Content-Type: multipart/mixed; boundary="-------------0301170832231" This is a multi-part message in MIME format. ---------------0301170832231 Content-Type: text/plain; name="03-22.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-22.keywords" Hardy inequality, Schrodinger operators, optimality ---------------0301170832231 Content-Type: application/postscript; name="+adimurthi-NODEA.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="+adimurthi-NODEA.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86f Copyright 2001 Radical Eye Software %%Title: +adimurthi-NODEA.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMMI12 CMR8 CMMI8 CMBX10 CMR7 CMR10 CMSY10 %%+ CMMI10 CMMI7 CMEX10 CMSY7 CMR6 CMR5 CMMI5 CMSY5 CMTI10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: %+ /usr/local/teTeX/bin/powerpc-apple-darwin-current/dvips -R -u %+ +pdftex.map -u +Ttbbold.map -u +wolfram.map -o %+ /tmp/altpdflatex.2797/+adimurthi-NODEA.ps +adimurthi-NODEA.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.01.17:1528 %%BeginProcSet: texc.pro %! 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All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 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 61 /slash put dup 82 /R put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 120 /x put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % 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 (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 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 61 /equal put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F D1F017CE45884D76EF2CB9BC5821FD25365DDEA1F9B0FF4CFF25B8E64D0747A3 7CAD14E0DBA3E3CA95F10F24B7D5D75451845F1FB7221D7794A860756CFBB3E7 704A52A22448C34812C3DBEDD41892577AABA7D555E9298C1A0F7DA638078167 F56E29672683C51CF1C003764A8E7AD9D8ADE77B4983F56FE2D12723AAD8BF36 682CFBB71B1D12210144D39DD841A971F71DB82AC6CD815987CDCF29ABC3CC96 5EEBD5D661F452C6E0C74F9ED8D0C5B3755551A172E0FE31EA02344176E32666 14B6853A1C303A5E818C2E455A6CF8FC9A66DC6E279101D61C523BD9DB8EB82F EAF4D7FDF6372383C0794C4568D079648689A199D4B65BA646CF95B7647E4BEC 83856C27A8EF177B3A686EDA6354FE9573E123C12EC4BA56A7E8BFB8F9B75147 9DD79A743968F36F7D0D479FA610F0816E6267E5CE327686A5485AB72201525C FB3B7CA10E1BF26E44C24E1696CB089CB0055BD692C89B237CF269F77A31DC81 0F4B75C8400ABCFDCEC6443CD0E81871CD71AA3064ABDE882C4C52322C27FA8B 41C689F827FB0F8AAF8022CF3C1F41C0B45601190C1328831857CBF9B1E7D1AA 246117E56D6B7938488055F4E63E2A1C8D57C17D213729C68349FEC2C3466F41 171E00413D39DF1F67BC15912F30775AFDF7FB3312587E20A68CF77AD3906040 842D63C45E19278622DD228C18ABDD024DD9613CDC0B109095DB0ADC3A3C0CB5 AB597D490189EA81239E39202CBC7A829EB9B313A8F962F7879D374ADF529BD0 5533EF977142F647AD2F5975BA7E340419116099B19ACCCC37C5512FE9CE3EA3 279C187D9C0F533CF4D6FB17A307995154DF46CB619A7125A248A5B024FCF3BC 607508E60FC2233992B352EB050248F0846A150FDE5E44F7516223279688586B 38779C5AF158C5BF4B711F49762AF37AD4AD3DF6EE54B995E857D45722E79436 43A27DA282750F8C92AD61AEA32EF05CB529725EF74A8ABB7296CA14E28DF73C 832BF11587F77DF685964CEC47B708AE2D80A2923D2A346A8EBB5F1830DBB451 BEB07519163EDA13001C6C2F2AEF4CB0D38DCE44CA3CE8767232ABEE09D68364 282BA23F9A7AB59671EC0C825AE658AC22D351186C53540A4BC3AC2879566A87 9E321966C807F7B83F245664342FA661F98E06B3C25DA53D64E4CFC353C08201 A3BA5C8FB1287D76EE2297A0590F2C69CA0099A6C87C09AA94D9A853827F8FA6 62519570A5A1E1F4CAB6E0A8F11B38D41B5A115AEE2577A2AC256676A1980DFA 4E7117CF65568617C25B968AB354716D3F38185F8C0618D8354E6A5996CB6814 7C4E960365408D7355AC7D6C7D5FD035CA02377EC66209ADB68C9B175C04D5BA 6477DEF603E82AB8DE62B7AC715A2E82F2F29B404DB92D470CB37D92AFFB4DFF 789821C03E2C3A59A905C528D0B50673D5397B9BF756573C8806E5F17414AA92 4CE621C7A176F45E93D07FD2556FC9D8B012CC1E9876314BD76F96C7C0095FE4 88B4E31B64818213E1D1C1A53FF55825E75A8196A3B315319E2F87BE795BA1F7 7F50C7C5D2A9CCE4D58FB81A19DDCA6A2C08E232469486C85524F91EF51BACAF 46293AC3B2C9BBDCF5849489F95983F3265AC67C8B04C7A168A633A58E2D8B37 AA24427776AEA661B13A3E1B377F0086D7412B432305B7590F40A228FC2C94AB E2748C76C91458C23C23A5E4231573A16735D6EE501007A38695FE541830D60D 42F6068FAFE9E47A515D504B615B93A1AA3BAEDE7147AAD2189DA970EB4A8A24 8EC7AECD960DA639E707A1F115AB1AA7B1909A1C9258D38F14E15C922FC3D8A1 094C6AC461C97F7A0A4C835E804AA33EB1668D8B0795681CBCE30E97CD933941 1B4D8D7C93249015DFABA76B9392D013CDDE4E6E7F6CF96025598A3A5287B328 BA5345EC813B0B6493CBE0AA435FC6CB8D71D7BB2726B4E3BCA2AA3D007C8776 8E65E9870D026F63089A1E824E3CD1936C634F2E82208429A946BA3AC5B2D93F 61C033586066851CFC8D7C723FA78337E215FE2F0CDD2E86F9C1DC6C1AA1F96B AAF96EC909590C6233F3D9F13361B9612471B10E1E1B582BEAAC3DF0873879A7 63864D461DB74DB3515902C67F39A68E4B05E4AA12EB9A9ECCC4F431476B62D1 ACB201448DD78850B1779A0B2636EAB65DE44C04C0A8161B063FD6822C1F72AD 0FA042BB6DA6F0666BDAD760FFF932861E483C5982C371047F51ADDF185B7450 3320EDFCC1D8950A0B9A222381C6C797418ADE4A61D0CA2C4B7ECF65DC900284 B8257D9DD2F17F56E2CA3ACB44602F86B35E78564709B0FA8667407C9C49A86F 9DE0B750E3ABEDF4B91E110E347FA2065921485FB99E9310425F3699ECC2F8B0 6202FFE3DD7F4FBB91B840AA08CF68F1522054A5B636162F12E63F8C477E10B1 E7FEF65FB94CE275D8787D1C55D382EB7D217DD39D4ABF1D621945D02B384042 3AF80426501840E2411408E0853A612A694B1AA5934CBB013B8521A795BFC29C C2E389FE3EA284B20B1EC805DA99CA197CDB34F267CCE0EF 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA17D1AFFF95F4224CF7ECEE45C BFB7C8C77C22A01C345078D28D3ECBF804CDC2FE5025FA0D05CCC5EFC0C4F87E CBED13DDDF8F34E404F471C6DD2E43331D73E89BBC71E7BF889F6293793FEF5A C9DD3792F032E37A364C70914843F7AA314413D022AE3238730B420A7E9D0CF5 D0E24F501451F9CDECE10AF7E14FF15C4F12F3FCA47DD9CD3C7AEA8D1551017D 23131C09ED104C052054520268A4FA3C6338BA6CF14C3DE3BAF2EA35296EE3D8 D6496277E11DFF6076FE64C8A8C3419FA774473D63223FFA41CBAE609C3D976B 93DFB4079ADC7C4EF07303F93808DDA9F651F61BCCF79555059A44CBAF84A711 6D98083CEF58230D54AD486C74C4A257FC703ACF918219D0A597A5F680B606E4 EF94ADF8BF91A5096A806DB64EC96636A98397D22A74932EB7346A9C4B5EE953 CB3C80AA634BFC28AA938C704BDA8DC4D13551CCFE2B2784BE8BF54502EBA9AF D49B79237B9C56310550BC30E9108BB06EAC755D6AA4E688EFE2A0AAB17F20FE 00CD0BFF1B9CB6BDA0FA3A29A3117388B6686657A150CE6421FD5D420F4F7FB5 B0DAA1BA19D638676E9CF159AC7325EF17B9F74E082BEF75E10A31C7011C0FFA 99B797CE549B5C45238DD0FADD6B99D233AC69282DF0D91EA2DBD08CE0083904 A6D968D5AE3BD159D01BDFF42D16111BC0A517C66B43972080D9DD4F3B9AE7FB 11B035CE715C1218B2D779761D8D7E9DEBE277531BD58F313EBD27E33BEF9DC5 50C7821A8BBC3B9FDF899D7EAA0B94493B97AFEAC503EB5ED7A7AB67C3039A0F BF0BA4B455D035FF3126F33A4DF59FF96752CF58DC0863924F6042B14DC2DEE4 099CAA764A8EFFEB78F7FFD276F0928BA32624C3A69ACEB8777A461298B201D3 4DA271476B8AC8966668D4E28A6FBA952F6D295AA80CC09B0CFD6BE840703B07 7ED93999EB5630C2C7CD9BA683ED457D1341C3AA702B5FA8593A51E8E9165FF6 1385693E19161FF1ADC1FE0BE16FD3AFCDC3916A309FCE6B3F2B362C9F958A16 821E1EB2DD899522DD3155C0B7A7F0876E4DA32FD30D8289266D9F05E77F1F1C D3BFD2D4D6C8B6FDE0E8957BDBBEC54458C70B96F4CD77E734EE0CF620684024 EB05DF0536FAFF10FC6F6050AB338D44EA9C38150D3779A550C505C06D417A71 62D41C2D23843E0A2A73F72D4B5286E3CB7B8695B2434479776A59CB3BC6D352 677CCABF4133E20F12D91CFD8978C09FB81DF801707194AB8FABA9A6599C20FA B36DB0A166490D20BB0BD53581A8715B099754E6FA0F7D2892F5573380DDFF1E F9C3DA9F74BDBEBBDFA5715AE23264F967BC7841CE58E5EE2764555978D904F6 E8DEF6ED66C478848FC35A9827ACEA1D7CE85C8626C162FF296869DE 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 68 /D put dup 92 /intersection put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 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 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 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 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 23 /nu put dup 26 /rho put dup 34 /epsilon 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 72 /H put dup 73 /I put dup 77 /M put dup 78 /N put dup 82 /R put dup 87 /W put dup 99 /c put dup 101 /e put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l 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 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 23 /nu put dup 26 /rho put dup 30 /phi put dup 31 /chi put dup 34 /epsilon put dup 39 /phi1 put dup 42 /arrowrighttophalf put dup 44 /arrowhookleft 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 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 82 /R put dup 83 /S put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 10 /Omega put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question 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 82 /R put dup 83 /S put dup 84 /T 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 120 /x put dup 121 /y put dup 122 /z put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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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 0 /Gamma 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 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 103 /g put dup 104 /h put dup 108 /l put dup 111 /o put dup 114 /r put dup 116 /t put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 44 /comma 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 65 /A put dup 69 /E put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 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 59 /comma put dup 112 /p put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 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 19 /acute 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 58 /colon put dup 64 /at 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 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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 87 /W put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 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 45 /hyphen put dup 65 /A put dup 72 /H put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e 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 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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667 614 a Fq(An)37 b(impro)m(v)m(ed)h(Hardy-Sob)s(olev)i (inequalit)m(y)e(in)g Fp(W)2865 578 y Fo(1)p Fn(;p)2997 614 y Fq(and)g(its)1123 764 y(application)h(to)e(Sc)m(hro)s(dinger)h (op)s(erator)1274 955 y Fm(Adim)m(urthi)1712 925 y Fl(1)1781 955 y Fm(and)32 b(Maria)g(J.)g(Esteban)2688 925 y Fl(2)523 1250 y Fm(1.)g(In)m(tro)s(duction.)c Fk(Let)g(0)23 b Fj(2)h Fk(\012)g Fj(\032)f Fi(I)-21 b(R)1755 1214 y Fh(n)1828 1250 y Fk(b)r(e)29 b(a)e(domain)h(and)g Fi(n)23 b Fj(\025)h Fk(3)p Fi(:)j Fk(Then)i(Hardy-Sob)r(olev)523 1400 y(inequalit)n(y)e (states)g(that)h(for)f(all)h Fi(u)22 b Fj(2)i Fi(W)1806 1360 y Fl(1)p Fh(;p)1794 1422 y Fl(0)1897 1400 y Fk(\(\012\))14 b(,)1382 1499 y Fg(Z)1382 1737 y Fl(\012)1479 1612 y Fj(jr)p Fi(u)p Fj(j)1642 1578 y Fh(p)1699 1612 y Fj(\000)1782 1495 y Fg(\022)1853 1556 y Fi(n)k Fj(\000)g Fi(p)p 1853 1593 194 4 v 1929 1669 a(p)2056 1495 y Fg(\023)2117 1512 y Fh(p)2169 1499 y Fg(Z)2169 1737 y Fl(\012)2276 1556 y Fj(j)p Fi(u)p Fj(j)2370 1526 y Fh(p)p 2276 1593 133 4 v 2276 1669 a Fj(j)p Fi(x)p Fj(j)2369 1645 y Fh(p)2441 1612 y Fj(\025)23 b Fk(0)f Fi(;)689 b Fk(\(1.1\))523 1917 y(and)696 1825 y Fg(\020)756 1880 y Fh(n)p Ff(\000)p Fh(p)p 756 1898 128 4 v 802 1945 a(p)893 1825 y Fg(\021)942 1842 y Fh(p)1020 1917 y Fk(is)39 b(the)h(b)r(est)g(constan)n(t)e(in)i (\(1.1\))f(and)g(it)h(is)f(nev)n(er)f(ac)n(hiev)n(ed.)g(In)i(view)f(of) 523 2066 y(this,)33 b(is)f(there)h(a)f(scop)r(e)g(of)g(impro)n(ving)g (this)h(inequalit)n(y)f(b)n(y)g(replacing)f(the)i(zero)f(term)g(b)n(y) 523 2216 y(some)i(non)n(trivial)g(functional)h(of)g Fi(u)g Fk(in)g(\(1\)?)g(Recen)n(tly)g(there)g(is)f(a)h(considerable)e(in)n (terest)523 2365 y(in)39 b(this)f(question)g(and)g(one)g(of)g(the)h (imp)r(ortan)n(t)f(impro)n(v)n(emen)n(ts)f(is)h(obtained)g(b)n(y)g (Brezis)523 2514 y(and)27 b(V\023)-42 b(azquez)27 b([7].)g(They)g(sho)n (w)n(ed)f(that)h(if)h(\012)f(is)g(a)g(b)r(ounded)h(domain,)f(then)g (there)g(exists)g(a)523 2664 y(constan)n(t)g Fi(C)i(>)23 b Fk(0)k(suc)n(h)g(that)h(for)f(all)h Fi(u)22 b Fj(2)i Fi(H)1938 2634 y Fl(1)1931 2684 y(0)1975 2664 y Fk(\(\012\))p Fi(;)1263 2780 y Fg(Z)1262 3018 y Fl(\012)1360 2893 y Fj(jr)p Fi(u)p Fj(j)1523 2859 y Fl(2)1578 2893 y Fj(\000)1661 2776 y Fg(\022)1733 2837 y Fi(n)18 b Fj(\000)g Fk(2)p 1733 2874 193 4 v 1808 2950 a(2)1935 2776 y Fg(\023)1996 2793 y Fl(2)2047 2780 y Fg(Z)2047 3018 y Fl(\012)2154 2837 y Fj(j)p Fi(u)p Fj(j)2248 2807 y Fl(2)p 2154 2874 131 4 v 2154 2950 a Fj(j)p Fi(x)p Fj(j)2247 2926 y Fl(2)2318 2893 y Fj(\025)23 b Fi(C)2485 2780 y Fg(Z)2484 3018 y Fl(\012)2582 2893 y Fj(j)p Fi(u)p Fj(j)2676 2859 y Fl(2)2713 2893 y Fi(:)569 b Fk(\(1.2\))523 3160 y(F)-7 b(urthermore)23 b(if)h Fi(\025)p Fk(\(\012\))h(denotes)f(the)g(b)r(est)h(c)n(hoice)e (for)g(the)i(constan)n(t)e Fi(C)30 b Fk(in)25 b(inequalit)n(y)e (\(1.2\),)523 3309 y(then)38 b Fi(\025)p Fk(\(\012\))h(is)f(nev)n(er)f (ac)n(hiev)n(ed.)f(Again)i(this)g(sho)n(ws)e(that)i(there)g(is)g(a)f (scop)r(e)g(of)h(further)523 3459 y(impro)n(ving)29 b(this)i(inequalit) n(y)f(and)g(in)h(this)f(direction)g(Brezis)g(and)g(V\023)-42 b(azquez)30 b([7])g(raised)f(the)523 3608 y(follo)n(wing)f(question:)g (\\what)g(is)h(the)g(b)r(est)g(p)r(ossible)g(reminder)f(term)h(that)g (one)f(can)h(exp)r(ect)523 3758 y(in)f(\(1.2\)?")763 3990 y(Recen)n(tly)35 b(this)g(question)f(w)n(as)g(answ)n(ered)f(in)i ([2])f(and)h([4,5],)f(where)g(the)h(follo)n(wing)523 4140 y(inequalit)n(y)27 b(w)n(as)g(pro)n(v)n(ed.)763 4372 y(Let)i(\012)h(b)r(e)f(a)g(b)r(ounded)g(domain)g(and)g(1)c Fi(<)g(p)h Fj(\024)f Fi(n:)k Fk(Let)g Fi(R)h Fk(b)r(e)g(large)e(enough) g(\(to)h(b)r(e)523 4522 y(precised)h(later\).)h(Then)g(there)g(exist)g (a)f(constan)n(t)g Fi(C)35 b(>)28 b Fk(0)j(dep)r(ending)g(on)g Fi(n;)14 b(p)30 b Fk(and)h Fi(R)e(>)f Fk(0)p 523 4587 1182 4 v 615 4641 a Fe(1)650 4664 y Fo(TIFR)55 b(Cen)n(tre,)g(I)r(ISc)g (Campus,)g(P)-6 b(.B.)53 b(Bo)n(x)i(1234,)g(Bangalore)h(-)e(560)h(012,) g(India.)f(E-mail:)523 4743 y(aditi@math.tifrbng.res.in)615 4799 y Fe(2)650 4823 y Fo(CEREMADE,)35 b(Univ)n(ersit)n(\023)-33 b(e)36 b(P)n(aris)g(IX-Dauphine,)g(Place)h(du)g(Mar)n(\023)-33 b(ec)n(hal)36 b(de)h(Lattre)g(de)g(T)-6 b(assign)n(y)g(.)523 4901 y(75775,)25 b(P)n(aris)d(Cedex)j(16,)f(F)-6 b(rance.)24 b(E-mail:)e(esteban@ceremade.dauphine.fr)1979 5150 y Fk(1)p eop %%Page: 2 2 2 1 bop 523 614 a Fk(suc)n(h)27 b(that)1019 734 y Fg(Z)1018 972 y Fl(\012)1116 847 y Fj(jr)p Fi(u)p Fj(j)1279 813 y Fh(p)1335 847 y Fj(\000)1418 730 y Fg(\022)1489 791 y Fi(n)19 b Fj(\000)f Fi(p)p 1489 828 194 4 v 1565 904 a(p)1692 730 y Fg(\023)1754 747 y Fh(p)1806 734 y Fg(Z)1805 972 y Fl(\012)1913 791 y Fj(j)p Fi(u)p Fj(j)2007 761 y Fh(p)p 1913 828 133 4 v 1913 904 a Fj(j)p Fi(x)p Fj(j)2006 880 y Fh(p)2078 847 y Fj(\025)k Fi(C)2244 734 y Fg(Z)2244 972 y Fl(\012)2572 791 y Fj(j)p Fi(u)p Fj(j)2666 761 y Fh(p)p 2351 828 573 4 v 2351 904 a Fj(j)p Fi(x)p Fj(j)2444 880 y Fh(p)2497 904 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2881 863 y Fh(\015)2957 847 y Fi(;)325 b Fk(\(1.3\))523 1149 y(for)27 b(ev)n(ery)f Fi(u)d Fj(2)g Fi(W)1108 1109 y Fl(1)p Fh(;p)1096 1171 y Fl(0)1200 1149 y Fk(\(\012\))28 b(if)g(and)f(only)g(if)523 1298 y(\(i\))h Fi(\015)g Fj(\025)23 b Fk(2)k(when)h(1)22 b Fi(<)h(p)g(<)f(n)14 b Fk(,)523 1448 y(\(ii\))28 b Fi(\015)g Fj(\025)23 b Fi(n)k Fk(when)h Fi(p)23 b Fk(=)f Fi(n:)763 1680 y Fk(Moreo)n(v)n(er)30 b(it)k(w)n(as)e(sho)n(wn)h(in)g([3])g(that)h(in)f(case)g(\(i\),)g Fi(C)39 b Fk(=)2686 1647 y Fl(1)p 2686 1661 34 4 v 2686 1709 a(4)2763 1680 y Fk(if)33 b Fi(p)f Fk(=)g(2)h(and)g Fi(\015)k Fk(=)32 b(2)523 1830 y(and)23 b Fi(C)29 b Fk(=)856 1762 y Fg(\000)904 1797 y Fh(n)p Ff(\000)p Fl(1)p 904 1811 127 4 v 946 1858 a Fh(n)1040 1762 y Fg(\001)1078 1780 y Fh(n)1146 1830 y Fk(in)23 b(case)f(\(ii\),)i(these)f(b)r(eing)g (the)h(b)r(est)f(p)r(ossible)g(constan)n(ts.)f(F)-7 b(urthermore,)523 1979 y(the)28 b(p)r(erturb)r(ed)f(eigen)n(v)-5 b(alue)26 b(problem)h(corresp)r(onding)e(to)i(the)g(Euler-Lagrange)d(equations) 523 2129 y(asso)r(ciated)d(to)h(\(1.3\))g(w)n(as)f(studied)h(in)h([3],) e(where)h(one)g(can)f(\014nd)i(the)f(conditions)g(on)g(the)g(p)r(er-) 523 2278 y(turb)r(ed)g(co)r(e\016cien)n(ts)f(to)h(guaran)n(tee)d(the)j (existence)g(of)f(an)h(eigen)n(v)-5 b(alue)20 b(of)i(the)g(corresp)r (onding)523 2427 y(op)r(erator)k(in)i Fi(H)1031 2397 y Fl(1)1024 2448 y(0)1068 2427 y Fk(\(\012\).)g(This)g(w)n(as)e (extended)i(to)f Fi(W)2136 2387 y Fl(1)p Fh(;p)2124 2450 y Fl(0)2228 2427 y Fk(\(\012\))h(b)n(y)f([10].)763 2660 y(Our)g(in)n(terest)g(in)h(this)g(direction)f(is)h(t)n(w)n(ofold:)523 2809 y(\(A\))112 b(What)27 b(is)h(the)g(corresp)r(onding)e(result)h(of) g(\(1.3\))h(in)g Fi(W)2465 2779 y Fl(1)p Fh(;p)2556 2809 y Fk(\(\012\))g(if)g(\012)g(is)f(b)r(ounded?)523 2959 y(\(B\))111 b(What)28 b(is)g(the)g(corresp)r(onding)d(result)i(if)i (\012)23 b(=)f Fi(I)-21 b(R)2307 2923 y Fh(n)2352 2959 y Fk(?)763 3191 y(Since)29 b(the)g(constan)n(t)g(functions)g(are)f(in)h Fi(W)2149 3161 y Fl(1)p Fh(;p)2240 3191 y Fk(\(\012\),)h(it)f(is)g (clear)e(that)j(some)e(negativ)n(e)523 3341 y(term)41 b(should)f(b)r(e)h(added)f(to)g(the)h(righ)n(t)f(hand)h(side)f(of)g (\(1.3\).)h(The)f(\014rst)h(result)f(in)h(this)523 3490 y(direction)31 b(w)n(as)g(obtained)g(b)n(y)g(Lieb)h(and)f(Y)-7 b(au)32 b([9])f(for)g(the)h(case)e Fi(p)g Fk(=)f(2)i(when)h(\012)f(is)h (a)f(ball.)523 3640 y(More)18 b(recen)n(tly)g(this)h(question)g(w)n(as) e(answ)n(ered)h(in)h([1])f(for)h(the)g(case)f Fi(p)23 b Fk(=)f(2)d(for)f(ev)n(ery)f(b)r(ounded)523 3789 y(domain)25 b(\012.)h(The)f(pro)r(of)g(in)h(the)f(ab)r(o)n(v)n(e)g(pap)r(er)g(hea)n (vily)f(dep)r(ends)i(on)f(\(1.3\))g(and)g(the)h(Hilb)r(ert)523 3938 y(space)g(nature)g(of)h Fi(W)1190 3898 y Fl(1)p Fh(;)p Fl(2)1178 3961 y(0)1280 3938 y Fk(\(\012\))p Fi(:)g Fk(This)g(metho)r(d)g(will)g(not)f(allo)n(w)g(an)g(extension)g(to)h Fi(W)3131 3908 y Fl(1)p Fh(;p)3249 3938 y Fk(spaces)523 4088 y(for)g Fi(p)c Fj(6)p Fk(=)g(2)p Fi(:)763 4320 y Fk(In)j(this)f(pap)r(er)g(w)n(e)h(answ)n(er)e(the)h(t)n(w)n(o)g(ab)r(o) n(v)n(e)f(questions)h(b)n(y)g(using)g(the)h(nature)f(of)g(the)523 4470 y(Green's)33 b(function)g(for)g(the)g(Laplace)f(op)r(erator.)f(In) j(Section)f(3,)f(w)n(e)h(extend)g(these)g(results)523 4619 y(to)23 b(arbitrary)e(compact)i(Riemannian)f(manifolds.)h(F)-7 b(urthermore)22 b(w)n(e)h(study)g(the)h(eigen)n(v)-5 b(alue)523 4769 y(problem)27 b(for)g(the)h(p)r(erturb)r(ed)g(Sc)n (hr\177)-42 b(odinger)26 b(op)r(erator.)1979 5150 y(2)p eop %%Page: 3 3 3 2 bop 763 614 a Fk(In)30 b(order)f(to)g(state)h(our)f(main)h (results,)g(let)g(us)g(in)n(tro)r(duce)f(the)h(follo)n(wing)f (notation:)523 764 y(let)42 b Fi(e)696 734 y Fl(\(0\))808 764 y Fk(=)22 b(1)p Fi(;)37 b(e)1036 734 y Fl(\(1\))1148 764 y Fk(=)22 b Fi(e;)37 b(e)1373 734 y Fl(\()p Fh(k)q Fl(\))1488 764 y Fk(=)23 b Fi(e)1615 734 y Fh(e)1646 709 y Fd(\()p Fc(k)q Fb(\000)p Fd(1\))1855 764 y Fk(for)51 b Fi(k)26 b Fj(\025)c Fk(1)p Fi(:)27 b Fk(Let)h Fi(a)23 b(>)g Fk(0)k(and)g(de\014ne)968 941 y(log)1075 961 y Fl(\(1\))1178 941 y Fi(a)137 b Fk(=)83 b(log\()p Fi(a)p Fk(\))p Fi(;)968 1050 y Fk(log)1075 1070 y Fl(\()p Fh(k)q Fl(\))1168 1050 y Fk(\()p Fi(a)p Fk(\))g(=)g(log\(log)1754 1070 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))1932 1050 y Fk(\()p Fi(a)p Fk(\)\))14 b Fi(;)60 b Fk(if)30 b Fi(a)23 b(>)g(e)2441 1019 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))2669 1050 y Fk(for)k Fi(k)f Fj(\025)c Fk(2)14 b Fi(;)968 1212 y Fk(log)1075 1175 y Fl(\()p Fh(k)q Fl(\))1168 1212 y Fk(\()p Fi(a)p Fk(\))83 b(=)1546 1133 y Fh(k)1525 1150 y Fg(Q)1507 1285 y Fh(j)s Fl(=1)1636 1212 y Fk(log)1743 1232 y Fl(\()p Fh(j)s Fl(\))1830 1212 y Fk(\()p Fi(a)p Fk(\))14 b Fi(;)60 b Fk(if)53 b Fi(a)23 b(>)g(e)2330 1182 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))2507 1212 y Fi(:)523 1463 y Fk(Finally)-7 b(,)28 b(w)n(e)f(de\014ne)h(the)g(space)f Fj(D)1615 1433 y Fl(1)p Fh(;)p Fl(2)1705 1463 y Fk(\()p Fi(I)-21 b(R)1816 1427 y Fh(n)1861 1463 y Fk(\))28 b(b)n(y:)1026 1696 y Fj(D)1092 1662 y Fl(1)p Fh(;)p Fl(2)1182 1696 y Fk(\()p Fi(I)-21 b(R)1294 1660 y Fh(n)1339 1696 y Fk(\))23 b(:=)g Fj(f)p Fi(u)f Fj(2)i Fi(L)1753 1662 y Fl(2)p Fh(N)q(=)p Fl(\()p Fh(N)6 b Ff(\000)p Fl(2\))2072 1696 y Fk(\()p Fi(I)-21 b(R)2184 1660 y Fh(N)2247 1696 y Fk(\))23 b(;)51 b Fj(r)p Fi(u)23 b Fj(2)g Fi(L)2651 1662 y Fl(2)2688 1696 y Fk(\()p Fi(I)-21 b(R)2799 1660 y Fh(N)2862 1696 y Fk(\))p Fj(g)14 b Fi(;)523 1928 y Fk(then,)28 b(our)f(main)h(results) f(are)f(the)i(follo)n(wing.)523 2161 y Fm(Theorem)j(1.1.)c Fk(Let)g Fi(k)f Fj(\025)d Fk(1)k(b)r(e)h(an)f(in)n(teger)g(and)h(1)22 b Fi(<)h(p)g Fj(\024)f Fi(n:)28 b Fk(Then,)763 2393 y(\(1\))40 b(Let)g(\012)f(b)r(e)i(a)e(b)r(ounded)h(domain)f(with)i(smo)r(oth)e(b)r (oundary)g(and)g(0)k Fj(2)h Fk(\012)p Fi(:)53 b Fk(If)523 2543 y Fi(R)24 b(>)e(e)736 2513 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))930 2543 y Fk(sup)928 2612 y Fh(x)p Ff(2)p Fl(\012)1071 2543 y Fj(j)p Fi(x)p Fj(j)p Fk(,)28 b(there)e(exist)g(constan)n(ts)g Fi(C)j Fk(=)23 b Fi(C)6 b Fk(\()p Fi(p;)14 b(n)p Fk(\))23 b Fi(>)g Fk(0)40 b(and)49 b Fi(\025)24 b Fk(=)f Fi(\025)p Fk(\(\012)p Fi(;)14 b(p;)g(R)q Fk(\))23 b Fi(<)f Fk(0)523 2701 y(suc)n(h)27 b(that)h(for)f(all)h Fi(u)22 b Fj(2)i Fi(W)1372 2671 y Fl(1)p Fh(;p)1463 2701 y Fk(\(\012\))p Fi(;)766 2820 y Fg(Z)765 3059 y Fl(\012)862 2933 y Fj(jr)p Fi(u)p Fj(j)1025 2899 y Fh(p)1147 2933 y Fj(\000)1294 2816 y Fg(\022)1365 2877 y Fi(n)19 b Fj(\000)f Fi(p)p 1365 2914 194 4 v 1441 2990 a(p)1568 2816 y Fg(\023)1629 2834 y Fh(p)1682 2820 y Fg(Z)1681 3059 y Fl(\012)1789 2877 y Fj(j)p Fi(u)p Fj(j)1883 2847 y Fh(p)p 1789 2914 133 4 v 1789 2990 a Fj(j)p Fi(x)p Fj(j)1882 2966 y Fh(p)1954 2933 y Fj(\025)1294 3295 y Fi(C)1373 3182 y Fg(Z)1373 3420 y Fl(\012)1470 3103 y Fg(0)1470 3249 y(B)1470 3303 y(@)1584 3191 y Fh(k)1543 3216 y Fg(X)1545 3393 y Fh(j)s Fl(=1)1937 3239 y Fk(1)p 1687 3276 544 4 v 1687 3306 a Fg(\020)1736 3398 y Fk(log)1843 3361 y Fl(\()p Fh(j)s Fl(\))1944 3398 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)2143 3306 y Fg(\021)2192 3323 y Fl(2)2240 3103 y Fg(1)2240 3249 y(C)2240 3303 y(A)2336 3239 y Fj(j)p Fi(u)p Fj(j)2430 3209 y Fh(p)p 2336 3276 133 4 v 2336 3352 a Fj(j)p Fi(x)p Fj(j)2429 3328 y Fh(p)2497 3295 y Fk(+)g Fi(\025)2644 3182 y Fg(Z)2623 3421 y Fh(@)t Fl(\012)2742 3295 y Fj(j)p Fi(u)p Fj(j)2836 3261 y Fh(p)2888 3295 y Fi(;)37 b Fk(\()p Fi(p)23 b(<)g(n)p Fk(\))90 b(\(1.4\))772 3746 y Fg(Z)772 3984 y Fl(\012)869 3859 y Fj(jr)p Fi(u)p Fj(j)1032 3825 y Fh(n)1160 3859 y Fj(\000)1308 3742 y Fg(\022)1379 3803 y Fi(n)18 b Fj(\000)g Fk(1)p 1379 3840 193 4 v 1450 3916 a Fi(n)1581 3742 y Fg(\023)1643 3746 y(Z)1642 3984 y Fl(\012)1915 3803 y Fj(j)p Fi(u)p Fj(j)2009 3773 y Fh(n)p 1749 3840 470 4 v 1749 3858 a Fg(\020)1799 3950 y Fj(j)p Fi(x)p Fj(j)c Fk(log)2051 3917 y Fh(R)p 2037 3931 78 4 v 2037 3979 a Ff(j)p Fh(x)p Ff(j)2125 3858 y Fg(\021)2174 3875 y Fh(n)2252 3859 y Fj(\025)1308 4207 y Fi(C)1387 4094 y Fg(Z)1386 4332 y Fl(\012)1470 4041 y Fg(0)1470 4190 y(@)1584 4104 y Fh(k)1542 4129 y Fg(X)1545 4305 y Fh(j)s Fl(=2)1941 4151 y Fk(1)p 1686 4188 552 4 v 1686 4206 a Fg(\020)1736 4298 y Fk(log)1843 4261 y Fl(\()p Fh(j)s Fl(\))1944 4298 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)2143 4206 y Fg(\021)2192 4223 y Fh(n)2247 4041 y Fg(1)2247 4190 y(A)2344 4151 y Fj(j)p Fi(u)p Fj(j)2438 4121 y Fh(n)p 2344 4188 139 4 v 2344 4264 a Fj(j)p Fi(x)p Fj(j)2437 4240 y Fh(n)2511 4207 y Fk(+)k Fi(\025)2644 4094 y Fg(Z)2624 4333 y Fh(@)t Fl(\012)2743 4207 y Fj(j)p Fi(u)p Fj(j)2837 4173 y Fh(n)2854 4207 y Fi(;)37 b Fk(\()p Fi(p)23 b Fk(=)f Fi(n)p Fk(\))125 b(\(1.5\))763 4600 y(\(2\))115 b(Let)32 b Fi(R)f(>)e(e)1364 4570 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))1573 4600 y Fk(and)j Fi(\032)p Fk(\()p Fi(x)p Fk(\))f(=)f(max)o(\()p Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fi(;)14 b(R)q Fj(j)p Fi(x)p Fj(j)p Fk(\))p Fi(:)32 b Fk(Then)g(there)g(exist)g(con-)523 4750 y(stan)n(ts)j Fi(C)41 b Fk(=)35 b Fi(C)6 b Fk(\()p Fi(p;)14 b(n)p Fk(\))36 b Fi(>)f Fk(0)g(and)g Fi(\025)g Fk(=)g Fi(\025)p Fk(\()p Fi(R)q Fk(\))i Fi(<)e Fk(0)f(suc)n(h)h(that)g (for)g(all)g Fi(u)f Fj(2)i Fi(W)3089 4720 y Fl(1)p Fh(;p)3180 4750 y Fk(\()p Fi(I)-21 b(R)3292 4713 y Fh(n)3337 4750 y Fk(\))p Fi(;)35 b Fk(if)523 4899 y Fi(p)23 b(<)g(n)p Fk(,)1979 5150 y(3)p eop %%Page: 4 4 4 3 bop 885 703 a Fg(Z)885 941 y Fl(\012)982 816 y Fj(jr)p Fi(u)p Fj(j)1145 782 y Fh(p)1267 816 y Fj(\000)1414 699 y Fg(\022)1485 760 y Fi(n)19 b Fj(\000)f Fi(p)p 1485 797 194 4 v 1561 873 a(p)1688 699 y Fg(\023)1749 716 y Fh(p)1802 703 y Fg(Z)1801 941 y Fl(\012)1908 760 y Fj(j)p Fi(u)p Fj(j)2002 730 y Fh(p)p 1908 797 133 4 v 1908 873 a Fj(j)p Fi(x)p Fj(j)2001 849 y Fh(p)2073 816 y Fj(\025)1414 1178 y Fi(C)1503 1065 y Fg(Z)1474 1303 y Fh(I)-18 b(R)1536 1284 y Fc(n)1609 986 y Fg(0)1609 1132 y(B)1609 1185 y(@)1724 1074 y Fh(k)1682 1099 y Fg(X)1685 1276 y Fh(j)s Fl(=1)2071 1121 y Fk(1)p 1826 1159 532 4 v 1826 1189 a Fg(\020)1875 1281 y Fk(log)1983 1244 y Fl(\()p Fh(j)s Fl(\))2083 1281 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2269 1189 y Fg(\021)2320 1206 y Fl(2)2367 986 y Fg(1)2367 1132 y(C)2367 1185 y(A)2464 1121 y Fj(j)p Fi(u)p Fj(j)2558 1091 y Fh(p)p 2464 1159 133 4 v 2464 1235 a Fj(j)p Fi(x)p Fj(j)2557 1211 y Fh(p)2624 1178 y Fk(+)18 b Fi(\025)2809 1065 y Fg(Z)2750 1307 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)2944 1178 y Fj(j)p Fi(u)p Fj(j)3038 1143 y Fh(p)3090 1178 y Fi(;)192 b Fk(\(1.6\))523 1453 y(and)27 b(if)i Fi(p)22 b Fk(=)h Fi(n)p Fk(,)922 1542 y Fg(Z)894 1780 y Fh(I)-18 b(R)955 1761 y Fc(n)1015 1655 y Fj(jr)p Fi(u)p Fj(j)1178 1620 y Fh(n)1292 1655 y Fj(\000)1426 1538 y Fg(\022)1497 1598 y Fi(n)18 b Fj(\000)g Fk(1)p 1497 1636 193 4 v 1568 1712 a Fi(n)1699 1538 y Fg(\023)1761 1555 y Fh(n)1788 1542 y Fg(Z)1760 1780 y Fh(I)-18 b(R)1821 1761 y Fc(n)2081 1598 y Fj(j)p Fi(u)p Fj(j)2175 1568 y Fh(n)p 1904 1636 494 4 v 1904 1712 a Fk(\()p Fj(j)p Fi(x)p Fj(j)14 b Fk(log)h Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2352 1671 y Fh(n)2430 1655 y Fj(\025)1439 1991 y Fi(C)1528 1878 y Fg(Z)1499 2116 y Fh(I)-18 b(R)1562 2097 y Fc(n)1635 1825 y Fg(0)1635 1974 y(@)1749 1888 y Fh(k)1707 1913 y Fg(X)1710 2089 y Fh(j)s Fl(=2)2084 1935 y Fk(1)p 1851 1972 508 4 v 1851 1990 a Fg(\020)1901 2082 y Fk(log)2008 2045 y Fl(\()p Fh(j)s Fl(\))2109 2082 y Fi(\032)p Fk(\()p Fi(x)p Fk(\))2263 1990 y Fg(\021)2313 2007 y Fh(n)2368 1825 y Fg(1)2368 1974 y(A)2465 1935 y Fj(j)p Fi(u)p Fj(j)2559 1905 y Fh(n)p 2465 1972 139 4 v 2465 2048 a Fj(j)p Fi(x)p Fj(j)2558 2024 y Fh(n)2632 1991 y Fk(+)18 b Fi(\025)2789 1878 y Fg(Z)2731 2120 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)2925 1991 y Fj(j)p Fi(u)p Fj(j)3019 1957 y Fh(n)3063 1991 y Fi(:)219 b Fk(\(1.7\))523 2349 y(Moreo)n(v)n(er,)22 b(if)j Fi(p)e Fk(=)f(2)p Fi(;)j Fk(then)g Fi(C)k Fk(=)23 b(1)p Fi(=)p Fk(4)g(is)h(the)h(b)r(est)g(constan)n(t)f(in)h (inequalities)f(\(1.4\))h(to)f(\(1.7\).)523 2498 y(F)-7 b(or)27 b(further)h(impro)n(v)n(emen)n(ts)e(of)h(these)h(inequalities)f (see)g(Section)h(3.)763 2731 y(Motiv)-5 b(ated)22 b(b)n(y)g(the)h (results)f(of)g([6],)g([3],)g([1],)h(w)n(e)f(also)f(study)h(the)h (eigen)n(v)-5 b(alue)21 b(problem)523 2880 y(for)27 b(the)i(corresp)r (onding)d(p)r(erturb)r(ed)i(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erator.)g(Our)i(results)f(also)g(extend)h(to)523 3030 y(the)g Fi(p)p Fk(-Laplacian)e(as)h(it)h(w)n(as)e(done)i(in)g ([S].)763 3262 y(Let)37 b Fi(B)t Fk(\()p Fi(r)r Fk(\))g(denote)g(the)f (ball)h(of)f(radius)f Fi(r)40 b Fk(cen)n(tered)35 b(at)h(the)h(origin)e (and)i(let)f Fi(k)j Fk(b)r(e)523 3412 y(some)29 b(in)n(teger,)g Fi(k)h Fj(\025)d Fk(1)i(\()p Fi(k)h Fj(\025)c Fk(2,)k(if)g Fi(n)d Fk(=)g(2\).)i(Let)h Fi(q)s(;)14 b(\021)s(;)g(f)39 b Fk(b)r(e)30 b(measurable)f(functions)h(on)g Fi(I)-21 b(R)3431 3375 y Fh(n)523 3561 y Fk(satisfying)27 b(the)h(follo)n(wing)e (h)n(yp)r(othesis:)792 3707 y(\()p Fi(H)893 3719 y Fl(1)930 3707 y Fk(\))474 b(0)83 b Fj(\024)g Fi(q)s Fk(\()p Fi(x)p Fk(\))24 b Fj(\024)36 b Fk(1)p Fi(;)h(\021)26 b Fj(2)d Fi(L)2289 3677 y Ff(1)2359 3707 y Fk(\()p Fi(I)-21 b(R)2470 3671 y Fh(n)2534 3707 y Fj(n)18 b Fi(B)t Fk(\()p Fi(r)r Fk(\)\))p Fi(;)85 b Fj(8)23 b Fi(r)j(>)d Fk(0)792 3964 y(\()p Fi(H)893 3976 y Fl(2)930 3964 y Fk(\))167 b(0)22 b Fj(\024)h Fi(\021)s Fk(\()p Fi(x)p Fk(\))125 b Fj(\024)1719 3932 y Fl(1)p 1719 3946 34 4 v 1719 3993 a(4)1810 3886 y Fh(k)1784 3902 y Fg(P)1775 4039 y Fh(l)p Fl(=1)2120 3932 y(1)p 1905 3946 463 4 v 1905 4000 a(\()p Ff(j)p Fh(x)p Ff(j)10 b Fl(log)2105 3980 y Fd(\()p Fc(l)p Fd(\))2185 4000 y Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))2335 3983 y Fd(2)2401 3964 y Fi(;)480 b Fk(if)30 b Fi(n)23 b Fj(\025)f Fk(3)p Fi(:)1561 4275 y Fj(\024)1719 4242 y Fl(1)p 1719 4256 34 4 v 1719 4304 a(4)1810 4196 y Fh(k)1784 4213 y Fg(P)1775 4350 y Fh(l)p Fl(=2)2120 4242 y(1)p 1905 4256 463 4 v 1905 4310 a(\()p Ff(j)p Fh(x)p Ff(j)10 b Fl(log)2105 4290 y Fd(\()p Fc(l)p Fd(\))2185 4310 y Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))2335 4293 y Fd(2)2401 4275 y Fi(;)480 b Fk(if)30 b Fi(n)23 b Fk(=)f(2)p Fi(:)523 4578 y Fk(\()p Fi(H)624 4590 y Fl(3)662 4578 y Fk(\))28 b Fi(f)j Fj(2)24 b Fi(L)930 4547 y Fh(N)q(=)p Fl(2)1054 4578 y Fk(\()p Fi(I)-21 b(R)1165 4541 y Fh(N)1228 4578 y Fk(\))19 b Fj(\\)g Fi(L)1410 4538 y Fh(q)1410 4603 y(loc)1497 4578 y Fk(\()p Fi(I)-21 b(R)1608 4541 y Fh(N)1690 4578 y Fj(n)18 b(f)p Fk(0)p Fj(g)p Fk(\),)26 b(for)h(some)g Fi(q)f(>)d(N)r(=)p Fk(2)k(and)523 4779 y(\()p Fi(i)p Fk(\))706 b(lim)14 b(sup)1371 4849 y Fb(j)p Fc(x)p Fb(j!)p Fd(0)1309 4912 y Fl(or)e Ff(j)p Fh(x)p Ff(j!1)1626 4779 y Fi(f)d Fk(\()p Fi(x)p Fk(\))14 b Fj(j)p Fi(x)p Fj(j)1894 4745 y Fl(2)1946 4779 y Fk(\(log)2085 4742 y Fl(\()p Fh(k)q Fl(+1\))2276 4779 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2462 4745 y Fl(2)2524 4779 y Fk(=)22 b(0)14 b Fi(;)1979 5150 y Fk(4)p eop %%Page: 5 5 5 4 bop 523 614 a Fk(and)27 b(there)h(exist)f(a)h(constan)n(t)e Fi(A)e(>)e Fk(0,)28 b(and)f(a)g(neigh)n(b)r(orho)r(o)r(d)g Fi(U)36 b Fk(of)28 b Fj(j)p Fi(x)p Fj(j)23 b Fk(=)g(1)k(suc)n(h)g(that) 523 847 y(\()p Fi(ii)p Fk(\))977 b Fi(f)9 b Fk(\()p Fi(x)p Fk(\))24 b Fj(\025)e Fi(A)47 b Fk(for)22 b Fi(x)i Fj(2)f Fi(U)g(:)523 1079 y Fk(F)-7 b(or)41 b Fi(q)s(;)14 b(\021)s(;)g(f)36 b Fk(and)28 b Fi(\032)f Fk(as)g(ab)r(o)n(v)n(e)f(and)i Fi(u)22 b Fj(2)i(D)1867 1049 y Fl(1)p Fh(;)p Fl(2)1957 1079 y Fk(\()p Fi(I)-21 b(R)2068 1043 y Fh(n)2114 1079 y Fk(\),)28 b(de\014ne)903 1445 y Fi(I)939 1457 y Fh(q)r(;\021)1032 1445 y Fk(\()p Fi(u)p Fk(\))23 b(=)1255 1200 y Fg(8)1255 1274 y(>)1255 1299 y(>)1255 1324 y(>)1255 1349 y(<)1255 1499 y(>)1255 1524 y(>)1255 1548 y(>)1255 1573 y(:)1452 1220 y(R)1421 1366 y Fh(I)-18 b(R)1482 1347 y Fc(n)1545 1195 y Fg(\020)1594 1287 y Fj(jr)p Fi(u)p Fj(j)1757 1257 y Fl(2)1813 1287 y Fj(\000)1896 1220 y Fg(\000)1944 1254 y Fh(n)p Ff(\000)p Fl(2)p 1944 1268 127 4 v 1990 1315 a(2)2080 1220 y Fg(\001)2118 1237 y Fl(2)2179 1250 y Fh(q)14 b(u)2262 1225 y Fd(2)p 2179 1268 116 4 v 2182 1315 a Ff(j)p Fh(x)p Ff(j)2260 1299 y Fd(2)2323 1287 y Fj(\000)k Fi(\021)s(u)2498 1257 y Fl(2)2535 1195 y Fg(\021)2667 1287 y Fk(if)90 b Fi(n)23 b Fj(\025)g Fk(3)14 b Fi(;)1394 1502 y Fg(R)1362 1648 y Fh(I)-18 b(R)1424 1629 y Fc(n)1486 1477 y Fg(\020)1536 1569 y Fj(jr)p Fi(u)p Fj(j)1699 1539 y Fl(2)1755 1569 y Fj(\000)1848 1536 y Fl(1)p 1848 1550 34 4 v 1848 1598 a(4)2040 1532 y Fh(q)14 b(u)2123 1507 y Fd(2)p 1901 1550 394 4 v 1901 1603 a Fl(\()o Ff(j)p Fh(x)p Ff(j)d Fl(log)h Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))2262 1580 y Fd(2)2323 1569 y Fj(\000)18 b Fi(\021)s(u)2498 1539 y Fl(2)2535 1477 y Fg(\021)2667 1569 y Fk(if)90 b Fi(n)23 b Fk(=)g(2)14 b Fi(;)3305 1445 y Fk(\(1.8\))1029 2034 y Fi(L)1086 2046 y Fh(q)r(;\021)1178 2034 y Fk(\()p Fi(u)p Fk(\))23 b(=)1401 1839 y Fg(8)1401 1914 y(>)1401 1939 y(<)1401 2088 y(>)1401 2113 y(:)1580 1927 y Fj(\000)p Fk(\001)p Fi(u)18 b Fj(\000)1863 1859 y Fg(\000)1911 1894 y Fh(n)p Ff(\000)p Fl(2)p 1911 1908 127 4 v 1958 1955 a(2)2047 1859 y Fg(\001)2085 1876 y Fl(2)2165 1889 y Fh(q)r(u)p 2146 1907 110 4 v 2146 1955 a Ff(j)p Fh(x)p Ff(j)2224 1939 y Fd(2)2284 1927 y Fj(\000)g Fi(\021)s(u)83 b Fk(if)90 b Fi(n)23 b Fj(\025)f Fk(3)14 b Fi(;)1516 2139 y Fj(\000)p Fk(\001)p Fi(u)k Fj(\000)1809 2107 y Fl(1)p 1809 2121 34 4 v 1809 2168 a(4)2023 2102 y Fh(q)r(u)p 1862 2120 394 4 v 1862 2173 a Fl(\()p Ff(j)p Fh(x)p Ff(j)10 b Fl(log)i Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))2223 2150 y Fd(2)2284 2139 y Fj(\000)18 b Fi(\021)s(u)83 b Fk(if)90 b Fi(n)23 b Fk(=)f(2)14 b Fi(;)3305 2034 y Fk(\(1.9\))1307 2508 y Fi(\025)1355 2520 y Fl(1)1392 2508 y Fk(\()p Fi(q)s(;)g(\021)s Fk(\))24 b(=)f(inf)1804 2337 y Fg(8)1804 2412 y(<)1804 2562 y(:)1877 2508 y Fi(I)1913 2520 y Fh(q)r(;\021)2007 2508 y Fk(\()p Fi(u)p Fk(\);)2165 2395 y Fg(Z)2137 2633 y Fh(I)-18 b(R)2199 2614 y Fc(n)2272 2508 y Fi(f)9 b(u)2370 2473 y Fl(2)2429 2508 y Fk(=)23 b(1)2559 2337 y Fg(9)2559 2412 y(=)2559 2562 y(;)2669 2508 y Fi(;)572 b Fk(\(1.10\))523 2757 y(and)27 b(consider)g(the)h(follo)n(wing)f(eigen)n(v)-5 b(alue)26 b(problem)1394 2864 y Fg(8)1394 2939 y(<)1394 3088 y(:)1510 2934 y Fi(L)1567 2946 y Fh(q)r(;\021)1659 2934 y Fi(\036)83 b Fk(=)g Fi(\025f)9 b(\036)84 b Fk(in)46 b Fi(I)-21 b(R)2364 2898 y Fh(n)2423 2934 y Fi(;)1659 3133 y(\036)83 b Fj(2)93 b(D)2005 3103 y Ff(0)2029 3133 y Fk(\()p Fi(I)-21 b(R)2140 3097 y Fh(n)2185 3133 y Fk(\))14 b Fi(;)97 b(\036)24 b Fj(6)p Fk(=)e(0)3264 3034 y(\(1.11\))523 3312 y(Then)28 b(w)n(e)f(ha)n(v)n(e)f(the)i(follo)n(wing)523 3545 y Fm(Theorem)j(1.2.)58 b Fk(Let)42 b(0)23 b Fi(<)f(s)h(<)g Fk(1)41 b(and)g Fi(q)s(;)14 b(\021)s(;)g(f)50 b Fk(satisfy)41 b(\()p Fi(H)2482 3557 y Fl(1)2520 3545 y Fk(\))28 b(to)f(\()p Fi(H)2782 3557 y Fl(3)2820 3545 y Fk(\).)h(Then,)523 3694 y(\(i\))g(The)g(sp)r(ectrum)g(of)41 b Fi(L)1337 3706 y Fh(sq)r(;s\021)1534 3694 y Fk(in)g Fi(L)1701 3664 y Fl(2)1738 3694 y Fk(\()p Fi(f)9 b Fk(\))42 b(is)27 b(discrete.)523 3926 y(\(ii\))20 b(Let)33 b Fi(\025)855 3938 y Fl(1)893 3926 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))37 b Fj(\024)g Fi(\025)1343 3938 y Fl(2)1380 3926 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))38 b Fj(\024)22 b Fi(:)14 b(:)g(:)43 b Fk(b)r(e)19 b(the)h(eigen)n(v)-5 b(alues)18 b(of)33 b Fi(L)2725 3938 y Fh(sq)r(;s\021)2913 3926 y Fk(and)42 b Fi(\036)3138 3938 y Fl(1)3175 3926 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))g(,)523 4076 y Fi(\036)572 4088 y Fl(2)610 4076 y Fk(\()p Fi(sq)s(;)g(s\021)s Fk(\))p Fi(;)g(:)g(:)g(:)67 b Fk(b)r(e)34 b(the)g(corresp)r(onding)e (eigenfunctions.)i(Then)g(after)f(a)g(normalization,)523 4225 y(w)n(e)27 b(ha)n(v)n(e:)523 4458 y Fi(')577 4470 y Fl(1)615 4458 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))23 b Fi(>)g Fk(0)f Fi(;)97 b(')1227 4470 y Fh(i)1255 4458 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))47 b Fj(\000)-14 b(!)1557 4479 y Fc(s)p Fb(!)p Fd(1)1724 4458 y Fi(')1778 4470 y Fh(i)1806 4458 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))84 b(in)23 b Fi(W)2257 4418 y Fl(1)p Fh(;r)2245 4483 y(loc)2346 4458 y Fk(\()p Fi(I)-21 b(R)2458 4422 y Fh(n)2503 4458 y Fk(\))83 b(for)27 b(all)h(1)22 b Fi(<)h(r)j(<)c Fk(2)h Fi(;)97 b(i)22 b Fj(\025)h Fk(1)14 b Fi(;)1318 4690 y(\025)1366 4702 y Fh(i)1394 4690 y Fk(\()p Fi(sq)s(;)g(s\021)s Fk(\))23 b Fj(\000)-14 b(!)1672 4711 y Fc(s)p Fb(!)p Fd(1)1839 4690 y Fi(\025)1887 4702 y Fh(i)1915 4690 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))84 b(for)27 b(all)64 b Fi(i)23 b Fj(\025)g Fk(1)14 b Fi(;)523 4890 y Fk(where)27 b(\()p Fi(\036)844 4902 y Fh(i)873 4890 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))p Fi(;)g(\025)1143 4902 y Fh(i)1171 4890 y Fk(\()p Fi(q)s(;)g(\021)s Fk(\)\))29 b(satis\014es)e(\(1.11\).)1979 5150 y(5)p eop %%Page: 6 6 6 5 bop 523 614 a Fk(\(iii\))28 b(Let)g Fi(q)j Fk(satisfy)625 922 y(lim)14 b(inf)662 975 y Fb(j)p Fc(x)p Fb(j!)p Fd(0)600 1038 y Fl(or)d Ff(j)p Fh(x)p Ff(j!1)894 726 y Fg(8)894 801 y(>)894 826 y(<)894 976 y(>)894 1000 y(:)1009 822 y Fk(\(log)j Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1350 780 y Fl(2)1401 730 y Fg(n)1456 822 y Fk(\(1)k Fj(\000)g Fi(q)s Fk(\()p Fi(x)p Fk(\)\))i Fj(\000)1961 782 y Fl(4)p Ff(j)p Fh(x)p Ff(j)2072 757 y Fd(2)p 1927 803 211 4 v 1927 851 a Fl(\()p Fh(n)p Ff(\000)p Fl(2\))2105 834 y Fd(2)2147 822 y Fi(\021)s Fk(\()p Fi(x)p Fk(\))2302 730 y Fg(o)2382 822 y Fi(>)2568 789 y Fl(3)p 2480 803 V 2480 851 a(\()p Fh(n)p Ff(\000)p Fl(2\))2658 834 y Fd(2)2714 822 y Fi(;)293 b Fk(if)30 b Fi(n)23 b Fj(\025)g Fk(3)14 b Fi(;)1009 1045 y Fk(\(log)g(log)g Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1469 1015 y Fl(2)1508 1045 y Fj(f)p Fk(\(1)k Fj(\000)g Fi(q)s Fk(\()p Fi(x)p Fk(\)\))h Fj(\000)f Fk(4)p Fj(j)p Fi(x)p Fj(j)2145 1015 y Fl(2)2183 1045 y Fk(\(log)c Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2522 1015 y Fl(2)2561 1045 y Fi(\021)s Fk(\()p Fi(x)p Fk(\))p Fj(g)23 b Fi(>)g Fk(3)14 b Fi(;)82 b Fk(if)30 b Fi(n)23 b Fk(=)g(2)p Fi(;)3264 1146 y Fk(\(1.12\))523 1378 y(then)28 b(after)f(a)h(normalization)e Fi(\036)1553 1390 y Fh(i)1581 1378 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))23 b Fj(\000)-14 b(!)1859 1399 y Fc(s)p Fb(!)p Fd(1)2026 1378 y Fi(\036)2075 1390 y Fh(i)2103 1378 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))29 b(in)f Fj(D)2480 1348 y Fl(1)p Fh(;)p Fl(2)2570 1378 y Fk(\()p Fi(I)-21 b(R)2681 1342 y Fh(n)2727 1378 y Fk(\))51 b(for)27 b(all)g Fi(i)c Fj(\025)f Fk(1.)523 1611 y(\(iv\))28 b(Let)g Fi(R)894 1623 y Fl(1)954 1611 y Fi(>)23 b Fk(0)k(and)h Fi(\021)e Fj(\025)c Fk(0)28 b(in)f Fi(B)t Fk(\()p Fi(R)1755 1623 y Fl(1)1793 1611 y Fk(\))h(or)f Fi(I)-21 b(R)2034 1574 y Fh(n)2097 1611 y Fj(n)18 b Fi(B)t Fk(\()p Fi(R)2319 1623 y Fl(1)2357 1611 y Fk(\))p Fi(:)28 b Fk(Let)g Fi(q)j Fk(satisfy)c(either)651 1900 y(sup)578 1973 y Fh(x)p Ff(2)p Fh(B)s Fl(\()p Fh(R)790 1981 y Fd(1)822 1973 y Fl(\))862 1730 y Fg(8)862 1804 y(<)862 1954 y(:)977 1793 y Fk(\(log)14 b Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1316 1763 y Fl(2)1355 1793 y Fk(\(\(1)k Fj(\000)g Fi(q)s Fk(\()p Fi(x)p Fk(\)\))i Fj(\000)1947 1760 y Fl(4)p 1858 1774 V 1858 1821 a(\()p Fh(n)p Ff(\000)p Fl(2\))2036 1805 y Fd(2)2079 1793 y Fj(j)p Fi(x)p Fj(j)2172 1763 y Fl(2)2209 1793 y Fi(\021)s Fk(\()p Fi(x)p Fk(\)\))25 b Fj(\024)e Fk(3)p Fi(=)p Fk(\()p Fi(n)17 b Fj(\000)h Fk(2\))2849 1763 y Fl(2)2992 1793 y Fk(if)90 b Fi(n)23 b Fj(\025)g Fk(3)14 b Fi(;)977 2005 y Fk(\(log)g(log)h Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1438 1975 y Fl(2)1476 2005 y Fk(\(\(1)k Fj(\000)f Fi(q)s Fk(\()p Fi(x)p Fk(\)\))h Fj(\000)f Fk(4\()p Fj(j)p Fi(x)p Fj(j)c Fk(log)h Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2458 1975 y Fl(2)2496 2005 y Fi(\021)s Fk(\()p Fi(x)p Fk(\)\))25 b Fj(\024)d Fk(3)155 b(if)90 b Fi(n)23 b Fk(=)g(2)14 b Fi(;)3264 2106 y Fk(\(1.13\))523 2338 y(or)724 2706 y(sup)583 2780 y Fh(x)p Ff(2)p Fh(I)-18 b(R)727 2761 y Fc(n)768 2780 y Ff(n)p Fh(B)s Fl(\()p Fh(R)931 2788 y Fd(1)963 2780 y Fl(\))1003 2436 y Fg(8)1003 2511 y(>)1003 2536 y(>)1003 2561 y(>)1003 2586 y(>)1003 2611 y(<)1003 2760 y(>)1003 2785 y(>)1003 2810 y(>)1003 2835 y(>)1003 2860 y(:)1118 2500 y Fk(\(log)15 b Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1458 2470 y Fl(2)1496 2500 y Fk(\(\(1)k Fj(\000)f Fi(q)s Fk(\()p Fi(x)p Fk(\)\))i Fj(\000)2088 2467 y Fl(4)p 2000 2481 V 2000 2528 a(\()p Fh(n)p Ff(\000)p Fl(2\))2178 2512 y Fd(2)2220 2500 y Fj(j)p Fi(x)p Fj(j)2313 2470 y Fl(2)2351 2500 y Fi(\021)s Fk(\()p Fi(x)p Fk(\)\))52 b Fj(\024)2536 2613 y Fk(3)p Fi(=)p Fk(\()p Fi(n)17 b Fj(\000)h Fk(2\))2876 2583 y Fl(2)2927 2613 y Fi(;)97 b Fk(if)30 b Fi(n)23 b Fj(\025)g Fk(3)14 b Fi(;)1118 2812 y Fk(\(log)h(log)f Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1579 2782 y Fl(2)1617 2812 y Fk(\(\(1)19 b Fj(\000)f Fi(q)s Fk(\()p Fi(x)p Fk(\)\))i Fj(\000)e Fk(4\()p Fj(j)p Fi(x)p Fj(j)c Fk(log)g Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2599 2782 y Fl(2)2637 2812 y Fi(\021)s Fk(\()p Fi(x)p Fk(\)\))25 b Fj(\024)2536 2912 y Fk(3)14 b Fi(;)96 b Fk(if)30 b Fi(n)23 b Fk(=)g(2)p Fi(;)3264 3012 y Fk(\(1.14\))523 3244 y(then)28 b(\(1.11\))f(do)r(es)g(not)h(admit)g(an)n(y)f(solution)g Fi(\036)c Fj(2)h(D)2215 3214 y Fl(1)p Fh(;)p Fl(2)2305 3244 y Fk(\()p Fi(I)-21 b(R)2416 3208 y Fh(n)2461 3244 y Fk(\),)28 b Fi(\036)c Fj(\025)f Fk(0.)523 3477 y(\(v\))28 b(Assume)g(that)g Fi(\021)j Fk(and)c Fi(f)36 b Fk(satisfy)27 b(the)h(stronger)e(assumption,)855 3769 y(lim)835 3822 y Fb(j)p Fc(x)p Fb(j!)p Fd(0)769 3885 y Fl(or)19 b Ff(j)p Fh(x)p Ff(j!1)1070 3598 y Fg(8)1070 3673 y(<)1070 3823 y(:)1307 3668 y Fk(\(log)14 b Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1647 3626 y Ff(\000)p Fl(1)1750 3668 y Fk(\()p Fi(\021)s Fk(\()p Fi(x)p Fk(\))20 b(+)e Fj(j)p Fi(f)9 b Fk(\()p Fi(x)p Fk(\))p Fj(j)p Fk(\))84 b(=)f(0)14 b Fi(;)59 b Fk(if)30 b Fi(n)23 b Fj(\025)g Fk(3)14 b Fi(;)1186 3877 y Fk(\(log)g(log)g Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1647 3835 y Ff(\000)p Fl(1)1750 3877 y Fk(\()p Fi(\021)s Fk(\()p Fi(x)p Fk(\))20 b(+)e Fj(j)p Fi(f)9 b Fk(\()p Fi(x)p Fk(\))p Fj(j)p Fk(\))84 b(=)f(0)14 b Fi(;)59 b Fk(if)30 b Fi(n)23 b Fk(=)g(2)14 b Fi(;)3264 3769 y Fk(\(1.15\))523 4062 y(and)39 b(let)g Fi(\036)876 4074 y Fl(1)p Fh(;s)1007 4062 y Fi(>)j Fk(0)c(b)r(e)i(the)f(normalized)f(eigenfunction)h (corresp)r(onding)e(to)i Fi(\025)3136 4074 y Fl(1)3174 4062 y Fk(\()p Fi(s;)14 b(\021)s Fk(\).)39 b(If)523 4211 y Fi(R)586 4223 y Fl(1)646 4211 y Fi(>)23 b Fk(0)p Fi(;)k Fk(there)h(exist)f(t)n(w)n(o)g(constan)n(ts)g Fi(C)1819 4223 y Fl(1)1856 4211 y Fi(;)14 b(C)1952 4223 y Fl(2)2013 4211 y Fi(>)22 b Fk(0)28 b(suc)n(h)f(that)h(for)f Fi(x)c Fj(2)h Fi(B)t Fk(\()p Fi(R)2975 4223 y Fl(1)3012 4211 y Fk(\))14 b Fi(;)565 4407 y Fk(\()p Fi(C)656 4419 y Fl(1)693 4407 y Fi(;)g(C)789 4419 y Fl(1)827 4407 y Fk(\))83 b Fj(\024)g(j)p Fi(x)p Fj(j)1193 4351 y Fc(n)p Fb(\000)p Fd(2)p 1193 4365 110 4 v 1234 4397 a(2)1313 4377 y Fl([1)p Ff(\000)1417 4332 y(p)p 1471 4332 117 3 v 1471 4377 a Fl(1)p Ff(\000)p Fh(s)q Fl(])1611 4407 y Fk(\()p Fi(\036)1692 4419 y Fl(1)p Fh(;s)1781 4407 y Fk(\()p Fi(x)p Fk(\))p Fi(;)14 b Fj(j)p Fi(x)p Fj(jjr)p Fi(\036)2163 4419 y Fl(1)p Fh(;s)2253 4407 y Fk(\()p Fi(x)p Fk(\))p Fj(j)p Fk(\))25 b Fj(\024)d Fk(\()p Fi(C)2622 4419 y Fl(2)2660 4407 y Fi(;)14 b(C)2756 4419 y Fl(2)2794 4407 y Fk(\))g Fi(;)83 b Fk(if)30 b Fi(n)23 b Fj(\025)f Fk(3)14 b Fi(;)565 4618 y Fk(\()p Fi(C)656 4630 y Fl(1)693 4618 y Fi(;)g(C)789 4630 y Fl(1)827 4618 y Fk(\))83 b Fj(\024)g Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1484 4565 y Fd(1)p 1484 4574 29 4 v 1484 4607 a(2)1522 4587 y Fl(\()p Ff(\000)p Fl(1+)1684 4542 y Ff(p)p 1739 4542 117 3 v 45 x Fl(1)p Ff(\000)p Fh(s)p Fl(\))1885 4618 y Fk(\()p Fi(\036)1966 4630 y Fl(1)p Fh(;s)2055 4618 y Fk(\()p Fi(x)p Fk(\))p Fi(;)g Fj(j)p Fi(x)p Fj(jjr)p Fi(\036)2437 4630 y Fl(1)p Fh(;s)2528 4618 y Fk(\()p Fi(x)p Fk(\))p Fj(j)p Fk(\))24 b Fj(\024)f Fk(\()p Fi(C)2897 4630 y Fl(2)2935 4618 y Fi(;)14 b(C)3031 4630 y Fl(2)3068 4618 y Fk(\))g Fi(;)83 b Fk(if)30 b Fi(n)23 b Fk(=)g(2)p Fi(;)3264 4718 y Fk(\(1.16\))1979 5150 y(6)p eop %%Page: 7 7 7 6 bop 523 614 a Fk(and)27 b(for)h Fi(x)23 b Fj(2)g Fi(I)-21 b(R)1039 578 y Fh(n)1103 614 y Fj(n)18 b Fi(B)t Fk(\()p Fi(R)1325 626 y Fl(1)1363 614 y Fk(\))c Fi(;)565 790 y Fk(\()p Fi(C)656 802 y Fl(1)693 790 y Fi(;)g(C)789 802 y Fl(1)827 790 y Fk(\))83 b Fj(\024)g(j)p Fi(x)p Fj(j)1193 734 y Fc(n)p Fb(\000)p Fd(2)p 1193 748 110 4 v 1234 780 a(2)1313 760 y Fl([1+)1416 715 y Ff(p)p 1470 715 117 3 v 1470 760 a Fl(1)p Ff(\000)p Fh(s)q Fl(])1647 790 y Fk(\()p Fi(\036)1728 802 y Fl(1)p Fh(;s)1817 790 y Fk(\()p Fi(x)p Fk(\))p Fi(;)14 b Fj(j)p Fi(x)p Fj(jjr)p Fi(\036)2199 802 y Fl(1)p Fh(;s)2289 790 y Fk(\()p Fi(x)p Fk(\))p Fj(j)p Fk(\))25 b Fj(\024)d Fk(\()p Fi(C)2658 802 y Fl(2)2696 790 y Fi(;)14 b(C)2792 802 y Fl(2)2830 790 y Fk(\))g Fi(;)60 b Fk(if)30 b Fi(n)23 b Fj(\025)f Fk(3)14 b Fi(;)565 1000 y Fk(\()p Fi(C)656 1012 y Fl(1)693 1000 y Fi(;)g(C)789 1012 y Fl(1)827 1000 y Fk(\))83 b Fj(\024)g Fk(\(log)14 b Fi(R)q Fj(j)p Fi(x)p Fj(j)p Fk(\))1432 970 y Ff(\000)1495 948 y Fd(1)p 1495 957 29 4 v 1495 990 a(2)1533 970 y Fl(\(1+)1643 925 y Ff(p)p 1697 925 117 3 v 1697 970 a Fl(1)p Ff(\000)p Fh(s)q Fl(\))1881 1000 y Fk(\()p Fi(\036)1962 1012 y Fl(1)p Fh(;s)2051 1000 y Fk(\()p Fi(x)p Fk(\))p Fi(;)28 b Fj(j)p Fi(x)p Fj(jjr)p Fi(\036)2447 1012 y Fl(1)p Fh(;s)2537 1000 y Fk(\()p Fi(x)p Fk(\))p Fj(j)p Fk(\))c Fj(\024)f Fk(\()p Fi(C)2906 1012 y Fl(2)2944 1000 y Fi(;)14 b(C)3040 1012 y Fl(2)3077 1000 y Fk(\))g Fi(;)61 b Fk(if)30 b Fi(n)22 b Fk(=)h(2)p Fi(:)3264 1101 y Fk(\(1.17\))523 1333 y Fm(2.)31 b(Pro)s(of)h(of)g(the)f(theorems.)763 1566 y Fk(First)g(recall)f(some)h(of)g(the)h(basic)f(results)f(whic)n(h)i(are) e(needed)h(in)h(the)g(pro)r(of)e(of)h(the)523 1715 y(theorems.)d(Let)g (1)c Fi(<)f(p)h(<)g Fj(1)k Fk(and)g Fi(M)33 b(>)24 b Fk(1)p Fi(:)j Fk(There)h(exist)g(constan)n(ts)f Fi(\013)2788 1727 y Fl(1)2826 1715 y Fi(;)14 b(\013)2916 1727 y Fl(2)2977 1715 y Fi(>)24 b Fk(0)k(suc)n(h)g(that)523 1865 y(for)f(all)g Fi(a;)14 b(b)23 b Fj(2)g Fi(I)-21 b(R)1062 1828 y Fh(n)1108 1865 y Fi(;)14 b Fj(j)p Fi(a)p Fj(j)22 b Fk(=)h(1)k(w)n(e)h(ha)n(v)n(e) 1084 2077 y Fj(j)p Fi(a)19 b Fk(+)f Fi(b)p Fj(j)1312 2042 y Fh(p)1368 2077 y Fj(\000)g Fk(1)g Fj(\000)g Fi(p)p Fj(h)p Fi(a;)c(b)p Fj(i)23 b(\025)g Fi(\013)1981 2089 y Fl(1)2018 2077 y Fj(j)p Fi(b)p Fj(j)2100 2042 y Fl(2)2155 2077 y Fk(+)18 b Fi(\013)2291 2089 y Fl(2)2329 2077 y Fj(j)p Fi(b)p Fj(j)2411 2042 y Fh(p)2463 2077 y Fi(;)97 b Fk(if)53 b Fi(p)23 b Fj(\025)f Fk(2)14 b Fi(;)390 b Fk(\(2.1\))894 2288 y Fj(j)p Fi(a)19 b Fk(+)f Fi(b)p Fj(j)1122 2254 y Fh(p)1178 2288 y Fj(\000)g Fk(1)g Fj(\000)g Fi(p)p Fj(h)p Fi(a;)c(b)p Fj(i)23 b(\025)1738 2171 y Fg(\032)1841 2238 y Fi(\013)1894 2250 y Fl(1)1932 2238 y Fj(j)p Fi(b)p Fj(j)2014 2208 y Fl(2)2135 2238 y Fk(if)90 b Fj(j)p Fi(b)p Fj(j)23 b(\024)f Fi(M)t(;)97 b Fk(1)23 b Fi(<)f(p)h Fj(\024)g Fk(2)14 b Fi(;)1841 2337 y(\013)1894 2349 y Fl(2)1932 2337 y Fj(j)p Fi(b)p Fj(j)2014 2307 y Fh(p)2135 2337 y Fk(if)90 b Fj(j)p Fi(b)p Fj(j)23 b(\025)f Fi(M)t(;)97 b Fk(1)23 b Fi(<)f(p)h(<)g Fk(2)14 b Fi(:)3305 2288 y Fk(\(2.2\))523 2475 y(F)-7 b(urthermore)24 b(if)i Fi(p)d Fk(=)f(2)p Fi(;)j Fk(then)h Fi(\013)1558 2487 y Fl(1)1618 2475 y Fk(=)d Fi(\013)1759 2487 y Fl(2)1819 2475 y Fk(=)1917 2443 y Fl(1)p 1917 2457 34 4 v 1917 2504 a(2)1985 2475 y Fk(is)i(the)h(b)r(est)g(c)n(hoice)e(in)h (\(2.1\)-\(2.2\).)g(Note)g(also)523 2625 y(that)35 b(a)f(simple)h (computation)g(sho)n(ws)e(that)i(if)g Fj(j)p Fi(S)2138 2595 y Fh(n)p Ff(\000)p Fl(1)2268 2625 y Fj(j)g Fk(denotes)f(the)h(v)n (olume)f(of)h(the)g(unit)523 2774 y(sphere)27 b Fi(S)838 2744 y Fh(n)p Ff(\000)p Fl(1)996 2774 y Fk(in)g Fi(I)-21 b(R)1171 2738 y Fh(n)1217 2774 y Fk(,)27 b(then)h(for)g(all)f Fi(k)f Fj(\025)c Fk(1)p Fi(;)765 2998 y Fk(div)916 2881 y Fg(\022)1033 2942 y Fi(x)p 987 2979 139 4 v 987 3055 a Fj(j)p Fi(x)p Fj(j)1080 3031 y Fh(n)1136 2881 y Fg(\023)1220 2998 y Fk(=)h Fj(\000)f Fk(div)j(\()1575 2942 y(log)1682 2905 y Fl(\()p Fh(k)q Fl(\))1789 2942 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p 1575 2979 413 4 v 1669 3055 a(j)p Fi(x)p Fj(j)1762 3031 y Fh(n)p Ff(\000)p Fl(2)1997 2998 y Fj(r)14 b Fk(log)2188 3018 y Fl(\()p Fh(k)q Fl(+1\))2378 2998 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))23 b(=)g Fj(j)p Fi(S)2799 2964 y Fh(n)p Ff(\000)p Fl(1)2929 2998 y Fj(j)14 b Fi(\016)3003 3010 y Fl(0)3040 2998 y Fi(;)242 b Fk(\(2.3\))523 3210 y(where)27 b Fi(\016)800 3222 y Fl(0)865 3210 y Fk(is)g(the)h(Dirac)f(distribution)h(at)g(the)g(origin.)523 3442 y Fm(Lemma)33 b(2.1.)28 b Fk(Let)h Fi(R)h Fk(and)f Fi(\032)g Fk(b)r(e)g(as)g(in)g(Theorem)f(1.1.)g(Let)i Fi(w)2555 3454 y Fl(1)2618 3442 y Fj(2)c Fi(C)2764 3412 y Fl(1)2801 3442 y Fk(\()p 2833 3376 60 4 v(\012\))p Fi(;)14 b(W)3040 3454 y Fl(1)3104 3442 y Fj(2)25 b Fi(C)3249 3412 y Fl(1)3243 3463 y(0)3287 3442 y Fk(\()p Fi(I)-21 b(R)3398 3406 y Fh(n)3444 3442 y Fk(\))523 3592 y(and)27 b(de\014ne)h(the)g(sequence)41 b Fj(f)p Fi(w)1526 3604 y Fh(i)1554 3592 y Fj(g)1596 3604 y Fh(i)1664 3592 y Fk(and)h Fj(f)p Fi(W)1960 3604 y Fh(i)1987 3592 y Fj(g)2029 3604 y Fh(i)2098 3592 y Fk(b)n(y)842 3804 y Fi(w)901 3816 y Fl(2)962 3804 y Fk(=)23 b(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1434 3769 y Ff(\000)p Fl(1)p Fh(=)p Fl(2)1590 3804 y Fi(w)1649 3816 y Fl(1)1701 3804 y Fi(;)37 b(:)14 b(:)g(:)f(;)h(w)1967 3816 y Fh(k)2032 3804 y Fk(=)22 b(\(log)2259 3824 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))2450 3804 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2681 3769 y Ff(\000)p Fl(1)p Fh(=)p Fl(2)2838 3804 y Fi(w)2897 3816 y Fh(k)q Ff(\000)p Fl(1)3037 3804 y Fi(;)14 b(:)g(:)g(:)853 4016 y(W)931 4028 y Fl(2)991 4016 y Fk(=)23 b(\(log)14 b Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1418 3981 y Ff(\000)p Fl(1)p Fh(=)p Fl(2)1576 4016 y Fi(W)1654 4028 y Fl(1)1705 4016 y Fi(;)g(:)g(:)g(:)g(;)g(W)1968 4028 y Fh(k)2032 4016 y Fk(=)23 b(\(log)2259 4036 y Fl(\()p Fh(k)q Ff(\000)p Fl(1\))2451 4016 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2637 3981 y Ff(\000)p Fl(1)p Fh(=)p Fl(2)2794 4016 y Fi(W)2872 4028 y Fh(k)q Ff(\000)p Fl(1)3012 4016 y Fi(;)14 b(:)g(:)g(:)523 4203 y Fk(The)28 b(follo)n(wing)e(iden)n(tities)i(hold:)568 4347 y Fg(Z)614 4535 y Fl(\012)689 4403 y Fj(jr)p Fi(w)840 4415 y Fl(1)878 4403 y Fj(j)901 4373 y Fl(2)p 689 4441 250 4 v 702 4517 a Fj(j)p Fi(x)p Fj(j)795 4493 y Fh(n)p Ff(\000)p Fl(2)971 4460 y Fk(=)1069 4403 y(1)p 1069 4441 42 4 v 1069 4517 a(4)1134 4347 y Fg(Z)1134 4585 y Fl(\012)1231 4293 y Fg(0)1231 4442 y(@)1346 4356 y Fh(k)1304 4381 y Fg(X)1306 4558 y Fh(j)s Fl(=1)1681 4403 y Fk(1)p 1448 4441 509 4 v 1448 4531 a(\(log)1587 4494 y Fl(\()p Fh(j)s Fl(\))1688 4531 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1919 4507 y Fl(2)1966 4293 y Fg(1)1966 4442 y(A)2082 4403 y Fi(w)2143 4373 y Fl(2)2141 4424 y(1)p 2063 4441 139 4 v 2063 4517 a Fj(j)p Fi(x)p Fj(j)2156 4493 y Fh(n)2230 4460 y Fj(\000)2323 4403 y Fk(1)p 2323 4441 42 4 v 2323 4517 a(2)2390 4347 y Fg(Z)2370 4585 y Fh(@)t Fl(\012)2488 4293 y Fg(0)2488 4442 y(@)2603 4356 y Fh(k)2561 4381 y Fg(X)2563 4558 y Fh(j)s Fl(=1)2831 4403 y Fj(h)p Fi(x)19 b Fj(\001)g Fi(\027)5 b Fj(i)p 2705 4441 472 4 v 2705 4531 a Fk(\(log)2844 4494 y Fl(\()p Fh(j)s Fl(\))2945 4531 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3186 4293 y Fg(1)3186 4442 y(A)3302 4403 y Fi(w)3363 4373 y Fl(2)3361 4424 y(1)p 3282 4441 139 4 v 3282 4517 a Fj(j)p Fi(x)p Fj(j)3375 4493 y Fh(n)1476 4801 y Fk(+)1555 4688 y Fg(Z)1554 4926 y Fl(\012)1662 4745 y Fk(\(log)1801 4708 y Fl(\()p Fh(k)q Fl(\))1908 4745 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))p Fj(jr)p Fi(w)2290 4757 y Fh(k)q Fl(+1)2415 4745 y Fj(j)2438 4715 y Fl(2)p 1662 4782 815 4 v 1957 4858 a Fj(j)p Fi(x)p Fj(j)2050 4834 y Fh(n)p Ff(\000)p Fl(2)2499 4801 y Fi(;)783 b Fk(\(2.4\))1979 5150 y(7)p eop %%Page: 8 8 8 7 bop 671 589 a Fg(Z)643 827 y Fh(I)-18 b(R)705 807 y Fc(n)788 645 y Fj(jr)p Fi(W)958 657 y Fl(1)995 645 y Fj(j)1018 615 y Fl(2)p 788 682 269 4 v 810 758 a Fj(j)p Fi(x)p Fj(j)903 735 y Fh(n)p Ff(\000)p Fl(2)1089 702 y Fk(=)1186 645 y(1)p 1186 682 42 4 v 1186 758 a(4)1261 589 y Fg(Z)1233 827 y Fh(I)g(R)1295 807 y Fc(n)1368 535 y Fg(0)1368 684 y(@)1482 598 y Fh(k)1440 623 y Fg(X)1443 799 y Fh(j)s Fl(=1)1796 645 y Fk(1)p 1584 682 465 4 v 1584 773 a(\(log)1724 736 y Fl(\()p Fh(j)s Fl(\))1824 773 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2010 749 y Fl(2)2059 535 y Fg(1)2059 684 y(A)2161 645 y Fi(W)2251 615 y Fl(2)2239 666 y(1)p 2155 682 139 4 v 2155 758 a Fj(j)p Fi(x)p Fj(j)2248 735 y Fh(n)2322 702 y Fj(\000)2415 645 y Fk(1)p 2415 682 42 4 v 2415 758 a(2)2467 702 y(\()2541 598 y Fh(k)2499 623 y Fg(X)2502 799 y Fh(j)s Fl(=1)2790 645 y Fk(1)p 2643 682 337 4 v 2643 773 a(\(log)2782 736 y Fl(\()p Fh(j)s Fl(\))2883 773 y Fi(R)q Fk(\))2989 702 y(\))3074 589 y Fg(Z)3016 830 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)3210 702 y Fi(W)3300 667 y Fl(2)3288 722 y(1)1493 1033 y Fk(+)1581 920 y Fg(Z)1553 1158 y Fh(I)g(R)1615 1139 y Fc(n)1674 1033 y Fk(\(log)1813 996 y Fl(\()p Fh(k)q Fl(\))1920 1033 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2116 977 y Fj(jr)p Fi(W)2286 989 y Fh(k)q Fl(+1)2413 977 y Fj(j)2436 947 y Fl(2)p 2117 1014 356 4 v 2183 1090 a Fj(j)p Fi(x)p Fj(j)2276 1066 y Fh(n)p Ff(\000)p Fl(2)2483 1033 y Fi(:)799 b Fk(\(2.5\))523 1363 y Fm(Pro)s(of.)27 b Fk(W)-7 b(e)29 b(will)f(only)f(pro)n(v)n(e)f(one)i(of)f(the)i(equalities)e(since)h (the)g(other)f(one)h(follo)n(ws)e(in)i(the)523 1512 y(same)f(w)n(a)n(y) -7 b(.)26 b(Let)h(us)g(pro)n(v)n(e)f(\(2.5\).)h(Let)g(us)g(start)g (with)h(the)f(case)g Fi(k)f Fk(=)c(1)27 b(and)g(then)h(\(2.5\))f(will) 523 1662 y(follo)n(w)g(b)n(y)g(induction)h(on)f Fi(k)s Fk(.)h(Since)1336 1911 y Fj(r)14 b Fk(log)g Fi(\032)p Fk(\()p Fi(x)p Fk(\))24 b(=)1806 1769 y Fg(\()1914 1847 y Fj(\000)2024 1814 y Fh(x)p 1988 1828 110 4 v 1988 1876 a Ff(j)p Fh(x)p Ff(j)2066 1859 y Fd(2)2191 1847 y Fk(if)90 b Fj(j)p Fi(x)p Fj(j)23 b Fi(<)g Fk(1)1960 1927 y Fh(x)p 1924 1941 V 1924 1989 a Ff(j)p Fh(x)p Ff(j)2002 1972 y Fd(2)2191 1960 y Fk(if)90 b Fj(j)p Fi(x)p Fj(j)23 b Fi(>)g Fk(1)14 b Fi(;)523 2165 y Fk(b)n(y)27 b(\(2.3\))h(and)f Fi(W)1076 2177 y Fl(2)1114 2165 y Fk(\(0\))c(=)g(0)k(w)n(e)g(ha)n(v)n (e)745 2303 y Fg(R)713 2449 y Fh(I)-18 b(R)775 2430 y Fc(n)848 2330 y Ff(jr)p Fh(W)985 2338 y Fd(1)1018 2330 y Ff(j)1038 2304 y Fd(2)p 848 2351 223 4 v 863 2398 a Ff(j)p Fh(x)p Ff(j)941 2382 y Fc(n)p Fb(\000)p Fd(2)1163 2370 y Fk(=)1334 2303 y Fg(R)1302 2449 y Fh(I)g(R)1364 2430 y Fc(n)1436 2330 y Ff(jr)p Fl(\(log)13 b Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))1785 2304 y Fd(1)p Fc(=)p Fd(2)1876 2330 y Fh(W)1938 2338 y Fd(2)1982 2330 y Fl(+)e(\(log)i Fh(\032)p Fl(\()p Fh(x)p Fl(\))2292 2304 y Fd(1)p Fc(=)p Fd(2)2383 2330 y Ff(r)p Fh(W)2500 2338 y Fd(2)2533 2330 y Ff(j)2553 2304 y Fd(2)p 1436 2351 1149 4 v 1915 2398 a Ff(j)p Fh(x)p Ff(j)1993 2382 y Fc(n)p Fb(\000)p Fd(2)1163 2552 y Fk(=)1320 2519 y Fl(1)p 1320 2533 34 4 v 1320 2581 a(4)1401 2485 y Fg(R)1369 2631 y Fh(I)-18 b(R)1431 2612 y Fc(n)1503 2510 y Fl(\(log)12 b Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))1776 2485 y Fb(\000)p Fd(1)1854 2510 y Fh(W)1925 2485 y Fd(2)1916 2526 y(2)p 1503 2533 455 4 v 1672 2581 a Ff(j)p Fh(x)p Ff(j)1750 2564 y Fc(n)1986 2552 y Fj(\000)2079 2519 y Fl(1)p 2079 2533 34 4 v 2079 2581 a(2)2189 2485 y Fg(R)2128 2635 y Ff(j)p Fh(x)p Ff(j)p Fh(<)p Fl(1)2298 2552 y Fj(h)2380 2519 y Fh(x)p 2340 2533 118 4 v 2340 2581 a Ff(j)p Fh(x)p Ff(j)2418 2564 y Fc(n)2468 2552 y Fi(;)i Fj(r)p Fi(W)2664 2522 y Fl(2)2652 2573 y(2)2701 2552 y Fj(i)1310 2750 y Fk(+)1385 2717 y Fl(1)p 1385 2731 34 4 v 1385 2778 a(2)1495 2683 y Fg(R)1434 2833 y Ff(j)p Fh(x)p Ff(j)p Fh(>)p Fl(1)1604 2750 y Fj(h)1686 2717 y Fh(x)p 1646 2731 118 4 v 1646 2778 a Ff(j)p Fh(x)p Ff(j)1724 2762 y Fc(n)1774 2750 y Fi(;)g Fj(r)p Fi(W)1970 2720 y Fl(2)1958 2770 y(2)2007 2750 y Fj(i)19 b Fk(+)2164 2683 y Fg(R)2133 2829 y Fh(I)-18 b(R)2194 2810 y Fc(n)2267 2710 y Fl(log)12 b Fh(\032)p Fl(\()p Fh(x)p Fl(\))p Ff(jr)p Fh(W)2625 2718 y Fd(2)2658 2710 y Ff(j)2678 2684 y Fd(2)p 2267 2731 444 4 v 2393 2778 a Ff(j)p Fh(x)p Ff(j)2471 2762 y Fc(n)p Fb(\000)p Fd(2)1163 2950 y Fk(=)1320 2917 y Fl(1)p 1320 2931 34 4 v 1320 2978 a(4)1401 2883 y Fg(R)1369 3028 y Fh(I)-18 b(R)1431 3009 y Fc(n)1663 2907 y Fh(W)1734 2882 y Fd(2)1725 2924 y(1)p 1503 2930 424 4 v 1503 2978 a Ff(j)p Fh(x)p Ff(j)1581 2962 y Fc(n)1621 2978 y Fl(\(log)12 b Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))1894 2962 y Fd(2)1955 2950 y Fj(\000)2091 2883 y Fg(R)2030 3032 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)2223 2909 y Ff(h)p Fh(x)p Ff(\001)p Fh(\027)t Ff(i)p 2223 2930 149 4 v 2238 2978 a(j)p Fh(x)p Ff(j)2316 2962 y Fc(n)2395 2950 y Fi(W)2485 2919 y Fl(2)2473 2970 y(2)2541 2950 y Fk(+)2647 2883 y Fg(R)2616 3028 y Fh(I)-18 b(R)2677 3009 y Fc(n)2750 2909 y Fl(\(log)12 b Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))p Ff(jr)p Fh(W)3160 2917 y Fd(2)3193 2909 y Ff(j)3213 2884 y Fd(2)p 2750 2930 496 4 v 2902 2978 a Ff(j)p Fh(x)p Ff(j)2980 2962 y Fc(n)p Fb(\000)p Fd(2)1163 3149 y Fk(=)1320 3117 y Fl(1)p 1320 3131 34 4 v 1320 3178 a(4)1401 3082 y Fg(R)1369 3228 y Fh(I)-18 b(R)1431 3209 y Fc(n)1663 3107 y Fh(W)1734 3082 y Fd(2)1725 3124 y(1)p 1503 3130 424 4 v 1503 3178 a Ff(j)p Fh(x)p Ff(j)1581 3161 y Fc(n)1621 3178 y Fl(\(log)12 b Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))1894 3161 y Fd(2)1955 3149 y Fj(\000)2105 3117 y Fl(1)p 2048 3131 148 4 v 2048 3178 a(log)g Fh(R)2272 3082 y Fg(R)2211 3232 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)2394 3149 y Fi(W)2484 3119 y Fl(2)2472 3170 y(1)2540 3149 y Fk(+)2646 3082 y Fg(R)2615 3228 y Fh(I)-18 b(R)2676 3209 y Fc(n)2749 3109 y Fl(\(log)12 b Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))p Ff(jr)p Fh(W)3159 3117 y Fd(2)3192 3109 y Ff(j)3212 3084 y Fd(2)p 2749 3130 496 4 v 2901 3178 a Ff(j)p Fh(x)p Ff(j)2979 3161 y Fc(n)p Fb(\000)p Fd(2)3254 3149 y Fi(:)523 3521 y Fk(No)n(w)27 b(observ)n(e)f(that)i Fj(r)p Fk(\(log)1397 3542 y Fl(\()p Fh(k)q Fl(\))1503 3521 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))c(=)1801 3429 y Fg(\020)2032 3489 y Fl(1)p 1861 3503 376 4 v 1861 3556 a(log)1947 3537 y Fd(\()p Fc(k)q Fb(\000)p Fd(1\))2112 3556 y Fh(\032)p Fl(\()p Fh(x)p Fl(\))2246 3429 y Fg(\021)2310 3521 y Fj(r)14 b Fk(log)g Fi(\032)p Fk(\()p Fi(x)p Fk(\))p Fi(:)29 b Fk(Hence,)652 3694 y Fg(R)620 3840 y Fh(I)-18 b(R)682 3820 y Fc(n)754 3719 y Ff(jr)p Fh(W)891 3728 y Fc(k)q Fd(+1)999 3719 y Ff(j)1019 3694 y Fd(2)1063 3719 y Fl(log)1148 3694 y Fd(\()p Fc(k)q Fd(\))1241 3719 y Fh(\032)p Fl(\()p Fh(x)p Fl(\))p 754 3741 611 4 v 964 3789 a Ff(j)p Fh(x)p Ff(j)1042 3773 y Fc(n)p Fb(\000)p Fd(2)1458 3761 y Fk(=)1629 3694 y Fg(R)1598 3840 y Fh(I)g(R)1659 3820 y Fc(n)1732 3720 y Fl(log)1818 3695 y Fd(\()p Fc(k)q Fd(\))1910 3720 y Fh(\032)p Fl(\()p Fh(x)p Fl(\))p 1732 3741 303 4 v 1787 3789 a Ff(j)p Fh(x)p Ff(j)1865 3773 y Fc(n)p Fb(\000)p Fd(2)2044 3761 y Fj(jr)p Fk(\(log)2276 3781 y Fl(\()p Fh(k)q Fl(+1\))2467 3761 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2653 3730 y Fl(1)p Fh(=)p Fl(2)2758 3761 y Fi(W)2836 3773 y Fh(k)q Fl(+2)2961 3761 y Fk(+)1606 3922 y(\(log)1745 3942 y Fl(\()p Fh(k)q Fl(+1\))1936 3922 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2122 3891 y Fl(1)p Fh(=)p Fl(2)2227 3922 y Fj(r)p Fi(W)2374 3934 y Fh(k)q Fl(+2)2500 3922 y Fk(\))2532 3891 y Fl(2)1458 4058 y Fk(=)1616 4025 y Fl(1)p 1616 4039 34 4 v 1616 4086 a(4)1696 3991 y Fg(R)1664 4137 y Fh(I)g(R)1726 4118 y Fc(n)1965 4008 y Fh(W)2036 3983 y Fd(2)2027 4025 y Fc(k)q Fd(+2)p 1799 4039 503 4 v 1799 4093 a Ff(j)p Fh(x)p Ff(j)1877 4076 y Fc(n)1928 4093 y Fl(log)2014 4073 y Fd(\()p Fc(k)q Fd(+1\))2177 4093 y Fh(\032)p Fl(\()p Fh(x)p Fl(\))2330 4058 y Fj(\000)2423 4025 y Fl(1)p 2423 4039 34 4 v 2423 4086 a(2)2533 3991 y Fg(R)2471 4141 y Ff(j)p Fh(x)p Ff(j)p Fh(<)p Fl(1)2641 4058 y Fj(h)2724 4025 y Fh(x)p 2684 4039 118 4 v 2684 4086 a Ff(j)p Fh(x)p Ff(j)2762 4070 y Fc(n)2811 4058 y Fi(;)14 b Fj(r)p Fi(W)3007 4028 y Fl(2)2995 4081 y Fh(k)q Fl(+2)3121 4058 y Fj(i)1606 4261 y Fk(+)1681 4228 y Fl(1)p 1680 4242 34 4 v 1680 4290 a(2)1790 4194 y Fg(R)1729 4344 y Ff(j)p Fh(x)p Ff(j)p Fh(>)p Fl(1)1899 4261 y Fj(h)1981 4228 y Fh(x)p 1941 4242 118 4 v 1941 4290 a Ff(j)p Fh(x)p Ff(j)2019 4273 y Fc(n)2069 4261 y Fi(;)g Fj(r)p Fi(W)2265 4231 y Fl(2)2253 4285 y Fh(k)q Fl(+2)2379 4261 y Fj(i)k Fk(+)2536 4194 y Fg(R)2504 4340 y Fh(I)-18 b(R)2566 4321 y Fc(n)2638 4220 y Fl(\(log)2750 4195 y Fd(\()p Fc(k)q Fd(+1\))2914 4220 y Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))p Ff(jr)p Fh(W)3201 4229 y Fc(k)q Fd(+2)3309 4220 y Ff(j)3329 4195 y Fd(2)p 2638 4242 723 4 v 2904 4290 a Ff(j)p Fh(x)p Ff(j)2982 4273 y Fc(n)p Fb(\000)p Fd(2)1458 4461 y Fk(=)1616 4428 y Fl(1)p 1616 4442 34 4 v 1616 4490 a(4)1696 4394 y Fg(R)1664 4540 y Fh(I)g(R)1726 4521 y Fc(n)2034 4419 y Fh(W)2105 4394 y Fd(2)2096 4435 y(1)p 1799 4442 576 4 v 1799 4496 a Ff(j)p Fh(x)p Ff(j)1877 4479 y Fc(n)1916 4496 y Fl(\(log)2028 4476 y Fd(\()p Fc(k)q Fd(+1\))2192 4496 y Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))2342 4479 y Fd(2)2402 4461 y Fj(\000)2655 4428 y Fl(1)p 2495 4442 352 4 v 2495 4496 a(\(log)2607 4476 y Fd(\()p Fc(k)q Fd(+1\))2771 4496 y Fh(R)p Fl(\))2924 4394 y Fg(R)2863 4544 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)3046 4461 y Fi(W)3136 4431 y Fl(2)3124 4482 y(1)3173 4461 y Fk(+)1629 4597 y Fg(R)1598 4743 y Fh(I)g(R)1659 4724 y Fc(n)1732 4623 y Fl(\(log)1844 4598 y Fd(\()p Fc(k)q Fd(+1\))2007 4623 y Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))p Ff(jr)p Fh(W)2294 4632 y Fc(k)q Fd(+2)2402 4623 y Ff(j)2422 4598 y Fd(2)p 1732 4645 723 4 v 1997 4693 a Ff(j)p Fh(x)p Ff(j)2075 4676 y Fc(n)p Fb(\000)p Fd(2)523 4901 y Fk(Hence)28 b(b)n(y)f(induction)h(\(2.5\))f(follo)n(ws.)g(This)h(pro)n(v)n(es)d (the)j(lemma.)820 b Fj(t)-55 b(u)1979 5150 y Fk(8)p eop %%Page: 9 9 9 8 bop 523 614 a Fm(Pro)s(of)32 b(of)f(Theorem)h(1.1.)26 b Fk(W)-7 b(e)28 b(will)g(pro)n(v)n(e)e(this)i(in)g(sev)n(eral)d (steps.)523 847 y Fm(Step)35 b(1.)29 b Fk(Let)i(2)c Fj(\024)f Fi(p)i(<)f(n)j Fk(and)g(0)c Fj(\024)h Fi(u)g Fj(2)h Fi(C)1950 817 y Fl(1)1988 847 y Fk(\()p 2020 780 60 4 v(\012\))p Fi(:)i Fk(Let)h Fi(v)f Fk(=)d Fj(j)p Fi(x)p Fj(j)2582 782 y Fc(n)p Fb(\000)p Fc(p)p 2583 797 113 4 v 2624 830 a(p)2709 847 y Fi(u;)j Fk(then)h Fi(v)s Fk(\(0\))c(=)g(0)j(and)523 996 y(from)d(\(2.1\),)565 1204 y Fj(jr)p Fi(u)p Fj(j)728 1174 y Fh(p)849 1204 y Fk(=)82 b Fj(j)19 b(\000)1121 1112 y Fg(\020)1180 1167 y Fh(n)p Ff(\000)p Fh(p)p 1180 1185 128 4 v 1227 1232 a(p)1318 1112 y Fg(\021)1381 1204 y Fj(j)p Fi(x)p Fj(j)1474 1167 y Ff(\000)1526 1173 y Fk(\()1569 1139 y Fc(n)p Fb(\000)p Fc(p)p 1569 1154 113 4 v 1609 1187 a(p)1691 1173 y Fk(\))1723 1167 y Ff(\000)p Fl(1)1842 1171 y Fh(x)p 1822 1185 78 4 v 1822 1232 a Ff(j)p Fh(x)p Ff(j)1923 1204 y Fi(v)j Fk(+)c Fj(j)p Fi(x)p Fj(j)2161 1167 y Ff(\000)2213 1173 y Fk(\()2256 1139 y Fc(n)p Fb(\000)p Fc(p)p 2256 1154 113 4 v 2296 1187 a(p)2378 1173 y Fk(\))2414 1204 y Fj(r)p Fi(v)s Fj(j)2549 1174 y Fh(p)849 1383 y Fk(=)996 1291 y Fg(\020)1056 1346 y Fh(n)p Ff(\000)p Fh(p)p 1056 1364 128 4 v 1102 1411 a(p)1193 1291 y Fg(\021)1243 1308 y Fh(p)1295 1383 y Fj(j)p Fi(x)p Fj(j)1388 1353 y Ff(\000)p Fh(n)1485 1383 y Fi(v)1528 1353 y Fh(p)1567 1262 y Fg(\014)1567 1312 y(\014)1567 1362 y(\014)1567 1412 y(\014)1624 1350 y Fh(x)p 1605 1364 78 4 v 1605 1411 a Ff(j)p Fh(x)p Ff(j)1710 1383 y Fj(\000)1793 1291 y Fg(\020)1899 1346 y Fh(p)p 1853 1364 128 4 v 1853 1411 a(n)p Ff(\000)p Fh(p)1990 1291 y Fg(\021)2063 1342 y Ff(j)p Fh(x)p Ff(jr)p Fh(v)p 2063 1364 168 4 v 2130 1411 a(v)2241 1262 y Fg(\014)2241 1312 y(\014)2241 1362 y(\014)2241 1412 y(\014)2269 1283 y Fh(p)849 1562 y Fj(\025)996 1469 y Fg(\020)1056 1524 y Fh(n)p Ff(\000)p Fh(p)p 1056 1542 128 4 v 1102 1590 a(p)1193 1469 y Fg(\021)1243 1487 y Fh(p)1329 1529 y(v)1364 1504 y Fc(p)p 1305 1543 118 4 v 1305 1590 a Ff(j)p Fh(x)p Ff(j)1383 1574 y Fc(n)1079 1619 y Fg(\032)1142 1736 y Fk(1)g Fj(\000)g Fi(p)1341 1644 y Fg(\020)1446 1699 y Fh(p)p 1400 1717 128 4 v 1400 1764 a(n)p Ff(\000)p Fh(p)1537 1644 y Fg(\021)1610 1696 y Ff(h)p Fh(x;)p Ff(r)p Fh(v)r Ff(i)p 1610 1717 202 4 v 1694 1764 a Fh(v)1841 1736 y Fk(+)g Fi(\013)1977 1748 y Fl(1)2028 1644 y Fg(\020)2134 1699 y Fh(p)p 2087 1717 128 4 v 2087 1764 a(n)p Ff(\000)p Fh(p)2224 1644 y Fg(\021)2274 1661 y Fl(2)2335 1696 y Ff(j)p Fh(x)p Ff(j)2413 1671 y Fd(2)2445 1696 y Ff(jr)p Fh(v)r Ff(j)2575 1671 y Fd(2)p 2335 1717 273 4 v 2437 1764 a Fh(v)2472 1748 y Fd(2)2636 1736 y Fk(+)g Fi(\013)2772 1748 y Fl(2)2823 1644 y Fg(\020)2929 1699 y Fh(p)p 2882 1717 128 4 v 2882 1764 a(n)p Ff(\000)p Fh(p)3019 1644 y Fg(\021)3069 1661 y Fh(p)3131 1696 y Ff(j)p Fh(x)p Ff(j)3209 1671 y Fc(p)3243 1696 y Ff(jr)p Fh(v)r Ff(j)3373 1671 y Fc(p)p 3131 1717 277 4 v 3235 1764 a Fh(v)3270 1748 y Fc(p)3418 1619 y Fg(\033)849 1927 y Fk(=)996 1835 y Fg(\020)1056 1890 y Fh(n)p Ff(\000)p Fh(p)p 1056 1908 128 4 v 1102 1956 a(p)1193 1835 y Fg(\021)1243 1852 y Fh(p)1329 1894 y(v)1364 1869 y Fc(p)p 1305 1908 118 4 v 1305 1956 a Ff(j)p Fh(x)p Ff(j)1383 1939 y Fc(n)1451 1927 y Fj(\000)1534 1835 y Fg(\020)1594 1890 y Fh(n)p Ff(\000)p Fh(p)p 1594 1908 128 4 v 1640 1956 a(p)1731 1835 y Fg(\021)1780 1852 y Fh(p)p Ff(\000)p Fl(1)1917 1927 y Fj(h)2000 1894 y Fh(x)p 1960 1908 118 4 v 1960 1956 a Ff(j)p Fh(x)p Ff(j)2038 1939 y Fc(n)2087 1927 y Fi(;)c Fj(r)p Fi(v)2236 1897 y Fh(p)2275 1927 y Fj(i)p Fk(+)1089 2059 y Fl(4)p Fh(\013)1165 2067 y Fd(1)p 1089 2074 109 4 v 1110 2122 a Fh(p)1144 2106 y Fd(2)1222 2001 y Fg(\020)1281 2056 y Fh(n)p Ff(\000)p Fh(p)p 1281 2074 128 4 v 1328 2122 a(p)1418 2001 y Fg(\021)1468 2019 y Fh(p)p Ff(\000)p Fl(2)1615 2053 y Ff(jr)p Fh(v)1725 2028 y Fd(\()p Fc(p=)p Fd(2\))1864 2053 y Ff(j)1884 2028 y Fd(2)p 1615 2074 302 4 v 1670 2122 a Ff(j)p Fh(x)p Ff(j)1748 2106 y Fc(n)p Fb(\000)p Fd(2)1945 2094 y Fk(+)k Fi(\013)2081 2106 y Fl(2)2142 2053 y Ff(jr)p Fh(v)r Ff(j)2272 2028 y Fc(p)p 2128 2074 194 4 v 2128 2122 a Ff(j)p Fh(x)p Ff(j)2206 2106 y Fc(n)p Fb(\000)p Fc(p)849 2260 y Fj(\025)996 2168 y Fg(\020)1056 2223 y Fh(n)p Ff(\000)p Fh(p)p 1056 2241 128 4 v 1102 2288 a(p)1193 2168 y Fg(\021)1243 2185 y Fh(p)1324 2227 y(u)1363 2202 y Fc(p)p 1305 2241 112 4 v 1305 2288 a Ff(j)p Fh(x)p Ff(j)1383 2272 y Fc(p)1445 2260 y Fj(\000)1528 2168 y Fg(\020)1588 2223 y Fh(n)p Ff(\000)p Fh(p)p 1588 2241 128 4 v 1634 2288 a(p)1725 2168 y Fg(\021)1774 2185 y Fh(p)p Ff(\000)p Fl(1)1912 2260 y Fj(h)1994 2227 y Fh(x)p 1954 2241 118 4 v 1954 2288 a Ff(j)p Fh(x)p Ff(j)2032 2272 y Fc(n)2082 2260 y Fi(;)c Fj(r)p Fi(v)2231 2230 y Fh(p)2269 2260 y Fj(i)19 b Fk(+)2413 2226 y Fl(4)p Fh(\013)2489 2234 y Fd(1)p 2413 2241 109 4 v 2434 2288 a Fh(p)2468 2272 y Fd(2)2545 2168 y Fg(\020)2605 2223 y Fh(n)p Ff(\000)p Fh(p)p 2605 2241 128 4 v 2651 2288 a(p)2742 2168 y Fg(\021)2792 2185 y Fh(p)p Ff(\000)p Fl(2)2939 2220 y Ff(jr)p Fh(v)3049 2195 y Fd(\()p Fc(p=)p Fd(2\))3188 2220 y Ff(j)3208 2195 y Fd(2)p 2939 2241 302 4 v 2994 2288 a Ff(j)p Fh(x)p Ff(j)3072 2272 y Fc(n)p Fb(\000)p Fd(2)3264 2260 y Fi(:)3305 2384 y Fk(\(2.6\))523 2534 y(W)-7 b(e)39 b(no)n(w)f(in)n(tegrate)f(the) i(ab)r(o)n(v)n(e)e(inequalit)n(y)h(o)n(v)n(er)f(\012,)i(taking)f Fi(w)2653 2546 y Fl(1)2732 2534 y Fk(=)j Fi(v)2881 2504 y Fh(p=)p Fl(2)3025 2534 y Fk(in)e(\(2.4\))f(and)523 2683 y(remem)n(b)r(ering)27 b(that)h Fi(v)s Fk(\(0\))23 b(=)g(0.)k(W)-7 b(e)28 b(th)n(us)f(obtain)585 2841 y Fg(R)581 2987 y Fl(\012)654 2908 y Fj(jr)p Fi(u)p Fj(j)817 2878 y Fh(p)874 2908 y Fj(\000)957 2816 y Fg(\020)1016 2871 y Fh(n)p Ff(\000)p Fh(p)p 1016 2889 128 4 v 1063 2936 a(p)1154 2816 y Fg(\021)1203 2833 y Fh(p)1284 2875 y(u)1323 2850 y Fc(p)p 1265 2889 112 4 v 1265 2936 a Ff(j)p Fh(x)p Ff(j)1343 2920 y Fc(p)1410 2908 y Fj(\025)c(\000)1577 2816 y Fg(\020)1636 2871 y Fh(n)p Ff(\000)p Fh(p)p 1636 2889 128 4 v 1682 2936 a(p)1773 2816 y Fg(\021)1823 2833 y Fh(p)p Ff(\000)p Fl(1)1960 2841 y Fg(R)1956 2987 y Fl(\012)2015 2908 y Fj(h)2097 2875 y Fh(x)p 2057 2889 118 4 v 2057 2936 a Ff(j)p Fh(x)p Ff(j)2135 2920 y Fc(n)2185 2908 y Fi(;)14 b Fj(r)p Fi(v)2334 2878 y Fh(p)2373 2908 y Fj(i)19 b Fk(+)2517 2874 y Fl(4)p Fh(\013)2593 2882 y Fd(1)p 2517 2889 109 4 v 2538 2936 a Fh(p)2572 2920 y Fd(2)2649 2816 y Fg(\020)2709 2871 y Fh(n)p Ff(\000)p Fh(p)p 2709 2889 128 4 v 2755 2936 a(p)2846 2816 y Fg(\021)2895 2833 y Fh(p)p Ff(\000)p Fl(2)3032 2841 y Fg(R)3072 2937 y Fl(\012)3147 2868 y Ff(jr)p Fh(w)3270 2876 y Fd(1)3303 2868 y Ff(j)3323 2842 y Fd(2)p 3147 2889 208 4 v 3155 2936 a Ff(j)p Fh(x)p Ff(j)3233 2920 y Fc(n)p Fb(\000)p Fd(2)585 3107 y Fj(\025)k(\000)752 3015 y Fg(\020)811 3070 y Fh(n)p Ff(\000)p Fh(p)p 811 3088 128 4 v 857 3136 a(p)948 3015 y Fg(\021)997 3032 y Fh(p)p Ff(\000)p Fl(1)1158 3107 y Fj(j)p Fi(S)1237 3077 y Fh(n)p Ff(\000)p Fl(1)1367 3107 y Fj(j)14 b Fi(v)1447 3077 y Fh(p)1485 3107 y Fk(\(0\))19 b Fj(\000)1693 3015 y Fg(\020)1752 3070 y Fh(n)p Ff(\000)p Fh(p)p 1752 3088 V 1799 3136 a(p)1889 3015 y Fg(\021)1939 3032 y Fh(p)p Ff(\000)p Fl(1)2092 3041 y Fg(R)2068 3187 y Fh(@)t Fl(\012)2186 3067 y Ff(h)p Fh(x)p Ff(\001)p Fh(\027)t Ff(i)p 2186 3088 149 4 v 2201 3136 a(j)p Fh(x)p Ff(j)2279 3119 y Fc(n)2344 3107 y Fi(v)2387 3077 y Fh(p)668 3341 y Fk(+)743 3307 y Fl(4)p Fh(\013)819 3315 y Fd(1)p 743 3322 109 4 v 764 3369 a Fh(p)798 3353 y Fd(2)875 3248 y Fg(\020)935 3304 y Fh(n)p Ff(\000)p Fh(p)p 935 3321 128 4 v 981 3369 a(p)1072 3248 y Fg(\021)1121 3266 y Fh(p)p Ff(\000)p Fl(2)1259 3199 y Fg(\()1335 3308 y Fl(1)p 1335 3322 34 4 v 1335 3369 a(4)1392 3274 y Fg(R)1388 3419 y Fl(\012)1461 3199 y Fg( )1566 3262 y Fh(k)1541 3278 y Fg(P)1527 3414 y Fh(j)s Fl(=1)1859 3308 y(1)p 1666 3322 419 4 v 1666 3376 a(\(log)1778 3356 y Fd(\()p Fc(j)r Fd(\))1865 3376 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2053 3359 y Fd(2)2095 3199 y Fg(!)2202 3298 y Fh(w)2252 3273 y Fd(2)2250 3315 y(1)p 2184 3321 118 4 v 2184 3369 a Ff(j)p Fh(x)p Ff(j)2262 3353 y Fc(n)2330 3341 y Fj(\000)2423 3308 y Fl(1)p 2423 3322 34 4 v 2423 3369 a(2)2496 3274 y Fg(R)2472 3420 y Fh(@)t Fl(\012)2581 3199 y Fg( )2685 3262 y Fh(k)2660 3278 y Fg(P)2646 3414 y Fh(j)s Fl(=1)2878 3300 y Ff(h)p Fh(x)p Ff(\001)p Fh(\027)t Ff(i)p 2785 3321 335 4 v 2785 3376 a Fl(log)2871 3356 y Fd(\()p Fc(j)r Fd(\))2958 3376 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)3130 3199 y Fg(!)3237 3298 y Fh(w)3287 3273 y Fd(2)3285 3315 y(1)p 3219 3321 118 4 v 3219 3369 a Ff(j)p Fh(x)p Ff(j)3297 3353 y Fc(n)3347 3199 y Fg(\))668 3590 y Fj(\025)k Fi(C)815 3602 y Fl(1)866 3523 y Fg(R)862 3669 y Fl(\012)935 3448 y Fg( )1040 3511 y Fh(k)1015 3527 y Fg(P)1001 3663 y Fh(j)s Fl(=1)1333 3557 y(1)p 1140 3571 419 4 v 1140 3625 a(\(log)1252 3605 y Fd(\()p Fc(j)r Fd(\))1339 3625 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))1527 3608 y Fd(2)1569 3448 y Fg(!)1677 3557 y Fh(u)1716 3532 y Fc(p)p 1658 3571 112 4 v 1658 3618 a Ff(j)p Fh(x)p Ff(j)1736 3602 y Fc(p)1798 3590 y Fk(+)c Fi(\025)1959 3523 y Fg(R)1936 3669 y Fh(@)t Fl(\012)2044 3590 y Fi(u)2092 3559 y Fh(p)2130 3590 y Fi(;)763 3980 y Fk(where)27 b Fi(C)1062 3992 y Fl(1)1123 3980 y Fk(=)1220 3946 y Fh(\013)1263 3954 y Fd(1)p 1220 3961 76 4 v 1224 4009 a Fh(p)1258 3992 y Fd(2)1320 3888 y Fg(\020)1379 3943 y Fh(n)p Ff(\000)p Fh(p)p 1379 3961 128 4 v 1426 4009 a(p)1516 3888 y Fg(\021)1566 3905 y Fh(p)p Ff(\000)p Fl(2)1717 3980 y Fk(,)763 4271 y(and)41 b Fj(\000)p Fi(\025)23 b Fk(=)45 b(sup)1162 4341 y Fh(x)p Ff(2)p Fh(@)t Fl(\012)1355 4230 y Ff(jh)p Fh(x)p Ff(\001)p Fh(\027)t Ff(ij)p 1355 4252 188 4 v 1390 4299 a(j)p Fh(x)p Ff(j)1468 4283 y Fc(n)1566 4129 y Fg(\()1633 4179 y(\020)1693 4234 y Fh(n)p Ff(\000)p Fh(p)p 1693 4252 128 4 v 1739 4299 a(p)1830 4179 y Fg(\021)1879 4196 y Fh(p)p Ff(\000)p Fl(1)2021 4271 y Fk(+)2114 4237 y Fl(2)p Fh(\013)2190 4245 y Fd(1)p 2114 4252 109 4 v 2135 4299 a Fh(p)2169 4283 y Fd(2)2246 4179 y Fg(\020)2306 4234 y Fh(n)p Ff(\000)p Fh(p)p 2306 4252 128 4 v 2352 4299 a(p)2443 4179 y Fg(\021)2493 4196 y Fh(p)p Ff(\000)p Fl(2)2669 4192 y Fh(k)2644 4208 y Fg(P)2630 4344 y Fh(j)s Fl(=1)2919 4238 y(1)p 2769 4252 335 4 v 2769 4306 a(log)2854 4286 y Fd(\()p Fc(j)r Fd(\))2942 4306 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)3113 4129 y Fg(\))3194 4271 y Fi(:)523 4550 y Fm(Step)c(2.)35 b Fk(Let)g Fi(p)h Fk(=)f Fi(n)h Fk(and)f(0)h Fj(\024)f Fi(u)h Fj(2)g Fi(C)1868 4520 y Fl(1)1906 4550 y Fk(\()p 1938 4483 60 4 v(\012\))g(and)f(de\014ne)g Fi(v)40 b Fk(=)35 b(\(log)15 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3057 4493 y Fd(1)p Fb(\000)p Fc(n)p 3057 4507 110 4 v 3093 4540 a(n)3181 4550 y Fi(u:)35 b Fk(Then)1979 5150 y(9)p eop %%Page: 10 10 10 9 bop 523 614 a Fi(v)s Fk(\(0\))23 b(=)g(0)k(and)h(from)f(\(2.1\))g (w)n(e)g(ha)n(v)n(e)565 811 y Fj(jr)p Fi(u)p Fj(j)728 781 y Fh(p)849 811 y Fk(=)82 b Fj(j)19 b(\000)1121 743 y Fg(\000)1169 778 y Fh(n)p Ff(\000)p Fl(1)p 1169 792 127 4 v 1211 839 a Fh(n)1305 743 y Fg(\001)1357 811 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1741 781 y Ff(\000)p Fl(1)p Fh(=n)1951 778 y(x)p 1915 792 110 4 v 1915 839 a Ff(j)p Fh(x)p Ff(j)1993 823 y Fd(2)2035 811 y Fi(v)21 b Fk(+)e(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2574 754 y Fc(n)p Fb(\000)p Fd(1)p 2574 768 V 2610 801 a Fc(n)2698 811 y Fj(r)p Fi(v)s Fj(j)2833 781 y Fh(n)849 979 y Fk(=)996 911 y Fg(\000)1044 946 y Fh(n)p Ff(\000)p Fl(1)p 1044 960 127 4 v 1087 1007 a Fh(n)1180 911 y Fg(\001)1218 929 y Fh(n)1287 938 y Fl(\(log)f Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))1599 913 y Fb(\000)p Fd(1)p 1287 960 388 4 v 1422 1007 a Ff(j)p Fh(x)p Ff(j)1500 991 y Fc(n)1699 979 y Fi(v)1742 949 y Fh(n)1788 858 y Fg(\014)1788 908 y(\014)1788 958 y(\014)1788 1008 y(\014)1845 946 y Fh(x)p 1825 960 78 4 v 1825 1007 a Ff(j)p Fh(x)p Ff(j)1931 979 y Fj(\000)2024 938 y Ff(j)p Fh(x)p Ff(j)p Fl(\()o(log)f Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))o Ff(r)p Fh(v)p 2024 960 479 4 v 2245 1007 a(v)2512 858 y Fg(\014)2512 908 y(\014)2512 958 y(\014)2512 1008 y(\014)2540 879 y Fh(n)849 1151 y Fj(\025)996 1083 y Fg(\000)1044 1118 y Fh(n)p Ff(\000)p Fl(1)p 1044 1132 127 4 v 1087 1179 a Fh(n)1180 1083 y Fg(\001)1242 1110 y Fl(\(log)g Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))1553 1085 y Fb(\000)p Fd(1)p 1242 1131 388 4 v 1377 1179 a Ff(j)p Fh(x)p Ff(j)1455 1163 y Fc(n)1654 1151 y Fi(v)1697 1120 y Fh(n)1756 1083 y Fg(\010)1805 1151 y Fk(1)18 b Fj(\000)g Fi(n)p Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))23 b Fj(h)p Fi(x;)2531 1118 y Ff(r)p Fh(v)p 2531 1132 91 4 v 2559 1179 a(v)2632 1151 y Fj(i)1006 1289 y Fk(+)p Fi(\013)1124 1301 y Fl(1)1175 1289 y Fj(j)p Fi(x)p Fj(j)1268 1259 y Fl(2)1320 1289 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1704 1259 y Fl(2)1765 1249 y Ff(jr)p Fh(v)r Ff(j)1895 1224 y Fd(2)p 1765 1270 163 4 v 1813 1318 a Fh(v)1848 1301 y Fd(2)1956 1289 y Fk(+)k Fi(\013)2092 1301 y Fl(2)2143 1289 y Fj(j)p Fi(x)p Fj(j)2236 1259 y Fh(n)2296 1289 y Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2680 1259 y Fh(n)2749 1249 y Ff(jr)p Fh(v)r Ff(j)2879 1224 y Fc(n)p 2749 1270 171 4 v 2797 1318 a Fh(v)2832 1301 y Fc(n)2930 1197 y Fg(o)849 1456 y Fj(\025)996 1388 y Fg(\000)1044 1423 y Fh(n)p Ff(\000)p Fl(1)p 1044 1437 127 4 v 1087 1484 a Fh(n)1180 1388 y Fg(\001)1218 1406 y Fh(n)1277 1363 y Fg(\020)1327 1456 y Fk(log)1471 1423 y Fh(R)p 1458 1437 78 4 v 1458 1484 a Ff(j)p Fh(x)p Ff(j)1545 1363 y Fg(\021)1595 1381 y Ff(\000)p Fl(1)1729 1423 y Fh(v)1764 1398 y Fc(n)p 1708 1437 118 4 v 1708 1484 a Ff(j)p Fh(x)p Ff(j)1786 1468 y Fc(n)1854 1456 y Fj(\000)1937 1388 y Fg(\000)1985 1423 y Fh(n)p Ff(\000)p Fl(1)p 1985 1437 127 4 v 2027 1484 a Fh(n)2121 1388 y Fg(\001)2159 1406 y Fh(n)2218 1456 y Fj(h)2280 1423 y Fh(x)p 2260 1437 78 4 v 2260 1484 a Ff(j)p Fh(x)p Ff(j)2347 1456 y Fi(;)g Fj(r)p Fi(v)2496 1425 y Fh(n)2542 1456 y Fj(i)996 1585 y Fk(+)1071 1551 y Fl(4)p Fh(\013)1147 1559 y Fd(1)p 1071 1566 109 4 v 1089 1613 a Fh(n)1130 1597 y Fd(2)1190 1585 y Fk(\()1232 1552 y Fh(n)p Ff(\000)p Fl(1)p 1232 1566 127 4 v 1274 1613 a Fh(n)1368 1585 y Fk(\))1400 1554 y Fh(n)1459 1585 y Fk(\(log)g Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))g Fj(jr)p Fk(\()p Fi(v)2024 1554 y Fh(n=)p Fl(2)2137 1585 y Fk(\))p Fj(j)2192 1554 y Fl(2)2248 1585 y Fk(+)k Fi(\013)2384 1597 y Fl(2)2436 1517 y Fg(\000)2484 1552 y Fh(n)p Ff(\000)p Fl(1)p 2484 1566 V 2526 1613 a Fh(n)2620 1517 y Fg(\001)2658 1535 y Fh(n)2717 1585 y Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3101 1554 y Fh(n)p Ff(\000)p Fl(1)3245 1585 y Fj(jr)p Fi(v)s Fj(j)3403 1554 y Fh(n)3449 1585 y Fi(:)3305 1685 y Fk(\(2.7\))523 1835 y(Hence)28 b(using)f(\(2.4\))g (as)g(in)h(Step)g(1)f(w)n(e)h(\014nally)f(\014nd)523 2004 y Fg(Z)522 2242 y Fl(\012)620 2000 y Fg(\022)681 2117 y Fj(jr)p Fi(u)p Fj(j)844 2083 y Fh(n)907 2117 y Fj(\000)991 2000 y Fg(\022)1062 2061 y Fi(n)18 b Fj(\000)g Fk(1)p 1062 2098 193 4 v 1133 2174 a Fi(n)1264 2000 y Fg(\023)1325 2017 y Fh(n)1639 2061 y Fi(u)1687 2030 y Fh(n)p 1394 2098 583 4 v 1394 2174 a Fj(j)p Fi(x)p Fj(j)1487 2150 y Fh(n)1547 2174 y Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1931 2133 y Fh(n)1987 2000 y Fg(\023)2071 2117 y Fj(\025)22 b Fi(C)2217 2129 y Fl(1)2269 2004 y Fg(Z)2268 2242 y Fl(\012)2366 1950 y Fg(0)2366 2099 y(@)2480 2013 y Fh(k)2438 2038 y Fg(X)2441 2215 y Fh(j)s Fl(=2)2820 2061 y Fk(1)p 2582 2098 517 4 v 2582 2188 a(\(log)2722 2151 y Fl(\()p Fh(j)s Fl(\))2822 2188 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3053 2164 y Fh(n)3109 1950 y Fg(1)3109 2099 y(A)3228 2061 y Fi(u)3276 2030 y Fh(n)p 3205 2098 139 4 v 3205 2174 a Fj(j)p Fi(x)p Fj(j)3298 2150 y Fh(n)3354 2117 y Fk(+)p Fi(\025)3482 2004 y Fg(Z)3462 2242 y Fh(@)t Fl(\012)3581 2117 y Fi(u)3629 2083 y Fh(n)3674 2117 y Fi(:)523 2483 y Fm(Step)28 b(3.)23 b Fk(Let)h(1)f Fi(<)f(p)h(<)g Fk(2)p Fi(:)g Fk(In)h(this)h(case)e(the)h(analysis)f(is)h(more)f (delicate)h(and)f(w)n(e)h(use)g(Sard's)523 2632 y(Lemma.)30 b(T)-7 b(o)30 b(b)r(egin)g(with)h(w)n(e)e(assume)h(that)g Fi(u)d Fj(2)h Fi(C)2207 2602 y Ff(1)2277 2632 y Fk(\()p 2309 2566 60 4 v(\012\))p Fi(:)j Fk(Let)f Fi(v)h Fk(=)c Fj(j)p Fi(x)p Fj(j)2872 2568 y Fc(n)p Fb(\000)p Fc(p)p 2872 2583 113 4 v 2913 2615 a(p)2998 2632 y Fi(u)j Fk(and)g(c)n(ho)r (ose)523 2782 y Fi(M)i(>)22 b Fk(1)28 b(suc)n(h)f(that)1082 3193 y Fi(M)1172 3158 y Fh(p)p Ff(\000)p Fl(1)1318 3193 y Fi(>)1416 3136 y Fk(2)14 b Fi(\013)1525 3148 y Fl(1)p 1416 3174 146 4 v 1416 3250 a Fi(p)g(\013)1525 3262 y Fl(2)1586 3076 y Fg(\022)1657 3136 y Fi(n)k Fj(\000)g Fi(p)p 1657 3174 194 4 v 1732 3250 a(p)1860 3076 y Fg(\023)1921 3093 y Fh(p)p Ff(\000)p Fl(2)2033 3193 y Fk(sup)2030 3279 y Fh(x)p Ff(2)p 2113 3229 48 4 v Fl(\012)2174 3026 y Fg(0)2174 3175 y(@)2288 3089 y Fh(k)2247 3114 y Fg(X)2249 3291 y Fh(j)s Fl(=1)2573 3136 y Fk(1)p 2390 3174 407 4 v 2390 3264 a(log)2498 3227 y Fl(\()p Fh(j)s Fl(\))2598 3264 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)2807 3026 y Fg(1)2807 3175 y(A)2894 3193 y Fi(:)763 3530 y Fk(By)36 b(p)r(erturbing)g Fi(v)s Fk(,)h(and)f(if)h(necessary)e Fi(@)5 b Fk(\012)36 b(also,)f(w)n(e)h(can)g(assume)g(that)h Fi(M)3269 3500 y Fl(2)3342 3530 y Fk(is)f(a)523 3679 y(regular)26 b(v)-5 b(alue)27 b(for)1156 3639 y Ff(jr)p Fh(v)r Ff(j)1286 3614 y Fd(2)1319 3639 y Ff(j)p Fh(x)p Ff(j)1397 3614 y Fd(2)p 1156 3660 273 4 v 1259 3708 a Fh(v)1294 3691 y Fd(2)1438 3679 y Fi(:)h Fk(Let)1409 3895 y(\012)1469 3865 y Fl(+)1608 3895 y Fk(=)1755 3803 y Fg(n)1810 3895 y Fi(x)c Fj(2)p 1959 3828 60 4 v 23 w Fk(\012)f(:)2099 3855 y Ff(j)p Fh(x)p Ff(j)18 b(jr)p Fh(v)r Ff(j)p 2099 3876 227 4 v 2194 3924 a Fh(v)2358 3895 y Fi(<)k(M)2535 3803 y Fg(o)1409 4044 y Fk(\012)1469 4014 y Ff(\000)1608 4044 y Fk(=)1755 3952 y Fg(n)1810 4044 y Fi(x)i Fj(2)p 1959 3978 60 4 v 23 w Fk(\012)f(:)2099 4004 y Ff(j)p Fh(x)p Ff(j)18 b(jr)p Fh(v)r Ff(j)p 2099 4025 227 4 v 2194 4073 a Fh(v)2358 4044 y Fj(\025)k Fi(M)2535 3952 y Fg(o)1409 4194 y Fk(\000)147 b(=)1755 4102 y Fg(n)1810 4194 y Fi(x)24 b Fj(2)p 1959 4127 60 4 v 23 w Fk(\012)f(:)2099 4154 y Ff(j)p Fh(x)p Ff(j)18 b(jr)p Fh(v)r Ff(j)p 2099 4175 227 4 v 2194 4222 a Fh(v)2358 4194 y Fk(=)k Fi(M)2535 4102 y Fg(o)523 4405 y Fk(F)-7 b(urthermore)31 b(w)n(e)i(can)f(assume)g (that)g(0)f Fj(2)h Fk(\012)1990 4375 y Fl(+)2077 4405 y Fk(and)h(that)g(\000)f(in)n(tersects)g Fi(@)5 b Fk(\012)32 b(transv)n(ersely)-7 b(.)523 4555 y(Hence,)28 b(from)f(\(2.2\),)h(w)n (e)f(ha)n(v)n(e)1958 5150 y(10)p eop %%Page: 11 11 11 10 bop 523 828 a Fj(jr)p Fi(u)p Fj(j)686 794 y Fh(p)747 828 y Fj(\025)835 711 y Fg(\022)906 772 y Fi(n)18 b Fj(\000)g Fi(p)p 906 809 194 4 v 982 885 a(p)1109 711 y Fg(\023)1170 728 y Fh(p)1222 828 y Fj(j)p Fi(x)p Fj(j)1315 794 y Ff(\000)p Fh(n)1413 828 y Fi(v)1456 794 y Fh(p)1508 711 y Fg(\032)1571 828 y Fk(1)g Fj(\000)g Fi(p)1770 711 y Fg(\022)1916 772 y Fi(p)p 1840 809 V 1840 885 a(n)h Fj(\000)f Fi(p)2043 711 y Fg(\023)2128 772 y Fj(h)p Fi(x;)c Fj(r)p Fi(v)s Fj(i)p 2128 809 262 4 v 2237 885 a Fi(v)2418 828 y Fk(+)k Fi(\013)2554 840 y Fl(2)2605 711 y Fg(\022)2752 772 y Fi(p)p 2676 809 194 4 v 2676 885 a(n)h Fj(\000)f Fi(p)2879 711 y Fg(\023)2941 728 y Fh(p)3003 772 y Fj(j)p Fi(x)p Fj(j)3096 742 y Fh(p)3135 772 y Fj(jr)p Fi(v)s Fj(j)3293 742 y Fh(p)p 3003 809 329 4 v 3126 885 a Fi(v)3169 861 y Fh(p)3342 828 y Fi(\037)3394 861 y Fd(\012)3434 849 y Fb(\000)1530 1125 y Fk(+)p Fi(\013)1648 1137 y Fl(1)1699 1008 y Fg(\022)1845 1069 y Fi(p)p 1770 1106 194 4 v 1770 1182 a(n)g Fj(\000)g Fi(p)1973 1008 y Fg(\023)2034 1025 y Fl(2)2095 1069 y Fj(j)p Fi(x)p Fj(j)2188 1039 y Fl(2)2226 1069 y Fj(jr)p Fi(v)s Fj(j)2384 1039 y Fl(2)p 2095 1106 327 4 v 2218 1182 a Fi(v)2261 1158 y Fl(2)2431 1125 y Fi(\037)2483 1158 y Fd(\012)2523 1146 y(+)2578 983 y Fg(\))523 1334 y Fk(In)n(tegrating)26 b(this)i(w)n(e)f(obtain)653 1453 y Fg(Z)652 1691 y Fl(\012)750 1566 y Fj(jr)p Fi(u)p Fj(j)913 1532 y Fh(p)969 1566 y Fj(\000)1052 1449 y Fg(\022)1123 1510 y Fi(n)19 b Fj(\000)f Fi(p)p 1123 1547 194 4 v 1199 1623 a(p)1327 1449 y Fg(\023)1388 1466 y Fh(p)1473 1510 y Fi(u)1521 1480 y Fh(p)p 1450 1547 132 4 v 1450 1623 a Fj(j)p Fi(x)p Fj(j)1543 1599 y Fh(p)1615 1566 y Fj(\025)k(\000)1781 1449 y Fg(\022)1852 1510 y Fi(n)c Fj(\000)g Fi(p)p 1852 1547 194 4 v 1928 1623 a(p)2055 1449 y Fg(\023)2116 1466 y Fl(\()p Fh(p)p Ff(\000)p Fl(1\))2307 1453 y Fg(Z)2287 1692 y Fh(@)t Fl(\012)2415 1510 y Fj(h)p Fi(x)i Fj(\001)e Fi(\027)5 b Fj(i)14 b Fi(v)2690 1480 y Fh(p)p 2415 1547 314 4 v 2503 1623 a Fj(j)p Fi(x)p Fj(j)2596 1599 y Fh(n)2781 1566 y Fk(+)41 b Fi(\013)2940 1578 y Fl(2)2998 1453 y Fg(Z)2972 1696 y Fl(\012)3019 1679 y Fb(\000)3125 1510 y Fj(jr)p Fi(v)s Fj(j)3283 1480 y Fh(p)p 3111 1547 225 4 v 3111 1623 a Fj(j)p Fi(x)p Fj(j)3204 1599 y Fh(n)p Ff(\000)p Fh(p)1443 1914 y Fk(+)1518 1858 y(4)p Fi(\013)1613 1870 y Fl(1)p 1517 1895 132 4 v 1544 1971 a Fi(p)1586 1947 y Fl(2)1673 1797 y Fg(\022)1744 1858 y Fi(n)18 b Fj(\000)g Fi(p)p 1744 1895 194 4 v 1820 1971 a(p)1947 1797 y Fg(\023)2008 1814 y Fh(p)p Ff(\000)p Fl(2)2109 1801 y Fg(Z)2085 2044 y Fl(\012)2132 2027 y Fd(+)2222 1858 y Fj(jr)p Fi(v)2357 1828 y Fh(p=)p Fl(2)2463 1858 y Fj(j)2486 1828 y Fl(2)p 2222 1895 302 4 v 2261 1971 a Fj(j)p Fi(x)p Fj(j)2354 1947 y Fh(n)p Ff(\000)p Fl(2)2533 1914 y Fi(:)749 b Fk(\(2.8\))523 2212 y(Let)28 b Fi(A)23 b Fk(=)855 2178 y Fl(4)p Fh(\013)931 2186 y Fd(1)p 855 2193 109 4 v 876 2240 a Fh(p)910 2224 y Fd(2)987 2119 y Fg(\020)1047 2175 y Fh(n)p Ff(\000)p Fh(p)p 1047 2193 128 4 v 1093 2240 a(p)1184 2119 y Fg(\021)1233 2137 y Fh(p)p Ff(\000)p Fl(2)1357 2212 y Fk(.)28 b(Then,)f(from)h(\(2.4\))f(w) n(e)g(ha)n(v)n(e)570 2384 y Fg(R)566 2530 y Fl(\012)639 2451 y Fj(jr)p Fi(u)p Fj(j)802 2421 y Fh(p)923 2451 y Fj(\000)1071 2359 y Fg(\020)1130 2414 y Fh(n)p Ff(\000)p Fh(p)p 1130 2432 V 1177 2479 a(p)1267 2359 y Fg(\021)1317 2376 y Fh(p)1398 2418 y(u)1437 2393 y Fc(p)p 1379 2432 112 4 v 1379 2479 a Ff(j)p Fh(x)p Ff(j)1457 2463 y Fc(p)1520 2451 y Fk(+)1603 2359 y Fg(\020)1662 2414 y Fh(n)p Ff(\000)p Fh(p)p 1662 2432 128 4 v 1709 2479 a(p)1799 2359 y Fg(\021)1849 2376 y Fh(p)p Ff(\000)p Fl(1)2002 2384 y Fg(R)1978 2530 y Fh(@)t Fl(\012)2096 2411 y Ff(h)p Fh(x)p Ff(\001)p Fh(\027)t Ff(i)p Fh(u)2284 2386 y Fc(p)p 2096 2432 223 4 v 2151 2479 a Ff(j)p Fh(x)p Ff(j)2229 2463 y Fc(p)2352 2451 y Fj(\025)22 b Fi(A)2535 2384 y Fg(R)2507 2534 y Fl(\012)2554 2518 y Fd(+)2633 2411 y Ff(jr)p Fh(v)2743 2386 y Fc(p=)p Fd(2)2838 2411 y Ff(j)2858 2386 y Fd(2)p 2633 2432 257 4 v 2666 2479 a Ff(j)p Fh(x)p Ff(j)2744 2463 y Fc(n)p Fb(\000)p Fd(2)2918 2451 y Fk(+)c Fi(\013)3054 2463 y Fl(2)3126 2384 y Fg(R)3097 2534 y Fl(\012)3144 2518 y Fb(\000)3240 2411 y Ff(jr)p Fh(v)r Ff(j)3370 2386 y Fc(p)p 3226 2432 194 4 v 3226 2479 a Ff(j)p Fh(x)p Ff(j)3304 2463 y Fc(n)p Fb(\000)p Fc(p)1071 2688 y Fj(\025)1168 2655 y Fh(A)p 1168 2669 50 4 v 1176 2717 a Fl(4)1262 2621 y Fg(R)1234 2771 y Fl(\012)1281 2755 y Fd(+)1350 2546 y Fg( )1455 2609 y Fh(k)1430 2626 y Fg(P)1416 2761 y Fh(j)s Fl(=1)1748 2655 y(1)p 1555 2669 419 4 v 1555 2723 a(\(log)1666 2703 y Fd(\()p Fc(j)r Fd(\))1754 2723 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))1942 2707 y Fd(2)1983 2546 y Fg(!)2097 2655 y Fh(v)2132 2630 y Fc(p)p 2073 2669 118 4 v 2073 2717 a Ff(j)p Fh(x)p Ff(j)2151 2700 y Fc(n)2219 2688 y Fj(\000)2312 2655 y Fh(A)p 2312 2669 50 4 v 2320 2717 a Fl(2)2471 2621 y Fg(R)2378 2771 y Fl(\012)2425 2755 y Fd(+)2472 2771 y Ff(\\)p Fh(@)t Fl(\012)2625 2546 y Fg( )2730 2609 y Fh(k)2704 2626 y Fg(P)2691 2761 y Fh(j)s Fl(=1)2923 2648 y Ff(h)p Fh(x;\027)t Ff(i)p 2830 2669 335 4 v 2830 2723 a Fl(log)2915 2703 y Fd(\()p Fc(j)r Fd(\))3003 2723 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)3174 2546 y Fg(!)3287 2655 y Fh(v)3322 2630 y Fc(p)p 3264 2669 118 4 v 3264 2717 a Ff(j)p Fh(x)p Ff(j)3342 2700 y Fc(n)1071 2937 y Fj(\000)1146 2904 y Fh(A)p 1145 2918 50 4 v 1153 2966 a Fl(2)1219 2870 y Fg(R)1218 3016 y Fl(\000)1288 2795 y Fg( )1393 2858 y Fh(k)1368 2875 y Fg(P)1354 3010 y Fh(j)s Fl(=1)1612 2897 y Ff(h)p Fh(x)p Ff(\001)p Fh(\027)t Ff(i)p 1493 2918 387 4 v 1493 2972 a Fl(\(log)1604 2952 y Fd(\()p Fc(j)r Fd(\))1692 2972 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))1889 2795 y Fg(!)2002 2904 y Fh(v)2037 2879 y Fc(p)p 1979 2918 118 4 v 1979 2966 a Ff(j)p Fh(x)p Ff(j)2057 2949 y Fc(n)2125 2937 y Fk(+)g Fi(\013)2261 2949 y Fl(2)2333 2870 y Fg(R)2304 3020 y Fl(\012)2351 3004 y Fb(\000)2447 2897 y Ff(jr)p Fh(v)r Ff(j)2577 2872 y Fc(p)p 2432 2918 194 4 v 2432 2966 a Ff(j)p Fh(x)p Ff(j)2510 2949 y Fc(n)p Fb(\000)p Fc(p)2636 2937 y Fi(:)523 3180 y Fk(Since)30 b Fi(\027)k Fk(is)29 b(the)h(outer)f(normal)f(on)h(\000)g(with)h(resp)r(ect)f(to)g (\012)2387 3150 y Fl(+)2442 3180 y Fk(,)h(it)g(is)f(the)h(inner)f (normal)f(on)h(\000)523 3329 y(with)f(resp)r(ect)f(to)h(\012)1158 3299 y Ff(\000)1242 3329 y Fk(and)f(w)n(e)g(ha)n(v)n(e)619 3475 y Fg(Z)594 3718 y Fl(\012)641 3701 y Fb(\000)709 3588 y Fj(h)797 3532 y Fi(x)p 751 3569 139 4 v 751 3645 a Fj(j)p Fi(x)p Fj(j)844 3621 y Fh(n)900 3588 y Fi(;)83 b Fj(r)1158 3421 y Fg(0)1158 3571 y(@)1272 3485 y Fh(k)1231 3509 y Fg(X)1233 3686 y Fh(j)s Fl(=1)1569 3532 y Fi(v)1612 3502 y Fh(p)p 1374 3569 472 4 v 1374 3660 a Fk(\(log)1514 3623 y Fl(\()p Fh(j)s Fl(\))1615 3660 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1856 3421 y Fg(1)1856 3571 y(A)1928 3588 y Fj(i)9 b Fk(=)1158 3930 y Fj(\000)1223 3817 y Fg(Z)1226 4055 y Fl(\000)1306 3763 y Fg(0)1306 3912 y(@)1420 3826 y Fh(k)1378 3851 y Fg(X)1381 4028 y Fh(j)s Fl(=1)1616 3873 y Fj(h)p Fi(x)19 b Fj(\001)g Fi(\027)5 b Fj(i)p 1522 3911 407 4 v 1522 4001 a Fk(log)1629 3964 y Fl(\()p Fh(j)s Fl(\))1730 4001 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)1939 3763 y Fg(1)1939 3912 y(A)2064 3873 y Fi(v)2107 3843 y Fh(p)p 2035 3911 139 4 v 2035 3987 a Fj(j)p Fi(x)p Fj(j)2128 3963 y Fh(n)2202 3930 y Fk(+)2330 3817 y Fg(Z)2239 4059 y Fl(\012)2286 4043 y Fb(\000)2336 4059 y Ff(\\)p Fh(@)t Fl(\012)2485 3763 y Fg(0)2485 3912 y(@)2600 3826 y Fh(k)2558 3851 y Fg(X)2560 4028 y Fh(j)s Fl(=1)2807 3873 y Fj(h)p Fi(x;)14 b(\027)5 b Fj(i)p 2702 3911 407 4 v 2702 4001 a Fk(log)2809 3964 y Fl(\()p Fh(j)s Fl(\))2910 4001 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)3118 3763 y Fg(1)3118 3912 y(A)3243 3873 y Fi(v)3286 3843 y Fh(p)p 3215 3911 139 4 v 3215 3987 a Fj(j)p Fi(x)p Fj(j)3308 3963 y Fh(n)3364 3930 y Fi(:)523 4198 y Fk(Let)38 b Fi(J)46 b Fk(denote)38 b(the)h(LHS)f(of)g(the)g(previous)f(inequalit)n(y)-7 b(.)38 b(Then)g(substituting)h(the)f(ab)r(o)n(v)n(e)523 4348 y(iden)n(tit)n(y)28 b(in)g(the)g(previous)e(inequalit)n(y)h(one)g (gets,)565 4582 y Fi(J)91 b Fj(\025)859 4549 y Fh(A)p 859 4563 50 4 v 867 4610 a Fl(4)952 4515 y Fg(R)925 4665 y Fl(\012)972 4649 y Fd(+)1041 4440 y Fg( )1146 4503 y Fh(k)1120 4520 y Fg(P)1107 4655 y Fh(j)s Fl(=1)1438 4549 y(1)p 1245 4563 419 4 v 1245 4617 a(\(log)1357 4597 y Fd(\()p Fc(j)r Fd(\))1444 4617 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))1632 4600 y Fd(2)1674 4440 y Fg(!)1788 4549 y Fh(v)1823 4524 y Fc(p)p 1764 4563 118 4 v 1764 4610 a Ff(j)p Fh(x)p Ff(j)1842 4594 y Fc(n)1910 4582 y Fk(+)2003 4549 y Fh(A)p 2003 4563 50 4 v 2011 4610 a Fl(2)2097 4515 y Fg(R)2069 4665 y Fl(\012)2116 4649 y Fb(\000)2187 4440 y Fg( )2292 4503 y Fh(k)2266 4520 y Fg(P)2253 4655 y Fh(j)s Fl(=1)2584 4549 y(1)p 2391 4563 419 4 v 2391 4617 a(\(log)2503 4597 y Fd(\()p Fc(j)r Fd(\))2591 4617 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2779 4600 y Fd(2)2820 4440 y Fg(!)2934 4549 y Fh(v)2969 4524 y Fc(p)p 2910 4563 118 4 v 2910 4610 a Ff(j)p Fh(x)p Ff(j)2988 4594 y Fc(n)3038 4582 y Fj(\000)859 4798 y Fh(A)p 859 4812 50 4 v 867 4859 a Fl(2)948 4764 y Fg(R)925 4910 y Fh(@)t Fl(\012)1033 4689 y Fg( )1138 4752 y Fh(k)1112 4769 y Fg(P)1099 4904 y Fh(j)s Fl(=1)1331 4791 y Ff(h)p Fh(x)p Ff(\001)p Fh(\027)t Ff(i)p 1238 4812 335 4 v 1238 4866 a Fl(log)1323 4846 y Fd(\()p Fc(j)r Fd(\))1411 4866 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)1582 4689 y Fg(!)1695 4798 y Fh(v)1730 4773 y Fc(p)p 1672 4812 118 4 v 1672 4859 a Ff(j)p Fh(x)p Ff(j)1750 4843 y Fc(n)1818 4831 y Fk(+)1921 4764 y Fg(R)1893 4914 y Fl(\012)1940 4898 y Fb(\000)2011 4689 y Fg(\()2088 4794 y Fh(pA)p 2088 4812 85 4 v 2114 4859 a Fl(2)2196 4689 y Fg( )2301 4752 y Fh(k)2275 4769 y Fg(P)2262 4904 y Fh(j)s Fl(=1)2551 4798 y(1)p 2400 4812 335 4 v 2400 4866 a(log)2486 4846 y Fd(\()p Fc(j)r Fd(\))2574 4866 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)2745 4689 y Fg(!)2835 4791 y Ff(h)p Fh(x)p Ff(\001r)p Fh(v)r Ff(i)11 b Fh(v)3083 4766 y Fc(p)p Fb(\000)p Fd(1)p 2835 4812 357 4 v 2954 4859 a Ff(j)p Fh(x)p Ff(j)3032 4843 y Fc(n)3220 4831 y Fk(+)18 b Fi(\013)3356 4843 y Fl(2)3417 4791 y Ff(jr)p Fh(v)r Ff(j)3547 4766 y Fc(p)p 3403 4812 194 4 v 3403 4859 a Ff(j)p Fh(x)p Ff(j)3481 4843 y Fc(n)p Fb(\000)p Fc(p)3606 4689 y Fg(\))3687 4831 y Fi(:)1958 5150 y Fk(11)p eop %%Page: 12 12 12 11 bop 523 615 a Fk(No)n(w)27 b(on)g(\012)887 585 y Ff(\000)943 615 y Fi(;)990 575 y Ff(j)p Fh(x)p Ff(j)11 b(jr)p Fh(v)r Ff(j)p 990 596 219 4 v 1082 644 a Fh(v)1242 615 y Fj(\025)23 b Fi(M)9 b Fk(.)27 b(Therefore)f(on)41 b(\012)2035 585 y Ff(\000)2091 615 y Fk(,)822 860 y Fi(\013)875 872 y Fl(2)937 819 y Ff(jr)p Fh(v)r Ff(j)1067 794 y Fc(p)p 923 841 194 4 v 923 888 a Ff(j)p Fh(x)p Ff(j)1001 872 y Fc(n)p Fb(\000)p Fc(p)1209 860 y Fk(+)1367 823 y Fh(pA)p 1367 841 85 4 v 1392 888 a Fl(2)1475 718 y Fg( )1580 781 y Fh(k)1554 797 y Fg(P)1540 933 y Fh(j)s Fl(=1)1856 827 y(1)p 1679 841 387 4 v 1679 895 a(\(log)1791 875 y Fd(\()p Fc(j)r Fd(\))1878 895 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2076 718 y Fg(!)2165 819 y Ff(h)p Fh(x)p Ff(\001r)p Fh(v)r Ff(i)11 b Fh(v)2413 794 y Fc(p)p Fb(\000)p Fd(1)p 2165 841 357 4 v 2285 888 a Ff(j)p Fh(x)p Ff(j)2363 872 y Fc(n)1209 1109 y Fj(\025)1367 1068 y Ff(jr)p Fh(v)r Ff(j)g Fh(v)1543 1043 y Fc(p)p Fb(\000)p Fd(1)p 1367 1090 285 4 v 1414 1137 a Ff(j)p Fh(x)p Ff(j)1492 1121 y Fc(n)p Fb(\000)p Fd(1)1676 967 y Fg(\()1742 1109 y Fi(\013)1795 1121 y Fl(2)1847 1017 y Fg(\020)1906 1068 y Ff(j)p Fh(x)p Ff(j)g(jr)p Fh(v)r Ff(j)p 1906 1090 219 4 v 1998 1137 a Fh(v)2135 1017 y Fg(\021)2184 1034 y Fh(p)p Ff(\000)p Fl(1)2326 1109 y Fj(\000)2409 1017 y Fg(\020)2469 1072 y Fh(pA)p 2469 1090 85 4 v 2494 1137 a Fl(2)2563 1017 y Fg(\021)2665 1030 y Fh(k)2640 1046 y Fg(P)2626 1182 y Fh(j)s Fl(=1)2916 1076 y(1)p 2765 1090 335 4 v 2765 1144 a(log)2851 1124 y Fd(\()p Fc(j)r Fd(\))2938 1144 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)3109 967 y Fg(\))1209 1358 y Fj(\025)1367 1318 y Ff(jr)p Fh(v)r Ff(j)g Fh(v)1543 1292 y Fc(p)p Fb(\000)p Fd(1)p 1367 1339 285 4 v 1414 1386 a Ff(j)p Fh(x)p Ff(j)1492 1370 y Fc(n)p Fb(\000)p Fd(1)1676 1216 y Fg(\()1742 1358 y Fi(\013)1795 1370 y Fl(2)1833 1358 y Fi(M)1923 1328 y Fh(p)p Ff(\000)p Fl(1)2064 1358 y Fj(\000)2147 1266 y Fg(\020)2207 1321 y Fh(pA)p 2207 1339 85 4 v 2232 1386 a Fl(2)2301 1266 y Fg(\021)2403 1279 y Fh(k)2378 1296 y Fg(P)2364 1431 y Fh(j)s Fl(=1)2654 1325 y(1)p 2503 1339 335 4 v 2503 1393 a(log)2589 1373 y Fd(\()p Fc(j)r Fd(\))2676 1393 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)2848 1216 y Fg(\))1209 1531 y Fj(\025)83 b Fk(0)p Fi(:)523 1689 y Fk(So,)731 1945 y Fi(J)31 b Fj(\025)906 1889 y Fi(A)p 906 1926 63 4 v 916 2002 a Fk(4)992 1832 y Fg(Z)991 2070 y Fl(\012)1089 1779 y Fg(0)1089 1928 y(@)1203 1842 y Fh(k)1161 1867 y Fg(X)1164 2043 y Fh(j)s Fl(=1)1539 1889 y Fk(1)p 1305 1926 509 4 v 1305 2017 a(\(log)1445 1980 y Fl(\()p Fh(j)s Fl(\))1545 2017 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1776 1993 y Fl(2)1824 1779 y Fg(1)1824 1928 y(A)1949 1889 y Fi(v)1992 1859 y Fh(p)p 1920 1926 139 4 v 1920 2002 a Fj(j)p Fi(x)p Fj(j)2013 1978 y Fh(n)2087 1945 y Fj(\000)2180 1889 y Fi(A)p 2180 1926 63 4 v 2190 2002 a Fk(2)2268 1832 y Fg(Z)2248 2071 y Fh(@)t Fl(\012)2366 1779 y Fg(0)2366 1928 y(@)2481 1842 y Fh(k)2439 1867 y Fg(X)2442 2043 y Fh(j)s Fl(=1)2677 1889 y Fj(h)p Fi(x)19 b Fj(\001)g Fi(\027)5 b Fj(i)p 2583 1926 407 4 v 2583 2017 a Fk(log)2690 1980 y Fl(\()p Fh(j)s Fl(\))2791 2017 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)3000 1779 y Fg(1)3000 1928 y(A)3125 1889 y Fi(v)3168 1859 y Fh(p)p 3096 1926 139 4 v 3096 2002 a Fj(j)p Fi(x)p Fj(j)3189 1978 y Fh(n)3245 1945 y Fi(:)523 2208 y Fk(This)28 b(pro)n(v)n(es)d(the)j(theorem.)1989 b Fj(t)-55 b(u)763 2440 y Fk(In)28 b(order)e(to)i(pro)n(v)n(e)d (Theorem)i(1.2,)g(w)n(e)g(need)h(a)f(few)h(more)f(lemmas.)523 2673 y Fm(Lemma)37 b(2.2.)30 b Fk(Let)i Fi(f)41 b Fk(satisfy)32 b(\()p Fi(H)1647 2685 y Fl(3)1684 2673 y Fk(\))p Fi(:)h Fk(Then)f(there)g(exists)f(a)h(constan)n(t)2857 2651 y Fg(e)2856 2673 y Fi(\025)f Fj(2)g Fi(I)-21 b(R)33 b Fk(suc)n(h)f(that)523 2822 y(for)27 b(all)g Fi(u)c Fj(2)g(D)980 2792 y Fl(1)p Fh(;)p Fl(2)1071 2822 y Fk(\()p Fi(I)-21 b(R)1182 2786 y Fh(n)1227 2822 y Fk(\))24 b(:)j(if)h Fi(n)23 b Fj(\025)g Fk(3,)608 2995 y Fg(Z)580 3233 y Fh(I)-18 b(R)642 3214 y Fc(n)715 2966 y Fg( )780 3108 y Fj(jr)p Fi(u)p Fj(j)943 3074 y Fl(2)999 3108 y Fj(\000)1082 2991 y Fg(\022)1153 3052 y Fi(n)18 b Fj(\000)g Fk(2)p 1153 3089 193 4 v 1229 3165 a(2)1356 2991 y Fg(\023)1417 3009 y Fl(2)1501 3052 y Fi(u)1549 3022 y Fl(2)p 1478 3089 131 4 v 1478 3165 a Fj(j)p Fi(x)p Fj(j)1571 3141 y Fl(2)1619 2966 y Fg(!)1707 3108 y Fj(\025)1805 3052 y Fk(1)p 1805 3089 42 4 v 1805 3165 a(4)1880 2995 y Fg(Z)1852 3233 y Fh(I)-18 b(R)1913 3214 y Fc(n)1986 2917 y Fg(0)1986 3063 y(B)1986 3116 y(@)2101 3005 y Fh(k)2059 3030 y Fg(X)2062 3206 y Fh(j)s Fl(=1)2432 3052 y Fk(1)p 2203 3089 500 4 v 2203 3119 a Fg(\020)2252 3212 y Fk(log)2360 3175 y Fl(\()p Fh(j)s Fl(\))2460 3212 y Fi(\032)p Fk(\()p Fi(x)p Fk(\))2614 3119 y Fg(\021)2665 3137 y Fl(2)2712 2917 y Fg(1)2712 3063 y(C)2712 3116 y(A)2832 3052 y Fi(u)2880 3022 y Fl(2)p 2809 3089 131 4 v 2809 3165 a Fj(j)p Fi(x)p Fj(j)2902 3141 y Fl(2)2968 3108 y Fk(+)3052 3086 y Fg(e)3051 3108 y Fi(\025)3123 2995 y Fg(Z)3095 3233 y Fh(I)g(R)3156 3214 y Fc(n)3229 3108 y Fi(f)9 b(u)3327 3074 y Fl(2)3377 3108 y Fi(;)3305 3336 y Fk(\(2.9\))523 3486 y(and)27 b(if)i Fi(n)22 b Fk(=)h(2,)640 3630 y Fg(Z)616 3874 y Fh(I)-18 b(R)677 3855 y Fd(2)742 3625 y Fg(\022)803 3743 y Fj(jr)p Fi(u)p Fj(j)966 3708 y Fl(2)1022 3743 y Fj(\000)1115 3686 y Fk(1)p 1115 3723 42 4 v 1115 3799 a(4)1363 3686 y Fi(u)1411 3656 y Fl(2)p 1176 3723 458 4 v 1176 3799 a Fj(j)p Fi(x)p Fj(j)1269 3775 y Fl(2)1321 3799 y Fk(log)14 b Fi(\032)p Fk(\()p Fi(x)p Fk(\))1596 3775 y Fl(2)1644 3625 y Fg(\023)1728 3743 y Fj(\025)1826 3686 y Fk(1)p 1826 3723 42 4 v 1826 3799 a(4)1897 3630 y Fg(Z)1873 3874 y Fh(I)-18 b(R)1934 3855 y Fd(2)1999 3576 y Fg(0)1999 3725 y(@)2113 3639 y Fh(k)2072 3664 y Fg(X)2074 3840 y Fh(j)s Fl(=2)2427 3686 y Fk(1)p 2215 3723 465 4 v 2215 3814 a(\(log)2355 3777 y Fl(\()p Fh(j)s Fl(\))2456 3814 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2642 3790 y Fl(2)2690 3576 y Fg(1)2690 3725 y(A)2809 3686 y Fi(u)2857 3656 y Fl(2)p 2786 3723 131 4 v 2786 3799 a Fj(j)p Fi(x)p Fj(j)2879 3775 y Fl(2)2946 3743 y Fk(+)3030 3721 y Fg(e)3029 3743 y Fi(\025)3101 3630 y Fg(Z)3073 3868 y Fh(I)g(R)3134 3848 y Fc(n)3207 3743 y Fi(f)9 b(u)3305 3708 y Fl(2)3341 3743 y Fi(:)3264 3953 y Fk(\(2.10\))523 4185 y Fm(Pro)s(of.)35 b Fk(It)i(is)e(enough)h(to)f(pro)n(v)n(e)f(\(2.9\))i(and)g(\(2.10\))f (will)h(follo)n(w)f(in)h(a)g(similar)f(w)n(a)n(y)-7 b(.)34 b(Ob-)523 4335 y(serv)n(e)24 b(that)h(b)n(y)f(assumption)g(\()p Fi(H)1555 4347 y Fl(3)1593 4335 y Fk(\)\()p Fi(i)p Fk(\),)h(the)h(em)n (b)r(edding)38 b Fj(D)2403 4304 y Fl(1)p Fh(;)p Fl(2)2494 4335 y Fk(\()p Fi(I)-21 b(R)2605 4298 y Fh(n)2650 4335 y Fk(\))24 b Fi(,)-14 b Fj(!)23 b Fi(L)2878 4304 y Fl(2)2915 4335 y Fk(\()p Fi(f)9 b Fk(\))39 b(is)24 b(compact.)523 4484 y(Moreo)n(v)n(er,)k(from)j(\()p Fi(H)1215 4496 y Fl(3)1253 4484 y Fk(\)\()p Fi(ii)p Fk(\),)g(for)g(ev)n(ery)e Fi(")g(>)f Fk(0)p Fi(;)j Fk(there)g(exists)g Fi(C)6 b Fk(\()p Fi(")p Fk(\))29 b Fi(>)g Fk(0)h(suc)n(h)h(that)h(for)e(all)523 4633 y Fi(u)23 b Fj(2)g(D)738 4603 y Fl(1)p Fh(;)p Fl(2)828 4633 y Fk(\()p Fi(I)-21 b(R)940 4597 y Fh(n)985 4633 y Fk(\),)1383 4670 y Fg(Z)1326 4912 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)1519 4783 y Fi(u)1567 4749 y Fl(2)1627 4783 y Fj(\024)23 b Fi(")1777 4670 y Fg(Z)1749 4908 y Fh(I)-18 b(R)1810 4889 y Fc(n)1883 4783 y Fj(jr)p Fi(u)p Fj(j)2046 4749 y Fl(2)2102 4783 y Fk(+)18 b Fi(C)6 b Fk(\()p Fi(")p Fk(\))2377 4670 y Fg(Z)2348 4908 y Fh(I)-18 b(R)2410 4889 y Fc(n)2483 4783 y Fi(f)9 b(u)2581 4749 y Fl(2)2632 4783 y Fi(:)1958 5150 y Fk(12)p eop %%Page: 13 13 13 12 bop 523 614 a Fk(Let)38 b Fi(s)i Fk(=)963 582 y Fl(1)p 875 596 208 4 v 875 643 a(\(1)p Ff(\000)p Fh(\025")p Fl(\))1093 614 y Fi(:)e Fk(Since)g Fi(\025)i(<)g Fk(0,)d Fi(s)j Fj(2)g Fk(\(0)p Fi(;)14 b Fk(1\))p Fi(:)37 b Fk(Then)h(from)g (\(1)p Fi(:)p Fk(6\))f(and)h(from)f(the)h(ab)r(o)n(v)n(e)523 764 y(inequalit)n(y)27 b(w)n(e)g(obtain)533 911 y Fg(Z)505 1149 y Fh(I)-18 b(R)566 1130 y Fc(n)639 1024 y Fj(jr)p Fi(u)p Fj(j)802 989 y Fl(2)856 1024 y Fj(\000)937 907 y Fg(\022)1008 968 y Fi(n)19 b Fj(\000)f Fk(2)p 1008 1005 193 4 v 1084 1081 a(2)1211 907 y Fg(\023)1272 924 y Fl(2)1310 1024 y Fi(s)1359 911 y Fg(Z)1331 1149 y Fh(I)-18 b(R)1392 1130 y Fc(n)1498 968 y Fi(u)1546 937 y Fl(2)p 1475 1005 131 4 v 1475 1081 a Fj(j)p Fi(x)p Fj(j)1568 1057 y Fl(2)1632 1024 y Fj(\000)1725 968 y Fi(s)p 1723 1005 42 4 v 1723 1081 a Fk(4)1798 911 y Fg(Z)1770 1149 y Fh(I)g(R)1832 1130 y Fc(n)1905 857 y Fg(0)1905 1006 y(@)2019 920 y Fh(k)1977 945 y Fg(X)1980 1122 y Fh(j)s Fl(=1)2333 968 y Fk(1)p 2121 1005 465 4 v 2121 1095 a(\(log)2261 1058 y Fl(\()p Fh(j)s Fl(\))2361 1095 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2547 1071 y Fl(2)2596 857 y Fg(1)2596 1006 y(A)2715 968 y Fi(u)2763 937 y Fl(2)p 2692 1005 131 4 v 2692 1081 a Fj(j)p Fi(x)p Fj(j)2785 1057 y Fl(2)2856 1024 y Fj(\025)23 b Fi(\025)14 b(C)6 b Fk(\()p Fi(")p Fk(\))3198 911 y Fg(Z)3169 1149 y Fh(I)-18 b(R)3232 1130 y Fc(n)3305 1024 y Fi(f)9 b(u)3403 989 y Fl(2)3453 1024 y Fi(:)3264 1228 y Fk(\(2.11\))523 1377 y(Let)28 b Fi(A)734 1389 y Fh(k)775 1377 y Fk(\()p Fi(x)p Fk(\))c(=)1008 1344 y Fl(1)p 1008 1358 34 4 v 1008 1406 a(4)1104 1298 y Fh(k)1078 1315 y Fg(P)1064 1450 y Fh(j)s Fl(=1)1377 1344 y(1)p 1203 1358 382 4 v 1203 1412 a(\(log)1315 1392 y Fd(\()p Fc(j)r Fd(\))1402 1412 y Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))1552 1396 y Fd(2)1645 1377 y Fk(and)691 1714 y Fi(\025)p Fk(\()p Fi(s)p Fk(\))g(=)f(inf)1069 1543 y Fg(8)1069 1618 y(<)1069 1767 y(:)1152 1600 y(Z)1124 1839 y Fh(I)-18 b(R)1186 1819 y Fc(n)1259 1714 y Fj(jr)p Fi(u)p Fj(j)1422 1679 y Fl(2)1477 1714 y Fj(\000)1560 1596 y Fg(\022)1631 1657 y Fi(n)18 b Fj(\000)g Fk(2)p 1631 1694 193 4 v 1707 1770 a(2)1834 1596 y Fg(\023)1895 1614 y Fl(2)1946 1714 y Fi(s)2018 1657 y(u)2066 1627 y Fl(2)p 1995 1694 131 4 v 1995 1770 a Fj(j)p Fi(x)p Fj(j)2088 1746 y Fl(2)2154 1714 y Fj(\000)g Fi(sA)2338 1726 y Fh(k)2379 1714 y Fk(\()p Fi(x)p Fk(\))2524 1657 y Fi(u)2572 1627 y Fl(2)p 2501 1694 V 2501 1770 a Fj(j)p Fi(x)p Fj(j)2594 1746 y Fl(2)2642 1714 y Fk(\);)2804 1600 y Fg(Z)2776 1839 y Fh(I)-18 b(R)2837 1819 y Fc(n)2910 1714 y Fi(f)9 b(u)3008 1679 y Fl(2)3068 1714 y Fk(=)22 b(1)3197 1543 y Fg(9)3197 1618 y(=)3197 1767 y(;)3285 1714 y Fi(:)523 1983 y Fk(F)-7 b(rom)29 b(\(2.11\))g(and)h(b)n(y)f(the) h(de\014nition)g(of)g Fi(\025)p Fk(\()p Fi(s)p Fk(\))p Fi(;)h Fk(there)e(exists)h(a)f(constan)n(t)g Fi(C)36 b Fk(indep)r(enden)n(t)523 2132 y(of)24 b Fi(s)g Fk(suc)n(h)g(that)g Fi(\025)14 b(C)6 b Fk(\()p Fi(")p Fk(\))24 b Fj(\024)f Fi(\025)p Fk(\()p Fi(s)p Fk(\))h Fi(<)e(C)6 b Fk(.)25 b(Moreo)n(v)n(er,)c(from)j(\(1.6\),)g(for)g(an)n(y)f Fi(\016)j(>)d Fk(0,)g(there)h(exists)523 2282 y Fi(C)6 b Fk(\()p Fi(\016)s Fk(\))24 b Fi(>)f Fk(0)k(suc)n(h)g(that)h(for)f (all)g Fi(u)c Fj(2)g(D)1697 2251 y Fl(1)p Fh(;)p Fl(2)1788 2282 y Fk(\()p Fi(I)-21 b(R)1899 2245 y Fh(n)1944 2282 y Fk(\),)636 2397 y Fg(R)604 2542 y Fh(I)j(R)666 2523 y Fc(n)728 2463 y Fj(jr)p Fi(u)p Fj(j)891 2433 y Fl(2)947 2463 y Fj(\000)1030 2396 y Fg(\000)1078 2431 y Fh(n)p Ff(\000)p Fl(2)p 1078 2445 127 4 v 1124 2492 a(2)1214 2396 y Fg(\001)1252 2413 y Fl(2)1303 2463 y Fi(s)1371 2431 y Fh(u)1410 2406 y Fd(2)p 1352 2445 110 4 v 1352 2492 a Ff(j)p Fh(x)p Ff(j)1430 2475 y Fd(2)1490 2463 y Fj(\000)18 b Fi(sA)1674 2475 y Fh(k)1715 2463 y Fk(\()p Fi(x)p Fk(\))1856 2431 y Fh(u)1895 2406 y Fd(2)p 1837 2445 V 1837 2492 a Ff(j)p Fh(x)p Ff(j)1915 2475 y Fd(2)2039 2463 y Fj(\025)83 b Fk(\(1)18 b Fj(\000)g Fi(s)p Fk(\))2471 2397 y Fg(R)2439 2542 y Fh(I)-18 b(R)2501 2523 y Fc(n)2563 2463 y Fj(jr)p Fi(u)p Fj(j)2726 2433 y Fl(2)2782 2463 y Fk(+)18 b Fi(s\025)3019 2397 y Fg(R)2958 2546 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)3141 2463 y Fi(u)3189 2433 y Fl(2)2039 2639 y Fj(\025)83 b Fk(\(1)18 b Fj(\000)g Fi(s)p Fk(\))2471 2572 y Fg(R)2439 2718 y Fh(I)-18 b(R)2501 2698 y Fc(n)2563 2639 y Fj(jr)p Fi(u)p Fj(j)2726 2609 y Fl(2)2782 2639 y Fk(+)18 b Fi(s\025\016)3029 2572 y Fg(R)2998 2718 y Fh(I)-18 b(R)3059 2698 y Fc(n)3122 2639 y Fj(jr)p Fi(u)p Fj(j)3285 2609 y Fl(2)3322 2639 y Fk(+)2270 2796 y Fi(C)6 b Fk(\()p Fi(")p Fk(\))p Fi(C)g Fk(\()p Fi(\016)s Fk(\))p Fi(s\025)2733 2729 y Fg(R)2700 2875 y Fh(I)-18 b(R)2763 2855 y Fc(n)2826 2796 y Fi(f)9 b(u)2924 2766 y Fl(2)2039 2953 y Fk(=)83 b(\(1)18 b Fj(\000)g Fi(s)g Fk(+)g Fi(s\025\016)s Fk(\))2699 2886 y Fg(R)2667 3032 y Fh(I)-18 b(R)2729 3012 y Fc(n)2792 2953 y Fj(jr)p Fi(u)p Fj(j)2955 2923 y Fl(2)2992 2953 y Fk(+)2270 3110 y Fi(C)6 b Fk(\()p Fi(")p Fk(\))14 b Fi(C)6 b Fk(\()p Fi(\016)s Fk(\))14 b Fi(s)g(\025)2761 3043 y Fg(R)2728 3189 y Fh(I)-18 b(R)2791 3169 y Fc(n)2853 3110 y Fi(f)9 b(u)2951 3080 y Fl(2)523 3333 y Fk(Hence)32 b(if)g Fj(f)p Fi(u)944 3345 y Fh(k)984 3333 y Fj(g)g Fk(is)f(a)h(minimizing)g(sequence)f(for)g Fi(\025)p Fk(\()p Fi(s)p Fk(\))p Fi(;)i Fk(then)f(from)g(the)g(ab)r(o)n (v)n(e)e(inequalit)n(y)523 3482 y Fj(f)p Fi(u)613 3494 y Fh(k)653 3482 y Fj(g)f Fk(is)h(b)r(ounded)h(and)e(hence)h Fi(u)1596 3494 y Fh(k)1664 3482 y Fi(*)d(u)1822 3494 y Fh(s)1887 3482 y Fk(w)n(eakly)h(in)j Fj(D)2329 3452 y Fl(1)p Fh(;)p Fl(2)2419 3482 y Fk(\()p Fi(I)-21 b(R)2530 3446 y Fh(n)2576 3482 y Fk(\))44 b(for)29 b(some)43 b Fi(u)3053 3494 y Fh(s)3088 3482 y Fk(.)30 b(Since)g(the)523 3632 y(em)n(b)r(edding)e Fj(D)1009 3601 y Fl(1)p Fh(;)p Fl(2)1122 3632 y Fi(,)-14 b Fj(!)24 b Fi(L)1295 3601 y Fl(2)1331 3632 y Fk(\()p Fi(f)9 b Fk(\))28 b(is)g(compact,)f Fi(u)1960 3644 y Fh(s)2022 3632 y Fk(is)h(a)f(minimizer)h(for)f Fi(\025)p Fk(\()p Fi(s)p Fk(\))h(and)g(satis\014es)1203 3726 y Fg(8)1203 3801 y(<)1203 3950 y(:)1318 3819 y Fj(\000)p Fk(\001)p Fi(u)1500 3831 y Fh(s)1553 3819 y Fj(\000)1636 3752 y Fg(\000)1685 3786 y Fh(n)p Ff(\000)p Fl(2)p 1685 3800 127 4 v 1731 3848 a(2)1820 3752 y Fg(\001)1859 3769 y Fl(2)1923 3785 y Fh(su)1993 3793 y Fc(s)p 1920 3800 110 4 v 1920 3848 a Ff(j)p Fh(x)p Ff(j)1998 3831 y Fd(2)2057 3819 y Fj(\000)2150 3785 y Fh(sA)2231 3794 y Fc(k)2268 3785 y Fh(u)2307 3793 y Fc(s)p 2150 3800 189 4 v 2190 3848 a Ff(j)p Fh(x)p Ff(j)2268 3831 y Fd(2)2372 3819 y Fk(=)23 b Fi(\025)p Fk(\()p Fi(s)p Fk(\))p Fi(f)9 b(u)2709 3831 y Fh(s)1342 3865 y Fg(R)1310 4011 y Fh(I)-18 b(R)1372 3992 y Fc(n)1434 3932 y Fi(f)9 b(u)1532 3902 y Fl(2)1532 3952 y Fh(s)1592 3932 y Fk(=)22 b(1)14 b Fi(:)523 4161 y Fk(F)-7 b(rom)18 b(\()p Fi(H)833 4173 y Fl(3)871 4161 y Fk(\),)g(c)n(ho)r(ose)f(0)23 b Fi(<)g(\016)j(<)c Fk(1)c(suc)n(h)g (that)h Fi(f)9 b Fk(\()p Fi(x)p Fk(\))24 b Fj(\025)e Fi(A)d Fk(for)f Fi(x)23 b Fj(2)h Fi(D)r Fk(\()p Fi(\016)s Fk(\))g(=)e Fj(f)p Fi(x)p Fk(;)14 b(1)k Fj(\000)g Fi(\016)26 b(<)d Fj(j)p Fi(x)p Fj(j)g Fi(<)g Fk(1)18 b(+)g Fi(\016)s Fj(g)13 b Fi(:)523 4310 y Fk(Let)31 b(0)c Fj(\024)h Fi(')g Fj(2)g Fi(C)1067 4280 y Ff(1)1061 4331 y Fl(0)1138 4310 y Fk(\()p Fi(D)r Fk(\()p Fi(\016)s Fk(\)\))j(suc)n(h)g(that)f Fi(')f Fk(=)e(1)j(on)g Fi(D)r Fk(\()p Fi(\016)s(=)p Fk(2\))p Fi(:)h Fk(Since)2702 4243 y Fg(R)2670 4389 y Fh(I)-18 b(R)2732 4370 y Fc(n)2795 4310 y Fi(f)9 b(u)2893 4280 y Fl(2)2893 4331 y Fh(s)2957 4310 y Fk(=)27 b(1,)3187 4243 y Fg(R)3136 4393 y Fh(D)r Fl(\()p Fh(\016)r Fl(\))3299 4310 y Fi(u)3347 4280 y Fl(2)3347 4331 y Fh(s)3411 4310 y Fj(\024)523 4492 y Fk(1)p Fi(=)-5 b(A:)26 b Fk(Hence)g(m)n(ultiply)g (the)g(ab)r(o)n(v)n(e)f(equation)g(b)n(y)h Fi(')2168 4461 y Fl(2)2205 4492 y Fi(u)2253 4504 y Fh(s)2314 4492 y Fk(and)g(in)n(tegrate)f(b)n(y)g(parts)g(to)h(obtain)1116 4668 y Fg(Z)1036 4910 y Fh(D)r Fl(\()p Fh(\016)r(=)p Fl(2\))1275 4781 y Fj(jr)p Fi(u)1415 4793 y Fh(s)1451 4781 y Fj(j)1474 4747 y Fl(2)1534 4781 y Fj(\024)1631 4668 y Fg(Z)1604 4906 y Fh(I)-18 b(R)1665 4887 y Fc(n)1738 4781 y Fi(')1792 4747 y Fl(2)1829 4781 y Fj(jr)p Fi(u)1969 4793 y Fh(s)2005 4781 y Fj(j)2028 4747 y Fl(2)2088 4781 y Fk(=)23 b Fi(O)2255 4589 y Fg(0)2255 4735 y(B)2255 4789 y(@)2356 4668 y(Z)2309 4910 y Fh(D)r Fl(\()p Fh(\016)r Fl(\))2482 4781 y Fi(u)2530 4747 y Fl(2)2530 4802 y Fh(s)2567 4589 y Fg(1)2567 4735 y(C)2567 4789 y(A)2662 4781 y Fk(=)g Fi(O)r Fk(\(1\))p Fi(:)320 b Fk(\(2.12\))1958 5150 y(13)p eop %%Page: 14 14 14 13 bop 523 677 a Fk(Hence)20 b Fi(u)810 689 y Fh(s)864 677 y Fk(is)g(uniformly)f(b)r(ounded)h(in)g Fj(D)1796 647 y Fl(1)p Fh(;)p Fl(2)1886 677 y Fk(\()p Fi(D)r Fk(\()p Fi(\016)s(=)p Fk(2\)\))g(.)g(Therefore)2640 535 y Fg(\()2760 610 y(R)2699 759 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)2882 677 y Fi(u)2930 647 y Fl(2)2930 697 y Fh(s)2967 535 y Fg(\))3034 786 y Fh(s)3089 677 y Fk(is)f(b)r(ounded)523 852 y(as)27 b Fi(s)c Fj(!)g Fk(1)14 b Fi(;)27 b Fk(whence)h(the)g(existence)f(of)41 b Fi(C)1853 864 y Fl(0)1914 852 y Fj(2)23 b Fi(I)-21 b(R)43 b Fk(suc)n(h)27 b(that)1647 1072 y(lim)p 1639 1092 131 4 v 1639 1140 a Fh(s)p Ff(!)p Fl(1)1807 1072 y Fi(\025)1908 959 y Fg(Z)1850 1201 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)2044 1072 y Fi(u)2092 1037 y Fl(2)2092 1092 y Fh(s)2152 1072 y Fj(\025)c Fi(C)2299 1084 y Fl(0)2336 1072 y Fi(:)523 1364 y Fk(Therefore)j(from)i(\(1.6\))f(w)n(e)g(ha)n(v)n(e)850 1561 y Fg(e)849 1583 y Fi(\025)c Fk(=)30 b(lim)p 1008 1604 V 1008 1651 a Fh(s)p Ff(!)p Fl(1)1152 1583 y Fi(\025)p Fk(\()p Fi(s)p Fk(\))84 b(=)90 b(lim)p 1535 1604 V 1535 1651 a Fh(s)p Ff(!)p Fl(1)1679 1466 y Fg(\032)1765 1516 y(R)1733 1662 y Fh(I)-18 b(R)1795 1643 y Fc(n)1857 1583 y Fj(jr)p Fi(u)1997 1595 y Fh(s)2033 1583 y Fj(j)2056 1553 y Fl(2)2111 1583 y Fj(\000)2194 1516 y Fg(\000)2243 1551 y Fh(n)p Ff(\000)p Fl(2)p 2243 1565 127 4 v 2289 1612 a(2)2378 1516 y Fg(\001)2417 1533 y Fl(2)2468 1583 y Fi(s)2530 1543 y Ff(j)p Fh(u)2589 1551 y Fc(s)2621 1543 y Ff(j)2641 1518 y Fd(2)p 2530 1564 144 4 v 2547 1612 a Ff(j)p Fh(x)p Ff(j)2625 1595 y Fd(2)2702 1583 y Fj(\000)2795 1543 y Fh(s)11 b(A)2887 1552 y Fc(k)2935 1543 y Ff(j)p Fh(u)2994 1551 y Fc(s)3026 1543 y Ff(j)3046 1518 y Fd(2)p 2795 1564 284 4 v 2882 1612 a Ff(j)p Fh(x)p Ff(j)2960 1595 y Fd(2)3088 1466 y Fg(\033)1387 1740 y Fj(\025)90 b Fk(lim)p 1535 1761 131 4 v 1535 1808 a Fh(s)p Ff(!)p Fl(1)1702 1740 y Fi(\025)1817 1673 y Fg(R)1756 1823 y Ff(j)p Fh(x)p Ff(j)p Fl(=1)1939 1740 y Fj(j)p Fi(u)2010 1752 y Fh(s)2045 1740 y Fj(j)2068 1710 y Fl(2)1387 1916 y Fj(\025)83 b Fi(C)1594 1928 y Fl(0)1631 1916 y Fi(:)523 2081 y Fk(This)28 b(pro)n(v)n(es)d(the)j(lemma.)2045 b Fj(t)-55 b(u)523 2314 y Fm(Pro)s(of)29 b(of)g(Theorem)g(1.2.)24 b Fk(By)h(Lemma)g(2.2)g(and)g(assumption)g(\(H2\),)h(for)f Fi(u)d Fj(2)i(D)3174 2284 y Fl(1)p Fh(;)p Fl(2)3264 2314 y Fk(\()p Fi(I)-21 b(R)3375 2277 y Fh(n)3420 2314 y Fk(\),)523 2463 y Fi(s)23 b Fj(2)g Fk(\(0)p Fi(;)14 b Fk(1\),)680 2584 y Fg(R)649 2730 y Fh(I)-18 b(R)710 2711 y Fc(n)773 2651 y Fj(jr)p Fi(u)p Fj(j)936 2621 y Fl(2)991 2651 y Fj(\000)1074 2584 y Fg(\000)1122 2618 y Fh(n)p Ff(\000)p Fl(2)p 1122 2632 127 4 v 1169 2680 a(2)1258 2584 y Fg(\001)1297 2601 y Fl(2)1358 2614 y Fh(sq)r(u)1460 2589 y Fd(2)p 1358 2632 136 4 v 1371 2680 a Ff(j)p Fh(x)p Ff(j)1449 2663 y Fd(2)1521 2651 y Fj(\000)18 b Fi(s\021)s(u)1735 2621 y Fl(2)1855 2651 y Fj(\025)83 b Fk(\(1)18 b Fj(\000)g Fi(s)p Fk(\))2287 2584 y Fg(R)2255 2730 y Fh(I)-18 b(R)2317 2711 y Fc(n)2379 2651 y Fj(jr)p Fi(u)p Fj(j)2542 2621 y Fl(2)2003 2859 y Fk(+)p Fi(s)2121 2741 y Fg(\022)2205 2792 y(R)2173 2937 y Fh(I)g(R)2235 2918 y Fc(n)2298 2859 y Fj(jr)p Fi(u)p Fj(j)2461 2828 y Fl(2)2516 2859 y Fj(\000)2599 2791 y Fg(\000)2647 2826 y Fh(n)p Ff(\000)p Fl(2)p 2647 2840 127 4 v 2694 2887 a(2)2783 2791 y Fg(\001)2821 2808 y Fl(2)2901 2826 y Fh(u)2940 2801 y Fd(2)p 2882 2840 110 4 v 2882 2887 a Ff(j)p Fh(x)p Ff(j)2960 2871 y Fd(2)3020 2859 y Fj(\000)3113 2824 y Fh(A)3163 2833 y Fc(k)3199 2824 y Fh(u)3238 2799 y Fd(2)p 3113 2839 158 4 v 3137 2887 a Ff(j)p Fh(x)p Ff(j)3215 2871 y Fd(2)3281 2741 y Fg(\023)1855 3028 y Fj(\025)83 b Fk(\(1)18 b Fj(\000)g Fi(s)p Fk(\))2287 2961 y Fg(R)2255 3107 y Fh(I)-18 b(R)2317 3087 y Fc(n)2379 3028 y Fj(jr)p Fi(u)p Fj(j)2542 2997 y Fl(2)2598 3028 y Fk(+)2682 3006 y Fg(e)2681 3028 y Fi(\025s)2806 2961 y Fg(R)2774 3107 y Fh(I)g(R)2836 3087 y Fc(n)2898 3028 y Fi(f)9 b(u)2996 2997 y Fl(2)3033 3028 y Fi(:)523 3275 y Fk(So,)40 b(since)893 3253 y(~)889 3275 y Fi(\025)k(<)f Fk(0)c(and)h Fi(s)j(<)g Fk(1,)53 b Fi(L)1709 3287 y Fh(sq)r(;s\021)1890 3275 y Fj(\000)1983 3253 y Fg(e)1982 3275 y Fi(\025f)48 b Fk(is)40 b(a)f(co)r(erciv)n(e)g (op)r(erator)f(and)h(hence)h(b)n(y)523 3424 y(the)e(sp)r(ectral)f (theorem,)h(it)g(has)f(an)h(in\014nite)g(discrete)f(sp)r(ectrum)h Fj(f)p Fi(\025)2801 3436 y Fh(i)2829 3424 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(g)3134 3436 y Fh(i)p Ff(2)p Fh(I)-18 b(N)3314 3424 y Fk(with)523 3574 y(corresp)r(onding)29 b(eigenfunctions)h Fj(f)p Fi(\036)1696 3586 y Fh(i)1723 3574 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(g)2028 3586 y Fh(i)p Ff(2)p Fh(I)-18 b(N)2171 3574 y Fi(:)31 b Fk(The)f(rest)g(of)h(the)f(pro)r(of)g(follo)n(ws)g(as)g(in)523 3723 y([3])i(except)g(for)f(the)i(con)n(v)n(ergence)d(of)i(the)g (eigenfunctions)g Fi(\036)2493 3735 y Fh(i)2521 3723 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))33 b(for)e Fi(i)f Fj(\025)g Fk(2)p Fi(:)i Fk(W)-7 b(e)33 b(will)523 3873 y(giv)n(e)27 b(the)h(pro)r(of)f(of)g(this)h(b)r(elo)n(w.)523 4105 y Fm(Step)k(1.)27 b Fk(F)-7 b(or)27 b Fi(r)f(>)c Fk(0)p Fi(;)27 b Fk(let)h(\012\()p Fi(r)r Fk(\))d(=)d Fi(I)-21 b(R)1709 4069 y Fh(n)1773 4105 y Fj(n)18 b Fi(B)t Fk(\()p Fi(r)r Fk(\))29 b(and)f(de\014ne)523 4399 y Fi(a)p Fk(\()p Fi(r)r Fk(\))c(=)155 b(inf)782 4460 y Fh(u)p Ff(2)p Fh(H)924 4440 y Fd(1)920 4480 y(0)957 4460 y Fl(\()p Fh(B)s Fl(\()p Fh(r)r Fl(\)\))1160 4204 y Fg(8)1160 4279 y(>)1160 4304 y(<)1160 4453 y(>)1160 4478 y(:)1261 4286 y(Z)1216 4528 y Fh(B)s Fl(\()p Fh(r)r Fl(\))1386 4399 y Fj(jr)p Fi(u)p Fj(j)1549 4365 y Fl(2)1604 4399 y Fj(\000)1687 4282 y Fg(\022)1758 4343 y Fi(n)18 b Fj(\000)g Fk(2)p 1758 4380 193 4 v 1834 4456 a(2)1961 4282 y Fg(\023)2022 4299 y Fl(2)2106 4343 y Fi(u)2154 4313 y Fl(2)p 2083 4380 131 4 v 2083 4456 a Fj(j)p Fi(x)p Fj(j)2176 4432 y Fl(2)2242 4399 y Fj(\000)2335 4343 y Fi(A)2397 4355 y Fh(k)2438 4343 y Fk(\()p Fi(x)p Fk(\))p 2335 4380 216 4 v 2377 4456 a Fj(j)p Fi(x)p Fj(j)2470 4432 y Fl(2)2574 4399 y Fi(u)2622 4365 y Fl(2)2659 4399 y Fk(;)2723 4286 y Fg(Z)2677 4528 y Fh(B)s Fl(\()p Fh(r)r Fl(\))2847 4399 y Fi(f)9 b Fk(\()p Fi(x)p Fk(\))14 b Fj(j)p Fi(u)p Fj(j)3116 4365 y Fl(2)3177 4399 y Fk(=)22 b(1)3306 4204 y Fg(9)3306 4279 y(>)3306 4304 y(=)3306 4453 y(>)3306 4478 y(;)562 4781 y Fi(b)p Fk(\()p Fi(r)r Fk(\))i(=)152 b(inf)813 4842 y Fh(u)p Ff(2)p Fh(H)955 4822 y Fd(1)951 4862 y(0)988 4842 y Fl(\(\012\()p Fh(r)r Fl(\)\))1185 4586 y Fg(8)1185 4661 y(>)1185 4686 y(<)1185 4835 y(>)1185 4860 y(:)1284 4668 y(Z)1241 4910 y Fl(\012\()p Fh(r)r Fl(\))1405 4781 y Fj(jr)p Fi(u)p Fj(j)1568 4747 y Fl(2)1623 4781 y Fj(\000)1707 4664 y Fg(\022)1778 4725 y Fi(n)18 b Fj(\000)g Fk(2)p 1778 4762 193 4 v 1853 4838 a(2)1980 4664 y Fg(\023)2041 4681 y Fl(2)2126 4725 y Fi(u)2174 4695 y Fl(2)p 2103 4762 131 4 v 2103 4838 a Fj(j)p Fi(x)p Fj(j)2196 4814 y Fl(2)2262 4781 y Fj(\000)2355 4725 y Fi(A)2417 4737 y Fh(k)2458 4725 y Fk(\()p Fi(x)p Fk(\))p 2355 4762 216 4 v 2397 4838 a Fj(j)p Fi(x)p Fj(j)2490 4814 y Fl(2)2594 4781 y Fi(u)2642 4747 y Fl(2)2678 4781 y Fk(;)2740 4668 y Fg(Z)2696 4910 y Fl(\012\()p Fh(r)r Fl(\))2861 4781 y Fi(f)9 b Fk(\()p Fi(x)p Fk(\))14 b Fj(j)p Fi(u)p Fj(j)3130 4747 y Fl(2)3190 4781 y Fk(=)23 b(1)3320 4586 y Fg(9)3320 4661 y(>)3320 4686 y(=)3320 4835 y(>)3320 4860 y(;)1958 5150 y Fk(14)p eop %%Page: 15 15 15 14 bop 523 614 a Fk(then)28 b(b)n(y)f(part)h(\(i\))g(of)f(h)n(yp)r (othesis)h(\()p Fi(H)1728 626 y Fl(3)1765 614 y Fk(\),)g(w)n(e)f(ha)n (v)n(e)1443 841 y(lim)1435 893 y Fh(r)r Ff(!)p Fl(0)1604 841 y Fi(a)p Fk(\()p Fi(r)r Fk(\))d(=)e Fj(1)p Fi(;)62 b Fk(lim)2005 891 y Fh(r)r Ff(!1)2207 841 y Fi(b)p Fk(\()p Fi(r)r Fk(\))24 b(=)f Fj(1)p Fi(:)523 1068 y Fk(Indeed,)28 b(b)n(y)f(the)h(de\014nition)g(of)g Fi(A)1600 1080 y Fh(k)1669 1068 y Fk(and)f(Theorem)g(1.1,)g(for)g(all)41 b Fi(u)22 b Fj(2)i Fi(H)2819 1038 y Fl(1)2812 1089 y(0)2856 1068 y Fk(\()p Fi(B)t Fk(\()p Fi(r)r Fk(\))14 b(,)29 b(w)n(e)e(ha)n(v)n(e)689 1199 y Fg(Z)643 1441 y Fh(B)s Fl(\()p Fh(r)r Fl(\))813 1220 y Fg(\020)863 1312 y Fj(jr)p Fi(u)p Fj(j)1026 1278 y Fl(2)1081 1312 y Fj(\000)1164 1195 y Fg(\022)1235 1256 y Fi(n)19 b Fj(\000)f Fk(2)p 1235 1293 193 4 v 1311 1369 a(2)1438 1195 y Fg(\023)1499 1213 y Fl(2)1583 1256 y Fi(u)1631 1226 y Fl(2)p 1560 1293 131 4 v 1560 1369 a Fj(j)p Fi(x)p Fj(j)1653 1345 y Fl(2)1720 1312 y Fj(\000)1813 1256 y Fi(A)1875 1268 y Fh(k)1916 1256 y Fk(\()p Fi(x)p Fk(\))c Fi(u)2089 1226 y Fl(2)p 1813 1293 314 4 v 1904 1369 a Fj(j)p Fi(x)p Fj(j)1997 1345 y Fl(2)2136 1220 y Fg(\021)2200 1312 y Fi(dx)23 b Fj(\025)g Fi(C)2507 1199 y Fg(Z)2461 1441 y Fh(B)s Fl(\()p Fh(r)r Fl(\))2942 1256 y Fi(u)2990 1226 y Fl(2)p 2641 1293 686 4 v 2641 1384 a Fj(j)p Fi(x)p Fj(j)2734 1360 y Fl(2)2772 1384 y Fk(\(log)2912 1347 y Fl(\()p Fh(k)q Fl(+1\))3102 1384 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))3288 1360 y Fl(2)1489 1663 y Fj(\025)1648 1607 y Fi(C)p 1609 1644 144 4 v 1609 1720 a(")p Fk(\()p Fi(r)r Fk(\))1826 1550 y Fg(Z)1781 1792 y Fh(B)s Fl(\()p Fh(r)r Fl(\))1950 1663 y Fi(f)9 b Fk(\()p Fi(x)p Fk(\))14 b Fj(j)p Fi(u)p Fk(\()p Fi(x)p Fk(\))p Fj(j)2330 1629 y Fl(2)2383 1663 y Fi(dx)g(;)523 1951 y Fk(where)48 b Fi(")p Fk(\()p Fi(r)r Fk(\))35 b(:=)f(sup)1208 1971 y Fh(B)s Fl(\()p Fh(r)r Fl(\))1363 1951 y Fi(f)9 b Fk(\()p Fi(x)p Fk(\))14 b Fj(j)p Fi(x)p Fj(j)1631 1920 y Fl(2)1683 1951 y Fk(\(log)1823 1913 y Fl(\()p Fh(k)q Fl(+1\))2014 1951 y Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2200 1920 y Fl(2)2252 1951 y Fi(:)34 b Fk(Hence,)57 b(lim)2600 2002 y Fh(r)r Ff(!)p Fl(0)2780 1951 y Fi(a)p Fk(\()p Fi(r)r Fk(\))35 b Fj(\025)42 b Fk(lim)3061 2002 y Fh(r)r Ff(!)p Fl(0)3283 1918 y Fh(C)p 3251 1932 116 4 v 3251 1979 a(")p Fl(\()p Fh(r)r Fl(\))3411 1951 y Fk(=)523 2100 y(+)p Fj(1)p Fk(,)27 b(b)n(y)h(h)n(yp)r(othesis)f(\()p Fi(H)1347 2112 y Fl(3)1384 2100 y Fk(\)\()p Fi(i)p Fk(\).)523 2332 y Fm(Step)37 b(2.)32 b Fk(Let)g Fi(u)1055 2344 y Fl(0)1092 2332 y Fk(\()p Fi(x)p Fk(\))g(=)e Fj(j)p Fi(x)p Fj(j)1423 2302 y Ff(\000)1486 2272 y Fd(\()p Fc(n)p Fb(\000)p Fd(2\))p 1486 2289 155 4 v 1549 2322 a(2)1655 2332 y Fk(\(log)14 b Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1994 2302 y Ff(\000)p Fh(\016)2076 2310 y Fd(1)2114 2332 y Fk(,)32 b(where)g(thanks)g(to)g (\(1.12\),)g Fi(\016)3100 2344 y Fl(1)3168 2332 y Fi(>)e Fk(1)p Fi(=)p Fk(2)h(is)523 2482 y(c)n(hosen)c(so)g(that)925 2709 y(lim)14 b(inf)962 2762 y Fb(j)p Fc(x)p Fb(j!)p Fd(0)900 2825 y Fl(or)e Ff(j)p Fh(x)p Ff(j!1)1180 2709 y Fk(\(log)j Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))1520 2675 y Fl(2)1572 2592 y Fg(\032)1634 2709 y Fk(\(1)j Fj(\000)g Fi(q)s Fk(\()p Fi(x)p Fk(\)\))i Fj(\000)2105 2653 y Fk(4)14 b Fj(j)p Fi(x)p Fj(j)2254 2623 y Fl(2)2305 2653 y Fi(\021)s Fk(\()p Fi(x)p Fk(\))p 2105 2690 357 4 v 2136 2766 a(\()p Fi(n)19 b Fj(\000)f Fk(2\))2394 2742 y Fl(2)2471 2592 y Fg(\033)2557 2709 y Fi(>)2654 2653 y Fk(4)p Fi(\016)2733 2665 y Fl(1)2770 2653 y Fk(\(1)g(+)g Fi(\016)2982 2665 y Fl(1)3019 2653 y Fk(\))p 2654 2690 398 4 v 2706 2766 a(\()p Fi(n)g Fj(\000)g Fk(2\))2963 2742 y Fl(2)3075 2709 y Fi(:)523 2992 y Fk(There)40 b(exist)g(t)n(w)n(o)f(constan)n(ts)g Fi(R)1597 3004 y Fl(1)1635 2992 y Fi(;)14 b(R)1735 3004 y Fl(2)1772 2992 y Fk(,)40 b Fi(R)1898 3004 y Fl(1)1979 2992 y Fi(<)k Fk(1)f Fi(<)h(R)2345 3004 y Fl(2)2423 2992 y Fk(suc)n(h)39 b(that)i(for)e(all)h Fi(\025)45 b(<)e Fk(0,)d(on)523 3141 y Fi(D)r Fk(\()p Fi(R)689 3153 y Fl(1)727 3141 y Fi(;)14 b(R)827 3153 y Fl(2)864 3141 y Fk(\))23 b(=)g Fj(f)o Fi(x)p Fk(;)14 b Fi(R)1195 3153 y Fl(1)1256 3141 y Fi(<)23 b Fj(j)p Fi(x)p Fj(j)g Fi(<)g(R)1611 3153 y Fl(2)1648 3141 y Fj(g)565 3343 y Fk(\()p Fi(L)654 3355 y Fh(q)r(;\021)764 3343 y Fj(\000)18 b Fi(\025f)9 b Fk(\))14 b Fi(u)1039 3355 y Fl(0)1159 3343 y Fk(=)1307 3276 y Fg(\000)1355 3311 y Fh(n)p Ff(\000)p Fl(2)p 1355 3325 127 4 v 1401 3372 a(2)1491 3276 y Fg(\001)1529 3293 y Fl(2)1751 3309 y Fh(u)1790 3317 y Fd(0)p 1590 3324 394 4 v 1590 3372 a Fl(\()p Ff(j)p Fh(x)p Ff(j)c Fl(log)j Fh(\032)p Fl(\()p Fh(x)p Fl(\)\))1952 3355 y Fd(2)2008 3251 y Fg(h)2047 3343 y Fk(\(log)h Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2386 3313 y Fl(2)2439 3251 y Fg(\020)2488 3343 y Fk(\(1)19 b Fj(\000)f Fi(q)s Fk(\()p Fi(x)p Fk(\)\))h Fj(\000)3048 3311 y Fl(4)p 2959 3325 211 4 v 2959 3372 a(\()p Fh(n)p Ff(\000)p Fl(2\))3137 3355 y Fd(2)3180 3343 y Fj(j)p Fi(x)p Fj(j)3273 3313 y Fl(2)3310 3343 y Fi(\021)3354 3251 y Fg(\021)1390 3493 y Fj(\000)1577 3460 y Fl(4)p 1488 3474 V 1488 3521 a(\()p Fh(n)p Ff(\000)p Fl(2\))1666 3505 y Fd(2)1709 3493 y Fi(\016)1746 3505 y Fl(1)1783 3493 y Fk(\(1)f(+)g Fi(\016)1995 3505 y Fl(1)2032 3493 y Fk(\))h Fj(\000)2239 3460 y Fl(4)12 b Fh(\025)p 2176 3474 V 2176 3521 a Fl(\()p Fh(n)p Ff(\000)p Fl(2\))2354 3505 y Fd(2)2396 3493 y Fk(\()p Fj(j)p Fi(x)p Fj(j)i Fk(log)h Fi(\032)p Fk(\()p Fi(x)p Fk(\)\))2843 3463 y Fl(2)2881 3493 y Fi(f)2931 3401 y Fg(i)1307 3617 y Fj(\025)23 b Fk(0)3470 3468 y Fi(:)523 3803 y Fk(No)n(w,)h(let)g Fi(\036)897 3815 y Fh(i)926 3803 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))24 b(b)r(e)h(the)g Fi(i)1492 3773 y Fl(th)1582 3803 y Fk(eigenfunction)g(in)f(\(1.11\))g(with)h Fj(k)p Fi(\036)2696 3815 y Fh(i)2723 3803 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(k)3028 3818 y Ff(D)3082 3801 y Fd(1)p Fc(;)p Fd(2)3189 3803 y Fk(=)22 b(1)i(and)523 3952 y(suc)n(h)34 b(that)43 b(lim)904 4004 y Fh(s)p Ff(!)p Fl(1)1048 3952 y Fi(\025)1096 3964 y Fh(i)1125 3952 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))35 b(=)f Fi(\025)1570 3964 y Fh(i)1598 3952 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))p Fi(:)35 b Fk(By)g(elliptic)g(regularit)n(y)-7 b(,)32 b Fi(\036)2708 3964 y Fh(i)2736 3952 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))36 b(is)e(uniformly)523 4102 y(b)r(ounded)28 b(in)g(a)f(neigh)n (b)r(ourho)r(o)r(d)g(of)g Fi(@)5 b(D)r Fk(\()p Fi(R)1905 4114 y Fl(1)1942 4102 y Fi(;)14 b(R)2042 4114 y Fl(2)2080 4102 y Fk(\))p Fi(:)28 b Fk(Let)1143 4329 y Fi(M)1224 4341 y Fl(1)1283 4329 y Fk(=)68 b(sup)1384 4398 y Fb(j)p Fc(x)p Fb(j)p Fd(=)p Fc(R)1541 4410 y Fd(1)1371 4463 y Fh(s)p Ff(2)p Fl(\(0)p Fh(;)p Fl(1\))1599 4329 y Fi(\036)1648 4341 y Fh(i)1676 4329 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fi(;)97 b(M)2140 4341 y Fl(2)2200 4329 y Fk(=)67 b(sup)2300 4398 y Fb(j)p Fc(x)p Fb(j)p Fd(=)p Fc(R)2457 4410 y Fd(2)2288 4463 y Fh(s)p Ff(2)p Fl(\(0)p Fh(;)p Fl(1\))2516 4329 y Fi(\036)2565 4341 y Fh(i)2593 4329 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))523 4630 y(and)27 b(let)905 4857 y Fi(W)983 4869 y Fl(1)1044 4857 y Fk(=)22 b Fi(\036)1180 4869 y Fh(i)1209 4857 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))19 b Fj(\000)1649 4801 y Fi(M)1730 4813 y Fl(1)p 1584 4838 250 4 v 1584 4914 a Fi(u)1632 4926 y Fl(0)1668 4914 y Fk(\()p Fi(R)1763 4926 y Fl(1)1801 4914 y Fk(\))1843 4857 y Fi(u)1891 4869 y Fl(0)1928 4857 y Fi(;)97 b(W)2126 4869 y Fl(2)2187 4857 y Fk(=)22 b Fi(\036)2323 4869 y Fh(i)2351 4857 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))19 b Fj(\000)2792 4801 y Fi(M)2873 4813 y Fl(2)p 2726 4838 V 2726 4914 a Fi(u)2774 4926 y Fl(0)2811 4914 y Fk(\()p Fi(R)2906 4926 y Fl(2)2944 4914 y Fk(\))2986 4857 y Fi(u)3034 4869 y Fl(0)3071 4857 y Fi(:)1958 5150 y Fk(15)p eop %%Page: 16 16 16 15 bop 523 614 a Fk(Then)28 b Fi(W)818 626 y Fl(1)879 614 y Fj(\024)22 b Fk(0)27 b(on)h Fi(@)5 b(B)t Fk(\()p Fi(R)1362 626 y Fl(1)1399 614 y Fk(\))28 b(and)f Fi(W)1698 626 y Fl(2)1759 614 y Fj(\024)c Fk(0)k(on)g Fi(@)5 b(B)t Fk(\()p Fi(R)2242 626 y Fl(2)2280 614 y Fk(\))p Fi(:)28 b Fk(F)-7 b(urthermore,)523 872 y(\()p Fi(L)612 884 y Fh(sq)r(;s\021)771 872 y Fj(\000)t Fi(\025)888 884 y Fh(i)915 872 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fi(f)9 b Fk(\))14 b Fi(W)1352 884 y Fl(1)1413 872 y Fj(\024)23 b(\000)p Fk(\(1)t Fj(\000)t Fi(s)p Fk(\))1809 816 y Fi(M)1890 828 y Fl(1)1940 816 y Fi(u)1988 828 y Fl(0)p 1792 853 250 4 v 1792 929 a Fi(u)1840 941 y Fl(0)1877 929 y Fk(\()p Fi(R)1972 941 y Fl(1)2009 929 y Fk(\))2066 730 y Fg( )2131 872 y Fi(\021)f Fk(+)2277 755 y Fg(\022)2348 816 y Fi(n)c Fj(\000)g Fk(2)p 2348 853 193 4 v 2424 929 a(2)2551 755 y Fg(\023)2612 772 y Fl(2)2718 816 y Fi(q)p 2673 853 131 4 v 2673 929 a Fj(j)p Fi(x)p Fj(j)2766 905 y Fl(2)2814 730 y Fg(!)2902 872 y Fj(\024)23 b Fk(0)83 b(in)23 b Fi(B)t Fk(\()p Fi(R)3369 884 y Fl(1)3407 872 y Fk(\))14 b Fi(;)523 1212 y Fk(\()p Fi(L)612 1224 y Fh(sq)r(;s\021)767 1212 y Fj(\000)p Fi(\025)880 1224 y Fh(i)908 1212 y Fk(\()p Fi(sq)s(;)g(s\021)s Fk(\))p Fi(f)9 b Fk(\))14 b Fi(W)1345 1224 y Fl(2)1406 1212 y Fj(\024)22 b(\000)p Fk(\(1)p Fj(\000)p Fi(s)p Fk(\))1794 1156 y Fi(M)1875 1168 y Fl(2)1926 1156 y Fi(u)1974 1168 y Fl(0)p 1778 1193 250 4 v 1778 1269 a Fi(u)1826 1281 y Fl(0)1862 1269 y Fk(\()p Fi(R)1957 1281 y Fl(2)1995 1269 y Fk(\))2051 1070 y Fg( )2117 1212 y Fi(\021)f Fk(+)2262 1095 y Fg(\022)2333 1156 y Fi(n)e Fj(\000)f Fk(2)p 2333 1193 193 4 v 2409 1269 a(2)2536 1095 y Fg(\023)2597 1112 y Fl(2)2704 1156 y Fi(q)p 2658 1193 131 4 v 2658 1269 a Fj(j)p Fi(x)p Fj(j)2751 1245 y Fl(2)2799 1070 y Fg(!)2888 1212 y Fj(\024)23 b Fk(0)82 b(in)23 b Fi(I)-21 b(R)3271 1176 y Fh(n)3317 1212 y Fj(n)p Fi(B)t Fk(\()p Fi(R)3521 1224 y Fl(2)3558 1212 y Fk(\))14 b Fi(:)523 1446 y Fk(No)n(w,)25 b(since)f Fi(W)1023 1410 y Fl(+)1011 1468 y(1)1101 1446 y Fj(2)g Fi(H)1256 1415 y Fl(1)1249 1466 y(0)1293 1446 y Fk(\()p Fi(B)t Fk(\()p Fi(R)1487 1458 y Fl(1)1525 1446 y Fk(\)\))h(and)g Fi(W)1863 1410 y Fl(+)1851 1468 y(2)1941 1446 y Fj(2)f Fi(H)2096 1415 y Fl(1)2089 1466 y(0)2133 1446 y Fk(\()p Fi(I)-21 b(R)2244 1409 y Fh(n)2302 1446 y Fj(n)13 b Fi(B)t Fk(\()p Fi(R)2519 1458 y Fl(2)2557 1446 y Fk(\)\),)25 b(if)h(w)n(e)e(m)n (ultiply)i(the)f(\014rst)523 1595 y(inequalit)n(y)e(b)n(y)f Fi(W)1104 1559 y Fl(+)1092 1617 y(1)1182 1595 y Fk(and)h(the)h(second)e (one)g(b)n(y)h Fi(W)2089 1559 y Fl(+)2077 1617 y(2)2167 1595 y Fk(and)g(w)n(e)g(in)n(tegrate)f(b)n(y)g(parts)h(w)n(e)f(obtain:) 651 1713 y Fg(R)577 1863 y Fh(B)s Fl(\()p Fh(R)706 1871 y Fd(1)738 1863 y Fl(\))772 1780 y Fk(\()p Fi(a)p Fk(\()p Fi(R)943 1792 y Fl(1)981 1780 y Fk(\))c Fj(\000)g Fi(\025)1162 1792 y Fh(i)1191 1780 y Fk(\()p Fi(sq)s(;)c(s\021)s Fk(\)\))g Fi(f)23 b Fj(j)p Fi(W)1677 1744 y Fl(+)1665 1802 y(1)1732 1780 y Fj(j)1755 1750 y Fl(2)1834 1780 y Fj(\024)2019 1713 y Fg(R)1945 1863 y Fh(B)s Fl(\()p Fh(R)2074 1871 y Fd(1)2107 1863 y Fl(\))2141 1780 y Fj(jr)p Fi(W)2323 1744 y Fl(+)2311 1802 y(1)2378 1780 y Fj(j)2401 1750 y Fl(2)2425 1780 y Fj(\000)1981 1889 y Fg(\020)2045 1914 y(\000)2093 1948 y Fh(n)p Ff(\000)p Fl(2)p 2093 1962 127 4 v 2139 2010 a(2)2229 1914 y Fg(\001)2267 1931 y Fl(2)2317 1944 y Fh(s)12 b(q)p 2300 1962 110 4 v 2300 2010 a Ff(j)p Fh(x)p Ff(j)2378 1993 y Fd(2)2410 1981 y Fj(\000)5 b Fi(s\021)21 b Fj(\000)d Fi(\025)2712 1993 y Fh(i)2740 1981 y Fk(\()p Fi(sq)s(;)c(s\021)s Fk(\))g Fi(f)3081 1889 y Fg(\021)3144 1981 y Fj(j)p Fi(W)3257 1945 y Fl(+)3245 2003 y(1)3312 1981 y Fj(j)3335 1951 y Fl(2)1834 2106 y Fk(=)2047 2040 y Fg(R)1973 2189 y Fh(B)s Fl(\()p Fh(R)2102 2197 y Fd(1)2134 2189 y Fl(\))2182 2106 y Fi(W)2272 2071 y Fl(+)2260 2129 y(1)2327 2106 y Fk(\()p Fi(L)2416 2118 y Fh(sq)r(;s\021)2590 2106 y Fj(\000)k Fi(\025)2721 2118 y Fh(i)2749 2106 y Fk(\()p Fi(sq)s(;)c(sn)p Fk(\))g Fi(f)22 b Fk(\))14 b Fi(W)3231 2071 y Fl(+)3219 2130 y Fh(i)1834 2282 y Fj(\024)82 b Fk(0)p Fi(:)523 2465 y Fk(But)39 b(from)f(Step)h(1,)g Fi(a)p Fk(\()p Fi(R)1349 2477 y Fl(1)1386 2465 y Fk(\))j Fj(!)f(1)e Fk(as)f Fi(R)1882 2477 y Fl(1)1960 2465 y Fj(!)k Fk(0.)c(So,)h(for)f Fi(R)2539 2477 y Fl(1)2615 2465 y Fk(small)g(enough,)52 b Fi(a)p Fk(\()p Fi(R)3316 2477 y Fl(1)3353 2465 y Fk(\))26 b Fj(\000)523 2615 y Fi(\025)571 2627 y Fh(i)599 2615 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))28 b Fi(>)f Fk(0)43 b(in)h Fi(B)t Fk(\()p Fi(R)1342 2627 y Fl(1)1380 2615 y Fk(\))g(and)30 b(hence)44 b Fi(W)1957 2579 y Fl(+)1945 2637 y(1)2039 2615 y Fk(=)27 b(0)43 b(a.e.)30 b(This)g(implies)g(that)44 b Fi(\036)3092 2627 y Fh(i)3121 2615 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))27 b Fj(\024)523 2764 y Fi(M)604 2776 y Fl(1)655 2764 y Fi(u)703 2776 y Fl(0)788 2764 y Fk(in)34 b Fi(B)t Fk(\()p Fi(R)1053 2776 y Fl(1)1091 2764 y Fk(\))p Fi(:)h Fk(No)n(w)e(apply)h(the)h(same)f(metho)r(d)h(to)48 b Fj(\000)p Fi(\036)2521 2776 y Fh(i)2617 2764 y Fk(to)34 b(obtain)g Fj(j)p Fi(\036)3062 2776 y Fh(i)3090 2764 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(j)35 b(\024)523 2914 y Fi(M)604 2926 y Fl(1)641 2914 y Fi(u)689 2926 y Fl(0)765 2914 y Fk(in)k Fi(B)t Fk(\()p Fi(R)1035 2926 y Fl(1)1073 2914 y Fk(\))p Fi(:)25 b Fk(Similarly)-7 b(,)25 b Fj(j)p Fi(\036)1592 2926 y Fh(i)1620 2914 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(j)24 b(\024)e Fi(M)2098 2926 y Fl(2)2135 2914 y Fi(u)2183 2926 y Fl(0)2245 2914 y Fk(on)j Fi(I)-21 b(R)2437 2877 y Fh(n)2496 2914 y Fj(n)13 b Fi(B)t Fk(\()p Fi(R)2713 2926 y Fl(2)2750 2914 y Fk(\))26 b(if)f Fi(R)2944 2926 y Fl(2)3007 2914 y Fk(is)g(su\016cien)n(tly)523 3063 y(large.)h(Hence)i(b)n(y)f (elliptic)h(regularit)n(y)-7 b(,)26 b(there)i(exists)f(a)g(constan)n(t) g Fi(M)32 b(>)22 b Fk(0)27 b(suc)n(h)h(that)1472 3296 y Fj(j)p Fi(\036)1544 3308 y Fh(i)1572 3296 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(j)24 b(\024)f Fi(M)9 b(u)2108 3308 y Fl(0)2227 3296 y Fk(in)83 b Fi(I)-21 b(R)2458 3259 y Fh(n)2504 3296 y Fi(:)763 3611 y Fk(Assume)28 b(that)g(up)g(to)g(subsequences,)f Fi(\036)2047 3623 y Fh(i)2075 3611 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))29 b(con)n(v)n(erges)c(w)n(eakly)i(to)h Fi(\036)3165 3623 y Fh(i)3193 3611 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))29 b(in)523 3760 y Fj(D)589 3730 y Fl(1)p Fh(;)p Fl(2)679 3760 y Fk(\()p Fi(I)-21 b(R)791 3724 y Fh(n)836 3760 y Fk(\))30 b(and)f(a.e.)g(in)h Fi(I)-21 b(R)1393 3724 y Fh(n)1468 3760 y Fk(as)28 b Fi(s)i Fk(go)r(es)e(to)i(1.)f(Since)h Fi(u)2289 3772 y Fl(0)2352 3760 y Fj(2)c(D)2499 3730 y Fl(1)p Fh(;)p Fl(2)2590 3760 y Fk(\()p Fi(I)-21 b(R)2701 3724 y Fh(n)2746 3760 y Fk(\),)30 b(b)n(y)f(the)h(dominated)523 3910 y(con)n(v)n(ergence)25 b(theorem)i(w)n(e)h(ha)n(v)n(e)1414 4142 y(lim)1406 4194 y Fh(s)p Ff(!)p Fl(1)1560 4029 y Fg(Z)1532 4267 y Fh(I)-18 b(R)1594 4248 y Fc(n)1677 4086 y Fi(\036)1726 4056 y Fl(2)1726 4108 y Fh(i)1763 4086 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p 1677 4123 351 4 v 1786 4199 a Fj(j)p Fi(x)p Fj(j)1879 4175 y Fl(2)2060 4142 y Fk(=)2157 4029 y Fg(Z)2130 4267 y Fh(I)-18 b(R)2191 4248 y Fc(n)2274 4086 y Fi(\036)2323 4056 y Fl(2)2323 4108 y Fh(i)2360 4086 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))p 2274 4123 273 4 v 2344 4199 a Fj(j)p Fi(x)p Fj(j)2437 4175 y Fl(2)2570 4142 y Fi(;)523 4425 y Fk(and)27 b(b)n(y)h(our)f(h)n (yp)r(othesis)g(on)g Fi(\021)k Fk(and)c Fi(f)9 b Fk(,)28 b(w)n(e)f(also)f(ha)n(v)n(e)1349 4657 y(lim)1342 4708 y Fh(s)p Ff(!)p Fl(1)1496 4544 y Fg(Z)1468 4782 y Fh(I)-18 b(R)1529 4763 y Fc(n)1602 4657 y Fi(s)14 b(\021)j(\036)1762 4623 y Fl(2)1762 4678 y Fh(i)1800 4657 y Fk(\()p Fi(sq)s(;)d(s\021)s Fk(\))23 b(=)2183 4544 y Fg(Z)2156 4782 y Fh(I)-18 b(R)2217 4763 y Fc(n)2290 4657 y Fi(\021)17 b(\036)2397 4623 y Fl(2)2397 4678 y Fh(i)2435 4657 y Fk(\()p Fi(q)s(;)d(\021)s Fk(\))g Fi(;)1958 5150 y Fk(16)p eop %%Page: 17 17 17 16 bop 1056 764 a Fk(lim)1048 815 y Fh(s)p Ff(!)p Fl(1)1202 651 y Fg(Z)1174 889 y Fh(I)-18 b(R)1236 870 y Fc(n)1309 764 y Fi(\025)1357 776 y Fh(i)1385 764 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))g Fi(f)22 b(\036)1774 730 y Fl(2)1774 784 y Fh(i)1812 764 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))24 b(=)2196 651 y Fg(Z)2169 889 y Fh(I)-18 b(R)2230 870 y Fc(n)2303 764 y Fi(\025)2351 776 y Fh(i)2379 764 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))g Fi(f)23 b(\036)2691 730 y Fl(2)2691 784 y Fh(i)2728 764 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))g Fi(:)763 1101 y Fk(Hence,)40 b(b)n(y)f(using)g(the)h (equations)f(satis\014ed)g(b)n(y)g Fi(\036)2447 1113 y Fh(i)2475 1101 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))40 b(and)g(b)n(y)f Fi(\036)3128 1113 y Fh(i)3156 1101 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))40 b(w)n(e)523 1250 y(obtain:)1240 1632 y(lim)1233 1683 y Fh(s)p Ff(!)p Fl(1)1377 1519 y Fg(Z)1423 1708 y Fh(I)-18 b(R)1485 1688 y Fc(n)1544 1632 y Fj(jr)p Fi(\036)1685 1644 y Fh(i)1713 1632 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))p Fj(j)1999 1598 y Fl(2)2060 1632 y Fk(=)2147 1519 y Fg(Z)2194 1708 y Fh(I)-18 b(R)2255 1688 y Fc(n)2314 1632 y Fj(jr)p Fi(\036)2455 1644 y Fh(i)2483 1632 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))p Fj(j)2691 1598 y Fl(2)2743 1632 y Fi(;)523 1831 y Fk(and)39 b(therefore,)80 b Fi(\036)1170 1843 y Fh(i)1198 1831 y Fk(\()p Fi(sq)s(;)14 b(s\021)s Fk(\))84 b Fj(\000)-14 b(!)1537 1852 y Fc(s)p Fb(!)p Fd(1)1742 1831 y Fi(\036)1791 1843 y Fh(i)1819 1831 y Fk(\()p Fi(q)s(;)14 b(\021)s Fk(\))40 b(strongly)d(in)i Fj(D)2550 1801 y Fl(1)p Fh(;)p Fl(2)2641 1831 y Fk(\()p Fi(I)-21 b(R)2752 1795 y Fh(n)2797 1831 y Fk(\))p Fi(:)40 b Fk(This)f(pro)n(v)n(es)d(the)523 1981 y(theorem.)2578 b Fj(t)-55 b(u)523 2213 y Fm(3.)34 b(Extensions.)29 b Fk(The)i(b)r(oundary)e(in)n(tegral)h(on)g(the)h(righ)n(t)e(hand)i(side) f(of)h(\(1.4\))f(and)g(\(1.5\))523 2362 y(can)g(b)r(e)g(replaced)g(b)n (y)g(an)g(in)n(tegral)f(o)n(v)n(er)f(the)j(whole)f(domain.)f(The)i(pro) r(of)e(follo)n(ws)h(exactly)523 2512 y(as)d(in)h(Lemma)f(2.2)g(and)g (hence)h(w)n(e)f(will)h(just)g(brie\015y)f(sk)n(etc)n(h)g(it.)523 2744 y Fm(1.)42 b(Replacemen)m(t)g(of)h(b)s(oundary)h(in)m(tegrals)e (in)g(\(1.4\))g(and)i(\(1.5\))e(b)m(y)h(in)m(tegrals)523 2894 y(o)m(v)m(er)33 b Fk(\012)p Fm(.)763 3126 y Fk(Let)28 b(\012)f(b)r(e)h(a)g(b)r(ounded)g(domain)f(of)g Fi(I)-21 b(R)1989 3090 y Fh(n)2062 3126 y Fk(and)1117 3417 y Fi(H)7 b Fk(\()p Fi(u)p Fk(\))23 b(=)1416 3222 y Fg(8)1416 3297 y(>)1416 3322 y(<)1416 3471 y(>)1416 3496 y(:)1531 3225 y(\020)1590 3281 y Fh(n)p Ff(\000)p Fh(p)p 1590 3298 128 4 v 1637 3346 a(p)1728 3225 y Fg(\021)1777 3243 y Fh(p)1829 3251 y Fg(R)1825 3397 y Fl(\012)1909 3277 y Ff(j)p Fh(u)p Ff(j)1988 3252 y Fc(p)p 1909 3298 114 4 v 1910 3346 a Ff(j)p Fh(x)p Ff(j)1988 3330 y Fc(p)2453 3318 y Fk(if)90 b Fi(n)23 b(>)g(p)14 b(;)1531 3421 y Fg(\000)1579 3456 y Fh(n)p Ff(\000)p Fl(1)p 1579 3470 127 4 v 1621 3517 a Fh(n)1715 3421 y Fg(\001)1753 3439 y Fh(n)1812 3422 y Fg(R)1808 3568 y Fl(\012)2066 3448 y Ff(j)p Fh(u)p Ff(j)2145 3423 y Fc(n)p 1891 3470 470 4 v 1891 3517 a Ff(j)p Fh(x)p Ff(j)1969 3501 y Fc(n)2009 3517 y Fl(\(log)e Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2320 3501 y Fc(n)2453 3489 y Fk(if)90 b Fi(n)23 b Fk(=)g Fi(p)14 b(;)1034 3923 y(A)p Fk(\()p Fi(u)p Fk(\))24 b(=)e Fi(C)1422 3678 y Fg(8)1422 3752 y(>)1422 3777 y(>)1422 3802 y(>)1422 3827 y(<)1422 3977 y(>)1422 4001 y(>)1422 4026 y(>)1422 4051 y(:)1537 3728 y(R)1533 3874 y Fl(\012)1645 3716 y Fh(k)1620 3732 y Fg(P)1606 3868 y Fh(j)s Fl(=1)1938 3762 y(1)p 1745 3776 419 4 v 1745 3830 a(\(log)1857 3810 y Fd(\()p Fc(j)r Fd(\))1944 3830 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2132 3813 y Fd(2)2184 3754 y Ff(j)p Fh(u)p Ff(j)2263 3729 y Fc(p)p 2184 3776 114 4 v 2185 3823 a Ff(j)p Fh(x)p Ff(j)2263 3807 y Fc(p)2536 3795 y Fk(if)90 b Fi(n)23 b(>)f(p)14 b(;)1537 3966 y Fg(R)1533 4112 y Fl(\012)1606 3891 y Fg( )1711 3954 y Fh(k)1686 3970 y Fg(P)1672 4106 y Fh(j)s Fl(=2)2008 4000 y(1)p 1811 4014 428 4 v 1811 4068 a(\(log)1922 4048 y Fd(\()p Fc(j)r Fd(\))2010 4068 y Fh(R=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2198 4051 y Fc(n)2258 3992 y Ff(j)p Fh(u)p Ff(j)2337 3967 y Fc(n)p 2258 4014 120 4 v 2259 4061 a Ff(j)p Fh(x)p Ff(j)2337 4045 y Fc(n)2387 3891 y Fg(!)2536 4033 y Fk(if)90 b Fi(n)23 b Fk(=)f Fi(p)14 b(;)523 4257 y Fk(where)27 b Fi(C)34 b Fk(is)28 b(the)g(constan)n(t)e(of)i(inequalit)n(y)f (\(1.3\).)523 4489 y Fm(Step)32 b(1.)27 b Fk(Let)h Fi(")22 b(>)h Fk(0)p Fi(;)k Fk(then)h(there)g(exist)f Fi(C)6 b Fk(\()p Fi(")p Fk(\))24 b Fi(>)e Fk(0)28 b(suc)n(h)f(that)1379 4609 y Fg(Z)1359 4847 y Fh(@)t Fl(\012)1478 4722 y Fj(j)p Fi(u)p Fj(j)1572 4687 y Fh(p)1633 4722 y Fj(\024)c Fi(")1774 4609 y Fg(Z)1773 4847 y Fl(\012)1870 4722 y Fj(jr)p Fi(u)p Fj(j)2033 4687 y Fh(p)2090 4722 y Fk(+)18 b Fi(C)6 b Fk(\()p Fi(")p Fk(\))2355 4609 y Fg(Z)2354 4847 y Fl(\012)2452 4722 y Fj(j)p Fi(u)p Fj(j)2546 4687 y Fh(p)2598 4722 y Fi(:)1958 5150 y Fk(17)p eop %%Page: 18 18 18 17 bop 523 614 a Fk(holds)40 b(for)f(all)g Fi(u)k Fj(2)h Fi(W)1299 584 y Fl(1)p Fh(;p)1390 614 y Fk(\(\012\))p Fi(:)d Fk(Substituting)g(this)f(in)g(\(1.4\))g(and)f(\(1.5\))h(yields)f (for)h Fi(s)j Fk(=)620 731 y Fl(1)p 533 745 208 4 v 533 792 a(\(1)p Ff(\000)p Fh(\025")p Fl(\))774 764 y Fi(<)22 b Fk(1)1206 815 y Fg(Z)1205 1053 y Fl(\012)1303 928 y Fj(jr)p Fi(u)p Fj(j)1466 894 y Fh(p)1522 928 y Fj(\000)c Fi(sH)7 b Fk(\()p Fi(u)p Fk(\))19 b Fj(\000)f Fi(sA)p Fk(\()p Fi(u)p Fk(\))23 b Fj(\025)g Fi(\025C)6 b Fk(\()p Fi(")p Fk(\))p Fi(s)2527 815 y Fg(Z)2526 1053 y Fl(\012)2624 928 y Fj(j)p Fi(u)p Fj(j)2718 894 y Fh(p)2770 928 y Fi(:)523 1173 y Fm(Step)32 b(2.)27 b Fk(Let)1073 1443 y Fi(\025)1121 1455 y Fh(s)1180 1443 y Fk(=)c(inf)1382 1273 y Fg(8)1382 1348 y(<)1382 1497 y(:)1456 1330 y(Z)1456 1568 y Fl(\012)1553 1443 y Fj(jr)p Fi(u)p Fj(j)1716 1409 y Fh(p)1773 1443 y Fj(\000)18 b Fi(sH)7 b Fk(\()p Fi(u)p Fk(\))18 b Fj(\000)g Fi(sA)p Fk(\()p Fi(u)p Fk(\);)2434 1330 y Fg(Z)2433 1568 y Fl(\012)2531 1443 y Fj(j)p Fi(u)p Fj(j)2625 1409 y Fh(p)2686 1443 y Fk(=)23 b(1)2816 1273 y Fg(9)2816 1348 y(=)2816 1497 y(;)2903 1443 y Fi(;)523 1723 y Fk(By)31 b(Step)i(1,)e Fi(\025)995 1735 y Fh(s)1061 1723 y Fj(\025)f Fi(\025)14 b(C)6 b Fk(\()p Fi(")p Fk(\))30 b Fi(>)g Fj(\0001)45 b Fk(.)32 b(Without)h(loss)d(of)i(generalit)n(y)-7 b(,)31 b(assume)g(that)h Fi(p)d(<)h(n)p Fk(.)523 1873 y(Then)k(from)g(Step)h (1)e(and)h(as)g(in)g(Lemma)g(2.2,)f(the)i(corresp)r(onding)d(op)r (erator)g(is)i(co)r(erciv)n(e)523 2022 y(and)29 b(hence)h(from)f(Ek)n (eland's)e(principle)j Fi(\025)1885 2034 y Fh(s)1950 2022 y Fk(is)f(ac)n(hiev)n(ed)f(b)n(y)h(some)g(some)g(function)h Fi(u)3283 2034 y Fh(s)3344 2022 y Fj(\025)25 b Fk(0)523 2172 y(whic)n(h)j(satis\014es)1086 2397 y Fj(\000)23 b Fk(div)h(\()p Fj(jr)p Fi(u)1483 2409 y Fh(s)1519 2397 y Fj(j)1542 2363 y Fh(p)p Ff(\000)p Fl(2)1665 2397 y Fj(r)p Fi(u)1782 2409 y Fh(s)1817 2397 y Fk(\))19 b Fj(\000)1951 2280 y Fg(\022)2022 2341 y Fi(n)g Fj(\000)f Fi(p)p 2022 2378 194 4 v 2098 2454 a(p)2225 2280 y Fg(\023)2286 2297 y Fh(p)2349 2341 y Fi(s)c Fj(j)p Fi(u)2473 2353 y Fh(s)2507 2341 y Fj(j)2530 2311 y Fh(p)p Ff(\000)p Fl(2)2668 2341 y Fi(u)2716 2353 y Fh(s)p 2349 2378 403 4 v 2484 2454 a Fj(j)p Fi(x)p Fj(j)2577 2430 y Fh(p)2779 2397 y Fj(\000)1137 2513 y Fg(0)1137 2662 y(@)1251 2576 y Fh(k)1209 2601 y Fg(X)1212 2778 y Fh(j)s Fl(=1)1532 2623 y Fi(C)20 b(s)p 1353 2661 477 4 v 1353 2751 a Fk(log)1460 2714 y Fl(\()p Fh(j)s Fl(\))1561 2751 y Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1792 2727 y Fl(2)1839 2513 y Fg(1)1839 2662 y(A)1936 2623 y Fj(j)p Fi(u)2007 2635 y Fh(s)2042 2623 y Fj(j)2065 2593 y Fh(p)p Ff(\000)p Fl(2)2202 2623 y Fi(u)2250 2635 y Fh(s)p 1936 2661 350 4 v 2044 2737 a Fj(j)p Fi(x)p Fj(j)2137 2713 y Fh(p)2318 2680 y Fk(=)j Fi(\025)2454 2692 y Fh(s)2490 2680 y Fj(j)p Fi(u)2561 2692 y Fh(s)2596 2680 y Fj(j)2619 2645 y Fh(p)p Ff(\000)p Fl(2)2756 2680 y Fi(u)2804 2692 y Fh(s)2839 2680 y Fi(:)523 2960 y Fk(Hence)34 b(b)n(y)g(regularit)n(y)-7 b(,)33 b Fj(f)p Fi(u)1390 2972 y Fh(s)1424 2960 y Fj(g)h Fk(is)g(b)r(ounded)g(in)h Fi(W)2129 2930 y Fl(1)p Fh(;p)2220 2960 y Fk(\(\012)23 b Fj(n)f Fi(B)t Fk(\()p Fi(r)r Fk(\)\))36 b(for)e(some)g Fi(r)i(>)e Fk(0)p Fi(:)g Fk(Hence)523 3110 y Fj(f)580 3043 y Fg(R)557 3189 y Fh(@)t Fl(\012)665 3110 y Fi(u)713 3080 y Fh(p)713 3130 y(s)751 3110 y Fj(g)27 b Fk(is)g(b)r(ounded.)h(F) -7 b(rom)28 b(\(1.4\))f(and)g(the)h(ab)r(o)n(v)n(e)f(equation,)1571 3395 y Fi(\025)1635 3282 y Fg(Z)1614 3521 y Fh(@)t Fl(\012)1733 3395 y Fi(u)1781 3361 y Fh(p)1781 3416 y(s)1842 3395 y Fj(\024)c Fi(\025)1978 3407 y Fh(s)2028 3282 y Fg(Z)2027 3520 y Fl(\012)2124 3395 y Fi(u)2172 3361 y Fh(p)2172 3416 y(s)2233 3395 y Fk(=)g Fi(\025)2369 3407 y Fh(s)2405 3395 y Fi(:)523 3671 y Fk(and)29 b(hence)h Fi(\025)967 3683 y Fh(s)1032 3671 y Fk(is)f(b)r(ounded)h(uniformly)-7 b(.)30 b(Clearly)-7 b(,)28 b(b)n(y)h(monotonicit)n(y)-7 b(,)29 b Fi(\025)2847 3683 y Fh(s)2909 3671 y Fj(!)d Fi(\025)3066 3683 y Fl(1)3133 3671 y Fk(as)j Fi(s)d Fj(!)g Fk(1)p Fi(:)523 3821 y Fk(So,)h(w)n(e)h(ha)n(v)n(e)e(the)i(follo)n (wing)f(inequalities)g(for)g(all)g Fi(u)c Fj(2)g Fi(W)2386 3791 y Fl(1)p Fh(;p)2477 3821 y Fk(\(\012\))1337 3933 y Fg(Z)1336 4171 y Fl(\012)1434 4046 y Fj(jr)p Fi(u)p Fj(j)1597 4012 y Fh(p)1653 4046 y Fj(\000)18 b Fi(H)7 b Fk(\()p Fi(u)p Fk(\))19 b Fj(\000)f Fi(A)p Fk(\()p Fi(u)p Fk(\))23 b Fj(\025)g Fi(\025)2359 4058 y Fl(1)2410 3933 y Fg(Z)2410 4171 y Fl(\012)2507 4046 y Fj(j)p Fi(u)p Fj(j)2601 4012 y Fh(p)2639 4046 y Fi(:)643 b Fk(\(3.1\))523 4409 y Fm(2.)39 b Fk(V\023)-42 b(azquez)39 b(and)g(Zuazua)g([11])g (\(see)g(also)g([8]\))g(pro)n(v)n(ed)f(the)i(follo)n(wing)e(inequalit)n (y)-7 b(.)40 b(Let)523 4559 y(1)23 b Fi(<)f(q)k(<)d Fk(2)p Fi(;)k Fk(then)h(there)g(exists)f(a)g(constan)n(t)g Fi(C)i(>)23 b Fk(0)k(suc)n(h)g(that)h(for)f(all)h Fi(u)22 b Fj(2)i Fi(H)3033 4529 y Fl(1)3026 4579 y(0)3070 4559 y Fk(\(\012\),)1177 4688 y Fg(Z)1177 4926 y Fl(\012)1274 4801 y Fj(jr)p Fi(u)p Fj(j)1437 4767 y Fl(2)1493 4801 y Fj(\000)1576 4684 y Fg(\022)1647 4745 y Fi(n)18 b Fj(\000)g Fk(2)p 1647 4782 193 4 v 1722 4858 a(2)1849 4684 y Fg(\023)1911 4701 y Fl(2)1962 4688 y Fg(Z)1961 4926 y Fl(\012)2092 4745 y Fi(u)2140 4715 y Fl(2)p 2068 4782 131 4 v 2068 4858 a Fj(j)p Fi(x)p Fj(j)2161 4834 y Fl(2)2232 4801 y Fj(\025)23 b Fi(C)d Fj(k)p Fi(u)p Fj(k)2531 4767 y Fl(2)2531 4824 y Fh(W)2602 4807 y Fd(1)p Fc(;q)2681 4824 y Fl(\(\012\))2799 4801 y Fi(:)483 b Fk(\(3.2\))1958 5150 y(18)p eop %%Page: 19 19 19 18 bop 523 614 a Fk(First)34 b(they)g(pro)n(v)n(e)e(the)j(ab)r(o)n (v)n(e)d(inequalit)n(y)i(when)g(\012)g(a)f(ball)h(and)g(for)f(radial)g (functions)h Fi(u)p Fk(.)523 764 y(Then)20 b(b)n(y)f(using)g(spherical) g(harmonics)f(they)i(decomp)r(ose)f(the)h(functions)g Fi(u)f Fk(in)n(to)g(their)h(radial)523 913 y(and)30 b(non)g(radial)f (part.)h(Next)g(they)h(pro)n(v)n(e)d(the)j(inequalit)n(y)f(for)f(non)h (radial)g(functions)g(and)523 1063 y(then)e(extend)g(it)g(to)f (artitrary)f(\012)p Fi(:)763 1295 y Fk(Here)31 b(w)n(e)h(can)f(also)g (obtain)g(an)h(extension)f(of)g(\(3.2\))h(b)n(y)f(using)h(the)g(term) 3186 1255 y Ff(jr)p Fh(v)r Ff(j)3316 1230 y Fc(p)p 3171 1276 194 4 v 3171 1324 a Ff(j)p Fh(x)p Ff(j)3249 1307 y Fc(n)p Fb(\000)p Fc(p)3407 1295 y Fk(in)523 1445 y(\(2.6\).)e(F)-7 b(or)30 b(the)h(sak)n(e)e(of)i(simplicit)n(y)g(assume)e(that)i(1)d Fi(<)f(p)h(<)f(n)45 b Fk(and)f Fi(k)31 b Fk(=)c(1.)j(Then)h(for)f(all) 523 1594 y Fi(u)23 b Fj(2)g Fi(W)762 1554 y Fl(1)p Fh(;p)750 1616 y Fl(0)853 1594 y Fk(\(\012\))p Fi(;)28 b Fk(\(2.4\))g(and)f (\(2.6\))h(imply)635 1713 y Fg(Z)634 1952 y Fl(\012)731 1709 y Fg(\022)793 1826 y Fj(jr)p Fi(u)p Fj(j)956 1792 y Fh(p)1012 1826 y Fj(\000)1095 1709 y Fg(\022)1166 1770 y Fi(n)19 b Fj(\000)f Fi(p)p 1166 1807 V 1242 1883 a(p)1369 1709 y Fg(\023)1431 1727 y Fh(p)1493 1770 y Fj(j)p Fi(u)p Fj(j)1587 1740 y Fh(p)p 1493 1807 133 4 v 1493 1883 a Fj(j)p Fi(x)p Fj(j)1586 1859 y Fh(p)1653 1826 y Fj(\000)1909 1770 y Fi(C)1968 1782 y Fl(1)p 1746 1807 422 4 v 1746 1883 a Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2130 1859 y Fl(2)2188 1770 y Fj(j)p Fi(u)p Fj(j)2282 1740 y Fh(p)p 2188 1807 133 4 v 2188 1883 a Fj(j)p Fi(x)p Fj(j)2281 1859 y Fh(p)2330 1709 y Fg(\023)2414 1826 y Fj(\025)22 b Fi(C)2560 1838 y Fl(2)2612 1713 y Fg(Z)2611 1952 y Fl(\012)2719 1770 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))p Fj(jr)p Fi(w)r Fj(j)3279 1740 y Fl(2)p 2719 1807 599 4 v 2906 1883 a Fj(j)p Fi(x)p Fj(j)2999 1859 y Fh(n)p Ff(\000)p Fl(2)3341 1826 y Fi(;)3305 2030 y Fk(\(3.3\))523 2180 y(where)27 b Fi(w)f Fk(=)c(\(log)15 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1320 2150 y Ff(\000)p Fl(1)p Fh(=)p Fl(2)1476 2180 y Fi(v)1519 2150 y Fh(p=)p Fl(2)1648 2180 y Fk(=)23 b(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2120 2150 y Ff(\000)p Fl(1)p Fh(=)p Fl(2)2276 2180 y Fj(j)p Fi(x)p Fj(j)2379 2122 y Fc(n)p Fb(\000)p Fc(p)p 2380 2137 113 4 v 2422 2170 a Fd(2)2506 2180 y Fi(u)2554 2150 y Fh(p=)p Fl(2)2659 2180 y Fi(:)28 b Fk(But)1010 2437 y Fj(jr)p Fi(u)1150 2403 y Fh(p=)p Fl(2)1255 2437 y Fj(j)23 b Fk(=)g Fi(O)1468 2295 y Fg( )1544 2381 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1928 2351 y Fl(1)p Fh(=)p Fl(2)p 1544 2418 489 4 v 1673 2512 a Fj(j)p Fi(x)p Fj(j)1776 2456 y Fc(n)p Fb(\000)p Fc(p)p 1777 2471 113 4 v 1819 2503 a Fd(2)2056 2437 y Fj(r)p Fi(w)22 b Fk(+)2299 2381 y(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2683 2351 y Fl(1)p Fh(=)p Fl(2)p 2299 2418 489 4 v 2386 2512 a Fj(j)p Fi(x)p Fj(j)2489 2456 y Fc(n)p Fb(\000)p Fc(p)p 2490 2471 113 4 v 2532 2503 a Fd(2)2612 2483 y Fl(+1)2811 2437 y Fi(w)2872 2295 y Fg(!)2966 2437 y Fi(;)523 2690 y Fk(therefore,)629 2855 y Fg(Z)628 3093 y Fl(\012)726 2968 y Fj(jr)p Fk(\()p Fi(u)898 2933 y Fh(p=)p Fl(2)1003 2968 y Fk(\))p Fj(j)1058 2933 y Fh(q)1118 2968 y Fk(=)23 b Fi(O)1285 2801 y Fg(0)1285 2950 y(@)1358 2855 y(Z)1357 3093 y Fl(\012)1455 2826 y Fg( )1531 2911 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1915 2870 y Fh(q)r(=)p Fl(2)2033 2911 y Fj(jr)p Fi(w)r Fj(j)2209 2881 y Fh(q)p 1531 2949 716 4 v 1725 3050 a Fj(j)p Fi(x)p Fj(j)1818 3015 y Fh(q)1850 3021 y Fk(\()1893 2987 y Fc(n)p Fb(\000)p Fc(p)p 1893 3002 113 4 v 1935 3035 a Fd(2)2015 3021 y Fk(\))2275 2968 y(+)2368 2911 y(\(log)g Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2752 2870 y Fh(q)r(=)p Fl(2)2870 2911 y Fj(j)p Fi(w)r Fj(j)2977 2881 y Fh(q)p 2368 2949 647 4 v 2485 3050 a Fj(j)p Fi(x)p Fj(j)2578 3015 y Fh(q)2610 3021 y Fk(\()2654 2987 y Fc(n)p Fb(\000)p Fc(p)p 2654 3002 113 4 v 2696 3035 a Fd(2)2776 3015 y Fl(+1)2860 3021 y Fk(\))3024 2826 y Fg(!)3090 2801 y(1)3090 2950 y(A)3176 2968 y Fi(:)106 b Fk(\(3.4\))763 3338 y(In)33 b(order)e(to)h(estimate)g(the)h(righ)n(t)f(hand)g(side)g(of)h(\(3.4\),) f(w)n(e)g(need)g(some)g(estimates)523 3487 y(regarding)25 b(the)j(w)n(eigh)n(ted)f(Hardy's)g(inequalit)n(y)-7 b(.)763 3719 y(Let)28 b Fi(w)e Fj(2)d Fi(C)1140 3689 y Ff(1)1134 3740 y Fl(0)1211 3719 y Fk(\(\012\))28 b(and)f Fi(k)f Fj(\025)d Fk(0)p Fi(;)14 b(\013)23 b Fj(\025)f Fk(0)p Fi(:)27 b Fk(Then,)665 3839 y Fg(Z)664 4077 y Fl(\012)772 3896 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1156 3866 y Fh(\013)1203 3896 y Fj(j)p Fi(w)r Fj(j)1310 3866 y Fh(q)p 772 3933 577 4 v 992 4009 a Fj(j)p Fi(x)p Fj(j)1085 3985 y Fh(k)1441 3952 y Fk(=)1602 3896 y(1)p 1598 3933 50 4 v 1598 4009 a Fi(n)1672 3839 y Fg(Z)1671 4077 y Fl(\012)1779 3896 y Fk(\(div)24 b Fi(x)p Fk(\)\(log)15 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2412 3866 y Fh(\013)2460 3896 y Fj(j)p Fi(w)r Fj(j)2567 3866 y Fh(q)p 1779 3933 826 4 v 2124 4009 a Fj(j)p Fi(x)p Fj(j)2217 3985 y Fh(k)1441 4242 y Fk(=)82 b Fj(\000)1668 4186 y Fi(q)p 1663 4223 50 4 v 1663 4299 a(n)1736 4129 y Fg(Z)1736 4367 y Fl(\012)1843 4186 y Fj(j)p Fi(w)r Fj(j)1950 4156 y Fh(q)r Ff(\000)p Fl(2)2073 4186 y Fi(w)r Fj(hr)p Fi(w)r(;)14 b(x)p Fj(i)p Fk(\(log)j Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2799 4156 y Fh(\013)p 1843 4223 1003 4 v 2277 4299 a Fj(j)p Fi(x)p Fj(j)2370 4275 y Fh(k)1588 4527 y Fk(+)1665 4471 y Fi(k)p 1663 4508 50 4 v 1663 4584 a(n)1736 4414 y Fg(Z)1736 4652 y Fl(\012)1843 4471 y Fj(j)p Fi(w)r Fj(j)1950 4441 y Fh(q)1988 4471 y Fk(\(log)d Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2372 4441 y Fh(\013)p 1843 4508 577 4 v 2064 4584 a Fj(j)p Fi(x)p Fj(j)2157 4560 y Fh(k)2448 4527 y Fk(+)2541 4471 y Fi(\013)p 2541 4508 54 4 v 2543 4584 a(n)2618 4414 y Fg(Z)2617 4652 y Fl(\012)2725 4471 y Fj(j)p Fi(w)r Fj(j)2832 4441 y Fh(q)2869 4471 y Fk(\(log)h Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3254 4441 y Fh(\013)p 2725 4508 577 4 v 2754 4584 a Fj(j)p Fi(x)p Fj(j)2847 4560 y Fh(k)2888 4584 y Fk(\(log)f Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3311 4527 y Fi(:)523 4825 y Fk(Let)30 b Fi(k)h(<)c(n)j Fk(and)g(c)n(ho)r(ose)f Fi(R)1413 4837 y Fl(0)1477 4825 y Fi(>)e Fk(0)j(suc)n(h)g(that)2023 4792 y Fh(\013)p 2023 4806 44 4 v 2024 4854 a(n)2120 4825 y Fk(sup)2118 4895 y Fh(x)p Ff(2)p Fl(\012)2426 4792 y(1)p 2271 4806 343 4 v 2271 4854 a(\(log)12 b Fh(R)2444 4862 y Fd(0)2477 4854 y Fh(=)p Ff(j)p Fh(x)p Ff(j)p Fl(\))2651 4825 y Fi(<)2753 4792 y Fl(1)p 2753 4806 34 4 v 2753 4854 a(2)2810 4758 y Fg(\000)2848 4825 y Fk(1)18 b Fj(\000)3003 4792 y Fh(k)p 3001 4806 42 4 v 3001 4854 a(n)3052 4758 y Fg(\001)3104 4825 y Fi(:)30 b Fk(Then)g(for)1958 5150 y(19)p eop %%Page: 20 20 20 19 bop 523 614 a Fi(R)24 b Fj(\025)e Fi(R)760 626 y Fl(0)798 614 y Fi(;)27 b Fk(the)h(ab)r(o)n(v)n(e)f(iden)n(tit)n(y)g (giv)n(es)788 791 y(1)p 788 828 42 4 v 788 904 a(2)854 730 y Fg(\022)915 847 y Fk(1)18 b Fj(\000)1070 791 y Fi(k)p 1068 828 50 4 v 1068 904 a(n)1128 730 y Fg(\023)1203 734 y(Z)1202 972 y Fl(\012)1309 791 y Fj(j)p Fi(w)r Fj(j)1416 761 y Fh(q)1454 791 y Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1838 761 y Fh(\013)p 1309 828 577 4 v 1530 904 a Fj(j)p Fi(x)p Fj(j)1623 880 y Fh(k)1979 847 y Fj(\024)2141 791 y Fi(q)p 2136 828 50 4 v 2136 904 a(n)2210 734 y Fg(Z)2209 972 y Fl(\012)2317 791 y Fj(j)p Fi(w)r Fj(j)2424 761 y Fh(q)r Ff(\000)p Fl(1)2546 791 y Fj(jr)p Fi(w)r Fj(j)p Fk(\(log)h Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))3107 761 y Fh(\013)p 2317 828 839 4 v 2626 904 a Fj(j)p Fi(x)p Fj(j)2719 880 y Fh(k)q Ff(\000)p Fl(1)1979 1218 y Fj(\024)2141 1162 y Fi(q)p 2136 1199 50 4 v 2136 1275 a(n)2210 1051 y Fg(0)2210 1201 y(@)2282 1105 y(Z)2282 1343 y Fl(\012)2389 1162 y Fj(j)p Fi(w)r Fj(j)2496 1132 y Fh(q)2534 1162 y Fk(\(log)f Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2918 1132 y Fh(\013)2965 1162 y Fk(\))p 2389 1199 609 4 v 2626 1275 a Fj(j)p Fi(x)p Fj(j)2719 1251 y Fh(k)3008 1051 y Fg(1)3008 1201 y(A)3090 1041 y Fc(q)q Fb(\000)p Fd(1)p 3090 1056 103 4 v 3127 1089 a Fc(q)2209 1417 y Fg(0)2209 1566 y(@)2282 1471 y(Z)2281 1709 y Fl(\012)2389 1527 y Fj(jr)p Fi(w)r Fj(j)2565 1497 y Fh(q)2602 1527 y Fk(\(log)h Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2987 1497 y Fh(\013)p 2389 1564 646 4 v 2602 1641 a Fj(j)p Fi(x)p Fj(j)2695 1617 y Fh(k)q Ff(\000)p Fh(q)3044 1417 y Fg(1)3044 1566 y(A)3127 1412 y Fd(1)p 3127 1421 29 4 v 3127 1454 a Fc(q)523 1871 y Fk(and)27 b(hence)h(there)f(exists)h(a) f(constan)n(t)g Fi(C)i Fk(=)23 b Fi(C)6 b Fk(\()p Fi(K)q(;)14 b(N)t(;)g(\013)p Fk(\))24 b Fi(>)e Fk(0)28 b(suc)n(h)f(that)1159 1990 y Fg(Z)1158 2228 y Fl(\012)1265 2047 y Fj(j)p Fi(w)r Fj(j)1372 2017 y Fh(q)1410 2047 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))1794 2017 y Fh(\013)p 1265 2084 577 4 v 1486 2160 a Fj(j)p Fi(x)p Fj(j)1579 2136 y Fh(k)1875 2103 y Fj(\024)22 b Fi(C)2041 1990 y Fg(Z)2041 2228 y Fl(\012)2148 2047 y Fj(jr)p Fi(w)r Fj(j)2324 2017 y Fh(q)2376 2047 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2760 2017 y Fh(\013)p 2148 2084 660 4 v 2368 2160 a Fj(j)p Fi(x)p Fj(j)2461 2136 y Fh(k)q Ff(\000)p Fh(q)2817 2103 y Fi(:)465 b Fk(\(3.5\))523 2401 y(Let)25 b Fi(k)h Fk(=)d Fi(q)880 2333 y Fg(\000)928 2364 y Fh(n)p Ff(\000)p Fh(p)p 928 2382 128 4 v 975 2429 a Fl(2)1083 2401 y Fk(+)18 b(1)1208 2333 y Fg(\001)1271 2401 y Fk(and)25 b Fi(q)j Fk(b)r(e)e(c)n(hosen)e(so)g(that)h Fi(k)h(<)d(n)p Fk(,)i(that)g(is,)g(1)e Fi(<)g(q)j(<)3063 2368 y Fl(2)p Fh(n)p 2994 2382 212 4 v 2994 2429 a(n)p Ff(\000)p Fh(p)p Fl(+2)3215 2401 y Fk(.)g(Then,)523 2550 y(\(3.4\))h(and)h(\(3.5\))f (with)h Fi(\013)c Fk(=)e Fi(q)s(=)p Fk(2)p Fi(;)27 b(R)d Fj(\025)f Fi(R)1846 2562 y Fl(0)1924 2550 y Fk(imply)1171 2719 y Fg(Z)1170 2957 y Fl(\012)1268 2832 y Fj(jr)p Fk(\()p Fi(u)1440 2798 y Fh(p=)p Fl(2)1545 2832 y Fk(\))p Fj(j)1600 2798 y Fh(q)1660 2832 y Fk(=)g Fi(O)1827 2666 y Fg(0)1827 2815 y(@)1900 2719 y(Z)1899 2957 y Fl(\012)2007 2776 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2391 2746 y Fh(q)r(=)p Fl(2)2495 2776 y Fj(jr)p Fi(w)r Fj(j)2671 2746 y Fh(q)p 2007 2813 702 4 v 2204 2913 a Fj(j)p Fi(x)p Fj(j)2297 2884 y Fh(q)2340 2854 y Fd(\()p Fc(n)p Fb(\000)p Fc(p)p Fd(\))p 2340 2871 158 4 v 2405 2904 a(2)2719 2666 y Fg(1)2719 2815 y(A)2805 2832 y Fi(:)523 3132 y Fk(Let)28 b(1)22 b Fi(<)h(q)j(<)d Fk(2)p Fi(;)1086 3099 y Fh(n)p 1086 3113 42 4 v 1090 3160 a(q)1160 3132 y Fi(>)1258 3095 y Fl(2)p Ff(\000)p Fh(p)p 1258 3112 120 4 v 1259 3160 a Fl(2)p Ff(\000)p Fh(q)1401 3132 y Fi(:)28 b Fk(The)f(ab)r(o)n(v) n(e)f(inequalit)n(y)i(yields)725 3355 y Fg(Z)725 3593 y Fl(\012)822 3468 y Fj(jr)p Fk(\()p Fi(u)994 3433 y Fh(p=)p Fl(2)1100 3468 y Fk(\))p Fj(j)1155 3433 y Fh(q)1275 3468 y Fk(=)82 b Fi(O)1502 3276 y Fg(0)1502 3422 y(B)1502 3475 y(@)1574 3351 y(\022)1645 3412 y Fk(\(log)15 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))p Fj(jr)p Fi(w)r Fj(j)2206 3381 y Fl(2)p 1645 3449 599 4 v 1833 3525 a Fj(j)p Fi(x)p Fj(j)1926 3501 y Fh(n)p Ff(\000)p Fl(2)2254 3351 y Fg(\023)2315 3365 y Fh(q)r(=)p Fl(2)2433 3301 y Fg(0)2433 3450 y(@)2505 3355 y(Z)2505 3593 y Fl(\012)2602 3468 y Fj(j)p Fi(x)p Fj(j)2705 3401 y Fd(\()p Fc(p)p Fb(\000)p Fd(2\))p Fc(q)p 2706 3418 178 4 v 2744 3451 a Fd(2)p Fb(\000)p Fc(q)2898 3468 y Fi(dx)2988 3301 y Fg(1)2988 3450 y(A)3071 3290 y Fd(2)p Fb(\000)p Fc(q)p 3071 3305 103 4 v 3108 3338 a Fd(2)3187 3276 y Fg(1)3187 3422 y(C)3187 3475 y(A)1275 3850 y Fk(=)82 b Fi(O)1502 3658 y Fg(0)1502 3804 y(B)1502 3857 y(@)1574 3683 y(0)1574 3832 y(@)1647 3737 y(Z)1646 3975 y Fl(\012)1754 3793 y Fk(\(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))23 b Fj(jr)p Fi(w)r Fj(j)2337 3763 y Fl(2)p 1754 3831 622 4 v 1953 3907 a Fj(j)p Fi(x)p Fj(j)2046 3883 y Fh(n)p Ff(\000)p Fl(2)2386 3683 y Fg(1)2386 3832 y(A)2458 3700 y Fh(q)r(=)p Fl(2)2562 3658 y Fg(1)2562 3804 y(C)2562 3857 y(A)2648 3850 y Fi(:)634 b Fk(\(3.6\))523 4157 y(Com)n(bining)27 b(this)h(with)g(\(3.3\))f(w)n(e)h(obtain)f(the)h (follo)n(wing:)523 4389 y Fm(Theorem)37 b(3.1.)30 b Fk(Let)j(1)d Fi(<)g(p)g(<)g(n;)e Fk(1)i Fi(<)g(q)k(<)c Fk(2)p Fi(;)2155 4357 y Fh(n)p 2155 4371 42 4 v 2159 4418 a(q)2236 4389 y Fi(>)2341 4352 y Fl(2)p Ff(\000)p Fh(p)p 2341 4370 120 4 v 2342 4418 a Fl(2)p Ff(\000)p Fh(q)2484 4389 y Fi(;)j Fk(then)f(there)g(exists)g Fi(R)3247 4401 y Fl(0)3316 4389 y Fk(suc)n(h)523 4539 y(that)c(for)f(all)g Fi(R)d Fj(\025)f Fi(R)1183 4551 y Fl(0)1220 4539 y Fk(,)731 4658 y Fg(Z)731 4896 y Fl(\012)828 4771 y Fj(jr)p Fi(u)p Fj(j)991 4737 y Fh(p)1048 4771 y Fj(\000)1131 4654 y Fg(\022)1202 4715 y Fi(n)18 b Fj(\000)g Fi(p)p 1202 4752 194 4 v 1278 4828 a(p)1405 4654 y Fg(\023)1466 4671 y Fh(p)1528 4715 y Fj(j)p Fi(u)p Fj(j)1622 4685 y Fh(p)p 1528 4752 133 4 v 1528 4828 a Fj(j)p Fi(x)p Fj(j)1621 4804 y Fh(p)1689 4771 y Fj(\000)1944 4715 y Fi(C)2003 4727 y Fl(1)p 1782 4752 422 4 v 1782 4828 a Fk(\(log)c Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fk(\))2166 4804 y Fl(2)2223 4715 y Fj(j)p Fi(u)p Fj(j)2317 4685 y Fh(p)p 2223 4752 133 4 v 2223 4828 a Fj(j)p Fi(x)p Fj(j)2316 4804 y Fh(p)2388 4771 y Fj(\025)23 b Fi(C)34 b Fj(k)p Fi(u)2659 4737 y Fh(p=)p Fl(2)2763 4771 y Fj(k)2805 4737 y Fl(2)2805 4805 y Fh(W)2876 4777 y Fd(1)p Fc(;q)2867 4825 y Fd(0)2957 4805 y Fl(\(\012\))3074 4771 y Fi(;)208 b Fk(\(3.7\))1958 5150 y(20)p eop %%Page: 21 21 21 20 bop 523 614 a Fk(for)27 b(all)g Fi(u)c Fj(2)g Fi(W)1004 574 y Fl(1)p Fh(;p)992 637 y Fl(0)1096 614 y Fk(\(\012\).)763 847 y(Notice)37 b(that)g(when)g Fi(p)h Fk(=)g(2)p Fi(;)f Fk(\(3.6\))f(is)h(Zuazua-V\023)-42 b(azquez's)35 b(inequalit)n(y)-7 b(.)36 b(A)i(similar)523 996 y(inequalit)n(y)20 b(can)g(b)r(e)h (obtained)g(when)f Fi(p)j Fk(=)g Fi(n:)d Fk(F)-7 b(or)20 b(that)h(purp)r(ose)f(one)g(m)n(ust)h(use)f(the)h(function)523 1146 y(log)14 b(log)g Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)28 b Fk(instead)f(of)h(log)14 b Fi(R)q(=)p Fj(j)p Fi(x)p Fj(j)p Fi(:)523 1378 y Fm(3.)33 b Fk(Let)h(\()p Fi(M)t(;)14 b(g)s Fk(\))34 b(b)r(e)h(a)e(compact)g(Riemannian)h(manifold)g(of)g (dimension)g Fj(\025)f Fk(2)g(and)h(let)g(\001)3437 1390 y Fh(g)523 1528 y Fk(denote)k(the)g(Laplace-Beltrami)e(op)r(erator)g (asso)r(ciated)h(to)h Fi(g)i Fk(and)e Fi(G)p Fk(\()p Fi(x;)14 b(y)s Fk(\),)39 b(the)f(Green's)523 1677 y(function)33 b(corresp)r(onding)e(to)i(the)g(op)r(erator)e Fj(\000)p Fk(\001)2121 1689 y Fh(g)2181 1677 y Fk(+)22 b(1)p Fi(:)32 b Fk(Assume)h(moreo)n(v)n(er)d(that)j(0)e Fj(2)h Fi(M)523 1826 y Fk(and)25 b(de\014ne)f Fi(G)p Fk(\()p Fi(x)p Fk(\))h(:=)d Fi(G)p Fk(\(0)p Fi(;)14 b(x)p Fk(\))p Fi(:)26 b Fk(Then)f(w)n(e)f(ha)n (v)n(e)g(the)h(follo)n(wing)e(Hardy-Sob)r(olev)g(inequalit)n(y)-7 b(.)763 2059 y(If)26 b(1)d Fi(<)g(p)g(<)f(dim)h(M)32 b Fk(=)22 b Fi(n)k Fk(and)g Fi(k)j Fk(is)d(a)f(nonnegativ)n(e)g(in)n (teger,)g(there)h(exist)g(constan)n(ts)523 2208 y Fi(C)j(>)23 b Fk(0)41 b(and)g Fi(\025)24 b(<)e Fk(0)41 b(suc)n(h)28 b(that)g(for)f(all)g Fi(u)c Fj(2)g Fi(W)2048 2178 y Fl(1)p Fh(;p)2139 2208 y Fk(\()p Fi(M)9 b Fk(\),)619 2378 y Fg(Z)607 2616 y Fh(M)715 2491 y Fj(jr)p Fi(u)p Fj(j)878 2456 y Fh(p)935 2491 y Fj(\000)1018 2374 y Fg(\022)1142 2434 y Fi(n)19 b Fj(\000)f Fi(p)p 1089 2472 300 4 v 1089 2548 a(p)p Fk(\()p Fi(n)h Fj(\000)f Fk(2\))1398 2374 y Fg(\023)1459 2391 y Fh(p)1512 2370 y Fg(\014)1512 2420 y(\014)1512 2470 y(\014)1512 2520 y(\014)1549 2434 y Fj(r)p Fi(G)p 1549 2472 135 4 v 1584 2548 a(G)1694 2370 y Fg(\014)1694 2420 y(\014)1694 2470 y(\014)1694 2520 y(\014)1721 2391 y Fh(p)1760 2491 y Fj(j)p Fi(u)p Fj(j)1854 2456 y Fh(p)1975 2491 y Fj(\000)82 b Fi(C)2202 2324 y Fg(0)2202 2473 y(@)2316 2387 y Fh(k)2274 2412 y Fg(X)2277 2589 y Fh(j)s Fl(=1)2585 2434 y Fk(1)p 2418 2472 376 4 v 2418 2562 a(\(log)2557 2525 y Fl(\()p Fh(j)s Fl(\))2658 2562 y Fi(G)p Fk(\))2755 2538 y Fl(2)2803 2324 y Fg(1)2803 2473 y(A)2889 2370 y(\014)2889 2420 y(\014)2889 2470 y(\014)2889 2520 y(\014)2927 2434 y Fj(r)p Fi(G)p 2927 2472 135 4 v 2962 2548 a(G)3071 2370 y Fg(\014)3071 2420 y(\014)3071 2470 y(\014)3071 2520 y(\014)3099 2391 y Fh(p)3137 2491 y Fj(j)p Fi(u)p Fj(j)3231 2456 y Fh(p)3293 2491 y Fj(\025)2122 2770 y Fi(\025)2184 2657 y Fg(Z)2172 2895 y Fh(M)2282 2770 y Fj(j)p Fi(u)p Fj(j)2376 2736 y Fh(p)2427 2770 y Fi(;)60 b Fk(if)30 b Fi(p)23 b(<)g(n)647 3135 y Fg(Z)636 3373 y Fh(M)744 3248 y Fj(jr)p Fi(u)p Fj(j)907 3214 y Fh(n)971 3248 y Fj(\000)1054 3131 y Fg(\022)1125 3192 y Fi(n)18 b Fj(\000)g Fk(1)p 1125 3229 193 4 v 1196 3305 a Fi(n)1328 3131 y Fg(\023)1389 3148 y Fh(n)1448 3128 y Fg(\014)1448 3178 y(\014)1448 3227 y(\014)1448 3277 y(\014)1485 3192 y Fj(r)p Fi(G)p 1485 3229 135 4 v 1520 3305 a(G)1630 3128 y Fg(\014)1630 3178 y(\014)1630 3227 y(\014)1630 3277 y(\014)1657 3148 y Fh(n)1703 3248 y Fj(j)p Fi(u)p Fj(j)1797 3214 y Fh(n)1925 3248 y Fj(\000)82 b Fi(C)2151 3081 y Fg(0)2151 3231 y(@)2265 3144 y Fh(k)2224 3169 y Fg(X)2226 3346 y Fh(j)s Fl(=1)2538 3192 y Fk(1)p 2368 3229 383 4 v 2368 3320 a(\(log)2507 3283 y Fl(\()p Fh(j)s Fl(\))2608 3320 y Fi(G)p Fk(\))2705 3296 y Fh(n)2761 3081 y Fg(1)2761 3231 y(A)2847 3128 y(\014)2847 3178 y(\014)2847 3227 y(\014)2847 3277 y(\014)2885 3192 y Fj(r)p Fi(G)p 2885 3229 135 4 v 2919 3305 a(G)3029 3128 y Fg(\014)3029 3178 y(\014)3029 3227 y(\014)3029 3277 y(\014)3057 3148 y Fh(n)3102 3248 y Fj(j)p Fi(u)p Fj(j)3196 3214 y Fh(n)3264 3248 y Fj(\025)2072 3528 y Fi(\025)2134 3415 y Fg(Z)2122 3653 y Fh(M)2231 3528 y Fj(j)p Fi(u)p Fj(j)2325 3493 y Fh(n)2384 3528 y Fi(;)60 b Fk(if)30 b Fi(p)23 b Fk(=)f Fi(n:)523 3893 y Fk(If)35 b Fi(M)43 b Fk(has)34 b(a)h(b)r(oundary)-7 b(,)33 b(then)j(a)e(b)r(oundary)g(in)n (tegral)f(app)r(ears)g(on)i(the)g(righ)n(t)e(hand)i(side)523 4042 y(of)25 b(the)g(ab)r(o)n(v)n(e)e(inequalit)n(y)-7 b(.)24 b(F)-7 b(urthermore)24 b(the)h(corresp)r(onding)e(p)r(erturb)r (ed)i(Euler-Lagrange)523 4192 y(equation)h(can)g(b)r(e)h(studied.)g (The)g(condition)f(to)h(b)r(e)g(put)g(on)f Fi(q)k Fk(in)d(order)e(to)h (pro)n(v)n(e)f(existence)523 4341 y(and)37 b(non)g(existence)g(results) f(remains)g(same)h(as)f(that)i(used)f(in)g(the)h(case)e(of)h(a)f(b)r (ounded)523 4491 y(domain)27 b(in)h Fi(I)-21 b(R)994 4455 y Fh(n)1039 4491 y Fi(:)763 4723 y Fm(Ac)m(kno)m(wledgemen)m(t.)22 b Fk(The)g(\014rst)h(author)e(w)n(ould)h(lik)n(e)g(to)g(thank)g (Olivier)g(Druet)g(for)523 4873 y(p)r(oin)n(ting)28 b(out)f(the)h (extension)f(to)h(compact)f(manifolds.)1958 5150 y(21)p eop %%Page: 22 22 22 21 bop 763 614 a Fk(Both)24 b(authors)f(ac)n(kno)n(wledge)f(funding) j(from)e(the)i(Indo-F)-7 b(renc)n(h)23 b(Cen)n(tre)h(for)f(promo-)523 764 y(tion)28 b(of)f(Adv)-5 b(anced)28 b(Researc)n(h,)e(under)i(pro)5 b(ject)27 b(1901-02.)763 996 y Fm(References)601 1295 y Fk([1])42 b(Adim)n(urthi,)30 b(Hardy)g(Sob)r(olev)g(inequalit)n(y)g (in)g Fi(H)2294 1265 y Fl(1)2331 1295 y Fk(\(\012\))h(and)f(its)h (applications,)e(to)h(ap-)731 1445 y(p)r(ear)d(in)g(Comm)n(unications)g (in)h(con)n(temp)r(orary)e(mathematics.)601 1660 y([2])42 b(Adim)n(urthi,)23 b(M.)g(Ramasw)n(am)n(y)e(and)h(N.)i(Chaudh)n(uri,)e (An)h(impro)n(v)n(ed)f(Hardy-Sob)r(olev)731 1810 y(inequalit)n(y)27 b(and)g(its)h(applications,)f Fa(Pr)l(o)l(c.)j(AMS,)58 b Fk(2001,)26 b(1-17.)601 2026 y([3])42 b(Adim)n(urthi)29 b(and)f(K.)h(Sandeep,)g(Existence)f(and)g(Non)h(existence)g(of)f (\014rst)h(eigen)n(v)-5 b(alue)731 2175 y(fo)30 b(p)r(erturb)r(ed)i (Hardy-Sob)r(olev)d(op)r(erator,)g(to)i(app)r(ear)f(in)h(Pro)r(c.)e(Ro) n(y)n(ar)g(So)r(c.)i(Edin,)731 2325 y(\(2001\).)601 2540 y([4])42 b(G.)29 b(Barbatis,)e(S.)j(Filipp)r(os)f(and)g(A.)g(T)-7 b(ertik)i(as,)28 b(Series)h(expansion)f(for)g Fi(L)3058 2510 y Fh(p)3096 2540 y Fk(-Hardy)g(in-)731 2690 y(equalities.)f (Preprin)n(t)f(2001.)601 2906 y([5])42 b(G.)35 b(Barbatis,)f(S.)h (Filippas)g(and)g(A.)h(T)-7 b(ertik)i(as,)35 b(A)g(uni\014ed)h(approac) n(h)d(to)i(impro)n(v)n(ed)731 3055 y Fi(L)788 3025 y Fh(p)825 3055 y Fk(-Hardy)27 b(inequalities)g(wih)h(b)r(est)g(constan)n (ts,)f(Preprin)n(t)f(2001.)601 3271 y([6])42 b(H.)19 b(Brezis,)f(M.)i(Marcus)e(and)h(I.)g(Shafrir,)g(External)f(functions)h (for)g(Hardy's)f(inequalit)n(y)731 3421 y(with)28 b(w)n(eigh)n(t,)f (Journal)f(of)h(F)-7 b(unct.)29 b(Anal.)f Fm(171,)e Fk(\(2000\),)g (177-)g(191.)601 3636 y([7])42 b(H.)25 b(Brezis)f(and)g(J.)h(L.)g (V\023)-42 b(azquez,)24 b(Blo)n(w)g(up)i(of)e(solutions)g(of)h(some)g (non)f(linear)g(elliptic)731 3786 y(problems,)i(pub.)j(lab.)e (D'Analyse)h(Num,)g(R.)g(96018,)d(\(1996\))601 4002 y([8])42 b(F.)32 b(Gazzola,)f(H.)i(Grunau)f(and)g(E.)g(Mitidieri,)g(Hardy)g (inequalities)g(with)g(reminder)731 4151 y(term,)27 b(Preprin)n(t)g (2001.)601 4367 y([9])42 b(E.)25 b(H.)i(Lieb)f(and)g(H.-T.)g(Y)-7 b(au,)27 b(The)f(Stabilit)n(y)g(and)g(Instabilit)n(y)g(of)g (Relativistic)g(Mat-)731 4516 y(ter,)h(Comm)n(un.)h(Math.)f(Ph)n(ys.)g (118)f(\(1988\))g(177-213.)560 4732 y([10])41 b(K.)28 b(Sandeep,)i(On)e(the)i(\014rst)f(eigenfunction)g(of)g(a)g(p)r(erturb)r (ed)g(Hardy-Sob)r(olev)f(op)r(er-)731 4882 y(ator,)e(Preprin)n(t)h (2001.)1958 5150 y(22)p eop %%Page: 23 23 23 22 bop 560 614 a Fk([11])41 b(J.)d(L.)h(V\023)-42 b(azquez)38 b(and)g(E.)g(Zuazua,)g(The)h(Hardy)f(inequalit)n(y)g(and)g (the)h(asymptotic)731 764 y(b)r(eha)n(viour)32 b(of)h(the)h(heat)g (equation)f(with)h(an)f(in)n(v)n(erse-square)d(p)r(oten)n(tial,)k(J.)f (F)-7 b(unct.)731 913 y(Anal.,)27 b Fm(173,)g Fk(no.1,)g(\(2000\))f (103-153.)1958 5150 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0301170832231--