Content-Type: multipart/mixed; boundary="-------------0501281204313" This is a multi-part message in MIME format. ---------------0501281204313 Content-Type: text/plain; name="05-37.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-37.keywords" Carnot group, Hardy inequality ---------------0501281204313 Content-Type: application/postscript; name="kombecarnot.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="kombecarnot.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: kombecarnot.dvi %%CreationDate: Fri Jan 28 11:56:43 2005 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMBX12 CMR10 CMCSC10 CMMI10 CMSY10 CMR12 CMMI12 CMEX10 %%+ MSBM7 CMMI6 CMR8 CMSY8 MSBM10 CMMI8 CMTI10 CMR9 CMTI12 CMR6 CMSY6 %%+ MSAM10 CMBX10 CMTT10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 -p1 -l8 kombecarnot %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2005.01.28:0958 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 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 readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 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 49 /one put dup 50 /two put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 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 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 67 /C put dup 70 /F put dup 72 /H put dup 76 /L put dup 77 /M put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 121 /y put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 50 /two 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 75 /K put dup 76 /L 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 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 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 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 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 67 /C put dup 71 /G put dup 72 /H put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 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 48 /prime put dup 49 /infinity put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 61 /equal put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 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 81 /Q put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /G put dup 72 /H put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 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 16 /parenleftBig put dup 17 /parenrightBig put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 8 /circleplus put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 71 /G put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 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 21 /lambda put dup 25 /pi put dup 26 /rho put dup 30 /phi put dup 32 /psi 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 66 /B put dup 67 /C put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 86 /V put dup 88 /X put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f 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 112 /p put dup 113 /q 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 /UniqueID 5087386 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 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 38 /ampersand 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 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D 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 82 /R put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (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 0 /Gamma put dup 1 /Delta put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W 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 readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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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 81 /Q put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 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 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 39 /quoteright 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 57 /nine 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 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 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L 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 89 /Y put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i 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 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 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Fp(j)p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))p Fp(j)24246 71866 y Fl(4)25066 72415 y Fs(+)295 b(16)p Fp(j)p Fq(z)59 b Fs(\()p Fq(x)p Fs(\))p Fp(j)30825 71866 y Fl(2)31351 70940 y Fo(\021)32277 70827 y Fe(2)p Fd(\000)p Fm(Q)p 32277 71036 1688 40 v 32918 71585 a Fe(4)p eop end %%Page: 6 6 TeXDict begin 6 5 bop -880 -714 a Fg(6)20957 b(ISMAIL)341 b(K)-28 b(OMBE)-880 1942 y Fs(is)434 b(a)g(fundamen)-36 b(tal)433 b(solution)h(for)g(the)f(op)36 b(erator)434 b Fp(\000)p Fs(\001)25531 2141 y Fn(G)26275 1942 y Fs(.)578 b(W)-108 b(e)434 b(note)f(that)-880 5200 y(\(3.7\))14283 b Fq(K)95 b Fs(\()p Fq(x)p Fs(\))369 b(=)20779 3725 y Fo(\020)21572 5200 y Fp(j)p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))p Fp(j)24738 4651 y Fl(4)25558 5200 y Fs(+)295 b(16)p Fp(j)p Fq(z)59 b Fs(\()p Fq(x)p Fs(\))p Fp(j)31317 4651 y Fl(2)31843 3725 y Fo(\021)32770 3661 y Fe(1)p 32770 3822 406 40 v 32770 4370 a(4)-880 8018 y Fs(de\014nes)475 b(a)i(homogeneous)f(norm)g(and)g Fq(Q)441 b Fs(=)h Fq(m)324 b Fs(+)f(2)p Fq(k)522 b Fs(is)477 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b(oin)-36 b(t)433 b Fq(x)369 b Fp(2)g Fj(G)295 b Fp(n)h(f)p Fs(0)p Fp(g)434 b Fs(where)f Fq(f)19141 34571 y Fk(0)19452 35053 y Fs(\()p Fq(K)95 b Fs(\()p Fq(x)p Fs(\)\))368 b Fp(6)p Fs(=)h(0.)448 37524 y(W)-108 b(e)434 b(no)-36 b(w)434 b(ha)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)436 b(theorem)d(on)h(the)f Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)432 b(group)i(:)-880 40066 y Fx(Theorem)557 b(3.2.)580 b Ff(L)-66 b(et)510 b Fj(G)h Ff(b)-66 b(e)510 b(an)h(H-typ)-66 b(e)510 b(gr)-66 b(oup)511 b(with)g(homo)-66 b(gene)g(ous)510 b(dimension)g Fq(Q)454 b Fs(=)g Fq(m)329 b Fs(+)h(2)p Fq(k)556 b Ff(and)-880 41615 y(let)464 b Fq(\013)378 b Fp(2)369 b Fj(R)465 b Ff(and)g Fq(\036)369 b Fp(2)f Fq(C)10773 41133 y Fk(1)10678 41944 y Fl(0)11770 41615 y Fs(\()p Fj(G)295 b Fp(n)g(f)p Fs(0)p Fp(g)p Fs(\))p Ff(.)598 b(Then)464 b(the)h(fol)66 b(lowing)464 b(ine)-66 b(quality)463 b(is)i(valid)g(:)-880 44931 y Fs(\(3.9\))11278 43123 y Fo(Z)12016 46130 y Fn(G)12982 44931 y Fq(K)14184 44383 y Fi(\013)14843 44931 y Fp(jr)16319 45130 y Fn(G)17063 44931 y Fq(\036)p Fp(j)18202 44383 y Fl(2)18728 44931 y Fq(dx)369 b Fp(\025)g Fs(\()22553 44033 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 22553 44626 5742 54 v 25098 45842 a(2)28427 44931 y(\))28933 44383 y Fl(2)29680 43123 y Fo(Z)30418 46130 y Fn(G)31384 44931 y Fq(K)32586 44383 y Fi(\013)33378 44033 y Fp(j)p Fq(v)48 b Fp(j)34793 43550 y Fl(2)p 33378 44626 1941 54 v 33484 45842 a Fq(K)34686 45459 y Fl(4)35451 44931 y Fq(\036)36221 44383 y Fl(2)36747 44931 y Fq(dx)-880 48352 y Ff(wher)-66 b(e)465 b Fq(K)95 b Fs(\()p Fq(x)p Fs(\))368 b(=)h(\()p Fp(j)p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))p Fp(j)11164 47870 y Fl(4)11984 48352 y Fs(+)295 b(16)p Fp(j)p Fq(z)59 b Fs(\()p Fq(x)p Fs(\))p Fp(j)17743 47870 y Fl(2)18269 48352 y Fs(\))18775 47870 y Fl(1)p Fi(=)p Fl(4)20242 48352 y Ff(.)597 b(Mor)-66 b(e)g(over,)463 b(the)i(c)-66 b(onstant)463 b Fs(\()35329 47768 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 35329 48046 3279 54 v 36733 48810 a(2)38740 48352 y Fs(\))39246 47870 y Fl(2)40237 48352 y Ff(is)i(sharp.)-880 50893 y(Pr)-66 b(o)g(of.)552 b Fs(The)434 b(pro)36 b(of)434 b(is)g(iden)-36 b(tical)434 b(to)f(the)g(Heisen)-36 b(b)36 b(erg)434 b(group)f(case.)16762 b Fc(\003)2244 54611 y Fs(4.)651 b Fr(Hard)-30 b(y-type)501 b(inequality)h(on)g(the)g (Carnot)g(gr)-30 b(oup)501 b(of)i(arbitrar)-89 b(y)502 b(step)448 56935 y Fs(In)542 b(this)g(section,)570 b(w)-36 b(e)542 b(consider)g(the)g(Carnot)g(group)g Fj(G)g Fs(of)h(an)-36 b(y)542 b(step)g Fq(k)587 b Fs(with)542 b(the)g(homogeneous)-880 58485 y(norm)531 b Fq(N)672 b Fs(=)535 b Fq(u)6615 58003 y Fl(1)p Fi(=)p Fl(\(2)p Fk(\000)p Fi(Q)p Fl(\))10816 58485 y Fs(asso)36 b(ciated)532 b(to)f(F)-108 b(olland's)531 b(solution)h Fq(u)f Fs(for)g(the)g(sub-Laplacian)f(\001)44635 58684 y Fn(G)45910 58485 y Fs([7].)871 b(W)-108 b(e)-880 60035 y(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)435 b(theorem:)-880 62576 y Fx(Theorem)715 b(4.1.)650 b Ff(L)-66 b(et)636 b Fj(G)i Ff(b)-66 b(e)636 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b(\(4.3\))g(and)f(using)g(the)g(fact)h(that)f Fq( )29871 31631 y Fl(2)30766 32113 y Fs(=)368 b Fq(N)33327 31631 y Fk(\000)p Fl(2)p Fi(\014)35158 32113 y Fq(\036)35928 31631 y Fl(2)36453 32113 y Fs(,)434 b(w)-36 b(e)434 b(get)g(the)f (follo)-36 b(wing)435 b(:)-880 35071 y Fo(Z)-142 38079 y Fn(G)823 36879 y Fq(N)2004 36331 y Fi(\013)2663 36879 y Fp(jr)4139 37078 y Fn(G)4884 36879 y Fq(\036)p Fp(j)6023 36331 y Fl(2)6548 36879 y Fq(dx)369 b Fp(\025)g Fs(\()p Fp(\000)p Fq(\014)12081 36331 y Fl(2)12831 36879 y Fp(\000)224 b Fq(\014)74 b Fs(\()p Fq(\013)232 b Fs(+)224 b Fq(Q)g Fp(\000)g Fs(2\))21585 35071 y Fo(Z)22323 38079 y Fn(G)23289 36879 y Fq(N)24470 36331 y Fi(\013)25261 35981 y Fp(jr)26737 36180 y Fn(G)27481 35981 y Fq(N)139 b Fp(j)29031 35499 y Fl(2)p 25261 36574 4296 54 v 26556 37791 a Fq(N)27737 37407 y Fl(2)29689 36879 y Fq(\036)30459 36331 y Fl(2)30985 36879 y Fq(dx)224 b Fp(\000)35262 35981 y Fq(\014)p 34014 36574 3305 54 v 34014 37791 a Fs(2)295 b Fp(\000)h Fq(Q)37672 35071 y Fo(Z)38410 38079 y Fn(G)39154 36879 y Fs(\(\001)40744 37078 y Fn(G)41488 36879 y Fq(u)p Fs(\))p Fq(N)43915 36331 y Fi(\013)p Fl(+)p Fi(Q)p Fk(\000)p Fl(2)47248 36879 y Fq(\036)48018 36331 y Fl(2)48544 36879 y Fq(dx:)-880 40266 y Fs(Since)433 b Fq(u)h Fs(is)g(the)f(fundamen)-36 b(tal)433 b(solution)h(of)g(sub-Laplacian)f(\001)30574 40465 y Fn(G)31751 40266 y Fs(on)h(the)f(Carnot)h(group)f Fj(G)p Fs(,)h(w)-36 b(e)434 b(get)11754 41789 y Fo(Z)12492 44796 y Fn(G)13236 43597 y Fs(\(\001)14826 43796 y Fn(G)15570 43597 y Fq(u)p Fs(\))p Fq(N)17997 43049 y Fi(\013)p Fl(+)p Fi(Q)p Fk(\000)p Fl(2)21331 43597 y Fq(\036)22101 43049 y Fl(2)22626 43597 y Fq(dx)369 b Fs(=)g Fq(N)26972 43049 y Fi(\013)p Fl(+)p Fi(Q)p Fk(\000)p Fl(2)30305 43597 y Fs(\(0\))p Fq(\036)32737 43049 y Fl(2)33262 43597 y Fs(\(0\))g(=)g(0)p Fq(:)-880 46984 y Fs(This)434 b(implies)g(that)7685 49399 y Fo(Z)8423 52407 y Fn(G)9389 51208 y Fq(N)10570 50659 y Fi(\013)11229 51208 y Fp(jr)12705 51407 y Fn(G)13449 51208 y Fq(\036)p Fp(j)14588 50659 y Fl(2)15113 51208 y Fq(dx)369 b Fp(\025)h Fs(\()p Fp(\000)p Fq(\014)20647 50659 y Fl(2)21467 51208 y Fp(\000)296 b Fq(\014)74 b Fs(\()p Fq(\013)303 b Fs(+)295 b Fq(Q)h Fp(\000)f Fs(2\))30578 49399 y Fo(Z)31316 52407 y Fn(G)32282 51208 y Fq(N)33463 50659 y Fi(\013)34255 50309 y Fp(jr)35731 50508 y Fn(G)36475 50309 y Fq(N)139 b Fp(j)38025 49827 y Fl(2)p 34255 50902 4296 54 v 35549 52119 a Fq(N)36730 51735 y Fl(2)38683 51208 y Fq(\036)39453 50659 y Fl(2)39978 51208 y Fq(dx:)-880 54323 y Fs(Cho)36 b(osing)20427 57110 y Fq(\014)443 b Fs(=)23117 56211 y(2)296 b Fp(\000)f Fq(Q)g Fp(\000)h Fq(\013)p 23117 56805 5764 54 v 25674 58021 a Fs(2)-880 59937 y(giv)-36 b(es)435 b(the)e(follo)-36 b(wing)435 b(sharp)e(inequalit)-36 b(y)-880 63412 y(\(4.5\))9942 61604 y Fo(Z)10680 64611 y Fn(G)11645 63412 y Fq(N)12826 62864 y Fi(\013)13485 63412 y Fp(jr)14961 63611 y Fn(G)15705 63412 y Fq(\036)p Fp(j)16844 62864 y Fl(2)17370 63412 y Fq(dx)369 b Fp(\025)g Fs(\()21195 62513 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 21195 63106 5742 54 v 23740 64323 a(2)27069 63412 y(\))27575 62864 y Fl(2)28322 61604 y Fo(Z)29060 64611 y Fn(G)30026 63412 y Fq(N)31207 62864 y Fi(\013)31998 62513 y Fp(jr)33474 62712 y Fn(G)34218 62513 y Fq(N)139 b Fp(j)35768 62031 y Fl(2)p 31998 63106 4296 54 v 33293 64323 a Fq(N)34474 63939 y Fl(2)36426 63412 y Fq(\036)37196 62864 y Fl(2)37722 63412 y Fq(dx:)49287 66773 y Fc(\003)-880 69736 y Fx(Remark)p Fs(.)560 b(In)377 b(the)g(ab)36 b(elian)377 b(case,)390 b(when)376 b Fj(G)369 b Fs(=)g Fj(R)23571 69254 y Fi(n)24575 69736 y Fs(with)377 b(the)g(ordinary)g(dilations,)389 b(one)378 b(has)f Fp(G)448 b Fs(=)368 b Fq(V)48413 69935 y Fl(1)49308 69736 y Fs(=)-880 71286 y Fj(R)79 70804 y Fi(n)1202 71286 y Fs(so)496 b(that)f Fq(Q)476 b Fs(=)e Fq(n)p Fs(.)766 b(No)-36 b(w)497 b(it)f(is)g(clear)g(that)g(the)f (inequalit)-36 b(y)497 b(\(4.1\))g(with)f(the)f(homogeneous)h(norm)-880 72835 y Fq(N)139 b Fs(\()p Fq(x)p Fs(\))368 b(=)h Fp(j)p Fq(x)p Fp(j)433 b Fs(and)g Fq(\013)378 b Fs(=)369 b(0)434 b(reco)-36 b(v)g(ers)434 b(the)f(Hardy's)h(inequalit)-36 b(y)434 b(\(1.1\).)p eop end %%Page: 8 8 TeXDict begin 8 7 bop -880 -714 a Fg(8)20957 b(ISMAIL)341 b(K)-28 b(OMBE)20847 1942 y Fr(References)-880 4046 y Fw([1])832 b(W.)355 b(Allegretto,)361 b Fh(Nonoscil)57 b(lation)382 b(the)-57 b(ory)383 b(of)h(el)57 b(liptic)383 b(e)-57 b(quations)384 b(of)f(or)-57 b(der)383 b(2n)g Fw(,)358 b(P)-31 b(aci\014c)357 b(journal)f(of)g(mathematics,)1119 5374 y Fb(64)370 b Fw(\(1976\),)i(no.)e(1,)g(1-16.)-880 6702 y([2])832 b(L.)443 b(D'Am)-31 b(brosio)445 b(,)462 b Fh(Some)j(Har)-57 b(dy)464 b(ine)-57 b(qualities)466 b(on)e(the)g(Heisenb)-57 b(er)g(g)466 b(gr)-57 b(oup)p Fw(,)462 b(Di\013eren)-31 b(tial)445 b(Equations)g Fb(40)f Fw(\(2004\))1119 8031 y(no.)370 b(4,)g(552-564,)-880 9359 y([3])832 b(Z.)457 b(Balogh,)480 b(I.)456 b(Holopainen)j(and)d(J.) g(T)-31 b(yson,)480 b Fh(Singular)d(solutions,)495 b(homo)-57 b(gene)g(ous)477 b(norms)f(and)g(quasic)-57 b(onformal)1119 10687 y(mappings)396 b(in)h(Carnot)e(gr)-57 b(oups)p Fw(,)369 b(Math.)i(Ann.)e(324)i(\(2002\),)h(no.)e(1,)g(159-186.)-880 12016 y([4])832 b(L.)381 b(Cap)31 b(ogna,)386 b(D.)380 b(Danielli)j(and)e(N.)g(Garofalo,)796 b Fh(Cap)-57 b(acitary)407 b(estimates)f(and)h(the)f(lo)-57 b(c)g(al)407 b(b)-57 b(ehavior)408 b(of)f(solutions)f(of)1119 13344 y(nonline)-57 b(ar)397 b(sub)-57 b(el)57 b(liptic)397 b(e)-57 b(quations)p Fw(,)370 b(Amer.)f(J.)h(Math.)g Fb(118)g Fw(\(1996\),)j(no.)d(6,)g (1153-1196)-880 14672 y([5])832 b(A.)271 b(Bon\014glioli,)294 b(E.)271 b(Lanconelli)h(and)f(F.)g(Uguzzoni,)292 b Fh(Uniform)306 b(Gaussian)f(estimates)g(of)h(the)f(fundamental)h(solutions)1119 16001 y(for)396 b(he)-57 b(at)396 b(op)-57 b(er)g(ators)395 b(on)h(Carnot)g(gr)-57 b(oups)p Fw(,)369 b(Adv.)h(Di\013eren)-31 b(tial)370 b(Equations)h Fb(7)f Fw(\(2002\),)i(1153-1192.)-880 17329 y([6])832 b(G.)432 b(B.)h(F)-92 b(olland,)450 b Fh(A)k(fundamental)h(solution)f(for)h(a)f(sub)-57 b(el)57 b(liptic)455 b(op)-57 b(er)g(ator)p Fw(,)448 b(Bull.)434 b(Amer.)f(Math.)g(So)31 b(c.)433 b Fb(79)h Fw(\(1973\),)1119 18657 y(373-376.)-880 19986 y([7])832 b(G.)389 b(B.)g(F)-92 b(olland,)395 b Fh(Sub)-57 b(el)57 b(liptic)416 b(estimates)e(and)g (function)h(sp)-57 b(ac)g(es)414 b(on)h(nilp)-57 b(otent)415 b(Lie)g(gr)-57 b(oups)p Fw(,)394 b(Arkiv.)c(f)31 b(\177)-584 b(ur)388 b(Math.)i Fb(13)1119 21314 y Fw(\(1975\),)372 b(161-207.)-880 22642 y([8])832 b(G.)446 b(B.)h(F)-92 b(olland)447 b(and)g(E.)g(Stein,)467 b(\\)446 b(Hardy)h(Spaces)f(on)h (Homogeneous)h(Groups",)466 b(Princeton)447 b(Univ)-31 b(ersit)g(y)448 b(Press,)1119 23971 y(Princeton,)370 b(NJ)-880 25299 y([9])832 b(B.)255 b(F)-92 b(ranc)-31 b(hi,)278 b(R.)256 b(Serapioni)g(and)f(F.)g(Serra)f(Cassano,)592 b Fh(R)-57 b(e)g(gular)292 b(Hyp)-57 b(ersurfac)g(es,)313 b(Intrinsic)292 b(Perimeter)f(and)g(Implicit)1119 26628 y(F)-85 b(unction)396 b(The)-57 b(or)g(em)397 b(in)g(Carnot)f(Gr)-57 b(oups)p Fw(,)369 b(Comm.)i(Anal.)g(Geom.)f Fb(11)g Fw(5)g(\(2003\),)i (909-944.)-880 27956 y([10])555 b(N.)327 b(Garofalo)i(and)d(E.)h (Lanconelli,)338 b Fh(F)-85 b(r)-57 b(e)g(quency)359 b(functions)e(on)g(the)g(Heisenb)-57 b(er)g(g)358 b(gr)-57 b(oup,)365 b(the)357 b(unc)-57 b(ertainty)358 b(principle)1119 29284 y(and)396 b(unique)h(c)-57 b(ontinuation)p Fw(,)371 b(Ann.)f(Inst.)f(F)-92 b(ourier\(Grenoble\))369 b Fb(40)h Fw(\(1990\),)j(313-356)-880 30613 y([11])555 b(J.)290 b(A.)h(Goldstein)f(and)g(Q.)g(S.)g(Zhang,)307 b Fh(On)323 b(a)g(de)-57 b(gener)g(ate)324 b(he)-57 b(at)323 b(e)-57 b(quation)324 b(with)f(a)f(singular)h(p)-57 b(otential)p Fw(,)306 b(J.)290 b(F)-92 b(unctional)1119 31941 y(Analysis,)370 b Fb(186)h Fw(\(2001\),)h(342-359.)-880 33269 y([12])555 b(J.)526 b(Heinonen)g(and)f(I.)g(Holopainen,)567 b Fh(Quasir)-57 b(e)g(gular)540 b(maps)f(on)g(Carnot)g(gr)-57 b(oups)p Fw(,)565 b(J.)525 b(Geom.)h(Anal.)g Fb(7)1130 b Fw(\(1997\),)1119 34598 y(109-148.)-880 35926 y([13])555 b(A.)262 b(Kaplan,)283 b Fh(F)-85 b(undamental)297 b(solutions)e(for)h(a)g(class)f(of)i(hyp) -57 b(o)g(el)57 b(liptic)297 b(PDE)e(gener)-57 b(ate)g(d)297 b(by)g(c)-57 b(omp)g(osition)297 b(of)g(quadr)-57 b(atic)1119 37254 y(forms)p Fw(,)369 b(T)-92 b(rans.)369 b(Amer.)h(Math.)g(So)31 b(c.)370 b Fb(258)p Fw(,)h(1)e(\(1980\),)k(147-153.)-880 38583 y([14])555 b(R.)332 b(Mon)-31 b(ti)334 b(and)e(F.)g(Serra)f (Cassano,)709 b Fh(Surfac)-57 b(es)363 b(me)-57 b(asur)g(es)362 b(in)h(Carnot-Car)-57 b(eth)-28 b(\023)-537 b(eo)-57 b(dory)360 b(sp)-57 b(ac)g(es)p Fw(,)340 b(Calc.)333 b(V)-92 b(ar.)332 b(P)-31 b(artial)1119 39911 y(Di\013er.)369 b(Equations)i Fb(13)f Fw(\(2001\),)i(339-376.)-880 41239 y([15])555 b(E.)627 b(Stein,)692 b(\\Harmonic)628 b(Analysis,)692 b(Real-V)-92 b(ariable)628 b(Metho)31 b(ds,)691 b(Orthgonalit)-31 b(y)-92 b(,)694 b(and)627 b(Oscillatory)h(In)-31 b(tegrals",)1119 42568 y(Princeton)370 b(Univ)-31 b(ersit)g(y)371 b(Press,)d(Princeton,) 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