 b(3,)31 b(229{272.)424
1790 y([OS)48 b(])32 b(Ow)m(en,)h(N.;)g(Stern)m(b)s(erg,)f(P)-8
b(.)33 b({)f Fc(Nonc)-5 b(onvex)35 b(variational)h(pr)-5
b(oblems)36 b(with)f(anisotr)-5 b(opic)619 1903 y(p)g(erturb)g(ations)p
Fd(,)33 b(Nonlinear)e(Anal.)41 b(T.M.A.)31 b(16)h(\(1991\),)h(705{719.)
478 2090 y([R)49 b(])26 b(Ro)m(wlinson,)i(J.)e(S.)g({)h
Fc(T)-7 b(r)i(anslation)32 b(of)d(J.)f(D.)g(van)i(der)f(Waals')h(\\The)
f(thermo)-5 b(dynamic)619 2203 y(the)g(ory)26 b(of)f(c)-5
b(apil)5 b(larity)27 b(under)e(the)g(hyp)-5 b(othesis)27
b(of)e(a)g(c)-5 b(ontinuous)26 b(variation)g(of)f(density")p
Fd(,)619 2316 y(J.)30 b(Statist.)42 b(Ph)m(ys.)e(20)31
b(\(1979\),)j(no.)40 b(2,)31 b(197{244.)454 2504 y([Se)49
b(])27 b(Serrin,)f(J.)h({)g Fc(Mathematic)-5 b(al)32
b(principles)e(of)g(classic)-5 b(al)31 b(\015uid)f(me)-5
b(chanics)p Fd(,)28 b(1959)h(Hand-)619 2617 y(buc)m(h)h(der)f(Ph)m
(ysik,)i(Springer-V)-8 b(erlag,)31 b(Berlin-G\177)-45
b(ottingen-Heidelb)s(erg.)494 2804 y([S)49 b(])36 b(Simader,)g(C.)f(G.)
i({)e Fc(Sob)-5 b(olev's)39 b(original)f(de\014nition)h(of)e(his)h(sp)
-5 b(ac)g(es)39 b(r)-5 b(evisite)g(d)38 b(and)h(a)619
2917 y(c)-5 b(omp)g(arison)46 b(with)d(nowadays)i(de\014nition)p
Fd(,)g(Boundary)40 b(v)-5 b(alue)42 b(problems)e(for)h(elliptic)619
3030 y(and)33 b(parab)s(olic)h(op)s(erators)g(\(Catania,)i(1998\).)53
b(Matematic)m(he)37 b(\(Catania\))f(54)e(\(1999\),)619
3143 y(suppl.,)29 b(149{178.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF