Content-Type: multipart/mixed; boundary="-------------0411031049801" This is a multi-part message in MIME format. ---------------0411031049801 Content-Type: text/plain; name="04-355.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-355.keywords" essential specrum, compactness criteria, Banach modules, Riemannian manifolds, Lipschitz manifolds, locally compact abelian groups, Dirac operators, decay preserving operators, ---------------0411031049801 Content-Type: application/postscript; name="dpo.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dpo.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: /myhome/science/papers/golenia/quasilocal/final/dpo/dpo.dvi %%Pages: 54 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Bold MSBM10 CMMI7 Times-Italic CMMI10 %%+ rsfs10 CMSY10 CMR10 CMSY8 CMR8 CMMI8 rsfs7 CMMI9 CMSY9 CMMI6 CMR9 %%+ CMEX10 CMMI12 CMSY6 CMR6 TeX-cmex8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -z %+ /myhome/science/papers/golenia/quasilocal/final/dpo/dpo.dvi -o %+ /myhome/science/papers/golenia/quasilocal/final/dpo/dpo.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.11.03:1714 %%BeginProcSet: texc.pro %! 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/FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 102 /f put readonly def /FontBBox {-29 -2957 1554 772} readonly def /UniqueID 4314416 def currentdict end currentfile eexec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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 0 /.notdef 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 C5FD550F53CE5583743B50B0F9B7C4F836DEF30FA8174C963E076E79FDC8BCFD B4200A34BB7F70036CDA62BC4315CA986216DDA60103A7878028EF310DD1168F 7500EC86258E8878D5251D65E97F4A7294A67A9FFFF6CBD79ECD609CB2809D85 4A6E1DC9230EC93C75FE0C8BA65806CF5E8A484BBB6F3A6B85A9BE84D7A589FD 1BDAF326CB000E9055739E06A7F333763CD85CBEBC6FC1B744207F9EB3EB1E61 E79D7AE4A771F28347C997D157866CE072FD968855DB1F5D267BB44B556DBB27 E578018E51672FF1F5492E0389DE055591745A0C965CF3E316D39E39731C883B D6164CBC18E366E132921581E5D1490E33B0A74B6069D02A90D08CC2E2EB8D95 7A214EA72682E98A3596DF730A93278820B2DC46E0DB164397A9EF81B462CBB2 94CC558461BE6F03408175195CFDAD7B2D8AEB114593E45B4B474D4EE395ECEF EE4716481F7DE57E2B5CAEE8D81A01800C801F8737584BABE4FF6742B51A6DFF B126BF14E2FDE0FB031D0A66A48F2A869610C235C8F39D2F73A02000FD042388 3320DDFC50A8804208FB2789DC5BC39B3E4273F6CB3C19ADB9E7D001D79B54D9 A41AD680F305CDDB71D7EEC67CA8A74DE8FD49FCBCC029CC07C49534505D96D9 B2F396D3F4EC5E4D180EB64536136267A90A9B5DFA13743DFE3DDADB3C8627CD 090476A8531BF0992BEF889165A18CB51E73A3870458ECC347B09B0A16497399 9E9B664D572A6BBC5AC465CC437A4C38B9CB6A6D355EE7058CEF248CAD2F9C9B B210746D3BB49CD90DFB1E0F05CFA2ED2B46E124DD48AACEA24EF498A1B4808A ABAD9FBBE90CC8A71CF9919C3CEFB8F2721E224957B4D39C5AE4B07103898917 44D4FEF89FF85616C430DB45332F0AD3E9244D0C54B5C59642D147506663E192 023631EF70CEDC95FF50DC107A30C0BA7B588331680AB0708B390205A41B1420 BBEE5D59E6CAA7E08F762FF22EE5C74074893F0508C0B0ACB7F545D995CF0001 E962E00EDBFF1EFC227C50A7B21B230008B5F971039D5B021C66FC4A1D708A99 6939CCD80A67AA69EC1A4B8F19A93D1400B224364F6F60BA9BC66E83311DCAE8 212F29E4512C3C827243E1EEE93285FEDF17E446D76B1FB61863B9F39666FFE1 B6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DCC57F7903449E07914FBE9E67 3C15C2153C061EB541F66C11E7EE77D5D77C0B11E1AC55101DA976CCACAB6993 EED1406FBB7FF30EAC9E90B90B2AF4EC7C273CA32F11A5C1426FF641B4A2FB2F 4E68635C93DB835737567FAF8471CBC05078DCD4E40E25A2F4E5AF46C234CF59 2A1CE8F39E1BA1B2A594355637E474167EAD4D97D51AF0A899B44387E1FD933A 323AFDA6BA740534A510B4705C0A15647AFBF3E53A82BF320DD96753639BE49C 2F79A1988863EF977B800C9DB5B42039C23EB86953713F730E03EA22FF7BB2C1 D97D33FD77B1BDCC2A60B12CF7805CFC90C5B914C0F30A673DF9587F93E47CEA 5932DD1930560C4F0D97547BCD805D6D854455B13A4D7382A22F562D7C55041F 0FD294BDAA1834820F894265A667E5C97D95FF152531EF97258F56374502865D A1E7C0C5FB7C6FB7D3C43FEB3431095A59FBF6F61CEC6D6DEE09F4EB0FD70D77 2A8B0A4984C6120293F6B947944BE23259F6EB64303D627353163B6505FC8A60 00681F7A3968B6CBB49E0420A691258F5E7B07B417157803FCBE9B9FB1F80FD8 CA0DA1186446DD565542BCCC7D339A1EB34C7F49246E8D72E987EB477C6DB757 99AF86CEBCD7605C487A00CD2CD093098182DC57B20D78ECE0BECF3A0BF88EBA C866DB19F34BBBED6634AFC0F08D2AFB2A92578A6F8B4ADCD6594737FF6EED7D 5B536DA9E3E2CADB40DB7C600EA4D100D33C3B92B1CF857E012C4EB370BA8295 55B50047CC8911C98FE1A7BA6CDEA82D34476286E7105F9286496EB9C6EA7FE8 F7977CCECFE788D1D5C22A36E7C23C478FB3547B57879BA0938E99DEEE75BC38 640D905C5E15FD0772B35EA556606D4F9F71D2EDAFC11020AB5C51F6C356D67F 39017621E36A45444220C29DFB24C94D6265CFA507B702EE7B661760C2C3E837 9711622B770C5713765DC9CEB57DF46B12F32892A791CA5198169220C88E364A 9FD9F56CEFB789E190293397BB78172DB48C7169A6D810D706A297391DEB6FA7 D518015739E310943395455FED78193578BC56C867D6249E93183377A5CA9CAC C5E1E43C9A5A6162EA49944D5515B767699B961A86AB75FAE2DFBA75D8C39D89 7C9EE5003A3CF212D6AFA8BD9F901D6127C894BDE30A21BA767F16D1D1D037AB BC507FDDE4C50713D1F252E4A74672C931C8F6B2ED65C57DDE401E122817C9DC 4103A87E91BB298C07B55B6C2A1AEBF626E28283166E89D17B979C3704E53E57 B0F800F562DDC1E8EB23121A4EFCA1BA08D3232F155F15A3C9E2527B004A6098 AEB44E2830AE726B6D724C7CE32C6697395F6C2E29E8D7919E93B9A1C9E17BF8 6E7D7FE3F2F7C20771DA823896F141733BFAC034920D3B6F3FFD44CD296FF1A1 59FD3A6A543795BB3AB0EF517E58ACF8300938CA4010DC5AB5197E4B9448 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 26 /braceleftbigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 99 /hatwider put dup 101 /tildewide put dup 102 /tildewider put dup 112 /radicalbig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: rsfs7 %!PS-AdobeFont-1.0: rsfs7 001.000 %%CreationDate: Sat Mar 21 18:45:46 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion of metafont curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs7) readonly def /FamilyName (rsfs7) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /G put dup 72 /H put dup 75 /K put dup 76 /L put dup 77 /M put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {16 -302 1349 728} readonly def currentdict end currentfile eexec D9D66F633B846A989B9974B0179FC6CC445BCF7C3C3333173232E3FDBFF43949 1DB866C39088C203DC22FDC758584860EC7BB67FDA28CC6208249060E18FAB32 204779B5C03C0493BBBBC95CF02692CC4DEAA8D2EA90B5C2E64374E92BCB8501 429B8FAE4A76C0C6B76D6FF7CF9A7D5EDFBCA0E959541C59BD05B7DE43D25D53 FC3DDA6EF0C2743978A6D03E19CCED4A11F2EA4BCC3110BE8B8D9E2772361969 C19258EFAFDC276CB1ADE9208A941A36D18F9FB1C33DEF76AA315DD8F2826A55 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 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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. 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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 cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. 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Fo(\013)1930 4236 y Ft(k)p Fv(J)2025 4250 y Fo(\013)2075 4236 y Fv(M)33 b Ft(\000)22 b Fv(M)10 b Ft(k)32 b Fs(=)e(lim)2692 4250 y Fo(\013)2756 4236 y Ft(k)p Fv(M)10 b(J)2949 4250 y Fo(\013)3022 4236 y Ft(\000)22 b Fv(M)10 b Ft(k)32 b Fs(=)f(0)448 4348 y Fy(for)22 b(all)f Fv(M)36 b Ft(2)25 b(M)p Fy(.)i(An)20 b(approximate)k(unit)e(e)o(xists)h(if)e(and)h(only)g (if)f(there)h(is)f(a)g(number)h Fv(C)27 b Fy(such)448 4461 y(that)e(for)g(each)h Fv(")h(>)g Fs(0)d Fy(and)h(for)g(each)g (\002nite)g(set)g Ft(F)36 b(\032)27 b(M)d Fy(there)i(is)e Fv(J)36 b Ft(2)27 b(M)d Fy(with)h Ft(k)p Fv(J)9 b Ft(k)28 b(\024)f Fv(C)448 4574 y Fy(and)f Ft(k)p Fv(J)9 b(M)32 b Ft(\000)22 b Fv(M)10 b Ft(k)29 b(\024)f Fv(")p Fy(,)d Ft(k)p Fv(M)10 b(J)32 b Ft(\000)21 b Fv(M)10 b Ft(k)29 b(\024)f Fv(")d Fy(for)h(all)f Fv(M)39 b Ft(2)28 b(F)9 b Fy(.)34 b(It)25 b(is)g(well)h(kno)n(wn)f(that)h(an)o(y)448 4687 y Fv(C)520 4654 y Fr(\003)559 4687 y Fy(-algebra)33 b(has)f(an)f(approximate)i(unit.)52 b(If)31 b Fu(H)56 b Fy(is)31 b(a)g(Banach)h(space,)h(we)d(shall)i(say)g(that)448 4800 y(a)26 b(Banach)h(subalgebra)i Ft(M)d Fy(of)g Ft(B)s Fs(\()p Fu(H)g Fs(\))g Fy(is)g Fw(non-de)l(g)o(ener)o(ate)31 b Fy(if)26 b(the)g(linear)i(subspace)g(of)e Fu(H)448 4913 y Fy(generated)g(by)e(the)g(elements)h Fv(M)10 b(u)p Fy(,)22 b(with)i Fv(M)35 b Ft(2)25 b(M)e Fy(and)h Fv(u)h Ft(2)g Fu(H)i Fy(,)22 b(is)h(dense)i(in)f Fu(H)i Fy(.)1920 5225 y(6)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 589 573 a Fy(In)29 b(vie)n(w)e(of)i(its)f(importance)i(in)e(our)h(paper)l(,)h(we)e(state)h (belo)n(w)f(the)h(Cohen-He)n(witt)g(f)o(ac-)448 686 y(torization)d (theorem)f([FD,)d(Ch.)h(V)-9 b(\2269.2].)448 873 y Fx(Theor)n(em)24 b(2.1)46 b Fw(Let)34 b Ft(C)39 b Fw(be)34 b(a)g(Banac)o(h)h(alg)o(ebr)o (a)h(with)f(an)f(appr)l(oximate)j(unit,)h(let)c Fu(E)51 b Fw(be)35 b(a)448 986 y(Banac)o(h)28 b(space)o(,)h(and)f(let)f Fv(Q)32 b Fs(:)h Ft(C)k(!)32 b(B)s Fs(\()p Fu(E)17 b Fs(\))27 b Fw(be)g(a)g(continuous)j(morphism.)41 b(Denote)28 b Fu(E)3260 1000 y Fq(0)3326 986 y Fw(the)448 1099 y(closed)36 b(linear)f(subspace)i(of)c Fu(E)51 b Fw(g)o(ener)o(ated)36 b(by)e(the)h(elements)g(of)f(the)g(form)g Fv(Q)p Fs(\()p Fv(')p Fs(\))p Fv(v)j Fw(with)448 1212 y Fv(')f Ft(2)e(C)f Fw(and)c Fv(v)38 b Ft(2)c Fu(E)18 b Fw(.)43 b(Then)29 b(for)g(eac)o(h)g Fv(u)34 b Ft(2)h Fu(E)1970 1226 y Fq(0)2037 1212 y Fw(ther)m(e)30 b(ar)m(e)e Fv(')36 b Ft(2)e(C)f Fw(and)c Fv(v)38 b Ft(2)c Fu(E)46 b Fw(suc)o(h)29 b(that)448 1325 y Fv(u)d Fs(=)f Fv(Q)p Fs(\()p Fv(')p Fs(\))p Fv(v)s Fw(.)589 1513 y Fy(No)n(w)e(we)f(introduce)k(the)e(frame)n(w)o(ork)h (in)e(which)h(we)f(shall)h(w)o(ork.)448 1700 y Fx(De\002nition)f(2.2)46 b Fw(A)37 b Fy(Banach)j(module)f Fw(is)f(a)g(couple)i Fs(\()p Fu(H)27 b Fv(;)15 b Ft(M)p Fs(\))39 b Fw(consisting)i(of)e(a)f (Banac)o(h)448 1813 y(space)c Fu(H)57 b Fw(and)33 b(a)f(non-de)l(g)o (ener)o(ate)37 b(Banac)o(h)32 b(subalg)o(ebr)o(a)j Ft(M)d Fw(of)g Ft(B)s Fs(\()p Fu(H)26 b Fs(\))32 b Fw(whic)o(h)g(has)g(an)448 1926 y(appr)l(oximate)26 b(unit.)j(If)22 b Fu(H)49 b Fw(is)23 b(a)f(Hilbert)i(space)g(and)f Ft(M)f Fw(is)h(a)f Fv(C)2508 1893 y Fr(\003)2547 1926 y Fw(-alg)o(ebr)o(a)j(of)e(oper)o (ator)o(s)i(on)448 2039 y Fu(H)i Fw(,)22 b(we)h(say)h(that)g Fu(H)50 b Fw(is)23 b(a)g Fy(Hilbert)h(module)p Fw(.)448 2227 y Fy(W)-7 b(e)30 b(shall)i(adopt)g(the)f(usual)h Fw(ab)n(us)g(de)f(langua)o(g)o(e)i Fy(and)f(say)f(that)g Fu(H)57 b Fy(is)30 b(a)h(Banach)g(module)448 2340 y(\(o)o(v)o(er)21 b Ft(M)p Fy(\).)27 b(The)19 b(distinguished)25 b(subalgebra)e Ft(M)c Fy(will)h(be)g(called)h Fw(multiplier)h(alg)o(ebr)o(a)g(of)d Fu(H)448 2452 y Fy(and,)30 b(when)e(required)i(by)f(the)f(clarity)i(of) e(the)g(presentation,)33 b(we)27 b(shall)i(denote)h(it)e Ft(M)p Fs(\()p Fu(H)f Fs(\))p Fy(.)448 2565 y(W)-7 b(e)32 b(are)h(only)g(interested)j(in)c(the)h(case)g(when)g Ft(M)f Fy(does)h(not)g(ha)n(v)o(e)h(a)e(unit:)48 b(the)33 b(operators)448 2678 y(from)25 b Ft(M)g 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Fy(and)f Fv(u)39 b Ft(2)g Fu(H)26 b Fy(,)32 b(is)f(dense)h(in)f Fu(H)c Fy(.)50 b(If)30 b Fu(H)57 b Fy(is)31 b(a)g(Hilbert)448 1137 y(space)24 b(and)f Fv(Q)f Fy(is)h(a)f Ft(\003)p Fy(-morphism,)j(we)d(say)h(that)g Fu(H)49 b Fy(is)22 b(a)h Fw(Hilbert)g Fv(X)7 b Fw(-module)p Fy(.)30 b(W)-7 b(e)22 b(shall)i(use)448 1250 y(the)h(notation)h Fv(')p Fs(\()p Fv(Q)p Fs(\))i Ft(\021)e Fv(Q)p Fs(\()p Fv(')p Fs(\))p Fy(.)31 b(The)24 b(Banach)h(module)g(structure)h(on)f Fu(H)50 b Fy(is)24 b(de\002ned)h(by)f(the)448 1363 y(closure)31 b Ft(M)e Fy(in)g Ft(B)s Fs(\()p Fu(H)d Fs(\))j Fy(of)g(the)g(set)h(of)f (operators)i(of)f(the)f(form)g Fv(')p Fs(\()p Fv(Q)p Fs(\))g Fy(with)g Fv(')37 b Ft(2)e Fv(C)3222 1377 y Fq(0)3261 1363 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)448 1476 y(In)28 b(the)g(case)h(of)e(a)h(Hilbert)g Fv(X)7 b Fy(-module)30 b(the)e(closure)h(is)f(not)g(needed)i(and)e(we)f(get)h(a)g(Hilbert)448 1589 y(module)i(structure)h(\(because)g(a)e Ft(\003)p Fy(-morphism)i(between)f(tw)o(o)e Fv(C)2549 1556 y Fr(\003)2588 1589 y Fy(-algebras)j(is)e(continuous)448 1702 y(and)d(its)f(range)h (is)f(a)g Fv(C)1162 1669 y Fr(\003)1201 1702 y Fy(-algebra\).)35 b(Banach)26 b Fv(X)7 b Fy(-modules)27 b(appear)g(naturally)g(in)e(dif)n (ferential)448 1815 y(geometry)g(as)f(spaces)h(of)f(sections)h(of)f(v)o (ector)h(\002ber)e(b)n(undles)j(o)o(v)o(er)d(a)h(manifold)h Fv(X)7 b Fy(,)23 b(and)h(this)448 1928 y(is)g(the)f(point)i(of)f (interest)h(for)f(us.)448 2140 y Fx(Remark)f(2.5)46 b Fy(In)26 b(the)h(case)g(of)f(a)g(Banach)h Fv(X)7 b Fy(-module,)28 b(Lemma)d(2.3)h(gi)n(v)o(es:)35 b(each)27 b Fv(u)k Ft(2)e Fu(H)448 2253 y Fy(can)d(be)g(written)g(as)g Fv(u)j Fs(=)g Fv( )s Fs(\()p Fv(Q)p Fs(\))p Fv(v)g Fy(with)d Fv( )32 b Ft(2)d Fv(C)1984 2267 y Fq(0)2024 2253 y Fs(\()p Fv(X)7 b Fs(\))26 b Fy(and)g Fv(v)32 b Ft(2)d Fu(H)e Fy(.)34 b(In)25 b(particular)l(,)k(we)c(de-)448 2366 y(duce)e(that)f Fw(the)g(morphism)g Fv(Q)f Fw(has)h(an)f(e)n(xtension,)k(also)d (denoted)i Fv(Q)p Fw(,)c(to)i(a)f(unital)i(continuous)448 2479 y(morphism)33 b(of)e Fv(C)1009 2494 y Fq(b)1053 2479 y Fs(\()p Fv(X)7 b Fs(\))32 b Fw(into)g Ft(B)s Fs(\()p Fu(H)27 b Fs(\))k Fy(which)h(is)f(uniquely)j(determined)g(by)e(the)g (follo)n(wing)448 2592 y(strong)37 b(continuity)g(property:)54 b(if)35 b Ft(f)p Fv(')1690 2606 y Fo(n)1738 2592 y Ft(g)f Fy(is)h(a)f(bounded)j(sequence)g(in)e Fv(C)2870 2607 y Fq(b)2913 2592 y Fs(\()p Fv(X)7 b Fs(\))35 b Fy(such)h(that)448 2705 y Fv(')507 2719 y Fo(n)584 2705 y Ft(!)29 b Fv(')c Fy(locally)j(uniformly)-6 b(,)27 b(then)g Fv(')1707 2719 y Fo(n)1754 2705 y Fs(\()p Fv(Q)p Fs(\))j Ft(!)f Fv(')p Fs(\()p Fv(Q)p Fs(\))d Fy(strongly)i(on)e Fu(H)g Fy(.)35 b(Indeed,)27 b(we)e(can)448 2818 y(de\002ne)e Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(u)j Fs(=)f(\()p Fv(' )s Fs(\)\()p Fv(Q)p Fs(\))p Fv(v)i Fy(for)22 b(each)g Fv(')k Ft(2)f Fv(C)2021 2833 y Fq(b)2064 2818 y Fs(\()p Fv(X)7 b Fs(\))p Fy(;)23 b(then)g(if)f Fv(e)2561 2832 y Fo(\013)2631 2818 y Fy(is)g(an)g(approximate)j(unit)448 2931 y(for)g Fv(C)643 2945 y Fq(0)683 2931 y Fs(\()p Fv(X)7 b Fs(\))25 b Fy(with)f Ft(k)p Fv(e)1132 2945 y Fo(\013)1182 2931 y Ft(k)k(\024)f Fs(1)d Fy(we)g(get)h Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(u)k Fs(=)e(lim\()p Fv('e)2329 2945 y Fo(\013)2380 2931 y Fs(\)\()p Fv(Q)p Fs(\))p Fv(u)d Fy(hence)i(the)f(de\002nition)h(is)448 3044 y(independent)h(of)d(the)g(f)o(actorization)j(of)c Fv(u)g Fy(and)h Ft(k)p Fv(')p Fs(\()p Fv(Q)p Fs(\))p Ft(k)j(\024)e(k)p Fv(Q)p Ft(k)15 b Fs(sup)g Ft(j)p Fv(')p Ft(j)p Fy(.)448 3256 y Fx(Remark)23 b(2.6)46 b Fy(If)25 b Fu(H)50 b Fy(is)25 b(a)f(Hilbert)i Fv(X)7 b Fy(-module)26 b(one)g(can)f(e)o(xtend)h(the)f(morphism)h(e)n(v)o(en)f(fur)n(-)448 3369 y(ther:)k Fv(Q)19 b Fw(canonically)24 b(e)n(xtends)f(to)d(a)g Ft(\003)p Fw(-morphism)j Fv(')i Ft(7!)g Fv(')p Fs(\()p Fv(Q)p Fs(\))c Fw(of)g Fv(B)5 b Fs(\()p Fv(X)i Fs(\))20 b Fw(into)h Ft(B)s Fs(\()p Fu(H)27 b Fs(\))20 b Fw(suc)o(h)448 3482 y(that)588 3449 y Fn(2)627 3482 y Fw(:)29 b(if)23 b Ft(f)p Fv(')863 3496 y Fo(n)911 3482 y Ft(g)h Fw(is)f(a)h(bounded)i (sequence)g(in)e Fv(B)5 b Fs(\()p Fv(X)i Fs(\))23 b Fw(and)h Fs(lim)2453 3496 y Fo(n)p Fr(!1)2656 3482 y Fv(')2715 3496 y Fo(n)2762 3482 y Fs(\()p Fv(x)p Fs(\))j(=)e Fv(')p Fs(\()p Fv(x)p Fs(\))f Fw(for)g(all)448 3595 y Fv(x)i Ft(2)e Fv(X)7 b Fw(,)22 b(then)g Fy(s-)q Fs(lim)1107 3609 y Fo(n)1169 3595 y Fv(')1228 3609 y Fo(n)1276 3595 y Fs(\()p Fv(Q)p Fs(\))j(=)g Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fw(.)k Fy(This)21 b(follo)n(ws)h(from)g(standard)i(inte)o(gration)g (theory)448 3708 y(see)33 b([Be,)e(Lo].)55 b(In)32 b(particular)l(,)37 b(a)32 b(separable)j(Hilbert)e Fv(X)7 b Fy(-module)34 b(is)e(essentially)j(a)d(direct)448 3821 y(inte)o(gral)25 b(of)f(Hilbert)g(spaces)h(o)o(v)o(er)f Fv(X)7 b Fy(,)23 b(see)g([Di)q(,)f(II.6.2],)i(b)n(ut)g(we)e(shall)j(not)f(need)g(this)g (f)o(act.)589 4033 y(The)j(class)h(of)g Fv(X)7 b Fy(-modules)29 b(is)e(more)g(general)i(than)f(it)f(appears)i(at)e(\002rst)g(sight.)41 b(Indeed,)448 4146 y(if)29 b Ft(C)k Fy(is)c(an)g(abelian)i Fv(C)1179 4113 y Fr(\003)1217 4146 y Fy(-algebra)g(then)f(one)f(has)h (a)e(canonical)k(identi\002cation)g Ft(C)40 b(\021)35 b Fv(C)3249 4160 y Fq(0)3288 4146 y Fs(\()p Ft(X)13 b Fs(\))448 4259 y Fy(where)27 b Ft(X)39 b Fy(is)27 b(the)g(spectrum)h (of)f Ft(C)5 b Fy(.)37 b(Ho)n(we)n(v)o(er)l(,)27 b(the)g(space)h Ft(X)39 b Fy(is)26 b(in)h(general)h(rather)g(compli-)448 4372 y(cated)e(so)g(it)e(is)h(not)h(really)g(useful)h(to)e(tak)o(e)g (it)g(into)h(account.)35 b(In)25 b(particular)l(,)k(this)c(happens)i (in)448 4485 y(the)d(follo)n(wing)h(class)g(of)e(e)o(xamples)i(of)e (interest)j(in)d(applications)k(\(see)d(Section)h(7\).)p 448 4666 1196 4 v 554 4721 a Fm(2)583 4753 y Fl(If)k Fi(X)36 b Fl(is)30 b(second)h(countable)g(then)f(this)g(property)h (determines)f(uniquely)i(the)e(e)o(xtension.)57 b(In)30 b(general,)448 4844 y(uniqueness)22 b(is)e(assured)h(by)g(the)f (property:)27 b(if)20 b Fi(U)32 b Fh(\032)23 b Fi(X)k Fl(is)20 b(open)h(then)2326 4829 y Fi(\037)2374 4852 y Fg(U)2426 4844 y Ff(\()p Fi(Q)p Ff(\))i(=)g(sup)2769 4863 y Fg(')2826 4844 y Fi(')p Ff(\()p Fi(Q)p Ff(\))p Fl(,)d(where)g Fi(')g Fl(runs)448 4936 y(o)o(v)o(er)g(the)f(set)f(of)h (continuous)i(functions)f(with)e(compact)i(support)g(such)f(that)g Ff(0)j Fh(\024)f Fi(')g Fh(\024)2779 4920 y Fi(\037)2827 4944 y Fg(U)2879 4936 y Fl(.)1920 5225 y Fy(8)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 448 573 a Fx(Example)23 b(2.7)47 b Fy(Let)22 b Fv(X)30 b Fy(be)23 b(a)g(set)g(and)h Ft(F)32 b Fy(a)23 b(\002lter)g(on)g Fv(X)7 b Fy(.)28 b(Let)23 b(us)g(choose)i(a)e Fv(C)2927 540 y Fr(\003)2966 573 y Fy(-algebra)i Ft(C)i Fy(of)448 686 y(bounded)f(comple)o(x)e (functions)i(on)e Fv(X)30 b Fy(\(with)23 b(the)h(sup)g(norm\))f(and)h (then)g(let)g Ft(C)2934 700 y Fq(0)2996 686 y Fy(be)f(the)h(set)f(of) 448 799 y Fv(')j Ft(2)f(C)i Fy(such)d(that)g Fs(lim)1168 813 y Fr(F)1244 799 y Fv(')i Fs(=)f(0)p Fy(.)j(Then)23 b Ft(C)1778 813 y Fq(0)1840 799 y Fy(is)g(a)f 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Fx(Example)f(2.8)47 b Fy(Let)23 b Fu(H)49 b Fy(be)24 b(a)f(Hilbert)h Fv(X)7 b Fy(-module)26 b(o)o(v)o(er)d(a)g(locally)j(compact)e(non-compact)448 1996 y(topological)35 b(space)e Fv(X)38 b Fy(and)32 b(let)f Ft(F)40 b Fy(be)32 b(a)f(\002lter)g(on)h Fv(X)38 b Fy(\002ner)31 b(than)i(the)e(Fr)5 b(\264)-35 b(echet)33 b(\002lter)-5 b(.)52 b(W)-7 b(e)448 2108 y(e)o(xtend)32 b(the)f(morphism)h Fv(Q)e Fy(to)h(all)f(of)h Fv(B)5 b Fs(\()p Fv(X)i Fs(\))31 b Fy(as)f(e)o(xplained)j(in)e(Remark)g(2.6)g(and)g(observ)o(e)448 2221 y(that)23 b(we)f(get)g(a)g(\002ner)g(Hilbert)h(module)g(structure) i(on)d Fu(H)49 b Fy(by)22 b(taking)i Ft(f)p Fv(')p Fs(\()p Fv(Q)p Fs(\))i Ft(j)g Fs(lim)3089 2235 y Fr(F)3166 2221 y Fv(')f Fs(=)g(0)p Ft(g)448 2334 y Fy(as)19 b(multiplier)h(algebra.)29 b(One)18 b(can)h(proceed)i(similarly)f(in)f(the)f(case)i(of)e(a)g (Banach)i Fv(X)7 b Fy(-module,)448 2447 y(it)24 b(suf)n(\002ces)g(to)f (replace)i Fv(B)5 b Fs(\()p Fv(X)i Fs(\))24 b Fy(by)f Fv(C)1626 2462 y Fq(b)1670 2447 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)589 2650 y(W)-7 b(e)23 b(no)n(w)g(gi)n(v)o(e)h(an)f(e)o(xample)i (of)e(a)g(non-topological)29 b(nature.)448 2854 y Fx(Example)23 b(2.9)47 b Fy(Let)25 b Fs(\()p Fv(X)r(;)15 b(\026)p Fs(\))27 b Fy(be)f(measure)h(space)h(with)e Fv(\026)p Fs(\()p Fv(X)7 b Fs(\))31 b(=)f Ft(1)p Fy(.)36 b(W)-7 b(e)25 b(de\002ne)i(the)f(class)448 2967 y(of)31 b(functions)i(which)e(\223v)n (anish)h(at)f(in\002nity\224)h(as)e(follo)n(ws.)51 b(Let)30 b(us)h(say)g(that)g(a)f(set)h Fv(F)52 b Ft(\032)38 b Fv(X)448 3079 y Fy(is)c(of)f Fw(co\002nite)i(measur)m(e)g Fy(if)e(its)h(complement)h Fv(F)2048 3046 y Fq(c)2116 3079 y Fy(is)e(of)h(\002nite)f(\(e)o(xterior\))j(measure.)60 b(The)448 3192 y(f)o(amily)32 b(of)f(sets)h(of)f(co\002nite)h(measure)g (is)f(clearly)i(a)e(\002lter)g Ft(F)2437 3206 y Fo(\026)2484 3192 y Fy(.)51 b(If)31 b Fv(')f Fy(is)h(a)g(function)i(on)f Fv(X)448 3305 y Fy(then)27 b Fs(lim)757 3319 y Fr(F)807 3327 y Fg(\026)868 3305 y Fv(')k Fs(=)f(0)c Fy(means)h(that)g(for)g 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3757 y Fy(and)24 b(each)f Fv(")j(>)f Fs(0)e Fy(we)f(ha)n(v)o(e)h Fv(N)36 b Fs(=)25 b Ft(f)p Fv(x)g Ft(j)h(j)p Fv(')p Fs(\()p Fv(x)p Fs(\))p Ft(j)h(\025)e Fv(")p Ft(g)h(2)f(N)2346 3771 y Fo(\026)2414 3757 y Fy(and)e(ess-sup)18 b Ft(j)p Fv(')h Ft(\000)3042 3739 y Fv(\037)3099 3771 y Fo(N)3166 3757 y Fv(')p Ft(j)26 b(\024)f Fv(")p Fy(.)448 3870 y(No)n(w)18 b(it)h(is)f(clear)i(that)g Fv(L)1200 3837 y Fq(2)1239 3870 y Fs(\()p Fv(X)7 b Fs(\))19 b Fy(and,)h(more)f(generally)-6 b(,)23 b(an)o(y)c(direct)h(inte)o(gral) g(of)f(Hilbert)h(spaces)448 3983 y(o)o(v)o(er)k Fv(X)7 b Fy(,)23 b(has)h(a)f(natural)i(Hilbert)f(module)h(structure)g(with)f Fv(B)2410 3997 y Fo(\026)2456 3983 y Fs(\()p Fv(X)7 b Fs(\))24 b Fy(as)g(multiplier)h(algebra.)589 4186 y(If)31 b Fu(H)57 b Fy(is)31 b(a)g(Banach)h(module)g(and)g(the)f(Banach)h (space)g Fu(H)57 b Fy(is)31 b(re\003e)o(xi)n(v)o(e)h(we)e(say)i(that) 448 4299 y Fu(H)60 b Fy(is)33 b(a)g Fw(r)m(e\003e)n(xive)j(Banac)o(h)e (module)p Fy(.)60 b(In)34 b(this)g(case)g(the)g(adjoint)h(Banach)g (space)g Fu(H)3304 4266 y Fr(\003)3376 4299 y Fw(is)448 4412 y(equipped)d(with)e(a)f(canonical)j(Banac)o(h)e(module)h(structur) m(e)p Fy(,)i(its)c(multiplier)j(algebra)f(being)448 4525 y Ft(M)p Fs(\()p Fu(H)708 4492 y Fr(\003)748 4525 y Fs(\))h(:=)g Ft(f)p Fv(A)1056 4492 y Fr(\003)1127 4525 y Ft(j)g Fv(A)g Ft(2)f(M)p Fs(\()p Fu(H)d Fs(\))p Ft(g)p Fy(.)39 b(This)27 b(is)f(a)h(closed)i(subalgebra)h(of)c Ft(B)s Fs(\()p Fu(H)3114 4492 y Fr(\003)3153 4525 y Fs(\))h Fy(which)448 4638 y(clearly)f(has)f(an)f(approximate)j(unit)e(and)g(the)g(linear)g (subspace)i(generated)g(by)d(the)h(elements)448 4751 y(of)f(the)g(form)g Fv(A)949 4718 y Fr(\003)989 4751 y Fv(v)s Fy(,)f(with)g Fv(A)k Ft(2)e(M)p Fs(\()p Fu(H)i Fs(\))d Fy(and)g Fv(v)29 b Ft(2)c Fu(H)2195 4718 y Fr(\003)2235 4751 y Fy(,)d(is)i(weak)2555 4718 y Fr(\003)2595 4751 y Fy(-dense,)h(hence)h(dense,)e(in)448 4863 y Fu(H)564 4830 y Fr(\003)603 4863 y Fy(.)43 b(Indeed,)31 b(if)d Fv(u)35 b Ft(2)f Fu(H)54 b Fy(and)29 b Ft(h)p Fv(u;)15 b(A)1730 4830 y Fr(\003)1770 4863 y Fv(v)s Ft(i)35 b Fs(=)f(0)29 b Fy(for)f(all)h(such)g Fv(A;)15 b(v)32 b Fy(then)d Fv(Au)34 b Fs(=)g(0)29 b Fy(for)f(all)448 4976 y Fv(A)e Ft(2)f(M)p Fs(\()p Fu(H)i Fs(\))c Fy(hence)i Fv(u)g Fs(=)g(0)e Fy(because)i(of)f(\(2.4\).)1920 5225 y(9)p eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 448 573 a Fx(Example)23 b(2.10)47 b Fy(F)o(or)27 b(each)i(real)f(number)h Fv(s)e Fy(let)h Fu(H)2137 540 y Fo(s)2207 573 y Fs(:=)33 b Fu(H)2452 540 y Fo(s)2489 573 y Fs(\()p Fp(R)2590 540 y Fo(n)2637 573 y Fs(\))27 b Fy(be)h(the)g(Hilbert)h(space)448 686 y(of)h(distrib)n(utions)k Fv(u)29 b Fy(on)h Fp(R)1302 653 y Fo(n)1378 686 y Fy(such)h(that)f Ft(k)p Fv(u)p Ft(k)1881 653 y Fq(2)1881 708 y Fo(s)1958 686 y Fs(:=)2091 613 y Fe(R)2152 686 y Fs(\(1)c(+)e Ft(j)p Fv(k)s Ft(j)2453 653 y Fq(2)2493 686 y Fs(\))2528 653 y Fo(s)2566 686 y Ft(j)s Fe(b)-54 b Fv(u)p Fs(\()p Fv(k)s Fs(\))p Ft(j)2788 653 y Fq(2)2828 686 y Fv(dk)41 b(<)c Ft(1)p Fy(,)30 b(where)452 799 y Fe(b)-55 b Fv(u)25 b Fy(is)h(the)g(F)o(ourier)g(transform)i(of)e Fv(u)p Fy(.)35 b(This)25 b(is)h(the)g(usual)h(Sobole)n(v)g(space)g(of)e (order)i Fv(s)e Fy(on)h Fp(R)3368 766 y Fo(n)3414 799 y Fy(.)448 912 y(The)21 b(algebra)i Fu(S)38 b Fy(of)21 b(Schw)o(artz)h(test)g(functions)i(on)d Fp(R)2152 879 y Fo(n)2219 912 y Fy(is)g(naturally)j(embedded)f(in)e Ft(B)s Fs(\()p Fu(H)3342 879 y Fo(s)3379 912 y Fs(\))p Fy(,)448 1024 y(a)33 b(function)j Fv(')44 b Ft(2)g Fu(S)50 b Fy(being)35 b(identi\002ed)g(with)f(the)f(operator)j(of)d (multiplication)k(by)d Fv(')f Fy(on)448 1137 y Fu(H)564 1104 y Fo(s)601 1137 y Fy(.)f(If)24 b(we)g(denote)i(by)f Ft(M)1360 1104 y Fo(s)1421 1137 y Fy(the)h(closure)g(of)f Fu(S)42 b Fy(in)24 b Ft(B)s Fs(\()p Fu(H)2372 1104 y Fo(s)2409 1137 y Fs(\))p Fy(,)g(then)i(clearly)g Fs(\()p Fu(H)3094 1104 y Fo(s)3131 1137 y Fv(;)15 b Ft(M)3280 1104 y Fo(s)3317 1137 y Fs(\))24 b Fy(is)448 1250 y(a)f(Banach)g (module)h(and)f(this)g(Banach)g(module)h(is)e(a)h(Hilbert)g(module)h (if)e(and)h(only)h(if)e Fv(s)j Fs(=)g(0)p Fy(.)448 1363 y(The)g(module)h(adjoint)h(to)e Fs(\()p Fu(H)1434 1330 y Fo(s)1470 1363 y Fv(;)15 b Ft(M)1619 1330 y Fo(s)1657 1363 y Fs(\))24 b Fy(is)h(identi\002ed)i(with)e Fs(\()p Fu(H)2506 1330 y Fr(\000)p Fo(s)2598 1363 y Fv(;)15 b Ft(M)2747 1330 y Fr(\000)p Fo(s)2839 1363 y Fs(\))p Fy(.)33 b(Note)25 b(that)g Ft(M)3400 1330 y Fo(s)448 1476 y Fy(can)j(be)f (realized)i(as)e(a)f(subalgebra)k(of)d Ft(M)1826 1443 y Fq(0)1898 1476 y Fs(=)k Fv(C)2065 1490 y Fq(0)2105 1476 y Fs(\()p Fp(R)2206 1443 y Fo(n)2253 1476 y Fs(\))p Fy(,)c(namely)h Ft(M)2741 1443 y Fo(s)2804 1476 y Fy(is)f(the)g (completion)448 1589 y(of)h Fu(S)45 b Fy(for)28 b(the)h(norm)f Ft(k)p Fv(')p Ft(k)1315 1603 y Fr(M)1399 1584 y Fg(s)1471 1589 y Fs(:=)33 b(sup)1737 1611 y Fr(k)p Fo(u)p Fr(k)1848 1619 y Fg(s)1882 1611 y Fq(=1)1991 1589 y Ft(k)p Fv('u)p Ft(k)2192 1603 y Fo(s)2231 1589 y Fy(,)28 b(and)g(then)h(we)e(ha)n(v)o (e)i Ft(M)3062 1556 y Fo(s)3132 1589 y Fs(=)k Ft(M)3345 1556 y Fr(\000)p Fo(s)448 1702 y Fy(isometrically)27 b(and)d Ft(M)1209 1669 y Fo(s)1271 1702 y Ft(\032)h(M)1476 1669 y Fo(t)1528 1702 y Fy(if)f Fv(s)h Ft(\025)g Fv(t)g Ft(\025)g Fs(0)e Fy(\(by)h(interpolation\).)448 1914 y Fx(De\002nition)f(2.11)46 b Fw(A)19 b(couple)i Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))19 b Fw(consisting)j(of)e(a)f(Hilbert) h(module)h Fu(H)45 b Fw(and)20 b(a)f(Hilbert)448 2027 y(space)31 b Fu(G)44 b Fw(suc)o(h)30 b(that)f Fu(G)52 b Ft(\032)35 b Fu(H)56 b Fw(continuously)33 b(and)d(densely)h(will)e (be)g(called)i(a)d Fy(Friedrichs)448 2140 y(module)p Fw(.)i(If)22 b Fu(H)49 b Fw(is)22 b(a)h(Hilbert)g Fv(X)7 b Fw(-module)25 b(o)o(ver)e(a)g(locally)h(compact)g(space)g Fv(X)7 b Fw(,)23 b(we)f(say)h(that)448 2253 y Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))f Fw(is)h(a)f Fy(Friedrichs)j Fv(X)7 b Fy(-module)p Fw(.)39 b(If)27 b Ft(M)p Fs(\()p Fu(H)g Fs(\))k Ft(\032)g(K)q Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))p Fw(,)g(we)f(say)h(that)g Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))448 2366 y Fw(is)d(a)f Fy(compact)i (Friedrichs)g(module)p Fw(.)448 2579 y Fy(In)36 b(the)g(situation)i(of) e(this)g(de\002nition)h(we)e(al)o(w)o(ays)i(identify)g Fu(H)62 b Fy(with)35 b(its)h(adjoint)h(space,)448 2692 y(which)22 b(gi)n(v)o(es)f(us)g(a)f(Gelf)o(and)i(triplet)h Fu(G)40 b Ft(\032)25 b Fu(H)52 b Ft(\032)25 b Fu(G)2111 2659 y Fr(\003)2150 2692 y Fy(.)i(If)21 b Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))20 b Fy(is)h(a)f(compact)i(Friedrichs)448 2804 y(module)32 b(then)f(each)h(operator)h Fv(M)40 b Fy(from)31 b Ft(M)p Fs(\()p Fu(H)c Fs(\))j Fy(e)o(xtends)i(to)f(a)f (compact)i(operator)h Fv(M)49 b Fs(:)448 2917 y Fu(H)60 b Ft(!)33 b Fu(G)792 2884 y Fr(\003)858 2917 y Fy(\(this)c(is)e(the)h (adjoint)i(of)d(the)h(compact)h(operator)h Fv(M)2555 2884 y Fr(\003)2627 2917 y Fs(:)j Fu(G)49 b Ft(!)32 b Fu(H)27 b Fy(\).)40 b(Thus)28 b(we)448 3030 y(shall)d(ha)n(v)o(e)f Ft(M)p Fs(\()p Fu(H)j Fs(\))f Ft(\032)f(K)q Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))21 b Ft(\\)e(K)q Fs(\()p Fu(H)28 b Fv(;)15 b Fu(G)2054 2997 y Fr(\003)2093 3030 y Fs(\))p Fy(.)448 3243 y Fx(Example)23 b(2.12)47 b Fy(W)l(ith)26 b(the)g(notations)i(of)d(Example)h(2.10,)g(if)f(we)f(set)i Fu(H)55 b Fs(=)29 b Fu(H)3067 3210 y Fq(0)3131 3243 y Fy(and)d(tak)o(e)448 3356 y Fv(s)h(>)f Fs(0)p Fy(,)e(then)h Fs(\()p Fu(H)1038 3323 y Fo(s)1075 3356 y Fv(;)15 b Fu(H)27 b Fs(\))d Fy(is)g(a)g(compact)i(Friedrichs)g(module)f(and)g(the)g (associated)i(Gelf)o(and)448 3469 y(triplet)d(is)e Fu(H)885 3436 y Fo(s)947 3469 y Ft(\032)j Fu(H)52 b Ft(\032)25 b Fu(H)1395 3436 y Fr(\000)p Fo(s)1487 3469 y Fy(.)j(Indeed,)23 b(if)f Fv(')k Ft(2)f Fv(C)2142 3483 y Fq(0)2181 3469 y Fs(\()p Fp(R)2282 3436 y Fo(n)2329 3469 y Fs(\))d Fy(then)h(the)g (operator)i(of)d(multipli-)448 3582 y(cation)j(by)f Fv(')f Fy(is)g(a)h(compact)g(operator)i Fu(H)1807 3549 y Fo(s)1870 3582 y Ft(!)f Fu(H)h Fy(.)448 3831 y Fj(2.2)99 b(Decay)25 b(impr)n(o)o(ving)g(operators)448 4005 y Fy(Let)g Fu(H)50 b Fy(and)26 b Fu(K)53 b Fy(be)25 b(Banach)h(spaces.)34 b(If)25 b Fu(K)53 b Fy(is)25 b(a)f(Banach)i(module)g(then)g(we)e(shall) i(denote)448 4118 y(by)j Ft(B)642 4085 y Fo(l)627 4142 y Fq(0)667 4118 y Fs(\()p Fu(H)e Fv(;)15 b Fu(K)30 b Fs(\))e Fy(the)h(norm)g(closed)i(linear)f(subspace)h(generated)g(by)e (the)g(operators)j Fv(M)10 b(T)j Fy(,)448 4231 y(with)24 b Fv(T)38 b Ft(2)25 b(B)s Fs(\()p Fu(H)i Fv(;)15 b Fu(K)29 b Fs(\))23 b Fy(and)h Fv(M)36 b Ft(2)25 b(M)p Fs(\()p Fu(K)30 b Fs(\))p Fy(.)e(W)-7 b(e)23 b(say)h(that)h(an)e(operator)j(in) e Ft(B)2990 4198 y Fo(l)2975 4255 y Fq(0)3015 4231 y Fs(\()p Fu(H)j Fv(;)15 b Fu(K)29 b Fs(\))23 b Fy(is)448 4344 y Fw(decay)28 b(impr)l(o)o(ving)p Fy(,)h(or)d Fw(left)g(vanishes)j (at)d(in\002nity)i Fy(\(with)f(respect)h(to)e Ft(M)p Fs(\()p Fu(K)k Fs(\))p Fy(,)c(if)g(this)h(is)f(not)448 4457 y(ob)o(vious)32 b(from)d(the)h(conte)o(xt\).)49 b(If)30 b Fv(J)1615 4471 y Fo(\013)1693 4457 y Fy(is)g(an)g (approximate)i(unit)e(for)g Ft(M)p Fs(\()p Fu(K)g Fs(\))p Fy(,)g(then)g(for)g(an)448 4570 y(operator)c Fv(S)k Ft(2)25 b(B)s Fs(\()p Fu(H)i Fv(;)15 b Fu(K)29 b Fs(\))23 b Fy(we)g(ha)n(v)o (e:)512 4774 y Fv(S)30 b Ft(2)25 b(B)759 4736 y Fo(l)744 4796 y Fq(0)784 4774 y Fs(\()p Fu(H)i Fv(;)15 b Fu(K)30 b Fs(\))83 b Ft(,)g Fs(lim)1421 4828 y Fo(\013)1522 4774 y Ft(k)p Fv(J)1617 4788 y Fo(\013)1667 4774 y Fv(S)25 b Ft(\000)20 b Fv(S)5 b Ft(k)26 b Fs(=)f(0)1151 b Fy(\(2.6\))1206 4939 y Ft(,)83 b Fv(S)30 b Fs(=)25 b Fv(M)10 b(T)36 b Fy(for)24 b(some)f Fv(M)36 b Ft(2)24 b(M)p Fs(\()p Fu(K)30 b Fs(\))23 b Fy(and)h Fv(T)38 b Ft(2)25 b(B)s Fs(\()p Fu(H)h Fv(;)15 b Fu(K)30 b Fs(\))p Fv(:)1897 5225 y Fy(10)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 448 573 a Fy(The)23 b(second)j(equi)n(v)n(alence)g(follo)n(ws)e(from)g(the)f(Cohen-He)n (witt)i(theorem)g(\(Theorem)f(2.1\).)589 686 y(If)34 b Fu(H)59 b Fy(is)34 b(a)f(Banach)i(module)g(then)f(one)g(can)g (similarly)h(de\002ne)g Ft(B)2809 653 y Fo(r)2794 710 y Fq(0)2846 686 y Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)29 b Fs(\))k Fy(as)h(the)448 799 y(norm)24 b(closed)i(linear)f(subspace)h (generated)g(by)e(the)h(operators)h Fv(T)13 b(M)33 b Fy(with)24 b Fv(T)38 b Ft(2)26 b(B)s Fs(\()p Fu(H)g Fv(;)15 b Fu(K)30 b Fs(\))448 912 y Fy(and)i Fv(M)48 b Ft(2)39 b(M)p Fs(\()p Fu(H)27 b Fs(\))p Fy(.)50 b(W)-7 b(e)30 b(say)h(that)g(the)g(elements)i(of)d Ft(B)2355 879 y Fo(r)2340 936 y Fq(0)2393 912 y Fs(\()p Fu(H)c Fv(;)15 b Fu(K)30 b Fs(\))g Fw(right)i(vanish)g(at)f(in-)448 1024 y(\002nity)p Fy(.)47 b(If)29 b(both)h Fu(H)55 b Fy(and)30 b Fu(K)57 b Fy(are)29 b(Banach)h(modules)h(then)f(both)g (spaces)g Ft(B)2913 992 y Fo(l)2898 1049 y Fq(0)2939 1024 y Fs(\()p Fu(H)d Fv(;)15 b Fu(K)29 b Fs(\))g Fy(and)448 1137 y Ft(B)523 1104 y Fo(r)508 1162 y Fq(0)561 1137 y Fs(\()p Fu(H)d Fv(;)15 b Fu(K)30 b Fs(\))23 b Fy(mak)o(e)h(sense)g (and)g(we)f(set)h Ft(B)1827 1151 y Fq(0)1866 1137 y Fs(\()p Fu(H)j Fv(;)15 b Fu(K)29 b Fs(\))d(=)f Ft(B)2401 1104 y Fo(l)2386 1162 y Fq(0)2426 1137 y Fs(\()p Fu(H)i Fv(;)15 b Fu(K)29 b Fs(\))21 b Ft(\\)f(B)2941 1104 y Fo(r)2926 1162 y Fq(0)2978 1137 y Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)29 b Fs(\))p Fy(.)589 1250 y(Some)23 b(simple)h(properties)j(of)c(these)i (spaces)g(are)e(described)k(belo)n(w)-6 b(.)448 1438 y Fx(Pr)n(oposition)25 b(2.13)46 b Fw(If)30 b Fu(K)59 b Fw(is)30 b(a)g(r)m(e\003e)n(xive)i(Banac)o(h)f(module)g(and)f Fv(S)43 b Ft(2)37 b(B)2887 1405 y Fo(l)2872 1462 y Fq(0)2912 1438 y Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)30 b Fs(\))g Fw(then)448 1551 y Fv(S)509 1518 y Fr(\003)571 1551 y Fw(belongs)c(to)d Ft(B)1045 1518 y Fo(r)1030 1575 y Fq(0)1083 1551 y Fs(\()p Fu(K)1230 1518 y Fr(\003)1269 1551 y Fv(;)15 b Fu(H)1425 1518 y Fr(\003)1465 1551 y Fs(\))p Fw(.)448 1738 y Fx(Pr)n(oof:)30 b Fy(W)-7 b(e)22 b(ha)n(v)o(e)i Fv(S)30 b Fs(=)25 b Fv(M)10 b(T)35 b Fy(with)23 b Fv(M)35 b Ft(2)25 b(M)p Fs(\()p Fu(K)k Fs(\))23 b Fy(and)g Fv(T)38 b Ft(2)25 b(B)s Fs(\()p 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b(These)29 b(are)g(closed)h(subspaces)i(of)c Ft(B)s Fs(\()p Fu(H)f Fv(;)15 b Fu(G)h Fs(\))p Fy(.)44 b(The)448 3301 y(ne)o(xt)24 b(result)h(is)e(ob)o(vious;)j(a)d(similar)h(assertion)i (holds)f(in)e(the)h(right)h(decay)f(preserving)j(case.)448 3480 y Fx(Pr)n(oposition)e(2.19)46 b Fw(Let)31 b Ft(f)p Fv(J)1365 3494 y Fo(\013)1416 3480 y Ft(g)g Fw(be)g(an)g(appr)l (oximate)j(unit)e(for)g Ft(M)p Fs(\()p Fu(H)27 b Fs(\))k Fw(and)h(let)f Fv(S)36 b Fw(be)31 b(an)448 3593 y(oper)o(ator)h(in)e Ft(B)s Fs(\()p Fu(H)c Fv(;)15 b Fu(K)30 b Fs(\))p Fw(.)47 b(Then)30 b Fv(S)k Fw(is)29 b(left)h(decay)i(pr)m(eserving)g(if)e(and)g (only)g(if)g(one)g(of)g(the)448 3706 y(following)c(conditions)g(is)d (satis\002ed:)448 3819 y Fy(\(1\))h Fv(S)5 b(J)688 3833 y Fo(\013)763 3819 y Ft(2)25 b(B)924 3786 y Fo(l)909 3843 y Fq(0)949 3819 y Fs(\()p Fu(H)i Fv(;)15 b Fu(K)30 b Fs(\))23 b Fw(for)h(all)f Fv(\013)p Fw(.)448 3932 y Fy(\(2\))k Fw(for)g(eac)o(h)g Fv(M)41 b Ft(2)31 b(M)p Fs(\()p Fu(H)c Fs(\))f 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Fu(K)29 b Fs(\))44 b Ft(\032)f(B)1671 540 y Fo(l)1656 595 y(q)1696 573 y Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)29 b Fs(\))k Fy(and)h Ft(B)2306 540 y Fo(r)2291 597 y Fq(0)2343 573 y Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)30 b Fs(\))43 b Ft(\032)g(B)2914 540 y Fo(r)2899 595 y(q)2952 573 y Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)29 b Fs(\))k Fy(b)n(ut)448 686 y(this)c(f)o(act)f(is)f(of)h(no)g (interest.)43 b(The)27 b(main)h(results)i(of)d(this)h(paper)h(depend)h (on)e(\002nding)g(other)l(,)448 799 y(more)c(interesting)k(e)o(xamples) d(of)f(decay)h(preserving)j(operators.)33 b(W)-7 b(e)23 b(shall)i(gi)n(v)o(e)f(in)g(the)h(rest)448 912 y(of)e(this)h (subsection)i(some)d(elementary)i(e)o(xamples)f(of)e(such)i(operators)i (and)d(in)g(Subsections)448 1024 y(2.4)h(and)g(7.1)f(more)h(subtle)g (ones.)589 1179 y(From)19 b(no)n(w)h(on)g(in)g(this)g(subsection)j Fv(X)j Fy(will)20 b(be)g(a)f(locally)i(compact)h(non-compact)g(topo-) 448 1292 y(logical)31 b(space.)45 b(The)28 b Fw(support)33 b Fy(supp)24 b Fv(u)35 b 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Fu(K)560 2632 y Fr(\003)531 2687 y Fq(c)617 2665 y Ft(\002)c Fu(H)794 2679 y Fq(c)851 2665 y Fy(of)22 b Fu(K)1060 2632 y Fr(\003)1117 2665 y Ft(\002)17 b Fu(H)48 b Fy(and)24 b(the)f(preceding)i (boundedness)h(notion)f(is)d(equi)n(v)n(alent)j(to)e(the)448 2778 y(continuity)i(of)d(this)h(form)f(for)g(the)h(topology)h(induced)h (by)d Fu(K)2424 2745 y Fr(\003)2478 2778 y Ft(\002)15 b Fu(H)27 b Fy(.)g(W)-7 b(e)21 b(similarly)j(de\002ne)448 2891 y(the)g(boundedness)j(of)d(the)g(commutator)h Fs([)p Fv(S;)15 b(')p Fs(\()p Fv(Q)p Fs(\)])p Fy(.)448 3057 y Fx(Pr)n(oposition)25 b(2.21)46 b Fw(Assume)37 b(that)g Fv(S)54 b Ft(2)48 b(B)s Fs(\()p Fu(H)27 b Fv(;)15 b Fu(K)29 b Fs(\))36 b Fw(and)h(let)f Fv(\022)51 b Fs(:)e Fv(X)57 b Ft(!)49 b Fs([1)p Fv(;)15 b Ft(1)p Fs([)36 b Fw(be)h(a)448 3170 y(continuous)30 b(function)e(suc)o(h)f(that)g Fs(lim)1680 3184 y Fo(x)p Fr(!1)1880 3170 y Fv(\022)s Fs(\()p Fv(x)p Fs(\))k(=)f Ft(1)p Fw(.)36 b(If)26 b Fv(\022)s Fs(\()p Fv(Q)p Fs(\))p Fv(S)5 b(\022)2706 3137 y Fr(\000)p Fq(1)2799 3170 y Fs(\()p Fv(Q)p Fs(\))26 b Fw(is)g(a)g(bounded)448 3283 y(oper)o(ator)k(then)e Fv(S)k Fw(is)c(left)g(decay)h(pr)m (eserving)o(.)43 b(If)28 b Fv(\022)2109 3250 y Fr(\000)p Fq(1)2202 3283 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)5 b(\022)s Fs(\()p Fv(Q)p Fs(\))27 b Fw(is)h(a)f(bounded)j(oper)o(ator)448 3396 y(then)25 b Fv(S)i Fw(is)d(right)g(decay)h(pr)m(eserving)o(.)448 3562 y Fx(Pr)n(oof:)36 b Fy(Let)25 b Fv(K)37 b Ft(\032)30 b Fv(X)j Fy(be)27 b(compact,)h(let)e Fv(U)40 b Ft(\032)30 b Fv(X)j Fy(be)27 b(a)e(neighbourhood)31 b(of)c(in\002nity)g(in)f Fv(X)7 b Fy(,)448 3675 y(and)26 b(let)g Fv(';)15 b( )33 b Ft(2)c Fv(C)1066 3690 y Fq(b)1109 3675 y Fs(\()p Fv(X)7 b Fs(\))26 b Fy(such)g(that)g(supp)f Fv(')k Ft(\032)f Fv(K)q(;)41 b Fy(supp)25 b Fv( )32 b Ft(\032)d Fv(U)34 b Fy(and)26 b Ft(j)p Fv(')p Ft(j)k(\024)f Fs(1)p Fv(;)15 b Ft(j)p Fv( )s Ft(j)30 b(\024)f Fs(1)p Fy(.)448 3788 y(Then)c Fv(\022)s(')f Fy(is)h(a)f(bounded)j(function)g(and)e Fv( )s(\022)1864 3755 y Fr(\000)p 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Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))h Fw(is)h(a)f(stable)j(F)-5 b(riedric)o(hs)30 b Fv(X)7 b Fw(-module)o(,)32 b(then)d Fu(G)44 b Fw(is)29 b(equipped)i(with)e(a)448 4976 y(canonical)f(Banac)o(h)e Fv(X)7 b Fw(-module)27 b(structur)m(e)p Fy(.)35 b(Then,)26 b(by)f(taking)i(adjoints,)g(we)d(get)i(a)f(natural)1897 5225 y(16)p eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 448 573 a Fy(Banach)37 b Fv(X)7 b Fy(-module)38 b(structure)h(on)d Fu(G)1728 540 y Fr(\003)1803 573 y Fy(too.)67 b(Our)36 b(de\002nitions)i(are)e (such)h(that)g(after)g(the)448 686 y(identi\002cations)27 b Fu(G)42 b Ft(\032)25 b Fu(H)53 b Ft(\032)25 b Fu(G)1489 653 y Fr(\003)1552 686 y Fy(the)f(restriction)j(to)c Fu(H)50 b Fy(of)24 b(the)g(operator)i Fv(V)2922 701 y Fo(k)2987 686 y Fy(acting)f(in)f Fu(G)3398 653 y Fr(\003)448 799 y Fy(is)29 b(just)h(the)f(initial)h Fv(V)1129 814 y Fo(k)1172 799 y Fy(.)44 b(Indeed,)32 b(we)c(ha)n(v)o(e)h Fv(V)1943 766 y Fr(\003)1922 826 y Fo(k)2017 799 y Fs(=)35 b Fv(V)2197 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Fv(V)2667 4769 y Fo(k)2711 4754 y Fv(dk)s(:)1897 5225 y Fy(17)p eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 448 573 a Fy(Since)30 b Fv(k)40 b Ft(7!)d Fv(T)947 588 y Fo(k)1019 573 y Fy(is)30 b(norm)g(continuous)j(on)d(the)g(compact)h(support)h(of)41 b Fe(b)-63 b Fv(')q Fy(,)30 b(for)g(each)h Fv(")37 b(>)g Fs(0)448 686 y Fy(we)25 b(can)g(construct,)j(with)d(the)g(help)h(of)f (a)g(partition)i(of)e(unity)-6 b(,)27 b(functions)h Fv(\022)2844 700 y Fo(i)2900 686 y Ft(2)f Fv(C)3053 700 y Fq(c)3089 686 y Fs(\()p Fv(X)3206 653 y Fr(\003)3246 686 y Fs(\))e Fy(and)448 799 y(operators)h Fv(S)865 813 y Fo(i)918 799 y Ft(2)f(B)s Fs(\()p Fu(H)h Fv(;)15 b Fu(K)29 b Fs(\))p Fy(,)23 b(such)g(that)h Ft(k)p Fv(T)1895 814 y Fo(k)1956 799 y Ft(\000)2044 730 y Fe(P)2140 757 y Fo(n)2140 826 y(i)p Fq(=1)2273 799 y Fv(\022)2316 813 y Fo(i)2344 799 y Fs(\()p Fv(k)s Fs(\))p Fv(S)2520 813 y Fo(i)2549 799 y Ft(k)i Fv(<)f(")d Fy(if)35 b Fe(b)-63 b Fv(')p Fs(\()p Fv(k)s Fs(\))26 b Ft(6)p 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Fr(\003)1909 3047 y Fo(k)1969 3024 y Fv(S)5 b(V)2083 3039 y Fo(k)2125 3024 y Fv(\022)s Fs(\()p Fv(k)s Fs(\))p Fv(dk)s(:)804 b Fy(\(2.12\))448 3218 y(The)26 b(inte)o(gral)h(is)e(well)h(de\002ned)g(because)i Fv(k)33 b Ft(7!)c Fv(V)2061 3185 y Fr(\003)2041 3246 y Fo(k)2101 3218 y Fv(S)5 b(V)2215 3233 y Fo(k)2287 3218 y Ft(2)29 b(B)s Fs(\()p Fu(H)d Fs(\))f Fy(is)h(a)f(bounded)j(strongly) 448 3331 y(continuous)34 b(map.)49 b(In)30 b(order)i(to)e(e)o(xplain)i (the)f(main)f(idea)h(of)g(the)g(proof)g(we)f(shall)h(mak)o(e)g(a)448 3444 y(formal)e(computation)i(in)l(v)n(olving)g(the)e(spectral)h (measure)f Fv(E)5 b Fs(\()p Fv(A)p Fs(\))35 b(=)2698 3425 y Fv(\037)2755 3458 y Fo(A)2812 3444 y Fs(\()p Fv(Q)p Fs(\))p Fy(,)29 b(see)f(Remark)448 3557 y(2.6)g(and)g(Lemma)e(2.27)i (\(we)f(shall)h(use)g(the)g(same)f(notation)j(for)d(the)h(spectral)h (measures)g(in)448 3670 y Fu(H)50 b Fy(and)24 b Fu(K)29 b Fy(\).)f(W)-7 b(e)22 b(ha)n(v)o(e)j(for)e Fv(k)29 b Ft(2)c Fv(X)1639 3637 y Fr(\003)1701 3670 y Fy(and)f Fv(')p Fs(\()p Fv(Q)p Fs(\))j Ft(2)d Fv(B)5 b Fs(\()p Fv(X)i Fs(\))778 3878 y Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(V)1053 3841 y Fr(\003)1032 3901 y Fo(k)1118 3878 y Fs(=)25 b Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(k)s Fs(\()p Fv(Q)p Fs(\))1607 3841 y Fr(\003)1673 3878 y Fs(=)g(\()p Fv(')p 1863 3804 51 4 v(k)t Fs(\)\()p Fv(Q)p Fs(\))h(=)2213 3755 y Fe(Z)2319 3878 y Fv(')p Fs(\()p Fv(x)p Fs(\))p 2500 3804 V Fv(k)5 b Fs(\()p Fv(x)p Fs(\))p Fv(E)g Fs(\()p Fv(dx)p Fs(\))p Fv(:)448 4100 y Fy(Note)31 b(also)g(that)g(for)f Fv(x;)15 b(y)42 b Ft(2)37 b Fv(X)g Fy(we)30 b(ha)n(v)o(e)p 1855 4026 V 31 w Fv(k)s Fs(\()p Fv(x)p Fs(\))p Fv(k)s Fs(\()p Fv(y)s Fs(\))40 b(=)d Fv(k)s Fs(\()p Ft(\000)p Fv(x)p Fs(\))p Fv(k)s Fs(\()p Fv(y)s Fs(\))j(=)d Fv(k)s Fs(\()p Fv(y)29 b Ft(\000)c Fv(x)p Fs(\))p Fy(.)49 b(Let)453 4189 y Fe(b)448 4213 y Fv(\022)s Fs(\()p Fv(x)p Fs(\))25 b(=)737 4140 y Fe(R)p 813 4134 173 4 v 813 4213 a Fv(k)s Fs(\()p Fv(x)p Fs(\))q Fv(\022)s Fs(\()p Fv(k)s Fs(\))p Fv(dk)h Fy(be)e(the)g(F)o(ourier)g(transform)h(of)e Fv(\022)s Fy(.)k(Then)d(if)f Fv(';)15 b( )30 b Ft(2)25 b Fv(B)5 b Fs(\()p Fv(X)i Fs(\))p Fy(:)593 4427 y Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(S)850 4442 y Fo(\022)890 4427 y Fv( )s Fs(\()p Fv(Q)p Fs(\))84 b(=)1332 4303 y Fe(Z)1382 4509 y Fo(X)1445 4490 y Fc(\003)1501 4427 y Fv(\022)s Fs(\()p Fv(k)s Fs(\))p Fv(dk)1780 4303 y Fe(Z)1830 4509 y Fo(X)1913 4303 y Fe(Z)1963 4509 y Fo(X)2046 4427 y Fv(')p Fs(\()p Fv(x)p Fs(\))p 2227 4353 51 4 v Fv(k)t Fs(\()p Fv(x)p Fs(\))p Fv(k)s Fs(\()p Fv(y)s Fs(\))p Fv( )s Fs(\()p Fv(y)s Fs(\))p Fv(E)5 b Fs(\()p Fv(dx)p Fs(\))p Fv(S)g(E)g Fs(\()p Fv(dy)s Fs(\))1178 4666 y(=)1332 4543 y Fe(Z)1382 4749 y Fo(X)1465 4543 y Fe(Z)1516 4749 y Fo(X)1603 4642 y Fe(b)1598 4666 y Fv(\022)r Fs(\()p Fv(x)21 b Ft(\000)f Fv(y)s Fs(\))p Fv(')p Fs(\()p Fv(x)p Fs(\))p Fv( )s Fs(\()p Fv(y)s Fs(\))p Fv(E)5 b Fs(\()p Fv(dx)p Fs(\))p Fv(S)g(E)g Fs(\()p Fv(dy)s Fs(\))p Fv(:)367 b Fy(\(2.13\))p 448 4798 1196 4 v 554 4853 a Fm(3)583 4885 y Fl(If)24 b Fi(X)30 b Fl(is)25 b(an)f(euclidean)i(space)f(and)h Fu(H)58 b Ff(=)32 b Fu(K)61 b Ff(=)32 b Fi(L)2080 4853 y Fb(2)2115 4885 y Ff(\()p Fi(X)6 b Ff(\))p Fl(,)25 b(the)g(ne)o(xt)f(condition)i (means)f(that)g(there)f(is)448 4976 y Fi(r)f(<)e Fh(1)e Fl(such)h(that)e(the)h(distrib)o(ution)g(k)o(ernel)h(of)f Fi(S)k Fl(satis\002es)18 b Fi(S)t Ff(\()p Fi(x;)13 b(y)s Ff(\))20 b(=)h(0)e Fl(for)g Fh(j)p Fi(x)e Fh(\000)g Fi(y)s Fh(j)k Fi(>)g(r)r Fl(.)1897 5225 y Fy(18)p eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 448 573 a Fy(W)-7 b(e)26 b(can)g(rigorously)j(justify)f(this)f(computation)i(and)d(gi)n (v)o(e)g(a)g(meaning)i(to)e(the)g(last)h(inte)o(gral)448 686 y(by)22 b(taking)h(into)g(account)g(that)g Fv(E)5 b Fs(\()p Fv(A)p Fs(\))p Fv(S)g(E)g Fs(\()p Fv(B)g Fs(\))23 b Fy(induces)g(a)f(\002nitely)g(additi)n(v)o(e)h(measure)g(on)f(the)448 799 y(algebra)33 b(generated)g(by)e(rectangles)i Fv(A)26 b Ft(\002)f Fv(B)35 b Fy(in)c Fv(X)i Ft(\002)25 b Fv(X)37 b Fy(\(note)32 b(that)2747 775 y Fe(b)2742 799 y Fv(\022)41 b Ft(2)d Fv(C)2990 813 y Fq(0)3030 799 y Fs(\()p Fv(X)7 b Fs(\))p Fy(\).)51 b(If)30 b Fv(S)448 912 y Fy(is)h(Hilbert-Schmidt)i (then)f(the)f(measure)i(is)d(in)h(f)o(act)h Fv(\033)s Fy(-additi)n(v)o(e)h(and)e(the)h(result)g(becomes)448 1024 y(ob)o(vious.)e(W)-7 b(e)21 b(shall,)i(ho)n(we)n(v)o(er)l(,)f(a)n (v)n(oid)i(these)e(questions)j(and)d(we)f(shall)i(directly)g(pro)o(v)o (e)f(only)448 1137 y(what)i(we)e(need.)30 b(Namely)-6 b(,)23 b(we)g(sho)n(w)g(the)h(follo)n(wing:)448 1400 y Fx(\()p Ft(\003)p Fx(\))593 1271 y Fe(\032)703 1348 y Fy(If)f(the)h(support)h(of)1319 1324 y Fe(b)1314 1348 y Fv(\022)40 b Fy(is)23 b(a)g(compact)i(set)f Fs(\003)f Fy(and)h(if)f(supp)h Fv(')d Ft(\\)f Fs(\(\003)h(+)e Fy(supp)25 b Fv( )s Fs(\))h(=)f Fp(?)703 1461 y Fy(then)f Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(S)1139 1476 y Fo(\022)1179 1461 y Fv( )s Fs(\()p Fv(Q)p Fs(\))i(=)f(0)p Fv(:)448 1717 y Fy(Observ)o(e)k(that)f(if)f(\()p Ft(\003)p Fy(\))h(holds)h(for)e 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Fe(b)1599 2411 y Fv(\022)r Fs(\()p Fv(x)21 b Ft(\000)e Fv(y)s Fs(\))p Fv(')p Fs(\()p Fv(x)p Fs(\))p Fv( )s Fs(\()p Fv(y)s Fs(\))p Ft(h)p Fv(g)s(;)c(E)5 b Fs(\()p Fv(dx)p Fs(\))p Fv(u)p Ft(ih)p Fv(u;)16 b(E)5 b Fs(\()p Fv(d)q(y)s Fs(\))p Fv(f)10 b Ft(i)p Fv(;)448 2656 y Fy(hence)25 b(\()p Ft(\003)p Fy(\))f(holds)g(for)g(such)h Fv(S)5 b Fy(.)589 2768 y(Finally)-6 b(,)29 b(note)e(that)g(if)g Fv(S)36 b Ft(2)31 b Fv(C)1573 2735 y Fq(u)1616 2768 y Fs(\()p Fv(Q)p Fs(\))26 b Fy(then)i Fv(S)j Fy(is)26 b(norm)h(limit)g (of)f(operators)k(of)c(the)h(form)448 2881 y Fv(S)504 2896 y Fo(\022)543 2881 y Fy(.)45 b(F)o(or)28 b(this)i(it)e(suf)n (\002ces)i(to)f(tak)o(e)h Fv(\022)38 b Fs(=)d Ft(j)p Fv(K)7 b Ft(j)1905 2848 y Fr(\000)p Fq(1)2000 2863 y Fv(\037)2057 2895 y Fo(K)2153 2881 y Fy(where)30 b Fv(K)35 b Fy(runs)30 b(o)o(v)o(er)f(the)g(set)g(of)g(open)448 2994 y(relati)n(v)o(ely)36 b(compact)e(neighbourhoods)39 b(of)33 b(the)g(neutral)i(element)g(of)e Fv(X)2827 2961 y Fr(\003)2867 2994 y Fy(,)i Ft(j)p Fv(K)7 b Ft(j)32 b Fy(being)j(the)448 3107 y(Haar)i(measure)h(of)e Fv(K)7 b Fy(.)67 b(Then,)40 b(by)d(approximating)j(con)l(v)o(eniently)g Fv(\022)e Fy(in)f Fv(L)2977 3074 y Fq(1)3052 3107 y Fy(norm,)j(one)448 3220 y(sho)n(ws)24 b(that)g Fv(S)k Fy(is)23 b(norm)h(limit)f(of)h (operators)i Fv(S)1947 3235 y Fo(\022)2008 3220 y Fy(such)f(that)2362 3196 y Fe(b)2357 3220 y Fv(\022)g Fy(has)f(compact)g(support.)p 3371 3212 67 67 v 448 3458 a Fx(Pr)n(oposition)h(2.34)46 b Fw(Assume)29 b(that)g Fv(X)36 b Fw(is)28 b(a)g(disjoint)j(union)f Fv(X)42 b Fs(=)35 b Ft([)2690 3472 y Fo(a)p Fr(2)p Fo(A)2831 3458 y Fv(X)2906 3472 y Fo(a)2976 3458 y Fw(of)28 b(Bor)m(el)h(sets)448 3570 y Fv(X)523 3584 y Fo(a)595 3570 y Fw(suc)o(h)h(that:)43 b(1\))30 b(ther)m(e)h(is)f(a)f(compact)j(set)e Fv(K)36 b Fw(suc)o(h)30 b(that)h(eac)o(h)g Fv(X)2705 3584 y Fo(a)2776 3570 y Fw(is)f(a)f(tr)o(anslate)k(of)d(a)448 3683 y(subset)24 b(of)f Fv(K)7 b Fw(,)22 b(and)h(2\))f(for)h(eac)o(h)g(compact)h (neighborhood)j Fs(\003)22 b Fw(of)h(the)g(origin,)h(the)f(number)g(of) 448 3796 y(sets)j Fv(X)684 3811 y Fo(b)740 3796 y Fs(+)21 b(\003)j Fw(whic)o(h)h(inter)o(sects)j(a)d(given)h Fv(X)1897 3810 y Fo(a)1963 3796 y Fw(is)f(bounded)j(by)d(a)f(constant)k (independent)g(of)448 3909 y Fv(a)p Fw(.)g(Then)c(a)f(\002nite)i(r)o (ang)o(e)f(oper)o(ator)i(is)d(of)g(class)i Fv(C)2050 3876 y Fq(u)2093 3909 y Fs(\()p Fv(Q)p Fs(\))p Fw(.)448 4097 y Fx(Pr)n(oof:)36 b Fy(It)26 b(suf)n(\002ces)h(to)g(pro)o(v)o(e)g (that)g(a)f(\002nite)g(range)i(operator)g Fv(S)j Fy(is)26 b(of)h(class)g Fv(C)2982 4064 y Fq(u)3025 4097 y Fs(\()p Fv(Q)p Fs(\))p Fy(.)37 b(Let)26 b Fs(\003)448 4210 y Fy(be)21 b(such)h(that)897 4191 y Fv(\037)954 4224 y Fo(H)1021 4210 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)1224 4191 y(\037)1282 4224 y Fo(K)1350 4210 y Fs(\()p Fv(Q)p Fs(\))k(=)f(0)c Fy(if)f Fv(H)r(;)15 b(K)27 b Fy(are)21 b(compact)h(sets)g(with)f Fs(\()p Fv(H)c Ft(\000)10 b Fv(K)d Fs(\))j Ft(\\)g Fs(\003)26 b(=)f Ft(;)p Fy(.)448 4323 y(Let)595 4304 y Fv(\037)651 4337 y Fo(a)718 4323 y Fy(be)h(the)g(characteristic)k(function)e(of)e Fv(X)1979 4337 y Fo(a)2046 4323 y Fy(and)g Fv(')2261 4337 y Fo(a)2328 4323 y Fy(that)g(of)g Fv(Y)2643 4337 y Fo(a)2714 4323 y Fs(=)j Fv(X)2889 4337 y Fo(a)2953 4323 y Fs(+)22 b(\003)p Fy(.)35 b(W)-7 b(e)25 b(can)448 4436 y(assume)f(that)f Fv(A)i Ft(\032)g Fv(X)k Fy(and)23 b(that)h Fv(X)1576 4450 y Fo(a)1643 4436 y Fs(=)h Fv(a)16 b Fs(+)h Fv(K)1968 4450 y Fo(a)2031 4436 y Fy(for)23 b(some)g Fv(K)2450 4450 y Fo(a)2517 4436 y Ft(\032)i Fs(\003)p Fy(.)j(W)-7 b(e)21 b(shall)j(abbre)n(viate)448 4530 y Fv(\037)505 4562 y Fo(a)577 4548 y Fs(=)678 4530 y Fv(\037)735 4562 y Fo(a)776 4548 y Fs(\()p Fv(Q)p Fs(\))i Fy(and)h Fv(')1160 4562 y Fo(a)1232 4548 y Fs(=)j Fv(')1392 4562 y Fo(a)1434 4548 y Fs(\()p Fv(Q)p Fs(\))p Fy(.)36 b(W)-7 b(e)26 b(ha)n(v)o(e)1974 4480 y Fe(P)2069 4575 y Fo(a)2126 4530 y Fv(\037)2183 4562 y Fo(a)2255 4548 y Fs(=)k(1)c Fy(strongly)i(on)e Fu(H)h Fy(,)f(cf.)f(Remark)448 4661 y(2.6,)e(and)g 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y(functions)d(on)d Fv(X)40 b Fy(and)34 b Fv(X)1303 3213 y Fr(\003)1375 3246 y Fy(respecti)n(v)o(ely)-6 b(.)61 b(Belo)n(w)32 b(the)i(space)g Fv(L)2592 3213 y Fq(2)2631 3246 y Fs(\()p Fv(X)7 b Fs(\))34 b Fy(is)f(equipped)i(with) 448 3359 y(its)24 b(natural)h(Hilbert)f Fv(X)7 b Fy(-module)26 b(structure.)448 3547 y Fx(Pr)n(oposition)f(2.35)46 b Fw(The)36 b Fv(C)1367 3514 y Fr(\003)1406 3547 y Fw(-alg)o(ebr)o(a)i(g) o(ener)o(ated)g(by)e(the)g(oper)o(ator)o(s)j Fv(')p Fs(\()p Fv(Q)p Fs(\))d Fw(and)g Fv( )s Fs(\()p Fv(P)13 b Fs(\))p Fw(,)448 3660 y(with)24 b Fv(')h Ft(2)g Fv(C)870 3627 y Fq(u)863 3687 y(b)913 3660 y Fs(\()p Fv(X)7 b Fs(\))24 b Fw(and)g Fv( )29 b Ft(2)c Fv(C)1494 3627 y Fq(u)1487 3687 y(b)1536 3660 y Fs(\()p Fv(X)1653 3627 y Fr(\003)1694 3660 y Fs(\))p Fw(,)e(consists)i(of)f(decay)g(pr)m(eserving)j(oper)o (ator)o(s.)448 3847 y Fx(Pr)n(oof:)k Fy(By)23 b(Proposition)j(2.20,)e Ft(B)1559 3861 y Fo(q)1597 3847 y Fs(\()p Fv(L)1694 3814 y Fq(2)1734 3847 y Fs(\()p Fv(X)7 b Fs(\)\))24 b Fy(is)g(a)f Fv(C)2164 3814 y Fr(\003)2203 3847 y Fy(-algebra,)j(hence)f(it)e(suf)n (\002ces)i(to)e(sho)n(w)448 3960 y(that)33 b(each)g Fv(')p Fs(\()p Fv(Q)p Fs(\))g Fy(and)g Fv( )s Fs(\()p Fv(P)13 b Fs(\))32 b Fy(is)g(decay)i(preserving.)58 b(F)o(or)31 b Fv(')p Fs(\()p Fv(Q)p Fs(\))i Fy(the)f(assertion)j(is)d(tri)n(vial) 448 4073 y(while)24 b(for)g Fv( )s Fs(\()p Fv(P)13 b Fs(\))24 b Fy(we)e(apply)j(Proposition)h(2.32.)p 3371 4065 V 448 4416 a Fz(3)120 b(Compact)29 b(perturbations)i(in)g(Banach)f (modules)448 4623 y Fy(In)22 b(this)g(section)h Fs(\()p Fu(G)15 b Fv(;)g Fu(H)27 b Fs(\))21 b Fy(will)g(al)o(w)o(ays)i(be)e(a)g Fw(compact)h(F)-5 b(riedric)o(hs)23 b(module)p Fy(,)g(see)e (De\002nition)448 4736 y(2.11.)57 b(As)32 b(usual,)j(we)d(associate)j (to)e(it)f(a)g(Gelf)o(and)i(triplet)g Fu(G)58 b Ft(\032)42 b Fu(H)68 b Ft(\032)42 b Fu(G)2964 4703 y Fr(\003)3035 4736 y Fy(and)33 b(we)f(set)448 4848 y Ft(k)23 b(\001)g(k)31 b Fs(=)f Ft(k)23 b(\001)g(k)902 4864 y Fk(H)999 4848 y Fy(.)37 b(W)-7 b(e)25 b(are)i(interested)i(in)d(criteria)j(which)d (ensure)i(that)f(an)g(operator)h Fv(B)i Fy(is)d(a)448 4961 y(compact)f(perturbation)i(of)c(an)h(operator)h Fv(A)p Fy(,)e(both)i(operators)h(being)e(unbounded)j(operators)1897 5225 y(20)p eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 448 573 a Fy(in)19 b Fu(H)44 b Fy(obtained)20 b(as)f(restrictions)i(of)e(some)f(bounded)j (operators)g Fu(G)40 b Ft(!)26 b Fu(G)2789 540 y Fr(\003)2828 573 y Fy(.)h(More)18 b(precisely)-6 b(,)448 686 y(the)25 b(follo)n(wing)h(is)f(a)f(general)j(assumption)g(\(suggested)h(by)c (the)h(statement)i(of)d(Theorem)h(2.1)448 799 y(in)f([OS)o(]\))f(which) h(will)g(al)o(w)o(ays)g(be)f(ful\002lled:)448 1117 y Fx(\()p Fv(AB)5 b Fx(\))689 930 y Fe(8)689 1012 y(<)689 1176 y(:)812 999 y Fv(A;)15 b(B)27 b Fy(are)d(closed)h(densely)h (de\002ned)e(operators)i(in)d Fu(H)50 b Fy(with)23 b Fv(\032)p Fs(\()p Fv(A)p Fs(\))e Ft(\\)f Fv(\032)p Fs(\()p Fv(B)5 b Fs(\))26 b Ft(6)p Fs(=)f Ft(;)834 1112 y 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2725 y(parently)32 b(more)d(general)j(form,)e(namely)h Fv(B)e Ft(\000)24 b Fv(A)37 b Fs(=)2218 2657 y Fe(P)2314 2683 y Fo(n)2314 2752 y(k)r Fq(=1)2462 2725 y Fv(S)2518 2740 y Fo(k)2560 2725 y Fv(T)2613 2740 y Fo(k)2684 2725 y Fy(with)30 b(operators)i Fv(S)3298 2740 y Fo(k)3377 2725 y Ft(2)448 2838 y(B)s Fs(\()p Fu(K)629 2853 y Fo(k)672 2838 y Fv(;)15 b Fu(G)784 2805 y Fr(\003)824 2838 y Fs(\))k Fy(and)i Fv(T)1082 2853 y Fo(k)1150 2838 y Ft(2)j(B)s Fs(\()p Fu(G)16 b Fv(;)f Fu(K)1528 2853 y Fo(k)1571 2838 y Fs(\))p Fy(.)27 b(But)20 b(we)f(are)h(reduced)i(to)e(the)g(stated)i(v)o(ersion)f(of)f (the)g(as-)448 2951 y(sumption)j(by)f(considering)j(the)c(Hilbert)h (module)h Fu(K)54 b Fs(=)25 b Ft(\010)p Fu(K)2454 2966 y Fo(k)2517 2951 y Fy(and)d Fv(S)30 b Fs(=)25 b Ft(\010)p Fv(S)2978 2966 y Fo(k)3021 2951 y Fv(;)15 b(T)38 b Fs(=)25 b Ft(\010)p Fv(T)3372 2966 y Fo(k)3414 2951 y Fy(.)448 3064 y(\(2\))k(If)f Fv(V)55 b Ft(2)35 b(K)q Fs(\()p Fu(G)16 b Fv(;)f Fu(G)1163 3031 y Fr(\003)1202 3064 y Fs(\))29 b Fy(and)g(if)f Fu(K)57 b Fy(is)28 b(an)h(in\002nite)g(dimensional)j (module,)e(then)g(there)f(are)448 3176 y(operators)h Fv(S)36 b Ft(2)c(B)s Fs(\()p Fu(K)c Fv(;)15 b Fu(G)1319 3143 y Fr(\003)1359 3176 y Fs(\))26 b Fy(and)i Fv(T)44 b Ft(2)31 b(K)q Fs(\()p Fu(G)17 b Fv(;)e Fu(K)29 b Fs(\))e Fy(such)g(that)h Fv(V)52 b Fs(=)31 b Fv(S)5 b(T)39 b Fy(\(the)27 b(proof)h(is)f(an)448 3289 y(easy)f(e)o(x)o(ercise\).)35 b(This)25 b(and)h(the)f(preceding)j(remark)e(sho)n(w)f(that)h(compact)g (contrib)n(utions)j(to)467 3379 y Fe(e)448 3402 y Fv(B)c Ft(\000)654 3379 y Fe(e)633 3402 y Fv(A)e Fy(are)h(tri)n(vially)h(co)o (v)o(ered)g(by)e(the)h(f)o(actorization)j(assumption.)589 3615 y(If)i Fv(A)e Fy(is)h(self-adjoint)k(then)d(the)g(conditions)i(on) e Fv(A)e Fy(in)i(assumption)h(\(AB\))e(are)g(satis\002ed)448 3728 y(if)g Ft(D)s Fs(\()p Fv(A)p Fs(\))35 b Ft(\032)f Fu(G)50 b Ft(\032)34 b(D)s Fs(\()p Ft(j)p Fv(A)p Ft(j)1319 3695 y Fq(1)p Fo(=)p Fq(2)1429 3728 y Fs(\))28 b Fy(densely)j(\(see)e 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y(ther)m(e)h(is)f(a)f (Hilbert)i(module)g Fu(K)52 b Fw(and)24 b(that)1862 4527 y Fe(e)1843 4550 y Fv(B)h Ft(\000)2049 4527 y Fe(e)2028 4550 y Fv(A)h Fs(=)g Fv(S)2280 4517 y Fr(\003)2319 4550 y Fv(T)36 b Fw(for)24 b(some)g Fv(S)31 b Ft(2)26 b(B)s Fs(\()p Fu(G)15 b Fv(;)g Fu(K)29 b Fs(\))24 b Fw(and)448 4663 y Fv(T)39 b Ft(2)26 b(B)702 4630 y Fo(l)687 4687 y Fq(0)727 4663 y Fs(\()p Fu(G)16 b Fv(;)f Fu(K)29 b Fs(\))24 b Fw(suc)o(h)g(that)h Fv(S)5 b Fs(\()p Fv(A)21 b Ft(\000)f Fv(z)t Fs(\))1755 4630 y Fr(\000)p Fq(1)1876 4663 y Ft(2)26 b(B)2038 4630 y Fo(r)2023 4685 y(q)2075 4663 y Fs(\()p Fu(H)h Fv(;)15 b Fu(K)30 b Fs(\))23 b Fw(for)h(some)g Fv(z)31 b Ft(2)26 b Fv(\032)p Fs(\()p Fv(A)p Fs(\))21 b Ft(\\)f Fv(\032)p Fs(\()p Fv(B)5 b Fs(\))p Fw(.)448 4775 y(Then)24 b Fv(B)j Fw(is)c(a)h(compact)g (perturbation)k(of)23 b Fv(A)g Fw(and)h Fv(\033)2109 4789 y Fq(ess)2200 4775 y Fs(\()p Fv(B)5 b Fs(\))26 b(=)f Fv(\033)2518 4789 y Fq(ess)2609 4775 y Fs(\()p Fv(A)p Fs(\))p Fw(.)1897 5225 y Fy(22)p 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y(erators)i(on)f Fu(H)49 b Fy(associated)27 b(to)c(them.)29 b(The)23 b(notation)j Ft(B)2253 1217 y Fo(l)2238 1275 y Fq(00)2312 1250 y Fs(\()p Fu(E)18 b Fv(;)d Fu(E)2524 1217 y Fr(\003)2564 1250 y Fs(\))23 b Fy(is)g(introduced)k(in)c(\(2.7\).)448 1427 y Fx(Theor)n(em)h(3.5)46 b Fw(Let)17 b Fs(\()p Fu(E)h Fv(;)d Fu(K)29 b Fs(\))18 b Fw(be)g(an)g(arbitr)o(ary)j(F)-5 b(riedric)o(hs)19 b(module)g(and)g(let)f Fv(D)28 b Ft(2)d(B)s Fs(\()p Fu(G)15 b Fv(;)g Fu(E)j Fs(\))p Fw(,)448 1540 y Fv(a;)d(b)26 b Ft(2)f(B)s Fs(\()p Fu(E)17 b Fv(;)e Fu(E)962 1507 y Fr(\003)1001 1540 y Fs(\))23 b Fw(and)h Fv(z)30 b Ft(2)25 b Fp(C)d Fw(suc)o(h)i(that:)448 1653 y Fy(\(1\))g Fw(The)f(oper)o(ator)o(s)j Fv(D)1188 1620 y Fr(\003)1227 1653 y Fv(aD)d Ft(\000)d Fv(z)27 b Fw(and)d Fv(D)1770 1620 y Fr(\003)1810 1653 y Fv(bD)f Ft(\000)d Fv(z)27 b Fw(ar)m(e)c(bijective)j(maps)d Fu(G)41 b Ft(!)25 b Fu(G)3077 1620 y Fr(\003)3116 1653 y Fw(,)448 1766 y Fy(\(2\))f Fv(a)c Ft(\000)g Fv(b)25 b Ft(2)g(B)961 1733 y Fo(l)946 1790 y Fq(00)1021 1766 y Fs(\()p Fu(E)17 b Fv(;)e Fu(E)1233 1733 y Fr(\003)1272 1766 y Fs(\))p Fw(,)448 1879 y Fy(\(3\))24 b Fv(D)s Fs(\(\001)766 1846 y Fr(\003)766 1901 y Fo(a)828 1879 y Ft(\000)h Fs(\026)-50 b Fv(z)t Fs(\))1000 1846 y Fr(\000)p Fq(1)1120 1879 y Ft(2)25 b(B)1281 1846 y Fo(r)1266 1901 y(q)1318 1879 y Fs(\()p Fu(H)i Fv(;)15 b Fu(K)29 b Fs(\))p Fw(.)448 1992 y(Then)24 b Fs(\001)729 2007 y Fo(b)786 1992 y Fw(is)f(a)g (compact)i(perturbation)i(of)d Fs(\001)1920 2006 y Fo(a)1961 1992 y Fw(.)448 2169 y Fx(Pr)n(oof:)30 b Fy(W)-7 b(e)22 b(gi)n(v)o(e)g(a)g(proof)i(independent)i(of)d(Theorem)g(3.2,)f (although)j(we)d(could)i(apply)g(this)448 2282 y(theorem.)30 b(Clearly)22 b Fs(\001)1161 2296 y Fo(a)1218 2282 y Ft(\000)15 b Fv(z)24 b Fy(and)f Fs(\001)1599 2297 y Fo(b)1648 2282 y Ft(\000)15 b Fv(z)24 b Fy(e)o(xtend)g(to)e(bijections)i Fu(G)41 b Ft(!)25 b Fu(G)2814 2249 y Fr(\003)2874 2282 y Fy(and)e(the)f(identity)614 2473 y Fv(R)k Fs(:=)f(\(\001)941 2487 y Fo(a)1003 2473 y Ft(\000)20 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y(to)g(zero)g(at)f(in\002nity)-6 b(.)589 1738 y(In)24 b(order)i(to)e(apply)h(Proposition)h(4.1)e(we)g(need)h (that)f Fv(D)2364 1705 y Fr(\003)2403 1738 y Fv(aD)g Ft(\000)c Fv(z)31 b Fs(:)26 b Fu(H)2881 1705 y Fo(m)2974 1738 y Ft(!)g Fu(H)3207 1705 y Fr(\000)p Fo(m)3351 1738 y Fy(be)448 1851 y(bijecti)n(v)o(e)j(for)e(some)g Fv(z)36 b Ft(2)c Fp(C)p Fy(,)26 b(and)i(similarly)g(for)f Fv(b)p Fy(.)39 b(A)26 b(standard)j(w)o(ay)e(of)f(checking)k(this)d(is)448 1964 y(to)d(require)h(the)f(follo)n(wing)h Fw(coer)m(civity)h (condition)p Fy(:)448 2184 y Fx(\()p Fv(C)7 b Fx(\))619 2056 y Fe(\032)729 2122 y Fy(there)24 b(are)g Fv(\026;)15 b(\027)31 b(>)25 b Fs(0)e Fy(such)i(that)f(for)f(all)h Fv(u)h Ft(2)g Fu(H)2272 2089 y Fo(m)2364 2122 y Fs(:)729 2167 y Fe(P)825 2262 y Fr(j)p Fo(\013)p Fr(j)p Fo(;)p Fr(j)p Fo(\014)s Fr(j\024)p Fo(m)1148 2235 y Fy(Re)d Ft(h)p Fv(P)1377 2202 y Fo(\013)1427 2235 y Fv(u;)15 b(a)1567 2250 y Fo(\013\014)1660 2235 y Fv(P)1731 2202 y Fo(\014)1778 2235 y Fv(u)p Ft(i)26 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j(to)d(sho)n(w)g(the)g(stability)j(of)d(the)g(essential)j(spectrum)448 4158 y(of)h(such)g(operators)i(under)f(perturbations)j(which)c(are)g (small)g(at)f(in\002nity)-6 b(.)29 b(W)-7 b(e)21 b(stress)i(that)f(the) 448 4271 y(dif)n(ferential)32 b(operators)f(co)o(v)o(ered)f(by)f(these) g(results)h(can)f(be)g(of)g(an)o(y)f(order)i(and)f(that)g(in)g(the)448 4383 y(usual)f(case)f(when)f(the)h(coef)n(\002cients)h(are)f(comple)o (x)g(measurable)i(functions)f(a)e(condition)j(of)448 4496 y(the)22 b(type)778 4473 y Fe(e)756 4496 y Fv(A)k Ft(2)f Fv(C)1008 4463 y Fq(u)1050 4496 y Fs(\()p Fv(Q)p Fs(;)15 b Fu(H)1314 4463 y Fo(w)1370 4496 y Fv(;)g Fu(H)1526 4463 y Fo(w)r Fr(\003)1618 4496 y Fs(\))21 b Fy(is)f(v)o(ery)i (general,)h(if)d(not)i(automatically)i(satis\002ed)e(\(see)448 4609 y(the)k(remark)f(at)g(the)h(end)f(of)g(this)h(subsection\).)37 b(Hence)25 b(the)h(only)g(condition)i(really)e(rele)n(v)n(ant)448 4722 y(in)32 b(this)g(conte)o(xt)h(is)1120 4699 y Fe(e)1100 4722 y Fv(B)e Ft(\000)1318 4699 y Fe(e)1297 4722 y Fv(A)40 b Ft(2)g(B)1581 4689 y Fo(l)1566 4747 y Fq(0)1606 4722 y Fs(\()p Fu(H)1757 4689 y Fo(w)1814 4722 y Fv(;)15 b Fu(H)1970 4689 y Fo(w)r Fr(\003)2062 4722 y Fs(\))31 b Fy(and)h(the)g(main)f(point)i(is)f(that)g(it)f(allo)n(ws)448 4835 y(perturbations)d(of)23 b(the)h(higher)h(order)g(coef)n (\002cients)g(e)n(v)o(en)f(in)f(the)h(non-smooth)i(case.)1897 5225 y(28)p eop end end %%Page: 29 29 TeXDict begin HPSdict begin 29 28 bop 589 573 a Fy(It)23 b(is)f(clear)h(that)g(these)g(results)h(can)f(be)f(used)h(to)f (establish)j(the)e(stability)h(of)e(the)h(essential)448 686 y(spectrum)j(of)e(pseudo-dif)n(ferential)29 b(operators)d(on)e (\002nite)g(dimensional)j(v)o(ector)e(spaces)g(o)o(v)o(er)448 799 y(local)g(\002elds)e(\(see)i([Sa,)d(T)-7 b(a]\))23 b(under)i(perturbations)i(of)d(the)g(same)f(order)-5 b(.)589 961 y(W)e(e)20 b(shall)i(gi)n(v)o(e)f(only)h(one)f(e)o(xplicit) h(e)o(xample)g(of)f(physical)h(interest,)h(that)e(of)g(Dirac)g(oper)n (-)448 1074 y(ators.)32 b(Let)24 b Fv(X)34 b Fs(=)27 b Fp(R)1096 1041 y Fo(n)1166 1074 y Fy(and)e(let)f Fv(\013)1493 1088 y Fq(0)1559 1074 y Ft(\021)j Fv(\014)5 b(;)15 b(\013)1811 1088 y Fq(1)1851 1074 y Fv(;)g(:)g(:)g(:)i(;)e(\013)2111 1088 y Fo(n)2182 1074 y Fy(be)24 b(symmetric)h(operators)i(on)e Fv(E)j Fy(such)448 1187 y(that)d Fv(\013)666 1201 y Fo(j)702 1187 y Fv(\013)760 1202 y Fo(k)824 1187 y Fs(+)20 b Fv(\013)973 1202 y Fo(k)1016 1187 y Fv(\013)1074 1201 y Fo(j)1136 1187 y Fs(=)26 b Fv(\016)1273 1202 y Fo(j)t(k)1348 1187 y Fy(.)j(Then)24 b(the)g(free)h(Dirac)f(operator)i(is)d Fv(D)29 b Fs(=)2744 1119 y Fe(P)2840 1145 y Fo(n)2840 1214 y(k)r Fq(=1)2988 1187 y Fv(\013)3046 1202 y Fo(k)3088 1187 y Fv(P)3146 1202 y Fo(k)3210 1187 y Fs(+)20 b Fv(m\014)448 1300 y Fy(for)26 b(some)f(real)h(number)h Fv(m)p Fy(.)33 b(The)25 b(natural)i(compact)f(stable)h(Friedrichs)g Fv(X)7 b Fy(-module)27 b(in)f(this)448 1413 y(conte)o(xt)c(is)e Fs(\()p Fu(H)965 1380 y Fq(1)p Fo(=)p Fq(2)1075 1413 y Fv(;)15 b Fu(H)27 b 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b(quite)i(general,)448 2854 y(compare)25 b(with)e([Ar)q(,)f(A)-10 b(Y,)22 b(Kl,)h(N1,)f(N2].) 448 3017 y Fx(Remark:)30 b Fy(In)25 b(order)g(to)f(clarify)i(the)e (relation)j(between)e(the)f(notion)i(of)e(uniform)i(hypoellip-)448 3130 y(ticity)20 b(introduced)i(abo)o(v)o(e)e(and)g(the)f(original)i (notion)g(of)d(hypoellipticity)24 b(due)19 b(to)g(H)8 b(\250)-38 b(ormander)l(,)448 3243 y(we)23 b(shall)i(consider)h(the)e (case)g(of)g(dif)n(ferential)j(operators)f(on)e Fp(R)2481 3210 y Fo(n)2550 3243 y Fy(\(which)h(is)e(identi\002ed)i(with)448 3356 y(its)f(dual)g(group)h(in)e(the)h(standard)i(w)o(ay\).)i(Assume)c (\002rst)f(that)h Fv(h)f Fy(is)g(a)g(polynomial)j(on)e Fp(R)3237 3323 y Fo(n)3306 3356 y Fy(and)448 3469 y(that)32 b Fv(A)40 b Fs(=)f Fv(h)p Fs(\()p Fv(P)13 b Fs(\))p Fy(.)52 b(Then)32 b(the)f(function)i(de\002ned)g(by)e Fv(w)r Fs(\()p Fv(k)s Fs(\))2400 3436 y Fq(4)2480 3469 y Fs(=)2591 3400 y Fe(P)2687 3496 y Fo(\013)2751 3469 y Ft(j)p Fv(h)2828 3436 y Fq(\()p Fo(\013)p Fq(\))2933 3469 y Fs(\()p Fv(k)s Fs(\))p Ft(j)3078 3436 y Fq(2)3149 3469 y Fy(satis\002es)448 3582 y Fv(w)r Fs(\()p Fv(k)600 3549 y Fr(0)642 3582 y Fs(+)17 b Fv(k)s Fs(\))26 b Ft(\024)f Fs(\(1)18 b(+)f Fv(c)p Ft(j)p Fv(k)1237 3549 y Fr(0)1261 3582 y Ft(j)p Fs(\))1321 3549 y Fo(m=)p Fq(2)1459 3582 y Fv(w)r Fs(\()p Fv(k)s Fs(\))p Fy(,)23 b(where)g Fv(c)f Fy(is)h(a)f(number)i(and)f Fv(m)f Fy(is)g(the)h(order)h(of)f Fv(h)p Fy(,)f(see)448 3694 y([Ho)q(,)30 b(Example)g(10.1.3].)48 b(No)n(w)28 b(the)i(\223form)g(domain\224)h(of)f(the)g(operator)i Fv(h)p Fs(\()p Fv(P)13 b Fs(\))30 b Fy(in)f Fv(L)3214 3661 y Fq(2)3254 3694 y Fs(\()p Fp(R)3355 3661 y Fo(n)3402 3694 y Fs(\))448 3807 y Fy(is)d(the)h(space)g Fu(G)46 b Fs(=)30 b Ft(D)s Fs(\()p Ft(j)p Fv(h)p Fs(\()p Fv(P)13 b Fs(\))p Ft(j)1452 3774 y Fq(1)p Fo(=)p Fq(2)1563 3807 y Fs(\))26 b Fy(and)h(this)g(domain)g(is)f(stable)h(under)h Fv(V)2845 3822 y Fo(k)2918 3807 y Fs(=)i(exp)15 b Fv(i)p Ft(h)p Fv(k)s(;)g(Q)p Ft(i)448 3920 y Fy(if)27 b(and)g(only)h(the)f 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Fo(n)1637 4372 y Fs(\))f Fy(is)h(not)g(dense)h (in)f Fp(C)p Fy(.)k(If)c Fv(n)i Fs(=)g(2)d Fy(then)i Fv(h)p Fs(\()p Fv(k)s Fs(\))i(=)f Fv(k)3202 4339 y Fq(4)3199 4396 y(1)3259 4372 y Fs(+)17 b Fv(k)3397 4339 y Fq(2)3394 4396 y(2)448 4485 y Fy(is)31 b(a)f(simple)i(e)o(xample)g(of)f (polynomial)i(which)e(satis\002es)h(all)f(these)h(conditions)i(b)n(ut)d (is)g(not)448 4598 y(elliptic.)g(See)22 b([GM,)g(Subsections)k (2.7-2.10])f(for)e(the)h(case)g(of)f(matrix)h(v)n(alued)g(functions)i Fv(h)p Fy(.)589 4711 y(In)36 b(the)g(v)n(ariable)h(coef)n(\002cient)g (case)f(the)g(notion)h(of)e(hypoellipticity)k(de\002ned)e(in)e([Ho,)448 4824 y(De\002nition)d(13.4.3])g(is)f(a)g(local)h(one)g(and)f(one)h(may) f(consider)i(dif)n(ferent)g(global)g(v)o(ersions.)448 4936 y(F)o(or)27 b(instance,)j([Ho)q(,)d(Theorem)h(13.4.4])g(suggests)i (that)e(the)g(notion)h(we)e(introduced)k(abo)o(v)o(e)1897 5225 y(29)p eop end end %%Page: 30 30 TeXDict begin HPSdict begin 30 29 bop 448 573 a Fy(is)35 b(natural)i(for)e(operators)j(of)d(uniform)h(constant)h(strength.)66 b(But)35 b(the)g(uniform)h(constant)448 686 y(strength)f(condition)h (is)d(not)g(satis\002ed)h(by)g(the)f(operators)i(with)e(polynomial)j (coef)n(\002cients,)448 799 y(for)31 b(e)o(xample,)h(hence)f(such)g (operators)h(are)f(not)f(uniformly)i(hypoelliptic)h(in)d(our)g(sense)h (in)448 912 y(general.)448 1202 y Fz(5)120 b(Riemannian)31 b(manif)m(olds)448 1409 y Fy(Let)37 b Fu(H)26 b Fv(;)15 b Fu(K)66 b Fy(be)37 b(tw)o(o)f(Hilbert)i(spaces)g(identi\002ed)h(with) e(their)g(adjoints)i(and)f Fs(d)d Fy(a)i(closed)448 1522 y(densely)29 b(de\002ned)f(operator)i(mapping)e Fu(H)53 b Fy(into)28 b Fu(K)h Fy(.)39 b(Let)27 b Fu(G)47 b Fs(=)32 b Ft(D)s Fs(\(d\))26 b Fy(equipped)k(with)d(the)448 1635 y(graph)e(norm,)e(so)h Fu(G)41 b Ft(\032)25 b Fu(H)49 b Fy(continuously)27 b(and)d(densely)i(and)e Fs(d)h Ft(2)g(B)s Fs(\()p Fu(G)15 b Fv(;)g Fu(K)29 b Fs(\))p Fy(.)589 1748 y(Then)36 b(the)f(quadratic)j(form)d Ft(k)p Fs(d)p Fv(u)p 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4939 y Ft(j)p Fv(')p Fs(\()p Fv(x)p Fs(\))21 b Ft(\000)f Fv(')p Fs(\()p Fv(y)s Fs(\))p Ft(j)p Fv(\032)p Fs(\()p Fv(x;)15 b(y)s Fs(\))2520 4901 y Fr(\000)p Fq(1)2617 4939 y Fv(:)1897 5225 y Fy(30)p eop end end %%Page: 31 31 TeXDict begin HPSdict begin 31 30 bop 448 573 a Fy(In)24 b(more)f(e)o(xplicit)i(terms,)f(we)f(require)832 744 y Ft(jh)p Fs(d)943 706 y Fr(\003)983 744 y Fv(u;)15 b(')p Fs(\()p Fv(Q)p Fs(\))p Fv(v)s Ft(i)1358 759 y Fk(H)1477 744 y Ft(\000)20 b(h)p Fv(u;)15 b(')p Fs(\()p Fv(Q)p Fs(\)d)p Fv(v)s Ft(i)2029 759 y Fk(K)2125 744 y Ft(j)25 b(\024)g Fv(C)d Fy(Lip)h Fv(')15 b Ft(k)p Fv(u)p Ft(k)2723 759 y Fk(K)2818 744 y Ft(k)p Fv(v)s Ft(k)2955 759 y Fk(H)448 915 y Fy(for)22 b(all)g Fv(u)j Ft(2)g(D)s Fs(\(d)1009 882 y Fr(\003)1048 915 y Fs(\))c Fy(and)h Fv(v)29 b Ft(2)c(D)s Fs(\()p Fv(d)p Fs(\))p Fy(.)i(Thus)22 b Ft(h)p Fs(d)1944 882 y Fr(\003)1983 915 y Fv(u;)15 b(')p Fs(\()p Fv(Q)p Fs(\))p Fv(v)s Ft(i)e(\000)g(h)p Fv(u;)i(')p Fs(\()p Fv(Q)p Fs(\)d)p Fv(v)s Ft(i)25 b 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Ft(2)48 b(B)s Fs(\()p Fu(H)26 b Fv(;)15 b(L)3002 1741 y Fq(2)3042 1774 y Fs(\()p Fv(X)3152 1788 y Fo(n)3200 1774 y Fs(\)\))36 b Fy(and)448 1887 y Fv(g)494 1854 y Fo(k)491 1910 y(n)569 1887 y Ft(2)31 b Fv(L)723 1854 y Fo(q)761 1887 y Fs(\()p Fv(X)871 1901 y Fo(n)918 1887 y Fs(\))p Fy(,)c(and)g(clearly)h(we)d(may)i(assume)g Fv(g)2086 1854 y Fo(k)2083 1910 y(n)2161 1887 y Ft(\025)k Fs(0)p Fy(.)37 b(Let)26 b Fv(g)2558 1901 y Fo(n)2636 1887 y Fs(=)k(sup)2874 1909 y Fo(k)2932 1887 y Fv(g)2978 1854 y Fo(k)2975 1910 y(n)3053 1887 y Ft(2)g Fv(L)3206 1854 y Fo(q)3244 1887 y Fs(\()p Fv(X)3354 1901 y Fo(n)3402 1887 y Fs(\))448 2000 y Fy(and)g Fv(S)664 2014 y Fo(n)747 2000 y Ft(2)36 b(B)s Fs(\()p Fu(H)27 b Fv(;)15 b(L)1160 1967 y Fq(2)1199 2000 y Fs(\()p Fv(X)1309 2014 y Fo(n)1357 2000 y Fs(;)g Fp(C)1463 1967 y Fo(n)1510 2000 y Fs(\)\))30 b Fy(be)f(the)h(operator)h(with)f(components)i Fs(\()p Fv(g)2935 1967 y Fo(k)2932 2022 y(n)2980 2000 y Fv(g)3026 1967 y Fr(\000)p Fq(1)3023 2022 y Fo(n)3120 2000 y Fs(\)\()p Fv(Q)p Fs(\))p Fv(R)3367 1967 y Fo(k)3366 2022 y(n)3414 2000 y Fy(.)448 2113 y(Then)24 b Fv(T)711 2127 y Fo(n)783 2113 y Fs(=)h Fv(g)922 2127 y Fo(n)969 2113 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)1167 2127 y Fo(n)1238 2113 y Fy(and)f(if)f(we)g (de\002ne)g Fv(R)1912 2127 y Fo(n)1985 2113 y Fs(=)i Fv(J)2140 2080 y Fr(\000)p Fq(1)2234 2113 y Fv(S)2290 2127 y Fo(n)2360 2113 y Fy(we)d(get)1115 2290 y Fv(g)1158 2304 y Fo(n)1205 2290 y Fs(\()p Fv(Q)p Fs(\))p Fv(R)1416 2304 y Fo(n)1489 2290 y Fs(=)j Fv(J)1644 2252 y Fr(\000)p Fq(1)1738 2290 y Fv(g)1781 2304 y Fo(n)1828 2290 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)2026 2304 y Fo(n)2099 2290 y Fs(=)g Fv(J)2254 2252 y Fr(\000)p Fq(1)2348 2290 y Fv(T)2401 2304 y Fo(n)2474 2290 y Fs(=)g(\005)2638 2304 y Fo(n)2685 2290 y Fv(T)8 b(:)448 2466 y Fy(Thus,)24 b(if)f(we)g(de\002ne)g Fv(g)29 b Fs(=)1295 2398 y Fe(P)1391 2493 y Fo(n)1453 2448 y Fv(\037)1510 2480 y Fo(X)1568 2488 y Fg(n)1615 2466 y Fv(g)1658 2480 y Fo(n)1728 2466 y Fy(and)24 b Fv(R)i Fs(=)2073 2398 y Fe(P)2184 2466 y Fs(\005)2252 2480 y Fo(n)2299 2466 y Fv(R)2368 2480 y Fo(n)2415 2466 y Fy(,)d(we)f(get)i Fv(T)38 b Fs(=)25 b Fv(g)s Fs(\()p Fv(Q)p Fs(\))p Fv(R)q Fy(.)p 3371 2458 67 67 v 589 2661 a(Our)31 b(purpose)j(in)d(the)h(rest)f(of)h(this)g(section)h(is)e(to)g (e)o(xtend)i(Theorem)e(6.1)g(\(in)h(the)g(case)448 2774 y Fv(X)50 b Fs(=)42 b Fp(R)752 2741 y Fo(n)799 2774 y Fy(\))32 b(to)h(more)f(general)j(classes)f(of)f(spaces)h(of)f (measurable)i(functions,)i(which)c(do)448 2886 y(not)26 b(seem)f(to)g(be)g(co)o(v)o(ered)i(by)e(the)g(results)i(e)o(xisting)g (in)e(the)h(literature,)h(cf.)e([Kr].)33 b(Our)25 b(proof)448 2999 y(follo)n(ws)f(closely)i(that)e(of)f(Maure)o(y)-6 b(.)30 b(W)-7 b(e)22 b(\002rst)h(recall)i(K)n(y)e(F)o(an')-5 b(s)23 b(Lemma,)f(see)i([DJT,)e(9.10].)448 3164 y Fx(Pr)n(oposition)j (6.3)46 b Fw(Let)36 b Ft(K)i Fw(be)f(a)g(compact)h(con)l(ve)n(x)h (subset)g(of)e(a)g(Hausdorf)n(f)i(topolo)o(gical)448 3277 y(vector)32 b(space)f(and)g(let)f Fu(F)41 b Fw(be)30 b(a)g(con)l(ve)n(x)i(set)e(of)g(functions)j Fv(F)50 b Fs(:)38 b Ft(K)h(!)p Fs(])25 b Ft(\000)f(1)p Fv(;)15 b Fs(+)p Ft(1)p Fs(])30 b Fw(suc)o(h)448 3390 y(that)25 b(eac)o(h)f Fv(F)39 b Ft(2)26 b Fu(F)35 b Fw(is)24 b(con)l(ve)n(x)i (and)e(lower)g(semicontinuous.)34 b(If)24 b(for)g(eac)o(h)g Fv(F)39 b Ft(2)26 b Fu(F)35 b Fw(ther)m(e)24 b(is)448 3503 y Fv(g)29 b Ft(2)c(K)e Fw(suc)o(h)h(that)f Fv(F)13 b Fs(\()p Fv(g)s Fs(\))27 b Ft(\024)d Fs(0)p Fw(,)f(then)g(ther)m(e)g (is)g Fv(g)29 b Ft(2)c(K)e Fw(suc)o(h)g(that)g Fv(F)13 b Fs(\()p Fv(g)s Fs(\))27 b Ft(\024)e Fs(0)d Fw(for)h(all)g Fv(F)38 b Ft(2)25 b Fu(F)12 b Fw(.)589 3668 y Fy(W)-7 b(e)18 b(need)h(a)f(second)i(general)g(f)o(act)e(that)h(we)f(state)h (belo)n(w)-6 b(.)27 b(Let)18 b Fs(\()p Fv(X)r(;)d(\026)p Fs(\))j Fy(be)h(a)e Fv(\033)s Fy(-\002nite)i(posi-)448 3781 y(ti)n(v)o(e)h(measure)h(space)g(and)f(let)f Fv(L)1464 3748 y Fq(0)1504 3781 y Fs(\()p Fv(X)7 b Fs(\))20 b Fy(be)g(the)f (space)i(of)f Fv(\026)p Fy(-equi)n(v)n(alence)i(classes)g(of)d(comple)o (x)448 3893 y(v)n(alued)30 b(measurable)g(functions)h(on)e Fv(X)34 b Fy(with)29 b(the)f(topology)j(of)d(con)l(v)o(er)n(gence)k(in) c(measure.)448 4006 y(Let)h Fu(L)45 b Fy(be)29 b(a)g(Banach)h(space)g (with)e Fu(L)53 b Ft(\032)35 b Fv(L)1927 3973 y Fq(0)1966 4006 y Fs(\()p Fv(X)7 b Fs(\))30 b Fy(linearly)h(and)e(continuously)k (and)c(such)448 4119 y(that)c(if)e Fv(f)35 b Ft(2)25 b Fv(L)914 4086 y Fq(0)953 4119 y Fs(\()p Fv(X)7 b Fs(\))p Fy(,)24 b Fv(g)29 b Ft(2)c Fu(L)41 b Fy(and)24 b Ft(j)p Fv(f)10 b Ft(j)25 b(\024)g(j)p Fv(g)s Ft(j)f Fy(\()p Fv(\026)p Fy(-a.e.\))g(then)h Fv(f)34 b Ft(2)26 b Fu(L)40 b Fy(and)24 b Ft(k)p Fv(f)10 b Ft(k)2991 4134 y Fk(L)3099 4119 y Ft(\024)25 b(k)p Fv(g)s Ft(k)3331 4134 y Fk(L)3414 4119 y Fy(.)448 4232 y(The)32 b(ne)o(xt)h(result)g(is)f(a)g(rather)i (straightforw)o(ard)i(consequence)g(of)c(Khinchin')-5 b(s)34 b(inequality)448 4345 y([DJT,)23 b(1.10])h(\(see)g(also)g([Pi,)e (Section)j(8]\).)448 4510 y Fx(Pr)n(oposition)g(6.4)46 b Fw(Ther)m(e)20 b(is)g(a)f(number)i Fv(C)7 b Fw(,)19 b(independent)k(of)d Fu(L)e Fw(,)h(suc)o(h)h(that)h(for)f(any)g (Hilbert)448 4623 y(space)25 b Fu(H)49 b Fw(and)24 b(any)g Fv(T)38 b Ft(2)25 b(B)s Fs(\()p Fu(H)i Fv(;)15 b Fu(L)j Fs(\))23 b Fw(the)g(following)j(inequality)g(holds)988 4800 y Ft(k)p Fs(\()1068 4731 y Fe(P)1165 4826 y Fo(j)1201 4800 y Ft(j)p Fv(T)13 b(u)1344 4814 y Fo(j)1381 4800 y Ft(j)1406 4762 y Fq(2)1446 4800 y Fs(\))1481 4762 y Fq(1)p Fo(=)p Fq(2)1591 4800 y Ft(k)1636 4815 y Fk(L)1744 4800 y Ft(\024)25 b Fv(C)7 b Ft(k)p Fv(T)13 b Ft(k)2068 4818 y Fr(B)r Fq(\()p Fk(H)20 b Fo(;)p Fk(L)13 b Fq(\))2365 4800 y Fs(\()2400 4731 y Fe(P)2496 4826 y Fo(j)2533 4800 y Ft(k)p Fv(u)2630 4814 y Fo(j)2667 4800 y Ft(k)2712 4762 y Fq(2)2752 4800 y Fs(\))2787 4762 y Fq(1)p Fo(=)p Fq(2)3217 4800 y Fy(\(6.21\))448 4976 y Fw(for)24 b(all)g(\002nite)g (families)h Ft(f)p Fv(u)1308 4990 y Fo(j)1345 4976 y Ft(g)e Fw(of)h(vector)o(s)h(in)e Fu(H)k Fw(.)1897 5225 y Fy(38)p eop end end %%Page: 39 39 TeXDict begin HPSdict begin 39 38 bop 589 573 a Fy(From)24 b(no)n(w)f(on)h(we)g(w)o(ork)g(in)g(a)f(setting)j(adapted)g(to)e(our)g (needs)h(in)f(Section)h(7,)f(although)448 686 y(it)g(is)f(clear)i(that) f(we)f(could)i(treat)g(by)f(the)g(same)g(methods)h(a)e(general)i (abstract)h(situation.)32 b(Let)448 799 y Fv(X)h Fs(=)25 b Fp(R)718 766 y Fo(n)782 799 y Fy(equipped)d(with)c(the)h(Lebesgue)h (measure,)h(denote)f Fv(Z)32 b Fs(=)25 b Fp(Z)2664 766 y Fo(n)2710 799 y Fy(,)19 b(and)g(for)g(each)g Fv(a)26 b Ft(2)f Fv(Z)448 912 y Fy(let)h Fv(K)641 926 y Fo(a)713 912 y Fs(=)j Fv(a)22 b Fs(+)g Fv(K)7 b Fy(,)25 b(where)h Fv(K)36 b Fs(=])22 b Ft(\000)g Fs(1)p Fv(=)p Fs(2)p Fv(;)15 b Fs(1)p Fv(=)p Fs(2])2014 879 y Fo(n)2064 912 y Fy(,)26 b(so)g(that)g Fv(K)2457 926 y Fo(a)2524 912 y Fy(is)g(a)f(unit)i(cube)f (centered)i(at)448 1024 y Fv(a)23 b Fy(and)g(we)g(ha)n(v)o(e)g Fv(X)33 b Fs(=)1195 956 y Fe(S)1271 1051 y Fo(a)p Fr(2)p Fo(Z)1427 1024 y Fv(K)1504 1038 y Fo(a)1568 1024 y Fy(disjoint)25 b(union.)30 b(Let)2265 1006 y Fv(\037)2322 1038 y Fo(a)2386 1024 y Fy(be)23 b(the)g(characteristic)k(function)448 1137 y(of)d Fv(K)624 1151 y Fo(a)690 1137 y Fy(and)g(if)g Fv(f)36 b Fs(:)27 b Fv(X)34 b Ft(!)27 b Fp(C)c Fy(let)h Fv(f)1531 1151 y Fo(a)1599 1137 y Fs(=)i Fv(f)10 b Ft(j)p Fv(K)1853 1151 y Fo(a)1895 1137 y Fy(.)30 b(W)-7 b(e)23 b(\002x)g(a)h(number)h Fs(1)i Fv(<)g(p)f(<)g Fs(2)e Fy(and)h(a)f(f)o (amily)448 1250 y Ft(f)p Fv(\025)546 1264 y Fo(a)588 1250 y Ft(g)633 1264 y Fo(a)p Fr(2)p Fo(Z)804 1250 y Fy(of)29 b(strictly)i(positi)n(v)o(e)f(numbers)h Fv(\025)1896 1264 y Fo(a)1973 1250 y Fv(>)36 b Fs(0)29 b Fy(and)g(we)g(de\002ne)g Fu(L)53 b Ft(\021)36 b Fv(`)2976 1217 y Fq(2)2976 1278 y Fo(\025)3021 1250 y Fs(\()p Fv(L)3118 1217 y Fo(p)3158 1250 y Fs(\))29 b Fy(as)g(the)448 1363 y(Banach)c(space)f(of)g(all)f (\(equi)n(v)n(alence)k(classes\))e(of)f(comple)o(x)g(functions)i Fv(f)32 b Fy(on)24 b Fv(X)30 b Fy(such)25 b(that)1231 1600 y Ft(k)p Fv(f)10 b Ft(k)1376 1615 y Fk(L)1483 1600 y Fs(:=)1605 1499 y Fe(\020)1677 1513 y(X)1674 1710 y Fo(a)p Fr(2)p Fo(Z)1826 1600 y Ft(k)p Fv(\025)1924 1614 y Fo(a)1966 1581 y Fv(\037)2023 1614 y Fo(a)2065 1600 y Fv(f)g Ft(k)2165 1562 y Fq(2)2165 1622 y Fo(L)2213 1603 y Fg(p)2253 1499 y Fe(\021)2307 1522 y Fq(1)p Fo(=)p Fq(2)2442 1600 y Fv(<)25 b Ft(1)p Fv(:)563 b Fy(\(6.22\))448 1881 y(Here)22 b Fv(L)708 1848 y Fo(p)772 1881 y Fs(=)j Fv(L)930 1848 y Fo(p)970 1881 y Fs(\()p Fv(X)7 b Fs(\))22 b Fy(b)n(ut)g(note)g(that,)g(by)f(identifying)2167 1862 y Fv(\037)2224 1895 y Fo(a)2265 1881 y Fv(f)35 b Ft(\021)25 b Fv(f)2486 1895 y Fo(a)2527 1881 y Fy(,)c(we)f(can)i(also)g(interpret) i Fu(L)448 1994 y Fy(as)i(a)f(con)l(v)o(eniently)k(normed)e(direct)f (sum)g(of)f(the)h(spaces)h Fv(L)2374 1961 y Fo(p)2413 1994 y Fs(\()p Fv(K)2525 2008 y Fo(a)2567 1994 y Fs(\))p Fy(,)f(see)f([DJT,)g(page)i(XIV].)448 2106 y(If)d Fv(\025)585 2120 y Fo(a)652 2106 y Fs(=)g(1)g Fy(for)f(all)h Fv(a)f Fy(we)g(set)g Fv(`)1419 2073 y Fq(2)1419 2134 y Fo(\025)1464 2106 y Fs(\()p Fv(L)1561 2073 y Fo(p)1601 2106 y Fs(\))j(=)f Fv(`)1796 2073 y Fq(2)1835 2106 y Fs(\()p Fv(L)1932 2073 y Fo(p)1972 2106 y Fs(\))p Fy(.)j(Observ)o(e)d(that)f Fv(`)2580 2073 y Fq(2)2619 2106 y Fs(\()p Fv(L)2716 2073 y Fq(2)2756 2106 y Fs(\))h(=)g Fv(L)2974 2073 y Fq(2)3014 2106 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)589 2219 y(Let)25 b Fv(q)j Fy(be)d(gi)n(v)o(en)h(by)1263 2184 y Fq(1)p 1263 2199 36 4 v 1263 2251 a Fo(p)1337 2219 y Fs(=)1446 2184 y Fq(1)p 1446 2199 V 1446 2251 a(2)1513 2219 y Fs(+)1615 2184 y Fq(1)p 1615 2199 V 1616 2251 a Fo(q)1660 2219 y Fy(,)f(so)h(that)g Fs(1)j Fv(<)f(p)h(<)f Fs(2)h Fv(<)g(q)i(<)e Ft(1)p Fy(.)k(W)-7 b(e)25 b(also)h(need)g(the)448 2332 y(space)f Fu(M)39 b Ft(\021)25 b Fv(`)945 2299 y Fr(1)945 2360 y Fo(\025)1020 2332 y Fs(\()p Fv(L)1117 2299 y Fo(q)1155 2332 y Fs(\))e Fy(de\002ned)i(by)e(the)h(condition)1350 2526 y Ft(k)p Fv(g)s Ft(k)1486 2541 y Fk(M)1607 2526 y Fs(:=)i(sup)1729 2604 y Fo(a)p 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Fo(q)2490 3008 y Fs(+)2606 2972 y Fq(1)p 2596 2987 57 4 v 2596 3039 a Fo(q)2630 3020 y Fc(0)2700 3008 y Fs(=)38 b(1)p Fy(,)31 b(de\002ned)g(by)g(the)448 3121 y(norm)1399 3233 y Ft(k)p Fv(h)p Ft(k)1541 3249 y Fk(M)1622 3257 y Fc(\003)1689 3233 y Fs(:=)1813 3147 y Fe(X)1811 3344 y Fo(a)p Fr(2)p Fo(Z)1963 3233 y Ft(k)p Fv(\025)2061 3196 y Fr(\000)p Fq(1)2061 3256 y Fo(a)2156 3215 y Fv(\037)2212 3247 y Fo(a)2254 3233 y Fv(h)p Ft(k)2351 3263 y Fo(L)2399 3244 y Fg(q)2429 3230 y Fc(0)2461 3233 y Fv(:)448 3482 y Fy(Belo)n(w)-6 b(,)19 b(when)f(we)f(speak)i(about)h Fv(w)1567 3449 y Fr(\003)1607 3482 y Fy(-topology)g(on)f Fu(M)31 b Fy(we)17 b(mean)i(the)f Fv(\033)s Fs(\()p Fu(M)d Fv(;)g Fu(M)3025 3496 y Fr(\003)3064 3482 y Fs(\))p Fy(-topology)-6 b(.)448 3595 y(Clearly)1251 3708 y Fu(M)1365 3670 y Fq(+)1350 3734 y(1)1449 3708 y Fs(=)24 b Ft(f)p Fv(g)30 b Ft(2)25 b Fu(M)39 b Ft(j)26 b Fv(g)j Ft(\025)24 b Fs(0)p Fv(;)15 b Ft(k)p Fv(g)s Ft(k)2325 3723 y Fk(M)2448 3708 y Ft(\024)25 b Fs(1)p Ft(g)448 3870 y Fy(is)f(a)f(con)l(v)o(e)o(x)i(compact)f (subset)h(of)f Fu(M)37 b Fy(for)24 b(the)f Fv(w)2018 3837 y Fr(\003)2058 3870 y Fy(-topology)-6 b(.)448 4048 y Fx(Lemma)23 b(6.5)46 b Fw(F)-10 b(or)24 b(eac)o(h)g Fv(f)34 b Ft(2)25 b Fu(L)40 b Fw(ther)m(e)24 b(is)g Fv(g)k Ft(2)d Fu(M)2117 4010 y Fq(+)2102 4074 y(1)2198 4048 y Fw(suc)o(h)f(that)h Ft(k)p Fv(f)10 b Ft(k)2696 4063 y Fk(L)2803 4048 y Fs(=)25 b Ft(k)p Fv(g)2990 4015 y Fr(\000)p Fq(1)3085 4048 y Fv(f)10 b Ft(k)3185 4068 y Fo(L)3233 4049 y Fb(2)3272 4048 y Fw(.)448 4227 y Fx(Pr)n(oof:)30 b Fy(W)-7 b(e)23 b(can)h(assume)g Fv(f)35 b Ft(\025)25 b Fs(0)p Fy(.)j(Since)c Fs(1)h(=)1976 4186 y Fo(p)p 1976 4206 36 4 v 1976 4258 a Fq(2)2042 4227 y Fs(+)2143 4186 y Fo(p)p 2143 4206 V 2144 4258 a(q)2188 4227 y Fy(,)e(we)g(ha)n(v)o(e:) 605 4458 y Ft(k)p Fv(f)695 4472 y Fo(a)737 4458 y Ft(k)782 4472 y Fo(L)830 4453 y Fg(p)896 4458 y Fs(=)i Ft(k)p Fv(f)1082 4472 y Fo(a)1123 4458 y Ft(k)1168 4410 y Fo(p=)p Fq(2)1168 4486 y Fo(L)1216 4467 y 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Fe(\001)3048 2350 y Fv(:)448 2641 y Fy(Our)30 b(purpose)j(is)d(to)h(apply)h (Proposition)h(6.3)d(with)g Fu(K)68 b Fs(=)38 b Fu(M)2517 2603 y Fq(+)2502 2667 y(1)2605 2641 y Fy(equipped)33 b(with)d(the)h Fv(w)3367 2608 y Fr(\003)3407 2641 y Fy(-)448 2754 y(topology)c(and)d Fu(F)35 b Fy(equal)25 b(to)f(the)g(set)g(of)g (all)g(functions)i Fv(F)2270 2768 y Fo(\013)2343 2754 y Fy(de\002ned)f(abo)o(v)o(e.)30 b(W)-7 b(e)23 b(sa)o(w)g(before)448 2867 y(that)37 b Fu(K)65 b Fy(is)36 b(a)g(con)l(v)o(e)o(x)i(compact)f (set.)68 b(From)36 b(the)g(second)i(representation)j(of)36 b Fv(F)3153 2881 y Fo(\013)3239 2867 y Fy(gi)n(v)o(en)448 2980 y(abo)o(v)o(e)e(it)e(follo)n(ws)i(that)f Fu(F)44 b Fy(is)33 b(a)f(con)l(v)o(e)o(x)j(set.)56 b(Each)33 b Fv(F)2279 2994 y Fo(\013)2361 2980 y Fy(is)g(a)f(con)l(v)o(e)o(x)i (function)h(because)448 3093 y Ft(k)p Fv(g)539 3060 y Fr(\000)p Fq(1)635 3093 y Fv(f)690 3060 y Fo(\013)738 3093 y Ft(k)783 3060 y Fq(2)783 3126 y Fo(L)831 3107 y Fb(2)899 3093 y Fs(=)998 3020 y 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Ft(1)p Fs(])30 b Fy(is)f(lo)n(wer)g(semicontinuous) k(on)d Fu(K)57 b Fy(if)29 b Fv(f)45 b Ft(2)36 b Fu(L)17 b Fv(;)e(f)45 b Ft(\025)36 b Fs(0)p Fy(.)448 4163 y(But)1370 4296 y Ft(k)p Fv(g)1461 4259 y Fr(\000)p Fq(1)1557 4296 y Fv(f)10 b Ft(k)1657 4259 y Fq(2)1657 4324 y Fo(L)1705 4305 y Fb(2)1768 4296 y Fs(=)1864 4210 y Fe(X)1911 4401 y Fo(a)2011 4173 y Fe(Z)2061 4379 y Fo(K)2121 4387 y Fg(a)2178 4296 y Fv(g)2224 4259 y Fr(\000)p Fq(2)2221 4319 y Fo(a)2319 4296 y Fv(f)2374 4259 y Fq(2)2364 4319 y Fo(a)2413 4296 y Fs(d)p Fv(x)448 4540 y Fy(and)28 b(the)g(set)g(of)f (lo)n(wer)h(semicontinuous)j(functions)f Fu(K)62 b Ft(!)32 b Fs([0)p Fv(;)15 b Ft(1)p Fs(])28 b Fy(is)g(stable)g(under)h(sums)448 4653 y(and)h(upper)h(bounds)h(of)d(arbitrary)j(f)o(amilies.)48 b(Hence)30 b(it)f(suf)n(\002ces)i(to)e(pro)o(v)o(e)h(that)g(each)h(map) 448 4766 y Fv(g)40 b Ft(7!)658 4693 y Fe(R)701 4798 y Fo(K)761 4806 y Fg(a)818 4766 y Fv(g)864 4733 y Fr(\000)p Fq(2)861 4789 y Fo(a)959 4766 y Fv(f)1014 4733 y 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b Fs(\))24 b Fy(and)g(let)1897 5225 y(43)p eop end end %%Page: 44 44 TeXDict begin HPSdict begin 44 43 bop 448 573 a Fv(f)43 b Ft(2)33 b Fv(C)695 587 y Fq(c)730 573 y Fs(\()p Fv(X)7 b Fs(\))29 b Fy(be)e(positi)n(v)o(e)j(and)e(not)g(zero.)43 b(Since)28 b Fv( )s Fs(\()p Fv(P)13 b Fs(\))p Fv(f)37 b Fy(is)28 b(essentially)j(the)d(con)l(v)n(olution)448 686 y(of)564 662 y Fe(b)548 686 y Fv( )f Fy(with)d Fv(f)10 b Fy(,)23 b(there)i(is)f(a)g(compact)i(set)f Fv(K)30 b Fy(with)24 b(non-empty)j(interior)f(such)f(that)g Fv( )s Fs(\()p Fv(P)13 b Fs(\))p Fv(f)37 b Ft(\025)448 799 y Fv(c)487 780 y(\037)545 813 y Fo(K)646 799 y Fy(with)d(a)g(number)h Fv(c)46 b(>)f Fs(0)p Fy(.)60 b(Let)34 b Fv(U)1772 813 y Fo(a)1847 799 y Fy(be)g(the)h(unitary)h(operator)g(of)e(translation)k (by)c Fv(a)448 912 y Fy(in)j Fv(L)617 879 y Fq(2)656 912 y Fs(\()p Fv(X)7 b Fs(\))p Fy(,)40 b(then)d Fv(U)1125 926 y Fo(a)1167 912 y Fv(f)59 b Ft(!)49 b Fs(0)36 b Fy(weakly)i(when)e Fv(a)50 b Ft(!)f(1)p Fy(,)39 b(hence)f Ft(k)p Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(U)2970 926 y Fo(a)3013 912 y Fv( )s Fs(\()p Fv(P)13 b Fs(\))p Fv(f)d Ft(k)50 b Fs(=)448 1024 y Ft(k)p Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv( )s Fs(\()p Fv(P)13 b Fs(\))p Fv(U)959 1038 y Fo(a)1004 1024 y Fv(f)d Ft(k)25 b(!)g Fs(0)p Fy(.)h(Since)19 b Fv(U)1631 992 y Fr(\003)1621 1047 y Fo(a)1670 1024 y Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv(U)1933 1038 y Fo(a)2001 1024 y Fs(=)25 b Fv(')p Fs(\()p Fv(Q)q Ft(\000)q Fv(a)p Fs(\))18 b Fy(we)g(get)g Ft(k)p Fv(')p Fs(\()p Fv(Q)q Ft(\000)q Fv(a)p Fs(\))3056 1006 y Fv(\037)3114 1038 y Fo(K)3182 1024 y Ft(k)26 b(!)f Fs(0)p Fy(,)448 1137 y(hence)g(\(1\))f(holds.)589 1250 y(Finally)-6 b(,)40 b(let)c(us)g(pro)o(v)o(e)h(that)f(\(1\))g Ft(\))g Fy(\(4\).)66 b(It)36 b(suf)n(\002ces)g(to)g(pro)o(v)o(e)h(that) f Fv(')p Fs(\()p Fv(Q)p Fs(\))p Fv( )s Fs(\()p Fv(P)13 b Fs(\))37 b Fy(is)448 1363 y(compact)25 b(if)873 1339 y Fe(b)857 1363 y Fv( )k Ft(2)c Fv(C)1096 1377 y Fq(c)1131 1363 y Fs(\()p Fv(X)7 b Fs(\))24 b Fy(and)g(for)g(this)g(it)f(suf)n (\002ces)i(that)2293 1339 y Fs(\026)2275 1363 y Fv( )s Fs(\()p Fv(P)13 b Fs(\))p Ft(j)p Fv(')p Ft(j)2587 1330 y Fq(2)2628 1363 y Fs(\()p Fv(Q)p Fs(\))p Fv( )s Fs(\()p Fv(P)g Fs(\))25 b Fy(be)e(compact.)448 1476 y(Since)34 b Fv(\030)46 b Fs(:=)d Ft(j)p Fv(')p Ft(j)1017 1443 y Fq(2)1100 1476 y Ft(2)g Fv(B)1273 1490 y Fq(w)1329 1476 y Fs(\()p Fv(X)7 b Fs(\))33 b Fy(and)h(since)g Fv( )s Fs(\()p Fv(P)13 b Fs(\))33 b Fy(is)g(the)g(operator)j(of)c(con)l(v)n (olution)38 b(by)33 b(a)448 1589 y(function)25 b(in)c Fv(\022)28 b Ft(2)d Fv(C)1086 1603 y Fq(c)1121 1589 y Fs(\()p Fv(X)7 b Fs(\))p Fy(,)23 b(we)e(are)h(reduced)i(to)e(pro)o (ving)i(that)e(the)h(inte)o(gral)g(operator)h Fv(S)i Fy(with)448 1702 y(k)o(ernel)i Fv(S)5 b Fs(\()p Fv(x;)15 b(y)s Fs(\))31 b(=)1103 1629 y Fe(R)1186 1678 y Fs(\026)1179 1702 y Fv(\022)r Fs(\()p Fv(z)c Ft(\000)21 b Fv(x)p Fs(\))p Fv(\030)t Fs(\()p Fv(z)t Fs(\))p Fv(\022)s Fs(\()p Fv(z)28 b Ft(\000)22 b Fv(y)s Fs(\))p Fv(dz)30 b Fy(is)c(compact.)38 b(If)26 b Fv(K)36 b Fs(=)30 b Fy(supp)25 b Fv(\022)i Fy(and)g Fs(\003)e Fy(is)448 1815 y(the)f(compact)h(set)f Fv(K)i Ft(\000)20 b Fv(K)7 b Fy(,)23 b(then)h(clearly)h(there)f(is)g(a) f(number)h Fv(C)29 b Fy(such)c(that)926 2058 y Ft(j)p Fv(S)5 b Fs(\()p Fv(x;)15 b(y)s Fs(\))p Ft(j)26 b(\024)f Fv(C)1456 1934 y Fe(Z)1506 2140 y Fo(K)1566 2148 y Fg(x)1624 2058 y Fv(\030)t Fs(\()p Fv(z)t Fs(\))p Fv(dz)1877 2039 y(\037)1935 2072 y Fq(\003)1989 2058 y Fs(\()p Fv(x)20 b Ft(\000)g Fv(y)s Fs(\))25 b Ft(\021)g Fv(\036)p Fs(\()p Fv(x)p Fs(\))2567 2039 y Fv(\037)2625 2072 y Fq(\003)2678 2058 y Fs(\()p Fv(x)c Ft(\000)f Fv(y)s Fs(\))448 2316 y Fy(where)i Fv(\036)j Ft(2)g Fv(C)921 2330 y Fq(0)960 2316 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)29 b(The)21 b(last)g(term)g(here)h (is)f(a)g(k)o(ernel)i(which)e(de\002nes)h(a)f(compact)h(operator)448 2429 y Fv(T)13 b Fy(.)68 b(Thus)37 b Fv(\021)s Fs(\()p Fv(Q)p Fs(\))p Fv(S)42 b Fy(is)37 b(a)f(Hilbert-Schmidt)j(operator)g (for)e(each)g Fv(\021)54 b Ft(2)49 b Fv(C)2868 2443 y Fq(c)2904 2429 y Fs(\()p Fv(X)7 b Fs(\))37 b Fy(and)g(from)448 2542 y(the)d(preceding)i(estimate)e(we)e(get)i Ft(k)p Fs(\()p Fv(S)f Ft(\000)27 b Fv(\021)s Fs(\()p Fv(Q)p Fs(\))p Fv(S)5 b Fs(\))p Fv(u)p Ft(k)45 b(\024)d(k)p Fs(\(1)29 b Ft(\000)e Fv(\021)s Fs(\()p Fv(Q)p Fs(\)\))p Fv(T)13 b Ft(j)p Fv(u)p Ft(jk)34 b Fy(for)g(each)448 2655 y Fv(u)29 b Ft(2)g Fv(L)681 2622 y Fq(2)720 2655 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)35 b(Thus)25 b Ft(k)p Fv(S)i Ft(\000)22 b Fv(\021)s Fs(\()p Fv(Q)p Fs(\))p Fv(S)5 b Ft(k)30 b(\024)e(k)p Fs(\(1)23 b Ft(\000)f Fv(\021)s Fs(\()p Fv(Q)p Fs(\)\))p Fv(T)13 b Ft(k)26 b Fy(and)g(the)f(right)i (hand)f(side)g(tends)448 2768 y(to)e(zero)g(if)f Fv(\021)29 b Ft(\021)c Fv(\021)1014 2782 y Fo(\013)1086 2768 y Fy(is)f(an)f (approximate)j(unit)e(for)g Fv(C)2113 2782 y Fq(0)2153 2768 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)p 3371 2760 67 67 v 589 2931 a(W)-7 b(e)27 b(shall)h(consider)h(no)n(w)d(a)h(general)i (class)f(of)f(\002lters)g(de\002ned)h(in)f(terms)g(of)g(the)h(metric) 448 3044 y(and)e(measure)g(space)g(structure.)36 b(W)-7 b(e)24 b(consider)j(only)f(the)f(case)h(of)f(an)g(Euclidean)i(space)f Fv(X)7 b Fy(,)448 3157 y(the)21 b(e)o(xtension)i(to)e(the)f(case)i(of)e (locally)i(compact)g(groups)g(or)f(metric)g(spaces)h(being)g(ob)o (vious.)448 3270 y(W)-7 b(e)29 b(set)g Fv(B)794 3284 y Fo(a)836 3270 y Fs(\()p Fv(r)s Fs(\))36 b(=)f Ft(f)p Fv(x)h Ft(2)g Fv(X)43 b Ft(j)36 b(j)p Fv(x)25 b Ft(\000)f Fv(a)p Ft(j)36 b Fv(<)f(r)s Ft(g)p Fy(,)30 b Fv(B)2124 3284 y Fo(a)2201 3270 y Fs(=)36 b Fv(B)2377 3284 y Fo(a)2418 3270 y Fs(\(1\))30 b Fy(and)g Fv(B)5 b Fs(\()p Fv(r)s Fs(\))35 b(=)h Fv(B)3122 3284 y Fq(0)3161 3270 y Fs(\()p Fv(r)s Fs(\))p Fy(.)45 b(T)-7 b(o)448 3382 y(each)25 b(function)g Fv(\027)31 b Fs(:)26 b Fv(X)32 b Ft(!)p Fs(]0)p Fv(;)15 b Ft(1)p Fs([)24 b Fy(such)g(that)g Fs(lim)15 b(inf)2139 3396 y Fo(a)p Fr(!1)2336 3382 y Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))26 b(=)f(0)e Fy(we)g(associate)j(a)d(set)g(of) 448 3495 y(subsets)j(of)d Fv(X)30 b Fy(as)24 b(follo)n(ws:)1023 3700 y Fu(N)1105 3714 y Fo(\027)1174 3700 y Fs(=)h Ft(f)p Fv(N)35 b Ft(\032)25 b Fv(X)33 b Ft(j)25 b Fs(lim)15 b(sup)1727 3772 y Fo(a)p Fr(!1)1971 3700 y Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))2140 3662 y Fr(\000)p Fq(1)2235 3700 y Ft(j)p Fv(N)30 b Ft(\\)20 b Fv(B)2513 3714 y Fo(a)2554 3700 y Ft(j)26 b Fv(<)f Ft(1g)p Fv(:)355 b Fy(\(7.27\))448 3949 y(Clearly)25 b Fu(F)821 3963 y Fo(\027)889 3949 y Fs(=)g Ft(f)p Fv(F)39 b Ft(\032)25 b Fv(X)32 b Ft(j)26 b Fv(F)1452 3916 y Fq(c)1513 3949 y Ft(2)f Fu(N)1681 3963 y Fo(\027)1724 3949 y Ft(g)e Fy(is)h(a)f(\002lter)g(on)h Fv(X)30 b Fy(\002ner)23 b(than)i(the)e(Fr)5 b(\264)-35 b(echet)25 b(\002lter)-5 b(.)448 4161 y Fx(Theor)n(em)24 b(7.5)46 b Fw(Let)22 b Fv(X)32 b Fs(=)25 b Fp(R)1389 4128 y Fo(n)1457 4161 y Fw(and)e(let)g Fv(\027)31 b Fs(:)25 b Fv(X)33 b Ft(!)p Fs(]0)p Fv(;)15 b Ft(1)p Fs([)22 b Fw(suc)o(h)h(that)g Fs(lim)15 b(inf)2903 4175 y Fo(a)p Fr(!1)3101 4161 y Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))26 b(=)f(0)448 4274 y Fw(and)35 b Fs(sup)755 4296 y Fr(j)p Fo(b)p Fr(\000)p Fo(a)p Fr(j\024)p Fo(r)1025 4274 y Fv(\027)6 b Fs(\()p Fv(b)p Fs(\))p Fv(=\027)g Fs(\()p Fv(a)p Fs(\))47 b Fv(<)e Ft(1)33 b Fw(for)i(eac)o(h)g(r)m(eal)g Fv(r)s Fw(.)60 b(If)34 b Fv(S)50 b Ft(2)45 b(B)s Fs(\()p Fv(L)2795 4241 y Fq(2)2835 4274 y Fs(\()p Fv(X)7 b Fs(\)\))35 b Fw(is)f(of)g(class)448 4387 y Fv(C)520 4354 y Fq(u)563 4387 y Fs(\()p Fv(Q)p Fs(\))23 b Fw(and)i(if)e Fv(S)30 b Ft(2)25 b(B)s Fs(\()p Fv(L)1293 4354 y Fo(p)1332 4387 y Fs(\()p Fv(X)7 b Fs(\)\))25 b Fw(for)e(some)h Fv(p)h(<)g Fs(2)p Fw(,)e(then)h Fv(S)k Fw(is)23 b(left)h Fu(F)2707 4401 y Fo(\027)2750 4387 y Fw(-decay)h(pr)m(eserving)o(.)448 4600 y Fx(Pr)n(oof:)52 b Fy(W)-7 b(e)33 b(can)i(approximate)i(in)d (norm)h(in)f Ft(B)s Fs(\()p Fv(L)2138 4567 y Fq(2)2177 4600 y Fs(\()p Fv(X)7 b Fs(\)\))35 b Fy(the)g(operator)h Fv(S)i Fy(by)d(operators)448 4713 y(which)30 b(are)f(in)g Ft(B)s Fs(\()p Fv(L)1097 4680 y Fq(2)1136 4713 y Fs(\()p Fv(X)7 b Fs(\)\))26 b Ft(\\)e(B)s Fs(\()p Fv(L)1594 4680 y Fo(p)1633 4713 y Fs(\()p Fv(X)7 b Fs(\)\))30 b Fy(and)f(ha)n(v)o(e)h (\002nite)f(range.)46 b(Indeed,)32 b(the)d(approxi-)448 4825 y(mation)24 b(procedure)i(\(2.12\))f(used)f(in)f(the)h(proof)g(of) f(Proposition)j(2.33)e(is)f(such)h(that)g(it)f(lea)n(v)o(es)448 4938 y Ft(B)s Fs(\()p Fv(L)608 4905 y Fq(2)647 4938 y Fs(\()p Fv(X)7 b Fs(\)\))23 b Ft(\\)d(B)s Fs(\()p Fv(L)1098 4905 y Fo(p)1137 4938 y Fs(\()p Fv(X)7 b Fs(\)\))25 b Fy(in)l(v)n(ariant)i(\(because)f Fv(V)2084 4953 y Fo(k)2151 4938 y Fy(are)e(isometries)i(in)e Fv(L)2838 4905 y Fo(p)2901 4938 y Fy(too\).)32 b(Since)24 b(the)1897 5225 y(44)p eop end end %%Page: 45 45 TeXDict begin HPSdict begin 45 44 bop 448 573 a Fy(set)32 b(of)g(left)g Fu(F)921 587 y Fo(\027)964 573 y Fy(-decay)h(preserving)i (operators)f(is)e(norm)f(closed)j(in)d Ft(B)s Fs(\()p Fv(L)2860 540 y Fq(2)2899 573 y Fs(\()p Fv(X)7 b Fs(\)\))p Fy(,)35 b(we)c(may)448 686 y(assume)c(in)f(the)g(rest)g(of)g(the)g (proof)h(that)f Fv(S)k Fy(is)c(of)g(\002nite)g(range.)36 b(According)28 b(to)e(Lemma)f(7.2,)448 799 y(it)h(suf)n(\002ces)h(to)f (sho)n(w)g(that,)h(for)f(a)g(gi)n(v)o(en)h(Borel)f(set)g Fv(N)40 b Ft(2)30 b Fu(N)2381 813 y Fo(\027)2450 799 y Fy(and)c(for)h(an)o(y)f(number)h Fv(")j(>)g Fs(0)p Fy(,)448 912 y(there)25 b(is)e(a)g(Borel)h(set)g Fv(M)35 b Ft(2)25 b Fu(N)1439 926 y Fo(\027)1505 912 y Fy(such)f(that)h Ft(k)1899 893 y Fv(\037)1956 926 y Fo(M)2031 907 y Fb(c)2066 912 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)2269 893 y(\037)2327 926 y Fo(N)2394 912 y Fs(\()p Fv(Q)p Fs(\))p Ft(k)h Fv(<)f(")p Fy(.)589 1024 y(In)i(the)f(rest)h(of)g(the)f(proof)i(we)d(shall)j (freely)f(use)g(the)g(notations)i(introduced)g(in)d(Section)448 1137 y(6)d(\(see)g(also)h(the)f(proof)g(of)g(Proposition)i(2.33\).)k (In)23 b(particular)l(,)j Fv(q)f Fy(is)d(de\002ned)i(by)3032 1102 y Fq(1)p 3032 1117 36 4 v 3032 1169 a Fo(p)3102 1137 y Fs(=)3208 1102 y Fq(1)p 3208 1117 V 3208 1169 a(2)3271 1137 y Fs(+)3369 1102 y Fq(1)p 3369 1117 V 3370 1169 a Fo(q)3414 1137 y Fy(.)448 1266 y(If)g Fv(f)34 b Ft(2)25 b Fv(L)759 1233 y Fq(2)798 1266 y Fs(\()p Fv(X)7 b Fs(\))24 b Fy(we)f(ha)n(v)o(e)712 1470 y Ft(k)757 1452 y Fv(\037)814 1484 y Fo(N)882 1470 y Fv(f)10 b Ft(k)982 1488 y Fo(L)1030 1470 y Fg(p)1066 1488 y Fq(\()p Fo(K)1153 1496 y Fg(a)1190 1488 y Fq(\))1247 1470 y Ft(\024)25 b(k)1388 1452 y Fv(\037)1445 1484 y Fo(N)1513 1470 y Ft(k)1558 1488 y Fo(L)1606 1470 y Fg(q)1640 1488 y Fq(\()p Fo(K)1727 1496 y Fg(a)1765 1488 y Fq(\))1797 1470 y Ft(k)p Fv(f)10 b Ft(k)1942 1490 y Fo(L)1990 1471 y Fb(2)2025 1490 y Fq(\()p Fo(K)2112 1498 y Fg(a)2149 1490 y Fq(\))2206 1470 y Ft(\024)25 b(j)p Fv(N)30 b Ft(\\)20 b Fv(K)2588 1484 y Fo(a)2630 1470 y Ft(j)2655 1432 y Fq(1)p Fo(=q)2764 1470 y Ft(k)p Fv(f)10 b Ft(k)2909 1490 y Fo(L)2957 1471 y Fb(2)2992 1490 y Fq(\()p Fo(K)3079 1498 y Fg(a)3117 1490 y Fq(\))3148 1470 y Fv(:)448 1674 y Fy(Since)30 b Fv(N)47 b Ft(2)36 b Fu(N)978 1688 y Fo(\027)1050 1674 y Fy(we)29 b(can)h(\002nd)g(a)f(constant)j Fv(c)d Fy(such)i(that)f Ft(j)p Fv(N)35 b Ft(\\)24 b Fv(K)2636 1688 y Fo(a)2678 1674 y Ft(j)37 b(\024)f Fv(c\027)6 b Fs(\()p Fv(a)p Fs(\))30 b Fy(\(note)h(that)448 1787 y(the)24 b(de\002nition)h(\(7.27\))f(does)g (not)g(in)l(v)n(olv)o(e)i(the)d(restriction)j(of)e Fv(\027)k Fy(to)23 b(bounded)j(sets\).)j(Thus,)24 b(if)448 1900 y(we)h(tak)o(e)i Fv(\025)808 1914 y Fo(a)879 1900 y Fs(=)j Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))1149 1867 y Fr(\000)p Fq(1)p Fo(=q)1338 1900 y Fy(for)26 b Fv(a)k Ft(2)f Fv(Z)36 b Ft(\021)30 b Fp(Z)1897 1867 y Fo(n)1943 1900 y Fy(,)c(we)f(get)2259 1882 y Fv(\037)2316 1914 y Fo(N)2383 1900 y Fv(f)39 b Ft(2)29 b Fu(L)43 b Fy(with)26 b(the)g(notations)i(of)448 2013 y(Section)e(6.)34 b(In)25 b(other)h(terms,)f(we)g(see)g(that)h(we) e(ha)n(v)o(e)2162 1995 y Fv(\037)2219 2027 y Fo(N)2286 2013 y Fs(\()p Fv(Q)p Fs(\))29 b Ft(2)f(B)s Fs(\()p Fv(L)2706 1980 y Fq(2)2745 2013 y Fs(\()p Fv(X)7 b Fs(\))p Fv(;)15 b Fu(L)k Fs(\))p Fy(.)33 b(Let)24 b Fv(T)41 b Fs(=)448 2126 y Fv(S)509 2107 y(\037)566 2140 y Fo(N)633 2126 y Fs(\()p Fv(Q)p Fs(\))20 b Fy(and)h(let)f(us)g(assume)h(that)f(we)f (also)i(ha)n(v)o(e)g Fv(S)30 b Ft(2)25 b(B)s Fs(\()p Fu(L)17 b Fs(\))p Fy(.)27 b(Then)20 b Fv(T)38 b Ft(2)25 b(B)s Fs(\()p Fv(L)3072 2093 y Fq(2)3111 2126 y Fs(\()p Fv(X)7 b Fs(\))p Fv(;)15 b Fu(L)k Fs(\))448 2239 y Fy(and)h(we)e(can)h (apply)i(the)e(Maure)o(y)h(type)f(f)o(actorization)k(theorem)d(Theorem) g(6.7,)f(where)h Fu(H)51 b Fs(=)448 2352 y Fv(L)510 2319 y Fq(2)550 2352 y Fs(\()p Fv(X)7 b Fs(\))p Fy(.)66 b(Thus)36 b(we)f(can)i(write)f Fv(T)61 b Fs(=)48 b Fv(g)s Fs(\()p Fv(Q)p Fs(\))p Fv(R)37 b Fy(for)g(some)f Fv(R)49 b Ft(2)f(B)s Fs(\()p Fv(L)2816 2319 y Fq(2)2855 2352 y Fs(\()p Fv(X)7 b Fs(\)\))37 b Fy(and)f(some)448 2465 y(function)d Fv(g)42 b Ft(2)c Fu(M)15 b Fy(,)31 b(which)g(means)g(that)g Fv(G)39 b Fs(:=)f(sup)2194 2486 y Fo(a)p Fr(2)p Fo(Z)2351 2465 y Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))2520 2432 y Fr(\000)p Fq(1)p Fo(=q)2684 2465 y Ft(k)p Fv(g)s Ft(k)2820 2483 y Fo(L)2868 2464 y Fg(q)2903 2483 y Fq(\()p Fo(K)2990 2491 y Fg(a)3028 2483 y Fq(\))3090 2465 y Fy(is)30 b(a)g(\002nite)448 2578 y(number)-5 b(.)30 b(If)23 b Fv(t)i(>)g Fs(0)e Fy(and)h Fv(M)36 b Fs(=)25 b Ft(f)p Fv(x)g Ft(j)h Fv(g)s Fs(\()p Fv(x)p Fs(\))g Fv(>)f(t)p Ft(g)e Fy(then)i(we)d(get)i(for)g(all)g Fv(a)h Ft(2)g Fv(Z)7 b Fy(:)890 2782 y Ft(j)p Fv(M)31 b Ft(\\)19 b Fv(K)1191 2796 y Fo(a)1233 2782 y Ft(j)26 b Fs(=)f Ft(k)1425 2763 y Fv(\037)1482 2796 y Fo(M)1561 2782 y Ft(k)1606 2738 y Fo(q)1606 2815 y(L)1654 2796 y Fg(q)1689 2815 y Fq(\()p Fo(K)1776 2823 y Fg(a)1814 2815 y Fq(\))1870 2782 y Ft(\024)g(k)p Fv(g)s(=t)p Ft(k)2180 2738 y Fo(q)2180 2815 y(L)2228 2796 y Fg(q)2265 2815 y Fq(\()p Fo(K)2352 2823 y Fg(a)2389 2815 y Fq(\))2446 2782 y Ft(\024)g Fs(\()p Fv(G=t)p Fs(\))2761 2744 y Fo(q)2801 2782 y Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))p Fv(:)589 2986 y Fy(Note)20 b(that)g(the)g(second)h(condition)h(imposed)f(on)e Fv(\027)24 b Fy(in)c(Theorem)g(7.5)f(ca)g(be)h(stated)g(as)g(fol-)448 3099 y(lo)n(ws:)33 b(there)26 b(is)g(an)f(increasing)j(strictly)g (positi)n(v)o(e)e(function)i Fv(\016)h Fy(on)c Fs([0)p Fv(;)15 b Ft(1)p Fs([)26 b Fy(such)g(that)g Fv(\027)6 b Fs(\()p Fv(b)p Fs(\))29 b Ft(\024)448 3212 y Fv(\016)s Fs(\()p Ft(j)p Fv(b)8 b Ft(\000)g Fv(a)p Ft(j)p Fs(\))p Fv(\027)e Fs(\()p Fv(a)p Fs(\))23 b Fy(for)d(all)g Fv(a;)15 b(b)p Fy(.)28 b(Indeed,)22 b(we)d(may)i(tak)o(e)g Fv(\016)s Fs(\()p Fv(r)s Fs(\))26 b(=)f(sup)2568 3234 y Fr(j)p Fo(b)p Fr(\000)p Fo(a)p Fr(j\024)p Fo(r)2837 3212 y Fv(\027)6 b Fs(\()p Fv(b)p Fs(\))p Fv(=\027)g Fs(\()p Fv(a)p Fs(\))p Fy(.)29 b(No)n(w)448 3325 y(let)h Fv(a)36 b Ft(2)f Fv(X)h Fy(and)30 b(let)f Fv(D)s Fs(\()p Fv(a)p Fs(\))g Fy(be)g(the)h(set)f(of) h Fv(b)36 b Ft(2)f Fv(Z)g Fy(such)30 b(that)g Fv(K)2556 3340 y Fo(b)2619 3325 y Fy(intersects)i Fv(B)3060 3339 y Fo(a)3101 3325 y Fy(.)45 b(Clearly)448 3438 y Fv(D)s Fs(\()p Fv(a)p Fs(\))23 b Fy(contains)h(at)f(most)g Fs(2)1323 3405 y Fo(n)1392 3438 y Fy(points)h Fv(b)e Fy(all)g(of)h(them)g (satisfying)i Ft(j)p Fv(b)17 b Ft(\000)f Fv(a)p Ft(j)26 b(\024)2849 3372 y(p)p 2925 3372 55 4 v 66 x Fv(n)16 b Fs(+)h(1)p Fy(.)28 b(Hence:)448 3655 y Ft(j)p Fv(M)22 b Ft(\\)12 b Fv(K)733 3669 y Fo(a)774 3655 y Ft(j)26 b(\024)970 3569 y Fe(X)921 3770 y Fo(b)p Fr(2)p Fo(D)r Fq(\()p Fo(a)p Fq(\))1165 3655 y Ft(j)p Fv(M)c Ft(\\)12 b Fv(K)1450 3670 y Fo(b)1484 3655 y Ft(j)25 b(\024)g Fs(2)1675 3618 y Fo(n)1784 3655 y Fs(sup)1738 3738 y Fo(b)p Fr(2)p Fo(D)r Fq(\()p Fo(a)p Fq(\))1967 3655 y Fs(\()p Fv(G=t)p Fs(\))2186 3618 y Fo(q)2226 3655 y Fv(\027)6 b Fs(\()p Fv(b)p Fs(\))25 b Ft(\024)g Fs(2)2552 3618 y Fo(n)2600 3655 y Fs(\()p Fv(G=t)p Fs(\))2819 3618 y Fo(q)2858 3655 y Fv(\016)s Fs(\()2936 3586 y Ft(p)p 3013 3586 V 3013 3655 a Fv(n)11 b Fs(+)h(1\))p Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))p Fv(;)448 3960 y Fy(which)24 b(pro)o(v)o(es)h (that)f Fv(M)33 b Fy(belongs)25 b(to)f Fu(N)1717 3974 y Fo(\027)1760 3960 y Fy(.)k(On)23 b(the)h(other)g(hand,)g(we)f(ha)n(v) o(e:)797 4164 y Ft(k)842 4145 y Fv(\037)900 4178 y Fo(M)975 4159 y Fb(c)1010 4164 y Fs(\()p Fv(Q)p Fs(\))p Fv(T)13 b Ft(k)26 b Fs(=)f Ft(k)1430 4145 y Fv(\037)1487 4178 y Fo(M)1562 4159 y Fb(c)1598 4164 y Fs(\()p Fv(Q)p Fs(\))p Fv(g)s Fs(\()p Fv(Q)p Fs(\))p Fv(R)q Ft(k)i(\024)e(k)2211 4145 y Fv(\037)2268 4178 y Fo(M)2343 4159 y Fb(c)2378 4164 y Fv(g)s Ft(k)2469 4178 y Fo(L)2517 4159 y Fc(1)2588 4164 y Ft(k)p Fv(R)q Ft(k)h(\024)f Fv(t)p Ft(k)p Fv(R)q Ft(k)p Fv(:)448 4368 y Fy(T)-7 b(o)23 b(\002nish)h(the)f(proof)i(of)e (the)h(theorem)h(it)e(suf)n(\002ces)h(to)g(tak)o(e)g Fv(t)h Fs(=)g Fv("=)p Ft(k)p Fv(R)q Ft(k)p Fy(.)589 4481 y(W)-7 b(e)30 b(still)h(ha)n(v)o(e)h(to)e(pro)o(v)o(e)h(that)g Fv(S)43 b Ft(2)38 b(B)s Fs(\()p Fu(L)17 b Fs(\))p Fy(.)50 b(Since)30 b Fv(S)35 b Fy(is)c(of)f(\002nite)h(range,)i(there)e(is)g(a) 448 4594 y(number)25 b Fv(r)g Fy(such)f(that)1163 4576 y Fv(\037)1220 4608 y Fo(a)1262 4594 y Fs(\()p Fv(Q)p Fs(\))1404 4576 y Fv(\037)1461 4609 y Fo(b)1496 4594 y Fs(\()p Fv(Q)p Fs(\))i(=)f(0)e Fy(if)g Ft(j)p Fv(a)e Ft(\000)e Fv(b)p Ft(j)26 b(\025)f Fv(r)s Fy(.)i(Then)d(for)g(an)o(y)f Fv(f)35 b Ft(2)25 b Fu(L)17 b Fy(:)536 4725 y Fe(X)583 4916 y Fo(a)682 4812 y Fv(\025)735 4774 y Fq(2)735 4834 y Fo(a)777 4812 y Ft(k)822 4793 y Fv(\037)879 4826 y Fo(a)921 4812 y Fv(S)5 b(f)10 b Ft(k)1082 4774 y Fq(2)1082 4834 y Fo(L)1130 4815 y Fg(p)1195 4812 y Fs(=)1291 4725 y Fe(X)1338 4916 y Fo(a)1438 4812 y Fv(\025)1491 4774 y Fq(2)1491 4834 y Fo(a)1532 4812 y Ft(k)1652 4725 y Fe(X)1592 4927 y Fr(j)p Fo(b)p Fr(\000)p Fo(a)p Fr(j)p Fo()f Fs(0)p Fy(,)448 2621 y(and)c(for)f(an)o(y)g Fv(N)43 b Ft(2)33 b Fu(N)1189 2635 y Fo(\027)1259 2621 y Fy(there)c(is)f Fv(M)43 b Ft(2)33 b Fu(N)1863 2635 y Fo(\027)1934 2621 y Fy(such)28 b(that)h Ft(k)2336 2602 y Fv(\037)2393 2635 y Fo(M)2468 2616 y Fb(c)2503 2621 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)2706 2602 y(\037)2764 2635 y Fo(N)2831 2621 y Fs(\()p Fv(Q)p Fs(\))p Ft(k)34 b(\024)f Fv(")p Fy(.)42 b(No)n(w)448 2734 y(let)30 b Fv(N)39 b Fy(be)29 b(a)g(Borel)h Fs(w)q Fy(-small)g(set,)h(i.e.)d(such)j(that)f Ft(j)p Fv(N)35 b Ft(\\)24 b Fv(B)2358 2748 y Fo(a)2399 2734 y Ft(j)37 b(!)f Fs(0)29 b Fy(if)h Fv(a)36 b Ft(!)g(1)p Fy(.)46 b(W)-7 b(e)28 b(shall)448 2846 y(pro)o(v)o(e)k(that)g(there)h (is)e(a)g(function)i Fv(\027)k Fy(with)31 b(the)h(properties)i (required)f(in)f(Theorem)g(7.5)f(and)448 2959 y(with)24 b Fs(lim)759 2973 y Fo(a)p Fr(!1)957 2959 y Fv(\027)6 b Fs(\()p Fv(a)p Fs(\))26 b(=)f(0)f Fy(such)g(that)h Fv(N)35 b Ft(2)25 b Fu(N)1942 2973 y Fo(\027)1986 2959 y Fy(.)j(This)c(\002nishes)g(the)g(proof)h(of)f(the)g(corollary)448 3072 y(because)i(the)e(relation)h Fv(M)35 b Ft(2)25 b Fu(N)1484 3086 y Fo(\027)1550 3072 y Fy(implies)g(no)n(w)e(that)h Fv(M)33 b Fy(is)23 b Fs(w)q Fy(-small.)589 3185 y(W)-7 b(e)29 b(construct)k Fv(\027)h Fy(as)c(follo)n(ws.)48 b(The)30 b(relation)h Fv(\022)s Fs(\()p Fv(r)s Fs(\))37 b(=)f(sup)2542 3207 y Fr(j)p Fo(a)p Fr(j\025)p Fo(r)2727 3185 y Ft(j)p Fv(N)f Ft(\\)24 b Fv(B)3014 3199 y Fo(a)3056 3185 y Ft(j)29 b Fy(de\002nes)i(a)448 3298 y(positi)n(v)o(e)25 b(decreasing)h(function)f(on)f Fs([0)p Fv(;)15 b Ft(1)p Fs([)23 b Fy(which)h(tends)g(to)f(zero)h(at)f(in\002nity)h(and)g(such)g (that)448 3411 y Ft(j)p Fv(N)32 b Ft(\\)21 b Fv(B)729 3425 y Fo(a)771 3411 y Ft(j)29 b(\024)f Fv(\022)s Fs(\()p Ft(j)p Fv(a)p Ft(j)p Fs(\))d Fy(for)g(all)h Fv(a)i Ft(2)h Fv(X)7 b Fy(.)33 b(W)-7 b(e)25 b(set)g Fv(\030)t Fs(\()p Fv(t)p Fs(\))k(=)g Fv(\022)s Fs(\(0\))24 b Fy(if)h Fs(0)k Ft(\024)g Fv(t)f(<)g Fs(1)d Fy(and)h(for)g Fv(k)31 b Ft(\025)e Fs(0)448 3524 y Fy(inte)o(ger)h(and)f Fs(2)931 3491 y Fo(k)1008 3524 y Ft(\024)34 b Fv(t)g(<)g Fs(2)1330 3491 y Fo(k)r Fq(+1)1490 3524 y Fy(we)28 b(de\002ne)h Fv(\030)t Fs(\()p Fv(t)p Fs(\))34 b(=)g(max)p Ft(f)p Fv(\030)t Fs(\(2)2498 3491 y Fo(k)r Fr(\000)p Fq(1)2632 3524 y Fs(\))p Fv(=)p Fs(2)p Fv(;)15 b(\022)s Fs(\(2)2923 3491 y Fo(k)2968 3524 y Fs(\))p Ft(g)p Fy(.)43 b(So)27 b Fv(\030)32 b Fy(is)c(a)448 3637 y(strictly)j(positi)n(v)o(e)g (decreasing)h(function)g(on)d Fs([0)p Fv(;)15 b Ft(1)p Fs([)30 b Fy(which)f(tends)i(to)e(zero)h(at)f(in\002nity)h(and)448 3750 y(such)25 b(that)f Fv(\022)j Ft(\024)e Fv(\030)t Fy(.)j(Moreo)o(v)o(er)l(,)d(if)e Fs(2)1579 3717 y Fo(k)1647 3750 y Ft(\024)i Fv(s)g(<)g Fs(2)1952 3717 y Fo(k)r Fq(+1)2108 3750 y Fy(and)f Fs(2)2307 3717 y Fo(k)r Fq(+)p Fo(p)2466 3750 y Ft(\024)h Fv(t)g(<)g Fs(2)2761 3717 y Fo(k)r Fq(+)p Fo(p)p Fq(+1)3007 3750 y Fy(then)809 3931 y Fv(\030)t Fs(\()p Fv(t)p Fs(\))h(=)f Fv(\030)t Fs(\(2)1202 3893 y Fo(k)r Fq(+)p Fo(p)1336 3931 y Fs(\))g Ft(\025)g Fv(\030)t Fs(\(2)1616 3893 y Fo(k)r Fq(+)p Fo(p)p Fr(\000)p Fq(1)1840 3931 y Fs(\))p Fv(=)p Fs(2)i Ft(\025)e Fv(:)15 b(:)g(:)26 b Ft(\025)f Fs(2)2360 3893 y Fr(\000)p Fo(p)2455 3931 y Fv(\030)t Fs(\(2)2579 3893 y Fo(k)2622 3931 y Fs(\))h(=)f(2)2824 3893 y Fr(\000)p Fo(p)2919 3931 y Fv(\030)t Fs(\()p Fv(s)p Fs(\))448 4111 y Fy(hence)20 b Fv(\030)t Fs(\()p Fv(s)p Fs(\))26 b Ft(\025)f Fv(\030)t 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Fy(if)448 4450 y Ft(j)p Fv(a)p Ft(j)c(\024)f(j)p Fv(b)p Ft(j)e Fy(and)h(if)g Ft(j)p Fv(a)p Ft(j)h Fv(>)g Ft(j)p Fv(b)p Ft(j)f Fy(then)1311 4617 y Fv(\027)6 b Fs(\()p Fv(b)p Fs(\))p 1307 4658 170 4 v 1307 4741 a Fv(\027)g Fs(\()p Fv(a)p Fs(\))1512 4679 y(=)1622 4617 y Fv(\030)t Fs(\()p Ft(j)p Fv(b)p Ft(j)p Fs(\))p 1617 4658 214 4 v 1617 4741 a Fv(\030)t Fs(\()p Ft(j)p Fv(a)p Ft(j)p Fs(\))1866 4679 y Ft(\024)1972 4617 y Fs(2)p Ft(j)p Fv(a)p Ft(j)p 1972 4658 145 4 v 1999 4741 a(j)p Fv(b)p Ft(j)2151 4679 y(\024)25 b Fs(2\(1)c(+)f Fv(r)s Fs(\))p Fv(:)448 4915 y Fy(Thus)k(the)g(second)h(condition)h(imposed)f(on)e Fv(\027)29 b Fy(in)23 b(Theorem)h(7.5)g(is)f(also)h(satis\002ed.)p 3371 4907 67 67 v 1897 5225 a(46)p eop end end %%Page: 47 47 TeXDict begin HPSdict begin 47 46 bop 448 573 a Fx(Remark)23 b(7.7)46 b Fy(W)-7 b(e)27 b(stress)i(that)g(we)e(shall)h(need)h(this)g (theorem)f(for)g(a)g(v)o(ery)g(simple)h(class)f(of)448 686 y(operators,)e(namely)e Fv(S)31 b Fs(=)25 b Fv( )s Fs(\()p Fv(P)13 b Fs(\))24 b Fy(with)f Fv( )s Fs(\()p Fv(k)s Fs(\))j(=)f Fv(k)2070 653 y Fo(\013)2120 686 y Fs(\()2155 618 y Fe(P)2251 713 y Fr(j)p Fo(\014)s Fr(j\024)p Fo(m)2470 686 y Fv(k)2520 653 y Fq(2)p Fo(\014)2603 686 y Fs(\))2638 653 y Fr(\000)p Fq(1)p Fo(=)p Fq(2)2826 686 y Fy(and)f Ft(j)p Fv(\013)p Ft(j)i(\024)f Fv(m)p Fy(.)589 888 y(As)e(a)f(\002nal)h(e)o(xample,)g(we)g(introduce)i(no)n (w)d(classes)j(of)e(v)n(anishing)i(at)e(in\002nity)h(functions)448 1001 y(of)j(a)f(more)g(topological)k(nature.)39 b(Let)25 b(us)i(\002x)f(a)g Fw(uniformly)i(discr)m(ete)g(set)g Fv(L)j Ft(\032)f Fv(X)7 b Fy(,)26 b(i.e.)g(a)g(set)448 1114 y(such)d(that)g Fs(inf)f Ft(j)p Fv(a)15 b Ft(\000)g Fv(b)p Ft(j)25 b Fv(>)g Fs(0)c Fy(where)i(the)f(in\002mum)g(is)g(tak)o (en)h(o)o(v)o(er)f(couples)i(of)e(distinct)i(points)448 1227 y Fv(a;)15 b(b)38 b Ft(2)f Fv(L)p Fy(.)47 b(Let)30 b Fv(L)1056 1241 y Fo(")1130 1227 y Fs(=)37 b Fv(L)25 b Fs(+)f Fv(B)5 b Fs(\()p Fv(")p Fs(\))30 b Fy(be)g(the)h(set)f(of)g (points)h(at)f(distance)i Fv(<)37 b(")30 b Fy(from)g Fv(L)p Fy(.)47 b(W)-7 b(e)448 1340 y(say)32 b(that)g(a)f(subset)i Fv(N)49 b Ft(\032)40 b Fv(X)e Fy(is)31 b Fv(L)p Fw(-thin)h Fy(if)f(for)h(each)g Fv(")40 b(>)f Fs(0)31 b Fy(there)h(is)g Fv(r)42 b(<)d Ft(1)31 b Fy(such)h(that)448 1453 y Fv(N)23 b Ft(n)12 b Fv(B)5 b Fs(\()p Fv(r)s Fs(\))26 b Ft(\032)f Fv(L)973 1467 y Fo(")1009 1453 y Fy(.)j(In)21 b(other)h(terms,)g Fv(N)31 b Fy(is)21 b Fv(L)p Fy(-thin)h(if)f(there)i(is)e(a)g(f)o(amily) h Ft(f)p Fv(\016)2811 1467 y Fo(a)2854 1453 y Ft(g)2899 1467 y Fo(a)p Fr(2)p Fo(L)3056 1453 y Fy(of)g(positi)n(v)o(e)448 1566 y(real)k(numbers)h(with)e Fv(\016)1173 1580 y Fo(a)1244 1566 y Ft(!)k Fs(0)c Fy(as)h Fv(a)i Ft(!)h(1)c Fy(such)h(that)g Fv(N)39 b Ft(\032)2411 1497 y Fe(S)2502 1566 y Fv(B)2571 1580 y Fo(a)2612 1566 y Fs(\()p Fv(\016)2687 1580 y Fo(a)2730 1566 y Fs(\))p Fy(.)34 b(The)25 b(complement)448 1678 y(of)h(such)h(a)f(set)g(will)f(be)h(called)i Fv(L)p Fw(-fat)p Fy(.)36 b(W)-7 b(e)25 b(denote)j Ft(F)2176 1692 y Fo(L)2253 1678 y Fy(the)f(f)o(amily)f(of)g Fv(L)p Fy(-f)o(at)h(sets,)f(we)g(note) 448 1791 y(that)k Ft(F)678 1805 y Fo(L)758 1791 y Fy(is)f(a)f(\002lter) h(on)g Fv(X)35 b Fy(contained)d(in)c Ft(F)1889 1805 y Fq(w)1974 1791 y Fy(and)h(\002ner)g(than)g(the)g(Fr)5 b(\264)-35 b(echet)30 b(\002lter)l(,)g(and)f(we)448 1904 y(denote)37 b Fv(B)794 1918 y Fo(L)846 1904 y Fs(\()p Fv(X)7 b Fs(\))35 b Fy(the)f(set)h(of)g(bounded)i(Borel)e(functions)i (such)e(that)g Fs(lim)2874 1918 y Fr(F)2924 1929 y Fg(L)2990 1904 y Fv(')46 b Fs(=)f(0)p Fy(.)62 b(So)448 2017 y Fv(')34 b Ft(2)e Fv(B)5 b Fs(\()p Fv(X)i Fs(\))28 b Fy(belongs)h(to)f Fv(B)1364 2031 y Fo(L)1416 2017 y Fs(\()p Fv(X)7 b Fs(\))28 b Fy(if)f(and)h(only)h(if)e(the)h(set)g Ft(fj)p Fv(')p Ft(j)34 b(\025)f Fv(\025)p Ft(g)27 b Fy(is)g Fv(L)p Fy(-thin)i(for)f (each)448 2130 y Fv(\025)e(>)f Fs(0)p Fy(.)448 2332 y Fx(Pr)n(oposition)g(7.8)46 b Fw(Let)31 b Fv(X)47 b Fs(=)40 b Fp(R)1524 2299 y Fo(n)1601 2332 y Fw(and)32 b(let)g Fv(S)j Fw(be)d(a)f(bounded)j(oper)o(ator)f(on)f Fv(L)3049 2299 y Fq(2)3088 2332 y Fs(\()p Fv(X)7 b Fs(\))32 b Fw(suc)o(h)448 2445 y(that)24 b(on)g(the)f(r)m(e)l(gion)i Fv(x)g Ft(6)p Fs(=)g Fv(y)h Fw(its)d(distrib)n(ution)k(k)o(ernel)e(is)e(a)g(function) i(satisfying)i(the)c(estimate)448 2558 y Ft(j)p Fv(S)5 b Fs(\()p Fv(x;)15 b(y)s Fs(\))p Ft(j)27 b(\024)e Fv(c)p Ft(j)p Fv(x)c Ft(\000)f Fv(y)s Ft(j)1193 2525 y Fr(\000)p Fo(m)1337 2558 y Fw(for)k(some)f Fv(m)i(>)g(n)p Fw(.)j(Then)23 b Fv(S)28 b Fw(is)23 b Ft(F)2418 2572 y Fo(L)2471 2558 y Fw(-decay)i(pr)m(eserving)o(.)448 2760 y Fx(Pr)n(oof:)42 b Fy(Let)28 b Fv(\022)38 b Ft(2)e Fv(C)1126 2775 y Fq(b)1169 2760 y Fs(\()p Fv(X)7 b Fs(\))30 b Fy(such)g(that)g Fv(\022)s Fs(\()p Fv(x)p Fs(\))35 b(=)h(0)29 b Fy(on)g(a)g(neighborhood)k(of)d (the)f(origin)i(and)448 2873 y Fv(S)504 2888 y Fo(\022)543 2873 y Fs(\()p Fv(x;)15 b(y)s Fs(\))43 b(=)e Fv(\022)s Fs(\()p Fv(x)26 b Ft(\000)g Fv(y)s Fs(\))p Fv(S)5 b Fs(\()p Fv(x;)15 b(y)s Fs(\))p Fy(.)56 b(If)32 b Fv(\030)t Fs(\()p Fv(x)p Fs(\))42 b(=)f Fv(\022)s Fs(\()p Fv(x)p Fs(\))p Ft(j)p Fv(x)p Ft(j)2279 2840 y Fr(\000)p Fo(m)2432 2873 y Fy(then)34 b(for)e(the)h(operator)h Fv(S)3291 2888 y Fo(\022)3361 2873 y Fy(of)448 2986 y(k)o(ernel)25 b Fv(S)753 3001 y Fo(\022)792 2986 y Fs(\()p Fv(x;)15 b(y)s Fs(\))24 b Fy(we)f(ha)n(v)o(e)h Ft(k)p Fv(S)1447 3001 y Fo(\022)1486 2986 y Fv(u)p Ft(k)i(\024)f Fv(c)p Ft(k)p Fv(\030)g Ft(\003)c(j)p Fv(u)p Ft(jk)j Fy(hence)h Ft(k)p Fv(S)2427 3001 y Fo(\022)2466 2986 y Ft(k)h(\024)f Fv(c)p Ft(k)p Fv(\030)t Ft(k)2806 3006 y Fo(L)2854 2987 y Fb(1)2917 2986 y Fy(By)e(choosing)j(a)448 3099 y(con)l(v)o(enient)31 b(sequence)f(of)d(functions)j Fv(\022)e Fy(we)f(see)h(that)g Fv(S)j Fy(is)c(the)h(norm)f(limit)h(of)f(a)g(sequence)448 3212 y(of)c(operators)i(which)e(besides)h(the)f(properties)j(from)c (the)h(statement)i(of)d(the)h(proposition)j(are)448 3325 y(such)k(that)g Fv(S)5 b Fs(\()p Fv(x;)15 b(y)s Fs(\))37 b(=)e(0)29 b Fy(if)g Ft(j)p Fv(x)c Ft(\000)f Fv(y)s Ft(j)36 b Fv(>)f(R)q Fs(\()p Fv(S)5 b Fs(\))p Fy(.)46 b(Since)29 b(the)h(set)f(of)g Ft(F)2730 3339 y Fo(L)2783 3325 y Fy(-decay)h(preserving)448 3438 y(operators)d(is)d(closed)i(in)e(norm)h (\(see)g(Subsection)h(2.3\),)e(we)g(may)g(assume)h(in)f(the)h(rest)g (of)f(the)448 3551 y(proof)c(that)f(the)f(k)o(ernel)i(of)e Fv(S)k Fy(has)d(this)g(property)-6 b(.)29 b(In)18 b(f)o(act,)i(in)e (order)h(to)g(simplify)g(the)g(notations)448 3664 y(and)24 b(without)h(loss)f(of)f(generality)-6 b(,)27 b(we)22 b(shall)j(assume)f Fv(S)5 b Fs(\()p Fv(x;)15 b(y)s Fs(\))26 b(=)f(0)e Fy(if)h Ft(j)p Fv(x)c Ft(\000)g Fv(y)s Ft(j)25 b Fv(>)g Fs(1)p Fy(.)589 3777 y(Let)g Fv(N)35 b Fy(be)25 b(an)g Fv(L)p Fy(-thin)h(Borel)g(set)f(and)h(let)f Fv(")k(>)f Fs(0)p Fy(.)34 b(W)-7 b(e)24 b(shall)i(construct)i(an)e Fv(L)p Fy(-f)o(at)f(Borel)448 3890 y(set)i(with)e Fv(F)43 b Ft(\032)30 b Fv(N)1046 3857 y Fq(c)1106 3890 y Fy(such)d(that)g Ft(k)1505 3871 y Fv(\037)1562 3904 y Fo(N)1629 3890 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)1832 3871 y(\037)1890 3904 y Fo(F)1948 3890 y Fs(\()p Fv(Q)p Fs(\))p Ft(k)k(\024)f Fv(")p Fy(.)35 b(Since)27 b(the)f(adjoint)i(operator)g Fv(S)3398 3857 y Fr(\003)448 4002 y Fy(has)c(the)g(same)g(properties)i (as)d Fv(S)5 b Fy(,)23 b(this)h(suf)n(\002ces)g(to)f(pro)o(v)o(e)h (that)h(it)e(is)g(decay)i(preserving.)589 4115 y(W)-7 b(e)23 b(shall)h(only)g(need)g(tw)o(o)f(simple)h(estimates.)30 b(First,)23 b(if)g Fv(\032)2452 4129 y Fo(x)2496 4115 y Fs(\()p Fv(G)p Fs(\))h Fy(is)f(the)g(distance)j(from)d(a)448 4228 y(Borel)h(set)g Fv(G)f Fy(to)g(a)h(point)g Fv(x)p Fy(,)f(then)1241 4338 y Fe(Z)1291 4544 y Fo(G)1510 4400 y Fv(dy)p 1375 4441 364 4 v 1375 4524 a Ft(j)p Fv(x)e Ft(\000)f Fv(y)s Ft(j)1637 4498 y Fq(2)p Fo(m)1774 4462 y Ft(\024)25 b Fv(C)7 b Fs(\()p Fv(m;)15 b(n)p Fs(\))p Fv(\032)2234 4476 y Fo(x)2278 4462 y Fs(\()p Fv(G)p Fs(\))2419 4424 y Fo(n)p Fr(\000)p Fq(2)p Fo(m)2619 4462 y Fv(:)573 b Fy(\(7.28\))448 4700 y(Then,)24 b(if)f Fv(B)828 4714 y Fq(0)867 4700 y Fv(;)15 b(B)28 b Fy(are)c(tw)o(o)f(balls)h(with)g (the)f(same)h(center)h(and)f(radiuses)h Fv(\016)i Fy(and)d Fv(\016)g Fs(+)c Fv(")p Fy(,)j(then)1181 4807 y Fe(Z)1232 5013 y Fo(B)1285 5022 y Fb(0)1339 4930 y Fv(\032)1386 4944 y Fo(x)1430 4930 y Fs(\()p Fv(B)1539 4893 y Fq(c)1574 4930 y Fs(\))1609 4893 y Fo(n)p Fr(\000)p Fq(2)p Fo(m)1809 4930 y Fv(dx)i Ft(\024)g Fv(C)7 b Fs(\()p Fv(m;)15 b(n)p Fs(\))p Fv(")2388 4893 y Fo(n)p Fr(\000)p Fq(2)p Fo(m)2588 4930 y Fv(\016)2631 4893 y Fo(n)2679 4930 y Fv(:)513 b Fy(\(7.29\))1897 5225 y(47)p eop end end %%Page: 48 48 TeXDict begin HPSdict begin 48 47 bop 448 573 a Fy(W)-7 b(e)28 b(shall)i(choose)h Fv(")k Fs(=)g Fv(\016)1302 540 y Fo(n=)p Fq(2)p Fo(m)1483 573 y Fy(.)44 b(Then)1765 554 y Fv(\037)1822 587 y Fo(B)1875 596 y Fb(0)1914 573 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)2117 554 y(\037)2175 587 y Fo(B)2231 568 y Fb(c)2267 573 y Fs(\()p Fv(Q)p Fs(\))28 b Fy(is)h(an)g(operator)i(with)e(inte)o(gral)448 686 y(k)o(ernel)c(and)f(we)f(can)h(estimate)h(its)e(Hilbert-Schmidt)j(norm) e(as)f(follo)n(ws:)499 910 y Ft(k)544 892 y Fv(\037)601 924 y Fo(B)654 933 y Fb(0)693 910 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)896 892 y(\037)954 924 y Fo(B)1010 906 y Fb(c)1046 910 y Fs(\()p Fv(Q)p Fs(\))p Ft(k)1233 873 y Fq(2)1233 933 y Fo(H)5 b(S)1431 910 y Fs(=)1584 787 y Fe(Z)1635 993 y Fo(X)g Fr(\002)p Fo(X)1835 892 y Fv(\037)1892 924 y Fo(B)1945 933 y Fb(0)1984 910 y Fs(\()p Fv(x)p Fs(\))p Ft(j)p Fv(S)g Fs(\()p Fv(x;)15 b(y)s Fs(\))p Ft(j)2427 873 y Fq(2)2469 892 y Fv(\037)2526 924 y Fo(B)2582 906 y Fb(c)2618 910 y Fs(\()p Fv(y)s Fs(\))p Fv(dxdy)1431 1158 y Ft(\024)82 b Fv(c)1638 1034 y Fe(Z)1689 1240 y Fo(B)1742 1249 y Fb(0)1796 1158 y Fv(dx)1910 1034 y Fe(Z)1961 1240 y Fo(B)2017 1221 y Fb(c)2212 1096 y Fv(dy)p 2078 1137 364 4 v 2078 1220 a Ft(j)p Fv(x)21 b Ft(\000)f Fv(y)s Ft(j)2340 1194 y Fq(2)p Fo(m)2477 1158 y Ft(\024)25 b Fv(C)2660 1034 y Fe(Z)2710 1240 y Fo(B)2763 1249 y Fb(0)2817 1158 y Fv(\032)2864 1172 y Fo(x)2908 1158 y Fs(\()p Fv(B)3017 1120 y Fq(c)3052 1158 y Fs(\))3087 1120 y Fo(n)p Fr(\000)p Fq(2)p Fo(m)3287 1158 y Fv(dx)1431 1366 y Ft(\024)82 b Fv(C)1656 1329 y Fr(0)1679 1366 y Fv(")1721 1329 y Fo(n)p Fr(\000)p Fq(2)p Fo(m)1921 1366 y Fv(\016)1964 1329 y Fo(n)2037 1366 y Fs(=)25 b Fv(C)2205 1329 y Fr(0)2228 1366 y Fv(\016)2271 1329 y Fo(\025)3217 1366 y Fy(\(7.30\))448 1570 y(where)f Fv(\025)h Fs(=)g Fv(n)922 1537 y Fq(2)961 1570 y Fv(=)p Fs(2)p Fv(m)h(>)f Fs(0)p Fy(.)589 1683 y(W)-7 b(e)23 b(can)g(assume)h(that)g Fv(N)35 b Fs(=)1532 1615 y Fe(S)1608 1710 y Fo(a)1665 1683 y Fv(B)1734 1697 y Fo(a)1776 1683 y Fs(\()p Fv(\016)1851 1697 y Fo(a)1893 1683 y Fs(\))p Fy(,)22 b(where)i(the)f(sequence)j(of)d(numbers)i Fv(\016)3181 1697 y Fo(a)3245 1683 y Fy(satis-)448 1796 y(\002es)g Fv(\016)639 1810 y Fo(a)710 1796 y Ft(!)k Fs(0)c Fy(as)g Fv(a)k Ft(!)f(1)p Fy(.)34 b(Denote)26 b Fv(N)1704 1810 y Fo(a)1774 1796 y Fs(=)j Fv(B)1943 1810 y Fo(a)1984 1796 y Fs(\()p Fv(\016)2059 1810 y Fo(a)2102 1796 y Fs(\))c Fy(and)g Fv(M)2405 1810 y Fo(a)2476 1796 y Fs(=)j Fv(B)2644 1810 y Fo(a)2686 1796 y Fs(\()p Fv(\016)2761 1810 y Fo(a)2825 1796 y Fs(+)21 b Fv(")2959 1810 y Fo(a)3001 1796 y Fs(\))p Fy(,)k(where)h(we)448 1925 y(choose)e Fv(")764 1939 y Fo(a)832 1925 y Fs(=)h Fv(\016)971 1877 y Fo(n=)p Fq(2)p Fo(m)968 1937 y(a)1173 1925 y Fy(as)d(abo)o(v)o(e.)29 b(Choose)23 b Fv(r)h Fy(such)f(that)g(the)g(balls)g Fv(N)2639 1939 y Fo(a)2702 1925 y Fy(are)g(pairwise)g(disjoint)448 2038 y(and)g Fv(\016)641 2052 y Fo(a)700 2038 y Fs(+)17 b Fv(")830 2052 y Fo(a)897 2038 y Fv(<)25 b Fs(1)d Fy(if)h Ft(j)p Fv(a)p Ft(j)j Fv(>)f(r)f Fy(and)f(let)g Fv(R)f Fy(such)i(that)2127 2019 y Fv(\037)2184 2052 y Fo(N)2240 2060 y Fg(a)2282 2038 y Fs(\()p Fv(Q)p Fs(\))p Fv(S)2485 2019 y(\037)2542 2056 y Fo(B)s Fq(\()p Fo(R)p Fq(\))2705 2037 y Fb(c)2743 2038 y Fs(\()p Fv(Q)p Fs(\))h(=)g(0)e Fy(if)f Ft(j)p Fv(a)p Ft(j)k(\024)f Fv(r)s Fy(.)448 2151 y(Let)d Fv(M)35 b Fs(=)810 2083 y Fe(S)901 2151 y Fv(M)989 2165 y 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y(v)n(alues)h(of)e Fv(\022)s Fy(.)27 b(The)22 b(operator)j Fv(P)1452 1669 y Fo(\013)1501 1702 y Fs(\()p Fv(A)1604 1717 y Fo(\022)1660 1702 y Fs(+)16 b Fv(z)t Fs(\))1828 1669 y Fr(\000)p Fq(1)1945 1702 y Fy(is)22 b(also)h(a)f(holomorphic)k(function)e(of)f Fv(z)j Fy(in)c(the)448 1815 y(re)o(gion)h(Re)g Fv(z)29 b(>)c Fs(0)p Fy(,)d(so)g(by)f(a)h(similar)g(ar)n(gument)i(we)d(see)h(that)g (it)f(suf)n(\002ces)i(to)f(consider)h Fv(z)30 b Ft(\025)25 b Fs(0)p Fy(.)448 1928 y(Belo)n(w)e(we)g(shall)h(tak)o(e)h Fv(z)k Fs(=)c(0)p Fy(,)e(the)h(ar)n(gument)h(in)f(general)h(is)e (identical.)589 2041 y(F)o(or)g(reasons)i(of)f(simplicity)-6 b(,)25 b(we)d(change)k(again)e(the)f(notations:)32 b(we)23 b(set)g Fv(b)j Fs(=)f Fv(\022)s Fs(\()p Fv(a)19 b Ft(\000)h Fs(1\))p Fy(,)448 2154 y(we)e(assume)h Ft(k)p Fv(b)p Ft(k)986 2172 y Fr(B)r Fq(\()p Fk(K)i Fq(\))1208 2154 y Fv(<)k Fs(1)p Fy(,)19 b(and)f(denote)i Fv(V)46 b Fs(=)25 b Fv(D)2072 2121 y Fr(\003)2111 2154 y Fv(bD)20 b Fy(and)f Fv(A)25 b Fs(=)g(\001+)p 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y(con)l(v)o(er)n(ges)27 b(in)c(norm,)h(so)f(it)h(suf)n(\002ces)g(that)g(each)g(of)g(them)f(be)h Ft(F)2464 3450 y Fq(w)2520 3436 y Fy(-decay)h(preserving.)32 b(But)884 3641 y Fv(P)955 3603 y Fo(\013)1005 3641 y Fv(S)5 b Fs(\()p Fv(S)g(V)20 b(S)5 b Fs(\))1331 3603 y Fo(k)1374 3641 y Fv(S)31 b Fs(=)25 b(\()p Fv(P)1663 3603 y Fo(\013)1712 3641 y Fv(S)5 b Fs(\)\()p Fv(S)g(D)1982 3603 y Fr(\003)2022 3641 y Fs(\))p Fv(b)p Fs(\()p Fv(D)s(S)g Fs(\))15 b Fv(:)g(:)g(:)j Fs(\()p Fv(S)5 b(D)2617 3603 y Fr(\003)2656 3641 y Fs(\))p Fv(b)p Fs(\()p Fv(D)s(S)g Fs(\))p Fv(S)448 3845 y Fy(and)30 b(each)g(f)o(actor)h(in)e(the)g (product)j(is)d Ft(F)1742 3859 y Fq(w)1798 3845 y Fy(-decay)i (preserving:)44 b(for)29 b Fv(b)g Fy(by)g(assumption,)k(and)448 3958 y(for)24 b Fv(P)648 3925 y Fo(\013)698 3958 y Fv(S)5 b Fy(,)22 b Fv(D)s(S)27 b Fy(and)d Fv(S)5 b(D)1258 3925 y Fr(\003)1320 3958 y Fy(due)24 b(to)g(Theorem)g(7.6.)p 3371 3950 67 67 v 589 4121 a(Belo)n(w)29 b(we)e(gi)n(v)o(e)i(the)g 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4911 y Fs(\))p Fw(.)1897 5225 y Fy(50)p eop end end %%Page: 51 51 TeXDict begin HPSdict begin 51 50 bop 448 573 a Fx(Pr)n(oof:)49 b Fy(W)-7 b(e)32 b(check)i(the)g(conditions)h(of)e(Theorem)h(3.5.)57 b(Because)34 b(of)e(the)i(coerci)n(vity)h(as-)448 686 y(sumptions,)29 b(condition)g(\(1\))d(is)g(ful\002lled,)i(and)f(\(2\))f (is)g(satis\002ed)h(by)g(Lemma)e(7.10.)37 b(The)26 b(part)448 799 y(of)g(condition)i(\(3\))e(in)l(v)n(olving)i(the)e(coef)n (\002cients)i(such)e(that)g Ft(j)p Fv(\013)p Ft(j)d Fs(+)e Ft(j)p Fv(\014)5 b Ft(j)30 b Fs(=)e Fv(m)p Fy(\))d(is)h(satis\002ed)g (by)448 912 y(de\002nition,)f(for)f(the)g(lo)n(wer)f(order)i(coef)n (\002cients)g(it)f(suf)n(\002ces)g(to)f(use)h(\(2.8\).)p 3371 904 67 67 v 448 1163 a Fx(Remark)f(7.12)47 b Fy(If)34 b Fv(b)1127 1178 y Fo(\013\014)1265 1163 y Ft(2)45 b Fv(B)5 b Fs(\()p Fv(X)i Fs(\))34 b Fy(and)h Fv(b)1835 1178 y Fo(\013\014)1955 1163 y Ft(\000)28 b Fv(a)2102 1178 y Fo(\013\014)2240 1163 y Ft(2)45 b Fv(B)2415 1177 y Fq(w)2471 1163 y Fs(\()p Fv(X)7 b Fs(\))35 b Fy(for)f(all)h Fv(\013;)15 b(\014)5 b Fy(,)37 b(then)e(the)448 1276 y(compactness)j(conditions)g(on)d(the)g(lo)n(wer)g(order)h(coef)n (\002cients)h(are)e(satis\002ed.)64 b(Indeed,)40 b(if)448 1389 y Fv(')31 b Ft(2)g Fv(B)699 1403 y Fq(w)755 1389 y Fs(\()p Fv(X)7 b Fs(\))26 b Fy(then)i Fv(')p Fs(\()p Fv(Q)p Fs(\))j(:)g Fu(H)1520 1356 y Fo(s)1587 1389 y Ft(!)f Fu(H)1824 1356 y Fr(\000)p Fo(t)1934 1389 y Fy(is)c(compact)i (if)e Fv(s;)15 b(t)31 b Ft(\025)f Fs(0)c Fy(and)h(one)g(of)f(them)g(is) 448 1502 y(not)e(zero,)g(see)g(Lemma)e(7.3.)448 1793 y Fz(A)120 b(A)m(ppendix)448 2000 y Fy(In)20 b(this)g(Appendix)i(we)d (discuss)i(some)f(elementary)i(abstract)g(f)o(acts)f(which)f(are)g (used)g(without)448 2113 y(comment)k(in)g(the)g(main)f(te)o(xt.)589 2226 y(Let)33 b Fs(\()p Fu(G)16 b Fv(;)f Fu(H)27 b Fs(\))33 b Fy(be)g(a)g(Friedrichs)j(couple)f(and)f Fu(G)59 b Ft(\032)43 b Fu(H)70 b Ft(\032)43 b Fu(G)2680 2193 y Fr(\003)2752 2226 y Fy(the)33 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b(that)f(Re)e Ft(h)p Fv(u;)15 b(S)5 b(u)p Ft(i)30 b(\025)e Fv(\026)p Ft(k)p Fv(u)p Ft(k)2345 4943 y Fq(2)2345 5004 y Fk(G)2430 4976 y Ft(\000)22 b Fv(\027)6 b Ft(k)p Fv(u)p Ft(k)2716 4943 y Fq(2)2716 5004 y Fk(H)2838 4976 y Fy(for)25 b(some)h(strictly)1897 5225 y(51)p eop end end %%Page: 52 52 TeXDict begin HPSdict begin 52 51 bop 448 573 a Fy(positi)n(v)o(e)35 b(constants)h Fv(\026;)15 b(\027)39 b Fy(and)33 b(all)h Fv(u)44 b Ft(2)f Fu(G)15 b Fy(.)58 b(Indeed,)37 b(replacing)f Fv(S)h Fy(by)d Fv(S)e Fs(+)27 b Fv(\027)6 b Fy(,)35 b(we)e(may)448 686 y(assume)h(Re)22 b Ft(h)p Fv(u;)15 b(S)5 b(u)p Ft(i)44 b(\025)d Fv(\026)p Ft(k)p Fv(u)p Ft(k)1499 653 y Fq(2)1499 714 y Fk(G)1563 686 y Fy(.)55 b(Since)33 b Fv(S)1936 653 y Fr(\003)2007 686 y Fy(v)o(eri\002es)g(the)g(same)g(estimate,)j (this)d(clearly)448 799 y(gi)n(v)o(es)c Ft(k)p Fv(S)5 b(u)p Ft(k)867 814 y Fk(G)927 795 y Fc(\003)1000 799 y Ft(\025)33 b Fv(\026)p Ft(k)p Fv(u)p Ft(k)1301 814 y Fk(G)1392 799 y Fy(and)28 b Ft(k)p Fv(S)1656 766 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(the)g(closed)h(graph)g(theorem\),)f(so)g(we)f(ha)n(v)o(e)h(a)f(scale) 807 4103 y Ft(D)s Fs(\()p Fv(A)p Fs(\))j Ft(\032)f Fu(G)41 b Ft(\032)24 b(D)s Fs(\()p Ft(j)p Fv(A)p Ft(j)1558 4065 y Fq(1)p Fo(=)p Fq(2)1669 4103 y Fs(\))i Ft(\032)f Fu(H)52 b Ft(\032)25 b(D)s Fs(\()p Ft(j)p Fv(A)p Ft(j)2289 4065 y Fq(1)p Fo(=)p Fq(2)2399 4103 y Fs(\))2434 4065 y Fr(\003)2499 4103 y Ft(\032)g Fu(G)2667 4065 y Fr(\003)2731 4103 y Ft(\032)g(D)s Fs(\()p Fv(A)p Fs(\))3038 4065 y Fr(\003)448 4259 y Fy(with)f(continuous)i(and)e(dense)h(embeddings)h(\(because)g Ft(D)s Fs(\()p Fv(A)p Fs(\))d Fy(is)h(dense)g(in)g Ft(D)s Fs(\()p Ft(j)p Fv(A)p Ft(j)3108 4226 y Fq(1)p Fo(=)p Fq(2)3218 4259 y Fs(\))p Fy(\).)448 4544 y Fz(Refer)n(ences)448 4751 y Fy([ABG])45 b(W)-8 b(.)24 b(Amrein,)h(A.)f(Boutet)i(de)f(Mon)l (v)o(el,)h(V)-12 b(.)24 b(Geor)n(gescu,)k Fv(C)2612 4765 y Fq(0)2651 4751 y Fw(-Gr)l(oups,)f(commutator)706 4863 y(methods)k(and)g(spectr)o(al)h(theory)g(of)e Fv(N)10 b Fw(-body)32 b(Hamiltonians)p Fy(,)i(Birkh)5 b(\250)-35 b(auser)l(,)34 b(Basel-)706 4976 y(Boston-Berlin,)25 b(1996.)1897 5225 y(52)p eop end end %%Page: 53 53 TeXDict begin HPSdict begin 53 52 bop 448 573 a Fy([Ar])102 b(M.)24 b(Arai,)i Fw(On)f(essential)j(self-adjointness,)j (distinguished)e(selfadjoint)g(e)n(xtensions)706 686 y(and)34 b(essential)h(spectrum)g(of)f(Dir)o(ac)f(oper)o(ator)o(s)j (with)d(matrix)h(valued)h(potentials)p Fy(,)706 799 y(Publ.)23 b(RIMS,)f(K)n(yoto)h(Uni)n(v)o(ersity)-6 b(,)25 b Fx(19)f Fy(\(1983\),)h(33\22657.)448 986 y([A)-10 b(Y])76 b(M.)20 b(Arai,)h(O.)e(Y)-9 b(amada,)22 b Fw(Essential)g(self-adjointness)k (and)c(in)l(variance)j(of)c(the)g(essen-)706 1099 y(tial)g(spectrum)g (for)g(Dir)o(ac)f(oper)o(ator)o(s)p Fy(,)j(Publ.)d(RIMS)f(K)n(yoto)h (Uni)n(v)o(ersity)-6 b(,)23 b Fx(18)d Fy(\(1982\),)706 1212 y(973\226985.)448 1400 y([Ba1])52 b(G.)27 b(Barbatis,)j Fw(Spectr)o(al)g(stability)g(under)g Fv(L)2127 1367 y Fo(p)2166 1400 y Fw(-perturbations)j(of)28 b(the)h(second-or)m(der)706 1513 y(coef)n(\002cients)p Fy(,)e(J.)22 b(Dif)n(f.)h(Equations,)i Fx(124)f Fy(\(1996\),)h(302-323.)448 1700 y([Ba2])52 b(G.)34 b(Barbatis,)39 b Fw(Spectr)o(al)f(of)d(weighted)j (Laplace-Beltr)o(ami)f(oper)o(ator)o(s)h(under)f Fv(L)3367 1667 y Fo(p)3407 1700 y Fw(-)706 1813 y(perturbations)43 b(of)c(the)h(Riemannian)g(metric)p Fy(,)k(J.)38 b(d'Analyse)j(Math)5 b(\264)-35 b(emaique,)45 b Fx(68)706 1926 y Fy(\(1996\),)25 b(256-276.)448 2114 y([Be])97 b(S.K.)30 b(Berberian,)36 b Fw(Notes)d(on)f(spectr)o(al)i(theory)p Fy(,)i(Princeton,)g(V)-10 b(an)32 b(Nostrand)i(Com-)706 2227 y(pan)o(y)-6 b(,)24 b(1966.)448 2414 y([DP])81 b(G.)26 b(De)i(Cecco,)h(G.)d(P)o(almieri,)j Fv(p)p Fw(-Ener)m(gy)g(of)f(a)f(curve)i(on)g Fv(LI)7 b(P)13 b Fw(-manifolds)30 b(and)e(on)706 2527 y(g)o(ener)o(al)d(metric) f(spaces)p Fy(,)h(preprint.)448 2715 y([Di])107 b(J.)45 b(Dixmier,)51 b Fw(Les)45 b(alg)1476 2716 y(\036)1471 2715 y(ebr)m(es)j(d'op)1881 2716 y(\264)1876 2715 y(er)o(ateur)o(s)h (dans)d(l'espace)i(Hilbertien)p Fy(,)53 b(P)o(aris,)706 2828 y(Gauthier)n(-V)-5 b(illars,)26 b(1969.)448 3015 y([DJT])46 b(J.)38 b(Diestel,)43 b(H.)38 b(Jarcho)n(w)-6 b(,)43 b(A.)37 b(T)-7 b(onge,)43 b Fw(Absolutely)e(summing)e(oper)o (ator)o(s)p Fy(,)45 b(Cam-)706 3128 y(bridge)20 b(studies)h(in)f(adv)n (anced)h(mathematics,)h(Cambridge)e(Uni)n(v)o(ersity)h(Press,)f(1995.) 448 3316 y([FD])81 b(J.M.G.)19 b(Fell)h(and)h(R.S.)d(Doran,)j Fw(Repr)m(esentations)k(of)20 b Ft(\003)p Fw(-alg)o(ebr)o(as,)k (locally)e(compact)706 3429 y(gr)l(oups,)j(and)f(Banac)o(h)g Ft(\003)p Fw(-alg)o(ebr)o(aic)i(b)n(undles,)g(V)-10 b(ol.)23 b(1)p Fy(,)f(Academic)j(Press,)e(1988.)448 3616 y([GM])51 b(V)-12 b(.)29 b(Geor)n(gescu,)34 b(M.)29 b(M)5 b(\013)-35 b(antoiu,)33 b Fw(On)d(the)h(spectr)o(al)h(theory)f(of)g(singular)h (Dir)o(ac)f(type)706 3729 y(hamiltonians)p Fy(,)26 b(J.)d(Operator)i (Theory)f Fx(46)g Fy(\(2001\),)g(289\226321.)448 3917 y([GW])46 b(K.)18 b(Gustafson,)j(J.)e(W)-7 b(eidmann,)21 b Fw(On)d(the)i(essential)i(spectrum)p Fy(,)f(J.)d(Math.)i(Anal.)f (Appl.)706 4030 y Fx(25)k Fy(\(1969\))i(121\226127.)448 4218 y([He])92 b(R.)28 b(Hempel,)i Fw(P)-7 b(erturbation)32 b(by)e(Quadr)o(atic)g(F)-10 b(orms)30 b(and)g(In)l(variance)i(of)e (Essential)706 4330 y(Spectr)o(a,)25 b Fy(Math.)e(Z.)f Fx(185)i Fy(\(1984\),)h(281\226289.)448 4518 y([Hi])107 b(M.)21 b(Hilsum,)i Fw(Structur)m(es)i(riemanniennes)h Fv(L)2155 4485 y Fo(p)2216 4518 y Fw(et)d Fv(K)7 b Fw(-homolo)o(gie)p Fy(,)25 b(Annals)e(of)f(Math.)706 4631 y Fx(149)p Fy(,)i(1007\2261022.) 448 4819 y([Ho])87 b(L.)19 b(H)8 b(\250)-38 b(ormander)l(,)22 b Fw(The)e(analysis)j(of)d(linear)i(partial)g(dif)n(fer)m(ential)i (oper)o(ator)o(s)f(vol.)d(I-IV)p Fy(,)706 4932 y(Springer)l(,)25 b(1983-1987.)1897 5225 y(53)p eop end end %%Page: 54 54 TeXDict begin HPSdict begin 54 53 bop 448 573 a Fy([Kl])107 b(M.)18 b(Klaus,)j Fw(Dir)o(ac)e(Oper)o(ator)o(s)i(with)f(Se)o(ver)o (al)g(Coulomb)h(Singularities)p Fy(,)j(Helv)-6 b(.)19 b(Phys.)706 686 y(Acta)k Fx(l53)h Fy(\(1980\),)h(463\226482.)448 873 y([Kr])102 b(J.L.)55 b(Kri)n(vine,)66 b Fw(Th)1362 874 y(\264)1357 873 y(eor)1482 874 y(\036)1477 873 y(emes)57 b(de)g(factorisation)k(dans)d(les)f(espaces)h(r)3141 874 y(\264)3136 873 y(eticul)3341 874 y(\264)3336 873 y(es)p Fy(,)706 986 y(S)5 b(\264)-35 b(eminaire)24 b(Maure)o(y-Schw)o (artz)i(1973/74,)f(Exp.)e(XXII-XXIII.)448 1174 y([L)-9 b(V])85 b(V)-12 b(.)21 b(Lisk)o(e)n(vich,)i(H.)e(V)-12 b(ogt,)22 b Fw(On)g Fv(L)1722 1141 y Fo(p)1761 1174 y Fw(-spectr)o(a)i(and)f(essential)h(spectr)o(a)g(of)e(second-or)m(der) 706 1287 y(elliptic)j(oper)o(ator)o(s)p Fy(,)h(Proc.)d(London)h(Math.)g (Soc.)f(\(3\))g Fx(80)h Fy(\(2000\),)h(590\226610.)448 1474 y([Lo])97 b(L.H.)19 b(Loomis,)i Fw(An)f(intr)l(oduction)25 b(to)c(abstr)o(act)i(harmonic)g(analysis)p Fy(,)g(V)-10 b(an)21 b(Nostrand,)706 1587 y(1953.)448 1775 y([Ma])77 b(B.)41 b(Maure)o(y)-6 b(,)47 b Fw(Th)1276 1776 y(\264)1271 1775 y(eor)m(emes)c(de)g(factorisation)j(pour)d(les)g(op)2697 1776 y(\264)2692 1775 y(er)o(ateur)o(s)i(lin)3135 1776 y(\264)3130 1775 y(eair)m(es)3399 1776 y(\036)3392 1775 y(a)706 1888 y(valeur)o(s)25 b(dans)g(les)e(espaces)j Fv(L)1681 1855 y Fo(p)1720 1888 y Fy(,)d(Ast)5 b(\264)-35 b(erisque)25 b Fx(11)p Fy(,)f(Soci)5 b(\264)-35 b(et)5 b(\264)-35 b(e)24 b(Math.)f(de)h(France.)448 2076 y([N1])87 b(G.)34 b(Nenciu,)39 b Fw(Self-adjointness)i(and)36 b(in)l(variance)j (of)c(the)h(essential)i(spectrum)f(for)706 2188 y(Dir)o(ac)21 b(oper)o(ator)o(s)i(de\002ned)g(as)f(quadr)o(atic)h(forms)p Fy(,)f(Comm.)d(Math.)i(Phys.)g Fx(48)g Fy(\(1976\),)706 2301 y(235\226247.)448 2489 y([N2])87 b(G.)29 b(Nenciu,)j Fw(Distinguished)h(self-adjoint)h(e)n(xtensions)f(for)e(Dir)o(ac)f (oper)o(ator)o(s)i(with)706 2602 y(potentials)e(dominated)f(by)d (multicenter)k(Coulomb)d(potentials)p Fy(,)k(Helv)-6 b(.)26 b(Phys.)g(Acta)706 2715 y Fx(50)d Fy(\(1977\),)i(1\2263.)448 2902 y([OS])81 b(E.M.)23 b(Ouhabaz,)j(P)-10 b(.)24 b(Stollmann,)i Fw(Stability)i(of)d(the)g(ensential)j(spectrum)e(of)f(second-)706 3015 y(or)m(der)i(comple)n(x)h(elliptic)h(oper)o(ator)o(s)p Fy(,)g(J.)d(Reine)h(Ange)n(w)-6 b(.)26 b(Math.)g Fx(500)i Fy(\(1998\),)g(113\226)706 3128 y(126.)448 3316 y([Pi])122 b(G.)19 b(Pisier)l(,)i Fw(F)-7 b(actorization)24 b(of)c(oper)o(ator)o (s)i(and)f(g)o(eometry)h(of)e(Banac)o(h)h(spaces)p Fy(,)h(Amer)-5 b(.)706 3429 y(Math.)23 b(Soc.,)g(Pro)o(vidence)i(R.I.,)d(CBMS)f Fx(60)p Fy(,)j(1985.)448 3616 y([Sa])107 b(L.)29 b(Salof)n(f-Coste,)34 b Fw(Op)1442 3617 y(\264)1437 3616 y(er)o(ateur)o(s)f(pseudo-dif)n(f) 2176 3617 y(\264)2171 3616 y(er)m(entiels)k(sur)31 b(un)g(corps)h (local)p Fy(,)h(th)5 b(\036)-35 b(ese)706 3729 y(de)23 b(troisi)5 b(\036)-35 b(eme)25 b(c)o(ycle,)f(Uni)n(v)o(ersit)5 b(\264)-35 b(e)25 b(Pierre)f(et)f(Marie)h(Curie,)g(1983.)448 3917 y([T)-7 b(a])109 b(M.)40 b(H.)h(T)-7 b(aibleson,)47 b Fw(F)-10 b(ourier)44 b(analysis)g(on)d(local)i(\002elds)p Fy(,)k(Princeton)d(Uni)n(v)o(ersity)706 4030 y(Press,)23 b(1975.)448 4218 y([T)-6 b(e])108 b(N.)22 b(T)-6 b(eleman,)23 b Fw(The)g(inde)n(x)i(of)e(signatur)m(e)j(oper)o(ator)o(s)g(on)d(Lipsc) o(hitz)i(manifolds)p Fy(,)g(Publ.)706 4330 y(Math.)e(I.H.E.S.)e Fx(58)i Fy(\(1983\),)i(39\22678.)448 4518 y([W)-7 b(a])79 b(N.)30 b(W)-7 b(ea)n(v)o(er)l(,)34 b Fw(Lipsc)o(hitz)f(alg)o(ebr)o(as) h(and)e(derivations)j(II:)c(e)n(xterior)j(dif)n(fer)m(entiation)p Fy(,)706 4631 y(preprint)25 b(1998.)448 4819 y([W)-7 b(e])79 b(J.)23 b(W)-7 b(eidmann,)24 b Fw(Linear)g(oper)o(ator)o(s)i (in)d(Hilbert)i(space)p Fy(,)f(Springer)l(,)h(1980.)1897 5225 y(54)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0411031049801--