Content-Type: multipart/mixed; boundary="-------------9903180412792" This is a multi-part message in MIME format. ---------------9903180412792 Content-Type: text/plain; name="99-80.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-80.keywords" perturbation theory, quantum field theory on curved spacetimes, extensions of distributions ---------------9903180412792 Content-Type: application/postscript; name="maiqftrfb.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="maiqftrfb.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: maiqftrfb.dvi %%Pages: 76 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR10 CMR5 CMR8 CMTT8 CMTI8 CMR7 CMCSC10 CMMI10 %%+ CMTI10 CMSY10 CMSY7 CMR6 CMMI7 MSBM10 EUSM10 MSAM10 EUFM10 CMEX10 %%+ MSBM7 CMMI5 cmex7 CMSY5 MSBM5 EUSM7 CMTT10 CMBX8 CMMI8 %%EndComments %DVIPSCommandLine: dvips maiqftrfb %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.03.17:1802 %%BeginProcSet: texc.pro %! 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fb7c75bcd58527e6b69bb79fe5a0f5e4c5846585b141fef211c99700c04f008d 005af1a5c77880edceebe16adf827a0cf22b562f480c41eb822eec5185d72635 3a874dff851ec08f847d0f2237f8b0765e122ac5e474e9d11485b793b3429977 b21a5b252b4874fb42512006a7472a9514992cdc38e069b29e7daddfdf5e2826 108f0e927f9256c67cca7cb0ead8f7c1852d3c10f40f839ee63799a93c8655bd 841226518b0e5cd4d107c3e56b13f6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 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 69 /E put dup 100 /d put dup 110 /n put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ecefaacd079732f12c29 315e4b9623a5752c6f1d8145869e120d910b2644887cea7e30b15676a92537c2 9d3aa80dc30082aba94b40990b82fb1a877e805e0c8c48f61e9f2edac05b944e e4d8084ec1d5cc517aaeec5b3ea379dd011eeb454cecab3ad2443c887c582789 72355673e503affe0394fc7db31de364e4f56c24033c7df2265c56445ec63a1d 5695a6041ea1b94407e1cdb7c5635603a4fd047e6edcaeb2d0da6c9e0e9396d5 1a4a58e8fdc1578730f992435560a6e2d3e3687703ee2f78f5896389ac8470bc 806169eb01762e89b6dc9adf857ead656620e2589aae722c37a2ed7a2941c360 b067ee34d8d5ca3bf68db725614d936bcb207781f4d4ec2ac67b13a5ad161f3f 059add7b5e3d904831e31c20c04546fae83ca93a35989e65c201756888f727b9 7e5313c9870ec96e4cec3901ea03a5c744754485e7d169bbc98bf872d0796e18 9d43b712950c3786257d8be06ab6080b9d9392313298327549a8a41c00a2cea3 690b4a333e45aa815a64facae1c2b44640860b8b8687afbefdbd5b4a541a7251 4ee7d3e0752af3e96a88c95d31fa16d34ba2f02fdb0088b165844f02611c734a dbcc2037bf741fdea7e8bcdc130c70c33772f777d8bcddf4611db99001ccff14 d7af2bb05defc3480bdab312a0eff2f375afb4e0df2f803c594b7c93f71ca4af 861ae1711932fef19bfe2f9b7adb69d68e5a70ea4d1a3d5bd85231b16993f65e fb37a150884bd46e127db02bf63a091bd83e202e87fbdbc41c7e7dafe8dba11b c792dab14a3af7919c950f3cedb295474be9f31d61a3390b073a121c78d7075e 946228e6069c10af003269fa2578515ee731515453dadce35f7e9cae998eaa0c f06c0f559d552475fe56a10a3d9457d0f83482543ba9f60951dd5a350aaa0811 ac8ccf3f615744f28fbaec59428d542d1ddf35fb1e09b9a212ab4ea8c7701513 bba3f76833da089c938d79421d1b45159d45a2d525095702fda1ecd308feb687 2fd5347bd5fda8eae43c0f291ed13477d0bcd55934055d1c13539ab0c416df67 d7dbc9376216da2070280ba90d031c58a0472c8d02e858e19c39280290bf5b3e dd8bd76e63e95ecc8b4382360001b90c37d9f39f19a159348b59b8955568d674 213a2cc5f9fc535ac1b61f926be4fc72eb3b04e97e34c2a5ad16e1472bbb9a48 da9ae6dbcf5b823eca5b65d98ae33a388666addfc930bef24e8bbd1094302a52 1b507b5fbe6512e20b3daa45dd1891fa3c20537d9cb1384d4517dc95cfb7f5e1 7163fc224583ee7060201a532f0356ace367998fdafe02f62b991f9cba12555a b612d29e3cfc6496e0b14e429f6c71b354da68d10ff0dbd68f8583cdad5ce935 41203aa5349c6e7197de1896cb984d1e377438fa9fb7a10bdea01f2b1e7ffe2f 64b492483c3cafc5a0045a0faab504998f94ebf241c92ec3b6ae74ac839a7b4d 629e0da30b6d8dfc76396dd72a2e37df6adcafadb568be8ddbf79120d16c310f 22ac56b8a88b9be0f5a21e38f1bdb7e6f275c7f0d6ba48b132736a0d79140b9d a6dcfddebd8cfe4f69ee561474674fe3f1e9d9869f22c41afd79a2bb66a82925 938b05df032662838ca2602158bebc398ef04999c7bca5109f7c56d456c8ea3f 1915d296be0907b7d988edee4e9a1a9ad10788c7956e198d30c63ac527b1add1 085d1a416067ce84b4ab1ff506460b4a1f91a34f5c03bf16c6f45e1207633827 1ffb86422bafdd312a5eaa15518e83cbcf2f39fb73d1004211b7cd657e88ad06 902fd2564acee7ab76dcb1360fbcf6ffd96df7bf11bab2e6b59387e6a8ccb9c6 50a77c705b0455c07cdd74a9a3f49a6e8b60e46fe7b22dc8400af3c6b8e7590a 206c90fd8e5f7576e59a559320313af586ad27f2248434131d614908459d62c0 2639af52f5769e9ae40565c0caea89bdb3fe7a655d5a8ec6145bd15f52c8384b 87fedbd0e77a07bc49854ee6494880e45b746d6159dfed27de71f3b25113663b 90c61de1335b603e74896eb268 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUSM7 %!PS-AdobeFont-1.1: EUSM7 2.1 %%CreationDate: 1992 Nov 20 17:36:46 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUSM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUSM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put readonly def /FontBBox{0 -250 1178 750}readonly def /UniqueID 5031994 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6c8752eeafc779b1f67b40274cb47e098485 f163a6178722fa0d273e24cf928ed45cf1bac4c48c315abade8dbb5eb13b7406 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bff27c12f747e7a1682f79a1cb6d94633eece5519d5457a79ad5115aafa06cff a772f21a186a0251e6a758c179240b55868004c9d0a8d7e8f6d15eb81b85a718 a47dc82d21572091811a2f04ffce099740d26fb485471fd3a92ef73d3ad3e0c4 db7ef3a696cabea9c5f4d074fa0d92704504f5d19ced02840889f9d419f32904 79613439bc2f79f5bdef8941250050322e56321942cfb6fb385f25e8c190f375 635aaa9bbc55c09937bb731855fb81611e07ed4ba01bdfbf59afee20feea0c94 a9158b92382f5770f660851e71c8bda874fb0474c92379b9e48d1a2804b3b347 e9c50ab81636702fad8ad6816d30bcad2ff42fa6dcc90f807efff5f13f561cd4 54722cb24688cd7dff37c4f4e904b28148b7e83de0f1c7a734679d8856e3bffc 0f1676351e09df6d836277658e8cf6a15203dd72427a17f7e63342b07441d8ac b3b4a3570815ebc974 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM5 %!PS-AdobeFont-1.1: MSBM5 2.1 %%CreationDate: 1992 Oct 17 08:36:07 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM5) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM5 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 77 /M put readonly def /FontBBox{0 -574 3027 1074}readonly def /UniqueID 5032013 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7c70c7f34bbdc30f9a4193ae07570fd1e7ec5dace025152958a0eaadef 7e26b18c0cd9f922ba7dc501b446382ae3806ac642f7458dc2fd15718c70f4cc dd70de091ddea7f3adea68ade8a8c4641ccb8818218ab11ac7739f39aba38efd b4439890eaf3748de69390b00901394c9790dbd96b2fb142199599dcd5755a28 4457be850e6601cab34724ce8a7812c3b2c05f398c0c7d134b8a68ba5a2b958a af48835f859b4c0d3521d6d8c45e55dd766e5f5cf6d5e873ab0513e60fda8bd4 3754a0b44b0425083e6c8b288d3461d538b146b49e98fa3c48243c291b6afc5e 39e996726f9f66a43101116f787cab705f0f6ac414643cd511cc3a7739c6aa63 711d1edb55ef9a36aa83ab3bbdfe90d8a7c22d8e60c0656781dfb925ec90bde9 bc96e2529aaaaf21aa102c9c48500047cc2219e4ae22599bf63f8de8f8b4766c fdc8975d16b0d1efa3f462360bfc6db72aa2e29a372c1161fda61f3d7af1f719 0ec1e4eff89086fcb7605dc12492b90a6e0a81d165957ba254955f88ac16d670 74136ef53f542a170c4a3433308b3ff613eeeb9982c3f549af393758f06e046b daf23f918f4638931a4a934be67dd73938421dd61cc9732c1a94c2caf83f0c3d ea098113bcf867c88a7ae30f6d53cab8ab8042e92c12b26a7b7d6d659de8cf31 fde3523e9d438a93284d0f8519f206fe31825f2c1040dad07800ab29bf894442 9f4a3995ce613728c5d8d0bed41b4b73043410387437d28ad3863a0a6c404288 4a829fbddda0a10b56f38fa13b4aca5d8754f25c0cde5db32078002b9b4ec05d ad9fd2c1ffacf89e407cef26c7d11d8ea0c6b29dfe6ff77507c7f8796d7f6839 52d17da5bcf5cad9b7f5216617efa91bd7c5e502ee6530ebe9797789598262aa 3d31fd36a357c3c73b9fb05f391e1e5f62248d8b1390ef18e2b32a561b1a2c44 cb0813c31f6fa8f1d54983d455f60b05e79759c98c75b9d01b1e9744ed17f2e4 fbf75772f71c58e297792f2d25cbf83e3a81986504cf4028ec159905eb21e482 ded76e76b513cf6da884980165d11078fa3322869fbd53b5cf921230e345141c f3eddf7b6a2800a51b63e64e7d0c7d86842d058fcc09e07b022fb12c9624c472 b0c80f4a5db67568546106e324f09f965bde54c9d8c4a4b4dee4993582b70ab0 08d53fae880f10c5c29348987e94f0ca609b52165201b4f30fe1f3e5a958a521 280a5fc877351fd6529c859dc15024a3d2c5764201de185c30aee300d6a9c481 c1f135336260d5f61414c8e370e0648a1528035a139f30605324c2be3305d29a 3f3c9b8a60387e8b055117f85430e5e1c3960f8c4dfa883fefe14005cc7f6e34 63f9f33d73ec826130aa0680c99f2ffdbe65f3e609dd9136e5cacc160946e550 4798399a3aab921b43fb6ed9b7557613cc622f7987a13b9367992477c2c70bb6 c957e5e2f4218b8e83f95a48ded333f40c4283493cc88c15173e77c43473f2ec 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b6f9385526e98445876548d0f2a09cfea14b84be3721b51afa2185 90c7b9ca643c7f4fe67cd8985f27855c40ae7ea592fadfbba29301071039d18a 8a6880cc0bff22fc8af50fd2802aff5a96381fb4bef240dd26b34a5461188ca0 c85e4b3f803f94dce14e8ddb2284e22abfe148560aff784967ed795332cc0995 80b762f864b2c15bc640a4cd7f6927c7dd95aef777fd226254b8e6e1b8ec6483 2750d9929627c4cecf9f5c8feb57a2c7ee5d1315593f3fa16c7c7292a594ce14 2c396d9b787665569dbd0f710822ae3155a3aa53d6f055cb19e897b235b65bce 4d507a4d41edc6a09dff1f737f2711478e6e998218b7f3e7c1aaf3938f68e9cf 9ea46e4a2904c459281cc156f5ce80d6356e357011cd0bc383318129bc8ce38a 758c05d1205192a457f4b1b31df97baec7611905cac0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: cmex7 %!FontType1-1.0: cmex7 1.1/12-Nov-94 %%VMusage: 20901 24442 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.1/12-Nov-94) readonly def /Notice (Copyright \(C\) 1994, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-B.) readonly def /FullName (cmex7) readonly def /FamilyName (cmex7) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Ascender 694 def /Descender -194 def /CapHeight 683 def /XHeight 431 def end readonly def /FontName /cmex7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /integraltext put dup 98 /hatwide put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-12 -2950 1626 750] readonly def /UniqueID 4701540 def /StrokeWidth 0 def currentdict end currentfile eexec 9b9c0887db83fb1ecd8335b0bb39cef0af64f656fc6e5c230cc9d3a7346aac7a 06bd9a40393ca15d3773a21e06b9b4254d3050e90726bbb09120935a8d448cdb c799d90205a21291254fa633cc0c2ed88781ef21a5d45b72445c284fa44f8f39 cf566976075a92e9e2947a6fc93d879c29eb26809ecb4409cbc8666526cced92 65de661a2f8b0a16be45a9da17eeaf89b0e023de2b0373dcab997d60b7d7dc1b 4f7e650a2a9f13ca0543f3b080ae33d65f2802b8c7be7b9f15e795d04a36b838 6bcb70812df09a1ea17874483f0676ba21a9106e5415f302ffdd6f28eb3b12ec 38faa584b3afd480024b6a36ccc3cc815f03801184d0c0d67b2b603dfca60009 3facf424541fd908f66157dacd8fd4a1c059fa55fc9b6474ee74fa5cdaeebc8c 0d8ffdc777c845b28551ab4ad66ef1d5e71736805957efb962da0af3b503e799 524784f09df6e12fee1c1d6d43c5a21d664e8b10eab5f186b1a9f8336799c777 4d61c7e3983dd4ac561835f3c0ea9fb92f4842a5ad13f9b86d5e54fb13d462bd 3759a8c8137caf522d7b668d169785652d9a4fdabb479e0325dd35e3234d2a2e 476728c764347c4f7e99e66c389365d6544104952e338b6a8df3cf37e7f1b5c2 cdbd25f405eb75c0a48f4cc6284947246a0fd585939be06fd7a5d388aa38e71f ced304f5e3750a9f8e11760502e66663e69c57c57a6df482e751cc921a35e53c d3895fa0d4033cdff7109095c56d6de146b6f486d539aa194d530a66f903277c 2dfa414425e493afcfb8f056afedf89fe97bad6f103294aa13d97159e92c9047 0968e453eb0ae7144c29dadfca1c017f60df18c9e08eb794bf20651dd9da6886 433eca55c97b92cacea5891f1e3dfe1fc82e1404365a0885a8d3f83bd9df1aaa 6f37a176c65b657f32c0aef6fc57258f025f897e7670f7d80b8add1d0efeac50 c29225888f4ca1b0f040afb26697cb811ef7212f42f1f168edbd6801b957b99c a83bc9b3e5273a8ff8204c92995312ebab85e17c4f283df85dc40cdd3cd744e3 d2ee9d876a7b1c1ba18dc0039df11fa1ee281956053b1e3eb9523164507bdcc3 db1b2f50bcf2e88fc92e150a220a09cec5dfc98bb2cd22461e559b5996ebb165 1bbed06955a7d4aa923d98459f41aa103da4b78db2dfa5e2c4305fd7474c5916 a0d7ad103d9f93325ebadb150689842393bd4a8fa853e89ef0a844aac6b82eaf 7b08bdb7878f4fea635d33befc12cced1e14010f4baab598da4d0d1d8bfa1e7d e1e9f9d7eaeb60ba6022df01e4b3db6428acb3ece6b3a82df13b9dc38ea90896 b07b950ad6782f575467b6dc77e83ab1a8beb037790f427bdc3d428ba911f945 07332488e53b37f8440c46b77f73dd9b26706cb34e6ec3cf30963a742b9d789d 8003179ef9d923495d327ad824c6a04e88820cee731e730e87950f1e30a96161 ac54f0bb816e6e3938eef3de99c2f5dfd20e6d3c15970a37a12bd3f8dcfa1a8d 2fef0d120b659d03bee79dd45c6f7c906af2ffa562c168d4eae0a2a313bc1b5a bc932147862f18e5c5b2777b0235f25f1b0016bd35abf86afb6ff15ccac6b079 4b9ba7d1cbb3b96bd045d6d1edf0a2bb633e5fd198c626fb569b1258cea5f3b7 875ed14a74d77c4854a8cbe50823c28cbc82abcf44d8e86d5e686aedd8895149 f6604235f910abf631d06a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % 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 (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 58 /period put dup 59 /comma put dup 67 /C put dup 71 /G put dup 73 /I put dup 99 /c put dup 101 /e put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put dup 116 /t put dup 120 /x put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdb81dd075 febcdeadaec04147e5c6afd2a0c44dff4307c6735b5424ba4a01953eb28b8a7c f076daf43c70ff1e8e64e4072e088c21596bd2f2ba9381ea788213a84b719755 e5506f68dbc7fc36b6b23b00bb5ccfba2cfa89a4221daf2c9ac51f16c7b65d91 8bdba0493bc3896a5f965a8bf05760a978b9cc00a16db96d19a1e4cc3b8a3ae4 9fb231bd7f439339f94cfc1df789424ddf4bde14bc80b8e34be362398efc8443 93f566436e33d3ab3ec1b376b7d5bf438ecdc2782b34fa476d922dc4a01c1c8a efaf988a0b6e88279e5374309a99d9660b26d11152c3b182e5b360734c1bdbf3 af6c062aed78581dfa245ed1a860a641e4ea404991c55141fd83eddb0b0bf5ee dfb3bed8b66e15a52c5526d4d3994aac12b0e3edc1bac9c33e73583f7840fc0b 1a83c92081bf06a7a8f7a8c7c0037617bd06f74e1fc58267d8d547ea7ffdec1f 5f60a396c8402fec57408ae59e954e84d4e10508a67ce9fd15fc98eeac7d615a 8ca3bad204c16ed24516a0f5368cd5fe2bd3d2da41803d10a4c002c6089fa738 73fd6aada7f35ae70102fb089af92511a862007f309085a22e3f2aeed5c2f14a 940bacf5e679a72d882811fe9fcfbd82f0991ff72448a206b849ed13ab339f72 abd74981291c06ef8cad2bb07ff416fde58d907e03f47bb2bb8330b5041d7d06 c881135006c39ae007a149c24b810fcf8cb3e36ca3d9b4c16a7151516009c4a7 ad1f3375eb2a194307607c42c9c2f37fcf7097331015d2f53ae3935c456ebc46 5e4b4b11f54cb82bfe4f5a960d2c11372a41fe2bdde4b3a01a51b6cdcc946d9b 3108f7af1200c0c7b2cf474f0a90c09d18fc21835080c08a7bfc1d52f5452bee 85ccc1b2b5167d0a0688de92c7923ee7a7b078b212a4aed90cfdf18664e6d26a 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1684aa5a4af63d34c3bbeac935f6736d3c28425d14d807fbe13a738829f7fb77 e13ac8d5026a43e22e3112be1cffc872b9ce53c474fa3af1eb55ed8e192f2d04 e962633a72d1dab10d47b99487e95c522159817f27b1d9a955d74aa9377aee4b c6e161943ec559512fa4fd0da137c8486c6c0bca56b87c0e3c7296b885291133 74b026503332c61d1e50a52cccbf15af62c5164b138491ccf62ff4e3134fb9b3 532d1711578c9f930fe9a8d6b72c8379c10cbfafd61cba48b82396e29d4217c4 2a3ded1b9b8a6f19c199e1f956b9470e94d8646c3a523b98696622c6109ff2d0 36b8d435c3d601d18bfc33cd88114b25ac885e975a8dd859688cabc682581a42 eea943866ee6f9ec7cbcde0636281153964dc8cd4dcd57847a61f48f36a94d70 f58f56fc14ed981ccf01b9b803aaf2b3bbaeac49293f5e1e82da364939af6137 89541cb41fc2b110c7c244d11ab0402ca4a4e9532124e2e1d7290d1ffd8c9644 a7b47561f4ddd802bb1e91e4c895cced6c12d5bd01fd250a4437184d8168d0d1 dc0720b784e3d3b1034d037cd9fcadbb14dc680dfe33819bef58939a9ce58322 b1c16bd16907a9c2c423a7ee960cb7d04c7ea2e1344a9e957c2e14b8e54a3cda c834b409c5a096134e08ffbfb0ba523e7ecf1222604c390e864d552f35b3bca4 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a1f9a09527c40867fedf316f9669890c42c24bff91a98e7c4e7206d9cad8ec8c 140ef9cfab86ff8c134ebd9e31a0a3bf3e92e0049ce5c4224c9fae5a58c8e9ae 3b74c42528967b34a34c95121ebc186878949eb648861843f1a0bbea2f379989 69bf138b8a94e900750d1f43238902b17631e4612060a5e6d25834697a549b7d 618190a18b6fbe1e06413a2272563dab6d0d158f977610b00bb50b933f66601a 12d60ff3ef4205d0326cdb5fbd9ed86d665d9157b28bc69de90ef662e9a7d6b5 5e573f1b51afc81a16d4b317fae524b77f76f7cc72eb05653fd4f84907da4a6e a4b3925aa19de7ab0d72eca3bcca26db12b44ca7555ca0006ac3522192b26678 ab476a6670c01df594e89e89026d1413f447fb20d44ddaa30378f2d03194f8c7 53854ad0e3b3b4c75b209b3f57f0efdb5b33b9f46d1a1438a147fcd09961e1a1 41900c9dc67395b9e1277070dbd8fde90382913c066c7a25660ed3248eddf815 dc781e204ab2eef75aa13da8dc77e42963bfca19e3c662b051ba594a36b0d7d9 77b0f6a2db1d5bd133ce718eca5cae85504c806343f0c41b4f7ff385cff7e00c 6f46d9c71a95d592034cdcae77e98680703ca58d082db3a8b3d79a5f77ef9dca 21c14a12d7ed467522c076b736f4ac99b070397191c9a51a0d283043ea5e4042 cf3ffcf0f99220de34d0f1f342cc6eff4fb5d412fcb43c898f93ed8b72ec6dd3 20bb1e2492a335853edf549af9f020750205400067d657ecaf352d580b91a5f2 602c1daf91d0c8622ccf937f296dd5ed0186fb7e50580cc46ed728b8e90d63e4 ab62156ef800369422384d7a17d208d67a199ae1531edc9ca85d875af8233a35 990f88e477bff8c3746a18d93775db3543ca71490569244d3d469af655af8409 4f4e6ccde429a6a3708b91e27c83135e1b5ea4813b5f29cf161ccf62872b99fd 373e20f470 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put dup 78 /N put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcf3fe48b07398eec882c325567e28b9eb4d699111343b1910da9117d 74d6e9de75247ad908f8ce2cfc49e0faa22dbf4d1abe2c035b8f8eee2d1967e0 7cb9b5d8de5e0b1c3a86c7a08a580e51be75a8afdaba2cda08f87d352652a3c7 9bda6bcd32fd59e8079fee3fe7fa161ff626461e85ff8b9b60e2e6bc72884c6f dbaa5e8ad5912206cdd26b204cb34a5403c4fe9cc6d0a2e6805a98624c56baf0 3e673abaa926e941739ba23126dd78ca82499f376d15bfc7cc70f1586f5f10a1 fe99eed1a1f44ace1fc2bb8017599cf6ff3def1dcd7c683d21b5652c6f3efc22 ba60e48ccf80098276ecada9d860ce661bf5c42997ea934c90c18a2775a03251 65c0757807fd7951dc105a497d258c395e663579074a2ea1ada0686f6c1b3c76 62aea5f48f9d03f2a88facb36aaa9fed698ddee110e699974cc15b188d37b021 1dc1b8e894534109528cd063c402991cd2f4dd59622cbae9c292325d5e3394d1 4a398a08cd5b325c256ca7d8c35140402911aceb92d27a3e6e54f9b147510234 2fc672254185fd024a9426933c18997ccef9e90cfa51e507d3dfd9f2e179e8cb 3b7f068551e3d608f6dc599cdd142ba14b112f2fd0852b7cc515f35e98367dbc 8a687b56b527442fea37987692818e14d47ee9183ad24633e3829fcf5d5b7954 504ae1aa68c4be6462acba2a147a9e424209bbe00b0ebfb18348d630f209ed65 74f86388831ea7a8a12a2cc9261ef4d4dec11535bd30fcd4730c178670eafa22 51e3cb17851f52c059191023175bd82df403301b6941417daf62b55258610bad 371cc19a8ad8bc6cca7f3e5a05873f96666c6c0e7f234258b5c1b61991ab0efa 89d2198ec0b63ccbde799c9db0aabde1d41f702e9df47121a268655606cf5614 69637068b5f41f2465128fd85f2ea79076a0d715a63f980e513c7d04f090abd5 b776eec54926886fdda8fef65d3beca7131e2dd7d9d0eff7ec036188995da1ac 2459385ee356ae2932109bc3e466e7c1480113a3f97f796e87fb9c550d03431e eb4da384a9511a44b61b3cc1c142b214fc2d47d92264d4d95a6f3e09b155426b 68baa17afdcaf57da6979f3fc5796eee586d0d42171020549b98bec81c46d994 90438b1a952939cea04f86c4f9cadca73747cbac7d7b818286552fca6d2d378e 6a5412ab85b9a25fe09b89e7628dfafb03d11275da3962c105dddfd81948adad 736279796ef683ace29d460ccb8950d28e47869afb7174 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 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fe3c35a5e4206f60b2b4f567b948f7ac54e3ed16b58e137958bb0c141e411b c1c47810add8abfba9f5ef95f96d064cbea94293bf05dc6979c6001553e168fc 75fc6cffc105d3eecca34913cebaad61f3a0949cc0c6768d5795ac7405b64118 0c28dd9d2fbba6debe3de789753f299b268de623c58dd73f912c935c525fabd5 a8a45699842cb930cc3a9fc92d950210de61d71d4457d71b3788acf2afe5ecce 8ac9e714628742820e52127239253ac17b853d8d03e0530d0f53ca9d2e0f7830 daa082f471eb27eb9f2b2f4b06aeccc1b80c0895128e0724fdfe44d7a47e72a1 eca3df3411b50056b6811b41d513597d04f5c2ef0794c38971ae5a662330983d d76e5bd67dbb1836cb7dd48bdb38b59309a82edfa5e1aa1a4ab74694c6d1975b 7c73d3c5be222139b25c7b2ad2f252004a7f1f092028da2845ef8fec8e10ef4f a24b69875b40fbe542e3a8da32f8de8f686c191989b59636f56d596d83735ddc 4647b7cf8a3666f02fefb4f4191515edd0fbd12a8848ae29a5f27182981e0da9 6f8d15449cdc7d47a0b331e29157a8e7265fbd4ba59b316889d764201e67ff81 e808a12bf2f08f4059721c776947462aa30a2131cce0fd37245f6a25d781aa7c b497c37464e28ab126832fa9021e961d42874e4224894a21ed10e768032cf471 616afb23741c910ad8f6d8f7cc93ca85e3471697e0569ab777eeb837ece03563 5ec267a61c0ed9e7fa12b42f954a4ccd770e16fd7130c0faaf7a0ba866d8481f 509bbd3adf7ea12c8abce65ac5839c2cfc76ec67b103c2c127c2cfa6a5bc67aa 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 119 /w put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d3c0b4188a0543ee88fa787fb10793268e415e389ff071cdd46bf6b260e 1e9c9ac037edba327c6379836655308ddf35e4a4488ad32d3ab301c67a2fa71e 8ae00daeead4271df276f5ce395a57e414f85237cf8f1be070309ed5c3078506 c9d38c2adeeac618d04c0be2ff891dcfc9ada7588efb91f739188a933e1d37bf 632cf042106df1bae0cdfb79c578e565014cead123bf856cc0c14983f5548a20 c0d45446e0dd2b017f528e46d394a88a2d5f86fc669ac7d489f1851c100b29a0 ee3ec82faf87eba944a57dc14ebc94f2445a8a21ee68eef1848d534bdd192ea1 181220351ff93c51e53fcd5ee6ec92bd644d121eac52bc163acb6948a7a4c256 7243c7113b1b5650aeeedb14ffd9b6ce33605907595230a06db01bfb510a460e 76598fb2bb67498f79f0f5b5c5fdb5515cf2257aad324ef1a877a296e2851d38 f8995123e527c4956247fc8ab910206a9ce8736563ff0d4cd5d00579d60bca51 650af348ac4d6422c4cd74d924b532d4a28ba95eff24aca4b3cd1d28e207c7f8 906f8dd99aa04c135fe46c1dc8ee89bd768df17240a70f1a96382defcc180b9a f1b6fa57a9b0a2440bae0797790f91733970b16e742e6341d95faa40c583dac5 13a84eb181d2946b5d0075f71db8a991dc1a809cd3cfec573514392567a82e75 946ccf6e14f4a968313b3b6d727c83018ed58b882b96db208ebd27f7d30ffbc8 ab11e10a0f81425a62bdf7eed9f618c746103dc2d1c3e1f37d27083f4068a30e 099c92685e320266ff8047e81140658a63a6b52e4cd63e9403c28a892c8d258e 39492535973e1c042174e75f168bd0716603d8e13d46403aef6948f0ac238e60 96b243d529c8e486731405a7c115ac4f1e97344f8896566437fbc66c619e36bf b654a6d3547134dca3be4606c086c908a6aa711306225b47dde63e4330d9e483 db7f04427bc14af311f1cee8a2c5096630361024498727269ef27ab9258aa563 034abd702f4aca7303fd878f1578adce4accad35afc06d3d1955b538c28bfcae ad19860f28ff29d07b3c5ae9e12f91079e4aac3b694c35b401dcd8b216ebe520 efc4b3baa2d3daaf1cfe8636ed97a186547baddd1fe921a0236da9726cace534 c93b348c048083a00c4eb1bbfb38ae11906a19cb9edeb19626f2024866c1a4dd 95444f9b6aa34b94237e44b063ddc3ed474887e4432f8dab135f1d4b0b2c59b4 1ea141ea5131d7c8b641b1933d85936412edb3e35fe8a0ea119552bf182427a5 6649c0ef4ac1e60163f95ab43a6e03426c52d598cb9e12b38cbc62f883b94a0a 00eb7cd1b6becdf2f89d3c911cf7af731287508e493e31efa03af8aa5137fafa 75b3e7475831fb2baa52c4ad9093b6d264a5a856c9ccd4918092e0e044e06cc4 1d7793a29df2847e010728231f804bf29311e53b591349d5f1d32d5c28ac6cea 2b7148b68a104b8849a40671c49b2aa933046556796b8935387480e080ac1af6 d086306e1c2a4e7ac8e0b6628f6b7d34527c3fdcb2c1fafd9af490bb9f582bb9 43742fa664dcc423c905fc893e1ef292fb59c9c05a2509f16744a36e1eb8a44b fe513b98f9591bb98971b923baf3c71c3c816120264cf642d920fa541fd40bb6 668aafa7930156fe9278c5adc1780ef947a48b97cd5941b54a9c02ade58eb748 d7a17907bd321c56c23767397dfc1206766818180adb709d6044b56b0a278de1 ea5cdec1378aca2d44c796efe097faf795ccdf3da99215200d1016668f56e0a1 08fe697be27cb4c97de8ee5c00a7a59b669ad07d764bfef8aaf100b1de846233 4a3bf9f83c3f3e771d3d77b726bc0f0d85e3544ba3bc3b66412db664a45daeee 3f323b90ec670841242cfffbda1f1f48cc8615078cd033afd70f0be32d35f043 8dd7aeecea268224f7b45ad25f1741597f1b962d6f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 4 /squaresolid put dup 22 /harpoonupright put dup 116 /fork put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa216e8652c0b082c3d1074364e9b6351 415c33a2bedd7d9cf61ea597cfefb53818334419bc8dc2e47c16decf92a8c937 aa19ff759cea11032e87203d6356b2f51c7625ce092974d621141b2257c1b227 a59ff5bb7b8f46e66bc1705261942859ddc51852a272bbcc4136c2b62c7c8eb7 e95a46ac06c4fa77357ebd4dfdc16df07b2d952cd3f2eacb444eab084d9b9837 794a24b2d21d0420f1c1630e31258eb672ac44a130ecee09ba37dff28fbe15d0 07f71940368afccbbe15523a3315008818274f4491a0c801504611dad8775e40 a5e5c8130003397f8a6199df92f038127ff3a6c57b5aa160db5a08bfcb5368d0 684e0fdcd51849b828b4dcb540e0454d8b110b3e450a4978fd60f96c2de9f240 d50288359cf5304ff8ffc18877d9728b90527c64bdc3ebfc330dfbd4bff69cf7 a9f6f17c5095ddc3a433dedd2dee81cc100630f04b672165c432b82290c37a26 9bf5f14e5173a076b3c489c2e0d9e97762f45b23cfc98e6d2bac970ce4e85de0 caf7d94c6d828d815bc410100cd2f8a859e445e9055b390b3cf23f4cd594a7d2 ed7b6e93a58dfb287e29b6b89a89fb0635481b92ea4fad700474b6798eb1ebc2 39d334de405e31dcfcf45899268bf398c20e958d14d2b28cbf8077b176e4ba0c 8553776b93aa69a0b5ef42de9946b75ecb28ee761d1b0e1d3dd37300669038c5 f10beca4b682d2e916ba5b00c90a527489ad0f7a1e77e7390507390401a7fcdb cf44259f2520dce0d37d6d97f5a82e906a8a251696c04bec8dced59a355e94ac 69db5b80ac0c4815a9b9b67ef68abce8a861ea6b17a374a6cad644b7 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUSM10 %!PS-AdobeFont-1.1: EUSM10 2.1 %%CreationDate: 1992 Nov 20 17:36:44 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUSM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUSM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 86 /V put dup 87 /W put dup 90 /Z put readonly def /FontBBox{-8 -194 963 738}readonly def /UniqueID 5031988 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6c8752eeafc779b1f67b40274cb47e098485 f163a6178722fa0d273e24cf928ed45cf1bac4c48c315abade8dbb5eb13b7406 4e6990a3375ed2429e582f08c774cafb99075933a44fe9c80ca469d4641c4a83 5661f4fde8d37a35e49045891a2c6bc439c85e6e646892edcc6a4497ac4d3794 164b31d0053a0229691d68e30086bdd14c6572dfb18f2f1fc063425b320345a4 5f2050692f5f4dec1b815ea3e6d134646b23e44feb7098902bd2f383e927c94e c339e5929744a75ea155cb54bc097d8e9e84f1a8d58f9fd26766f0a34d424a39 4894b9d53b162f3a18098120397d5a00426aeb4d373bc48548ff7380976ca750 36ac450db2b884156fa907d3d1c108d7d16fb06bddf4260b4368eeaf24839d44 bcf63dbf55e26f3be2cb47c1caa5dffbcc0cd0437d70517ede599ecb7a392947 af93d0353b82878f80e091ca6794502bb5a8ef1e51e8d118ceb2d7ac3afdb9d2 5e0eb36c872c09a6f13d7cf0936ae9a2c571dfe4e1dfbe0f57cfe20ab610e2ac 6b9784bfccffa7769eaf7ffc3203c4ef954fcadb5cbae527e83aab07d6732538 55d17ac99f5ab1e237277e92d115d34a0bee1db41186ebf61b1af53110527b17 2c0fa77784f860850631ac92dc34027dab1b8d5dbfc137712badb1b91470c103 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701d6486d09a12d9beb4254907e365ec4e39135ca882b0ae0bafa63b65c868cd e67de8e2def44f79f0cca440432afc3e92f11062573565d4768a241e5132610c d2c658c4ef01e2fe2f5f19aa8a82f784449c4f50d70d8bf095de06dd0436ce25 a3cf37ba7f036f42ea8ac272a87d6892535dda9b100728d20e2d8f103a632cb6 b540a4cc2757664ecc075c3ecce5735362bd771c4957bb5de46186695dbb76e4 49186528c5102e11436f90342fdce07ca68c7e2742a225e786ed3742bbc0d9eb de7fbcb527ee83ecae7b55e78a9c7c2a8ca19dc04e328b4fd0c0870dbc465e61 7707036eb59427d307e16fd01b5ec2efccbeb823363348182323412af980a5d4 2f804863dfcae332f696867a59a2ed7e801e43720ef62b149f339dd324ebeb94 1dfcad2d6dbb615eb71445c5eb70a2458cc5bb177461c7ce3790b4c8e80781f1 75c35660b6a4416709fb977396d66be0b6df2c8d8e0a5d2a5657f925e0d42713 7ad18e43710a1aa9edf2f41ae3edc28aab05a14aed539de5c253f66482ea48b2 c7aa7d9a99f286dc31607dfda220c8da5a659b0508887e2281f505144a99eeef abd47a02ccd72c596291e590a9d7b6b2c82d094c39a50e0f51fb76ba8471ae81 1567a909dc5f3d6510455bc4e0dd1a04e33f2821b2de3c3729ce3d60d9448e1c 10bb070e19e2b0a884db7fc2f47118c0b448a7f6837b6f3183b2816bc26d74e3 609548aa32132ab9f8192afa3df0d141f79b94e6853b5b3052598e3926662ad8 2e14d0be1ca54c51431d6c4ec15065ba557b2870b6f8ed651572158bbaf38e1b aacf511422336ddfd922416cf0b7590de6609bcde6a89de7d4d2a9cac3091171 ef0acdd4df44386f2401553ea90b281c6908c3a206daebf7e8634b6264d796ae 46e4e719e56b7a98508a4b12fade416d072182adee9ef2e00025975a2b44fb37 01ce771b9efa263fcdffbc1a9e1144d1ed206b28e77fb51d7b1735a85cf8a301 571ffc33abd2dbf59e3abdf9790f898ec1579ac018fd2708cd1c397868109ab5 f3cc5732d234d160f142991243c7f4c85c0bc940d1d43c705dbf1d3845f96abd 9707af0803ad3f4e455ee27eb2d913e84f18d649effc6240f7b1411e11a6f5f5 67b3ef5612038f56a07c3bfae2513905b0ab3f5cdc4808bf581d14e1f3b0e771 c0e7df3cdc1a80002989f77eef09da90ec20cf2b1153598e7ef4524fe4bd2ce3 eb3e8e41041c42 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put dup 78 /N put dup 82 /R put dup 91 /hatwide put dup 122 /Digamma put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c45b254635c3352c04b4ac72a87290d2af4efd37444b86b0c9 1bf2c59722ff131313573f1588ef7847d6ceb4039b88d705991cf395047c80cc f446d9653679909637f060c23d48a4859c355922dd1a0a5e45b161f31f2fb7d8 9bfd29f41f4f605eab1141739e8de5769676f9485d778e8e403d3749f4ea95ed 289e7d9594513341de4c13ed4932bc3b035a029bcbe630d140bc6d31d7aca599 980827e490e575f1c1e8dea461d12d1733dc9f82501f6687a6efc96042b92dda afe13d984541121bbbd30599d82fa1b51f3500bd740c46e1fc8e7f6e4c0dd331 38014a79ed7f87758a26cd7849bd9d14069c4a8d705fa58b9c705db9f5f3a8ee 9a3e7be83eb3b7be78c88202d95a85470028a5085ea4e14d6c26ecffb3bef6c8 a21a5854f95ddb81f5d1df79fc92758f3220d4d02edd4987abc404d96749d619 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b7972ec2d28e612df66833022d84581f19ab1b8ba1522cc0ece8889aef4d49ba 49f8c5dde1b1b322c1d14e1588d63c725610ef6e586093f45c26453f614e886a 7fc5c1e4d87ea2ddb36fa1a70976a04a1a560c5e7e0170763040f6ff4ba662b6 188cd86aa7938e3f45b40c5f7e65c33ac27860919846c9f72835382be0c8e3c9 6d30062903feacadf4e945997d3da04dcabbb22821833a6393c6f24cae4c0901 0de4cf0a764ea8fdcb377505988bc20e33afe4f3b98b3f0b3d77aae2b0972893 913f2c574cc8eaefa6a82041a92a9bed546bc0d0c6a96233c2edda0ca48ca555 7b20477a8dc7bd9745d847968c2ae08c4de5a2c61ac0ae0cd23f8e1462acdf1e 2084f4d7a547a97e2b2689f2c98d650ebd0a2c7ba326234d0687c26d3d36d6b5 a6d1aa25ac1559b53c4f5a34d0b99125bfb76e6563671702fa5809aaedb4a8ec 5c3a95a87c99cb590bdaee38be2b78ffc8ea2253a2ebe13e3e01d2b9da3907c7 0ca6402927f63ae7bca0aa7d87636c002b3f19e5ebe923199501f4e7a9901324 4dd8746d102d75d74f86e8d8ec0f5c68dce82c8abc17710414ab2680bdb36ef4 ba055ae54896bc5d6dce7d3c48e96a10b3f5852e6ca7797155662a1a018b57bf e1f5047400ef717288b12ccbcec6e3340a13005243e6075aee9c355480 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 30 /phi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 75 /K put dup 77 /M put dup 78 /N put dup 80 /P put dup 84 /T put dup 85 /U put dup 86 /V put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d98f2e44ef37ad46 282d7a587974734c2b1e24d8418fba0841578cb551332e6f777773f1b3155cda 5c13080030cd6402a22174611509221f2b3ac19fb60920218837144746f81792 43cd14528d48c032d279f6d8a87d2eb484f1b977e07c9f343d925c4e29440e4d 7fa894f7b3aed4f35967cf0754f375a178c830bf375ed5712c0598dda2182320 99fd8a45277a5d36139c90411cdf1b382d711f4c8011df724a07333c5421040c 06c6bd0c5bfb17cc2aa404c135fece394c5e77f735945b7dee71367ddb6c0e18 57d9eff272051cf196ea6b066d9ce7babfe387d9119d18a5e39a309f31a783c3 0df3e61b5fbca7d2ec0c81e76c4bd5d8ee71005814bff982fc50a47d68cb3864 afc1374db9f708ca964ab5fbb144195867aeb5679ebc7cd16b322565cb125578 b6cbc34fb303222fec3740bfb667314a484686e28e05f65d34729b4015f84cd6 902fda338a4ed9fd2d03ad583454063b8cce0e526aef028fa2a2ebda1c84efc4 a4187c1cd50d1b83673fb9366ec4ea2e9fd8b06add2d388a820830d8c80a8eb9 e352c1c4b790fbf012264e3bd9bc05f000bef2ce0f883f73a4e60f32326dc0dc cf5ddd265c01b8b6fcc824baae99dd71f6022c56f8c4bf4922dc87216ae40978 d88c27bdd2fcd76a1b58f6276384e36accae91ea54d5355687dc09012f16d7c4 d77308a14413b8e4fb9ca4cc8e51ebef577d92697082a0efde58d9e69bfe7d83 5b344cfb5e04c8243b63dd244ef74704d2a4237dea61be26868c933980f1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 10 /circlemultiply put dup 18 /reflexsubset put dup 20 /lessequal put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 67 /C put dup 71 /G put dup 74 /J put dup 75 /K put dup 92 /intersection put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b872d4574b0d1aed26b0304fa60b0902cbdb70a85b6e3e5bc7f68f080d 37288ca0c054e1484c033c9915e7a03a6daa49a29202b44f8f7c1c1aae982fc9 a20bd1dd40af0962ec86cd796a262e81609c97dd99243d30a01ce5be4db72182 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5a713834ff4f831cd0c166a0a5b71a450acdc73580a1b548f29b7f3c6c581c7d defd066596041d4c8f810a990fec8608472b69bfc650600afc64c0281830c6ac 5c1eef05818dba78601bc4d5bc1c2c3cf2862ae76207053b40535a0a6057a6ca 41fb0e5932bbd23c2adff16a08c48d44f53bdbf4871622041ca292fb92b51838 b236376185582783976b63ed132b028a41ab53fe22bcd426aa324ded5cc1cbd6 fc700516fd0272c34b62dbdb2b96765d2724e29813da8bd14992317d5187b9a6 ceb233b2c99ab5145f80063661b845622c5f0f59e2d2027143268f279479a177 8777658860649eff85a54857bc27f709878c41828b14fb6924d3f50037fe8c1e ea92ec9f571577ce71084fdd70e7f5a3981f848d297f1aaa8c650285fbeb5f02 0335f76f858aec90c857cda74ef44dc515e8fae06c5744d315705bc50b46fd2e df5f1c5976c3bb81c549b203cb89a5d84ede287ffb18f625146eb6e1dd 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 35 /arrowdown put dup 39 /similarequal put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 63 /perpendicular put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 79 /O put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 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677bf51f1d074d54c057eea4218c4b8c83ece254f8072db8cec57fe22181207c 092b45ec47fb01d4c8fbf1952226eedb79ef9e75eb27fd7da9f4462e8f559a71 f26bfdae6f20631b39b8bd33f62c6ad7ac62d4e94c73d8674d9af66a361ddd91 4931edbef03a405d1b50ba778490d48afb940375cc2e5e122840dfb9817fa3ea cb2b134d748a1b019176b1acd214021544 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 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eb1fc1c6108a4b907491c0b06ad540f85505913df731cb1538453a6b82559908 3e166cc432f339fdbcd19a7c2f8a19ad89cfe59f8a4a01b290c9ad4fc99f7e55 b49a7dbf09924e14052c5c5a7a2066391e573f70555752dfb62d4f01caa50d00 eed01791ca9e376ca2e43d8c1ad176c3b3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 54 /six put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 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1173001ac73dca57b7bfac24754d7b18b13f1c57a7c3364f9f09e37e7ebade6f 8d1b7ed753b39a09ae38179d983ff19d39cd268e122a8e26cae1788abf7a4686 8b51 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 6 /Sigma put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 23 /ring put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def 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d3e17f59b599ddd2690707aee0f40a189d3437be41ae458bbfbce396d69fdd04 0c07abcc2fe3a9e3ce982626ae1ac30e92ec26144f905ce1593b4f14 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 cbec82a5ca50f0598502b434ecd94531b4a65181b5b3afcb650904b15157737a 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7a0efb4c83b45c068fc88617524a86254f16aa9d71205e8cd730d869ef94c6fb ba26c9ed031d6f5c877273befaa51ef6cdab40dd95c2ead783f3208ff2ef0d80 27669e2e77c2c1e0815cbcdd199c465f17d43773b74cd8ae94b7bad5770ee4ef 2af33910daa846eb21262d16fa 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 116 /t put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % 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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 68 /D put dup 80 /P put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put readonly def /FontBBox{-35 -250 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT8 %!PS-AdobeFont-1.1: CMTT8 1.0 %%CreationDate: 1991 Aug 20 16:46:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 64 /at put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X put dup 89 /Y put dup 97 /a put dup 99 /c put dup 101 /e put dup 103 /g put dup 104 /h put dup 109 /m put dup 112 /p put dup 113 /q put dup 114 /r put dup 116 /t put readonly def /FontBBox{-5 -232 545 699}readonly def /UniqueID 5000830 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 68 /D put dup 78 /N put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 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y(where)h(rigorous)e(renormalization)g(ideas)i(w)n (ere)g(used)h(as)e(fundamen)n(tal)i(inputs.)555 1222 y(One)i(of)g(the)g(aims)f(of)h(the)g(authors)f(is)h(to)f(put)i(forw)n (ard)d(a)h(form)n(ulation)g(of)h(p)r(erturbation)456 1421 y(theory)37 b(whic)n(h)h(satis\014es)g(the)h(needs)f(of)g (axiomatic)f(\014eld)i(theory)-7 b(,)40 b(m)n(uc)n(h)e(in)h(the)g (sense)f(of)456 1621 y([53)o(],)26 b(and)f(is)g(at)g(the)h(same)f(time) h(applicable)f(to)g(phenomenology)-7 b(.)34 b(In)26 b(distinction)g(to) f(earlier)456 1820 y(approac)n(hes)i(w)n(e)j(giv)n(e)f(a)h(purely)g(lo) r(cal)f(form)n(ulation)g(whic)n(h)h(is)g(meaningful)h(also)e(on)g(curv) n(ed)456 2019 y(space-times.)555 2218 y(Our)e(principal)g(result)h(is:) 456 2522 y Fs(Main)j(Theorem)g(.)41 b Ft(A)n(l)t(l)31 b(p)l(olynomial)t(ly)j(inter)l(acting)d(\(sc)l(alar\))g(quantum)f (\014eld)h(the)l(ories)g(on)456 2721 y(smo)l(oth)39 b(curve)l(d)g(glob) l(al)t(ly)i(hyp)l(erb)l(olic)g(sp)l(ac)l(e-times)e(of)g(dimension)h (four)f(fol)t(low)i(the)e(same)456 2920 y(p)l(erturb)l(ative)30 b(classi\014c)l(ation)h(as)f(on)g(Minkowski)h(sp)l(ac)l(e-time.)555 3223 y Fv(Before)h(starting)f(the)i(description)e(of)h(our)g(claim)g(w) n(e)g(con)n(tin)n(ue)g(the)g(description)g(of)g(the)456 3422 y(in)n(terpla)n(y)26 b(b)r(et)n(w)n(een)i(p)r(erturbation)f (theory)g(and)g(rigorous)e(metho)r(ds.)555 3622 y(One)33 b(of)g(the)h(most)f(puzzling)g(things)h(in)f(ph)n(ysics)g(is)g(that)h (all)f(the)g(attempts)h(to)f(include)456 3821 y(gra)n(vit)n(y)22 b(in)i(the)g(renormalization)e(program)g(failed:)35 b(More)23 b(recen)n(t)h(prop)r(osals)e(lo)r(ok)h(for)h(theo-)456 4020 y(ries)h(of)h(a)f(di\013eren)n(t)h(kind)g(lik)n(e)g(string)f (theory)g([28)o(])h(or)f(its)h(generalizations)e(whic)n(h)h(are)g(hop)r (ed)456 4219 y(to)i(describ)r(e)g(all)h(kno)n(wn)f(forces,)f(or)h(Ash)n (tek)-5 b(ar)27 b(program)e([1].)555 4419 y(Because)32 b(of)h(the)g(large)f(di\013erence)h(b)r(et)n(w)n(een)g(the)g(Planc)n(k) f(scale)g(\()p Fr(\031)g Fv(10)2882 4388 y Fq(\000)p Fy(33)3036 4419 y Fv(cm\),)j(where)456 4618 y(\\quan)n(tum)21 b(gra)n(vit)n(y")d(e\013ects)k(are)f(exp)r(ected)g(to)h(b)r(ecome)f (imp)r(ortan)n(t,)h(and)g(the)g(scale)e(relev)-5 b(an)n(t)456 4817 y(for)20 b(the)i(Standard)f(Mo)r(del)g([31])g(\()p Fr(\031)i Fv(10)1669 4787 y Fq(\000)p Fy(17)1812 4817 y Fv(cm\),)g(a)e(reasonable)e(appro)n(ximation)h(should)h(b)r(e)h(to) 456 5016 y(consider)28 b(gra)n(vit)n(y)f(as)i(a)g Ft(classic)l(al)i Fv(bac)n(kground)d(\014eld)i(and)f(therefore)f(in)n(v)n(estigate)h (quan)n(tum)456 5216 y(\014eld)k(theory)g(in)g(curv)n(ed)g(space-time.) 53 b(This)34 b(Ansatz)f(already)f(led)h(to)g(in)n(teresting)g(results,) p eop %%Page: 6 6 6 5 bop 456 226 a Fy(6)836 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(the)h(most)f(famous)g(b)r(eing)h (the)g(Ha)n(wking)e(radiation)h(of)g(blac)n(k-holes)f([32)o(].)43 b(A)30 b(lo)r(ok)f(through)456 624 y(the)j(literature)f(\(see,)i(e.g.,) g([60)o(,)f(3)o(,)g(25)o(]\),)i(ho)n(w)n(ev)n(er,)d(sho)n(ws)g(that)h (predominan)n(tly)f Ft(fr)l(e)l(e)h Fv(\014eld)456 824 y(theories)c(w)n(ere)f(treated)i(on)f(curv)n(ed)g(ph)n(ysical)g(bac)n (kgrounds.)38 b(In)29 b(fact,)h(most)e(of)h(the)g(pap)r(ers)456 1023 y(on)c(in)n(teracting)g(quan)n(tum)h(\014eld)g(theories)f(on)h (curv)n(ed)f(space-times)g(deal)h(with)g(the)h(\(lo)r(cally\))456 1222 y(Euclidean)f(case)f(and)i(discuss)f(renormalization)f(only)h(for) g(particular)f(F)-7 b(eynman)27 b(diagrams.)456 1421 y(W)-7 b(e)34 b(are)f(a)n(w)n(are)e(of)j(only)f(one)h(attempt)g(to)g(a) f(complete)h(pro)r(of)f(of)h(renormalizabilit)n(y)-7 b(,)33 b(that)456 1621 y(giv)n(en)27 b(b)n(y)h(Bunc)n(h)g([13])g(for)g (the)h(case)e(of)h Fu(\025')1852 1591 y Fy(4)1890 1621 y Fv(-mo)r(del.)39 b(His)29 b(attempt)g(w)n(as)e(ho)n(w)n(ev)n(er)g (con\014ned)456 1820 y(to)g(the)h(rather)e(sp)r(ecial)h(case)g(of)g (real)f(analytic)h(space-times)g(whic)n(h)g(allo)n(w)f(an)h(analytic)g (con-)456 2019 y(tin)n(uation)21 b(to)g(the)h(\(lo)r(cally\))g (Euclidean)f(situation.)35 b(It)21 b(is)h(in)n(teresting)f(to)g(note)h (that)g(the)g(main)456 2218 y(tec)n(hnical)28 b(to)r(ol)g(of)g(his)g (pap)r(er)g(is)g(a)g(kind)g(of)h(lo)r(cal)e(F)-7 b(ourier)28 b(transformation)e(and)i(that)h(some)456 2418 y(of)g(his)h (mathematical)f(claims)h(can)f(b)r(e)h(justi\014ed)h(in)e(the)i(framew) n(ork)c(of)j(Hadamard)f(\\para-)456 2617 y(metrices,")h(b)r(oth)i(of)f (whic)n(h)f(b)r(elong)h(to)g(the)g(p)r(o)n(w)n(erful)f(tec)n(hniques)h (of)f(microlo)r(cal)g(analysis)456 2816 y(that)d(w)n(e)h(use)f(in)h (this)g(pap)r(er.)555 3015 y(The)33 b(situation)g(is)f(then)i (uncertain)e(for)g(general)g(smo)r(oth)g(space-times)g(with)i(the)f (Lore-)456 3215 y(n)n(tzian)c(signature)g(for)g(the)i(metric.)43 b(Here,)31 b(to)e(our)h(kno)n(wledge,)f(more)g(or)g(less)g(nothing)h (has)456 3414 y(b)r(een)e(done.)555 3613 y(Wh)n(y)g(is)f(the)h(problem) f(of)h(renormalization)d(so)i(di\016cult)i(on)e(curv)n(ed)g (space-times?)555 3812 y(Again)f(the)g(precise)g(p)r(ersp)r(ectiv)n(e)g (gained)f(from)h(the)g(rigorous)e(approac)n(h)g(is)i(helpful.)37 b(The)456 4012 y(main)22 b(problem)f(is)h(the)h(absence)e(of)h (translation)f(in)n(v)-5 b(ariance)21 b(whic)n(h)h(in)g(the)h(rigorous) c(sc)n(hemes)456 4211 y(pla)n(ys)25 b(a)g(decisiv)n(e)g(r^)-42 b(ole.)35 b(In)27 b(general,)d(no)i(global)f(\(time-lik)n(e\))h (Killing)f(v)n(ector)g(\014eld)h(exists)f(\(no)456 4410 y(energy-momen)n(tum)34 b(op)r(erator\),)j(so)e(there)g(is)h(no)f (canonical)g(notion)h(of)f(a)h(v)-5 b(acuum)36 b(state,)456 4610 y(whic)n(h)20 b(is)g(a)f(cen)n(tral)g(ob)5 b(ject)20 b(in)h(most)f(form)n(ulations)e(of)j(quan)n(tum)f(\014eld)g(theory;)i (the)e(sp)r(ectrum)456 4809 y(condition)37 b(\(p)r(ositivit)n(y)g(of)h (the)g(energy-momen)n(tum)e(op)r(erator\))g(can)h(not)g(b)r(e)h(form)n (ulated.)456 5008 y(There)25 b(is)g(no)g(general)f(connection)h(b)r(et) n(w)n(een)g(quan)n(tum)h(\014eld)g(theories)e(on)h(Riemannian)h(and)456 5207 y(Loren)n(tzian)32 b(space-times)g(corresp)r(onding)g(to)h(the)h (Osterw)n(alder-Sc)n(hrader)c(Theorem)i([47)o(],)p eop %%Page: 7 7 7 6 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)615 b(7)456 425 y Fv(and)24 b(the)h(meaning)f(of)g(path)h(in)n(tegrals)e(for)h(quan)n(tum)g (\014eld)h(theories)f(on)g(curv)n(ed)f(space-times)456 624 y(is)j(unclear.)36 b(As)27 b(a)g(result,)g(the)g(rigorous)e(framew) n(orks)f(describ)r(ed)j(b)r(efore)g(cannot)f(simply)h(b)r(e)456 824 y(generalized,)34 b(and)g(the)g(more)f(formal)h(description)f (based)h(on)g(Euclidean)f(ideas)g(and)h(path)456 1023 y(in)n(tegrals)26 b(do)r(es)h(not)h(help)f(m)n(uc)n(h.)555 1222 y(On)f(the)g(other)f(hand,)i(ph)n(ysically)e(motiv)-5 b(ated)26 b(b)n(y)f(Einstein)h(Equiv)-5 b(alence)25 b(Principle)g([59)o (],)456 1421 y(a)k(quic)n(k)g(lo)r(ok)g(at)h(the)g(p)r(ossible)g(ultra) n(violet)e(\(short)i(distance\))g(div)n(ergences)e(indicates)h(that)456 1621 y(they)22 b(are)f(of)h(the)g(same)g(nature)f(as)g(on)h(Mink)n(o)n (wski)f(space,)h(so)g(no)f(real)g(obstruction)h(for)f(renor-)456 1820 y(malization)j(on)g(curv)n(ed)h(space-times)f(is)g(visible.)36 b(Despite)26 b(of)f(the)g(in)n(terest)g(in)g(its)g(o)n(wn)f(righ)n(t,) 456 2019 y(renormalization)19 b(on)i(curv)n(ed)g(space-time)g(migh)n(t) g(also)g(trigger)f(a)h(conceptual)g(revisitation)g(of)456 2218 y(renormalization)26 b(theory)i(on)g(Mink)n(o)n(wski)f(space)h(in) h(the)g(ligh)n(t)g(of)f(the)h(principle)g(of)f(lo)r(calit)n(y)456 2418 y([18)o(].)555 2617 y(T)-7 b(o)18 b(dev)n(elop)g(p)r(erturbation)g (theory)g(in)h(a)f(form)h(whic)n(h)f(is)h(suitable)f(to)h(extensions)f (to)g(curv)n(ed)456 2816 y(Loren)n(tzian)36 b(space-times,)k(w)n(e)e (mainly)g(rely)g(on)g(a)g(construction)f(giv)n(en)h(b)n(y)g(Epstein)g (and)456 3015 y(Glaser)26 b([20)o(])i(at)g(the)g(b)r(eginning)g(of)g (the)g(sev)n(en)n(ties)f(and)g(on)h(some)f(impro)n(v)n(emen)n(ts)g (suggested)456 3215 y(later)f(b)n(y)h(Stora)f([54)o(].)37 b(This)27 b(construction)f(mak)n(es)g(precise)g(older)h(ideas)f(of)h (St)r(\177)-44 b(uc)n(k)n(elb)r(erg)26 b([56)o(])456 3414 y(and)32 b(Bogoliub)r(o)n(v)e([5].)51 b(In)32 b(spite)g(of)h(its)f (elegance)f(it)i(w)n(as)e(widely)h(ignored)f(\(compare,)i(e.g.,)456 3613 y(its)25 b(neglection)f(in)i(sev)n(eral)d(b)r(o)r(oks)i(on)f(quan) n(tum)i(\014eld)f(theory)-7 b(,)25 b(with)h(the)f(exception)g(of)g([38) o(]\).)456 3812 y(Recen)n(tly)-7 b(,)24 b(it)f(w)n(as)f(further)h(dev)n (elop)r(ed)g(and)f(applied)h(to)g(gauge)f(\014eld)h(theories)g(b)n(y)f (Sc)n(harf)h(and)456 4012 y(his)k(collab)r(orators)e([50)o(])1239 3982 y Fy(1)1304 4012 y Fv(\(after)i(earlier)f(w)n(ork)h(b)n(y)g([4)o (]\).)555 4211 y(W)-7 b(e)34 b(o\013er)f(some)g(in)n(tuitiv)n(e)h (explanations)e(of)i(the)g(ideas)f(b)r(ehind)h(the)g(approac)n(h)e(of)h (St)r(\177)-44 b(u-)456 4410 y(c)n(k)n(elb)r(erg-Bogoliub)r(o)n (v-Epstein-Glaser)17 b(\(w)n(e)23 b(refrain)g(from)g(using)g(an)g (acron)n(ym)e(for)i(this)h(and)456 4610 y(simply)i(call)f(it)i(the)f (Epstein)g(and)g(Glaser)f(approac)n(h\).)35 b(F)-7 b(or)25 b(simplicit)n(y)i(w)n(e)e(discuss)h(the)g(case)456 4809 y(of)h(\015at)h(Mink)n(o)n(wskian)e(space-time.)p 456 4958 499 4 v 456 5026 a Fp(1)490 5050 y FB(Note)j(that)h(the)f(term)g (\\\014nite")g(in)f(Sc)n(harf)5 b('s)28 b(b)r(o)r(ok)h(refers)f(to)h (the)g(fact)g(that)g(in)f(the)i(Epstein)f(and)f(Glaser)456 5133 y(approac)n(h)d(\(as)f(in)g(the)h(similar)d(BPHZ)h(metho)r(d\))j (no)e(regularization)h(is)e(necessary)-6 b(.)33 b(It)24 b(do)r(es)h(not)g(mean)f(that)456 5216 y(the)g(indeterminacy)h (connected)h(with)e(the)g(div)n(ergence)i(of)d(naiv)n(e)h(p)r (erturbation)h(theory)f(disapp)r(ears.)p eop %%Page: 8 8 8 7 bop 456 226 a Fy(8)836 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)555 425 y Fv(The)19 b(basic)f(idea)g(is)h(that,) h(in)f(the)g(asymptotic)f(past)h(and)f(future,)j(the)e(in)n(teracting)f (quan)n(tum)456 624 y(\014elds)35 b(approac)n(h,)h(in)f(a)g(sense)g(to) h(b)r(e)f(sp)r(eci\014ed,)j Ft(fr)l(e)l(e)e Fv(\014elds,)i(i.e.)60 b(\014elds)35 b(satisfying)g(linear)456 824 y(h)n(yp)r(erb)r(olic)c (equations)f(of)i(motion.)48 b(F)-7 b(or)31 b(free)g(\014elds)g(there)h (exists)f(a)g(precise)f(construction)456 1023 y(whic)n(h)j(can)g(b)r(e) h(used)f(for)g(a)g(p)r(erturbativ)n(e)g(description)g(of)h(in)n (teracting)e(\014elds.)55 b(No)n(w,)34 b(in)g(a)456 1222 y(translationally)23 b(in)n(v)-5 b(arian)n(t)24 b(theory)-7 b(,)25 b(the)h(in)n(teracting)e(\014elds)h(approac)n(h)e(the)j (asymptotic)f(free)456 1421 y(\014elds)e(only)h(in)g(a)f(rather)g(w)n (eak)g(sense)g(\(LSZ-asymptotic)g(condition)h([34)o(]\).)36 b(Moreo)n(v)n(er,)22 b(Haag)456 1621 y(Theorem)32 b([29)o(])h(forbids)f (the)h(construction)f(of)h(in)n(teracting)f(\014elds)h(in)g(the)g(v)-5 b(acuum)33 b(Hilb)r(ert)456 1820 y(space)d(of)h(the)g(time-0)f(free)h (\014elds.)47 b(In)31 b(the)g(Epstein)g(and)f(Glaser)g(sc)n(heme,)h (these)g(problems)456 2019 y(are,)36 b(in)f(a)g(\014rst)g(step,)j (circum)n(v)n(en)n(ted)c(b)n(y)h(c)n(ho)r(osing)f(in)n(teractions)g (whic)n(h)h(tak)n(e)g(place)g(only)456 2218 y(in)f(a)g(b)r(ounded)h (region)e(of)h(space-time.)57 b(Then)35 b(the)f(scattering)g(op)r (erator)e(can)i(b)r(e)h(de\014ned)456 2418 y(in)29 b(the)g(in)n (teraction)e(picture)i(as)f(the)h(time)g(ev)n(olution)f(op)r(erator)f (from)h(the)h(past,)g(b)r(efore)f(the)456 2617 y(in)n(teraction)e(w)n (as)h(switc)n(hed)g(on,)h(to)f(the)h(future,)g(after)f(the)h(in)n (teraction)f(w)n(as)g(switc)n(hed)g(o\013.)555 2816 y(A)i(lo)r(calized) f(in)n(teraction)g(here)h(is)f(though)n(t)h(to)f(b)r(e)h(a)g(smo)r(oth) f(function)i(of)e(time,)i Fu(t)p Fv(,)f(with)456 3015 y(compact)i(supp)r(ort)g(with)h(v)-5 b(alues)31 b(in)h(the)g(lo)r(cal)f (op)r(erators)f(asso)r(ciated)g(with)i(the)g(free)f(\014eld.)456 3215 y(In)g(the)g(simplest)h(case)e(it)h(is)g Fu(H)1458 3227 y Fy(in)n(t)1542 3215 y Fv(\()p Fu(t)p Fv(\))e(=)g Fu(')p Fv(\()p Fu(f)1886 3227 y Fo(t)1915 3215 y Fv(\))j(where)e Fu(')i Fv(is)f(a)f(free)h(\014eld,)h(i.e.)48 b(an)31 b(op)r(erator-)456 3414 y(v)-5 b(alued)30 b(distribution)h(on)g(a)f (Hilb)r(ert)i(space,)f(and)f(where)h Fu(f)2354 3426 y Fo(t)2383 3414 y Fv(\()p Fu(x)p Fv(\))e(=)f Fu(\016)s Fv(\()p Fu(x)2735 3384 y Fy(0)2794 3414 y Fr(\000)20 b Fu(t)p Fv(\))p Fu(f)9 b Fv(\()p Fu(x)p Fv(\))32 b(for)e(some)456 3613 y(test)i(function)h Fu(f)9 b Fv(.)51 b(The)33 b Fu(S)5 b Fv(-matrix)31 b(is)h(then)h(an)f(op)r(erator-v)-5 b(alued)30 b(functional)j Fu(S)5 b Fv(\()p Fu(f)k Fv(\))32 b(on)g(the)456 3812 y(test)38 b(function)h(space.)69 b(The)38 b(functional)h(equation)e(for)h(the)h(ev)n(olution)f(op)r (erator)e(implies)456 4012 y(a)g(factorization)f(prop)r(ert)n(y)g(for)i (the)g Fu(S)5 b Fv(-matrix)35 b(if)i(the)g(supp)r(ort)f(of)h(the)g(in)n (teraction)f(\(as)g(a)456 4211 y(function)20 b(of)g(time\))h(consists)e (of)h(disjoin)n(t)g(in)n(terv)-5 b(als.)34 b(In)20 b(the)h(case)e(ab)r (o)n(v)n(e)f(with)j(the)f(in)n(teraction)456 4410 y(b)r(eing)27 b(a)g(linear)g(function)i(of)e(the)h(free)f(\014eld)h(w)n(e)g(ev)n(en)f (\014nd)h(the)g(factorization,)1475 4749 y Fu(S)5 b Fv(\()p Fu(f)1604 4761 y Fy(1)1660 4749 y Fv(+)18 b Fu(f)1784 4761 y Fy(2)1821 4749 y Fv(\))23 b(=)g Fu(S)5 b Fv(\()p Fu(f)2093 4761 y Fy(1)2130 4749 y Fv(\))p Fu(S)g Fv(\()p Fu(f)2291 4761 y Fy(2)2328 4749 y Fv(\))28 b Fu(;)927 b Fv(\(1\))p eop %%Page: 9 9 9 8 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)615 b(9)456 425 y Fv(whenev)n(er)26 b(there)g(exists)h(some)g Fu(t)c Fr(2)g Fn(R)k Fv(suc)n(h)g(that)g (supp\()p Fu(f)2296 437 y Fy(1)2333 425 y Fv(\))d Fr(\032)e(f)p Fu(x)p Fr(j)p Fu(x)2635 395 y Fy(0)2696 425 y Fr(\025)h Fu(t)p Fr(g)j Fv(and)h(supp\()p Fu(f)3287 437 y Fy(2)3324 425 y Fv(\))d Fr(\032)456 624 y(f)p Fu(x)p Fr(j)p Fu(x)615 594 y Fy(0)688 624 y Fr(\024)35 b Fu(t)p Fr(g)p Fv(,)h(and)f(where)g Fu(f)1377 636 y Fy(1)1449 624 y Fv(and)f Fu(f)1658 636 y Fy(2)1730 624 y Fv(need)i(not)f(b)r(e)g(smo)r(oth)g(at)g(the)g(h)n (yp)r(ersurface)f Fu(x)3307 594 y Fy(0)3380 624 y Fv(=)456 824 y Fu(t)p Fv(.)54 b(This)33 b(stronger)e(factorization)h(prop)r(ert) n(y)g(is)h(not)h(exp)r(ected)f(to)g(hold)h(for)e(more)h(singular)456 1023 y(in)n(teractions.)i(Instead)28 b(w)n(e)f(require)f(the)i(follo)n (wing)f(consequence)f(of)i(\(1\),)1063 1362 y Fu(S)5 b Fv(\()p Fu(f)1192 1374 y Fy(1)1247 1362 y Fv(+)18 b Fu(f)1371 1374 y Fy(2)1426 1362 y Fv(+)g Fu(f)1550 1374 y Fy(3)1587 1362 y Fv(\))24 b(=)e Fu(S)5 b Fv(\()p Fu(f)1859 1374 y Fy(1)1915 1362 y Fv(+)18 b Fu(f)2039 1374 y Fy(2)2076 1362 y Fv(\))p Fu(S)5 b Fv(\()p Fu(f)2237 1374 y Fy(2)2274 1362 y Fv(\))2306 1327 y Fq(\000)p Fy(1)2395 1362 y Fu(S)g Fv(\()p Fu(f)2524 1374 y Fy(2)2579 1362 y Fv(+)18 b Fu(f)2703 1374 y Fy(3)2740 1362 y Fv(\))28 b Fu(;)515 b Fv(\(2\))456 1700 y(to)25 b(hold)h(for)f(test)i(functions)f Fu(f)1421 1712 y Fy(1)1458 1700 y Fu(;)14 b(f)1536 1712 y Fy(2)1572 1700 y Fu(;)g(f)1650 1712 y Fy(3)1687 1700 y Fv(,)27 b(whenev)n(er)d(the)j(supp)r(orts)e(of)h Fu(f)2713 1712 y Fy(1)2776 1700 y Fv(and)f Fu(f)2976 1712 y Fy(3)3039 1700 y Fv(can)h(b)r(e)g(sep-)456 1900 y(arated)k(b)n(y)i(a)f(Cauc)n(h)n (y)g(surface)g(suc)n(h)g(that)h(supp\()p Fu(f)2116 1912 y Fy(1)2153 1900 y Fv(\))g(lies)g(in)g(the)g(future)g(and)g(supp\()p Fu(f)3274 1912 y Fy(3)3311 1900 y Fv(\))g(in)456 2099 y(the)24 b(past)g(of)h(this)f(surface.)35 b(T)-7 b(ogether)23 b(with)i(the)g(normalization)e(condition)h Fu(S)5 b Fv(\(0\))22 b(=)h Fs(1)h Fv(\(iden-)456 2298 y(tit)n(y)30 b(op)r(erator)f(on)h (Hilb)r(ert)h(space\))e(it)i(implies)g(the)f(\014rst)g(men)n(tioned)h (w)n(eak)n(er)d(factorization)456 2497 y(condition)f(in)h(the)g(case)f Fu(f)1277 2509 y Fy(2)1337 2497 y Fv(=)22 b(0.)555 2697 y(The)40 b(functional)f(equation)g(\(2\))g(has)g(an)g(in)n(teresting)g (prop)r(ert)n(y;)44 b(if)c Fu(S)k Fv(is)39 b(a)g(functional)456 2896 y(solving)g(it)i(w)n(e)f(get)h(other)f(solutions)g Fu(S)1772 2908 y Fo(f)1856 2896 y Fv(b)n(y)g(de\014ning)h Fu(S)2362 2908 y Fo(f)2404 2896 y Fv(\()p Fu(g)s Fv(\))k(=)g Fu(S)5 b Fv(\()p Fu(f)k Fv(\))2836 2866 y Fq(\000)p Fy(1)2925 2896 y Fu(S)c Fv(\()p Fu(f)36 b Fv(+)26 b Fu(g)s Fv(\))41 b(\(the)456 3095 y Ft(r)l(elative)d Fu(S)5 b Fv(-matrices\))37 b(where)g Fu(f)47 b Fv(is)37 b(an)h(arbitrary)d(test)j(function.)68 b(In)38 b(particular)e(w)n(e)i(get)456 3294 y(comm)n(utativit)n(y)27 b(in)h(case)e(supp\()p Fu(g)1529 3306 y Fy(1)1567 3294 y Fv(\))h(and)h(supp\()p Fu(g)2031 3306 y Fy(2)2068 3294 y Fv(\))g(are)f(space-lik)n(e)f(separated,)1136 3633 y Fu(S)1187 3645 y Fo(f)1230 3633 y Fv(\()p Fu(g)1302 3645 y Fy(1)1358 3633 y Fv(+)18 b Fu(g)1481 3645 y Fy(2)1518 3633 y Fv(\))23 b(=)g Fu(S)1712 3645 y Fo(f)1755 3633 y Fv(\()p Fu(g)1827 3645 y Fy(2)1864 3633 y Fv(\))p Fu(S)1947 3645 y Fo(f)1990 3633 y Fv(\()p Fu(g)2062 3645 y Fy(1)2099 3633 y Fv(\))h(=)e Fu(S)2293 3645 y Fo(f)2336 3633 y Fv(\()p Fu(g)2408 3645 y Fy(1)2445 3633 y Fv(\))p Fu(S)2528 3645 y Fo(f)2571 3633 y Fv(\()p Fu(g)2643 3645 y Fy(2)2681 3633 y Fv(\))28 b Fu(:)574 b Fv(\(3\))456 3972 y(Th)n(us)23 b(the)h(relativ)n(e)e Fu(S)5 b Fv(-matrices)22 b(satisfy)h(the)h(lo)r (calit)n(y)f(condition)g(required)f(for)h(lo)r(cal)g(observ-)456 4171 y(ables.)36 b(They)27 b(serv)n(e)g(as)g(generating)f(functionals)h (for)g(the)h(in)n(teracting)f(\014elds.)555 4370 y(Unfortunately)-7 b(,)34 b(a)f(construction)e(of)i Fu(S)5 b Fv(\()p Fu(f)k Fv(\))33 b(in)g(four)f(dimensions)g(is)h(kno)n(wn)f(only)g(in)h(the)456 4570 y(case)38 b(of)h(in)n(teraction)f(Hamiltonians)g(whic)n(h)h(are)f (linear)g(or)g(quadratic)g(in)h(the)g(free)g(\014eld,)456 4769 y(but)34 b(in)g(t)n(w)n(o)f(dimensions)g(W)-7 b(rezinski)33 b([63)o(])h(pro)n(v)n(ed)e(that,)j(at)f(least)f(in)h(the)g(particular)e (case)456 4968 y(of)i(factorizable)e Fu(f)9 b Fv(,)36 b Fu(f)9 b Fv(\()p Fu(t;)14 b(x)p Fv(\))34 b(=)f Fu(g)s Fv(\()p Fu(t)p Fv(\))p Fu(h)p Fv(\()p Fu(x)p Fv(\),)k(suc)n(h)c(a)h (construction)f(is)h(p)r(ossible)g(for)f Fu(')3180 4938 y Fy(4)3218 4968 y Fv(.)56 b(One)456 5167 y(therefore)23 b(mainly)h(relies)g(on)g(the)h(\\in\014nitesimal")f(description)g(of)g (the)h(lo)r(cal)f Fu(S)5 b Fv(-matrix)24 b Fu(S)5 b Fv(\()p Fu(g)s Fv(\))p eop %%Page: 10 10 10 9 bop 456 226 a Fy(10)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(b)n(y)19 b(studying)i(its)f(formal)f(p)r(o)n(w)n(er)g(series)g([6])h(expansion)f (in)i(terms)f(of)g(the)h(\\coupling)e(constan)n(t")456 624 y Fu(g)s Fv(.)36 b(The)27 b(connection)f(with)i(the)f(usual)g(form) n(ulation)f(ma)n(y)g(b)r(e)i(done)e(via)h(the)g Ft(adiab)l(atic)32 b(limit)p Fv(,)456 824 y(i.e.)43 b(the)30 b(limit)g(for)f Fu(S)5 b Fv(\()p Fu(g)s Fv(\))30 b(when)g Fu(g)f Fr(!)e Fv(1)i(all)g(o)n(v)n(er)f(space-time)h(is)h(the)g Fu(S)5 b Fv(-matrix,)29 b(or)g(in)h(cases)456 1023 y(where)j(the)h(limit)g (for)f Fu(S)5 b Fv(\()p Fu(g)s Fv(\))34 b(do)r(es)f(not)h(exist)f(due)h (to)f(infrared)g(div)n(ergences,)h(the)g(limit)g(for)456 1222 y(the)25 b(v)-5 b(acuum)25 b(exp)r(ectation)f(v)-5 b(alues)24 b(of)h Fu(S)1731 1234 y Fo(g)1770 1222 y Fv(\()p Fu(f)9 b Fv(\),)25 b Fu(g)h Fr(!)d Fv(1,)i(is)g(the)g(generating)e (functional)i(for)f(the)456 1421 y(time-ordered)i(correlation)g (functions.)555 1621 y(The)36 b(description)e(giv)n(en)h(so)g(far)g (emphasizes)f(the)i(fact)g(that)f(the)h(Epstein)f(and)h(Glaser)456 1820 y(metho)r(d)i(is)g(lo)r(cal)f(in)i(spirit,)h(and)e(it)h(migh)n(t)e (b)r(e)i(a)e(fa)n(v)n(orite)g(candidate)h(for)f(dev)n(eloping)g(a)456 2019 y(renormalization)23 b(theory)h(on)h(curv)n(ed)g(space-times.)35 b(A)26 b(closer)e(insp)r(ection,)i(ho)n(w)n(ev)n(er,)e(sho)n(ws)456 2218 y(that)29 b(also)f(in)h(this)g(metho)r(d)g(translation)f(in)n(v)-5 b(ariance)28 b(pla)n(ys)f(an)i(imp)r(ortan)n(t)g(r^)-42 b(ole,)28 b(b)r(oth)h(con-)456 2418 y(ceptually)k(and)h(tec)n(hnically) -7 b(,)35 b(and)f(it)g(will)g(require)f(a)g(lot)h(of)g(w)n(ork)f(to)g (replace)g(it)h(b)n(y)g(other)456 2617 y(structures.)g(A)22 b(similar)g(problem)f(w)n(as)g(studied)i(b)n(y)e(Dosc)n(h)h(and)g(M)r (\177)-44 b(uller)22 b([16)o(].)35 b(These)22 b(authors)456 2816 y(dev)n(elop)r(ed)31 b(the)h(Epstein)g(and)g(Glaser)e(metho)r(d)j (on)e(Mink)n(o)n(wski)g(space)g(for)g(quan)n(tum)h(elec-)456 3015 y(tro)r(dynamics)g(with)j(external)e(time)h(indep)r(enden)n(t)g (electromagnetic)f(\014elds.)55 b(Their)33 b(use)h(of)456 3215 y(the)22 b(Hadamard)f(parametrices)g(for)h(the)h(Dirac)e(op)r (erator)g(is)h(already)f(m)n(uc)n(h)h(in)h(the)f(spirit)h(of)f(a)456 3414 y(lo)r(cal)28 b(form)n(ulation)g(of)g(p)r(erturbation)g(theory;)h (b)n(y)f(the)i(assumption)e(of)h(time)g(indep)r(endence)456 3613 y(of)f(the)g(external)f(\014elds,)i(ho)n(w)n(ev)n(er,)d(time)i (translation)f(in)n(v)-5 b(ariance)27 b(still)h(pla)n(ys)g(a)f(crucial) g(r^)-42 b(ole)456 3812 y(in)25 b(their)h(approac)n(h.)34 b(As)26 b(a)f(matter)g(of)h(fact,)g(it)g(will)g(turn)f(out)h(that)g Ft(micr)l(olo)l(c)l(al)j(analysis)e Fv([35)o(])456 4012 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Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(11)555 425 y Fv(The)35 b(pap)r(er)f(is)g(organized)f(as)g(follo)n(ws:)50 b(After)35 b(this)g(in)n(tro)r(duction,)g(Section)g(2)f(pro)n(vides)456 624 y(some)19 b(useful)h(grounding)f(concepts)g(and)h(\014xes)f(the)h (notations.)34 b(Moreo)n(v)n(er,)18 b(w)n(e)i(presen)n(t)f(a)g(new)456 824 y(construction)j(of)g(Wic)n(k)h(p)r(olynomials)f(whic)n(h)g(ma)n(y) g(b)r(e)h(of)g(indep)r(enden)n(t)h(in)n(terest.)35 b(In)22 b(Section)456 1023 y(3)33 b(w)n(e)h(state)f(the)h(\014rst)g(principles) g(b)n(y)f(whic)n(h)h(w)n(e)f(build)i(up)f(the)g(p)r(erturbativ)n(e)f (metho)r(d)i(on)456 1222 y(smo)r(oth)27 b(curv)n(ed)g(globally)f(h)n (yp)r(erb)r(olic)h(space-times.)36 b(The)27 b(most)h(imp)r(ortan)n(t)f (c)n(hange)f(w.r.t.)456 1421 y(the)h(Epstein)f(and)h(Glaser)f(metho)r (d)h(is)g(a)f(c)n(haracterization)e(of)j(the)g(singularit)n(y)e (structure)i(of)456 1621 y(the)34 b(time-ordered)e(n)n(umerical)h 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(abstract)f(algebraic)g(form)n(ulation)h(of)g(quan)n(tum)h(\014eld)g (theory)-7 b(.)54 b(In)456 4809 y(fact,)33 b(w)n(e)f(are)f(able)h(to)g (de\014ne)h(a)f(unique)g(family)h(\(net,)h(precosheaf)d(etc.\))51 b(of)32 b Fr(\003)p Fv(-algebras)e(of)456 5008 y(observ)-5 b(ables)26 b(on)i(globally)f(h)n(yp)r(erb)r(olic)h(space-times)g(via)f (the)i(idea)f(of)g(the)h(lo)r(cal)f(relativ)n(e)f Fu(S)5 b Fv(-)456 5207 y(matrices.)35 b(Section)25 b(8)f(describ)r(es)h(this)g (construction)f(whic)n(h)h(seems)g(to)f(b)r(e)i(widely)f(unkno)n(wn,)p eop %%Page: 12 12 12 11 bop 456 226 a Fy(12)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(in)34 b(spite)g(of)g(the)h (fact)f(that)g(it)h(ma)n(y)e(already)g(b)r(e)h(found,)i(in)f(a)e (preliminary)g(form,)j(in)e([52)o(].)456 624 y(This)27 b(Section)h(partly)f(justify)h(the)g(rather)f(abstract)f(starting)h(p)r (oin)n(t)h(of)f(Section)h(2.)555 824 y(An)33 b(outlo)r(ok,)g(Section)f (9,)h(concludes)f(the)h(pap)r(er.)50 b(Finally)-7 b(,)34 b(w)n(e)e(stress)f(that)i(the)f(pro)r(ce-)456 1023 y(dure)27 b(w)n(orks)g(for)g(general)g(\014eld)h(theories)f(but)i(for)f (simplicit)n(y)g(w)n(e)g(stic)n(k)f(to)h(the)h(notationally)456 1222 y(easiest)h(case)g(of)h(a)g(single)f(scalar)g(\(massiv)n(e\))g (\014eld)h(theory)f(with)i(self)f(in)n(teractions)f(without)456 1421 y(deriv)-5 b(ativ)n(es.)549 1778 y(2.)45 b Fw(General)31 b(Theor)-6 b(y)32 b(of)g(Quantized)f(Fields)g(and)g(Micr)n(olocal)g (Anal)-6 b(ysis)555 2028 y Fv(In)27 b(order)f(to)g(\014x)h(our)f (notations)g(w)n(e)g(recall)g(some)g(basic)g(geometrical)f(concepts.)37 b(F)-7 b(urther)456 2227 y(details)26 b(ma)n(y)g(b)r(e)h(found)g(in)f (some)g(b)r(o)r(oks)g(on)g(general)g(relativit)n(y)f(and)i(Loren)n (tzian)e(geometry)456 2426 y(\(see,)34 b(for)f(instance,)h([59)o(])f (and)g([2]\).)54 b(W)-7 b(e)33 b(shall)g(w)n(ork)e(on)i(a)g Ft(sp)l(ac)l(e-time)h Fv(\()p Fu(M)t(;)14 b(g)s Fv(\),)34 b(where)f(b)n(y)456 2625 y(this)28 b(w)n(e)g(mean)h(that)g Fu(M)37 b Fv(is)28 b(a)g(connected,)h(Hausdor\013,)f(b)r(oundaryless)f (top)r(ological)g(space)h(of)456 2825 y(pure)34 b(dimension)h Fu(d)g Fr(\025)g Fv(2)g(whic)n(h)f(\()p Fu(i)p Fv(\))h(is)g (paracompact,)g(\()p Fu(ii)p Fv(\))g(is)g(equipp)r(ed)g(with)g(a)g(smo) r(oth)456 3024 y(structure,)c(\()p Fu(iii)p Fv(\))f(is)g(endo)n(w)n(ed) g(with)i(a)e(Loren)n(tzian)f(metric)i Fu(g)2430 3036 y Fo(ab)2499 3024 y Fv(,)h(i.e.)46 b(a)30 b(smo)r(oth)h(2-cotensor)456 3223 y(of)j(signature)e(\(1)p Fu(;)14 b(d)23 b Fr(\000)f Fv(1\),)36 b(i.e.)56 b(\(+)p Fu(;)14 b Fr(\000)p Fu(;)g Fr(\001)g(\001)g(\001)27 b Fu(;)14 b Fr(\000)p Fv(\))34 b(and)f(\()p Fu(iv)s Fv(\))i(is)f(orien)n(ted)f(and)h(time)g(orien)n (ted.)456 3422 y(Giv)n(en)k(the)g(metric)g(w)n(e)g(ha)n(v)n(e)f(a)h (canonically)f(asso)r(ciated)g(deriv)-5 b(ativ)n(e,)40 b(namely)d(the)i(Levi-)456 3622 y(Civita)33 b(deriv)-5 b(ativ)n(e)33 b(denoted)g(b)n(y)h Fr(r)f Fv(and)h(an)f(asso)r(ciated)f (curv)-5 b(ature)33 b(tensor)g Fu(R)3031 3591 y Fo(a)3030 3645 y(bcd)3127 3622 y Fv(,)j(with)e Fu(R)456 3821 y Fv(the)28 b(scalar)f(curv)-5 b(ature.)39 b(The)28 b(notion)g(of)g (totally)g(geo)r(desic)g(submanifold,)h(i.e.,)f(that)h(one)f(for)456 4020 y(whic)n(h)f(all)g(tangen)n(tial)g(geo)r(desics)g(sta)n(y)f(on)i (the)g(submanifold,)f(is)h(used)f(in)h(Section)g(6.)555 4219 y(Some)f(w)n(ords)e(on)h(notations.)36 b(Sometimes)26 b(w)n(e)h(write)f(a)g(zero)g(section)g(of)g(a)h(v)n(ector)e(bundle)456 4419 y Fm(B)j Fv(as)f Fr(f)p Fv(0)p Fr(g)g Fv(some)g(other)h(times)g (to)g(precise)f(that)i(it)f(b)r(elongs)g(to)f(that)i(bundle)f(w)n(e)g (shall)g(write)456 4618 y Fu(Z)6 b Fv(\()p Fm(B)p Fv(\).)40 b(Ho)n(w)n(ev)n(er,)27 b(in)i(order)f(to)g(a)n(v)n(oid)f(an)n(y)h (abuse,)h(w)n(e)f(use)h(the)g(notation)2886 4595 y(_)2866 4618 y Fm(B)g Fv(to)f(denote)h(the)456 4817 y(bundle)d(depriv)n(ed)f (from)g(its)h(zero)e(section,)i(i.e.,)g Fm(B)5 b Fr(n)g Fu(Z)h Fv(\()p Fm(B)p Fv(\))o(.)37 b(W)-7 b(e)26 b(shall)f(also)g(use)g (the)h(notation)456 5016 y Fu(M)546 4986 y Fo(n)590 5016 y Fv(,)j(whenev)n(er)e(w)n(e)h(treat)g(the)h Fu(n)p Fv(-th)f(order)f (cartesian)g(pro)r(duct)h(of)g(a)g(manifold)g Fu(M)9 b Fv(,)29 b(and)f(b)n(y)456 5216 y(\001)525 5228 y Fo(K)589 5216 y Fv(,)i(where)f Fu(K)j Fr(\022)26 b(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Fr(g)p Fv(,)29 b(w)n(e)g(mean)h(the)g (\(smo)r(oth,)g(closed\))f(submanifold)h(of)g Fu(M)3271 5185 y Fo(n)3345 5216 y Fv(for)p eop %%Page: 13 13 13 12 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(13)456 425 y Fv(whic)n(h)27 b(an)n(y)g(of)g(its)g(p)r(oin)n(ts)h(\()p Fu(p)1383 437 y Fy(1)1420 425 y Fu(;)14 b(:)g(:)g(:)g(;)g(p)1647 437 y Fo(n)1692 425 y Fv(\))27 b(are)g(suc)n(h)g(that)h Fu(p)2299 437 y Fo(k)2334 445 y FC(1)2393 425 y Fv(=)23 b Fu(p)2523 437 y Fo(k)2558 445 y FC(2)2622 425 y Fv(for)k(an)n(y)f(pair)h Fu(k)3119 437 y Fy(1)3180 425 y Fr(6)p Fv(=)22 b Fu(k)3310 437 y Fy(2)3375 425 y Fv(in)456 624 y Fu(K)6 b Fv(.)555 824 y(The)35 b(causalit)n(y)f(principle)h(pla)n(ys)f(a)g(crucial)h(r^) -42 b(ole)34 b(in)h(our)f(construction.)58 b(Therefore)34 b(w)n(e)456 1023 y(restrict)23 b(our)f(space-times)h(to)h(b)r(e)g Ft(glob)l(al)t(ly)k(hyp)l(erb)l(olic)p Fv(.)38 b(This)23 b(means)h(that)f Fu(M)33 b Fv(is)23 b(homeomor-)456 1222 y(phic)28 b(to)g Fn(R)19 b Fr(\002)f Fv(\006)28 b(where)f(\006)h(is)g (a)g(\()p Fu(d)19 b Fr(\000)f Fv(1\)-dimensional)27 b(top)r(ological)g (submanifold)h(of)g Fu(M)37 b Fv(and)456 1421 y(for)21 b(eac)n(h)h Fu(t)h Fr(2)h Fn(R)p Fv(,)f Fr(f)p Fu(t)p Fr(g)8 b(\002)g Fv(\006)21 b(is)h(a)g(\(spacelik)n(e\))g(Cauc)n(h)n(y)f (surface.)35 b(A)22 b(Cauc)n(h)n(y)g(surface)f(is)h(a)g(subset)456 1621 y(of)30 b Fu(M)39 b Fv(whic)n(h)30 b(ev)n(ery)f(inextendible)i (non)f(spacelik)n(e)f(curv)n(e)g(in)n(tersects)h(exactly)f(once.)45 b(Giv)n(en)456 1820 y(a)29 b(subset)h Fu(S)35 b Fv(of)30 b Fu(M)38 b Fv(w)n(e)30 b(de\014ne)g(the)g(causal)f(future/past)h(sets) f Fu(J)2512 1790 y Fq(\006)2568 1820 y Fv(\()p 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(de\014ne)h(the)g(future/past)f(ligh)n(t-cones)f Fu(V)2024 2828 y Fq(\006)2108 2816 y Fv(o)n(v)n(er)f(these)j(spaces)e(\(based)i (on)f Fu(p)p Fv(\))g(in)h(the)456 3015 y(usual)f(w)n(a)n(y)-7 b(.)555 3215 y(Quan)n(tum)30 b(\014eld)g(theories)f(on)g(more)h (general)e(spaces)h(p)r(ose)g(consistency)g(problems)h(\(see,)456 3414 y(e.g.,)i(Ha)n(wking's)e(\\Chronology)e(Protection)i(Conjecture")h ([33)o(])h(or)e(the)i(div)n(ergence)e(of)h(the)456 3613 y(energy)k(momen)n(tum)i(tensor)f(at)g(the)h(Cauc)n(h)n(y)f(horizon)f (observ)n(ed)g(in)i([40)o(]\).)65 b(W)-7 b(e)37 b(remark,)456 3812 y(ho)n(w)n(ev)n(er,)25 b(that)i(since)g(our)f(constructions)g (will)h(b)r(e)g(purely)g(lo)r(cal,)f(one)h(can)g(as)f(w)n(ell)h (consider)456 4012 y(a)g(globally)f(h)n(yp)r(erb)r(olic)h(submanifold)h (of)f Ft(any)i Fv(Loren)n(tzian)d(space-time.)555 4211 y(In)d(man)n(y)f(concrete)g(cases,)h(exact)f(solutions)g(of)h(the)g (Einstein)g(equation,)g(lik)n(e)g(Mink)n(o)n(wski,)456 4410 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Fv(satisfying)f(the)h(\(generalized\))e (Klein-Gordon)g(equation)h(of)h(motion)1525 963 y(\()p Fl(\003)1622 975 y Fo(g)1679 963 y Fv(+)18 b Fu(m)1835 929 y Fy(2)1891 963 y Fv(+)g Fu(\024R)q Fv(\))p Fu(')23 b Fv(=)g(0)k Fu(;)963 b Fv(\(4\))456 1302 y(where)19 b Fl(\003)753 1314 y Fo(g)812 1302 y Fv(is)h(the)h(d'Alem)n(b)r(ertian) g(\(or)e(Laplace-Beltrami\))g(op)r(erator)g(w.r.t.)34 b(the)21 b(Loren)n(tzian)456 1501 y(metric)32 b Fu(g)s Fv(,)g Fu(m)f Fr(\025)f Fv(0)i(and)f Fu(\024)g Fr(2)g Fn(R)p Fv(,)i(one)f(ma)n(y)f(asso)r(ciate)g(an)g(algebra)g(of)h(observ) -5 b(ables)30 b(de\014ned)456 1700 y(in)i(the)h(follo)n(wing)e(w)n(a)n (y:)45 b(Let)32 b Fu(E)1484 1712 y Fo(r)r(et)1610 1700 y Fv(resp.)50 b Fu(E)1893 1712 y Fo(adv)2036 1700 y Fv(b)r(e)32 b(the)h(retarded)e(resp.)50 b(adv)-5 b(anced)32 b(Green)456 1900 y(functions)c(of)g(the)h(Klein-Gordon)d(op)r(erator)g(whic)n(h)j (are)e Ft(uniquely)h Fv(de\014ned)g(on)g(globally)f(h)n(y-)456 2099 y(p)r(erb)r(olic)18 b(space-times,)h(and)g(let)g Fu(E)28 b Fv(=)22 b Fu(E)1725 2111 y Fo(r)r(et)1819 2099 y Fr(\000)p Fu(E)1945 2111 y Fo(adv)2055 2099 y Fv(.)34 b(Then)19 b(w)n(e)f(consider)f(the)i(unital)g Fr(\003)p Fv(-algebra)456 2298 y Fk(A)34 b Fv(whic)n(h)g(is)g(generated)f(b)n(y)i (the)f(sym)n(b)r(ols)g Fu(')p Fv(\()p Fu(f)9 b Fv(\),)37 b Fu(f)43 b Fr(2)34 b Fm(D)p Fv(\()p Fu(M)9 b Fv(\))35 b(\(space)f(of)g(complex-v)-5 b(alued)456 2497 y(smo)r(oth)27 b(and)g(compactly)g(supp)r(orted)h(functions\),)g(with)g(the)g(follo)n (wing)f(relations:)555 2721 y(1.)41 b(The)28 b(map)f Fu(f)32 b Fr(7!)23 b Fu(')p Fv(\()p Fu(f)9 b Fv(\))28 b(is)g(linear,)555 2920 y(2.)41 b Fu(')p Fv(\()p Fu(f)9 b Fv(\))829 2890 y Fq(\003)891 2920 y Fv(=)23 b Fu(')p Fv(\()1083 2898 y(\026)1065 2920 y Fu(f)9 b Fv(\),)555 3119 y(3.)41 b([)p Fu(')p Fv(\()p Fu(f)9 b Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(g)s Fv(\)])24 b(=)f Fu(iE)5 b Fv(\()p Fu(f)27 b Fr(\012)18 b Fu(g)s Fv(\))p Fs(1)p Fu(;)180 b Fr(8)p Fu(f)t(;)14 b(g)25 b Fr(2)f Fm(D)p Fv(\()p Fu(M)9 b Fv(\),)555 3319 y(4.)41 b Fu(')p Fv(\(\()p Fl(\003)844 3331 y Fo(g)902 3319 y Fv(+)18 b Fu(m)1058 3288 y Fy(2)1113 3319 y Fv(+)g Fu(\024R)q Fv(\))p Fu(f)9 b Fv(\))23 b(=)g(0)p Fu(;)179 b Fr(8)p Fu(f)32 b Fr(2)23 b Fm(D)p Fv(\()p Fu(M)9 b Fv(\),)456 3542 y(where)21 b(the)h(sym)n(b)r(ol)f([)p Fu(')p Fv(\()p Fu(f)9 b Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(g)s Fv(\)])22 b(stands)f(for)g Fu(')p Fv(\()p Fu(f)9 b Fv(\))p Fu(')p Fv(\()p Fu(g)s Fv(\))d Fr(\000)g Fu(')p Fv(\()p Fu(g)s Fv(\))p Fu(')p Fv(\()p Fu(f)j Fv(\))24 b(and)2845 3520 y(\026)2828 3542 y Fu(f)30 b Fv(means)21 b(complex)456 3741 y(conjugation.)45 b(A)32 b(state)e(is,)i(b)n(y)e (de\014nition,)i(a)f(linear)f(functional)h Fu(!)j Fv(on)c Fk(A)h Fv(\(the)g Ft(exp)l(e)l(ctation)456 3940 y(value)p Fv(\))25 b(whic)n(h)g(is)f(p)r(ositiv)n(e)g(\(i.e.)37 b Fu(!)s Fv(\()p Fu(a)1622 3910 y Fq(\003)1660 3940 y Fu(a)p Fv(\))23 b Fr(\025)g Fv(0\))h(and)h(normalized)e(\()p Fu(!)s Fv(\()p Fs(1)p Fv(\))g(=)g(1\).)36 b(It)25 b(is)f(uniquely)456 4140 y(determined)31 b(b)n(y)g(a)g(sequence)f(of)h(m)n(ultilinear)g (functionals)g Fu(!)2434 4152 y Fo(n)2479 4140 y Fv(,)h Fu(n)d Fv(=)g(0)p Fu(;)14 b Fv(1)p Fu(;)g(:)g(:)g(:)44 b Fv(\(the)32 b Fu(n)p Ft(-p)l(oint)456 4339 y(functions)p Fv(\))c(on)f(the)h(test)g(function)g(space)f Fm(D)p Fv(\()p Fu(M)9 b Fv(\),)1285 4678 y Fu(!)1337 4690 y Fo(n)1382 4678 y Fv(\()p Fu(f)1455 4690 y Fy(1)1492 4678 y Fu(;)14 b(:)g(:)g(:)27 b(;)14 b(f)1731 4690 y Fo(n)1776 4678 y Fv(\))23 b(=)g Fu(!)s Fv(\()p Fu(')p Fv(\()p Fu(f)2133 4690 y Fy(1)2171 4678 y Fv(\))14 b Fr(\001)g(\001)g(\001)f Fu(')p Fv(\()p Fu(f)2454 4690 y Fo(n)2500 4678 y Fv(\)\))28 b Fu(:)723 b Fv(\(5\))456 5016 y(W)-7 b(e)28 b(only)f(consider)g (states)g(whose)h Fu(n)p Fv(-p)r(oin)n(t)f(functions)h(are)f (distributions)h(and)g(restrict)f(fur-)456 5216 y(thermore)j(our)h (atten)n(tion)g(to)g(the)h(states)f(called)g Ft(quasi-fr)l(e)l(e)p Fv(,)i(namely)-7 b(,)32 b(those)f(states)g(whose)p eop %%Page: 15 15 15 14 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(15)456 425 y Fv(only)32 b(non)g(trivial)h Fu(n)p Fv(-p)r(oin)n(t)f(functions)h(ha)n(v)n(e)e Fu(n)i Fv(ev)n(en)f(and)h(are)f(generated)f(in)i(terms)f(of)h(the)456 624 y(2-p)r(oin)n(t)d(functions)h(\(see,)h(e.g.)46 b([29)o(]\).)i (Among)30 b(them)i(a)e(distinguished)h(class)f(is)h(formed)g(b)n(y)456 824 y(the)37 b(so)e(called)i(Hadamard)e(states)h(\(see,)j(e.g.)63 b([14)o(,)37 b(41)o(]\).)64 b(They)37 b(are)e(though)n(t)i(to)f(b)r(e)h (the)456 1023 y(appropriate)22 b(analogue)g(of)h(the)h(concept)g(of)f (the)i Ft(vacuum)e Fv(whic)n(h)h(has)f(no)g(direct)h(coun)n(terpart)456 1222 y(on)31 b(generic)f(space)h(times.)49 b(In)32 b(fact,)h(they)e (are)g(quasifree)f(states)h(whose)g(2-p)r(oin)n(t)g(functions)456 1421 y(ha)n(v)n(e)23 b(a)h(prescrib)r(ed)g(short-distance)f(b)r(eha)n (viour)h(whic)n(h)g(is)h(partially)e(motiv)-5 b(ated)25 b(b)n(y)f(the)h(fact)456 1621 y(that)h(it)h(allo)n(ws)e(the)i (de\014nition)g(of)f(the)h(exp)r(ectation)f(v)-5 b(alue)26 b(of)h(the)f(energy-momen)n(tum)f(ten-)456 1820 y(sor)k(\(see,)h(e.g.) 44 b([60)o(]\).)h(As)30 b(\014rst)g(observ)n(ed)f(b)n(y)h(Radzik)n(o)n (wski)e([49)o(])i(the)h(2-p)r(oin)n(t)e(functions)i(of)456 2019 y(Hadamard)26 b(states)h(can)g(b)r(e)h(c)n(haracterized)e(in)i (terms)f(of)h(their)f(w)n(a)n(v)n(e)f(fron)n(t)h(set.)555 2218 y(T)-7 b(o)34 b(discuss)h(this)g(c)n(haracterization)d(w)n(e)i (need)g(to)h(en)n(ter)f(in)n(to)g(the)h(realm)f(of)h(microlo)r(cal)456 2418 y(analysis.)62 b(W)-7 b(e)36 b(giv)n(e)g(some)g(motiv)-5 b(ations)36 b(to)g(the)h(basic)f(notions)f(of)i(w)n(a)n(v)n(e)d(fron)n (t)i(sets)h(and)456 2617 y(presen)n(t)f(those)i(basic)f(results)g(whic) n(h)g(are)g(used)g(throughout)g(the)h(pap)r(er.)66 b(W)-7 b(e)38 b(lea)n(v)n(e)e(the)456 2816 y(reader)e(the)j(task)e(to)h(lo)r (ok)g(further)g(in)n(to)g(the)g(large)f(existing)h(literature)f([35)o (].)63 b(Ph)n(ysicists)456 3015 y(migh)n(t,)33 b(for)f(concreteness,)h (start)f(from)g(the)h(w)n(ell-written)f(short)g(exp)r(osition)g(of)h (Junk)n(er)e(in)456 3215 y([39)o(],)25 b(where)f(they)g(can)g(\014nd)h (de\014nitions)g(and)f(results)g(ab)r(out)g(pseudo)r(di\013eren)n(tial) g(op)r(erators,)456 3414 y(whic)n(h)j(w)n(e)g(hold)h(as)f(kno)n(wn.)555 3613 y(W)-7 b(e)26 b(shall)f(denote)h(b)n(y)f Fm(E)p Fv(\()p Fn(R)1409 3583 y Fo(n)1455 3613 y Fv(\))h(the)g(space)f(of)g (complex-v)-5 b(alued)25 b(smo)r(oth)g(functions)h(and)g(b)n(y)456 3812 y Fm(E)506 3782 y Fq(0)529 3812 y Fv(\()p Fn(R)621 3782 y Fo(n)667 3812 y Fv(\))i(its)g(dual)f(space,)g(i.e.,)h(the)g (space)f(of)g(compactly)g(supp)r(orted)h(distributions.)555 4012 y(It)22 b(is)f(a)f(standard)g(result)h(in)g(distribution)g(theory) g(that)g Fu(u)i Fr(2)g Fm(E)2506 3982 y Fq(0)2530 4012 y Fv(\()p Fn(R)2622 3982 y Fo(n)2667 4012 y Fv(\))f(is)e(a)h(smo)r(oth) g(function)456 4211 y(i\013)36 b(its)g(F)-7 b(ourier)35 b(transform)j Fj(b)-49 b Fu(u)36 b Fv(deca)n(ys)f(rapidly)g(in)h(F)-7 b(ourier)35 b(dual)h(space)f Fn(R)2905 4223 y Fo(n)2950 4211 y Fv(,)k(i.e.)62 b(for)35 b(an)n(y)456 4410 y(in)n(teger)h Fu(N)46 b Fv(there)38 b(exists)f(a)g(constan)n(t)f Fu(C)1797 4422 y Fo(N)1898 4410 y Fv(suc)n(h)h(that)h Fr(j)s Fj(b)-49 b Fu(u)p Fv(\()p Fu(k)s Fv(\))p Fr(j)40 b(\024)f Fu(C)2692 4422 y Fo(N)2755 4410 y Fv(\(1)25 b(+)f Fr(j)p Fu(k)s Fr(j)p Fv(\))3067 4380 y Fq(\000)p Fo(N)3220 4410 y Fv(for)37 b(all)456 4610 y Fu(k)29 b Fr(2)629 4588 y Fv(_)610 4610 y Fn(R)670 4622 y Fo(n)715 4610 y Fv(,)i(where)e Fn(R)1071 4622 y Fo(n)1121 4610 y Fr(n)5 b(f)p Fv(0)p Fr(g)1339 4562 y Fu(:)1318 4610 y Fv(=)1428 4588 y(_)1410 4610 y Fn(R)1470 4622 y Fo(n)1515 4610 y Fv(.)43 b(In)30 b(case)f Fu(u)h Fv(is)f(not)h(smo)r(oth)g(the)g(F)-7 b(ourier)29 b(transform)f(ma)n(y)456 4809 y(still)i(rapidly)e(deca)n(y)h(in)h (certain)f(directions.)42 b(W)-7 b(e)30 b(ma)n(y)f(describ)r(e)h(this)g (set)f(of)h(directions)f(b)n(y)456 5008 y(an)35 b(op)r(en)g(cone)g(in) 1107 4987 y(_)1088 5008 y Fn(R)1148 5020 y Fo(n)1229 5008 y Fv(and)g(de\014ne)g(\006\()p Fu(u)p Fv(\))h(as)e(its)i (complemen)n(t)f(in)2680 4987 y(_)2661 5008 y Fn(R)2721 5020 y Fo(n)2767 5008 y Fv(.)60 b(It)35 b(is)g(easy)g(to)g(see)456 5207 y(that)f(\006\()p Fu(\036u)p Fv(\))g Fr(\032)f Fv(\006\()p Fu(u)p Fv(\))i(when)f Fu(u)f Fr(2)h Fm(E)1645 5177 y Fq(0)1669 5207 y Fv(\()p Fn(R)1761 5177 y Fo(n)1806 5207 y Fv(\))h(and)e Fu(\036)i Fr(2)f Fm(D)p Fv(\()p Fn(R)2374 5177 y Fo(n)2419 5207 y Fv(\).)56 b(This)34 b(prop)r(ert)n(y)f (suggests)g(a)p eop %%Page: 16 16 16 15 bop 456 226 a Fy(16)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(strategy)c(for)g(the)i (general)e(case)h(in)g(whic)n(h)g Fu(u)g Fv(is)h(not)f(of)g(compact)g (supp)r(ort.)36 b(So,)27 b(considering)456 624 y Fu(u)44 b Fr(2)i Fm(D)718 594 y Fq(0)741 624 y Fv(\()p Fn(R)833 594 y Fo(n)879 624 y Fv(\))41 b(and)g(a)f(p)r(oin)n(t)h Fu(x)46 b Fr(2)f Fn(R)1692 594 y Fo(n)1778 624 y Fv(in)c(the)h(supp)r (ort)e(of)h Fu(u)p Fv(,)j(supp\()p Fu(u)p Fv(\))i Fr(\032)e(O)r Fv(,)h Fr(O)e Fv(op)r(en)456 824 y(subset)27 b(of)h Fn(R)865 794 y Fo(n)910 824 y Fv(,)g(w)n(e)f(\014rst)g(lo)r(calize)g Fu(u)g Fv(via)g(m)n(ultiplication)h(with)g(some)f Fu(\036)c Fr(2)h Fm(D)p Fv(\()p Fn(R)3000 794 y Fo(n)3045 824 y Fv(\))k(suc)n(h)f(that)456 1023 y Fu(\036)p Fv(\()p Fu(x)p Fv(\))41 b Fr(6)p Fv(=)e(0)f(and)f(then)h(consider)f(the)h(F)-7 b(ourier)37 b(transform)f(of)i Fu(\036u)p Fv(,)i(no)n(w)d(a)h (distribution)g(of)456 1222 y(compact)26 b(supp)r(ort.)37 b(W)-7 b(e)28 b(then)g(de\014ne)f(the)h(set)g(\006)2031 1234 y Fo(x)2072 1222 y Fv(\()p Fu(u)p Fv(\))2228 1175 y Fu(:)2208 1222 y Fv(=)22 b Fr(\\)2350 1234 y Fo(\036)2395 1222 y Fv(\006\()p Fu(\036u)p Fv(\))28 b(where)f(the)g(in)n(tersection) 456 1421 y(is)37 b(tak)n(en)f(w.r.t.)66 b(all)37 b(smo)r(oth)g (functions)g(of)h(compact)e(supp)r(ort)h Fu(\036)h Fv(suc)n(h)f(that)h Fu(\036)p Fv(\()p Fu(x)p Fv(\))i Fr(6)p Fv(=)f(0.)456 1621 y(This)27 b(ma)n(y)f(b)r(e)i(called)f(the)g(set)g(of)g Ft(singular)j(dir)l(e)l(ctions)e Fv(of)f Fu(u)g Fv(o)n(v)n(er)e Fu(x)p Fv(.)38 b(It)27 b(is)g(empt)n(y)g(whenev)n(er)456 1820 y Fu(x)32 b(=)-51 b Fr(2)24 b Fv(singsupp)o(\()p Fu(u)p Fv(\).)555 2019 y(Hence,)h(\014nally)-7 b(,)25 b(w)n(e)f(de\014ne)h(the)f(\()p Ft(smo)l(oth)p Fv(\))h Ft(wave)j(fr)l(ont)f(set)d Fv(for)f Fu(u)g Fr(2)g Fm(D)2789 1989 y Fq(0)2813 2019 y Fv(\()p Fn(R)2905 1989 y Fo(n)2950 2019 y Fv(\))i(as)e(WF)q(\()p Fu(u)p Fv(\))3401 1972 y Fu(:)3380 2019 y Fv(=)456 2218 y Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(\))31 b Fr(2)h Fn(R)870 2188 y Fo(n)937 2218 y Fr(\002)1042 2197 y Fv(_)1024 2218 y Fn(R)1084 2230 y Fo(n)1162 2218 y Fr(j)h Fu(k)h Fr(2)e Fv(\006)1442 2230 y Fo(x)1484 2218 y Fv(\()p Fu(u)p Fv(\))p Fr(g)p Fv(.)51 b(This)33 b(set)g(is)f(readily)g(seen)h(to)f(b)r(e)h(closed)f (and)h(conic,)456 2418 y(where)27 b(the)h(last)f(means)g(that)h(if)g Fu(k)e Fr(2)d Fv(\006)1713 2430 y Fo(x)1755 2418 y Fv(\()p Fu(u)p Fv(\))28 b(so)f(do)g(an)n(y)g Fu(\025k)k Fv(for)c(all)g Fu(\025)d(>)e Fv(0.)555 2617 y(It)h(is)g(no)n(w)f(crucial)g(that)h(the) g(notion)f(of)h(the)g(w)n(a)n(v)n(e)e(fron)n(t)h(set)h(can)f(b)r(e)i (lifted)f(to)g(an)n(y)f(smo)r(oth)456 2816 y(manifold)f Fm(M)f Fv(where)h(it)h(is)f(in)n(v)-5 b(arian)n(tly)20 b(de\014ned)h(as)f(a)h(subset)g(of)g(the)h(cotangen)n(t)e(bundle)3267 2795 y(_)3242 2816 y Fu(T)3303 2786 y Fq(\003)3340 2816 y Fm(M)p Fv(.)456 3015 y(This)j(co)n(v)-5 b(ariance)22 b(under)i(co)r(ordinate)f(transformations)f(is)h(what)h(giv)n(es)f(to)h (the)g(de\014nition)g(its)456 3215 y(real)i(tec)n(hnical)h(p)r(o)n(w)n (er.)555 3414 y(Among)32 b(the)g(results)g(whic)n(h)f(will)i(b)r(e)f (imp)r(ortan)n(t)g(for)f(us)h(w)n(e)f(men)n(tion)h(that)h(deriv)-5 b(ativ)n(es)456 3613 y(do)26 b(not)h(enlarge)e(the)i(w)n(a)n(v)n(e)e (fron)n(t)h(set)h(of)f(a)g(distribution,)h(i.e.)37 b(WF)q(\()p Fu(@)5 b(u)p Fv(\))22 b Fr(\022)h Fv(WF)q(\()p Fu(u)p Fv(\),)k(and)f(the)456 3812 y(follo)n(wing)g(criterion)h(called)g (H\177)-42 b(ormander)26 b(criterion)h(for)g(m)n(ultiplication)h(of)f (distributions:)456 4012 y Fs(M1.)40 b(Pro)s(duct)p Fv(.)52 b(Pic)n(king)31 b(t)n(w)n(o)g(distributions)h Fu(u)2077 4024 y Fy(1)2114 4012 y Fu(;)14 b(u)2199 4024 y Fy(2)2266 4012 y Fr(2)32 b Fm(D)2422 3982 y Fq(0)2445 4012 y Fv(\()p Fm(M)p Fv(\),)h(the)g Ft(p)l(ointwise)h Fv(pro)r(duct)661 4211 y Fu(u)709 4223 y Fy(1)746 4211 y Fu(u)794 4223 y Fy(2)859 4211 y Fv(exists)28 b(as)f(a)g Ft(b)l(ona-\014de)i Fv(distribution)f(whenev)n(er)f(WF)q(\()p Fu(u)2649 4223 y Fy(1)2686 4211 y Fv(\))19 b(+)f(WF\()p Fu(u)3039 4223 y Fy(2)3076 4211 y Fv(\))29 b(do)r(es)e(not)661 4410 y(in)n(tersect)41 b(the)g(zero)f(section)h Fu(Z)6 b Fv(\()p Fu(T)1802 4380 y Fq(\003)1839 4410 y Fm(M)p Fv(\),)45 b(i.e.,)f(if)e(for)e(all)h(co)n(v)n(ectors)e Fu(k)2968 4422 y Fo(i)3041 4410 y Fr(2)46 b Fv(WF)q(\()p Fu(u)3362 4422 y Fo(i)3389 4410 y Fv(\),)661 4610 y Fu(i)27 b Fv(=)g(1)p Fu(;)14 b Fv(2,)30 b(based)g(o)n(v)n(er)e(the)i(same)g(p)r(oin)n(t)g (one)g(\014nds)g(that)h Fu(k)2554 4622 y Fy(1)2611 4610 y Fv(+)20 b Fu(k)2739 4622 y Fy(2)2804 4610 y Fr(6)p Fv(=)27 b(0.)44 b(Moreo)n(v)n(er,)28 b(if)661 4809 y(WF)q(\()p Fu(u)881 4821 y Fo(i)908 4809 y Fv(\))c Fr(\032)e Fv(\000)1103 4821 y Fo(i)1131 4809 y Fv(,)28 b Fu(i)22 b Fv(=)h(1)p Fu(;)14 b Fv(2,)26 b(then)j(WF\()p Fu(u)1900 4821 y Fy(1)1937 4809 y Fu(u)1985 4821 y Fy(2)2022 4809 y Fv(\))23 b Fr(\032)g Fv(\000)2217 4821 y Fy(1)2273 4809 y Fr([)18 b Fv(\000)2398 4821 y Fy(2)2454 4809 y Fr([)h Fv(\(\000)2612 4821 y Fy(1)2668 4809 y Fv(+)f(\000)2803 4821 y Fy(2)2840 4809 y Fv(\).)555 5008 y(W)-7 b(e)24 b(shall)e(also)g(refer)h(frequen)n(tly) g(to)g(a)f(certain)h(con)n(tin)n(uit)n(y)f(prop)r(ert)n(y)g(in)i (microlo)r(cal)e(anal-)456 5207 y(ysis)40 b(whic)n(h)h(in)g(the)g(b)r (o)r(dy)g(of)f(the)i(pap)r(er)e(is)h(sometimes)f(called)g(\\H\177)-42 b(ormander)39 b(\(pseudo\))p eop %%Page: 17 17 17 16 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(17)456 425 y Fv(top)r(ology)-7 b(.")34 b(It)27 b(has)g(to)f(do)g(with)h(the)h(notion)e(of)g(con)n(v)n (ergen)n(t)f(sequences)h(whic)n(h)g(resp)r(ect)h(also)456 624 y(w)n(a)n(v)n(e)f(fron)n(t)h(set)g(prop)r(erties:)456 824 y Fs(M2.)40 b(Con)m(tin)m(uit)m(y)p Fv(.)46 b(Let)31 b Fm(D)1397 794 y Fq(0)1397 847 y Fy(\000)1442 824 y Fv(\()p Fm(M)p Fv(\))1637 777 y Fu(:)1616 824 y Fv(=)d Fr(f)p Fu(v)i Fr(2)f Fm(D)1974 794 y Fq(0)1997 824 y Fv(\()p Fm(M)p Fv(\))i Fr(j)f Fv(WF)q(\()p Fu(v)s Fv(\))f Fr(\032)e Fv(\000)p Fr(g)p Fv(,)k(where)f(\000)h(is)f(a)g(closed)661 1023 y(conic)k(set)g(in)1145 1002 y(_)1120 1023 y Fu(T)1181 993 y Fq(\003)1218 1023 y Fm(M)p Fv(.)56 b(A)35 b(sequence)f Fr(f)p Fu(u)1917 1035 y Fo(i)1943 1023 y Fr(g)1985 1035 y Fo(i)p Fq(2)p Fi(N)2133 1023 y Fr(\032)g Fm(D)2301 993 y Fq(0)2301 1046 y Fy(\000)2346 1023 y Fv(\()p Fm(M)p Fv(\))g(con)n(v)n(erges)e(to)i Fu(u)g Fr(2)g Fm(D)3253 993 y Fq(0)3253 1046 y Fy(\000)3298 1023 y Fv(\()p Fm(M)p Fv(\))661 1222 y(in)27 b(the)g(sense)e(of)i(the)f(H\177)-42 b(ormander)25 b(\(pseudo\))i(top)r(ology)e(whenev)n(er)g(the)i(follo)n (wing)e(t)n(w)n(o)661 1421 y(prop)r(erties)i(hold)g(true:)514 1828 y(\(a\))41 b Fu(u)709 1840 y Fo(i)760 1828 y Fr(!)23 b Fu(u)k Fv(w)n(eakly)1189 1798 y Fq(\003)1253 1828 y Fv(\(i.e.)38 b(in)28 b Fm(D)1595 1798 y Fq(0)1618 1828 y Fv(\()p Fm(M)p Fv(\)\),)509 2227 y(\(b\))42 b(for)18 b(an)n(y)g(prop)r(erly)f(supp)r(orted)h(pseudo)r(di\013eren)n(tial)g (op)r(erator)f Fu(A)i Fv(suc)n(h)f(that)g Fu(\026)p Fv(supp\()p Fu(A)p Fv(\))p Fr(\\)661 2426 y Fv(\000)23 b(=)g Fr(;)p Fv(,)g(w)n(e)g(ha)n(v)n(e)e(that)i Fu(Au)1501 2438 y Fo(i)1552 2426 y Fr(!)g Fu(Au)f Fv(in)h(the)g(sense)f(of)h Fm(E)p Fv(\()p Fm(M)p Fv(\).)36 b(\()p Fu(\026)p Fv(supp\()p Fu(A)p Fv(\))24 b(is)f(the)g(pro)5 b(jec-)661 2625 y(tion)29 b(on)n(to)e(the)i(second)f(comp)r(onen)n(t)g(of)g(the)h(w)n(a)n(v)n(e)d (fron)n(t)i(set)h(of)f(the)h(Sc)n(h)n(w)n(artz)d(k)n(ernel)661 2825 y(of)i Fu(A)p Fv(.\))555 3024 y(A)g(last)f(prop)r(ert)n(y)f(is)h (connected)g(with)g(the)h(sequen)n(tial)e(con)n(tin)n(uit)n(y)-7 b(,)27 b(in)h(the)f(sense)g(of)g Fs(M2)p Fv(,)456 3223 y(of)g(the)h(op)r(eration)f(of)g(restriction)g(of)g(a)h(distribution)f (to)h(a)f(submanifold:)456 3422 y Fs(M3.)40 b(T)-8 b(race)p Fv(.)50 b(Let)32 b Fm(N)e Fr(\032)f Fm(M)j Fv(denote)f(a)g (submanifold,)h(and)g(let)f Fu(u)e Fr(2)h Fm(D)2779 3392 y Fq(0)2803 3422 y Fv(\()p Fm(M)p Fv(\).)48 b(Then)32 b Fu(u)f Fv(can)661 3622 y(b)r(e)j(restricted)f(to)g(the)h(submanifold) g Fm(N)h Fv(whenev)n(er)d(WF)q(\()p Fu(u)p Fv(\))h(do)r(es)h(not)f(in)n (tersect)g(the)661 3821 y(conormal)28 b(bundle)i Fu(N)1373 3791 y Fq(\003)1410 3821 y Fm(N)h Fv(of)e Fm(N)q Fv(.)42 b(Moreo)n(v)n(er,)27 b(if)j(WF\()p Fu(u)p Fv(\))c Fr(\032)f Fv(\000,)30 b(with)g(\000)f(a)f(closed)h(conic)661 4020 y(set)g(suc)n(h)f(that)h(\000)19 b Fr(\\)g Fu(N)1382 3990 y Fq(\003)1420 4020 y Fm(N)26 b Fv(=)e Fr(;)p Fv(,)29 b(then)g(the)g(op)r(erator)d(of)j(restriction)e(\(trace\))i Fu(\015)k Fv(can)28 b(b)r(e)661 4219 y(lifted)f(as)e(a)g(sequen)n (tially)f(con)n(tin)n(uous)h(op)r(erator,)f(in)i(the)g(sense)f(of)g Fs(M2)p Fv(,)h(from)f Fm(D)3253 4189 y Fq(0)3253 4242 y Fy(\000)3298 4219 y Fv(\()p Fm(M)p Fv(\))661 4419 y(to)j Fm(D)832 4388 y Fq(0)855 4419 y Fv(\()p Fm(N)q Fv(\).)555 4618 y(F)-7 b(or)31 b(later)g(purp)r(ose,)h(it)f(is)h(con)n(v)n(enien)n (t)e(to)h(ha)n(v)n(e)f(a)h(co)r(ordinate)f(dep)r(enden)n(t)i(form)n (ulation)456 4817 y(of)d Fs(M2)p Fv(\(b\))h(b)n(y)f(using)g(F)-7 b(ourier)29 b(transforms.)41 b(Namely)-7 b(,)30 b(let)g Fu(x)2401 4829 y Fy(0)2465 4817 y Fr(2)c Fm(M)k Fv(and)f(let)h Fu(V)48 b Fv(b)r(e)30 b(an)f(op)r(en)456 5016 y(conical)e(neigh)n(b)r (ourho)r(o)r(d)h(of)h(\000)1450 5028 y Fo(x)1488 5036 y FC(0)1524 5016 y Fv(,)g(where)f(the)h(last)g(denotes)f(a)g(set)h(of)g (co)n(v)n(ectors)d(asso)r(ciated)456 5216 y(to)36 b(the)g(p)r(oin)n(t)g Fu(x)989 5228 y Fy(0)1027 5216 y Fv(.)62 b(Cho)r(ose)36 b(a)f(c)n(hart)h(\()p Fu(';)14 b(U)9 b Fv(\))36 b(at)g Fu(x)2125 5228 y Fy(0)2199 5216 y Fv(suc)n(h)g(that)g(\000)2635 5228 y Fo(x)2714 5216 y Fr(\032)h Fu(V)55 b Fv(for)36 b(all)f Fu(x)j Fr(2)g Fu(U)9 b Fv(.)p eop %%Page: 18 18 18 17 bop 456 226 a Fy(18)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(Let)h Fu(\037)e Fr(2)g Fm(D)p Fv(\()p Fu(U)9 b Fv(\))31 b(with)g Fu(\037)p Fv(\()p Fu(x)1323 437 y Fy(0)1361 425 y Fv(\))d Fr(6)p Fv(=)g(0.)45 b(Then)30 b(the)h(F)-7 b(ourier)30 b(transform)f(of)i Fu(\037u)p Fv(,)f Fu(u)e Fr(2)g Fm(D)3143 395 y Fq(0)3143 448 y Fy(\000)3188 425 y Fv(\()p Fm(M)p Fv(\),)k(is)456 624 y(strongly)26 b(decreasing)g(in)i(the)g(complemen)n(t)f(of)h Fu(V)19 b Fv(,)27 b(and)1307 963 y(sup)1302 1037 y Fo(k)9 b(=)-42 b Fq(2)q Fo(V)1451 963 y Fr(j)p Fv(\()t Fj(d)-124 b Fu(\037u)1606 975 y Fo(i)1652 963 y Fr(\000)26 b Fj(c)-91 b Fu(\037u)o Fv(\)\()p Fu(k)s Fv(\))p Fr(j)p Fv(\(1)19 b(+)f Fr(j)p Fu(k)s Fr(j)p Fv(\))2299 929 y Fo(N)2386 963 y Fr(!)23 b Fv(0)k Fu(;)754 b Fv(\(6\))456 1302 y(for)24 b(all)h Fu(N)32 b Fr(2)23 b Fn(N)j Fv(if)f Fu(u)1077 1314 y Fo(i)1128 1302 y Fr(!)e Fu(u)h Fv(in)i Fm(D)1470 1272 y Fq(0)1470 1325 y Fy(\000)1515 1302 y Fv(\()p Fm(M)p Fv(\).)36 b(If,)26 b(on)f(the)h(con)n(trary)-7 b(,)24 b(the)h(ab)r(o)n(v)n(e)f(con)n(v)n(ergence)f(holds)456 1501 y(true)k(for)g(all)g(c)n(hoices)g(of)h Fu(x)1295 1513 y Fy(0)1332 1501 y Fv(,)g Fu(V)19 b Fv(,)28 b(\()p Fu(';)14 b(U)9 b Fv(\))28 b(and)f Fu(\037)p Fv(,)h(w)n(e)f(obtain)g Fs(M2)p Fv(\(b\).)555 1700 y(After)i(this)g(disgression)e(in)n(to)h (microlo)r(cal)g(analysis)f(w)n(e)h(brie\015y)g(describ)r(e)h(Radzik)n (o)n(wski's)456 1900 y(c)n(haracterization)e(of)j(Hadamard)f(states)h ([49)o(].)45 b(The)31 b(idea)e(is)i(to)f(use)g(w)n(a)n(v)n(e)e(fron)n (t)i(sets)g(for)g(a)456 2099 y(form)n(ulation)22 b(of)g(a)h(sp)r (ectral)f(condition.)36 b(The)23 b(an)n(tisymmetric)f(part)g(of)h(the)h (2-p)r(oin)n(t)e(function)456 2298 y(is)27 b(the)h(comm)n(utator)e (function)j Fu(E)5 b Fv(.)37 b(Its)27 b(w)n(a)n(v)n(e)f(fron)n(t)h(set) h(is)980 2637 y(WF)q(\()p Fu(E)5 b Fv(\))24 b(=)e Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(;)g Fu(x)1649 2603 y Fq(0)1673 2637 y Fu(;)g Fr(\000)p Fu(k)1821 2603 y Fq(0)1843 2637 y Fv(\))24 b Fr(2)2003 2616 y Fv(_)1977 2637 y Fu(T)2038 2603 y Fq(\003)2075 2637 y Fu(M)2165 2603 y Fy(2)2230 2637 y Fr(j)k Fv(\()p Fu(x;)14 b(k)s Fv(\))23 b Fr(\030)g Fv(\()p Fu(x)2665 2603 y Fq(0)2689 2637 y Fu(;)14 b(k)2772 2603 y Fq(0)2795 2637 y Fv(\))p Fr(g)28 b Fu(:)418 b Fv(\(7\))456 2976 y(Here)23 b(the)g(equiv)-5 b(alence)23 b(relation)g Fr(\030)g Fv(means)g(that)g(there)h(exists)f(a)g(n)n(ull)g (geo)r(desic)f(from)i Fu(x)f Fv(to)h Fu(x)3421 2946 y Fq(0)456 3175 y Fv(suc)n(h)j(that)i Fu(k)h Fv(is)e(coparallel)f(to)h (the)g(tangen)n(t)f(v)n(ector)g(of)h(the)g(geo)r(desic)g(and)f Fu(k)2925 3145 y Fq(0)2977 3175 y Fv(is)g(its)i(parallel)456 3374 y(transp)r(ort)24 b(from)i Fu(x)1063 3386 y Fy(1)1126 3374 y Fv(to)g Fu(x)1273 3386 y Fy(2)1311 3374 y Fv(.)36 b(F)-7 b(or)25 b(coinciding)g(p)r(oin)n(ts,)h(the)h(relation)d(is)i (de\014ned)g(as)f(consisting)456 3573 y(of)f(the)h(degenerate)f (\(i.e.,)i(only)e(one)h(p)r(oin)n(t\))g(geo)r(desic)f(at)g Fu(x)g Fv(=)e Fu(x)2488 3543 y Fq(0)2537 3573 y Fv(whic)n(h)j(has)f(co) n(v)n(ector)e Fu(k)28 b Fv(still)456 3773 y(along)k(the)i(b)r(oundary)f (of)h(the)g(ligh)n(t-cone)e(and)i Fu(k)2062 3743 y Fq(0)2118 3773 y Fr(\021)f Fu(k)s Fv(.)55 b(W)-7 b(e)34 b(remark,)g(for)f(later)g (purp)r(ose,)456 3972 y(that)e(since)f(only)h(ligh)n(t-lik)n(e)f(co)n (v)n(ectors)e(are)i(presen)n(t,)h(one)f(can)h(restrict)f Fu(E)5 b Fv(,)32 b(and,)f(whenev)n(er)456 4171 y(lo)r(cal)d(co)r (ordinates)f(are)h(c)n(hosen,)g(its)h(deriv)-5 b(ativ)n(e)28 b(w.r.t.)40 b(time)2477 4150 y(_)2448 4171 y Fu(E)5 b Fv(,)29 b(to)g(an)n(y)f(spacelik)n(e)g(Cauc)n(h)n(y)456 4370 y(h)n(yp)r(ersurface.)555 4570 y(As)c(a)f(result)g(of)g([49)o(,)g (43)o(],)i(the)f(2-p)r(oin)n(t)e(function)i(of)f(a)g(Hadamard)f(state)i (has)e(a)h(w)n(a)n(v)n(e)f(fron)n(t)456 4769 y(set)27 b(whic)n(h)h(is)f(just)h(the)g(p)r(ositiv)n(e)g(frequency)f(part)g(of)g (WF)q(\()p Fu(E)5 b Fv(\),)1071 5108 y(WF)q(\()p Fu(!)1295 5120 y Fy(2)1332 5108 y Fv(\))23 b(=)g Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(;)g Fu(x)1763 5073 y Fq(0)1787 5108 y Fu(;)g Fr(\000)p Fu(k)1935 5073 y Fq(0)1957 5108 y Fv(\))24 b Fr(2)f Fv(WF\()p Fu(E)5 b Fv(\))29 b Fr(j)55 b Fu(k)26 b Fr(2)p 2615 5041 67 4 v 24 w Fu(V)2681 5120 y Fy(+)2736 5108 y Fr(g)i Fu(:)509 b Fv(\(8\))p eop %%Page: 19 19 19 18 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(19)555 425 y Fv(Since)30 b(\(8\))g(restricts)e(the)i(singular)e(supp)r(ort)i(of)f Fu(!)2150 437 y Fy(2)2187 425 y Fv(\()p Fu(x)2266 437 y Fy(1)2304 425 y Fu(;)14 b(x)2388 437 y Fy(2)2426 425 y Fv(\))30 b(to)f(p)r(oin)n(ts)h Fu(x)2890 437 y Fy(1)2957 425 y Fv(and)f Fu(x)3167 437 y Fy(2)3235 425 y Fv(whic)n(h)456 624 y(are)i(n)n(ull)h(related,)h Fu(!)1125 636 y Fy(2)1195 624 y Fv(is)f(smo)r(oth)g(for)g(all)g(other)g(p)r(oin)n(ts.)51 b(The)32 b(smo)r(othness)g(for)g Ft(sp)l(ac)l(e-like)456 824 y Fv(related)d(p)r(oin)n(ts)i(is)f(kno)n(wn)g(to)g(b)r(e)h(true)f (for)g(quan)n(tum)g(\014eld)h(theories)e(on)h(Mink)n(o)n(wski)f(space) 456 1023 y(satisfying)d(the)h(sp)r(ectrum)g(condition)g(b)n(y)f(the)i (Bargmann-Hall-Wigh)n(tman)c(Theorem)i([55)o(].)456 1222 y(F)-7 b(or)19 b(time-lik)n(e)g(related)g(p)r(oin)n(ts,)i(ho)n(w)n(ev)n (er,)e(a)g(similar)g(general)f(prediction)h(on)g(the)h(smo)r(othness) 456 1421 y(do)r(es)27 b(not)g(exist.)555 1621 y(Another)22 b(deep)g(result)f(from)h(Radzik)n(o)n(wski)e([49)o(])i(sho)n(ws)e(that) i(the)g(Duistermaat-H\177)-42 b(orman-)456 1820 y(der)26 b([17)o(])h(distinguished)g(parametrices)e(for)h(the)h(Klein-Gordon)e (equation)h(are)g(nothing)h(else)456 2019 y(than)f(the)h(\(St)r(\177) -44 b(uc)n(k)n(elb)r(erg-\)F)-7 b(eynman{an)n(ti-F)g(eynman)25 b(propagators)e(\(up)k(to)g Fu(C)2974 1989 y Fq(1)3045 2019 y Fv(\))f(for)g(quasi-)456 2218 y(free)f(Hadamard)f(states.)36 b(W)-7 b(e)26 b(recall)f(that)h(the)g(time-ordered)e(2-p)r(oin)n(t)h (function)i Fu(E)3123 2230 y Fo(F)3204 2218 y Fv(arising)456 2418 y(from)g Fu(!)704 2430 y Fy(2)768 2418 y Fv(is)h(giv)n(en)f(b)n(y) 1218 2756 y Fu(iE)1308 2768 y Fo(F)1363 2756 y Fv(\()p Fu(x)1442 2768 y Fy(1)1480 2756 y Fu(;)14 b(x)1564 2768 y Fy(2)1602 2756 y Fv(\))23 b(=)g Fu(!)1797 2768 y Fy(2)1834 2756 y Fv(\()p Fu(x)1913 2768 y Fy(1)1950 2756 y Fu(;)14 b(x)2034 2768 y Fy(2)2072 2756 y Fv(\))19 b(+)f Fu(E)2267 2768 y Fo(r)r(et)2360 2756 y Fv(\()p Fu(x)2439 2768 y Fy(1)2477 2756 y Fu(;)c(x)2561 2768 y Fy(2)2599 2756 y Fv(\))28 b Fu(:)456 3095 y Fv(Its)f(w)n(a)n(v)n(e)f(fron)n(t)h(set)h ([49)o(])g(is)1595 3434 y(WF\()p Fu(E)1827 3446 y Fo(F)1883 3434 y Fv(\))23 b(=)g Fu(O)e Fr([)e Fu(D)29 b(;)456 3773 y Fv(where)e(the)h(o\013-diagonal)d(piece)j(is)f(giv)n(en)g(b)n(y)-7 b(,)547 4111 y Fu(O)26 b Fv(=)c Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(;)g Fu(x)1011 4077 y Fq(0)1035 4111 y Fu(;)g Fr(\000)p Fu(k)1183 4077 y Fq(0)1206 4111 y Fv(\))23 b Fr(2)1365 4090 y Fv(_)1339 4111 y Fu(T)1400 4077 y Fq(\003)1438 4111 y Fu(M)1528 4077 y Fy(2)1592 4111 y Fr(j)28 b Fv(\()p Fu(x;)14 b(k)s Fv(\))24 b Fr(\030)e Fv(\()p Fu(x)2027 4077 y Fq(0)2051 4111 y Fu(;)14 b(k)2134 4077 y Fq(0)2157 4111 y Fv(\))p Fu(;)g(x)24 b Fr(6)p Fv(=)f Fu(x)2432 4077 y Fq(0)2455 4111 y Fu(;)14 b(k)26 b Fr(2)p 2640 4045 67 4 v 24 w Fu(V)2707 4123 y Fq(\006)2790 4111 y Fv(if)i Fu(x)c Fr(2)f Fu(J)3069 4077 y Fq(\006)3125 4111 y Fv(\()p Fu(x)3204 4077 y Fq(0)3228 4111 y Fv(\))p Fr(g)28 b Fu(;)456 4450 y Fv(and)f(the)h(diagonal)e(one)h(b)n(y)-7 b(,)1074 4789 y Fu(D)25 b Fv(=)d Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(;)g Fu(x;)g Fr(\000)p Fu(k)s Fv(\))23 b Fr(2)1850 4768 y Fv(_)1825 4789 y Fu(T)1886 4755 y Fq(\003)1923 4789 y Fu(M)2013 4755 y Fy(2)2077 4789 y Fr(j)28 b Fu(x)c Fr(2)f Fu(M)t(;)14 b(k)26 b Fr(2)2572 4768 y Fv(_)2546 4789 y Fu(T)2607 4755 y Fq(\003)2595 4809 y Fo(x)2645 4789 y Fu(M)9 b Fr(g)26 b Fu(:)p eop %%Page: 20 20 20 19 bop 456 226 a Fy(20)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)555 425 y Fv(No)n(w,)38 b(one)d(can)g(see)g(wh)n(y)h(in)g(naiv)n(e)f(p)r(erturbation)g(theory)g (w)n(e)g(ma)n(y)g(\014nd)h(div)n(ergences.)456 624 y(Indeed,)31 b(the)g(p)r(erturbativ)n(e)f(expansion)f(in)i(terms)f(of)h(F)-7 b(eynman)30 b(graphs)f(in)i(p)r(osition)f(space)456 824 y(leads)i(to)h(p)r(oin)n(t)n(wise)f(pro)r(ducts)g(of)h(F)-7 b(eynman)33 b(propagators.)49 b(But)34 b(these)e(pro)r(ducts)h(do)f (not)456 1023 y(satisfy)22 b(H\177)-42 b(ormander)21 b(criterion)h(for)g(m)n(ultiplication)h(of)g(distributions)f(since)h (co)n(v)n(ectors)d(based)456 1222 y(on)27 b(the)h(diagonal)e(piece)i Fu(D)h Fv(can)e(add)h(up)g(to)f(zero.)456 1549 y(2.2.)46 b Fs(A)c(new)g(construction)g(of)g(Wic)m(k)g(p)s(olynomials.)d Fv(In)d(a)g(previous)f(pap)r(er)h([8)o(])h(w)n(e)456 1749 y(constructed)c(Wic)n(k)g(p)r(olynomials)f(as)h(op)r(erator-v)-5 b(alued)31 b(distributions.)55 b(W)-7 b(e)33 b(considered)g(a)456 1948 y(\014xed)28 b(Hadamard)f(state)h Fu(!)j Fv(and)d(the)g(induced)h (GNS)g(represen)n(tation)d(\()p Fr(H)2815 1960 y Fo(!)2864 1948 y Fu(;)14 b(\031)2948 1960 y Fo(!)2996 1948 y Fu(;)g Fv(\012)3093 1960 y Fo(!)3141 1948 y Fv(\))28 b(for)g(the)456 2147 y Fr(\003)p Fv(-algebra)17 b Fk(A)i Fv(and)h(found)g(the)g(Wic)n (k)g(p)r(olynomials)f(as)g(op)r(erator-v)-5 b(alued)18 b(distributions)i(on)f(the)456 2346 y(dense)24 b(cyclic)h(domain)f (generated)g(b)n(y)g(\012)1745 2358 y Fo(!)1793 2346 y Fv(.)36 b(W)-7 b(e)25 b(recall)f(that)h(b)n(y)g(a)f(GNS)i(triple)e (\()p Fr(H)3086 2358 y Fo(!)3135 2346 y Fu(;)14 b(\031)3219 2358 y Fo(!)3267 2346 y Fu(;)g Fv(\012)3364 2358 y Fo(!)3412 2346 y Fv(\))456 2546 y(w)n(e)37 b(mean)h(a)g(complex)g(Hilb)r(ert)h (space)e Fr(H)1835 2558 y Fo(!)1883 2546 y Fv(,)k(a)d(represen)n (tation)f Fu(\031)2628 2558 y Fo(!)2714 2546 y Fv(of)h Fk(A)g Fv(b)n(y)g(un)n(b)r(ounded)456 2745 y(op)r(erators)28 b(on)i Fr(H)1014 2757 y Fo(!)1062 2745 y Fv(,)i(and)e(\014nally)g(b)n (y)h(\012)1714 2757 y Fo(!)1790 2745 y Fr(2)d(H)1943 2757 y Fo(!)2022 2745 y Fv(the)i(cyclic)h(v)n(ector)e(represen)n(ting)g (the)i(state)456 2944 y Fu(!)f Fv(for)d(whic)n(h)h(one)f(has)g(the)h (connection)f(equation)1259 3283 y Fu(!)s Fv(\()p Fu(A)p Fv(\))c(=)g(\(\012)1643 3295 y Fo(!)1691 3283 y Fu(;)14 b(\031)1775 3295 y Fo(!)1823 3283 y Fv(\()p Fu(A)p Fv(\)\012)2009 3295 y Fo(!)2058 3283 y Fv(\))28 b Fu(;)180 b Fr(8)p Fu(A)24 b Fr(2)f Fk(A)k Fu(:)555 3622 y Fv(The)34 b(dep)r(endence)g(of) g(the)g(construction)f(of)h(Wic)n(k)g(p)r(olynomials)f(on)g(the)h(c)n (hoice)f(of)h(the)456 3821 y(Hadamard)29 b(state)i(led)g(to)g(t)n(w)n (o)f(problems:)43 b(The)31 b(\014rst)g(one)f(is)h(due)h(to)e(the)i(con) n(v)n(en)n(tion)d(that)456 4020 y(the)22 b(exp)r(ectation)h(v)-5 b(alue)22 b(of)g(a)g(Wic)n(k)h(p)r(olynomial)e(v)-5 b(anishes)22 b(in)h(the)g(c)n(hosen)e(Hadamard)h(state.)456 4219 y(Other)g(c)n (hoices)h(lead)g(to)g(a)g(\014nite)h(rede\014nition,)g(a)f(problem)g(w) n(ell)g(kno)n(wn)f(from)h(the)h(de\014nition)456 4419 y(of)31 b(the)h(exp)r(ectation)f(v)-5 b(alue)32 b(of)f(the)h(energy)e (momen)n(tum)i(tensor)f([61)o(].)49 b(Since)31 b(w)n(e)g(shall)g(not) 456 4618 y(discuss)h(\014nite)h(renormalizations)d(in)j(this)f(pap)r (er,)i(w)n(e)e(do)g(not)h(treat)f(this)h(problem)f(at)g(the)456 4817 y(momen)n(t.)66 b(The)37 b(other)g(problem)f(is)i(of)f(a)g(more)f (tec)n(hnical)h(nature:)56 b(The)37 b(smeared)g(Wic)n(k)456 5016 y(p)r(olynomials)26 b(are)g(un)n(b)r(ounded)i(op)r(erators.)35 b(W)-7 b(e)28 b(kno)n(w)e(from)h(the)h(w)n(ork)e(of)i(V)-7 b(erc)n(h)27 b([57)o(])g(that,)456 5216 y(lo)r(cally)-7 b(,)35 b(i.e.)58 b(in)34 b(b)r(ounded)h(regions)e(of)h(space)g(time,)j (di\013eren)n(t)d(Hadamard)g(states)g(lead)g(to)p eop %%Page: 21 21 21 20 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(21)456 425 y Fv(equiv)-5 b(alen)n(t)27 b(represen)n(tations.)35 b(But)27 b(this)h(theorem)f(do)r (es)g(not)g(guaran)n(tee)f(that)h(the)h(domains)456 624 y(of)34 b(de\014nition)g(for)g(di\013eren)n(t)g(c)n(hoices)f(of)h(the)g (cyclic)g(v)n(ector)f(coincide.)56 b(W)-7 b(e)34 b(therefore)f(giv)n(e) 456 824 y(here)20 b(a)g(new)g(de\014nition)h(whic)n(h)f(dep)r(ends)h (only)f(on)g(the)h(represen)n(tation)d(but)j(not)g(on)f(the)h(cyclic) 456 1023 y(v)n(ector.)34 b(Its)23 b(restriction)g(to)g(the)h(cyclic)f (subspaces)f(coincides)h(with)h(the)g(previous)e(de\014nition.)555 1222 y(It)31 b(is)f(w)n(ell)g(kno)n(wn)f(that)i(the)g(op)r(erators)d Fu(')p Fv(\()p Fu(f)9 b Fv(\))31 b(\(no)n(w,)f(represen)n(tativ)n(es)e (under)i Fu(\031)3153 1234 y Fo(!)3232 1222 y Fv(of)g(the)456 1421 y(abstract)18 b(elemen)n(ts)h(of)g(the)h Fr(\003)p Fv(-algebra)d Fk(A)i Fv(in)h(subSection)f(2.1\))g(for)g(a)f(real)h(v)-5 b(alued)19 b(test)h(function)456 1621 y(are)h(essen)n(tially)g(self)h (adjoin)n(t)g(on)g(the)g(cyclic)g(domain)g(generated)f(b)n(y)h(\012) 2700 1633 y Fo(!)2748 1621 y Fv(,)h(and)f(that)h(the)f(W)-7 b(eyl)456 1820 y(op)r(erators)33 b Fu(W)12 b Fv(\()p Fu(f)d Fv(\))36 b(=)f(exp\()p Fu(i')p Fv(\()p Fu(f)9 b Fv(\))1527 1790 y Fq(\003\003)1600 1820 y Fv(\))35 b(\(where)g(no)n(w)g(the)h Fr(\003)p Fv(-op)r(eration)d(denotes)i(the)h (Hilb)r(ert)456 2019 y(space)28 b(adjoin)n(t\))h(satisfy)g(the)h(W)-7 b(eyl)30 b(relation)e Fu(W)12 b Fv(\()p Fu(f)d Fv(\))p Fu(W)j Fv(\()p Fu(g)s Fv(\))26 b(=)f(exp\()p Fr(\000)2674 1987 y Fo(i)p 2669 2001 34 4 v 2669 2048 a Fy(2)2712 2019 y Fu(E)5 b Fv(\()p Fu(f)29 b Fr(\012)19 b Fu(g)s Fv(\)\))p Fu(W)12 b Fv(\()p Fu(f)28 b Fv(+)19 b Fu(g)s Fv(\).)456 2218 y(The)27 b(exp)r(ectation)h(v)-5 b(alue)27 b(in)h(the)g(giv)n(en)f(Hadamard)f(state)i(is)1406 2557 y Fu(!)s Fv(\()p Fu(W)12 b Fv(\()p Fu(f)d Fv(\)\))24 b(=)e(exp\()p Fr(\000)2074 2501 y Fv(1)p 2074 2538 42 4 v 2074 2614 a(2)2126 2557 y Fu(!)2178 2569 y Fy(2)2215 2557 y Fv(\()p Fu(f)t(;)14 b(f)9 b Fv(\)\))28 b Fu(:)844 b Fv(\(9\))456 2896 y(Let)29 b(:)n Fu(W)12 b Fv(\()p Fu(f)d Fv(\))n(:)901 2849 y Fu(:)880 2896 y Fv(=)25 b(exp\()1139 2863 y Fy(1)p 1139 2877 34 4 v 1139 2925 a(2)1182 2896 y Fu(!)1234 2908 y Fy(2)1271 2896 y Fv(\()p Fu(f)t(;)14 b(f)9 b Fv(\)\))p Fu(W)j Fv(\()p Fu(f)d Fv(\),)30 b(and)f(de\014ne)g (for)g(\011)c Fr(2)g(H)2529 2908 y Fo(!)2607 2896 y Fv(the)k(v)n (ector-v)-5 b(alued)27 b(func-)456 3095 y(tion)g(\011\()p Fu(f)9 b Fv(\))23 b(=)k(:)9 b Fu(W)j Fv(\()p Fu(f)d Fv(\))g(:)24 b(\011)j(.)456 3470 y Fs(De\014nition)j(2.1.)40 b Ft(We)25 b(say)g(that)g Fv(\011\()p Fu(f)9 b Fv(\))24 b Ft(is)h Fv(in\014nitely)d(often)g(di\013eren)n(tiable)i Ft(at)h Fu(f)31 b Fv(=)23 b(0)h Ft(if)h(ther)l(e)456 3670 y(exists)37 b(for)h(every)h(inte)l(ger)e Fu(n)g Fr(\025)g Fv(0)g Ft(a)h(symmetric)l(al)h(ve)l(ctor-value)l(d)f(distribution)g Fu(\016)3158 3639 y Fo(n)3203 3670 y Fv(\011)p Fu(=\016)s(f)3400 3639 y Fo(n)456 3869 y Ft(on)d Fm(D)p Fv(\()p Fu(M)771 3839 y Fo(n)816 3869 y Fv(\))p Ft(,)j(and)d(c)l(ontinuous)g(seminorms)h Fu(p)1970 3881 y Fo(n)2050 3869 y Ft(on)g(the)f(test)g(function)g(sp)l (ac)l(e)h Fm(D)p Fv(\()p Fu(M)9 b Fv(\))36 b Ft(with)456 4068 y Fu(p)498 4080 y Fo(n)p Fy(+1)650 4068 y Fr(\025)22 b Fu(p)779 4080 y Fo(n)824 4068 y Ft(,)30 b(such)g(that)514 4318 y Fv(\(a\))41 b Ft(if)31 b Fu(p)784 4330 y Fy(0)821 4318 y Fv(\()p Fu(h)p Fv(\))23 b(=)g(0)29 b Ft(then)h Fv(\011\()p Fu(h)p Fv(\))23 b(=)g(\011\(0\))29 b Ft(,)509 4517 y Fv(\(b\))42 b Ft(if)31 b Fu(h)23 b Fr(!)g Fv(0)29 b Ft(with)h Fu(p)1212 4529 y Fo(n)1257 4517 y Fv(\()p Fu(h)p Fv(\))24 b Fr(6)p Fv(=)f(0)29 b Ft(then)1282 4711 y Fj(\015)1282 4761 y(\015)1282 4810 y(\015)1282 4860 y(\015)1282 4910 y(\015)1328 4856 y Fv(\011\()p Fu(h)p Fv(\))19 b Fr(\000)1646 4752 y Fo(n)1607 4777 y Fj(X)1614 4956 y Fo(l)p Fy(=0)1754 4800 y Fv(1)p 1750 4837 50 4 v 1750 4913 a Fu(l)r Fv(!)1820 4800 y Fu(\016)1860 4770 y Fo(l)1885 4800 y Fv(\011)p 1820 4837 131 4 v 1827 4913 a Fu(\016)s(f)1917 4889 y Fo(l)1960 4856 y Fv(\()p Fu(h)2040 4822 y Fq(\012)p Fo(l)2117 4856 y Fv(\))2149 4711 y Fj(\015)2149 4761 y(\015)2149 4810 y(\015)2149 4860 y(\015)2149 4910 y(\015)2239 4856 y Fu(p)2281 4868 y Fo(n)2326 4856 y Fv(\()p Fu(h)p Fv(\))2438 4822 y Fq(\000)p Fo(n)2559 4856 y Fr(\000)-15 b(!)24 b Fv(0)29 b Fu(;)661 5195 y Ft(wher)l(e)i Fr(k)47 b(\001)i(k)29 b Ft(stands)h(for)g(the)g(Hilb)l (ert)g(sp)l(ac)l(e)g(norm)g(in)g Fr(H)2538 5207 y Fo(!)2586 5195 y Ft(.)p eop %%Page: 22 22 22 21 bop 456 226 a Fy(22)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)555 425 y Fv(The)f(k)n(ernel)f(of)g (the)h(functional)g(deriv)-5 b(ativ)n(e)27 b(can)g(b)r(e)h(written,) 1400 708 y Fu(\016)1440 678 y Fo(n)1485 708 y Fv(\011)p 1170 745 611 4 v 1170 821 a Fu(\016)s(f)9 b Fv(\()p Fu(x)1339 833 y Fy(1)1376 821 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fu(\016)s(f)9 b Fv(\()p Fu(x)1702 833 y Fo(n)1748 821 y Fv(\))1813 764 y(=)23 b Fu(i)1930 730 y Fo(n)1998 764 y Fv(:)9 b Fu(')p Fv(\()p Fu(x)2163 776 y Fy(1)2201 764 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fu(')p Fv(\()p Fu(x)2491 776 y Fo(n)2537 764 y Fv(\))9 b(:)24 b(\011)j Fu(:)557 b Fv(\(10\))456 1103 y(The)27 b(righ)n(t)g(hand)h(side)f(of)h(Eqn.)f (\(10\))g(de\014nes)h(what)f(is)h(called)f(a)g(Wic)n(k)h(monomial.)555 1302 y(W)-7 b(e)32 b(w)n(an)n(t)e(to)h(\014nd)g(those)f(v)n(ectors)g (on)g(whic)n(h)h(the)h(Wic)n(k)e(monomials)g(can)h(b)r(e)g(restricted) 456 1501 y(to)23 b(partial)f(diagonals.)34 b(In)24 b(view)f(of)g(the)h (criterion)e Fs(M3)h Fv(for)g(the)h(restriction)e(of)i(distributions,) 456 1700 y(w)n(e)j(de\014ne)h(as)f(the)h Ft(micr)l(olo)l(c)l(al)j (domain)g(of)g(smo)l(othness)d Fv(the)g(follo)n(wing)e(set:)559 1991 y Fr(D)f Fv(=)736 1874 y Fj(\032)798 1991 y Fv(\011)e Fr(2)g(H)1034 2003 y Fo(!)1110 1991 y Fr(j)28 b Fv(\011\()p Fu(f)9 b Fv(\))27 b(is)h(in\014nitely)g(often)g(di\013eren)n(tiable)f (at)h Fu(f)j Fv(=)23 b(0,)k(and)g(for)h(ev)n(ery)798 2255 y Fu(n)23 b Fr(2)g Fn(N)28 b Fv(the)g(w)n(a)n(v)n(efron)n(t)d(set) j(of)1793 2199 y Fu(\016)1833 2169 y Fo(n)1878 2199 y Fv(\011)p 1793 2236 150 4 v 1800 2312 a Fu(\016)s(f)1890 2288 y Fo(n)1980 2255 y Fv(is)g(con)n(tained)e(in)i(the)g(set)798 2514 y Fr(f)p Fv(\()p Fu(x)919 2526 y Fy(1)957 2514 y Fu(;)14 b(k)1037 2526 y Fy(1)1074 2514 y Fv(;)g Fu(:)g(:)g(:)g Fv(;)g Fu(x)1306 2526 y Fo(n)1351 2514 y Fu(;)g(k)1431 2526 y Fo(n)1476 2514 y Fv(\))24 b Fr(2)1636 2493 y Fv(_)1610 2514 y Fu(T)1671 2479 y Fq(\003)1708 2514 y Fu(M)1798 2479 y Fo(n)1871 2514 y Fr(j)k Fu(k)1965 2526 y Fo(i)2015 2514 y Fr(2)p 2094 2447 67 4 v 24 w Fu(V)2161 2526 y Fq(\000)2217 2514 y Fu(;)41 b(i)23 b Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Fr(g)2739 2397 y Fj(\033)2828 2514 y Fu(:)3297 2448 y Fv(\(11\))555 2803 y(The)28 b(v)n(ector-v)-5 b(alued)27 b(distributions)h(\(10\))f(with)i(\011)23 b Fr(2)i(D)30 b Fv(can)e(b)r(e)g(restricted)g(to)g(all)f(partial)456 3002 y(diagonals,)36 b(and)g(giv)n(e)g(all)g(p)r(ossible)g(Wic)n(k)g(p) r(olynomials.)62 b(Moreo)n(v)n(er,)35 b(according)g(to)h Fs(M1)p Fv(,)456 3201 y(they)30 b(ma)n(y)g(also)f(b)r(e)i(m)n (ultiplied)g(b)n(y)f(distributions)h(whose)e(w)n(a)n(v)n(efron)n(t)g (sets)h(do)g(not)g(con)n(tain)456 3400 y(elemen)n(ts)35 b(\()p Fu(x)882 3412 y Fy(1)920 3400 y Fu(;)14 b(k)1000 3412 y Fy(1)1037 3400 y Fv(;)g Fu(:)g(:)g(:)g Fv(;)g Fu(x)1269 3412 y Fo(n)1314 3400 y Fu(;)g(k)1394 3412 y Fo(n)1439 3400 y Fv(\))36 b Fr(2)1624 3379 y Fv(_)1598 3400 y Fu(T)1659 3370 y Fq(\003)1697 3400 y Fu(M)1787 3370 y Fo(n)1831 3400 y Fv(,)i Fu(k)1935 3412 y Fo(i)1998 3400 y Fr(2)p 2089 3334 V 36 w Fu(V)2156 3412 y Fy(+)2211 3400 y Fu(;)49 b(i)35 b Fv(=)h(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Fv(.)59 b(The)35 b(domain)g Fr(D)j Fv(is)456 3600 y(in)n(v)-5 b(arian)n(t)26 b(under)i(application)e(of)i(W)-7 b(eyl)28 b(op)r(erators)d(and)j(smeared)e(Wic)n(k)i(p)r(olynomials.)555 3799 y(W)-7 b(e)28 b(state)g(a)f(crucial)g(prop)r(ert)n(y:)456 4322 y Fs(Lemma)k(2.2.)40 b Ft(L)l(et)23 b Fv(\010)g Fr(2)h(H)1353 4334 y Fo(!)1425 4322 y Ft(induc)l(e)f(some)i(quasi-fr)l (e)l(e)f(Hadamar)l(d)h(state)e Fu(!)2896 4292 y Fq(0)2919 4322 y Ft(.)37 b(Then)24 b Fv(\010)f Fr(2)h(D)r Ft(.)456 4846 y(Pr)l(o)l(of.)43 b Fv(The)27 b(main)h(p)r(oin)n(t)g(rests)f(on)g (the)h(v)-5 b(alidit)n(y)27 b(of)h(Leibniz)g(rule.)36 b(Indeed,)28 b(w)n(e)f(can)h(write,)813 5185 y(\010\()p Fu(f)9 b Fv(\))23 b(=)g(exp)o(\()1266 5129 y(1)p 1266 5166 42 4 v 1266 5242 a(2)1318 5185 y(\()p Fu(!)1402 5197 y Fy(2)1439 5185 y Fv(\()p Fu(f)t(;)14 b(f)9 b Fv(\))19 b Fr(\000)f Fu(!)1792 5151 y Fq(0)1789 5205 y Fy(2)1826 5185 y Fv(\()p Fu(f)t(;)c(f)9 b Fv(\)\))14 b(exp\()2237 5129 y(1)p 2237 5166 V 2237 5242 a(2)2289 5185 y Fu(!)2344 5151 y Fq(0)2341 5205 y Fy(2)2377 5185 y Fv(\()p Fu(f)t(;)g(f)9 b Fv(\))19 b(+)f Fu(i')p Fv(\()p Fu(f)9 b Fv(\))2872 5151 y Fq(\003\003)2944 5185 y Fv(\)\010)28 b Fu(;)p eop %%Page: 23 23 23 22 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(23)456 425 y Fv(and)27 b(di\013eren)n(tiate)h(w.r.t.)36 b Fu(f)9 b Fv(.)37 b(The)28 b(general)e Fu(n)p Fv(-th)h(order)g(deriv)-5 b(ativ)n(e)26 b(giv)n(es)908 708 y Fu(\016)948 678 y Fo(n)993 708 y Fv(\010)p 908 745 146 4 v 913 821 a Fu(\016)s(f)1003 797 y Fo(n)1063 764 y Fv(\()p Fu(h)1143 730 y Fq(\012)p Fo(n)1240 764 y Fv(\))d(=)1627 685 y Fj(X)1383 867 y Fo(I)5 b Fq(\022f)p Fy(1)p Fo(;:::)o(;n)p Fq(g)p Fo(;)p Fq(j)p Fo(I)g Fq(j)26 b Fv(ev)n(en)2006 764 y Fu(\037)p Fv(\()p Fu(h;)14 b(h)p Fv(\))2255 730 y Fq(j)p Fo(I)5 b Fq(j)p Fo(=)p Fy(2)2437 708 y Fu(\016)2477 678 y Fq(j)p Fo(I)2531 653 y Fh(c)2562 678 y Fq(j)2586 708 y Fv(\010)2646 678 y Fq(0)p 2437 745 232 4 v 2454 822 a Fu(\016)s(f)2544 798 y Fq(j)p Fo(I)2598 782 y Fh(c)2628 798 y Fq(j)2679 764 y Fv(\()p Fu(h)2759 730 y Fq(\012j)p Fo(I)2865 705 y Fh(c)2896 730 y Fq(j)2919 764 y Fv(\))28 b Fu(;)295 b Fv(\(12\))456 1108 y(where,)33 b(\010)785 1078 y Fq(0)808 1108 y Fv(\()p Fu(f)9 b Fv(\))31 b(=)f(exp\()1217 1075 y Fy(1)p 1217 1089 34 4 v 1217 1136 a(2)1260 1108 y Fu(!)1315 1078 y Fq(0)1312 1128 y Fy(2)1349 1108 y Fv(\()p Fu(f)t(;)14 b(f)9 b Fv(\))22 b(+)f Fu(i')p Fv(\()p Fu(f)9 b Fv(\))1850 1078 y Fq(\003\003)1922 1108 y Fv(\)\010,)34 b Fu(\037)d Fv(=)f Fu(!)2301 1120 y Fy(2)2360 1108 y Fr(\000)21 b Fu(!)2501 1078 y Fq(0)2498 1128 y Fy(2)2567 1108 y Fv(is)32 b(a)g(smo)r(oth)g(function)h(on)456 1307 y Fu(M)546 1277 y Fy(2)610 1307 y Fv(and)28 b(a)f(solution)g(of)g(the)h(Klein-Gordon)e (equation)h(in)h(b)r(oth)g(en)n(tries.)555 1506 y(No)n(w,)i(\010)830 1476 y Fq(0)853 1506 y Fv(\()p Fu(h)p Fv(\))g(satis\014es)f(the)h (estimate)f(in)h(De\014nition)g(2.1)f(with)h(the)g(seminorms)e Fu(p)3196 1476 y Fq(0)3196 1527 y Fo(n)3241 1506 y Fv(\()p Fu(h)p Fv(\))f(=)456 1635 y Fj(p)p 539 1635 287 4 v 71 x Fu(!)594 1677 y Fq(0)591 1728 y Fy(2)628 1706 y Fv(\()p Fu(h;)14 b(h)p Fv(\))41 b(for)g(all)h Fu(n)p Fv(,)i(and)e(the)g(n)n (umerical)e(prefactor)h(with)h(the)g(seminorms)e Fu(p)3176 1718 y Fo(n)3221 1706 y Fv(\()p Fu(h)p Fv(\))47 b(=)456 1834 y Fj(p)p 539 1834 542 4 v 71 x Fv(\()p Fu(!)623 1917 y Fy(2)678 1905 y Fv(+)18 b Fu(!)816 1876 y Fq(0)813 1927 y Fy(2)850 1905 y Fv(\)\()p Fu(h;)c(h)p Fv(\))q(,)32 b(hence)f(for)f(the)i(whole)e(expression)g(w)n(e)h(ma)n(y)f(also)g(use) h(the)g(seminorms)456 2104 y Fu(p)498 2116 y Fo(n)543 2104 y Fv(.)45 b(Actually)-7 b(,)32 b(as)d(w)n(as)h(sho)n(wn)g(b)n(y)g (V)-7 b(erc)n(h)30 b(in)h([57)o(],)g(there)f(exist)h(t)n(w)n(o)f(p)r (ositiv)n(e)g(constan)n(ts)f Fu(A)456 2303 y Fv(and)e Fu(B)t Fv(,)h(suc)n(h)f(that)992 2642 y Fu(A!)1106 2654 y Fy(2)1144 2642 y Fv(\()p Fu(f)t(;)14 b(g)s Fv(\))23 b Fr(\024)f Fu(!)1498 2608 y Fq(0)1495 2663 y Fy(2)1532 2642 y Fv(\()p Fu(f)t(;)14 b(g)s Fv(\))23 b Fr(\024)g Fu(B)t(!)1951 2654 y Fy(2)1988 2642 y Fv(\()p Fu(f)t(;)14 b(g)s Fv(\))28 b Fu(;)180 b(f)t(;)14 b(g)25 b Fr(2)f Fm(D)p Fv(\()p Fu(M)9 b Fv(\))27 b Fu(;)456 2981 y Fv(hence)j(all)f (these)h(seminorms)f(are)g(equiv)-5 b(alen)n(t.)44 b(W)-7 b(e)31 b(conclude)e(that)h(\010\()p Fu(f)9 b Fv(\))31 b(is)f(in\014nitely)g(dif-)456 3180 y(feren)n(tiable)d(at)g Fu(f)32 b Fv(=)22 b(0.)555 3379 y(The)37 b(w)n(a)n(v)n(e)f(fron)n(t)g (sets)h(of)g(the)g(functional)g(deriv)-5 b(ativ)n(es)36 b(of)h(\010\()p Fu(f)9 b Fv(\))37 b(and)g(\010)2956 3349 y Fq(0)2980 3379 y Fv(\()p Fu(f)9 b Fv(\))37 b(coincide,)456 3578 y(since)27 b Fu(\037)h Fv(is)f(smo)r(oth.)37 b(Using)27 b(the)h(form)n(ula)1383 3797 y Fj(\015)1383 3847 y(\015)1383 3896 y(\015)1383 3946 y(\015)1439 3861 y Fu(\016)1479 3831 y Fo(n)1524 3861 y Fv(\010)1584 3831 y Fq(0)p 1439 3898 169 4 v 1456 3974 a Fu(\016)s(f)1546 3950 y Fo(n)1617 3917 y Fv(\()p Fu(h)1697 3883 y Fq(\012)p Fo(n)1794 3917 y Fv(\))1826 3797 y Fj(\015)1826 3847 y(\015)1826 3896 y(\015)1826 3946 y(\015)1873 3817 y Fy(2)1933 3917 y Fv(=)23 b Fu(n)p Fv(!)k Fu(!)2176 3883 y Fq(0)2173 3938 y Fy(2)2210 3917 y Fv(\()p Fu(h;)14 b(h)p Fv(\))2407 3883 y Fo(n)2480 3917 y Fu(;)794 b Fv(\(13\))456 4256 y(and)27 b(the)h(information)f(on)g(the)h(w)n(a)n(v)n(e)e(fron)n(t)h (set)h(of)g(Hadamard)e(states,)h(w)n(e)g(\014nd)657 4595 y(WF)810 4478 y Fj(\022)881 4539 y Fu(\016)921 4508 y Fo(n)967 4539 y Fv(\010)p 881 4576 146 4 v 886 4652 a Fu(\016)s(f)976 4628 y Fo(n)1037 4478 y Fj(\023)1121 4595 y Fv(=)22 b Fr(f)p Fv(\()p Fu(x)1329 4607 y Fy(1)1367 4595 y Fu(;)14 b(k)1447 4607 y Fy(1)1484 4595 y Fv(;)g Fu(:)g(:)g(:)g Fv(;)g Fu(x)1716 4607 y Fo(n)1761 4595 y Fu(;)g(k)1841 4607 y Fo(n)1887 4595 y Fv(\))23 b Fr(2)2046 4574 y Fv(_)2020 4595 y Fu(T)2081 4560 y Fq(\003)2119 4595 y Fu(M)2209 4560 y Fo(n)2281 4595 y Fr(j)28 b Fu(k)2375 4607 y Fo(i)2426 4595 y Fr(2)23 b Fu(@)5 b(V)2601 4607 y Fq(\000)2657 4595 y Fu(;)42 b(i)23 b Fv(=)f(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Fr(g)27 b Fu(;)3297 4794 y Fv(\(14\))456 5062 y(so)g(\010)c Fr(2)g(D)r Fv(.)2572 b Fl(\004)p eop %%Page: 24 24 24 23 bop 456 226 a Fy(24)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)555 425 y Fv(W)-7 b(e)35 b(use)f(the)h(so)e(called)h(\(generalized\))f(Wic)n(k)i(expansion)e (form)n(ula)g(whic)n(h)h(is)g(the)h(basic)456 624 y(com)n(binatorial)23 b(form)n(ula)i(for)h(p)r(erturbation)f(theory)-7 b(.)35 b(Let)26 b(us)g(denote)g(b)n(y)f Fm(L)h Fv(an)n(y)f(Wic)n(k)h(p)r(oly-) 456 824 y(nomial)37 b(on)g Fr(D)42 b(\032)e(H)1144 836 y Fo(!)1192 824 y Fv(,)g(and)e(w)n(e)f(de\014ne)h(the)g(\\deriv)-5 b(ativ)n(e")36 b(of)h(a)g(Wic)n(k)h(p)r(olynomial)f(with)456 1023 y(resp)r(ect)27 b(to)g Fu(')h Fv(to)g(b)r(e)g Fu(@)5 b Fm(L)p Fu(=@)g(')p Fv(.)36 b(W)-7 b(e)28 b(can)g(c)n(haracterize)d (it)j(b)n(y)f(the)h(follo)n(wing)f(result:)456 1348 y Fs(Lemma)k(2.3.)40 b Ft(Ther)l(e)31 b(is)g(a)g(unique)f(Wick)h(p)l (olynomial)i Fu(@)5 b Fm(L)p Fu(=@)g(')30 b Ft(which)i(satis\014es)e (the)h(e)l(qua-)456 1547 y(tion,)1410 1886 y Fv([)p Fm(L)p Fv(\()p Fu(x)p Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(y)s Fv(\)])25 b(=)1948 1829 y Fu(@)5 b Fm(L)p 1948 1866 109 4 v 1951 1943 a Fu(@)g(')2066 1886 y Fv(\()p Fu(x)p Fv(\))p Fu(E)g Fv(\()p Fu(x;)14 b(y)s Fv(\))32 b Fu(;)807 b Fv(\(15\))456 2224 y Ft(in)29 b(the)h(sense)g(of)h(op)l(er)l(ator-value)l(d)g (distributions)f(on)g Fr(D)r Ft(.)456 2549 y(Pr)l(o)l(of.)43 b Fv(By)27 b(linearit)n(y)g(it)i(is)f(su\016cien)n(t)g(to)g(pro)n(v)n (e)e(it)i(for)g(an)n(y)f(Wic)n(k)h(monomial)f(e.g.)37 b(:)9 b Fu(')3231 2519 y Fo(n)3277 2549 y Fv(\()p Fu(x)p Fv(\))g(:)q(.)456 2748 y(It)28 b(is)g(ob)n(vious,)e(that)i Fu(n)p Fv(:)9 b Fu(')1269 2718 y Fo(n)p Fq(\000)p Fy(1)1400 2748 y Fv(\()p Fu(x)p Fv(\))g(:)29 b(satis\014es)e(\(15\),)h(hence)g(w) n(e)f(only)h(need)g(to)f(pro)r(of)h(that)g(if)g Fu(A)456 2947 y Fv(is)23 b(an)n(y)h(Wic)n(k)f(p)r(olynomial)h(for)f(whic)n(h)h Fu(A)p Fv(\()p Fu(x)p Fv(\))p Fu(E)5 b Fv(\()p Fu(x;)14 b(y)s Fv(\))25 b(=)d(0)i(this)g(means)g(that)g Fu(A)f Fr(\021)g Fv(0.)35 b(But)24 b(this)456 3147 y(follo)n(ws)29 b(from)h(the)g(fact)h(that)f(for)g(ev)n(ery)f Fu(x)f Fr(2)g Fu(M)39 b Fv(w)n(e)29 b(can)h(\014nd)h(some)f(test)g(function)h Fu(f)39 b Fv(suc)n(h)456 3346 y(that)31 b(the)h(\(smo)r(oth\))f (solution)g Fu(E)5 b Fv(\()p Fu(x;)14 b(f)9 b Fv(\))32 b(of)f(the)g(Klein-Gordon)f(equation)h(do)r(es)f(not)i(v)-5 b(anish)456 3545 y(at)27 b Fu(x)p Fv(.)2753 b Fl(\004)555 3826 y Fv(No)n(w,)36 b(let)f(us)g(consider,)h(for)e(an)n(y)g(Wic)n(k)h (p)r(olynomial)f Fm(L)p Fv(,)j(the)e(\014elds)g Fm(L)2877 3796 y Fy(\()p Fo(j)s Fy(\))2999 3826 y Fv(=)g Fu(@)3148 3796 y Fo(j)3182 3826 y Fm(L)p Fu(=@)5 b(')3386 3796 y Fo(j)3421 3826 y Fv(,)456 4026 y Fu(j)28 b Fr(2)23 b Fn(N)p Fv(,)i(whic)n(h,)g(b)n(y)g(induction,)g(are)f(uniquely)g (de\014ned)h(according)e(to)h(the)h(previous)f(Lemma.)456 4350 y Fs(Theorem)31 b(2.4)g Fv(\(Generalized)c(Wic)n(k)g(expansion)g (Theorem\))p Fs(.)40 b Ft(L)l(et)f Fm(L)2735 4362 y Fo(k)2776 4350 y Ft(,)i Fu(k)h Fv(=)c(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Ft(,)41 b(b)l(e)456 4550 y(Wick)30 b(p)l(olynomials.)41 b(The)31 b(fol)t(lowing)h(r)l(elation)e(holds:)603 4840 y Fm(L)662 4852 y Fy(1)700 4840 y Fv(\()p Fu(x)779 4852 y Fy(1)816 4840 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fm(L)1032 4852 y Fo(n)1078 4840 y Fv(\()p Fu(x)1157 4852 y Fo(n)1203 4840 y Fv(\))23 b(=)1399 4762 y Fj(X)1346 4938 y Fo(j)1373 4946 y FC(1)1406 4938 y Fo(;:::)o(;j)1532 4946 y Fh(n)1572 4840 y Fv(\(\012)1664 4852 y Fo(!)1713 4840 y Fu(;)14 b Fm(L)1809 4797 y Fy(\()p Fo(j)1862 4805 y FC(1)1895 4797 y Fy(\))1809 4863 y(1)1925 4840 y Fv(\()p Fu(x)2004 4852 y Fy(1)2042 4840 y Fv(\))g Fr(\001)g(\001)g(\001)g Fm(L)2258 4806 y Fy(\()p Fo(j)2311 4814 y Fh(n)2352 4806 y Fy(\))2258 4861 y Fo(n)2382 4840 y Fv(\()p Fu(x)2461 4852 y Fo(n)2507 4840 y Fv(\)\012)2599 4852 y Fo(!)2648 4840 y Fv(\))k Fr(\002)2401 5143 y(\002)2494 5087 y Fv(:)9 b Fu(')2580 5057 y Fo(j)2607 5065 y FC(1)2644 5087 y Fv(\()p Fu(x)2723 5099 y Fy(1)2761 5087 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fu(')2972 5057 y Fo(j)2999 5065 y Fh(n)3045 5087 y Fv(\()p Fu(x)3124 5099 y Fo(n)3170 5087 y Fv(\))9 b(:)p 2494 5124 741 4 v 2703 5200 a Fu(j)2737 5212 y Fy(1)2775 5200 y Fv(!)14 b Fr(\001)g(\001)g(\001)f Fu(j)2956 5212 y Fo(n)3002 5200 y Fv(!)3274 5143 y Fu(;)3297 5185 y Fv(\(16\))p eop %%Page: 25 25 25 24 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(25)456 425 y Ft(wher)l(e)27 b(the)f(summations)g(over)h(the)g Fu(j)1629 437 y Fo(k)1670 425 y Ft('s)f(go)h(fr)l(om)g(0)g(to)f(the)g(or)l(der)h(of)h(the)e(c)l (orr)l(esp)l(onding)i Fm(L)3378 437 y Fo(k)3419 425 y Ft(.)555 832 y Fv(F)-7 b(or)26 b(a)g(pro)r(of)g(see)g([34)o(])h(or)e (just)i(use)g(the)f(previous)g(notion)g(of)g(di\013eren)n(tiabilit)n(y) g(and)h(apply)456 1031 y(induction.)52 b(Note)33 b(that)g(the)g(pro)r (ducts)g(in)g(the)g(Theorem)f(ab)r(o)n(v)n(e)f(exist)i(b)r(ecause)f (the)h(w)n(a)n(v)n(e)456 1231 y(fron)n(t)f(sets)h(of)h(their)f(exp)r (ectation)g(v)-5 b(alues)33 b(satisfy)g(H\177)-42 b(ormander)32 b(criterion)g Fs(M1)g Fv(due)i(to)f(the)456 1430 y(con)n(v)n(exit)n(y) 26 b(of)h(the)h(forw)n(ard)e(ligh)n(t)h(cone.)555 1629 y(The)d(w)n(a)n(v)n(e)d(fron)n(t)i(sets)g(for)g(the)g(Wigh)n(tman)h 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b(criterion,)i(and)g(its)g(w)n(a)n(v)n(e)e(fron)n(t)i (set)g(is)456 2625 y(con)n(tained)d(in)h(\(WF)q(\()p Fu(!)1177 2637 y Fy(2)1214 2625 y Fv(\))12 b(+)g(WF\()p Fu(!)1558 2637 y Fy(2)1595 2625 y Fv(\)\))g Fr([)g Fv(WF)q(\()p Fu(!)1962 2637 y Fy(2)1999 2625 y Fv(\))2075 2578 y Fu(:)2054 2625 y Fv(=)23 b Fn(z)2207 2637 y Fy(2)2244 2625 y Fv(.)35 b(The)25 b(set)f Fn(z)2661 2637 y Fy(2)2722 2625 y Fv(will)g(b)r(e)h (instrumen)n(tal)456 2825 y(for)c(some)h(results)g(b)r(elo)n(w.)34 b(No)n(w,)23 b(WF)q(\()p Fu(!)1736 2837 y Fy(2)1773 2825 y Fv(\))8 b(+)g(WF\()p Fu(!)2109 2837 y Fy(2)2146 2825 y Fv(\))23 b(con)n(tains)e(directions)h(whic)n(h)g(lie)g Ft(inside)456 3024 y Fv(the)32 b(ligh)n(t-cone)g(as)f(it)i(is)g(clear)e (b)n(y)h(adding)g(up)h(t)n(w)n(o)e(co)n(v)n(ectors)f Fu(k)2564 3036 y Fy(1)2624 3024 y Fv(+)21 b Fu(k)2753 3036 y Fy(2)2823 3024 y Fv(for)32 b(p)r(oin)n(ts)g(on)g(the)456 3223 y(diagonal.)46 b(One)31 b(th)n(us)g(sees)g(ho)n(w)f(already)g(the) h(smallest)g(p)r(ossible)g(non-linearit)n(y)f(ma)n(y)g(giv)n(e)456 3422 y(rise)d(to)g(additional)g(singular)g(directions)g(w.r.t.)36 b(those)28 b(already)e(presen)n(t)h(in)h(the)g(w)n(a)n(v)n(e)e(fron)n (t)456 3622 y(sets)19 b(of)g(the)h(Wigh)n(tman)f(functions)h(for)f(the) g(original)f(\014eld)i Fu(')p Fv(.)34 b(Another)20 b(imp)r(ortan)n(t)f (remark)f(is)456 3821 y(that)k Fn(z)695 3833 y Fy(2)753 3821 y Fv(is)g(an)g(in)n(v)n(olutiv)n(e)e(closed)i(cone,)g(i.e.)35 b(is)22 b(a)f(closed)g(cone)h(whic)n(h)g(is)f(stable)h(under)g(sums,) 456 4020 y(and,)27 b(as)g(a)g(straigh)n(tforw)n(ard)e(result,)i(it)h (giv)n(es)f(that)h(\()p Fu(!)2201 4032 y Fy(2)2238 4020 y Fv(\()p Fu(x;)14 b(y)s Fv(\)\))2462 3990 y Fo(n)2535 4020 y Fv(still)28 b(has)f(WF)q(\()p Fu(!)3072 3990 y Fo(n)3069 4041 y Fy(2)3117 4020 y Fv(\))c Fr(\032)g Fn(z)3325 4032 y Fy(2)3362 4020 y Fv(.)555 4219 y(The)36 b(general)f(structure)g (for)h(m)n(ulti-p)r(oin)n(t)g(exp)r(ectation)g(v)-5 b(alues)36 b(of)f(Wic)n(k)h(p)r(olynomials)456 4419 y(can)25 b(b)r(e)i(found)f(as) f(follo)n(ws.)35 b(F)-7 b(or)26 b(a)f(more)h(compact)f(notation)g(some) h(de\014nitions)g(from)g(graph)456 4618 y(theory)d(are)g(used:)35 b(Let)24 b Fr(G)1262 4630 y Fo(n)1331 4618 y Fv(denote)g(the)g(set)g (of)g(all)f(\014nite)i(nonorien)n(ted)e(graphs)f(with)j(v)n(ertices)456 4817 y Fu(V)43 b Fv(=)25 b Fr(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)26 b(;)14 b(n)p Fr(g)28 b Fv(and)h(let)g Fu(E)1388 4787 y Fo(G)1473 4817 y Fv(denote)f(the)i(set)e(of)h(edges)f(of)h(a)f(giv)n (en)g(graph)g Fu(G)p Fv(.)40 b(Moreo)n(v)n(er,)456 5016 y(for)34 b(an)n(y)h(v)n(ertex)f Fu(i)i Fr(2)g Fu(V)54 b Fv(w)n(e)35 b(denote)g(b)n(y)g Fu(E)1866 4986 y Fo(G)1861 5038 y(i)1957 5016 y Fv(the)h(subset)f(of)g(edges)g(whic)n(h)g(b)r (elong)g(to)g(the)456 5216 y(v)n(ertex)23 b Fu(i)p Fv(,)i(p)r(ossibly)f (empt)n(y)-7 b(,)26 b(and)e(b)n(y)h Fr(j)p Fu(E)1725 5185 y Fo(G)1720 5237 y(i)1781 5216 y Fr(j)f Fv(their)h(n)n(um)n(b)r (er)f(and)h(similarly)f(b)n(y)g Fu(E)2997 5185 y Fo(G)2992 5237 y(ij)3077 5216 y Fv(the)h(subset)p eop %%Page: 26 26 26 25 bop 456 226 a Fy(26)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(of)e(edges)g(connecting)g (p)r(oin)n(ts)g Fu(i)g Fv(and)h Fu(j)5 b Fv(,)27 b(with)h(the)g(ob)n (vious)e(relation)h Fr(j)p Fu(E)2769 395 y Fo(G)2764 447 y(i)2825 425 y Fr(j)c Fv(=)2959 363 y Fj(P)3046 450 y Fo(j)3095 425 y Fr(j)p Fu(E)3184 395 y Fo(G)3179 447 y(ij)3240 425 y Fr(j)p Fv(.)37 b(F)-7 b(or)456 624 y(an)n(y)26 b(edge)h Fu(e)c Fr(2)g Fu(E)33 b Fv(connecting)27 b(p)r(oin)n(ts)h Fu(i)f Fv(and)g Fu(j)33 b Fv(w)n(e)27 b(use)g(the)h(\\source)e(and)h (range")f(notation,)456 824 y(i.e.)37 b Fu(i)22 b Fv(=)h Fu(s)p Fv(\()p Fu(e)p Fv(\))28 b(and)f Fu(j)h Fv(=)23 b Fu(r)r Fv(\()p Fu(e)p Fv(\),)29 b(whenev)n(er)d Fu(i)d(<)f(j)5 b Fv(.)555 1023 y(It)36 b(is)g(su\016cien)n(t,)i(b)n(y)e(linearit)n(y) -7 b(,)37 b(to)f(restrict)f(ourselv)n(es)f(to)i(the)g(treatmen)n(t)g 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Fu(j)34 b Fv(with)c Fu(m)2057 1832 y Fo(j)2122 1820 y Fv(edges)e(are)h(saturated,)f(i.e.)43 b Fr(j)p Fu(E)3118 1790 y Fo(G)3113 1842 y(j)3174 1820 y Fr(j)26 b Fv(=)f Fu(m)3386 1832 y Fo(j)3421 1820 y Fv(.)456 2019 y(Moreo)n(v)n(er,)d (follo)n(wing)h([8)o(],)i(w)n(e)f(call)g(a)g(triple)g(\()p Fu(x;)14 b(\015)5 b(;)14 b(k)s Fv(\))24 b(an)g Ft(immersion)h Fv(of)f(an)n(y)g(graph)f Fu(G)g Fr(\032)g(G)3399 2031 y Fo(n)456 2218 y Fv(in)n(to)36 b(the)h(manifold)g Fu(M)45 b Fv(whenev)n(er,)38 b(\()p Fu(a)p Fv(\))g Fu(x)g Fv(:)h Fu(V)57 b Fr(!)38 b Fu(M)45 b Fv(is)37 b(a)f(map)h(from)f(all)h(v)n (ertices)e Fu(i)i Fv(of)456 2418 y Fu(G)f Fv(to)g(p)r(oin)n(ts)g Fu(x)972 2430 y Fo(i)1036 2418 y Fv(of)g Fu(M)9 b Fv(;)40 b(\()p Fu(b)p Fv(\))c Fu(\015)41 b Fv(maps)35 b(edges)g Fu(e)i Fr(2)h Fu(E)2202 2388 y Fo(G)2294 2418 y Fv(to)d(n)n(ull)i(geo)r (desics)d Fu(\015)2985 2430 y Fo(e)3057 2418 y Fv(connecting)456 2617 y(p)r(oin)n(ts)d Fu(x)756 2632 y Fo(s)p Fy(\()p Fo(e)p Fy(\))906 2617 y Fv(and)g 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Fo(m)2079 4791 y FC(1)2111 4783 y Fo(;:::)o(;m)2269 4791 y Fh(n)2017 4833 y Fo(n)2343 4813 y Ft(is)f(ge)l(ometric)l(al)t(ly)i(b)l(ounde)l(d)e(as)1537 5152 y Fv(WF)q(\()p Fu(!)1764 5117 y Fo(m)1823 5125 y FC(1)1855 5117 y Fo(;:::)o(;m)2013 5125 y Fh(n)1761 5172 y Fo(n)2057 5152 y Fv(\))24 b Fr(\032)e Fn(z)2265 5164 y Fo(n)2340 5152 y Fu(:)934 b Fv(\(18\))p eop %%Page: 27 27 27 26 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(27)456 425 y Ft(Pr)l(o)l(of.)43 b Fv(Let)27 b(us)g(de\014ne)h(for)e(notational)h(purp)r(ose)2058 404 y Fj(b)2044 425 y Fr(G)2093 437 y Fo(n)2183 378 y Fu(:)2162 425 y Fv(=)2263 404 y Fj(b)2250 425 y Fr(G)2299 437 y Fo(n)2344 425 y Fv(\()p Fu(m)2449 437 y Fy(1)2487 425 y Fu(;)14 b(:)g(:)g(:)f(;)h(m)2744 437 y Fo(n)2789 425 y Fv(\).)37 b(As)28 b(follo)n(ws)e(from)456 624 y(Theorem)g(2.4)h (the)h(exp)r(ectation)g(v)-5 b(alue)27 b(is,)628 963 y Fu(!)683 929 y Fo(m)742 937 y FC(1)774 929 y Fo(;:::)11 b(;m)944 937 y Fh(n)680 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Fo(n)1164 1723 y Fv(\))1240 1676 y Fu(:)1219 1723 y Fv(=)1318 1644 y Fj(Y)1307 1822 y Fo(e)p Fq(2)p Fo(G)1449 1723 y Fu(!)1501 1735 y Fy(2)1538 1723 y Fv(\()p Fu(x)1617 1738 y Fo(s)p Fy(\()p Fo(e)p Fy(\))1736 1723 y Fu(;)g(x)1820 1738 y Fo(r)r Fy(\()p Fo(e)p Fy(\))1941 1723 y Fv(\))1219 2020 y(=)23 b Fu(!)1359 2032 y Fy(2)1396 2020 y Fv(\()p Fu(x)1475 2032 y Fy(1)1513 2020 y Fu(;)14 b(x)1597 2032 y Fy(2)1634 2020 y Fv(\))1666 1986 y Fq(j)p Fo(E)1738 1960 y Fh(G)1735 2002 y FC(1)p Fh(;)p FC(2)1815 1986 y Fq(j)1839 2020 y Fu(!)1891 2032 y Fy(2)1928 2020 y Fv(\()p Fu(x)2007 2032 y Fy(1)2045 2020 y Fu(;)g(x)2129 2032 y Fy(3)2166 2020 y Fv(\))2198 1986 y Fq(j)p Fo(E)2270 1960 y Fh(G)2267 2002 y FC(1)p Fh(;)p FC(3)2347 1986 y Fq(j)2384 2020 y Fr(\001)g(\001)g(\001)g Fu(!)2547 2032 y Fy(2)2584 2020 y Fv(\()p Fu(x)2663 2032 y Fo(n)p Fq(\000)p Fy(1)2794 2020 y Fu(;)g(x)2878 2032 y Fo(n)2923 2020 y Fv(\))2955 1986 y Fq(j)p Fo(E)3027 1960 y Fh(G)3024 2002 y(n)p Ff(\000)p FC(1)p Fh(;n)3194 1986 y Fq(j)3218 2020 y Fu(:)456 2289 y Fv(In)26 b(the)g(last)g(equalit)n(y)g(the)g(2-p) r(oin)n(t)g(distributions)g(should)g(b)r(e)g(understo)r(o)r(d)g(as)g (distributions)456 2488 y(on)h(the)h(pro)r(duct)f(manifold)h Fu(M)1459 2458 y Fo(n)1504 2488 y Fv(.)37 b(Hence)28 b(their)f(w)n(a)n(v)n(e)f(fron)n(t)h(set)h(is)f(giv)n(en,)g(when)h Fu(i)23 b(<)f(j)5 b Fv(,)28 b(b)n(y)-7 b(,)456 2827 y(WF\()p Fu(!)682 2774 y Fq(j)p Fo(E)754 2749 y Fh(G)751 2790 y(i;j)823 2774 y Fq(j)679 2849 y Fy(2)847 2827 y Fv(\))23 b Fr(\032)g(f)p Fv(\()p Fu(x)1111 2839 y Fy(1)1148 2827 y Fu(;)14 b Fv(0;)g Fu(:)g(:)g(:)f Fv(;)h Fu(x)1458 2839 y Fo(i)1486 2827 y Fu(;)g(k)1566 2839 y Fo(i;j)1644 2827 y Fv(;)g Fu(:)g(:)g(:)g Fv(;)g Fu(x)1876 2839 y Fo(j)1911 2827 y Fu(;)g Fr(\000)p Fu(k)2056 2839 y Fo(i;j)2134 2827 y Fv(;)g Fu(:)g(:)g(:)g Fv(;)g Fu(x)2366 2839 y Fo(n)2411 2827 y Fu(;)g Fv(0\))p Fr(j)p Fv(\()p Fu(x)2624 2839 y Fo(i)2652 2827 y Fu(;)g(k)2732 2839 y Fo(i;j)2811 2827 y Fv(;)g Fu(x)2895 2839 y Fo(j)2930 2827 y Fu(;)g Fr(\000)p Fu(k)3075 2839 y Fo(i;j)3153 2827 y Fv(\))23 b Fr(2)h Fn(z)3352 2839 y Fy(2)3389 2827 y Fr(g)p Fu(:)555 3166 y Fv(It)i(is)f(straigh)n(tforw)n(ard)e(to)i(see)g(from)g(the)h (last)f(expression)f(that)h(the)h(claim)f(of)g(the)h(Prop)r(o-)456 3365 y(sition)h(is)h(correct.)35 b(Indeed)28 b(a)f(general)f(co)n(v)n (ector)g Fu(k)2069 3377 y Fo(i)2124 3365 y Fv(will)i(ha)n(v)n(e)e(the)i (follo)n(wing)f(expression,)1125 3704 y Fu(k)1168 3716 y Fo(i)1219 3704 y Fv(=)22 b Fr(\000)p Fu(k)1414 3716 y Fy(1)p Fo(;i)1513 3704 y Fr(\000)c(\001)c(\001)g(\001)k(\000)g Fu(k)1837 3716 y Fo(i)p Fq(\000)p Fy(1)p Fo(;i)2012 3704 y Fv(+)g Fu(k)2138 3716 y Fo(i;i)p Fy(+1)2311 3704 y Fv(+)h Fr(\001)14 b(\001)g(\001)k Fv(+)g Fu(k)2636 3716 y Fo(i;n)2752 3704 y Fu(;)522 b Fv(\(20\))456 4043 y(where)31 b(some)f(of)i(the)g Fu(k)1200 4055 y Fo(l;m)1335 4043 y Fv(ma)n(y)f(b)r(e)h(zero.)47 b(No)n(w,)32 b(as)f(follo)n(ws)f(from)h (Eqn.)g(\(8\),)i(to)e(an)n(y)g(edge)456 4242 y Fu(e)e 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4678 y Fv(=)979 4574 y Fq(1)952 4599 y Fj(X)952 4777 y Fo(k)q Fy(=0)1097 4621 y Fv(\()p Fu(i\025)p Fv(\))1238 4591 y Fo(k)p 1097 4658 183 4 v 1153 4735 a Fu(k)s Fv(!)1303 4565 y Fj(Z)1349 4753 y Fi(M)1422 4678 y Fr(\001)14 b(\001)g(\001)1533 4565 y Fj(Z)1579 4753 y Fi(M)1652 4678 y Fu(T)e Fv(\()p Fm(L)1804 4690 y Fo(int)1897 4678 y Fv(\()p Fu(x)1976 4690 y Fy(1)2014 4678 y Fv(\))p Fm(L)2105 4690 y Fo(int)2199 4678 y Fv(\()p Fu(x)2278 4690 y Fy(2)2316 4678 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)2532 4690 y Fo(int)2626 4678 y Fv(\()p Fu(x)2705 4690 y Fo(k)2747 4678 y Fv(\)\))2861 4574 y Fo(k)2826 4599 y Fj(Y)2825 4776 y Fo(i)p Fy(=1)2947 4678 y Fu(d)2990 4643 y Fy(4)3027 4678 y Fu(x)3074 4690 y Fo(i)3130 4678 y Fu(:)456 5016 y Fv(Hence,)22 b(the)f(p)r(erturbativ) n(e)e(solution)h(to)g(scattering)g(theory)g(is)g(reduced)g(to)g (quadratures)f(once)456 5216 y(one)i(\014nds)i(the)f(general)f (solution)h(of)g(the)h(time)g(ordering)d(op)r(eration)h(inside)i(the)f (in)n(tegral.)34 b(F)-7 b(or)p eop %%Page: 29 29 29 28 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(29)456 425 y Fv(noncoinciding)26 b(p)r(oin)n(ts)i(the)g(solution)f(is)h(giv)n(en)e(b)n(y)i(the)g(follo)n (wing)e(expression,)503 716 y Fu(T)12 b Fv(\()p Fm(L)655 728 y Fo(int)749 716 y Fv(\()p Fu(x)828 728 y Fy(1)866 716 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)1082 728 y Fo(int)1176 716 y Fv(\()p Fu(x)1255 728 y Fo(n)1301 716 y Fv(\)\))23 b(=)851 886 y Fj(X)763 1066 y Fo(\031)r Fq(2)p Fd(P)896 1074 y FC(1)p Fh(;:::)o(;n)1073 965 y Fu(\022)r Fv(\()p Fu(x)1193 980 y Fo(\031)r Fy(\(1\))1343 965 y Fr(\000)18 b Fu(x)1473 980 y Fo(\031)r Fy(\(2\))1603 965 y Fv(\))c Fr(\001)g(\001)g(\001)g Fu(\022)r Fv(\()p Fu(x)1880 980 y Fo(\031)r Fy(\()p Fo(n)p Fq(\000)p Fy(1\))2122 965 y Fr(\000)k Fu(x)2252 980 y Fo(\031)r Fy(\()p Fo(n)p Fy(\))2391 965 y Fv(\))p Fm(L)2482 977 y Fo(int)2576 965 y Fv(\()p Fu(x)2655 980 y Fo(\031)r Fy(\(1\))2786 965 y Fv(\))c Fr(\001)g(\001)g(\001)g Fm(L)3002 977 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y(where,)25 b(ho)n(w)n(ev)n(er,)e(the)j(exp)r (ectation)f(v)-5 b(alue)25 b(in)g(the)h(righ)n(t-hand)e(side)h(is)g Ft(a)j(priori)f Fv(not)e(de\014ned)456 1733 y(all)i(o)n(v)n(er)e Fn(M)826 1703 y Fo(n)872 1733 y Fv(,)i(but)h(only)f(o)n(v)n(er)f Fn(M)1512 1703 y Fo(n)1562 1733 y Fr(n)5 b Fv(\006)27 b(where)g(\006)g(is)g(the)h(union)g(of)f(all)g(diagonals)f(in)h Fn(M)3228 1703 y Fo(n)3274 1733 y Fv(.)37 b(So,)456 1932 y(the)27 b(main)g(problem)g(is)g(to)g(giv)n(e)f(a)h(mathematical)g (meaning)f(to)h(this)h(form)n(ula)e(on)h(all)f(p)r(oin)n(ts.)456 2745 y(3.1.)46 b Fs(F)-8 b(orm)m(ulation)31 b(of)f(the)h(lo)s(cal)e Fu(S)5 b Fs(-matrix.)41 b Fv(The)27 b(general)e(starting)h(idea,)g(due) h(to)f(Bo-)456 2944 y(goliub)r(o)n(v,)g(is)h(to)h(consider)e(the)i (usual)f(Gell-Mann)g(and)g(Lo)n(w)g(form)n(ulation)f(of)i(the)g Fu(S)5 b Fv(-matrix)456 3143 y(but)24 b(supplemen)n(ted)f(b)n(y)g(the)h (h)n(yp)r(othesis)f(necessary)e(to)j(the)f(implemen)n(tation)h(of)f (the)h(causal-)456 3343 y(it)n(y)g(principle.)36 b(In)25 b(one)g(strok)n(e)e(one)h(\014nds)h(also)f(the)h(solution)g(to)f(the)i (problem)e(of)h(the)g(correct)456 3542 y(treatmen)n(t)37 b(of)g(the)h(op)r(erator-v)-5 b(alued)35 b(distributions.)65 b(Cho)r(osing)37 b(as)f(the)i(in)n(teraction)e(La-)456 3741 y(grangian)g Fm(L)867 3753 y Fo(int)961 3741 y Fv(\()p Fu(x)p Fv(\))41 b(=)f Fm(L)p Fv(\()p Fu(x)p Fv(\))p Fu(\021)s Fv(\()p Fu(x)p Fv(\),)k(a)37 b(Wic)n(k)h(p)r(olynomial)f Fm(L)i Fv(m)n(ultiplied)g(b)n(y)f(a)f(space-time)456 3940 y(function)j(of)f(compact)g(supp)r(ort)h Fu(\021)46 b Fr(2)e Fm(D)p Fv(\()p Fu(M)9 b Fv(\))39 b(\(considered)g(as)g(a)g (generalized)f(\\coupling)456 4140 y(constan)n(t"\),)c(w)n(e)g Ft(de\014ne)g Fv(the)g(lo)r(cal)g Fu(S)5 b Fv(-matrix)33 b Fu(S)2033 4152 y Fo(\025)2076 4140 y Fv(\()p Fu(\021)s Fv(\))i(as)e(a)h(formal)f(p)r(o)n(w)n(er)g(series)f(\(see,)k(for)456 4339 y(instance,)27 b([6]\))h(in)f(the)h(coupling)g(strength)f Fu(\025)p Fv(,)798 4678 y Fu(S)849 4690 y Fo(\025)893 4678 y Fv(\()p Fu(\021)s Fv(\))1045 4631 y Fu(:)1025 4678 y Fv(=)22 b Fs(1)d Fv(+)1291 4574 y Fq(1)1264 4599 y Fj(X)1262 4775 y Fo(n)p Fy(=1)1410 4621 y Fv(\()p Fu(i\025)p Fv(\))1551 4591 y Fo(n)p 1410 4658 187 4 v 1467 4735 a Fu(n)p Fv(!)1621 4565 y Fj(Z)1667 4753 y Fo(M)1736 4737 y Fh(n)1795 4678 y Fu(T)12 b Fv(\()p Fm(L)1947 4690 y Fo(int)2041 4678 y Fv(\()p Fu(x)2120 4690 y Fy(1)2158 4678 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)2374 4690 y Fo(int)2468 4678 y Fv(\()p Fu(x)2547 4690 y Fo(n)2593 4678 y Fv(\)\))p Fu(d\026)2750 4690 y Fy(1)2802 4678 y Fr(\001)g(\001)g(\001)f Fu(d\026)3005 4690 y Fo(n)3078 4678 y Fu(;)196 b Fv(\(23\))456 5016 y(where)23 b Fu(d\026)h Fv(is)g(the)g(natural)f(in)n(v)-5 b(arian)n(t)23 b(v)n(olume)g(measure) g(on)g(the)i(globally)d(h)n(yp)r(erb)r(olic)i(space-)456 5216 y(time)37 b(\()p Fu(M)t(;)14 b(g)s Fv(\),)40 b(and)d Fs(1)g Fv(is)g(the)h(Hilb)r(ert)f(space)g(iden)n(tit)n(y)g(op)r (erator.)64 b(One)37 b(can)f(enlarge)g(the)p eop %%Page: 31 31 31 30 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(31)456 425 y Fv(de\014nition)28 b(e.g.)36 b(b)n(y)27 b(using)h(a)f(more)g(general)f(Lagrangian,)1610 764 y Fu(\025)p Fm(L)1717 776 y Fo(int)1835 764 y Fv(=)1972 660 y Fo(l)1923 685 y Fj(X)1922 864 y Fo(k)q Fy(=1)2057 764 y Fu(\021)2098 776 y Fo(k)2139 764 y Fm(L)2198 776 y Fo(k)2267 764 y Fu(;)1007 b Fv(\(24\))456 1103 y(with)28 b(Wic)n(k)g(p)r(olynomials)f Fm(L)1381 1115 y Fo(k)1450 1103 y Fv(and)h(asso)r(ciating)e(to)i(eac)n(h)f(a)h(di\013eren)n(t)g (\\coupling)f(constan)n(t")456 1302 y Fu(\021)497 1314 y Fo(k)564 1302 y Fr(2)f Fm(D)p Fv(\()p Fu(M)9 b Fv(\),)30 b(where)e(the)i(additional)f(\\Lagrangians")c Fm(L)2312 1314 y Fo(k)2383 1302 y Fv(are)j(de\014ned)i(as)e(terms)h(lik)n(e)g (cur-)456 1501 y(ren)n(ts,)24 b(external)f(\014elds)h(etc.,)h (including)f(in)g(particular)f(all)g(deriv)-5 b(ativ)n(es)23 b(of)h(the)g(basic)g(in)n(terac-)456 1700 y(tion)h(Lagrangian.)34 b(Using)25 b(this)g(extended)h(Lagrangian)d(w)n(e)i(ma)n(y)g(replace)f (the)i(time-ordered)456 1900 y(op)r(erator)f(in)j(Eqn.)f(\(23\))h(b)n (y)-7 b(,)564 2238 y Fu(T)12 b Fv(\()p Fm(L)716 2250 y Fo(int)810 2238 y Fv(\()p Fu(x)889 2250 y Fy(1)927 2238 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)1143 2250 y Fo(int)1237 2238 y Fv(\()p Fu(x)1316 2250 y Fo(n)1362 2238 y Fv(\)\))23 b(=)1598 2160 y Fj(X)1537 2338 y Fo(k)1572 2346 y FC(1)1605 2338 y Fo(;:::)o(;k)1739 2346 y Fh(n)1793 2238 y Fu(T)12 b Fv(\()p Fm(L)1945 2250 y Fo(k)1980 2258 y FC(1)2017 2238 y Fv(\()p Fu(x)2096 2250 y Fy(1)2134 2238 y Fv(\))i Fr(\001)g(\001)g(\001)f Fm(L)2349 2250 y Fo(k)2384 2258 y Fh(n)2430 2238 y Fv(\()p Fu(x)2509 2250 y Fo(n)2555 2238 y Fv(\)\))p Fu(\021)2660 2250 y Fo(k)2695 2258 y FC(1)2733 2238 y Fv(\()p Fu(x)2812 2250 y Fy(1)2850 2238 y Fv(\))h Fr(\001)g(\001)g(\001)f Fu(\021)3047 2250 y Fo(k)3082 2258 y Fh(n)3128 2238 y Fv(\()p Fu(x)3207 2250 y Fo(n)3253 2238 y Fv(\))28 b Fu(;)456 2577 y Fv(where)h(the)i (summation)f(o)n(v)n(er)e(the)j Fu(k)s Fv('s)f(go)f(from)h(1)g(to)g Fu(l)r Fv(,)g(the)h(n)n(um)n(b)r(er)f(of)g(the)h(terms)f(in)g(the)456 2776 y(extended)d(Lagrangian.)555 2976 y(W)-7 b(e)24 b(remark)d(that,)k(ev)n(en)n(tually)-7 b(,)23 b(the)g(test)h (function\(s\))g Fu(\021)i Fv(should)d(b)r(e)h(sen)n(t)f(to)g(a)f (\014xed)i(v)-5 b(alue)456 3175 y(o)n(v)n(er)21 b(all)i(space-time.)35 b(This)23 b(pro)r(cedure,)h(kno)n(wn)e(as)h(adiabatic)f(limit,)j(amoun) n(ts)e(to)g(treat)g(the)456 3374 y(infrared)29 b(nature)h(of)g(the)g (theory)-7 b(.)44 b(Some)30 b(studies)g(of)g(this)h(limit)g(in)f(the)g (case)g(of)g(Mink)n(o)n(wski)456 3573 y(space-time)h(ha)n(v)n(e)f(b)r (een)i(p)r(erformed)f(b)n(y)h(Epstein)f(and)h(Glaser)e(themselv)n(es)i ([21)o(].)49 b(It)32 b(is)f(not)456 3773 y(clear)39 b(ho)n(w)g(to)h (generalize)f(their)h(study)h(to)f(curv)n(ed)f(spaces.)74 b(It)40 b(is)g(therefore)g(gratifying)456 3972 y(that)d(all)f(lo)r(cal) g(prop)r(erties)g(of)g(the)h(theory)f(are)g(already)f(obtained)h(via)g (the)h(construction)456 4171 y(of)32 b(the)h(lo)r(cal)e Fu(S)5 b Fv(-matrices,)33 b(and)f(this)h(p)r(oin)n(t)f(of)g(view)h (migh)n(t)f(also)f(b)r(e)i(useful)g(in)f(cases)g(\(lik)n(e)456 4370 y(non)i(ab)r(elian)h(gauge)e(theories\))i(where)f(due)h(to)g (infrared)f(problems)g(the)h Fu(S)5 b Fv(-matrix)34 b(in)h(the)456 4570 y(adiabatic)26 b(limit)j(do)r(es)e(not)h(prop)r(erly)e(exist.)p eop %%Page: 32 32 32 31 bop 456 226 a Fy(32)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(3.2.)46 b Fs(De\014ning)g(prop)s(erties.)40 b Fv(Our)f(main)h(goal)e(is)i(the)g (inductiv)n(e)g(construction)f(of)h(the)456 624 y(time-ordered)26 b(pro)r(ducts)h(of)h(Wic)n(k)f(p)r(olynomials,)1546 963 y Fu(T)12 b Fv(\()p Fm(L)1698 975 y Fy(1)1735 963 y Fv(\()p Fu(x)1814 975 y Fy(1)1852 963 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)2068 975 y Fo(n)2114 963 y Fv(\()p Fu(x)2193 975 y Fo(n)2238 963 y Fv(\)\))29 b Fu(:)456 1302 y Fv(F)-7 b(ollo)n(wing)26 b(Epstein)h(and)h(Glaser)e(w)n(e)i(require)e(the)i (follo)n(wing)f(prop)r(erties:)473 1613 y Fs(P1.)41 b(W)-8 b(ell-p)s(osedness)p Fv(.)49 b(The)33 b(sym)n(b)r(ols)66 b Fu(T)12 b Fv(\()p Fm(L)2043 1625 y Fy(1)2080 1613 y Fv(\()p Fu(x)2159 1625 y Fy(1)2197 1613 y Fv(\))i Fr(\001)g(\001)g (\001)g Fm(L)2413 1625 y Fo(n)2458 1613 y Fv(\()p Fu(x)2537 1625 y Fo(n)2583 1613 y Fv(\)\))34 b(are)e(w)n(ell)h(de\014ned)g(op)r (e-)661 1812 y(rator-v)-5 b(alued)38 b(distributions)i(on)f(the)i (GNS-Hilb)r(ert)f(space)g Fr(H)2727 1824 y Fo(!)2815 1812 y Fv(i.e.)73 b(\(m)n(ultilinear,)661 2011 y(strongly)35 b(con)n(tin)n(uous\))g(maps)h Fm(D)p Fv(\()p Fu(M)1864 1981 y Fo(n)1909 2011 y Fv(\))h Fr(!)g Fc(End)n Fv(\()p Fr(D)r Fv(\))g(where)f Fr(D)j(\032)d(H)2918 2023 y Fo(!)3003 2011 y Fv(is)f(the)i(dense)661 2211 y(subspace)27 b(\(microlo)r(cal)g (domain)g(of)g(smo)r(othness\))h(de\014ned)g(in)f(\(11\).)473 2522 y Fs(P2.)41 b(Symmetry)p Fv(.)74 b(An)n(y)40 b(time-ordered)f(pro) r(duct)80 b Fu(T)12 b Fv(\()p Fm(L)2392 2534 y Fy(1)2429 2522 y Fv(\()p Fu(x)2508 2534 y Fy(1)2546 2522 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)2762 2534 y Fo(n)2807 2522 y Fv(\()p Fu(x)2886 2534 y Fo(n)2932 2522 y Fv(\)\))41 b(is)f(symmet-)661 2721 y(ric)33 b(under)g(p)r(erm)n(utations)g(of)g (indices,)h(i.e.)54 b(the)33 b(action)g(of)g(the)h(p)r(erm)n(utation)f (group)661 2920 y Fm(P)716 2935 y Fq(f)p Fy(1)p Fo(;:::)o(;n)p Fq(g)989 2920 y Fv(of)27 b(the)h(index)g(set)g Fr(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)e(;)i(n)p Fr(g)27 b Fv(giv)n(es,)1030 3259 y Fu(T)12 b Fv(\()p Fm(L)1182 3274 y Fo(\031)r Fy(\(1\))1311 3259 y Fv(\()p Fu(x)1390 3274 y Fo(\031)r Fy(\(1\))1521 3259 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)1737 3274 y Fo(\031)r Fy(\()p Fo(n)p Fy(\))1875 3259 y Fv(\()p Fu(x)1954 3274 y Fo(\031)r Fy(\()p Fo(n)p Fy(\))2093 3259 y Fv(\)\))24 b Fr(\021)e Fu(T)12 b Fv(\()p Fm(L)2420 3271 y Fy(1)2457 3259 y Fv(\()p Fu(x)2536 3271 y Fy(1)2574 3259 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)2790 3271 y Fo(n)2836 3259 y Fv(\()p Fu(x)2915 3271 y Fo(n)2961 3259 y Fv(\)\))28 b Fu(;)661 3597 y Fv(for)f(an)n(y)g Fu(\031)g Fr(2)c Fm(P)1152 3612 y Fq(f)p Fy(1)p Fo(;:::)o(;n)p Fq(g)1397 3597 y Fv(,)k(in)h(the)g(sense)f(of)h(distributions.)555 3908 y(This)38 b(symmetry)f(prop)r(ert)n(y)f(corresp)r(onds)g(to)h(the) h(fact)g(that)g(the)g(time-ordered)e(pro)r(d-)456 4108 y(ucts)29 b(are)f(functional)h(deriv)-5 b(ativ)n(es)28 b(of)h(the)g(lo)r(cal)g Fu(S)5 b Fv(-matrix.)40 b(More)28 b(crucial)g(is)h(the)h(follo)n(wing)456 4307 y(causalit)n(y)c(prop)r (ert)n(y)-7 b(,)27 b(whic)n(h)g(follo)n(ws)g(from)g(the)h(Eqn.)f (\(2\);)473 4618 y Fs(P3.)41 b(Causalit)m(y)p Fv(.)d(Consider)27 b(an)n(y)h(set)g(of)g(p)r(oin)n(ts)f(\()p Fu(x)2163 4630 y Fy(1)2201 4618 y Fu(;)14 b(:)g(:)g(:)g(;)g(x)2433 4630 y Fo(n)2479 4618 y Fv(\))24 b Fr(2)g Fu(M)2704 4588 y Fo(n)2777 4618 y Fv(and)j(an)n(y)h(full)g(parti-)661 4817 y(tion)j(of)f(the)g(set)h Fr(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)e(;)i(n)p Fr(g)30 b Fv(in)n(to)g(t)n(w)n(o)f(non)h(empt)n (y)h(subsets)f Fu(I)37 b Fv(and)30 b Fu(I)2920 4787 y Fo(c)2984 4817 y Fv(suc)n(h)g(that)h(no)661 5016 y(p)r(oin)n(t)24 b Fu(x)921 5028 y Fo(i)972 5016 y Fv(with)g Fu(i)f Fr(2)g Fu(I)30 b Fv(is)24 b(in)f(the)h(past)f(of)g(the)h(p)r(oin)n(ts)f Fu(x)2361 5028 y Fo(j)2420 5016 y Fv(with)h Fu(j)k Fr(2)23 b Fu(I)2788 4986 y Fo(c)2822 5016 y Fv(,)h(i.e.)36 b Fu(x)3058 5028 y Fo(i)3118 5016 y Fu(=)-51 b Fr(2)23 b Fu(J)3241 4986 y Fq(\000)3297 5016 y Fv(\()p Fu(x)3376 5028 y Fo(j)3412 5016 y Fv(\))661 5216 y(for)j(an)n(y)g Fu(i)c Fr(2)i Fu(I)33 b Fv(and)26 b Fu(j)i Fr(2)c Fu(I)1486 5185 y Fo(c)1520 5216 y Fv(.)36 b(Then)27 b(the)g(time-ordered)e (distributions)h(are)f(required)h(to)p eop %%Page: 33 33 33 32 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(33)661 425 y Fv(satisfy)28 b(the)g(follo)n(wing)e(factorization)g(prop)r(ert)n(y)1084 764 y Fu(T)12 b Fv(\()p Fm(L)1236 776 y Fy(1)1273 764 y Fv(\()p Fu(x)1352 776 y Fy(1)1390 764 y Fv(\))i Fr(\001)g(\001)g (\001)g Fm(L)1606 776 y Fo(n)1651 764 y Fv(\()p Fu(x)1730 776 y Fo(n)1776 764 y Fv(\)\))24 b(=)f Fu(T)12 b Fv(\()2045 685 y Fj(Y)2047 863 y Fo(i)p Fq(2)p Fo(I)2164 764 y Fm(L)2223 776 y Fo(i)2251 764 y Fv(\()p Fu(x)2330 776 y Fo(i)2358 764 y Fv(\)\))28 b Fu(T)12 b Fv(\()2560 685 y Fj(Y)2543 863 y Fo(j)s Fq(2)p Fo(I)2652 847 y Fh(c)2697 764 y Fm(L)2756 776 y Fo(j)2792 764 y Fv(\()p Fu(x)2871 776 y Fo(j)2906 764 y Fv(\)\))29 b Fu(:)555 1103 y Fv(In)40 b(the)h(Epstein)f(and)f (Glaser)g(sc)n(heme)h(on)g(Mink)n(o)n(wski)e(space-time)i(one)f (requires,)j(in)456 1302 y(addition,)36 b(translation)d(co)n(v)-5 b(ariance)33 b(of)h(the)h(time-ordered)e(pro)r(ducts.)58 b(If)35 b(the)f(free)h(\014eld)g(is)456 1501 y(among)d(the)h(p)r (ossible)g(terms)f(in)i(the)f(Lagrangian)e(one)h(can)h(sho)n(w)f(that)h (the)h(time-ordered)456 1700 y(pro)r(ducts)41 b(are)f(sums)i(of)f(p)r (oin)n(t)n(wise)g(pro)r(ducts)g(of)h(Wic)n(k)f(p)r(olynomials)g(with)h (translation)456 1900 y(in)n(v)-5 b(arian)n(t)18 b(n)n(umerical)i (distributions.)34 b(\(Suc)n(h)20 b(pro)r(ducts)g(exist)g(due)g(to)g (Theorem)f(0)h(of)g(Epstein)456 2099 y(and)28 b(Glaser,)f(an)h(easy)f (pro)r(of)h(of)g(whic)n(h)g(follo)n(ws)g(from)g(our)f(microlo)r(cal)g (c)n(haracterization)f(of)456 2298 y(the)i(domain)f(of)g(de\014nition)h (of)g(Wic)n(k)f(pro)r(ducts.\))37 b(Moreo)n(v)n(er,)25 b(the)j(condition,)563 2637 y([)p Fu(T)12 b Fv(\()p Fm(L)738 2649 y Fy(1)775 2637 y Fv(\()p Fu(x)854 2649 y Fy(1)892 2637 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)1108 2649 y Fo(n)1153 2637 y Fv(\()p Fu(x)1232 2649 y Fo(n)1278 2637 y Fv(\)\))p Fu(;)g(')p Fv(\()p Fu(y)s Fv(\)])24 b(=)1715 2533 y Fo(n)1676 2558 y Fj(X)1682 2735 y Fo(i)p Fy(=1)1810 2637 y Fu(T)12 b Fv(\()p Fm(L)1962 2649 y Fy(1)1999 2637 y Fv(\()p Fu(x)2078 2649 y Fy(1)2116 2637 y Fv(\))i Fr(\001)g(\001)g(\001)2282 2581 y Fu(@)5 b Fm(L)2390 2593 y Fo(i)p 2282 2618 136 4 v 2299 2694 a Fu(@)g(')2428 2637 y Fv(\()p Fu(x)2507 2649 y Fo(i)2535 2637 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fm(L)2751 2649 y Fo(n)2797 2637 y Fv(\()p Fu(x)2876 2649 y Fo(n)2922 2637 y Fv(\)\))p Fu(E)5 b Fv(\()p Fu(x)3131 2649 y Fo(i)3160 2637 y Fu(;)14 b(y)s Fv(\))27 b Fu(;)3297 2836 y Fv(\(25\))456 3106 y(\014xes)39 b(the)h(co)r(e\016cien)n(ts)g(to)f(b)r(e)i(v)-5 b(acuum)39 b(exp)r(ectation)h(v)-5 b(alues)39 b(of)h(time-ordered)f (pro)r(ducts)456 3305 y(of)32 b(those)g(Wic)n(k)g(p)r(olynomials)g (whic)n(h)g(are)f(of)h(lo)n(w)n(er)f(order)g(w.r.t.)51 b(the)33 b(c)n(hosen)e(in)n(teracting)456 3504 y(Lagrangian)j(\(from)j (no)n(w)g(on,)j(w)n(e)c(shall)h(call)g(them)h Ft(sub)p Fv(-Wic)n(k)f(p)r(olynomials\),)i(hence)e(the)456 3703 y(problem)25 b(is)i(reduced)f(to)g(a)g(problem)g(for)g(n)n(umerical)f (distributions.)37 b(Unfortunately)-7 b(,)26 b(in)h(the)456 3903 y(case)21 b(of)h(a)g(curv)n(ed)f(space-time,)i(w)n(e)f(ha)n(v)n(e) f(not)h(y)n(et)g(determined)h(the)f(class)g(of)g(\014elds)g(whic)n(h)g (are)456 4102 y(relativ)n(ely)29 b(lo)r(cal)i(to)g(the)h(scalar)d(free) i(\014eld,)h(i.e.,)h(what)e(is)g(kno)n(wn)f(in)i(the)f(literature)g(as) f(the)456 4301 y(Borc)n(hers')e(class)j([22)o(].)47 b(One)31 b(also)f(needs)h(a)f(replacemen)n(t)g(of)h(the)h(condition)f(of)g (translation)456 4500 y(co)n(v)-5 b(ariance.)59 b(Our)35 b(idea)g(is)h(to)f(imp)r(ose)h(a)f(condition)h(on)f(the)h(time-ordered) f(distributions)456 4700 y(whic)n(h)h(in)g(a)g(sense)g(emplo)n(ys)f(b)r (oth)i(ideas)f(of)g(in)n(v)-5 b(ariance)35 b(and)h(sp)r(ectralit)n(y)f (crucial)h(in)g(the)456 4899 y(Mink)n(o)n(wskian)f(case.)64 b(Since,)39 b(as)d(emphasized)h(in)g(the)g(previous)f(section,)j(sp)r (ectralit)n(y)d(for)456 5098 y(us)e(means)g(w)n(a)n(v)n(e)e(fron)n(t)i (sets)g(prop)r(erties)f(w)n(e)h(no)n(w)g(lo)r(ok)f(for)h(a)g(condition) g(whic)n(h)g(\014xes)g(the)p eop %%Page: 34 34 34 33 bop 456 226 a Fy(34)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(prop)r(erties)f(of)g(the)i (singularities)d(of)i(the)g(time-ordered)f(distributions.)40 b(W)-7 b(e)30 b(use)e(the)i(graph)456 624 y(theoretic)d(de\014nitions)g (of)h(Section)g(2.)473 884 y Fs(P4.)41 b(Sp)s(ectralit)m(y)p Fv(.)55 b(F)-7 b(or)33 b(the)h(exp)r(ectation)g(v)-5 b(alue)33 b Fu(t)2195 896 y Fo(n)2273 884 y Fr(2)h Fm(D)2431 854 y Fq(0)2454 884 y Fv(\()p Fu(M)2576 854 y Fo(n)2621 884 y Fv(\),)i Fu(n)d Fr(\025)f Fv(2,)j(of)e(an)n(y)g(time-)661 1083 y(ordered)27 b(pro)r(duct)g(it)h(holds,)1775 1422 y(WF\()p Fu(t)1976 1434 y Fo(n)2022 1422 y Fv(\))23 b Fr(\032)g Fv(\000)2217 1388 y Fy(to)2217 1442 y Fo(n)2308 1422 y Fu(;)456 1761 y Fv(where)769 2035 y(\000)821 2001 y Fy(to)821 2056 y Fo(n)907 2035 y Fv(=)995 1918 y Fj(\032)1057 2035 y Fv(\()p Fu(x)1136 2047 y Fy(1)1174 2035 y Fu(;)14 b(k)1254 2047 y Fy(1)1291 2035 y Fv(;)g Fu(:)g(:)g(:)g Fv(;)g Fu(x)1523 2047 y Fo(n)1569 2035 y Fu(;)g(k)1649 2047 y Fo(n)1694 2035 y Fv(\))23 b Fr(2)1853 2014 y Fv(_)1828 2035 y Fu(T)1889 2001 y Fq(\003)1926 2035 y Fu(M)2016 2001 y Fo(n)2144 2035 y Fr(j)83 b(9)g Fv(a)27 b(graph)g Fu(G)c Fr(2)g(G)2899 2047 y Fo(n)2973 2035 y Fv(and)1057 2299 y(an)k(immersion)g(\()p Fu(x;)14 b(\015)5 b(;)14 b(k)s Fv(\))28 b(of)g Fu(G)g Fv(in)g(whic)n(h)f Fu(k)2446 2311 y Fo(e)2510 2299 y Fv(is)g(future)h(directed)1057 2549 y(whenev)n(er)f Fu(x)1469 2564 y Fo(s)p Fy(\()p Fo(e)p Fy(\))1620 2549 y Fu(=)-51 b Fr(2)23 b Fu(J)1743 2514 y Fq(\000)1800 2549 y Fv(\()p Fu(x)1879 2564 y Fo(r)r Fy(\()p Fo(e)p Fy(\))1999 2549 y Fv(\))28 b(and)g(suc)n(h)f(that,)1057 2798 y Fu(k)1100 2810 y Fo(i)1151 2798 y Fv(=)1326 2719 y Fj(X)1239 2901 y Fo(m)p Fy(:)p Fo(s)p Fy(\()p Fo(m)p Fy(\)=)p Fo(i)1547 2798 y Fu(k)1590 2810 y Fo(m)1653 2798 y Fv(\()p Fu(x)1732 2810 y Fo(i)1760 2798 y Fv(\))19 b Fr(\000)1964 2719 y Fj(X)1894 2901 y Fo(n)p Fy(:)p Fo(r)r Fy(\()p Fo(n)p Fy(\)=)p Fo(i)2168 2798 y Fu(k)2211 2810 y Fo(n)2256 2798 y Fv(\()p Fu(x)2335 2810 y Fo(i)2364 2798 y Fv(\))2396 2681 y Fj(\033)2486 2798 y Fu(:)555 3129 y Fv(This)35 b(ma)n(y)e(b)r(e)i(motiv)-5 b(ated)35 b(b)n(y)f(the)g(fact)h(that,)h(for)e(non)h(coinciding)e(p)r(oin)n(ts,)j Fu(t)3120 3141 y Fo(n)3200 3129 y Fv(can)e(b)r(e)456 3328 y(expressed)22 b(in)h(terms)g(of)g(the)g(usual)g(F)-7 b(eynman)23 b(graphs)f(and)h(for)f(the)i(set)f(of)g(coinciding)g(p)r (oin)n(ts)456 3527 y(w)n(e)34 b(ha)n(v)n(e)f(an)h(in\014nitesimal)h (remnan)n(t)f(of)h(translation)e(in)n(v)-5 b(ariance,)35 b(since)f(all)h(co)n(v)n(ectors)d(at)456 3727 y(coinciding)27 b(p)r(oin)n(ts)g(sum)h(up)g(to)f(zero.)555 3926 y(W)-7 b(e)26 b(can)f(no)n(w)f(form)n(ulate)h(a)g(microlo)r(cal)f(v)n(ersion)f (of)j(Theorem)e(0)h(of)g(Epstein)g(and)g(Glaser.)456 4323 y Fs(Theorem)31 b(3.1)g Fv(\(Microlo)r(cal)26 b(Theorem)h(0\))p Fs(.)41 b Ft(If)30 b Fs(P4)g Ft(holds)h(for)f Fu(t)2550 4335 y Fo(n)2625 4323 y Ft(then)1309 4661 y Fu(t)1339 4673 y Fo(n)1384 4661 y Fv(\()p Fu(x)1463 4673 y Fy(1)1501 4661 y Fu(;)14 b(:)g(:)g(:)g(;)g(x)1733 4673 y Fo(n)1778 4661 y Fv(\):)9 b Fu(')1896 4627 y Fo(l)1917 4635 y FC(1)1955 4661 y Fv(\()p Fu(x)2034 4673 y Fy(1)2072 4661 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fu(')2283 4627 y Fo(l)2304 4635 y Fh(n)2349 4661 y Fv(\()p Fu(x)2428 4673 y Fo(n)2474 4661 y Fv(\))9 b(:)30 b Fu(;)456 5000 y Ft(is)j(a)g(wel)t(l)h(de\014ne) l(d)g(op)l(er)l(ator-value)l(d)g(distribution,)h(for)f(any)f Fu(n)g Ft(and)g(any)h(choic)l(e)g(of)g(indic)l(es)456 5199 y Fu(l)481 5211 y Fy(1)518 5199 y Fu(;)14 b(:)g(:)g(:)f(;)h(l)727 5211 y Fo(n)772 5199 y Ft(,)30 b(on)g(the)g(dense)g(invariant)h(domain) g Fr(D)h Ft(in)e(the)g(Hilb)l(ert)f(sp)l(ac)l(e)i Fr(H)2865 5211 y Fo(!)2913 5199 y Ft(.)p eop %%Page: 35 35 35 34 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(35)456 425 y Ft(Pr)l(o)l(of.)43 b Fv(Let)34 b(\011)f Fr(2)h(D)r Fv(.)56 b(The)34 b(v)n(ector-v)-5 b(alued)33 b(distribution)h(:)9 b Fu(')2442 395 y Fo(l)2463 403 y FC(1)2500 425 y Fv(\()p Fu(x)2579 437 y Fy(1)2617 425 y Fv(\))14 b Fr(\001)g(\001)g(\001)g Fu(')2828 395 y Fo(l)2849 403 y Fh(n)2894 425 y Fv(\()p Fu(x)2973 437 y Fo(n)3019 425 y Fv(\))9 b(:\011)34 b(is)g(a)f(re-)456 624 y(striction)i(of)g Fu(\016)934 594 y Fo(l)960 624 y Fv(\011)p Fu(=\016)s(f)1157 594 y Fo(l)1181 624 y Fv(,)j Fu(l)g Fv(=)1406 562 y Fj(P)1494 583 y Fo(n)1494 649 y(i)p Fy(=1)1619 624 y Fu(l)1644 636 y Fo(i)1672 624 y Fv(,)f(to)f(a)f(partial)g(diagonal)f(with)i(w)n(a)n(v)n(e)e(fron)n(t) i(set)f(con-)456 824 y(tained)40 b(in)832 761 y Fj(S)901 849 y Fo(x)p Fq(2)p Fo(M)1053 832 y Fh(n)1098 824 y Fr(f)p Fu(x)p Fr(g)26 b(\002)p 1347 757 67 4 v 27 w Fu(V)1414 771 y Fq(\002)p Fo(n)1414 844 y Fq(\000)1511 824 y Fv(.)75 b(Since)41 b(\000)1891 794 y Fy(to)1891 844 y Fo(n)1994 824 y Fv(do)r(es)f(not)h(con)n(tain)e(elemen)n(ts)i(of)f(the)h(form)456 1023 y(\()p Fu(x)535 1035 y Fy(1)573 1023 y Fu(;)14 b(k)653 1035 y Fy(1)690 1023 y Fu(;)g(:)g(:)g(:)g(;)g(x)922 1035 y Fo(n)967 1023 y Fu(;)g(k)1047 1035 y Fo(n)1092 1023 y Fv(\))26 b(with)h Fu(k)1381 1035 y Fo(i)1431 1023 y Fr(2)p 1510 956 V 24 w Fu(V)1577 1035 y Fy(+)1632 1023 y Fv(,)f Fu(i)d Fv(=)f(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Fv(,)26 b(the)g(pro)r(duct)g(is)f(a)h(w)n(ell)f(de\014ned)h(v)n(ector-) 456 1222 y(v)-5 b(alued)29 b(distribution,)h(and)g(after)f(smearing)f (with)i(some)f(test)h(function)g(one)g(obtains)f(again)456 1421 y(a)e(v)n(ector)f(in)i Fr(D)r Fv(.)2420 b Fl(\004)555 1806 y Fv(In)36 b(particular,)h(form)n(ula)e(\(22\))g(mak)n(es)g(sense) g(ev)n(erywhere,)h(pro)n(vided)f(the)h(exp)r(ectation)456 2005 y(v)-5 b(alues)37 b(of)i(all)f(time-ordered)f(pro)r(ducts)h(of)g (sub-Wic)n(k)g(p)r(olynomials)f(satisfy)h Fs(P4)p Fv(.)69 b(More-)456 2205 y(o)n(v)n(er,)35 b(ev)n(ery)f(expansion)g(in)n(to)h(a) f(sum)h(of)h(pro)r(ducts)e(of)h(Wic)n(k)g(p)r(olynomials)g(b)n(y)f(n)n (umerical)456 2404 y(distributions)27 b(whic)n(h)h(satis\014es)e (\(25\))i(is)f(of)h(this)g(form:)473 2668 y Fs(P5.)41 b(Causal)32 b(Wic)m(k)g(Expansion)p Fv(.)847 2959 y Fu(T)12 b Fv(\()p Fm(L)999 2971 y Fy(1)1036 2959 y Fv(\()p Fu(x)1115 2971 y Fy(1)1153 2959 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)t Fv(\()p Fu(x)1452 2971 y Fo(n)1499 2959 y Fv(\)\))23 b(=)1727 2880 y Fj(X)1674 3057 y Fo(j)1701 3065 y FC(1)1734 3057 y Fo(;:::)o(;j)1860 3065 y Fh(n)1901 2959 y Fv(\(\012)1993 2971 y Fo(!)2041 2959 y Fu(;)14 b(T)e Fv(\()p Fm(L)2230 2916 y Fy(\()p Fo(j)2283 2924 y FC(1)2315 2916 y Fy(\))2230 2981 y(1)2346 2959 y Fv(\()p Fu(x)2425 2971 y Fy(1)2462 2959 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)2678 2925 y Fy(\()p Fo(j)2731 2933 y Fh(n)2773 2925 y Fy(\))2678 2979 y Fo(n)2803 2959 y Fv(\()p Fu(x)2882 2971 y Fo(n)2928 2959 y Fv(\)\)\012)3052 2971 y Fo(!)3100 2959 y Fv(\))19 b Fr(\002)2364 3262 y(\002)2457 3205 y Fv(:)9 b Fu(')2543 3175 y Fo(j)2570 3183 y FC(1)2608 3205 y Fv(\()p Fu(x)2687 3217 y Fy(1)2725 3205 y Fv(\))14 b Fr(\001)g(\001)g(\001)f Fu(')2935 3175 y Fo(j)2962 3183 y Fh(n)3008 3205 y Fv(\()p Fu(x)3087 3217 y Fo(n)3133 3205 y Fv(\))c(:)p 2457 3243 741 4 v 2666 3319 a Fu(j)2700 3331 y Fy(1)2738 3319 y Fv(!)14 b Fr(\001)g(\001)g(\001)g Fu(j)2920 3331 y Fo(n)2965 3319 y Fv(!)3235 3262 y Fu(:)3297 3303 y Fv(\(26\))812 3572 y(4.)45 b Fw(Inductive)32 b(Constr)n(uction)f(up)h(to)f(the)g (Small)g(Dia)n(gonal)555 3821 y Fv(The)j(prop)r(erties)e(de\014ned)i (in)f(the)h(previous)e(section)h(allo)n(w)g(us)g(to)g(set)h(up)f(an)g (inductiv)n(e)456 4020 y(pro)r(cedure)38 b(in)i(the)g(spirit)f(of)h (Epstein)f(and)h(Glaser.)71 b(W)-7 b(e)40 b(rely)f(on)g(a)h(v)-5 b(ariation)38 b(of)i(their)456 4219 y(construction)25 b(prop)r(osed)f(b)n(y)i(Stora)f([54)o(].)37 b(W)-7 b(e)26 b(start)f(with)h(a)g(linear)f(space)g Fm(W)h Fv(of)g(Wic)n(k)g(p)r (oly-)456 4419 y(nomials)e(whic)n(h)h(con)n(tains)e(all)i(resp)r(ectiv) n(e)f(sub-Wic)n(k)g(p)r(olynomials)g(and)h(w)n(an)n(t)f(to)h(de\014ne)g (the)456 4618 y(time-ordered)f(pro)r(ducts)i Fu(T)12 b Fv(\()p Fm(L)1438 4630 y Fy(1)1475 4618 y Fv(\()p Fu(x)1554 4630 y Fy(1)1592 4618 y Fv(\))i Fr(\001)g(\001)g(\001)g Fm(L)t Fv(\()p Fu(x)1891 4630 y Fo(n)1937 4618 y Fv(\)\))27 b(as)e(a)h(family)g(of)g(op)r(erator-v)-5 b(alued)24 b(distribu-)456 4817 y(tions)k(whic)n(h)g(are)g(m)n(ultilinear)g(in)h (the)f(en)n(tries)g Fm(L)2028 4829 y Fo(i)2081 4817 y Fr(2)c Fm(W)29 b Fv(and)f(satisfy)g(the)h(prop)r(erties)f Fs(P1-5)p Fv(.)456 5016 y(W)-7 b(e)33 b(start)g(the)g(induction)g(b)n (y)g(setting)g Fu(T)12 b Fv(\()p Fs(1)o Fv(\))33 b(=)e Fs(1)i Fv(and)g Fu(T)12 b Fv(\()p Fm(L)p Fv(\))32 b(=)g Fm(L)i Fv(and)e(assume)h(that)g(the)456 5216 y(time-ordered)27 b(pro)r(ducts)i(for)f(1)c Fu(<)h(l)h Fr(\024)f Fu(n)19 b Fr(\000)f Fv(1)29 b(factors)e(ha)n(v)n(e)h(b)r(een)h(constructed)f (and)h(satisfy)p eop %%Page: 36 36 36 35 bop 456 226 a Fy(36)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(all)36 b(the)h(de\014ning) g(prop)r(erties.)62 b(In)37 b(a)f(\014rst)h(step,)i(w)n(e)d(aim)g(at)h (constructing)f(time-ordered)456 624 y(pro)r(ducts)c(of)h Fu(n)f Fv(factors)g(on)g Fu(M)1473 594 y Fo(n)1522 624 y Fr(n)5 b Fv(\001)1638 636 y Fo(n)1716 624 y Fv(where)32 b(\001)2030 636 y Fo(n)2108 624 y Fv(is)g(the)h(small)g(diagonal)e (submanifold)i(of)456 824 y Fu(M)546 794 y Fo(n)590 824 y Fv(,)28 b(i.e.)37 b(the)28 b(set)g(of)f(p)r(oin)n(ts)h(\()p Fu(x)1480 836 y Fy(1)1518 824 y Fu(;)14 b(:)g(:)g(:)f(;)h(x)1749 836 y Fo(n)1795 824 y Fv(\))28 b(with)g(the)g(prop)r(ert)n(y)e Fu(x)2573 836 y Fy(1)2634 824 y Fv(=)d Fu(x)2769 836 y Fy(2)2830 824 y Fv(=)f Fr(\001)14 b(\001)g(\001)23 b Fv(=)g Fu(x)3172 836 y Fo(n)3217 824 y Fv(.)555 1023 y(W)-7 b(e)24 b(use)e(the)i(space-time)e(notion)h(of)g(causalit)n(y)e (in)j(order)d(to)i(de\014ne)g(a)g(certain)f(partition)h(of)456 1222 y(unit)n(y)k(for)h Fu(M)890 1192 y Fo(n)939 1222 y Fr(n)5 b Fv(\001)1055 1234 y Fo(n)1099 1222 y Fv(:)38 b(Let)27 b(us)h(denote)f(b)n(y)h Fr(J)43 b Fv(the)28 b(family)g(of)f(all)h(non-empt)n(y)f(prop)r(er)g(subsets)456 1421 y Fu(I)35 b Fv(of)29 b(the)g(index)f(set)h Fr(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)e(;)i(n)p Fr(g)28 b Fv(and)h(de\014ne,)g (accordingly)-7 b(,)27 b(the)i(sets)f Fr(C)2745 1433 y Fo(I)2807 1421 y Fv(=)d Fr(f)p Fv(\()p Fu(x)3018 1433 y Fy(1)3055 1421 y Fu(;)14 b(:)g(:)g(:)g(;)g(x)3287 1433 y Fo(n)3332 1421 y Fv(\))25 b Fr(2)456 1621 y Fu(M)546 1591 y Fo(n)618 1621 y Fr(j)j Fu(x)716 1633 y Fo(i)777 1621 y Fu(=)-51 b Fr(2)24 b Fu(J)901 1591 y Fq(\000)957 1621 y Fv(\()p Fu(x)1036 1633 y Fo(j)1072 1621 y Fv(\))p Fu(;)14 b(i)23 b Fr(2)h Fu(I)7 b(;)14 b(j)28 b Fr(2)c Fu(I)1536 1591 y Fo(c)1570 1621 y Fr(g)j Fv(for)h(an)n(y)f Fu(I)j Fr(2)24 b(J)16 b Fv(.)38 b(Note)28 b(that)g(the)g(de\014ning)g (relation)f(for)456 1820 y(the)g Fr(C)642 1832 y Fo(I)680 1820 y Fv('s)g(is)g(related)f(to)h(causalit)n(y)f(on)h Fu(M)36 b Fv(and)27 b Ft(not)g Fv(on)f Fu(M)2310 1790 y Fo(n)2355 1820 y Fv(.)37 b(It)27 b(is)g(fairly)g(easy)f(to)h(sho)n(w) f(that)456 2175 y Fs(Lemma)31 b(4.1.)40 b Ft(L)l(et)29 b Fu(M)39 b Ft(b)l(e)30 b(a)g(glob)l(al)t(ly)i(hyp)l(erb)l(olic)g(sp)l (ac)l(e-time,)e(then)g(it)g(holds)1624 2435 y Fj([)1603 2613 y Fo(I)5 b Fq(2J)1752 2514 y Fr(C)1796 2526 y Fo(I)1857 2514 y Fv(=)22 b Fu(M)2034 2480 y Fo(n)2084 2514 y Fr(n)5 b Fv(\001)2200 2526 y Fo(n)2274 2514 y Fu(:)456 2853 y Ft(Pr)l(o)l(of.)43 b Fv(The)28 b(inclusion)g Fr([)1289 2865 y Fo(I)1327 2853 y Fr(C)1371 2865 y Fo(I)1433 2853 y Fr(\032)c Fu(M)1612 2823 y Fo(n)1661 2853 y Fr(n)5 b Fv(\001)1777 2865 y Fo(n)1850 2853 y Fv(is)28 b(ob)n(vious.)38 b(The)28 b(opp)r(osite)g(inclusion)g(is)g(pro)n(v)n(ed)456 3052 y(as)23 b(follo)n(ws.)34 b(Consider)23 b(an)n(y)g(set)g(of)h(p)r (oin)n(ts)g(\()p Fu(x)1894 3064 y Fy(1)1932 3052 y Fu(;)14 b(:)g(:)g(:)f(;)h(x)2163 3064 y Fo(n)2209 3052 y Fv(\))24 b(suc)n(h)f(that)h Fu(x)2671 3064 y Fo(i)2722 3052 y Fr(6)p Fv(=)f Fu(x)2857 3064 y Fo(j)2916 3052 y Fv(for)g(some)g Fu(i)g Fr(6)p Fv(=)f Fu(j)5 b Fv(,)456 3251 y(then)28 b(the)g(p)r(oin)n(ts)f Fu(x)1084 3263 y Fo(i)1140 3251 y Fv(and)h Fu(x)1349 3263 y Fo(j)1412 3251 y Fv(can)f(b)r(e)h (separated)f(b)n(y)g(a)g(Cauc)n(h)n(y)g(surface)f(\006)i(as)f(follo)n (ws)g(from)456 3451 y(the)i(global)f(h)n(yp)r(erb)r(olicit)n(y)g (assumption.)41 b(One)28 b(ma)n(y)h(c)n(ho)r(ose)f(it)h(as)f(to)h(con)n (tain)f(none)h(of)g(the)456 3650 y(p)r(oin)n(ts)k Fu(x)758 3662 y Fo(k)800 3650 y Fv(,)i Fu(k)i Fv(=)c(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(n)p Fv(.)55 b(Hence,)36 b(de\014ning)e Fu(I)40 b Fv(=)33 b Fr(f)p Fu(k)j Fr(j)e Fu(x)2387 3662 y Fo(k)2462 3650 y Fr(2)g Fu(J)2605 3620 y Fy(+)2660 3650 y Fv(\(\006\))p Fr(g)g Fv(and)g(noting)f(that)456 3849 y Fu(I)d Fr(2)23 b(J)43 b Fv(w)n(e)28 b(\014nd)g(\()p Fu(x)1067 3861 y Fy(1)1105 3849 y Fu(;)14 b(:)g(:)g(:)f(;)h(x)1336 3861 y Fo(n)1382 3849 y Fv(\))23 b Fr(2)h(C)1560 3861 y Fo(I)1597 3849 y Fv(.)1760 b Fl(\004)555 4169 y Fv(W)-7 b(e)28 b(use)g(the)g(short)e(hand)i(notations)1135 4507 y Fu(T)1196 4473 y Fo(I)1233 4507 y Fv(\()p Fu(x)1312 4519 y Fo(I)1351 4507 y Fv(\))23 b(=)g Fu(T)12 b Fv(\()1587 4429 y Fj(Y)1589 4607 y Fo(i)p Fq(2)p Fo(I)1706 4507 y Fm(L)1765 4519 y Fo(i)1793 4507 y Fv(\()p Fu(x)1872 4519 y Fo(i)1901 4507 y Fv(\)\))p Fu(;)180 b(x)2215 4519 y Fo(I)2277 4507 y Fv(=)22 b(\()p Fu(x)2443 4519 y Fo(i)2472 4507 y Fu(;)14 b(i)22 b Fr(2)i Fu(I)7 b Fv(\))28 b Fu(:)532 b Fv(\(27\))456 4846 y(The)27 b(\014rst)h(step)f(no)n(w)g(is)h(to)f (set)h(on)f(an)n(y)g Fr(C)1777 4858 y Fo(I)1477 5185 y Fu(T)1526 5197 y Fo(I)1563 5185 y Fv(\()p Fu(x)p Fv(\))1719 5138 y Fu(:)1698 5185 y Fv(=)c Fu(T)1847 5151 y Fo(I)1884 5185 y Fv(\()p Fu(x)1963 5197 y Fo(I)2002 5185 y Fv(\))28 b Fu(T)2123 5151 y Fo(I)2157 5126 y Fh(c)2191 5185 y Fv(\()p Fu(x)2270 5197 y Fo(I)2304 5181 y Fh(c)2340 5185 y Fv(\))g Fu(;)874 b Fv(\(28\))p eop %%Page: 37 37 37 36 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(37)456 425 y Fv(as)26 b(an)g(op)r(erator-v)-5 b(alued)24 b(distribution)j(since)f(according)f (to)i(the)g(induction)g(h)n(yp)r(othesis)f(and)456 624 y(the)i(fact)f(that)h Fu(I)35 b Fv(is)27 b(prop)r(er)g(this)h(is)f(a)h (w)n(ell)f(de\014ned)h(op)r(eration)e(on)i Fm(D)p Fv(\()p Fr(C)2763 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FC(2)1830 4532 y Fv(\()p Fu(x)1909 4544 y Fo(I)1943 4525 y Fh(c)1938 4563 y FC(1)1975 4544 y Fq(\\)p Fo(I)2049 4552 y FC(2)2086 4532 y Fv(\))28 b Fu(T)2207 4498 y Fo(I)2241 4473 y Fh(c)2236 4515 y FC(1)2271 4498 y Fq(\\)p Fo(I)2350 4473 y Fh(c)2345 4515 y FC(2)2385 4532 y Fv(\()p Fu(x)2464 4544 y Fo(I)2498 4525 y Fh(c)2493 4563 y FC(1)2530 4544 y Fq(\\)p Fo(I)2609 4525 y Fh(c)2604 4563 y FC(2)2644 4532 y Fv(\))g Fu(;)3297 4397 y Fv(\(29\))456 4817 y(and)20 b(similarly)g(for)h Fu(T)1125 4787 y Fo(I)1154 4795 y FC(2)1211 4817 y Fv(and)f Fu(T)1426 4787 y Fo(I)1460 4762 y Fh(c)1455 4804 y FC(2)1495 4817 y Fv(.)34 b(No)n(w)21 b(the)g(terms)g Fu(T)2157 4787 y Fo(I)2186 4795 y FC(1)2218 4787 y Fq(\\)p Fo(I)2297 4762 y Fh(c)2292 4804 y FC(2)2352 4817 y Fv(and)g Fu(T)2568 4787 y Fo(I)2602 4762 y Fh(c)2597 4804 y FC(1)2632 4787 y Fq(\\)p Fo(I)2706 4795 y FC(2)2764 4817 y Fv(comm)n(ute.)35 b(Namely)-7 b(,)456 5016 y(they)34 b(are)f(based)g(on)h(m)n(utually)g (space-lik)n(e)e(p)r(oin)n(ts,)j(th)n(us)f(using)g(the)g(Wic)n(k)g (expansion)f(for)456 5216 y(these)28 b(terms)f(this)h(follo)n(ws)f (from)h(lo)r(cal)f(comm)n(utativit)n(y)h(for)f(the)h(Wic)n(k)g(p)r (olynomials)f(of)h(the)p eop %%Page: 38 38 38 37 bop 456 226 a Fy(38)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(free)e(scalar)f(\014eld)i Fu(')p Fv(.)37 b(Hence)28 b(from)f(de\014nition)h(\(28\))f(w)n(e)h(get) f(on)g Fr(C)2552 437 y Fo(I)2581 445 y FC(1)2636 425 y Fr(\\)19 b(C)2754 437 y Fo(I)2783 445 y FC(2)941 814 y Fu(T)990 826 y Fo(I)1019 834 y FC(1)1078 814 y Fv(=)k Fu(T)1227 779 y Fo(I)1256 787 y FC(1)1288 779 y Fq(\\)p Fo(I)1362 787 y FC(2)1399 814 y Fu(T)1460 779 y Fo(I)1494 754 y Fh(c)1489 796 y FC(1)1524 779 y Fq(\\)p Fo(I)1598 787 y FC(2)1634 814 y Fu(T)1695 779 y Fo(I)1724 787 y FC(1)1756 779 y Fq(\\)p Fo(I)1835 754 y Fh(c)1830 796 y FC(2)1870 814 y Fu(T)1931 779 y Fo(I)1965 754 y Fh(c)1960 796 y FC(1)1995 779 y Fq(\\)p Fo(I)2074 754 y Fh(c)2069 796 y FC(2)2137 814 y Fu(;)166 b Fv(\(Eqn.\(28)o(\))19 b(+)f Fs(P3)p Fv(\))p Fu(;)1078 1063 y Fv(=)23 b Fu(T)1227 1029 y Fo(I)1256 1037 y FC(2)1292 1063 y Fu(T)1353 1029 y Fo(I)1387 1003 y Fh(c)1382 1045 y FC(2)1449 1063 y Fu(;)854 b Fv(\(Eqn.\(29)o(\)\))p Fu(;)1078 1312 y Fv(=)23 b Fu(T)1215 1324 y Fo(I)1244 1332 y FC(2)1308 1312 y Fu(;)995 b Fv(\(Eqn.\(28)o(\)\))p Fu(:)3380 1651 y Fl(\004)555 1968 y Fv(Let)26 b(no)n(w)g Fr(f)p Fu(f)957 1980 y Fo(I)994 1968 y Fr(g)1036 1980 y Fo(I)5 b Fq(2J)1201 1968 y Fv(b)r(e)26 b(a)g(lo)r(cally)f(\014nite)i(smo)r(oth)f(partition)f(of)h(unit)n(y)g (of)g Fu(M)2979 1938 y Fo(n)3029 1968 y Fr(n)5 b Fv(\001)3145 1980 y Fo(n)3215 1968 y Fv(sub)r(or-)456 2167 y(dinate)27 b(to)h Fr(fC)897 2179 y Fo(I)934 2167 y Fr(g)976 2179 y Fo(I)5 b Fq(2J)1115 2167 y Fv(.)37 b(W)-7 b(e)28 b(formally)e (de\014ne,)i(follo)n(wing)f(\(27\))g(and)g(\(28\),)1301 2472 y Fy(0)1339 2506 y Fu(T)11 b Fv(\()p Fm(L)1490 2518 y Fy(1)1528 2506 y Fv(\()p Fu(x)1607 2518 y Fy(1)1645 2506 y Fv(\))j Fu(:)g(:)g(:)f Fm(L)1860 2518 y Fo(n)1906 2506 y Fv(\()p Fu(x)1985 2518 y Fo(n)2031 2506 y Fv(\)\))2139 2459 y Fu(:)2118 2506 y Fv(=)2214 2427 y Fj(X)2206 2605 y Fo(I)5 b Fq(2J)2355 2506 y Fu(f)2396 2518 y Fo(I)2461 2506 y Fu(T)2510 2518 y Fo(I)2576 2506 y Fu(:)698 b Fv(\(30\))456 2845 y(Hence,)27 b(w)n(e)h(get)f(our)g(\014rst)g(crucial)g(result,)g (namely)-7 b(,)456 3198 y Fs(Theorem)31 b(4.3.)40 b Ft(The)30 b(symb)l(ols)1522 3168 y Fy(0)1559 3198 y Fu(T)40 b Ft(ar)l(e)29 b(wel)t(l)g(de\014ne)l(d)g(op)l(er)l(ator-value)l(d)h(distributions)g (over)456 3398 y Fu(M)546 3367 y Fo(n)595 3398 y Fr(n)5 b Fv(\001)711 3410 y Fo(n)785 3398 y Ft(which)32 b(satisfy)e(the)g (de\014ning)g(pr)l(op)l(erties)h(on)f Fr(D)25 b(\032)e(H)2477 3410 y Fo(!)2525 3398 y Ft(.)456 3751 y(Pr)l(o)l(of.)43 b Fv(W)-7 b(e)40 b(\014rst)g(pro)n(v)n(e)e(that)i(the)h(de\014nition)f (do)r(es)g(not)f(dep)r(end)i(on)f(the)g(c)n(hoice)f(of)h(the)456 3950 y(partition)33 b(of)h(unit)n(y)-7 b(.)57 b(Indeed,)36 b(let)e Fr(f)p Fu(f)1690 3920 y Fq(0)1681 3973 y Fo(I)1718 3950 y Fr(g)1760 3962 y Fo(I)5 b Fq(2J)1933 3950 y Fv(b)r(e)34 b(another)f(suc)n(h)h(partition.)56 b(Consider)33 b Fu(x)h Fr(2)456 4150 y Fu(M)546 4119 y Fo(n)595 4150 y Fr(n)5 b Fv(\001)711 4162 y Fo(n)756 4150 y Fv(,)30 b(and)g(let)g Fr(K)f Fv(=)d Fr(f)p Fu(I)34 b Fr(2)27 b(J)46 b(j)30 b Fu(x)d Fr(2)g(C)1827 4162 y Fo(I)1865 4150 y Fr(g)p Fv(.)44 b(Then)30 b(there)g(exists)f(a)h(neigh)n(b)r(ourho)r(o)r(d)f Fu(V)49 b Fv(of)456 4349 y Fu(x)28 b Fv(suc)n(h)g(that)g Fu(V)42 b Fr(\032)23 b(\\)1132 4361 y Fo(I)5 b Fq(2K)1266 4349 y Fr(C)1310 4361 y Fo(I)1348 4349 y Fv(,)28 b(and)g(supp\()p Fu(f)1805 4361 y Fo(I)1843 4349 y Fv(\))g(and)g(supp\()p Fu(f)2318 4319 y Fq(0)2309 4372 y Fo(I)2347 4349 y Fv(\))g(do)g(not)g (meet)g Fu(V)47 b Fv(for)27 b(all)g Fu(I)40 b(=)-51 b Fr(2)24 b(K)q Fv(.)456 4548 y(In)j(this)h(case)1233 4668 y Fj(X)1225 4847 y Fo(I)5 b Fq(2J)1360 4747 y Fv(\()p Fu(f)1433 4759 y Fo(I)1490 4747 y Fr(\000)18 b Fu(f)1623 4713 y Fq(0)1614 4768 y Fo(I)1652 4747 y Fv(\))28 b Fu(T)1761 4759 y Fo(I)1794 4747 y Fl(\026)1829 4759 y Fo(V)1886 4747 y Fv(=)1978 4668 y Fj(X)1974 4847 y Fo(I)5 b Fq(2K)2103 4747 y Fv(\()p Fu(f)2176 4759 y Fo(I)2232 4747 y Fr(\000)18 b Fu(f)2365 4713 y Fq(0)2356 4768 y Fo(I)2394 4747 y Fv(\))28 b Fu(T)2503 4759 y Fo(I)2536 4747 y Fl(\026)2571 4759 y Fo(V)2651 4747 y Fu(:)456 5016 y Fv(Ho)n(w)n(ev)n(er,)44 b(on)e Fu(V)62 b(T)1121 5028 y Fo(I)1201 5016 y Fv(is)42 b(indep)r(enden)n(t)i(of)e(the)h(c)n(hoice)f(of)g Fu(I)55 b Fr(2)48 b(K)q Fv(.)82 b(Since)3018 4954 y Fj(P)3106 5041 y Fo(I)5 b Fq(2K)3253 5016 y Fu(f)3294 5028 y Fo(I)3380 5016 y Fv(=)456 5153 y Fj(P)543 5240 y Fo(I)g Fq(2K)691 5216 y Fu(f)741 5185 y Fq(0)732 5238 y Fo(I)807 5216 y Fv(=)37 b(1)e(on)h Fu(V)19 b Fv(,)39 b(w)n(e)c(arriv)n(e)g(at)h(the)g (conclusion.)62 b(F)-7 b(urthermore,)37 b(an)f(insp)r(ection)h(of)p eop %%Page: 39 39 39 38 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(39)456 425 y Fv(the)26 b(form)n(ula)f(readily)g(giv)n(es)g(that)i(the)f(op)r(erator)2028 395 y Fy(0)2066 425 y Fu(T)37 b Fv(is)26 b(de\014ned)g(on)g(the)g (domain)g Fr(D)r Fv(,)h(the)f(mi-)456 624 y(crolo)r(cal)g(domain)i(of)h (de\014nition)f(of)h(the)g(Wic)n(k)f(monomials,)f(b)r(ecause)h(of)h (induction.)39 b(Hence)456 824 y(prop)r(ert)n(y)26 b Fs(P1)p Fv(.)555 1023 y(As)42 b(far)e(as)h(the)h(symmetry)e(prop)r(ert) n(y)g Fs(P2)i Fv(is)f(concerned)f(w)n(e)h(just)h(observ)n(e)e(that)h (the)456 1222 y(p)r(erm)n(uted)25 b(distribution)1277 1192 y Fy(0)1314 1222 y Fu(T)1374 1175 y Fo(\031)1419 1222 y Fv(\()p Fu(x)1498 1234 y Fy(1)1536 1222 y Fu(;)14 b(:)g(:)g(:)27 b(;)14 b(x)1781 1234 y Fo(n)1827 1222 y Fv(\))23 b(=)1970 1192 y Fy(0)2007 1222 y Fu(T)11 b Fv(\()p Fm(L)2158 1237 y Fo(\031)r Fy(\(1\))2289 1222 y Fv(\()p Fu(x)2368 1237 y Fo(\031)r Fy(\(1\))2499 1222 y Fv(\))j Fu(:)g(:)g(:)g Fm(L)2715 1237 y Fo(\031)r Fy(\()p Fo(n)p Fy(\))2853 1222 y Fv(\()p Fu(x)2932 1237 y Fo(\031)r Fy(\()p Fo(n)p Fy(\))3071 1222 y Fv(\)\),)26 b(has)e(the)456 1421 y(expansion,)1319 1726 y Fy(0)1356 1760 y Fu(T)1416 1709 y Fo(\031)1484 1760 y Fv(=)1572 1681 y Fj(X)1572 1859 y Fo(I)5 b Fq(2)p Fo(J)1707 1760 y Fu(f)1757 1726 y Fo(\031)1748 1781 y(I)1801 1760 y Fu(T)1862 1726 y Fo(\031)1850 1781 y(I)1929 1760 y Fv(=)2018 1681 y Fj(X)2017 1859 y Fo(I)g Fq(2)p Fo(J)2152 1760 y Fu(f)2202 1726 y Fo(\031)2193 1783 y(\031)r Fy(\()p Fo(I)g Fy(\))2351 1760 y Fu(T)2412 1726 y Fo(\031)2400 1783 y(\031)r Fy(\()p Fo(I)g Fy(\))2558 1760 y Fu(;)456 2099 y Fv(where)25 b(w)n(e)h(used)g(the)g(fact)g(that)h(the)f(set)g Fu(J)34 b Fv(is)26 b(in)n(v)-5 b(arian)n(t)25 b(under)h(p)r(erm)n(utations,)g (but)g Fu(T)3238 2069 y Fo(\031)3226 2126 y(\031)r Fy(\()p Fo(I)5 b Fy(\))3380 2099 y Fv(=)456 2298 y Fu(T)505 2310 y Fo(I)581 2298 y Fv(and)39 b Fr(f)p Fu(f)846 2268 y Fo(\031)837 2325 y(\031)r Fy(\()p Fo(I)5 b Fy(\))967 2298 y Fr(g)1009 2310 y Fo(I)g Fq(2)p Fo(J)1172 2298 y Fv(is)39 b(a)f(partition)g(of)h(unit)n(y)g(sub)r(ordinate)f(to)g Fr(fC)2702 2310 y Fo(I)2740 2298 y Fr(g)2782 2310 y Fo(I)5 b Fq(2)p Fo(J)2906 2298 y Fv(,)42 b(so)c(symmetry)456 2497 y(follo)n(ws)25 b(from)g(the)h(result)g(of)g(the)g(previous)f (paragraph)f(ab)r(out)i(the)g(indep)r(endence)h(of)3233 2467 y Fy(0)3271 2497 y Fu(T)37 b Fv(on)456 2697 y(the)28 b(c)n(hoice)e(of)i(the)g(partition)f(of)h(unit)n(y)-7 b(.)555 2896 y(Causalit)n(y)25 b Fs(P3)i Fv(follo)n(ws)e(from)h(an)g (argumen)n(t)f(similar)h(to)g(the)g(one)g(used)h(for)e(the)i(indep)r (en-)456 3095 y(dence)h(from)h(the)g(partition)f(of)h(unit)n(y)-7 b(.)41 b(Indeed,)29 b(tak)n(en)f(an)n(y)g(p)r(oin)n(t)h Fu(x)c Fr(2)h Fu(M)2858 3065 y Fo(n)2907 3095 y Fr(n)5 b Fv(\001)3023 3107 y Fo(n)3068 3095 y Fv(,)29 b(as)f(b)r(efore)456 3294 y Fu(x)23 b Fr(2)h Fu(V)41 b Fr(\032)23 b(\\)837 3306 y Fo(I)5 b Fq(2K)971 3294 y Fm(C)1021 3306 y Fo(I)1059 3294 y Fv(.)37 b(F)-7 b(rom)27 b(\(30\),)1498 3599 y Fy(0)1536 3633 y Fu(T)12 b Fv(\()p Fu(x)p Fv(\))23 b(=)1823 3554 y Fj(X)1818 3732 y Fo(I)5 b Fq(2K)1962 3633 y Fu(f)2003 3645 y Fo(I)2040 3633 y Fv(\()p Fu(x)p Fv(\))p Fu(T)2200 3645 y Fo(I)2239 3633 y Fv(\()p Fu(x)p Fv(\))29 b Fu(:)456 3972 y Fv(Since)34 b Fu(T)728 3984 y Fo(I)771 3972 y Fl(\026)806 3984 y Fo(V)897 3972 y Fv(do)r(es)g(not)g(dep)r(end)g(an)n (ymore)f(on)g Fu(I)41 b Fr(2)34 b(K)h Fv(and)2436 3910 y Fj(P)2523 3997 y Fo(I)5 b Fq(2K)2671 3972 y Fu(f)2712 3984 y Fo(I)2783 3972 y Fv(=)33 b(1)h(o)n(v)n(er)e Fu(V)52 b Fv(hence)456 4141 y Fy(0)493 4171 y Fu(T)12 b Fv(\()p Fu(x)p Fv(\))23 b Fr(\021)g Fu(T)825 4183 y Fo(I)862 4171 y Fv(\()p Fu(x)p Fv(\))29 b(whic)n(h)e(from)h(\(28\))f (satis\014es)g(causalit)n(y)f(b)n(y)h(de\014nition.)555 4370 y(No)n(w,)35 b(w)n(e)e(w)n(an)n(t)g(to)g(sho)n(w)g(that)h(prop)r (ert)n(y)e Fs(P4)i Fv(holds)f(on)g Fu(M)2543 4340 y Fo(n)2592 4370 y Fr(n)5 b Fv(\001)2708 4382 y Fo(n)2753 4370 y Fv(.)55 b(It)34 b(is)f(su\016cien)n(t)h(to)456 4570 y(c)n(hec)n(k)25 b(that)i(this)g(prop)r(ert)n(y)f(is)g(satis\014ed)g(for)g(eac)n(h)g Fu(T)2118 4582 y Fo(I)2182 4570 y Fv(on)h Fr(C)2341 4582 y Fo(I)2378 4570 y Fv(.)37 b(W)-7 b(e)27 b(apply)f(Wic)n(k)h(Theorem)f (to)456 4769 y(the)g(comp)r(onen)n(ts)g(of)g(the)h(pro)r(duct)f(in)h (the)g(de\014nition)f(\(28\).)36 b(It)27 b(can)f(b)r(e)h(easily)e(c)n (hec)n(k)n(ed)g(that)456 4968 y(the)h(distributions)h Fu(t)1112 4980 y Fo(I)1176 4968 y Fv(in)g(the)g(Wic)n(k)f(expansion)g (of)g Fu(T)2154 4980 y Fo(I)2218 4968 y Fv(are)f(sums)i(of)f(terms)g (of)h(the)g(follo)n(wing)p eop %%Page: 40 40 40 39 bop 456 226 a Fy(40)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(form)1152 764 y Fu(f)1193 776 y Fo(I)1231 764 y Fv(\()p Fu(x)p Fv(\))p Fu(t)1372 730 y Fo(I)1411 764 y Fv(\()p Fu(x)1490 776 y Fo(I)1529 764 y Fv(\))p Fu(t)1591 730 y Fo(I)1625 705 y Fh(c)1660 764 y Fv(\()p Fu(x)1739 776 y Fo(I)1773 760 y Fh(c)1809 764 y Fv(\))19 b Fr(\001)2009 685 y Fj(Y)1901 867 y Fy(\()p Fo(i;j)s Fy(\))p Fq(2)p Fo(I)5 b Fq(\002)p Fo(I)2191 850 y Fh(c)2237 764 y Fu(!)2289 776 y Fy(2)2326 764 y Fv(\()p Fu(x)2405 776 y Fo(i)2433 764 y Fu(;)14 b(x)2517 776 y Fo(j)2552 764 y Fv(\))2584 730 y Fo(a)2620 738 y Fh(i;j)2725 764 y Fu(;)549 b Fv(\(31\))456 1111 y(with)25 b Fu(a)686 1123 y Fo(i;j)787 1111 y Fr(2)f Fn(N)926 1123 y Fy(0)963 1111 y Fv(,)h(and)g(where)g Fu(t)1438 1081 y Fo(I)1476 1111 y Fu(;)14 b(t)1543 1081 y Fo(I)1577 1055 y Fh(c)1637 1111 y Fv(are)24 b(exp)r(ectation)g(v)-5 b(alues)25 b(of)g(lo)n(w)n(er)e(order)h(time-ordered)456 1310 y(pro)r(ducts.)555 1509 y(The)32 b(w)n(a)n(v)n(e)e(fron)n(t)h(set) h(of)f(\(31\))g(is)h(con)n(tained)f(in)h(the)g(con)n(v)n(ex)e(com)n (bination)h(of)g(the)h(w)n(a)n(v)n(e)456 1708 y(fron)n(t)h(sets)g(of)g (its)h(factors.)53 b(Hence)33 b(it)h(is)f(giv)n(en)g(in)h(terms)f(of)g (immersions)g(of)g(graphs)f(with)456 1908 y(v)n(ertex)j(sets)i Fu(I)7 b(;)14 b(I)1011 1878 y Fo(c)1045 1908 y Fv(,)39 b(resp.,)g(and)d(of)h Fu(a)1659 1920 y Fo(i;j)1774 1908 y Fv(graphs)e(with)j(v)n(ertex)e(sets)g Fr(f)p Fu(i;)14 b(j)5 b Fr(g)p Fv(.)63 b(All)38 b(these)e(im-)456 2107 y(mersions)g(satisfy)i(the)g(condition)f(in)h Fs(P4)p Fv(,)i(the)f(\014rst)e(t)n(w)n(o)g(b)n(y)g(assumption,)j(and)e(the)g (last)456 2306 y(ones)32 b(b)r(ecause)g(of)h(the)g(de\014nition)g(of)f Fr(C)1723 2318 y Fo(I)1794 2306 y Fv(and)h(the)g(prop)r(erties)e(of)i (the)g(w)n(a)n(v)n(e)e(fron)n(t)i(set)f(of)h(a)456 2505 y(Hadamard)22 b(state.)35 b(The)23 b(union)g(of)h(these)f(graphs)f(is)h (a)g(graph)f(with)h(v)n(ertices)g Fr(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)e(;)i(n)p Fr(g)p Fv(,)24 b(and)456 2705 y(an)n(y)i(con)n(v) n(ex)f(com)n(bination)h(of)h(the)g(comp)r(onen)n(ts)f(is)h(giv)n(en)f (b)n(y)g(an)h(admissible)f(immersion)h(of)456 2904 y(this)g(graph.)555 3103 y(Finally)-7 b(,)29 b(prop)r(ert)n(y)f Fs(P5)h Fv(follo)n(ws)e (from)i(expression)e(\(28\))h(b)n(y)g(a)h(straigh)n(tforw)n(ard)c (applica-)456 3302 y(tion)i(of)h(the)g(\(generalized\))f(Wic)n(k)g (Theorem.)1461 b Fl(\004)553 3771 y Fv(5.)45 b Fw(Steinmann)32 b(Scaling)f(Degree)h(and)f(the)h(Extension)f(of)g(Distributions)555 4020 y Fv(W)-7 b(e)32 b(no)n(w)e(w)n(an)n(t)h(to)g(extend)1463 3990 y Fy(0)1501 4020 y Fu(T)11 b Fv(\()p Fm(L)1652 4032 y Fy(1)1690 4020 y Fv(\()p Fu(x)1769 4032 y Fy(1)1807 4020 y Fv(\))j Fu(:)g(:)g(:)f Fm(L)2022 4032 y Fo(n)2068 4020 y Fv(\()p Fu(x)2147 4032 y Fo(n)2193 4020 y Fv(\)\))32 b(to)f(the)g(whole)g Fu(M)2869 3990 y Fo(n)2914 4020 y Fv(.)48 b(As)31 b(discussed)456 4219 y(b)r(efore)d(the)g(problem)g (can)g(b)r(e)g(reduced)g(to)g(the)h(extension)f(of)g(the)g(n)n (umerical)g(time-ordered)456 4419 y(distributions)941 4388 y Fy(0)978 4419 y Fu(t)p Fv(\()p Fu(x)1087 4431 y Fy(1)1125 4419 y Fu(;)14 b(:)g(:)g(:)28 b(;)14 b(x)1371 4431 y Fo(n)1416 4419 y Fv(\))1492 4371 y Fu(:)1472 4419 y Fv(=)22 b(\(\012)1651 4431 y Fo(!)1700 4419 y Fu(;)1737 4388 y Fy(0)1774 4419 y Fu(T)11 b Fv(\()p Fm(L)1925 4431 y Fy(1)1963 4419 y Fv(\()p Fu(x)2042 4431 y Fy(1)2080 4419 y Fv(\))j Fu(:)g(:)g(:)f Fm(L)2295 4431 y Fo(n)2341 4419 y Fv(\()p Fu(x)2420 4431 y Fo(n)2466 4419 y Fv(\)\)\012)2590 4431 y Fo(!)2639 4419 y Fv(\).)555 4618 y(The)26 b(extension)g(can)g(b) r(e)g(p)r(erformed)g(in)g(t)n(w)n(o)f(steps.)37 b(First)2440 4588 y Fy(0)2477 4618 y Fu(t)26 b Fv(is)g(extended)g(b)n(y)g(con)n(tin) n(uit)n(y)456 4817 y(to)33 b(the)i(subspace)e(of)h(test-functions)g (whic)n(h)g(v)-5 b(anish)34 b(on)f(\001)2390 4829 y Fo(n)2470 4817 y Fv(up)h(to)g(a)f(certain)g(order,)i(and)456 5016 y(then)22 b(it)g(is)g(arbitrarily)e(de\014ned)i(on)g(a)g(complemen)n (tary)e(subspace.)35 b(It)22 b(is)g(this)g(last)f(step)i(whic)n(h)456 5216 y(corresp)r(onds)16 b(to)j(the)g(metho)r(d)g(of)g(coun)n(terterms) e(in)i(the)g(classical)e(pro)r(cedure)h(of)g(p)r(erturbativ)n(e)p eop %%Page: 41 41 41 40 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(41)456 425 y Fv(renormalization.)34 b(The)26 b(extension)g(of)1712 395 y Fy(0)1749 425 y Fu(t)g Fv(b)n(y)h(con)n(tin)n(uit)n(y)e(requires)g(some)h(top)r(ology)f (on)i(test-)456 624 y(function)d(space.)35 b(The)23 b(seminorms)g(used) g(b)n(y)h(Epstein)f(and)h(Glaser)e(in)i(their)f(pap)r(er)h(are)e(quite) 456 824 y(complicated,)34 b(and)f(their)g(generalization)e(to)i(curv)n (ed)g(space-times)f(app)r(ears)g(to)h(b)r(e)h(rather)456 1023 y(in)n(v)n(olv)n(ed.)68 b(W)-7 b(e)39 b(found)g(it)g(preferable)f (therefore)g(to)g(apply)h(a)f(di\013eren)n(t)h(metho)r(d)g(already)456 1222 y(in)n(tro)r(duced)26 b(b)n(y)h(Steinmann)h([53)o(],)f(namely)g (the)g(concept)g(of)g Ft(sc)l(aling)j(de)l(gr)l(e)l(e)e Fv(at)f(a)g(p)r(oin)n(t)g(of)g(a)456 1421 y(distribution)j(\(see)h (also)e([16)o(]\).)46 b(Its)30 b(generalization)f(to)h(curv)n(ed)g (space-time)g(is)g(v)n(ery)f(similar)456 1621 y(to)c(the)g(concept)g (of)g(the)h(scaling)e(limit)i(as)e(in)n(tro)r(duced)h(b)n(y)g(Haag,)g (Narnhofer)f(and)h(Stein)g([30)o(])456 1820 y(and)j(further)h(dev)n (elop)r(ed)f(b)n(y)g(F)-7 b(redenhagen)28 b(and)h(Haag)e([24)o(].)40 b(A)29 b(similar)f(tec)n(hnique)h(is)g(used)456 2019 y(in)e([50].)555 2218 y(On)33 b(Mink)n(o)n(wski)g(space,)h(b)n(y)f (translation)f(in)n(v)-5 b(ariance,)34 b(the)g(distribution)f(is)h(in)f (terms)h(of)456 2418 y(relativ)n(e)28 b(co)r(ordinates)g(ev)n(erywhere) g(de\014ned)i(up)f(to)h(the)g(origin,)f(and)g(there)g(the)h(concept)g (of)456 2617 y(the)37 b Ft(sc)l(aling)i(de)l(gr)l(e)l(e)f Fv(at)e(a)h(p)r(oin)n(t)g(leads)f(to)h(a)g(rather)f(smo)r(oth)g(and)h (economic)f(metho)r(d)i(of)456 2816 y(renormalization,)31 b(see)i(e.g.)51 b([48)o(])33 b(where)f(the)h(relation)f(to)h (di\013eren)n(tial)f(renormalization)f(is)456 3015 y(elab)r(orated.)i (On)19 b(a)g(curv)n(ed)f(space-time)h(one)g(needs)g(the)h(corresp)r (onding)e(notion)h(for)f(a)h(scaling)456 3215 y(degree)33 b(with)i(resp)r(ect)f(to)g(the)h(submanifold)f(\001)2010 3227 y Fo(n)2056 3215 y Fv(,)i(and)e(one)g(also)g(needs)g(some)g (uniformit)n(y)456 3414 y(of)j(the)g(singularit)n(y)e(along)h(\001)1432 3426 y Fo(n)1515 3414 y Fv(as)g(w)n(ell)h(as)f(con)n(trol)g(of)h(the)g (w)n(a)n(v)n(e)e(fron)n(t)i(sets)g(during)f(the)456 3613 y(extension)27 b(pro)r(cess.)555 3812 y(Our)h(strategy)g(will)h(b)r(e)g (that,)h(at)e(\014rst)h(w)n(e)f(in)n(tro)r(duce)h(this)g(impro)n(v)n (emen)n(t)e(in)i(the)h(case)e(of)456 4012 y Fn(R)516 3982 y Fo(n)561 4012 y Fv(,)38 b(then)e(w)n(e)g(discuss)g(the)g(case)f (of)h(manifolds.)62 b(There)35 b(w)n(e)h(try)g(to)g(set)g(up)g(a)f(pro) r(cedure)456 4211 y(whic)n(h)27 b(allo)n(ws)f(to)i(restrict)f(the)h (discussion)f(to)g(the)h(p)r(oin)n(t)n(wise)f(case.)456 5016 y(5.1.)46 b Fs(The)31 b(scaling)e(degree.)40 b Fv(F)-7 b(or)26 b(simplicit)n(y)-7 b(,)26 b(w)n(e)g(w)n(ork)f(at)g(\014rst)h (on)g Fn(R)2782 4986 y Fo(d)2821 5016 y Fv(.)36 b(Hence,)27 b(consider)456 5216 y(a)d(distribution)h Fu(t)e Fr(2)g Fm(D)1172 5185 y Fq(0)1195 5216 y Fv(\()p Fn(R)1287 5185 y Fo(d)1326 5216 y Fv(\).)37 b(Let)24 b(the)i(action)e(of)g(the)h(p)r (ositiv)n(e)g(reals)e(\(dilations\))i(b)r(e)g(de\014ned)p eop %%Page: 42 42 42 41 bop 456 226 a Fy(42)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(via)e(the)h(map)1192 814 y(\003)23 b(:)g Fn(R)1379 826 y Fy(+)1452 814 y Fr(\002)18 b Fm(D)p Fv(\()p Fn(R)1696 779 y Fo(d)1735 814 y Fv(\))28 b Fr(\000)-14 b(!)55 b Fm(D)p Fv(\()p Fn(R)2145 779 y Fo(d)2184 814 y Fv(\))1434 1063 y(\()p Fu(\025;)14 b(\036)p Fv(\))163 b Fr(7\000)-14 b(!)55 b Fu(\036)2033 1029 y Fo(\025)2121 1016 y Fu(:)2101 1063 y Fv(=)22 b Fu(\025)2236 1029 y Fq(\000)p Fo(d)2327 1063 y Fu(\036)p Fv(\()p Fu(\025)2456 1029 y Fq(\000)p Fy(1)2574 1063 y Fr(\001)28 b Fv(\))g Fu(;)456 1402 y Fv(and)f(obtain,)g(b)n(y)h(pull-bac)n(k,)f(the)h(map)f (o)n(v)n(er)f(distributions)h Fu(t)c Fr(2)h Fm(D)2588 1371 y Fq(0)2611 1402 y Fv(\()p Fn(R)2703 1371 y Fo(d)2742 1402 y Fv(\))k(as,)1474 1740 y(\(\003)1564 1706 y Fq(\003)1602 1740 y Fu(t)p Fv(\)\()p Fu(\036)p Fv(\))1822 1693 y Fu(:)1802 1740 y Fv(=)22 b Fu(t)1919 1752 y Fo(\025)1963 1740 y Fv(\()p Fu(\036)p Fv(\))2121 1693 y Fu(:)2100 1740 y Fv(=)h Fu(t)p Fv(\()p Fu(\036)2299 1706 y Fo(\025)2343 1740 y Fv(\))28 b Fu(;)456 2079 y Fv(where)f(this)h(op)r(eration)e(in)i (case)f Fu(t)c Fr(2)g Fu(L)1690 2049 y Fy(1)1690 2103 y(lo)r(c)1777 2079 y Fv(\()p Fn(R)1869 2049 y Fo(d)1908 2079 y Fv(\))28 b(is)f(giv)n(en)g(b)n(y)g(the)h(explicit)g(form)n(ula,) 1156 2418 y Fu(t)1186 2430 y Fo(\025)1230 2418 y Fv(\()p Fu(\036)p Fv(\))c(=)1455 2305 y Fj(Z)1552 2418 y Fu(t)p Fv(\()p Fu(\025x)p Fv(\))p Fu(\036)p Fv(\()p Fu(x)p Fv(\))p Fu(d)1944 2383 y Fo(d)1985 2418 y Fu(x)29 b(;)180 b Fr(8)p Fu(\036)23 b Fr(2)h Fm(D)p Fv(\()p Fn(R)2622 2383 y Fo(d)2660 2418 y Fv(\))k Fu(:)554 b Fv(\(32\))456 2756 y(The)29 b(map)g(\003)g(is)h(clearly)e(con)n(tin)n(uous)g(w.r.t.)42 b(the)30 b(top)r(ology)e(of)i Fm(D)p Fv(\()p Fn(R)2653 2726 y Fo(d)2691 2756 y Fv(\))g(and)f(w)n(e)g(shall)g(some-)456 2956 y(times)35 b(use)g(the)h(previous)e(form)n(ula)h(\(32\),)i(b)n(y)e (the)g(usual)g(abuse)g(of)g(notation,)i(also)d(in)i(the)456 3155 y(general)26 b(case.)555 3354 y(W)-7 b(e)34 b(sa)n(y)e(that)h Fu(t)h Fv(has)e Ft(sc)l(aling)k(de)l(gr)l(e)l(e)e Fv(sd\()p Fu(t)p Fv(\))f(=)f Fu(!)k Fv(w.r.t.)54 b(the)33 b(origin)f(in)i Fn(R)2974 3324 y Fo(d)3012 3354 y Fv(,)h(if)f Fu(!)i Fv(is)d(the)456 3553 y(in\014m)n(um)28 b(of)f(all)h Fu(!)1046 3523 y Fq(0)1092 3553 y Fr(2)23 b Fn(R)28 b Fv(for)f(whic)n(h,)1695 3892 y(lim)1699 3946 y Fo(\025)p Fq(#)p Fy(0)1824 3892 y Fu(\025)1872 3858 y Fo(!)1916 3833 y Ff(0)1943 3892 y Fu(t)1973 3904 y Fo(\025)2039 3892 y Fv(=)c(0)k Fu(;)1078 b Fv(\(33\))456 4231 y(holds)27 b(in)h(the)g(sense)f(of)g Fm(D)1289 4201 y Fq(0)1313 4231 y Fv(\()p Fn(R)1405 4201 y Fo(d)1444 4231 y Fv(\).)555 4430 y(It)32 b(should)f(b)r(e)h(clear)f (from)g(the)h(de\014nition)g(that)g(ev)n(ery)e(distribution)h Fu(t)f Fr(2)g Fm(D)3033 4400 y Fq(0)3056 4430 y Fv(\()p Fn(R)3148 4400 y Fo(d)3187 4430 y Fv(\))i(has)f(a)456 4629 y(scaling)i(degree)g Fu(!)j Fr(2)e Fv([)p Fr(\0001)p Fu(;)14 b Fv(+)p Fr(1)p Fv([.)56 b(If)34 b(the)h(distribution)f(is)g (not)g(de\014ned)g(at)g(the)h(p)r(oin)n(t)f(w)n(e)456 4829 y(w)n(an)n(t)29 b(to)g(c)n(hec)n(k,)g(then)i(the)f(scaling)e (degree)h(migh)n(t)g(also)g(b)r(e)h(equal)f(to)h(+)p Fr(1)p Fv(.)42 b(W)-7 b(e)30 b(giv)n(e)f(some)456 5028 y(examples.)p eop %%Page: 43 43 43 42 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(43)456 425 y Fs(Examples)31 b(.)555 662 y Fv(1.)41 b(T)-7 b(rivial)27 b(example.)37 b(Ev)n(ery)25 b Fu(\036)f Fr(2)f Fm(E)p Fv(\()p Fn(R)1818 632 y Fo(d)1858 662 y Fv(\))28 b(has)f(sd\()p Fu(\036)p Fv(\))d Fr(\024)e Fv(0.)555 861 y(2.)41 b(Dirac)26 b(measure.)36 b(Let)26 b Fu(\026)d Fr(2)h Fm(E)1590 831 y Fq(0)1614 861 y Fv(\()p Fn(R)1706 831 y Fo(d)1745 861 y Fv(\))i(with)h Fu(\026)p Fv(\()p Fu(\036)p Fv(\))d(=)f Fu(\036)p Fv(\(0\),)k Fu(\036)d Fr(2)f Fm(E)p Fv(\()p Fn(R)2764 831 y Fo(d)2804 861 y Fv(\),)k(then)g(sd\()p Fu(\026)p Fv(\))c(=)g Fu(d)p Fv(.)555 1060 y(3.)41 b(F)-7 b(eynman)33 b(propagator.)49 b(In)32 b(the)h(case)f(of)g(a)g(free)h(massiv)n(e)e(scalar)g(\014eld)h (whic)n(h)h(is)f(co-)661 1259 y(v)-5 b(arian)n(t)25 b(under)g (translation)g(on)g(Mink)n(o)n(wski)f(space-time,)i(the)g(F)-7 b(eynman)25 b(propagator)661 1459 y(can)j(b)r(e)g(written)f(as)1380 1797 y Fu(E)1441 1809 y Fo(F)1497 1797 y Fv(\()p Fu(x)p Fv(\))d(=)e(\(2)p Fu(\031)s Fv(\))1875 1763 y Fq(\000)p Fo(d)1980 1684 y Fj(Z)2235 1741 y Fu(e)2274 1711 y Fo(ip)p Fq(\001)p Fo(x)p 2087 1778 455 4 v 2087 1854 a Fu(p)2129 1830 y Fy(2)2184 1854 y Fr(\000)c Fu(m)2340 1830 y Fy(2)2396 1854 y Fv(+)g Fu(i\017)2551 1797 y(d)2594 1763 y Fo(d)2633 1797 y Fu(p)28 b(;)661 2136 y Fv(from)g(whic)n(h)f(it)h(is)g(readily)e (seen)i(that)f(sd\()p Fu(E)2068 2148 y Fo(F)2124 2136 y Fv(\))c(=)g Fu(d)c Fr(\000)f Fv(2.)555 2335 y(4.)41 b(Homogeneous)29 b(distributions.)45 b(If)31 b Fu(t)d Fr(2)g Fm(D)2010 2305 y Fq(0)2033 2335 y Fv(\()p Fn(R)2125 2305 y Fo(d)2164 2335 y Fv(\))j(is)f(homogeneous)f(of)h(order)f Fu(\013)i Fv(at)f(the)661 2535 y(origin,)d(i.e.)37 b Fu(t)1093 2547 y Fo(\025)1159 2535 y Fv(=)23 b Fu(\025)1295 2505 y Fo(\013)1343 2535 y Fu(t)p Fv(,)k(then)i(sd)o(\()p Fu(t)p Fv(\))24 b(=)f Fr(\000)p Fu(\013)p Fv(.)555 2734 y(5.)41 b(In\014nite)32 b(degree.)46 b(The)31 b(smo)r(oth)g(function)g Fu(x)e Fr(!)g Fv(exp\(1)p Fu(=x)p Fv(\),)j Fu(x)d Fr(2)g Fn(R)2838 2746 y Fy(+)2893 2734 y Fv(,)j(is)f(not)g(de\014ned)661 2933 y(at)d(the)g(origin)e(and)i(its)f(scaling)g(degree)g(w.r.t.)36 b(the)28 b(origin)f(is)g(clearly)g(in\014nite.)555 3277 y(As)i(inferred)g(from)f(the)i(4th)f(example,)g(the)g(scaling)f(degree) g(ma)n(y)g(b)r(e)i(seen)e(as)h(a)f(general-)456 3477 y(ization)d(of)g(the)h(notion)g(of)f(the)h(degree)f(of)h(homogeneit)n (y)-7 b(.)35 b(Actually)-7 b(,)26 b(our)f(extension)g(metho)r(d)456 3676 y(is)d(similar)f(to)h(the)h(extension)f(to)g(all)g(space)g(of)g(a) g(homogeneous)e(distribution)j(as)e(discussed)h(in)456 3875 y(H\177)-42 b(ormander's)23 b(b)r(o)r(ok)i([35)o(])g(whic)n(h)g (on)g(the)g(other)f(hand)h(is)g(also)f(quite)h(similar)f(to)h(the)h (Epstein)456 4074 y(and)j(Glaser)f(pro)r(cedure)h(of)g(distribution)h (splitting)g([20)o(].)42 b(The)30 b(fact)f(that)h(homogenous)e(ex-)456 4274 y(tensions)f(not)i(alw)n(a)n(ys)d(exist)i(is)h(the)f(mathematical) g(origin)f(of)i(the)f(logarithmic)f(corrections)456 4473 y(to)k(scaling)g(found)h(in)g(renormalization.)48 b(Here,)32 b(a)g(discussion)f(ab)r(out)h(space-time)f(symme-)456 4672 y(tries)25 b(and)g(their)g(implemen)n(tation)h(after)f (renormalization)f(is)h(absen)n(t.)36 b(It)25 b(will)h(b)r(e)g(presen)n (ted)456 4871 y(in)h([11].)456 5216 y Fs(Lemma)k(5.1.)40 b Ft(The)31 b(sc)l(aling)f(de)l(gr)l(e)l(e)h(ob)l(eys)f(the)g(fol)t (lowing)i(pr)l(op)l(erties:)p eop %%Page: 44 44 44 43 bop 456 226 a Fy(44)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)511 425 y Fv(\()p Fu(a)p Fv(\))42 b Ft(L)l(et)30 b Fu(t)23 b Fr(2)g Fm(D)1005 395 y Fq(0)1028 425 y Fv(\()p Fn(R)1120 395 y Fo(d)1159 425 y Fv(\))30 b Ft(have)h Fv(sd\()p Fu(t)p Fv(\))24 b(=)e Fu(!)33 b Ft(at)c Fv(0)h Ft(,)g(then)711 624 y Fv(1.)41 b Ft(L)l(et)30 b Fu(\013)23 b Fr(2)g Fn(N)1175 594 y Fo(d)1244 624 y Ft(b)l(e)30 b(any)g(multiindex,)g(then)f Fv(sd\()p Fu(@)2282 594 y Fo(\013)2330 624 y Fu(t)p Fv(\))23 b Fr(\024)g Fu(!)e Fv(+)d Fr(j)p Fu(\013)p Fr(j)30 b Ft(.)711 824 y Fv(2.)41 b Ft(L)l(et)30 b Fu(\013)23 b Fr(2)g Fn(N)1175 794 y Fo(d)1244 824 y Ft(b)l(e)30 b(any)g(multiindex,) g(then)f Fv(sd\()p Fu(x)2280 794 y Fo(\013)2328 824 y Fu(t)p Fv(\))24 b Fr(\024)e Fu(!)f Fr(\000)d(j)p Fu(\013)p Fr(j)31 b Ft(.)711 1023 y Fv(3.)41 b Ft(L)l(et)30 b Fu(f)h Fr(2)24 b Fm(E)p Fv(\()p Fn(R)1254 993 y Fo(d)1293 1023 y Fv(\))p Ft(,)30 b(then)g Fv(sd\()p Fu(f)9 b(t)p Fv(\))23 b Fr(\024)g Fv(sd\()p Fu(t)p Fv(\))30 b Ft(.)520 1222 y Fv(\()p Fu(b)p Fv(\))41 b Ft(F)-6 b(or)30 b Fu(t)846 1234 y Fo(i)897 1222 y Fr(2)23 b Fm(D)1044 1192 y Fq(0)1068 1222 y Fv(\()p Fn(R)1160 1192 y Fo(d)1195 1200 y Fh(i)1225 1222 y Fv(\))p Fu(;)14 b(i)23 b Fv(=)f(1)p Fu(;)14 b Fv(2)29 b Ft(we)h(have)h Fv(sd\()p Fu(t)2038 1234 y Fy(1)2094 1222 y Fr(\012)18 b Fu(t)2207 1234 y Fy(2)2244 1222 y Fv(\))23 b(=)g(sd\()p Fu(t)2528 1234 y Fy(1)2565 1222 y Fv(\))c(+)f(sd\()p Fu(t)2840 1234 y Fy(2)2878 1222 y Fv(\))30 b Ft(.)456 1653 y(Pr)l(o)l(of.)43 b Fv(The)28 b(\014rst)g(t)n(w)n(o)f(cases)g(in)h(\()p Fu(a)p Fv(\))g(as)g(w)n(ell)f (as)h(\()p Fu(b)p Fv(\))g(are)f(straigh)n(tforw)n(ard.)35 b(The)28 b(third)g(case)456 1852 y(in)h(\()p Fu(a)p Fv(\))h(follo)n(ws) f(from)g(the)g(fact)h(that,)g(b)n(y)f(the)h(Banac)n(h-Steinhaus)e (principle,)i(a)f(con)n(v)n(ergen)n(t)456 2052 y(sequence)k(of)g (distributions)g(is)h(uniformly)f(b)r(ounded.)55 b(Hence,)35 b(for)e(ev)n(ery)f Fu(!)2950 2022 y Fq(0)3006 2052 y Fu(>)h Fv(sd\()p Fu(t)p Fv(\))h(and)456 2251 y(ev)n(ery)26 b(compact)h(set)h Fu(K)g Fr(\032)23 b Fn(R)1384 2221 y Fo(d)1450 2251 y Fv(there)28 b(is)f(some)g(p)r(olynomial)g Fu(P)39 b Fv(suc)n(h)28 b(that)1225 2590 y Fr(j)p Fu(\025)1296 2555 y Fo(!)1340 2530 y Ff(0)1367 2590 y Fu(t)1397 2602 y Fo(\025)1441 2590 y Fv(\()p Fu(\036)p Fv(\))p Fr(j)c(\024)31 b Fv(sup)1689 2659 y Fo(x)p Fq(2)p Fo(K)1845 2590 y Fr(j)p Fu(P)12 b Fv(\()p Fu(@)5 b Fv(\))p Fu(\036)p Fv(\()p Fu(x)p Fv(\))p Fr(j)24 b(\021)f(jj)p Fu(\036)p Fr(jj)2482 2602 y Fq(1)p Fo(;P)2652 2590 y Fu(:)622 b Fv(\(34\))456 2928 y(Hence,)27 b(for)g Fu(f)32 b Fr(2)24 b Fm(E)p Fv(\()p Fn(R)1146 2898 y Fo(d)1185 2928 y Fv(\),)k(w)n(e)f(ha)n(v)n(e)1238 3267 y Fr(j)p Fv(\()p Fu(f)9 b(t)p Fv(\))1405 3279 y Fo(\025)1449 3267 y Fv(\()p Fu(\036)p Fv(\))p Fr(j)24 b Fv(=)f Fr(j)p Fu(t)1750 3279 y Fo(\025)1793 3267 y Fv(\()p Fu(f)1866 3279 y Fo(\025)1910 3267 y Fu(\036)p Fv(\))p Fr(j)g(\024)g Fu(\025)2173 3233 y Fo(!)2217 3208 y Ff(0)2244 3267 y Fr(jj)p Fu(f)2331 3279 y Fo(\025)2374 3267 y Fu(\036)p Fr(jj)2469 3279 y Fq(1)p Fo(;P)2639 3267 y Fu(:)635 b Fv(\(35\))456 3606 y(The)37 b(statemen)n(t)h(follo)n (ws)e(no)n(w)h(from)g(the)h(b)r(oundedness)g(of)f(the)h(sequence)f Fr(jj)p Fu(f)3052 3618 y Fo(\025)3096 3606 y Fu(\036)p Fr(jj)3191 3618 y Fq(1)p Fo(;P)3370 3606 y Fv(as)456 3805 y Fu(\025)23 b Fr(!)g Fv(0.)2682 b Fl(\004)456 4219 y Fv(5.2.)46 b Fs(Extensions)c(of)g(distributions)f(to)h(a)h(p)s(oin)m (t.)d Fv(W)-7 b(e)37 b(no)n(w)g(w)n(an)n(t)f(to)h(sho)n(w)e(ho)n(w)i (to)456 4419 y(extend)i(a)g(distribution)g Fu(t)k Fr(2)g Fm(D)1522 4388 y Fq(0)1545 4419 y Fv(\()p Fn(R)1637 4388 y Fo(d)1680 4419 y Fr(n)5 b(f)p Fv(0)p Fr(g)n Fv(\))40 b(to)f(all)g(space)f(b)n(y)h(using)g(the)h(concept)f(of)g(the)456 4618 y(scaling)30 b(degree.)48 b(The)32 b(scaling)e(degree)h(can)g (easily)g(b)r(e)h(de\014ned)g(for)f(suc)n(h)g(distributions)h(b)n(y)456 4817 y(restricting)37 b(the)j(test)f(functions)g(appropriately)-7 b(.)69 b(Equiv)-5 b(alen)n(tly)e(,)41 b(w)n(e)e(can)f(also,)j(for)d (eac)n(h)456 5016 y Fu(\037)j Fr(2)h Fm(E)p Fv(\()p Fn(R)788 4986 y Fo(d)828 5016 y Fv(\))d(with)g(0)j Fr(62)g Fv(supp\()p Fu(\037)p Fv(\))d(lo)r(ok)f(at)h(the)g(b)r(eha)n(viour)f(of)h(the)g (sequences)f Fu(\037t)3191 5028 y Fo(\025)3234 5016 y Fv(,)k(no)n(w)456 5216 y(considered)26 b(as)h(sequences)g(in)h Fm(D)1507 5185 y Fq(0)1530 5216 y Fv(\()p Fn(R)1622 5185 y Fo(d)1661 5216 y Fv(\).)p eop %%Page: 45 45 45 44 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(45)555 425 y Fv(There)31 b(are)g(three)g(p)r(ossible)g(cases;)i(when)f(the)g(scaling)e(degree)h (is)g(+)p Fr(1)p Fv(;)i(in)f(this)g(case)f(no)456 624 y(extension)k(to)h(a)f(distribution)h(on)g Fn(R)k Fv(exists;)g(when)c (the)g(scaling)f(degree)g Fu(!)j Fv(is)e(\014nite,)j(but)456 824 y Fu(!)31 b Fr(\025)d Fu(d)p Fv(;)33 b(then)e(a)f(\014nite)i (dimensional)e(set)h(of)g(extensions)f(exists;)i(or)e(otherwise)g Fu(!)h(<)d(d)p Fv(.)47 b(W)-7 b(e)456 1023 y(\014rst)27 b(study)h(the)g(third)f(case.)456 1397 y Fs(Theorem)k(5.2.)40 b Ft(L)l(et)33 b Fu(t)1228 1409 y Fy(0)1295 1397 y Fr(2)d Fm(D)1449 1367 y Fq(0)1472 1397 y Fv(\()p Fn(R)1564 1367 y Fo(d)1608 1397 y Fr(n)5 b(f)p Fv(0)p Fr(g)n Fv(\))33 b Ft(have)i(sc)l(aling)f(de)l(gr)l(e)l(e)g Fu(!)f(<)c(d)34 b Ft(w.r.t.)50 b(the)34 b(origin.)456 1596 y(Ther)l(e)39 b(exists)g(a)g(unique)f Fu(t)h Fr(2)h Fm(D)1530 1566 y Fq(0)1553 1596 y Fv(\()p Fn(R)1645 1566 y Fo(d)1684 1596 y Fv(\))f Ft(with)g(sc)l(aling)h(de)l(gr)l(e)l(e)f Fu(!)i Ft(such)e(that)g Fu(t)p Fv(\()p Fu(\036)p Fv(\))h(=)f Fu(t)3268 1608 y Fy(0)3305 1596 y Fv(\()p Fu(\036)p Fv(\))p Ft(,)456 1795 y Fu(\036)23 b Fr(2)h Fm(D)p Fv(\()p Fn(R)768 1765 y Fo(d)811 1795 y Fr(n)5 b(f)p Fv(0)p Fr(g)n Fv(\))p Ft(.)456 2169 y(Pr)l(o)l(of.)43 b Fv(The)20 b(uniqueness)g(is)g(easy)-7 b(.)33 b(Indeed,)22 b(the)f(di\013erence)f(among)f(t)n(w)n(o)g(p)r (ossible)h(extensions)456 2369 y(w)n(ould)26 b(b)r(e)h(a)g (distribution)g(with)g(supp)r(ort)g(at)g Fr(f)p Fv(0)p Fr(g)p Fv(.)35 b(By)27 b(a)f(w)n(ell)h(kno)n(wn)f(structural)g(Theorem) 456 2568 y(of)f(distribution)g(theory)f(this)h(last)g(is)g(giv)n(en)f (b)n(y)g Fu(P)12 b Fv(\()p Fu(@)5 b Fv(\))p Fu(\016)29 b Fv(where)24 b Fu(P)37 b Fv(is)25 b(a)f(p)r(olynomial)h(of)g(degree) 456 2767 y(deg)o(\()p Fu(P)12 b Fv(\))39 b(and)f Fu(\016)k Fv(is)c(Dirac)g(measure)f(at)h(the)h(origin.)68 b(But)39 b(this)f(distribution)h(has)e(scaling)456 2967 y(degree)26 b(equal)h(to)h Fu(d)18 b Fv(+)g(deg\()p Fu(P)12 b Fv(\),)28 b(hence)g(a)f(con)n(tradiction.)555 3166 y(Let)f(us)g(no)n(w)f (consider)f(a)i(smo)r(oth)f(function)h(of)g(compact)f(supp)r(ort)h Fu(#)g Fv(suc)n(h)f(that)h Fu(#)d Fv(=)g(1)i(in)456 3365 y(a)i(neigh)n(b)r(ourho)r(o)r(d)f(of)i(the)g(origin.)36 b(Set)28 b Fu(#)1790 3377 y Fo(\025)1833 3365 y Fv(\()p Fu(x)p Fv(\))1989 3318 y Fu(:)1968 3365 y Fv(=)23 b Fu(#)p Fv(\()p Fu(\025x)p Fv(\))p Fu(;)14 b(\025)25 b Fr(2)e Fn(R)28 b Fv(and)1384 3704 y Fu(t)1414 3669 y Fy(\()p Fo(n)p Fy(\))1555 3657 y Fu(:)1534 3704 y Fv(=)23 b(\(1)18 b Fr(\000)g Fu(#)1846 3716 y Fy(2)1879 3700 y Fh(n)1924 3704 y Fv(\))p Fu(t)1986 3716 y Fy(0)2051 3704 y Fu(;)180 b(n)23 b Fr(2)g Fn(N)28 b Fu(;)456 4042 y Fv(where)33 b(no)n(w)g Fu(t)911 4012 y Fy(\()p Fo(n)p Fy(\))1042 4042 y Fv(is)h(a)g(sequence)f(of)h(distributions)g(de\014ned)g(on)g (the)g(whole)g Fn(R)3016 4012 y Fo(d)3054 4042 y Fv(.)56 b(W)-7 b(e)35 b(wish)456 4242 y(to)28 b(sho)n(w)h(that)g(the)g (sequence)g(con)n(v)n(erges)d(in)k(the)f(w)n(eak)2234 4212 y Fq(\003)2300 4242 y Fv(top)r(ology)f(of)h Fm(D)2808 4212 y Fq(0)2831 4242 y Fv(\()p Fn(R)2923 4212 y Fo(d)2962 4242 y Fv(\).)42 b(Because)28 b(of)456 4441 y(the)k(sequen)n(tial)e (completeness)h(of)h Fm(D)1665 4411 y Fq(0)1688 4441 y Fv(\()p Fn(R)1780 4411 y Fo(d)1819 4441 y Fv(\))g(it)g(is)f (su\016cien)n(t)h(to)f(pro)n(v)n(e)f(that)i(it)g(is)f(a)h(Cauc)n(h)n(y) 456 4640 y(sequence.)k(Let)28 b Fu(\036)23 b Fr(2)h Fm(D)p Fv(\()p Fn(R)1293 4610 y Fo(d)1332 4640 y Fv(\))k(and)f(lo)r(ok)g(at) 1171 4936 y(\()p Fu(t)1233 4902 y Fy(\()p Fo(n)p Fy(+1\))1433 4936 y Fr(\000)18 b Fu(t)1546 4902 y Fy(\()p Fo(n)p Fy(\))1643 4936 y Fv(\)\()p Fu(\036)p Fv(\))24 b(=)f(\()p Fu(\036t)2011 4948 y Fy(0)2049 4936 y Fv(\)\()p Fu(#)2162 4948 y Fy(2)2195 4932 y Fh(n)2259 4936 y Fr(\000)18 b Fu(#)2391 4951 y Fy(2)2424 4935 y Fh(n)p FC(+1)2540 4936 y Fv(\))1812 5185 y(=)23 b(2)1942 5151 y Fq(\000)p Fo(nd)2073 5185 y Fv(\()p Fu(\036t)2184 5197 y Fy(0)2222 5185 y Fv(\))2254 5201 y Fy(2)2287 5184 y Ff(\000)p Fh(n)2377 5185 y Fv(\()p Fu(#)c Fr(\000)f Fu(#)2609 5197 y Fy(2)2646 5185 y Fv(\))28 b Fu(:)3297 5050 y Fv(\(36\))p eop %%Page: 46 46 46 45 bop 456 226 a Fy(46)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(According)c(to)i(Lemma)g (5.1,)f(\()p Fu(a)p Fv(\)3)p Fu(:)p Fv(,)h(this)h(sequence)e(is)h(ma)5 b(jorized,)26 b(for)g(ev)n(ery)g Fu(!)3021 395 y Fq(0)3067 425 y Fr(2)h Fv(])p Fu(!)s(;)14 b(d)p Fv([)27 b(b)n(y)456 624 y(const.)g(2)738 594 y Fo(n)p Fy(\()p Fo(!)849 569 y Ff(0)871 594 y Fq(\000)p Fo(d)p Fy(\))987 624 y Fv(,)h(hence)g(it)g (is)f(summable)h(as)f(required.)36 b(The)27 b(limit)1253 963 y Fu(t)p Fv(\()p Fu(\036)p Fv(\))1441 916 y Fu(:)1420 963 y Fv(=)52 b(lim)1508 1013 y Fo(n)p Fq(!1)1695 963 y Fu(t)1725 929 y Fy(\()p Fo(n)p Fy(\))1822 963 y Fv(\()p Fu(\036)p Fv(\))29 b Fu(;)180 b Fr(8)p Fu(\036)23 b Fr(2)h Fm(D)p Fv(\()p Fn(R)2525 929 y Fo(d)2564 963 y Fv(\))k Fu(;)456 1302 y Fv(then)35 b(de\014nes)h(an)f(extension)g(of)g Fu(t)1563 1314 y Fy(0)1600 1302 y Fv(.)61 b(It)35 b(is)g(ob)n(vious)g (that)g(the)h(scaling)e(degree)h(of)g Fu(t)g Fv(is)g(not)456 1501 y(smaller)26 b(than)i Fu(!)s Fv(.)37 b(It)28 b(remains)e(to)i(pro) r(of)f(that)h(it)g(is)f(not)h(bigger)e(than)i Fu(!)s Fv(.)555 1700 y(Pic)n(k)f Fu(\036)c Fr(2)h Fm(D)p Fv(\()p Fn(R)1053 1670 y Fo(d)1092 1700 y Fv(\))k(and)f(consider)g(the)h(follo) n(wing)e(expression:)1294 2039 y Fu(t)1324 2051 y Fo(\025)1367 2039 y Fv(\()p Fu(\036)p Fv(\))f(=)51 b(lim)1592 2089 y Fo(n)p Fq(!1)1779 2039 y Fu(\025)1827 2005 y Fq(\000)p Fo(d)1919 2039 y Fu(t)1949 2051 y Fy(0)1986 2039 y Fv(\(\(1)18 b Fr(\000)g Fu(#)2242 2051 y Fy(2)2275 2035 y Fh(n)2320 2039 y Fv(\))p Fu(\036)2401 2055 y Fo(\025)2440 2038 y Ff(\000)p FC(1)2523 2039 y Fv(\))28 b Fu(:)456 2378 y Fv(Let)f Fu(R)q(;)14 b(\017)23 b(>)f Fv(0)28 b(b)r(e)g(suc)n(h)f (that)h(supp\()p Fu(\036)p Fv(\))c Fr(\032)f(f)p Fu(x;)14 b Fr(j)p Fu(x)p Fr(j)23 b Fu(<)g(R)q Fr(g)j Fv(and)i Fu(#)p Fv(\()p Fu(x)p Fv(\))c(=)f(1)k(for)g Fr(j)p Fu(x)p Fr(j)d Fu(<)e(\017)p Fv(.)37 b(Then,)1471 2717 y(\(1)19 b Fr(\000)f Fu(#)p Fv(\(2)1770 2682 y Fo(n)1815 2717 y Fu(x)p Fv(\)\))p Fu(\036)p Fv(\()p Fu(\025)2055 2682 y Fq(\000)p Fy(1)2146 2717 y Fu(x)p Fv(\))24 b(=)e(0)28 b Fu(;)456 3055 y Fv(whenev)n(er)e(2)862 3025 y Fq(\000)p Fo(n)959 3055 y Fu(\017)d(>)f(\025R)q Fv(.)37 b(Let)28 b(us)g(c)n(ho)r(ose)e Fu(n)1844 3067 y Fo(\025)1910 3055 y Fr(2)e Fn(N)j Fv(suc)n(h)h(that)g(2)2486 3025 y Fq(\000)p Fo(n)2579 3034 y Fh(\025)2621 3055 y Fu(\017)23 b(>)g(\025R)h(>)e Fv(2)3030 3025 y Fq(\000)p Fy(\()p Fo(n)3149 3034 y Fh(\025)3188 3025 y Fy(+1\))3302 3055 y Fu(\017)p Fv(.)555 3255 y(W)-7 b(e)28 b(ha)n(v)n(e,)1089 3556 y Fu(t)1119 3568 y Fo(\025)1163 3556 y Fv(\()p Fu(\036)p Fv(\))c(=)1441 3452 y Fq(1)1414 3477 y Fj(X)1388 3651 y Fo(n)p Fy(=)p Fo(n)1521 3660 y Fh(\025)1574 3556 y Fu(\025)1622 3521 y Fq(\000)p Fo(d)1713 3556 y Fu(t)1743 3568 y Fy(0)1780 3556 y Fv(\(\()p Fu(#)1893 3568 y Fy(2)1926 3551 y Fh(n)1990 3556 y Fr(\000)18 b Fu(#)2122 3571 y Fy(2)2155 3555 y Fh(n)p FC(+1)2271 3556 y Fv(\))p Fu(\036)2352 3571 y Fo(\025)2391 3555 y Ff(\000)p FC(1)2474 3556 y Fv(\))1300 3863 y(=)1441 3760 y Fq(1)1414 3785 y Fj(X)1388 3959 y Fo(n)p Fy(=)p Fo(n)1521 3968 y Fh(\025)1560 3863 y Fv(\(2)1634 3829 y Fo(n)1679 3863 y Fu(\025)p Fv(\))1759 3829 y Fq(\000)p Fo(d)1850 3863 y Fv(\()p Fu(t)1912 3875 y Fy(0)1949 3863 y Fv(\))1981 3879 y Fy(2)2014 3862 y Ff(\000)p Fh(n)2105 3863 y Fv(\(\()p Fu(#)h Fr(\000)f Fu(#)2369 3875 y Fy(2)2406 3863 y Fv(\))p Fu(\036)2487 3879 y Fy(2)2520 3862 y Ff(\000)p Fh(n)2607 3879 y Fo(\025)2646 3862 y Ff(\000)p FC(1)2728 3863 y Fv(\))28 b Fu(:)3297 3718 y Fv(\(37\))555 4194 y(The)36 b(set)f Fr(f)p Fv(\()p Fu(#)23 b Fr(\000)g Fu(#)1154 4206 y Fy(2)1192 4194 y Fv(\))p Fu(\036)1273 4206 y Fo(\026)1318 4194 y Fu(;)14 b(\026)36 b(<)f Fv(const.)p Fr(g)g Fv(is)g(b)r(ounded)h (in)f Fm(D)p Fv(\()p Fn(R)2534 4164 y Fo(d)2578 4194 y Fr(n)5 b(f)p Fv(0)p Fr(g)m Fv(\).)61 b(Hence)35 b(for)g(ev)n(ery)456 4394 y Fu(!)511 4363 y Fq(0)557 4394 y Fu(>)22 b(!)31 b Fv(w)n(e)c(\014nd)h(a)f(constan)n(t)g Fu(c)c(>)g Fv(0)k(suc)n(h)g (that)1041 4732 y Fr(j)p Fv(\()p Fu(t)1126 4744 y Fy(0)1163 4732 y Fv(\))1195 4748 y Fy(2)1228 4731 y Ff(\000)p Fh(n)1318 4732 y Fv(\(\()p Fu(#)20 b Fr(\000)e Fu(#)1583 4744 y Fy(2)1620 4732 y Fv(\))p Fu(\036)1701 4748 y Fy(2)1734 4731 y Ff(\000)p Fh(n)1820 4748 y Fo(\025)1859 4731 y Ff(\000)p FC(1)1941 4732 y Fv(\))p Fr(j)24 b(\024)50 b Fu(c)28 b Fv(2)2241 4698 y Fo(n!)2326 4673 y Ff(0)2352 4732 y Fu(;)180 b(n)23 b Fr(\025)f Fu(n)2765 4744 y Fo(\025)2836 4732 y Fu(:)p eop %%Page: 47 47 47 46 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(47)456 425 y Fv(Inserting)27 b(this)g(estimate)h(bac)n(k)f(to)g(Eqn.)g(\(37\))h(w)n(e)f(ha)n(v)n(e) 987 731 y Fr(j)p Fu(t)1040 743 y Fo(\025)1083 731 y Fv(\()p Fu(\036)p Fv(\))p Fr(j)d Fu(<)51 b(c)27 b(\025)1470 697 y Fq(\000)p Fo(d)1628 627 y Fq(1)1601 652 y Fj(X)1575 826 y Fo(n)p Fy(=)p Fo(n)1708 835 y Fh(\025)1761 731 y Fv(2)1803 697 y Fq(\000)p Fo(n)p Fy(\()p Fo(d)p Fq(\000)p Fo(!)2053 671 y Ff(0)2074 697 y Fy(\))2127 731 y Fv(=)51 b Fu(c)27 b(\025)2354 697 y Fq(\000)p Fo(d)2487 675 y Fv(2)2529 645 y Fq(\000)p Fo(n)2622 654 y Fh(\025)2660 645 y Fy(\()p Fo(d)p Fq(\000)p Fo(!)2817 619 y Ff(0)2839 645 y Fy(\))p 2455 712 446 4 v 2455 789 a Fv(1)18 b Fr(\000)g Fv(2)2640 765 y Fq(\000)p Fy(\()p Fo(d)p Fq(\000)p Fo(!)2849 749 y Ff(0)2870 765 y Fy(\))1243 1050 y Fr(\024)1546 993 y Fu(c)p 1341 1030 V 1341 1108 a Fv(1)g Fr(\000)g Fv(2)1526 1084 y Fq(\000)p Fy(\()p Fo(d)p Fq(\000)p Fo(!)1735 1067 y Ff(0)1756 1084 y Fy(\))1796 1050 y Fu(\025)1844 1015 y Fq(\000)p Fo(d)1949 932 y Fj(\022)2020 993 y Fv(2)p Fu(R)p 2020 1030 106 4 v 2056 1106 a(\017)2135 932 y Fj(\023)2196 950 y Fo(d)p Fq(\000)p Fo(!)2327 925 y Ff(0)2367 1050 y Fu(\025)2415 1015 y Fo(d)p Fq(\000)p Fo(!)2546 990 y Ff(0)2596 1050 y Fr(\024)k Fu(c)2719 1015 y Fq(0)2743 1050 y Fu(\025)2791 1015 y Fq(\000)p Fo(!)2887 990 y Ff(0)456 1339 y Fv(for)27 b(some)g(constan)n(t)g Fu(c)1162 1308 y Fq(0)1208 1339 y Fu(>)22 b Fv(0.)37 b(This)28 b(pro)n(v)n(es)d(the)j(assertion.)1050 b Fl(\004)555 1945 y Fv(W)-7 b(e)37 b(no)n(w)f(deal)g(with)i(the)f(extension)f(pro)r (cedure)f(in)i(case)f(a)g(distribution)h(has)f(a)g(\014nite)456 2144 y(scaling)28 b(degree)g Fu(!)h Fr(\025)c Fu(d)p Fv(.)43 b(This)29 b(extension)g(pro)r(cedure)f(corresp)r(onds)g(to)h (renormalization)e(in)456 2343 y(other)22 b(sc)n(hemes.)34 b(T)-7 b(o)23 b(adhere)f(more)g(to)h(the)g(standard)f(notation)g(w)n(e) h(in)n(tro)r(duce)f(the)i(degree)e(of)456 2543 y(singularit)n(y)j Fu(\032)953 2496 y(:)932 2543 y Fv(=)e Fu(!)d Fr(\000)e Fu(d)p Fv(.)37 b(This)27 b(is)g(the)h(analog)d(of)j(the)f(degree)f(of)i (div)n(ergence)d(of)j(a)e(F)-7 b(eynman)456 2742 y(diagram.)555 2941 y(Let)28 b Fm(D)773 2953 y Fo(\032)811 2941 y Fv(\()p Fn(R)903 2911 y Fo(d)942 2941 y Fv(\))g(b)r(e)f(the)h(set)f(of)g(all)g (smo)r(oth)g(functions)h(of)f(compact)f(supp)r(ort)h(whic)n(h)h(v)-5 b(anish)456 3140 y(of)26 b(order)g Fu(\032)h Fv(at)f(the)i(origin)d (and)i(let)g Fu(W)39 b Fv(b)r(e)27 b(a)f(pro)5 b(jection)26 b(from)h Fm(D)p Fv(\()p Fn(R)2641 3110 y Fo(d)2680 3140 y Fv(\))g(on)n(to)f Fm(D)2994 3152 y Fo(\032)3032 3140 y Fv(\()p Fn(R)3124 3110 y Fo(d)3163 3140 y Fv(\).)37 b(Since)456 3340 y(the)29 b(orthogonal)e(complemen)n(t)h(of)h Fm(D)1646 3352 y Fo(\032)1685 3340 y Fv(\()p Fn(R)1777 3309 y Fo(d)1816 3340 y Fv(\))g(consists)f(of)h(the)g(deriv)-5 b(ativ)n(es)28 b(of)h(the)g Fu(\016)s Fv(-function)456 3539 y(up)f(to)f(order)f Fu(\032)p Fv(,)i Fu(W)40 b Fv(is)27 b(of)h(the)g(form)1444 3878 y Fu(W)12 b(\036)24 b Fv(=)e Fu(\036)d Fr(\000)1870 3799 y Fj(X)1845 3981 y Fq(j)p Fo(\013)p Fq(j\024)p Fo(\032)2028 3878 y Fk(w)2092 3890 y Fo(\013)2140 3878 y Fu(@)2189 3843 y Fo(\013)2236 3878 y Fu(\036)p Fv(\(0\))28 b Fu(;)855 b Fv(\(38\))456 4216 y(with)28 b Fk(w)709 4228 y Fo(\013)784 4216 y Fv(b)r(eing)g(smo)r(oth) f(functions)h(of)f(compact)h(supp)r(ort)f(suc)n(h)g(that)h Fu(@)2809 4186 y Fo(\013)2856 4216 y Fk(w)2920 4228 y Fo(\014)2965 4216 y Fv(\(0\))23 b(=)g Fu(\016)3222 4186 y Fo(\013)3219 4240 y(\014)3269 4216 y Fv(.)456 4801 y Fs(Theorem)31 b(5.3.)40 b Ft(L)l(et)24 b Fu(t)1219 4813 y Fy(0)1280 4801 y Fr(2)f Fm(D)1427 4771 y Fq(0)1450 4801 y Fv(\()p Fn(R)1542 4771 y Fo(d)1586 4801 y Fr(n)5 b(f)p Fv(0)p Fr(g)n Fv(\))25 b Ft(have)h(a)f(\014nite)f(sc)l(aling)i (de)l(gr)l(e)l(e)f Fu(!)h Fr(\025)d Fu(d)p Ft(.)37 b(Then)25 b(ther)l(e)456 5000 y(exist)30 b(extensions)g Fu(t)24 b Fr(2)h Fm(D)1255 4970 y Fq(0)1278 5000 y Fv(\()p Fn(R)1370 4970 y Fo(d)1409 5000 y Fv(\))30 b Ft(of)i Fu(t)1600 5012 y Fy(0)1667 5000 y Ft(with)f(the)g(same)g(sc)l(aling)g(de)l(gr)l (e)l(e,)g(and,)h(given)f Fu(W)12 b Ft(,)31 b(they)456 5199 y(ar)l(e)f(uniquely)g(determine)l(d)g(by)g(their)h(values)f(on)g (the)g(test)f(functions)g Fk(w)2746 5211 y Fo(\013)2794 5199 y Ft(.)p eop %%Page: 48 48 48 47 bop 456 226 a Fy(48)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Ft(Pr)l(o)l(of.)43 b Fv(An)n(y)32 b Fu(\036)g Fr(2)g Fm(D)p Fv(\()p Fn(R)1227 395 y Fo(d)1266 425 y Fv(\))h(can)g(b)r(e)g(uniquely)g(decomp)r(osed)f (as)g Fu(\036)g Fv(=)f Fu(\036)2751 437 y Fy(1)2810 425 y Fv(+)22 b Fu(\036)2946 437 y Fy(2)3016 425 y Fv(where)33 b Fu(\036)3311 437 y Fy(1)3380 425 y Fv(=)456 562 y Fj(P)543 649 y Fq(j)p Fo(\013)p Fq(j\024)p Fo(\032)730 624 y Fk(w)794 636 y Fo(\013)842 624 y Fu(@)891 594 y Fo(\013)938 624 y Fu(\036)p Fv(\(0\))28 b(and)f Fu(\036)1331 636 y Fy(2)1392 624 y Fr(2)d Fm(D)1540 636 y Fo(\032)1578 624 y Fv(\()p Fn(R)1670 594 y Fo(d)1709 624 y Fv(\).)37 b Fu(\036)1850 636 y Fy(2)1916 624 y Fv(has)27 b(the)h(form,)1464 963 y Fu(\036)1513 975 y Fy(2)1551 963 y Fv(\()p Fu(x)p Fv(\))c(=)1858 884 y Fj(X)1774 1066 y Fq(j)p Fo(\013)p Fq(j)p Fy(=[)p Fo(\032)p Fy(]+1)2077 963 y Fu(x)2124 929 y Fo(\013)2172 963 y Fu( )2226 975 y Fo(\013)2273 963 y Fv(\()p Fu(x)p Fv(\))29 b Fu(;)456 1307 y Fv(with)f Fu( )699 1319 y Fo(\013)769 1307 y Fr(2)c Fm(D)p Fv(\()p Fn(R)1009 1277 y Fo(d)1047 1307 y Fv(\).)38 b(W)-7 b(e)28 b(set)1292 1646 y Fr(h)p Fu(t;)14 b(\036)p Fr(i)24 b Fv(=)1668 1567 y Fj(X)1583 1749 y Fq(j)p Fo(\013)p Fq(j)p Fy(=[)p Fo(\032)p Fy(]+1)1873 1646 y Fr(h)p Fu(x)1952 1612 y Fo(\013)2000 1646 y Fu(t)2030 1658 y Fy(0)2067 1646 y Fu(;)14 b( )2158 1658 y Fo(\013)2206 1646 y Fr(i)k Fv(+)g Fr(h)p Fu(t;)c(\036)2487 1658 y Fy(1)2525 1646 y Fr(i)28 b Fu(:)689 b Fv(\(39\))456 1985 y(Since,)23 b(b)n(y)f(Lemma)g(5.1,)g Fu(x)1291 1955 y Fo(\013)1339 1985 y Fu(t)1369 1997 y Fy(0)1428 1985 y Fv(has)g(scaling)f(degree)g(equal)h(to)f Fu(\032)7 b Fr(\000)g Fv([)p Fu(\032)p Fv(])g Fr(\000)g Fv(1)g(+)g Fu(d)24 b Fv(whic)n(h)e(is)g(strictly)456 2184 y(smaller)k(than)i Fu(d)g Fv(this)g(term)f(has)g(a)h(unique)f(extension)h(b)n(y)f(Theorem) g(5.2.)555 2383 y(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)f(that)i Fu(t)g Fv(has)f(the)h(same)f(scaling)f(degree)h(as)g Fu(t)2495 2395 y Fy(0)2532 2383 y Fv(.)37 b(W)-7 b(e)28 b(write,)993 2722 y Fu(t)1023 2734 y Fo(\025)1066 2722 y Fv(\()p Fu(\036)p Fv(\))c(=)f(\()p Fu(t)1353 2734 y Fy(0)1409 2722 y Fr(\016)18 b Fu(W)12 b Fv(\))1591 2734 y Fo(\025)1635 2722 y Fv(\()p Fu(\036)p Fv(\))19 b Fr(\000)1875 2643 y Fj(X)1850 2825 y Fq(j)p Fo(\013)p Fq(j\024)p Fo(\032)2033 2722 y Fu(t)p Fv(\()p Fk(w)2159 2734 y Fo(\013)2207 2722 y Fv(\))14 b(\()p Fu(@)2334 2688 y Fo(\013)2381 2722 y Fu(\036)p Fv(\))h(\(0\))p Fu(\025)2631 2688 y Fq(\000)p Fo(d)p Fq(\000j)p Fo(\013)p Fq(j)2884 2722 y Fu(:)456 3061 y Fv(The)33 b(second)f(term)h(clearly)e(has)i(scaling)f(degree)g (less)g(or)g(equal)g(to)h Fu(\032)22 b Fv(+)g Fu(d)32 b Fv(=)f Fu(!)s Fv(.)53 b(The)33 b(\014rst)456 3260 y(term)27 b(can)g(b)r(e)h(written)g(in)g(the)g(form)1260 3599 y(\()p Fu(t)1322 3611 y Fy(0)1359 3599 y Fv(\))1391 3611 y Fo(\025)1435 3599 y Fv(\()p Fu(W)12 b(\036)p Fv(\))19 b(+)f(\()p Fu(t)1802 3611 y Fy(0)1840 3599 y Fv(\))1872 3611 y Fo(\025)1916 3599 y Fv(\(\()p Fu(W)12 b(\036)2119 3614 y Fo(\025)2158 3598 y Ff(\000)p FC(1)2241 3599 y Fv(\))2273 3611 y Fo(\025)2335 3599 y Fr(\000)18 b Fu(W)12 b(\036)p Fv(\))28 b Fu(:)456 3937 y Fv(By)c(assumption,)h(the)h(\014rst)f(term)f(has)h(scaling)f (degree)g Fu(!)s Fv(.)36 b(T)-7 b(o)24 b(analyze)g(the)i(second)e(term) h(w)n(e)456 4137 y(write)771 4475 y(\(\()p Fu(W)12 b(\036)974 4491 y Fo(\025)1013 4475 y Ff(\000)p FC(1)e Fv(\))1128 4487 y Fo(\025)1191 4475 y Fr(\000)18 b Fu(W)12 b(\036)p Fv(\)\()p Fu(x)p Fv(\))24 b(=)1693 4397 y Fj(X)1668 4579 y Fq(j)p Fo(\013)p Fq(j\024)p Fo(\032)1851 4383 y Fj(\020)1900 4475 y Fk(w)1964 4487 y Fo(\013)2012 4475 y Fv(\()p Fu(x)p Fv(\))19 b Fr(\000)f Fu(\025)2273 4441 y Fq(\000j)p Fo(\013)p Fq(j)2412 4475 y Fk(w)2476 4487 y Fo(\013)2524 4475 y Fv(\()p Fu(\025x)p Fv(\))2683 4383 y Fj(\021)2748 4475 y Fv(\()p Fu(@)2829 4441 y Fo(\013)2876 4475 y Fu(\036)p Fv(\))d(\(0\))27 b Fu(:)p eop %%Page: 49 49 49 48 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(49)456 425 y Fv(\(Note)28 b(that)f(\()p Fk(w)964 437 y Fo(\013)1012 425 y Fv(\()p Fr(\001)p Fv(\))19 b Fr(\000)f Fu(\025)1249 395 y Fq(\000j)p Fo(\013)p Fq(j)1388 425 y Fk(w)1452 437 y Fo(\013)1500 425 y Fv(\()p Fu(\025)28 b Fr(\001)p Fv(\)\))c Fr(2)f Fm(D)1866 437 y Fo(\032)1905 425 y Fv(\()p Fn(R)1997 395 y Fo(d)2036 425 y Fv(\)\).)37 b(Using)28 b(the)g(iden)n(tit)n(y)-7 b(,)623 727 y(\()p Fk(w)719 739 y Fo(\013)767 727 y Fv(\()p Fu(x)p Fv(\))20 b Fr(\000)e Fu(\025)1029 692 y Fq(\000j)p Fo(\013)p Fq(j)1168 727 y Fk(w)1232 739 y Fo(\013)1279 727 y Fv(\()p Fu(\025x)p Fv(\)\))25 b(=)1583 614 y Fj(Z)1666 634 y Fy(1)1629 802 y Fo(\025)1752 670 y Fu(d)p 1727 708 94 4 v 1727 784 a(d\026)1830 727 y(\026)1880 692 y Fq(\000j)p Fo(\013)p Fq(j)2019 727 y Fk(w)2083 739 y Fo(\013)2130 727 y Fv(\()p Fu(\026x)p Fv(\))k Fu(d\026)1495 990 y Fv(=)1583 877 y Fj(Z)1666 898 y Fy(1)1629 1066 y Fo(\025)1717 990 y Fv(\()p Fu(\026x)1846 1002 y Fo(k)1887 990 y Fv(\()p Fu(@)1963 1002 y Fo(k)2005 990 y Fk(w)2069 1002 y Fo(\013)2116 990 y Fv(\)\()p Fu(\026x)p Fv(\))20 b Fr(\000)e(j)p Fu(\013)p Fr(j)p Fk(w)2575 1002 y Fo(\013)2623 990 y Fv(\()p Fu(\026x)p Fv(\)\))d Fu(\026)2881 956 y Fq(\000j)p Fo(\013)p Fq(j\000)p Fy(1)3133 990 y Fu(d\026)28 b(;)3297 1051 y Fv(\(40\))456 1321 y(w)n(e)e(get,)g(after)g(a)h(momen)n (t)f(of)g(re\015ection)g(for)g(the)h(exc)n(hange)e(of)i(the)g(order)e (b)r(et)n(w)n(een)h(in)n(tegra-)456 1520 y(tion)h(and)h(dualit)n(y)-7 b(,)27 b(that,)803 1811 y(\()p Fu(t)865 1823 y Fy(0)902 1811 y Fv(\))934 1823 y Fo(\025)978 1811 y Fv(\()q(\()p Fu(W)12 b(\036)1182 1827 y Fo(\025)1221 1810 y Ff(\000)p FC(1)1303 1811 y Fv(\))1335 1823 y Fo(\025)1397 1811 y Fr(\000)18 b Fu(W)12 b(\036)p Fv(\))24 b(=)1049 1984 y Fj(X)1024 2166 y Fq(j)p Fo(\013)p Fq(j\024)p Fo(\032)1207 2063 y Fv(\()p Fu(@)1288 2029 y Fo(\013)1335 2063 y Fu(\036)p Fv(\))15 b(\(0\))1551 1950 y Fj(Z)1634 1970 y Fy(1)1597 2139 y Fo(\025)1685 2063 y Fu(\026)1735 2029 y Fq(\000)p Fo(d)p Fq(\000j)p Fo(\013)p Fq(j\000)p Fy(1)2045 2063 y Fv(\()p Fu(t)2107 2075 y Fy(0)2145 2063 y Fv(\))2177 2079 y Fo(\025\026)2256 2062 y Ff(\000)p FC(1)2338 2063 y Fv(\(\()p Fu(x)2449 2075 y Fo(k)2491 2063 y Fu(@)2535 2075 y Fo(k)2594 2063 y Fr(\000)j(j)p Fu(\013)p Fr(j)p Fv(\))p Fk(w)2872 2075 y Fo(\013)2921 2063 y Fv(\))p Fu(d\026)28 b(:)456 2399 y Fv(The)i(in)n(tegrand)f(can)h(b)r(e)g (estimated)g(according)f(to)h(Lemma)g(5.1.)43 b(Indeed,)32 b(for)d(an)n(y)h Fu(!)3248 2369 y Fq(0)3298 2399 y Fu(>)d(!)456 2598 y Fv(w)n(e)g(ha)n(v)n(e,)1080 2866 y Fj(\014)1080 2916 y(\014)1107 2937 y Fv(\()p Fu(t)1169 2949 y Fy(0)1207 2937 y Fv(\))1239 2953 y Fo(\025\026)1318 2936 y Ff(\000)p FC(1)1414 2937 y Fv(\(\()p Fu(x)1525 2949 y Fo(k)1567 2937 y Fu(@)1611 2949 y Fo(k)1670 2937 y Fr(\000)18 b(j)p Fu(\013)p Fr(j)p Fv(\))p Fk(w)1948 2949 y Fo(\013)1997 2937 y Fv(\))2057 2866 y Fj(\014)2057 2916 y(\014)2108 2937 y Fr(\024)k Fv(const.)2422 2870 y Fj(\000)2460 2937 y Fu(\025)2508 2903 y Fq(\000)p Fy(1)2597 2937 y Fu(\026)2647 2870 y Fj(\001)2685 2886 y Fo(!)2729 2861 y Ff(0)2797 2937 y Fu(;)456 3276 y Fv(and)27 b(therefore)609 3494 y Fj(\014)609 3544 y(\014)609 3594 y(\014)609 3644 y(\014)637 3501 y(Z)720 3522 y Fy(1)683 3690 y Fo(\025)771 3614 y Fu(\026)821 3580 y Fq(\000)p Fo(d)p Fq(\000j)p Fo(\013)p Fq(j\000)p Fy(1)1131 3614 y Fv(\()p Fu(t)1193 3626 y Fy(0)1231 3614 y Fv(\))1263 3630 y Fo(\025\026)1342 3614 y Ff(\000)p FC(1)c Fv(\(\()p Fu(x)1549 3626 y Fo(k)1591 3614 y Fu(@)1635 3626 y Fo(k)1694 3614 y Fr(\000)18 b(j)p Fu(\013)p Fr(j)p Fv(\))p Fk(w)1972 3626 y Fo(\013)2021 3614 y Fv(\))c Fu(d\026)2160 3494 y Fj(\014)2160 3544 y(\014)2160 3594 y(\014)2160 3644 y(\014)2211 3614 y Fr(\024)22 b Fv(const.)28 b Fu(\025)2587 3580 y Fq(\000)p Fo(!)2683 3555 y Ff(0)2747 3558 y Fv(1)18 b Fr(\000)g Fu(\025)2938 3528 y Fo(!)2982 3503 y Ff(0)3005 3528 y Fq(\000)p Fo(d)p Fq(\000j)p Fo(\013)p Fq(j)p 2747 3595 483 4 v 2777 3671 a Fu(!)2832 3647 y Fq(0)2873 3671 y Fr(\000)g Fu(d)h Fr(\000)f(j)p Fu(\013)p Fr(j)3268 3614 y Fu(;)456 3953 y Fv(whic)n(h)27 b(pro)n(v)n(es)f(the)i(assertion.)1943 b Fl(\004)555 4419 y Fv(The)21 b(exp)r(ert)f(reader)e(can)i(no)n(w)g (pro)r(ceed)g(from)f(this)i(p)r(oin)n(t)f(to)g(study)h(the)f (renormalizabilit)n(y)456 4618 y(of)g(an)n(y)h(theory)f(whic)n(h)h (admits)g(space-time)f(translation)g(co)n(v)-5 b(ariance.)32 b(The)21 b(am)n(biguit)n(y)f(of)h(the)456 4817 y(extension)h(is)h(giv)n (en)g(b)n(y)g(terms)g(lo)r(calized)f(o)n(v)n(er)g(the)h(origin.)34 b(The)24 b(co)r(e\016cien)n(ts)e(of)i(these)f(terms)456 5016 y(can)36 b(b)r(e)h(\014xed)g(b)n(y)g(additional)f(requiremen)n (ts,)i(as)e(customary)f(in)i(p)r(erturbativ)n(e)g(quan)n(tum)456 5216 y(\014eld)27 b(theory)-7 b(.)37 b(W)-7 b(e)28 b(refer)f(the)h (reader)e(to)h([11)o(])h(for)f(more)g(details.)p eop %%Page: 50 50 50 49 bop 456 226 a Fy(50)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)555 425 y Fv(During)35 b(this)g(pro)r(cess,)h(one)f(needs)f(estimates)h(on)g(the)g(scaling)f (degrees)g(of)h(the)g(arising)456 624 y(distributions,)h(corresp)r (onding)d(to)i(the)g(p)r(o)n(w)n(er)e(coun)n(ting)i(rules.)57 b(In)35 b(addition)g(to)g(Lemma)456 824 y(5.1)29 b(estimates)h(on)g (scaling)f(degrees)g(of)i(pro)r(ducts)f(of)g(distributions)g(\(pro)n (vided)g(they)g(exist\))456 1023 y(are)i(required.)53 b(These)33 b(can)g(b)r(e)h(obtained)f(b)n(y)g(explicit)h(calculations)e (\(see)h(e.g.)54 b(the)34 b(analo-)456 1222 y(gous)d(estimates)g(in)i ([20)o(]\).)50 b(Muc)n(h)32 b(more)g(elegan)n(t)f(is)h(a)f(general)g (metho)r(d)h(whic)n(h)g(exploits)g(a)456 1421 y(microlo)r(cal)24 b(v)n(ersion)g(of)h(the)h(scaling)e(degree.)35 b(This)26 b(tec)n(hnique)f(is)g(actually)g(necessary)f(if)i(one)456 1621 y(w)n(an)n(ts)k(to)h(generalize)f(the)h(metho)r(ds)h(ab)r(o)n(v)n (e)d(to)i(generic)g(manifolds.)47 b(W)-7 b(e)31 b(shall)g(describ)r(e)g (it)456 1820 y(in)c(the)h(next)g(section.)567 2111 y(6.)45 b Fw(Surf)-7 b(a)n(ces)32 b(of)f(Unif)n(orm)h(Singularity)e(and)i(the)f (Micr)n(olocal)g(Scaling)1850 2311 y(Degree)555 2560 y Fv(The)24 b(generalization)e(of)i(the)g(previous)f(pro)r(cedure)g(to) g(the)h(case)f(of)h(submanifolds)g(is)g(what)456 2759 y(w)n(e)30 b(really)f(need)i(in)g(the)g(treatmen)n(t)f(of)h(p)r (erturbation)f(theory)f(on)i(curv)n(ed)e(spaces.)45 b(Indeed,)456 2958 y(the)32 b(description)f(giv)n(en)g(in)h(Section)f(4)h(led)g(to)f (the)h(notion)g(of)f(a)h(scaling)e(degree)h(w.r.t.)49 b(the)456 3157 y(small)29 b(diagonal)f(\001)1078 3169 y Fo(n)1152 3157 y Fv(of)i(the)g(top)r(ological)d(pro)r(duct)j Fu(M)2221 3127 y Fo(n)2266 3157 y Fv(.)42 b(Here)29 b(w)n(e)g(classify) g(the)h(b)r(eha)n(viour)456 3357 y(of)36 b(distributions)g(near)g(some) g(surface)f(b)n(y)i(a)f(microlo)r(cal)f(v)n(ersion)g(of)h(the)h (scaling)e(degree.)456 3556 y(W)-7 b(e)31 b(in)n(tro)r(duce)f(t)n(w)n (o)g(di\013eren)n(t)h(notions.)45 b(The)31 b(\014rst)g(one,)g(the)g (\(microlo)r(cal\))f(scaling)g(degree)456 3755 y(at)37 b(a)h(surface,)h(in)n(v)n(olv)n(es)d(only)i(the)g(surface)f(under)h (consideration,)h(the)f(second)f(one,)j(the)456 3955 y(transv)n(ersal)19 b(scaling)h(degree,)i(in)n(v)n(olv)n(es)d(a)j (\014bration)e(of)i(the)g(surface)e(b)n(y)i(transv)n(ersal)d(surfaces.) 456 4154 y(The)i(\014rst)f(notion)h(b)r(eha)n(v)n(es)f(v)n(ery)f (nicely)i(under)g(tensor)f(pro)r(ducts)h(and)g(restrictions,)g(whereas) 456 4353 y(the)37 b(second)g(one)g(admits)h(an)f(easy)f(generalization) g(of)h(the)h(extension)f(pro)r(cedure.)65 b(As)38 b(a)456 4552 y(matter)27 b(of)h(fact,)f(the)h(notions)f(can)h(b)r(e)g(sho)n(wn) e(to)i(b)r(e)g(equiv)-5 b(alen)n(t.)456 4817 y(6.1.)46 b Fs(Scaling)30 b(degrees)e(at)j(submanifolds.)39 b Fv(Let)26 b Fm(M)f Fv(b)r(e)h(a)g(smo)r(oth)f(paracompact)f(mani-)456 5016 y(fold)h(of)f(dimension)h Fu(d)g Fv(and)g Fu(t)g Fv(b)r(e)g(a)g(distribution)g(in)g Fm(D)2170 4986 y Fq(0)2193 5016 y Fv(\()p Fm(M)p Fv(\).)36 b(Let)25 b Fm(N)g Fr(\032)d Fm(M)j Fv(b)r(e)g(a)f(submanifold)456 5216 y(suc)n(h)j(that)h(the)g(w)n (a)n(v)n(e)e(fron)n(t)i(set)g(of)f Fu(t)h Fv(is)g(orthogonal)d(to)j (the)g(tangen)n(t)g(bundle)g Fu(T)12 b Fm(N)28 b Fv(of)g Fm(N)q Fv(,)g(i.e.)p eop %%Page: 51 51 51 50 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(51)456 425 y Fv(for)27 b(\()p Fu(x;)14 b(k)s Fv(\))23 b Fr(2)h Fv(WF\()p Fu(t)p Fv(\))29 b(with)f Fu(k)e Fr(2)d Fu(T)1538 395 y Fq(\003)1526 446 y Fo(x)1575 425 y Fm(M)p Fu(;)14 b(x)23 b Fr(2)h Fm(N)q Fv(,)1468 764 y Fr(h)p Fu(k)s(;)14 b(\030)t Fr(i)23 b Fv(=)g(0)k Fu(;)180 b Fr(8)p Fu(\030)27 b Fr(2)d Fu(T)2275 776 y Fo(x)2316 764 y Fm(N)29 b Fu(:)865 b Fv(\(41\))456 1103 y(Under)30 b(these)h(circumstances,)f Fu(t)g Fv(can)h(b)r(e)g (restricted)e(to)i(a)f(su\016cien)n(tly)g(small)g(submanifold)456 1302 y Fm(C)d Fr(\032)h Fm(M)i Fv(whic)n(h)g(in)n(tersects)g Fm(N)i Fv(in)e(a)g(single)g(p)r(oin)n(t)h Fu(x)2117 1314 y Fy(0)2155 1302 y Fv(,)g(suc)n(h)f(that)h(the)g(in)n(tersection)e(of)i (their)456 1501 y(tangen)n(t)h(spaces)g(at)h Fu(x)1177 1513 y Fy(0)1248 1501 y Fv(is)g(trivial)f(and)h(their)g(sum)g(spans)f (the)h(whole)g(tangen)n(t)g(space)f(\(the)456 1700 y(submanifolds)e Fm(C)i Fv(and)f Fm(N)h Fv(are)e Ft(tr)l(ansversal)p Fv(,)j(see)e(e.g.) 47 b([27)o(],)32 b(sym)n(b)r(olically)e Fm(C)f Fl(t)g Fm(N)q Fv(\).)48 b(This)32 b(is)456 1900 y(due)23 b(to)g(the)h(fact)f (that)h(WF\()p Fu(t)p Fv(\))g(do)r(es)f(not)g(in)n(tersect)g(the)h (conormal)d(bundle)j Fu(N)2931 1870 y Fq(\003)2969 1900 y Fm(C)f Fv(=)g Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(\))23 b Fr(2)456 2099 y Fu(T)517 2069 y Fq(\003)554 2099 y Fm(M)p Fr(jh)p Fu(k)s(;)14 b(\030)t Fr(i)28 b Fv(=)g(0)p Fu(;)14 b Fr(8)p Fu(\030)32 b Fr(2)d Fu(T)1293 2111 y Fo(x)1334 2099 y Fm(C)p Fr(g)i Fv(of)g Fm(C)p Fv(.)46 b(Namely)-7 b(,)32 b(for)e Fu(k)h Fr(2)e Fu(T)2348 2069 y Fq(\003)2336 2119 y Fo(x)2374 2127 y FC(0)2410 2099 y Fm(M)p Fv(,)i(\()p Fu(x)2625 2111 y Fy(0)2663 2099 y Fu(;)14 b(k)s Fv(\))29 b Fr(2)f Fv(WF)q(\()p Fu(t)p Fv(\))j(w)n(e)g(ha)n(v)n(e)456 2298 y Fr(h)p Fu(k)s(;)14 b(\030)t Fr(i)30 b Fv(=)g(0)i(for)f Fu(\030)j Fr(2)d Fu(T)1178 2310 y Fo(x)1216 2318 y FC(0)1252 2298 y Fm(N)q Fv(,)i(hence)f Fr(h)p Fu(k)s(;)14 b(\030)t Fr(i)31 b(6)p Fv(=)f(0)h(for)g(some)h Fu(\030)i Fr(2)d Fu(T)2543 2310 y Fo(x)2581 2318 y FC(0)2616 2298 y Fm(C)p Fv(,)j(th)n(us)e(\()p Fu(x)2989 2310 y Fy(0)3027 2298 y Fu(;)14 b(k)s Fv(\))39 b Fu(=)-51 b Fr(2)30 b Fu(N)3333 2268 y Fq(\003)3371 2298 y Fm(C)p Fv(.)456 2497 y(But)d(WF)q(\()p Fu(t)p Fv(\))18 b Fr(\\)g Fu(N)1021 2467 y Fq(\003)1059 2497 y Fm(C)27 b Fv(is)g(a)g(closed)f(conical)h(set)g(in)2060 2476 y(_)2034 2497 y Fu(T)2095 2467 y Fq(\003)2083 2521 y Fd(C)2133 2497 y Fm(M)p Fv(,)g(hence)g(its)g(complemen)n(t)g(is)h(an) e(op)r(en)456 2697 y(conical)k(neigh)n(b)r(ourho)r(o)r(d)g(of)1431 2676 y(_)1406 2697 y Fu(T)1467 2667 y Fq(\003)1455 2717 y Fo(x)1493 2725 y FC(0)1529 2697 y Fm(M)p Fv(,)i(in)f(particular)f(it) i(con)n(tains)f(a)g(set)2801 2676 y(_)2776 2697 y Fu(T)2837 2667 y Fq(\003)2825 2720 y Fo(U)2870 2728 y FC(0)2906 2697 y Fm(M)g Fv(where)g Fu(U)3320 2709 y Fy(0)3389 2697 y Fv(is)456 2896 y(an)23 b(op)r(en)g(neigh)n(b)r(ourho)r(o)r(d)f(of)h Fu(x)1463 2908 y Fy(0)1524 2896 y Fv(in)h Fm(C)p Fv(.)35 b(By)23 b(c)n(ho)r(osing)f Fm(C)h Fv(=)g Fu(U)2402 2908 y Fy(0)2462 2896 y Fv(w)n(e)g(arriv)n(e)e(at)j(the)f(conclusion.)456 3095 y(So)k(w)n(e)g(pro)n(v)n(ed,)456 3458 y Fs(Lemma)k(6.1.)40 b Ft(L)l(et)24 b Fu(t)f Fr(2)g Fm(D)1322 3428 y Fq(0)1345 3458 y Fv(\()p Fm(M)p Fv(\))i 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b(=)f(\()p Fu(x;)14 b(x)p Fv(\),)29 b Fu(x)23 b Fr(2)h Fm(N)q Fv(;)498 1359 y(\()p Fu(ii)p Fv(\))41 b Fu(\013)p Fv(\()p Fu(T)12 b Fm(N)20 b Fr(\\)f Fu(U)9 b Fv(\))23 b Fr(\032)f Fm(N)e Fr(\002)e Fm(N)q Fv(;)469 1558 y(\()p Fu(iii)p Fv(\))41 b Fu(\013)p Fv(\()p Fu(x;)14 b(\030)t Fv(\))25 b Fr(2)e(f)p Fu(x)p Fr(g)18 b(\002)g Fm(M)p Fv(,)27 b Fu(x)d Fr(2)f Fm(N)q Fu(;)14 b(\030)28 b Fr(2)23 b Fu(T)1811 1570 y Fo(x)1853 1558 y Fm(M)p Fv(;)483 1757 y(\()p Fu(iv)s Fv(\))42 b Fu(d)704 1769 y Fo(\030)741 1757 y Fu(\013)p Fv(\()p Fu(x;)14 b Fr(\001)p Fv(\))-5 b Fl(\026)995 1769 y Fo(\030)r Fy(=0)1117 1757 y Fv(=)22 b(id)1273 1769 y Fo(T)1312 1777 y Fh(x)1350 1769 y Fd(M)1422 1757 y Fv(.)555 2093 y(A)30 b(concrete)f(example)g(of)h(suc)n(h)f(a)h(map)f Fu(\013)h Fv(can)g(b)r(e)g(de\014ned,)g(whenev)n(er)f(w)n(e)g(consider) g(the)456 2292 y(manifold)23 b Fm(M)h Fv(endo)n(w)n(ed)f(with)h(a)g (\(semi-\)Riemannian)g(metric,)g(in)g(terms)g(of)g(the)g(exp)r(onen)n (tial)456 2492 y(map,)i(namely)-7 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)s Fr(i)35 b Fv(=)1324 3222 y Fj(R)1393 3289 y Fu(\036)24 b Fr(\001)f(h)p Fu(t;)14 b( )s Fr(i)35 b Fv(for)f(test-densities)g Fu(\036)h Fr(2)g Fm(D)2618 3301 y Fy(1)2655 3289 y Fv(\()p Fm(N)q Fv(\))g(and)g Fu( )i Fr(2)e Fm(D)3238 3301 y Fy(1)3275 3289 y Fv(\()p Fm(M)p Fv(\).)456 3488 y(Since)30 b Fu(U)39 b Fv(is)31 b(starshap)r(ed,)f Fu(\025)1351 3458 y Fq(\000)p Fy(1)1440 3488 y Fu(U)37 b Fr(\033)27 b Fu(U)39 b Fv(for)30 b(0)d Fu(<)g(\025)h Fr(\024)f Fv(1,)k(hence)f Fu(t)2540 3458 y Fo(\013)2540 3511 y(\025)2618 3488 y Fv(can)g(b)r(e)h (considered)e(as)h(a)456 3687 y(distribution)d(on)h Fm(D)1093 3699 y Fy(1)1130 3687 y Fv(\()p Fu(U)9 b Fv(\).)555 3886 y(As)28 b(a)f(preliminary)g(step)g(w)n(e)h(ha)n(v)n(e)e(the)i(follo)n (wing)456 4458 y Fs(Prop)s(osition)i(6.2.)40 b Ft(F)-6 b(or)26 b(any)g Fu(t)d Fr(2)g Fm(D)1668 4428 y Fq(0)1692 4458 y Fv(\()p Fm(M)p Fv(\))i Ft(which)j(satis\014es)e(the)f(hyp)l (othesis)j(of)f(L)l(emma)f(6.1,)456 4657 y(ther)l(e)j(exists)h(a)g (close)l(d)h(c)l(onic)f(set)g Fv(\000)23 b Fr(\032)1738 4636 y Fv(_)1712 4657 y Fu(T)1773 4627 y Fq(\003)1810 4657 y Fu(U)39 b Ft(such)30 b(that)527 4993 y Fv(\()p Fu(i)p Fv(\))41 b(\000)23 b Fr(?)g Fu(T)12 b Fv(\()p Fu(T)g Fm(N)18 b Fr(\\)h Fu(U)9 b Fv(\))p Ft(;)498 5192 y Fv(\()p Fu(ii)p Fv(\))41 b(WF)q(\()p Fu(t)863 5162 y Fo(\013)863 5216 y(\025)910 5192 y Fv(\))24 b Fr(\032)e Fv(\000)p Ft(.)p eop %%Page: 53 53 53 52 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(53)456 425 y Ft(Pr)l(o)l(of.)43 b Fv(Since)25 b Fu(\013)h Fv(maps)e Fu(T)12 b Fm(N)i Fr(\\)g Fu(U)34 b Fv(in)n(to)25 b Fm(N)15 b Fr(\002)f Fm(N)q Fv(,)25 b(its)h(deriv)-5 b(ativ)n(e)24 b Fu(\013)2504 437 y Fq(\003)2565 425 y Fv(:)f Fu(T)12 b(U)31 b Fr(!)23 b Fu(T)12 b Fv(\()p Fm(N)j Fr(\002)f Fm(M)p Fv(\))24 b(maps)456 624 y Fu(T)12 b Fv(\()p Fu(T)g Fm(N)g Fr(\\)f Fu(U)e Fv(\))23 b(in)n(to)h Fu(T)12 b Fv(\()p Fm(N)g Fr(\002)f Fm(N)q Fv(\).)36 b(But)24 b(WF\()p Fu(t)p Fv(\))g Fr(?)f Fu(T)12 b Fm(N)24 b Fv(implies)g(WF)q(\()p Fs(1)11 b Fr(\012)g Fu(t)p Fv(\))23 b Fr(?)f Fu(T)12 b Fv(\()p Fm(N)g Fr(\002)f Fm(N)q Fv(\),)25 b(hence)702 963 y(WF)q(\()p Fu(t)904 929 y Fo(\013)951 963 y Fv(\))f(=)e Fu(\013)1147 929 y Fq(\003)1186 963 y Fv(WF\(\()p Fs(1)d Fr(\012)f Fu(t)p Fv(\))-5 b Fl(\026)1631 978 y Fo(\013)p Fy(\()p Fo(U)6 b Fy(\))1782 963 y Fv(\))24 b Fr(?)e Fu(\013)1978 929 y Fq(\000)p Fy(1)1978 984 y Fq(\003)2068 963 y Fv(\()p Fu(T)2149 978 y Fo(\013)p Fy(\()p Fo(U)6 b Fy(\))2299 963 y Fv(\()p Fm(N)20 b Fr(\002)e Fm(N)q Fv(\)\))24 b(=)f Fu(T)12 b Fv(\()p Fu(T)g Fm(N)18 b Fr(\\)h Fu(U)9 b Fv(\))28 b Fu(:)456 1302 y Fv(No)n(w,)975 1641 y(WF)q(\()p Fu(t)1177 1606 y Fo(\013)1177 1661 y(\025)1224 1641 y Fv(\))c(=)e Fr(f)p Fv(\()p Fu(x;)14 b(\030)t Fv(;)g Fu(k)s Fv(\))24 b Fr(2)f Fu(T)1843 1606 y Fq(\003)1831 1663 y Fy(\()p Fo(x;\030)r Fy(\))1976 1641 y Fv(\()p Fu(U)9 b Fv(\))p Fr(j)p Fv(\()p Fu(x;)14 b(\025\030)t Fv(;)g Fu(k)s Fv(\))25 b Fr(2)e Fv(WF)q(\()p Fu(t)2753 1606 y Fo(\013)2800 1641 y Fv(\))p Fr(g)28 b Fu(:)456 1979 y Fv(Here,)40 b(w)n(e)e(iden)n (ti\014ed)g(the)h(cotangen)n(t)e(spaces)g(at)h(the)g(p)r(oin)n(ts)g(\() p Fu(x;)14 b(\030)t Fv(\))40 b(and)e(\()p Fu(x;)14 b(\025\030)t Fv(\))39 b(b)n(y)f(the)456 2179 y(isomorphism)23 b(induced)i(b)n(y)f (the)h(di\013eomorphism)f Fu(U)32 b Fr(!)23 b Fu(\025U)9 b Fv(,)25 b(\()p Fu(x;)14 b(\030)2590 2148 y Fq(0)2614 2179 y Fv(\))23 b Fr(!)h Fv(\()p Fu(x;)14 b(\025\030)2980 2148 y Fq(0)3004 2179 y Fv(\),)26 b Fu(\030)3125 2148 y Fq(0)3171 2179 y Fr(2)d Fu(T)3298 2191 y Fo(x)3340 2179 y Fm(M)p Fv(.)456 2378 y(No)n(w,)33 b(let)f Fu(\030)k Fr(2)31 b Fu(T)1005 2390 y Fo(x)1046 2378 y Fm(N)j Fv(and)e Fu(\021)j Fr(2)c Fu(T)1521 2393 y Fy(\()p Fo(x;\030)r Fy(\))1666 2378 y Fm(N)q Fv(.)52 b(W)-7 b(e)33 b(ma)n(y)e(iden)n(tify)i Fu(\021)j Fv(with)d(a)f(v)n(ector)f(in)i Fu(T)3195 2393 y Fy(\()p Fo(x;\025\030)r Fy(\))3379 2378 y Fm(N)456 2577 y Fv(and)e(observ)n(e)g(that)h(it)g(is)g(orthogonal)e(to)i(WF\()p Fu(t)2008 2547 y Fo(\013)2056 2577 y Fv(\))g(and)g(hence)g(also)f(to)g (WF)q(\()p Fu(t)2999 2547 y Fo(\013)2999 2601 y(\025)3046 2577 y Fv(\).)51 b(W)-7 b(e)32 b(no)n(w)456 2776 y(set)27 b(\000)c(=)p 748 2704 551 4 v 23 w Fr([)803 2788 y Fy(0)p Fo(<\025)p Fq(\024)p Fy(1)1017 2776 y Fv(WF\()p Fu(t)1218 2748 y Fo(\013)1218 2801 y(\025)1266 2776 y Fv(\),)28 b(where)f(the)g(closure)g(is)g(p)r(erformed)g(within)2773 2755 y(_)2747 2776 y Fu(T)2808 2746 y Fq(\003)2845 2776 y Fu(U)9 b Fv(.)37 b(It)28 b(remains)f(to)456 2976 y(pro)n(v)n(e)f(\()p Fu(i)p Fv(\).)555 3175 y(Let)k(\()p Fu(x;)14 b(\030)858 3187 y Fo(n)904 3175 y Fv(;)g Fu(k)984 3187 y Fo(n)1029 3175 y Fv(\))27 b Fr(2)f Fv(WF)q(\()p Fu(t)1371 3145 y Fo(\013)1371 3198 y(\025)1410 3206 y Fh(n)1455 3175 y Fv(\))k(b)r(e)g(a)f(con)n(v)n(ergen)n(t)e(sequence)i(in)h Fu(T)2626 3145 y Fq(\003)2663 3175 y Fu(U)38 b Fv(with)30 b(limit)h(\()p Fu(x;)14 b(\030)t Fv(;)g Fu(k)s Fv(\),)456 3374 y Fu(k)35 b Fr(6)p Fv(=)e(0)g(and)g Fu(\030)k Fr(2)d Fu(T)1085 3386 y Fo(x)1126 3374 y Fm(N)q Fv(.)55 b(There)33 b(is)h(a)f(corresp)r(onding)e(sequence)i(\()p Fu(x;)14 b(\025)2733 3386 y Fo(n)2780 3374 y Fu(\030)2816 3386 y Fo(n)2861 3374 y Fv(;)g Fu(k)2941 3386 y Fo(n)2986 3374 y Fv(\))34 b Fr(2)f Fv(WF\()p Fu(t)3341 3344 y Fo(\013)3389 3374 y Fv(\).)456 3573 y(Let)g Fu(\025)f Fr(2)h Fv([0)p Fu(;)14 b Fv(1])32 b(b)r(e)i(a)e(limit)i(p)r(oin)n(t)f(of)h(the)f(b)r (ounded)g(sequence)g Fr(f)p Fu(\025)2629 3585 y Fo(n)2674 3573 y Fr(g)2716 3585 y Fo(n)p Fq(2)p Fi(N)2848 3573 y Fv(.)53 b(Then,)35 b(there)e(is)456 3773 y(a)f(subsequence)g(con)n(v) n(erging)f(to)i(\()p Fu(x;)14 b(\025\030)t Fv(;)g Fu(k)s Fv(\))33 b Fr(2)f Fv(WF\()p Fu(t)2170 3743 y Fo(\013)2218 3773 y Fv(\).)53 b(But)33 b Fu(\025\030)k Fr(2)32 b Fu(T)2753 3785 y Fo(x)2794 3773 y Fm(N)q Fv(,)j(hence)e(w)n(e)g(ha)n(v)n(e)456 3972 y Fu(k)g Fr(?)e Fu(T)677 3987 y Fy(\()p Fo(x;\025\030)r Fy(\))861 3972 y Fv(\()p Fu(T)12 b Fm(N)q Fv(\).)51 b(If)33 b(w)n(e)f(again)f(iden)n(tify)i(the)f(tangen)n(t)g(spaces)f(at)h(\()p Fu(x;)14 b(\025\030)t Fv(\))34 b(and)e(\()p Fu(x;)14 b(\030)t Fv(\))34 b(w)n(e)456 4171 y(obtain)27 b(the)h(desired)f (result.)2014 b Fl(\004)555 4475 y Fv(Cho)r(osing)27 b(\014rst)g(an)n(y)g(map)g Fu(\013)d Fr(2)f Fm(Z)28 b Fv(w)n(e)f(are)f(ready)h(for)g(the)h(follo)n(wing:)456 4817 y Fs(De\014nition)i(6.3.)40 b Ft(A)29 b(distribution)h Fu(t)23 b Fr(2)h Fm(D)1827 4787 y Fq(0)1850 4817 y Fv(\()p Fm(M)p Fv(\))29 b Ft(has)h(the)g(micr)l(olo)l(c)l(al)g(sc)l(aling)g(de) l(gr)l(e)l(e)g Fu(!)i Ft(at)d(a)456 5016 y(submanifold)i Fm(N)h Ft(w.r.t.)40 b(a)31 b(close)l(d)g(c)l(onic)l(al)g(set)f Fv(\000)2018 5028 y Fy(0)2079 5016 y Fr(\022)2201 4995 y Fv(_)2167 5016 y Fu(N)2243 4986 y Fq(\003)2281 5016 y Fm(N)q Ft(,)h(symb)l(olic)l(al)t(ly)i Fu(!)26 b Fv(=)e Fu(\026)p Fv(sd)3156 4979 y Fy(\000)3197 4987 y FC(0)3156 5042 y Fd(N)3234 5016 y Fv(\()p Fu(t;)14 b(\013)p Fv(\))p Ft(,)456 5216 y(if,)p eop %%Page: 54 54 54 53 bop 456 226 a Fy(54)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)527 425 y Fv(\()p Fu(i)p Fv(\))41 b Ft(ther)l(e)e(exists)g(a)g(close)l(d)h(c)l(onic)g(set)f Fv(\000)g Fr(\032)2032 404 y Fv(_)2006 425 y Fu(T)2067 395 y Fq(\003)2105 425 y Fv(\()p Fu(T)2186 437 y Fd(N)2245 425 y Fm(M)p Fv(\))g Ft(with)g(the)g(pr)l(op)l(erties)h(state)l(d)f(in) 661 624 y(Pr)l(op)l(osition)32 b(6.2,)f(with)g(the)e(\014rst)g(one)h(r) l(eplac)l(e)l(d)h(by)g Fv(\000)-5 b Fl(\026)2422 636 y Fo(T)9 b Fd(N)2529 624 y Fr(\032)23 b Fu(\013)2670 594 y Fq(\003)2708 624 y Fv(\()p Fu(Z)6 b Fv(\()p Fu(T)2896 594 y Fq(\003)2934 624 y Fm(N)q Fv(\))19 b Fr(\002)f Fv(\000)3185 636 y Fy(0)3222 624 y Fv(\))p Ft(,)498 824 y Fv(\()p Fu(ii)p Fv(\))41 b Fu(!)33 b Ft(is)d(the)g(in\014mum)f(of)h (al)t(l)h(those)f Fu(!)1787 794 y Fq(0)1840 824 y Ft(for)h(which,)1795 1162 y Fv(lim)1799 1217 y Fo(\025)p Fq(#)p Fy(0)1924 1162 y Fu(\025)1972 1128 y Fo(!)2016 1103 y Ff(0)2043 1162 y Fu(t)2073 1128 y Fo(\013)2073 1183 y(\025)2143 1162 y Fv(=)23 b(0)29 b Fu(;)972 b Fv(\(42\))661 1501 y Ft(in)30 b(the)g(sense)g(of)g(the)g(H\177)-42 b(ormander)31 b(top)l(olo)l(gy)g(on)f Fm(D)2305 1471 y Fq(0)2305 1524 y Fy(\000)2350 1501 y Fv(\()p Fu(T)2431 1513 y Fd(N)2490 1501 y Fm(M)p Fv(\))p Ft(.)555 1828 y Fv(No)n(w,)h(dep)r(ending)f(on)g (the)h(p)r(osition)f(of)g(\000)1904 1840 y Fy(0)1972 1828 y Fv(one)g(can)f(giv)n(e)h(di\013eren)n(t)g(re\014ned)g(v)n (ersions)f(of)456 2028 y(the)37 b(microlo)r(cal)f(scaling)g(degree.)65 b(Indeed,)40 b(when)d(the)h(inclusion)e(in)2798 2007 y(_)2765 2028 y Fu(N)2841 1997 y Fq(\003)2879 2028 y Fm(N)j Fv(is)e(prop)r(er)f(w)n(e)456 2227 y(sp)r(eak)31 b(of)h(the)g Ft(strict)g Fv(microlo)r(cal)e(scaling)h(degree.)48 b(When)33 b(\000)2439 2239 y Fy(0)2506 2227 y Fr(\021)2634 2206 y Fv(_)2601 2227 y Fu(N)2677 2197 y Fq(\003)2715 2227 y Fm(N)q Fv(,)g(w)n(e)f(call)f(it)h(simply)456 2426 y(the)24 b(scaling)e(degree)h(at)g(the)h(submanifold,)h(sym)n(b)r (olically)d(sd)2393 2438 y Fd(N)2452 2426 y Fv(\()p Fu(t)p Fv(\).)37 b(Moreo)n(v)n(er,)21 b(when)j(\000)3255 2438 y Fy(0)3315 2426 y Fv(=)f Fr(;)456 2625 y Fv(w)n(e)k(sp)r(eak)g(of)g (the)h Ft(smo)l(oth)h Fv(scaling)d(degree.)555 2825 y(The)g (de\014nition)g(seems)g(to)f(dep)r(end)i(on)e(the)h(c)n(hoice)f(of)h (the)g(map)g Fu(\013)d Fr(2)h Fm(Z)p Fv(.)35 b(In)26 b(our)f(concrete)456 3024 y(case)40 b(w)n(e)h(could)g(mak)n(e)f(use)h (of)h(the)f(metric)g(to)g(c)n(ho)r(ose)f(a)h(canonical)f (di\013eomorphism)h Fu(\013)456 3223 y Fv(in)34 b(terms)g(of)h(the)f (exp)r(onen)n(tial)g(map,)i(but)f(for)f(reasons)f(whic)n(h)h(will)h(b)r (ecome)f(clear)f(in)i(the)456 3422 y(follo)n(wing)26 b(it)i(is)g(helpful)g(to)g(pro)r(of)f(its)g(indep)r(endence.)555 3622 y(Before)36 b(coming)f(to)h(that)h(p)r(oin)n(t)f(w)n(e)g(sho)n(w)g (an)g(example)f(for)h(the)h(computation)f(of)g(the)456 3821 y(scaling)25 b(degree)g(whic)n(h)i(is)f(relev)-5 b(an)n(t)26 b(for)f(the)i(ph)n(ysical)f(discussion,)g(namely)-7 b(,)26 b(the)h(generaliza-)456 4020 y(tion)i(of)f(the)i(example)e (\(3\))h(in)g(subSection)g(5.1,)g(the)g(F)-7 b(eynman)29 b(propagator)d Fu(E)2994 4032 y Fo(F)3049 4020 y Fv(,)k(to)e(curv)n(ed) 456 4219 y(space-time.)49 b(The)32 b(F)-7 b(eynman)32 b(propagator)d(is)j(considered)f(as)g(a)h(distribution)g(on)g Fu(M)e Fr(\002)21 b Fu(M)9 b Fv(,)456 4419 y(and)22 b(w)n(e)g(are)f(in) n(terested)i(in)f(the)h(microlo)r(cal)e(scaling)h(degree)f(at)h(the)h (diagonal)e(\001)3013 4431 y Fy(2)3074 4419 y Fr(\032)h Fu(M)17 b Fr(\002)8 b Fu(M)h Fv(.)456 4618 y(Indeed,)22 b(the)f(w)n(a)n(v)n(e)e(fron)n(t)h(set)g(of)h Fu(E)1542 4630 y Fo(F)1618 4618 y Fv(is)g(orthogonal)d(to)i(the)h(tangen)n(t)f (bundle)h(of)g(the)g(diagonal.)456 4817 y(W)-7 b(e)26 b(c)n(ho)r(ose)e Fu(\013)f Fv(:)h Fu(T)1030 4829 y Fy(\001)1085 4837 y FC(2)1120 4817 y Fu(M)1210 4787 y Fy(2)1270 4817 y Fr(!)f Fv(\001)1445 4829 y Fy(2)1497 4817 y Fr(\002)14 b Fu(M)1666 4787 y Fy(2)1726 4817 y Fr(')23 b Fu(M)g Fr(\002)14 b Fu(M)2087 4787 y Fy(2)2149 4817 y Fv(as)25 b Fu(\013)p Fv(\()p Fu(x;)14 b(\030)2454 4829 y Fy(1)2493 4817 y Fu(;)g(\030)2566 4829 y Fy(2)2604 4817 y Fv(\))23 b(=)f(\()p Fu(x;)14 b Fv(exp)2990 4837 y Fo(x)3046 4817 y Fu(\030)3082 4829 y Fy(1)3119 4817 y Fu(;)g Fv(exp)3283 4837 y Fo(x)3339 4817 y Fu(\030)3375 4829 y Fy(2)3412 4817 y Fv(\))456 5016 y(and)24 b(obtain)g(as)g(on)h(Mink)n(o)n(wski)e (space)h(sd)1789 5028 y Fy(\001)1844 5036 y FC(2)1880 5016 y Fv(\()p Fu(E)1973 5028 y Fo(F)2029 5016 y Fv(\))f(=)g Fu(d)13 b Fr(\000)g Fv(2.)34 b(A)25 b(similar)f(result)h(holds)f(for)g (the)456 5216 y(2-p)r(oin)n(t)j(Wigh)n(tman)g(distribution)h Fu(!)1657 5228 y Fy(2)1694 5216 y Fv(.)37 b(So,)p eop %%Page: 55 55 55 54 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(55)456 425 y Fs(Lemma)31 b(6.4.)40 b Ft(The)32 b(micr)l(olo)l(c)l(al)h(sc)l(aling)f(de)l(gr)l(e) l(es)f(of)h(the)f(2-p)l(oint)h(Wightman)g(distribution)456 624 y Fu(!)508 636 y Fy(2)572 624 y Ft(at)c(the)f(diagonal)j Fv(\001)1203 636 y Fy(2)1268 624 y Ft(w.r.t.)39 b Fv(\000)1552 636 y Fy(0)1612 624 y Fv(=)22 b Fr(f)p Fv(\()p Fu(x;)14 b(k)s Fv(;)g Fu(x;)g Fr(\000)p Fu(k)s Fv(\))p Fr(j)p Fu(x)24 b Fr(2)f Fu(M)t(;)14 b(k)26 b Fr(2)e Fu(@)5 b(V)2706 636 y Fy(+)2761 624 y Fu(;)14 b(k)26 b Fr(6)p Fv(=)c(0)p Fr(g)27 b Ft(is)h(given)g(by)456 824 y Fu(\026)p Fv(sd)584 787 y Fy(\000)625 795 y FC(0)584 848 y Fy(\001)639 856 y FC(2)676 824 y Fv(\()p Fu(!)760 836 y Fy(2)797 824 y Fv(\))23 b(=)g Fu(d)18 b Fr(\000)g Fv(2)p Ft(,)30 b(wher)l(e)h Fu(d)e Ft(is)i(the)f(dimension)g(of)h(the)f(sp)l(ac)l(e-time.)456 1198 y Fv(6.2.)46 b Fs(In)m(v)-5 b(ariance)37 b(and)f(prop)s(erties)d (for)j(the)f(scaling)f(degrees.)40 b Fv(Let)31 b(us)f(then)h(c)n(ho)r (ose)456 1398 y(t)n(w)n(o)26 b(maps)i Fu(\013)883 1410 y Fy(1)920 1398 y Fu(;)14 b(\013)1010 1410 y Fy(2)1070 1398 y Fr(2)24 b Fm(Z)j Fv(and)g(state)h(the)g(follo)n(wing:)456 1772 y Fs(Prop)s(osition)i(6.5.)40 b Ft(L)l(et)30 b Fu(t)23 b Fr(2)h Fm(D)1507 1742 y Fq(0)1531 1772 y Fv(\()p Fm(M)p Fv(\))p Ft(.)40 b(L)l(et)29 b Fu(!)1937 1784 y Fo(i)1989 1772 y Fv(=)23 b Fu(\026)p Fv(sd)2206 1735 y Fy(\000)2247 1743 y FC(0)2206 1798 y Fd(N)2283 1772 y Fv(\()p Fu(t;)14 b(\013)2435 1784 y Fo(i)2463 1772 y Fv(\))p Ft(,)31 b Fu(i)24 b Fv(=)f(1)p Fu(;)14 b Fv(2)29 b Ft(b)l(e)h(the)h(micr)l(olo)l (c)l(al)456 1972 y(sc)l(aling)k(de)l(gr)l(e)l(es)g(w.r.t.)54 b Fm(N)36 b Ft(and)f Fv(\000)1583 1984 y Fy(0)1655 1972 y Ft(r)l(esp.)53 b(for)35 b(the)g(two)g(arbitr)l(arily)h(chosen)g (maps.)54 b(Then)456 2171 y Fu(!)508 2183 y Fy(1)568 2171 y Fv(=)22 b Fu(!)707 2183 y Fy(2)744 2171 y Ft(.)456 2546 y(Pr)l(o)l(of.)43 b Fv(It)28 b(is)f(simple)h(to)f(c)n(hec)n(k)g (that)1142 2884 y Fu(t)1172 2847 y Fo(\013)1215 2855 y FC(2)1172 2909 y Fo(\025)1252 2884 y Fv(\()p Fu(\036)p Fv(\))d(=)f Fu(t)1507 2847 y Fo(\013)1550 2855 y FC(1)1507 2909 y Fo(\025)1586 2884 y Fv(\()p Fu(\036)c Fr(\016)f Fu(\014)1797 2849 y Fq(\000)p Fy(1)1793 2909 y Fo(\025)1887 2884 y Fv(\))28 b Fu(;)180 b Fr(8)p Fu(\036)23 b Fr(2)h Fm(D)2416 2896 y Fy(1)2453 2884 y Fv(\()p Fu(T)2534 2896 y Fd(N)2593 2884 y Fm(M)p Fv(\))k Fu(;)539 b Fv(\(43\))456 3223 y(where)33 b Fu(\014)749 3235 y Fo(\025)793 3223 y Fv(\()p Fu(x;)14 b(\030)t Fv(\))34 b(=)g Fu(\025)1162 3193 y Fq(\000)p Fy(1)1251 3223 y Fu(\014)t Fv(\()p Fu(x;)14 b(\025\030)t Fv(\))36 b(and)e Fu(\014)k Fv(=)33 b Fu(\013)1978 3188 y Fq(\000)p Fy(1)1978 3245 y(1)2090 3223 y Fr(\016)22 b Fu(\013)2207 3235 y Fy(2)2245 3223 y Fv(.)56 b(No)n(w,)35 b(assume)e Fu(!)2889 3235 y Fy(1)2960 3223 y Fv(is)h(the)g(scaling)456 3422 y(degree)29 b(for)i Fu(t)f Fv(w.r.t.)47 b Fm(N)32 b Fv(and)f(\000)1462 3434 y Fy(0)1530 3422 y Fv(asso)r(ciated)e(with)j Fu(\013)2174 3434 y Fy(1)2211 3422 y Fv(.)46 b(W)-7 b(e)32 b(should)e(pro)r(of)g(that)h(Eqn.)g(\(42\))456 3622 y(for)c Fu(\013)636 3634 y Fy(2)701 3622 y Fv(con)n(v)n(erges)e(in)j(the)g (sense)f(of)g Fm(D)1689 3591 y Fq(0)1712 3622 y Fv(\()p Fu(T)1793 3634 y Fd(N)1853 3622 y Fm(M)p Fv(\))g(as)g(w)n(ell,)g(at)h (the)g(same)f(rate)g(as)g Fu(\025)c Fr(#)g Fv(0.)555 3821 y(The)k(con)n(v)n(ergence)d(in)i(the)h(sense)f(of)g(distributions) g(is)h(simple.)36 b(Indeed,)27 b(if)g(supp\()p Fu(\036)p Fv(\))d Fr(\032)f Fu(K)6 b Fv(,)456 4020 y Fu(K)42 b Fv(a)37 b(compact)f(subset,)k(then)d(there)g(exists)f(a)h Fu(\025)2064 4032 y Fy(0)2139 4020 y Fv(suc)n(h)f(that)i Fu(\014)2572 4032 y Fo(\025)2615 4020 y Fv(\(supp)q(\()p Fu(\036)p Fv(\)\))i Fr(\032)e Fu(K)k Fv(for)37 b(all)456 4219 y Fu(\025)31 b Fr(\024)f Fu(\025)678 4231 y Fy(0)715 4219 y Fv(.)51 b(Hence)32 b(it)g(su\016ces)g(b)n(y)g(Banac)n (h-Steinhaus)e(principle)i(to)g(pro)r(of)g(that)g(the)g(family)456 4419 y Fr(f)p Fu(\036)24 b Fr(\016)h Fu(\014)689 4383 y Fq(\000)p Fy(1)685 4444 y Fo(\025)778 4419 y Fr(j)p Fv(0)39 b Fu(<)g(\025)g Fr(\024)g Fu(\025)1225 4431 y Fy(0)1263 4419 y Fr(g)e Fv(is)g(b)r(ounded,)j(uniformly)d(in)h Fu(\025)p Fv(,)i(in)d Fm(D)2590 4431 y Fy(1)2628 4419 y Fv(\()p Fu(K)6 b Fv(\))37 b(w.r.t.)66 b(the)37 b(family)456 4618 y(of)26 b(con)n(tin)n(uous)f(seminorms)h(whic)n(h)g(giv)n(es)f (the)i(appropriate)e(F)-7 b(r)n(\023)-39 b(ec)n(het)25 b(top)r(ology)-7 b(.)35 b(This)26 b(c)n(hec)n(k)456 4817 y(pro)r(ceeds)20 b(easily)h(from)g(the)h(c)n(hain)f(rule)h(and)f(the)h (v)n(eri\014able)e(fact)i(that)g(the)g(only)f(con)n(tribution)456 5016 y(comes)32 b(from)h(the)h(0-th)f(and)g(1-st)f(order)g(deriv)-5 b(ativ)n(es)32 b(w.r.t.)54 b Fu(\014)2513 4981 y Fq(\000)p Fy(1)2509 5041 y Fo(\025)2602 5016 y Fv(.)g(In)33 b(the)h(limit)g(they) f(are)456 5216 y(the)28 b(only)h(terms)f(whic)n(h)g(surviv)n(e)f (giving)h(resp.)39 b(the)29 b(iden)n(tit)n(y)f(map)h(on)f(the)h(bundle) g(and)f(the)p eop %%Page: 56 56 56 55 bop 456 226 a Fy(56)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(deriv)-5 b(ativ)n(e)29 b(of)h(the)g(iden)n(tit)n(y)g(map.)44 b(Hence,)31 b(w)n(e)f(get)g(the)g(same)g(rate)f(of)h(con)n(v)n(ergence)e(as)h(far) 456 624 y(as)e(plain)g(distribution)h(con)n(v)n(ergence)d(is)i (concerned.)555 824 y(A)37 b(little)g(bit)g(tric)n(kier)f(is)g(the)h (con)n(v)n(ergence)d(in)j(the)f(sense)g(of)h(seminorms)e(for)h Fs(M2)g Fv(\()p Fu(b)p Fv(\).)456 1023 y(Starting)29 b(again)g(from)g(Eqn.)h(\(43\),)g(via)f(the)h(m)n(ultiplication)g(of)g (a)f(smo)r(oth)h(test)g(function)h(of)456 1222 y(compact)25 b(supp)r(ort)g Fu( )k Fv(suc)n(h)c(that)h Fu( )g Fr(\021)d Fv(1)i(on)g(a)h(small)f(neigh)n(b)r(ourho)r(o)r(d)g(of)g(supp\()p Fu(\036)p Fv(\),)i(w)n(e)f(ha)n(v)n(e)456 1421 y(that)e Fu( )s(t)h Fv(is)f(of)g(compact)g(supp)r(ort)h(and)f(then,)h(b)n(y)f(a) g(partition)g(of)h(unit)n(y)f(with)h(functions)g(with)456 1621 y(supp)r(ort)f(on)g(c)n(harts,)g(that)h(w)n(e)f(are)f(w)n(orking)g (on)h Fn(R)2054 1591 y Fo(\016)2103 1621 y Fr(\002)12 b Fn(R)2240 1591 y Fo(d)2278 1621 y Fv(,)26 b(where)d Fu(\016)28 b Fv(is)c(the)h(dimension)g(of)f(the)456 1820 y(submanifold)d Fm(N)q Fv(.)36 b(No)n(w,)22 b(let)g(us)g(m)n(ultiply)g (the)g(test)h(function)f Fu(\036)g Fv(with)g(the)h(term)e(exp\()p Fu(i)p Fr(h)p Fu(k)s(;)36 b Fr(\001)22 b(i)p Fv(\),)456 2019 y(and)27 b(use)g(in)n(v)n(erse)g(F)-7 b(ourier)26 b(transform)h(to)g(get,)1267 2284 y Fj(d)1248 2310 y Fu(\036t)1327 2273 y Fo(\013)1370 2281 y FC(2)1327 2335 y Fo(\025)1407 2310 y Fv(\()p Fu(k)s Fv(\))c(=)g Fu(\014)1675 2322 y Fo(\025)1707 2273 y Fq(\003)1745 2310 y Fu(t)1775 2273 y Fo(\013)1818 2281 y FC(1)1775 2335 y Fo(\025)1855 2310 y Fv(\()p Fu(\036)28 b Fv(exp\()p Fu(i)p Fr(h)p Fu(k)s(;)42 b Fr(\001)27 b(i)p Fv(\))q(\))1540 2559 y(=)1628 2446 y Fj(Z)1730 2536 y Fn([)1725 2559 y Fu( )s(t)1812 2522 y Fo(\013)1855 2530 y FC(1)1812 2584 y Fo(\025)1891 2559 y Fv(\()p Fu(p)p Fv(\))p Fu(I)2033 2571 y Fo(\036)2078 2559 y Fv(\()p Fu(p;)14 b(k)s Fv(;)g Fu(\014)2319 2571 y Fo(\025)2362 2559 y Fv(\))p Fu(d)2437 2525 y Fo(d)p Fy(+)p Fo(\016)2560 2559 y Fu(p)27 b(;)456 2836 y Fv(where,)1112 3175 y Fu(I)1148 3187 y Fo(\036)1193 3175 y Fv(\()p Fu(p;)14 b(k)s Fv(;)g Fu(\014)1434 3187 y Fo(\025)1477 3175 y Fv(\))23 b(=)1620 3062 y Fj(Z)1717 3175 y Fu(e)1756 3141 y Fq(\000)p Fo(i)p Fy(\()p Fq(h)p Fo(\014)1922 3150 y Fh(\025)1960 3141 y Fy(\()p Fo(\030)r Fy(\))p Fo(;p)p Fq(i\000h)p Fo(\030)r(;k)q Fq(i)p Fy(\))2350 3175 y Fu(\036)p Fv(\()p Fu(\030)t Fv(\))29 b Fu(d)2575 3141 y Fo(d)p Fy(+)p Fo(\016)2697 3175 y Fu(\030)j(;)456 3514 y Fv(where)d(in)i(all)f (these)h(expressions)d(the)j(co)r(ordinates)e Fu(\030)35 b Fv(are)29 b(the)i(lo)r(cal)f(co)r(ordinates)f(of)h Fu(T)3304 3526 y Fd(N)3363 3514 y Fm(M)456 3713 y Fv(and)d Fu(k)k Fv(and)c Fu(p)g Fv(are)g(their)h(dual)f(co)r(ordinates.)555 3912 y(W)-7 b(e)28 b(use)g(the)g(idea)g(of)g(the)g(pro)r(of)f(for)g (the)h(stationary)f(phase)g(Theorem,)g(see)g(for)h(instance)456 4111 y(Theorem)40 b(7.7.1)f(in)i(H\177)-42 b(ormander)40 b(b)r(o)r(oks)g([35)o(].)77 b(Because)40 b(of)h Fu(\014)2557 4123 y Fo(\025)2646 4111 y Fr(!)k Fv(id)c(for)g Fu(\025)k Fr(!)h Fv(0)40 b(the)456 4311 y(oscillatory)26 b(in)n(tegral)i Fu(I)1196 4323 y Fo(\036)1269 4311 y Fv(falls)g(o\013)h(rapidly)f (outside)g(of)h(an)n(y)e(conical)h(neigh)n(b)r(ourho)r(o)r(d)g(of)g (the)456 4510 y(diagonal)k Fu(p)i Fv(=)f Fu(k)k Fv(in)e Fn(R)1212 4480 y Fo(d)p Fy(+)p Fo(\016)1357 4510 y Fr(\002)22 b Fn(R)1504 4480 y Fo(d)p Fy(+)p Fo(\016)1626 4510 y Fv(,)36 b(uniformly)e(for)f Fu(\025)i Fv(su\016cien)n(tly)f(small,)i (i.e.)56 b(for)34 b(ev)n(ery)456 4709 y Fu(\017)22 b(>)h Fv(0)k(there)h(exists)f(a)g Fu(\025)1228 4721 y Fy(0)1289 4709 y Fu(>)22 b Fv(0)28 b(suc)n(h)f(that)h(for)f(ev)n(ery)f Fu(N)32 b Fr(2)23 b Fn(N)1102 5048 y Fv(sup)1040 5118 y Fy(0)p Fo(<\025<\025)1255 5126 y FC(0)1399 5048 y Fv(sup)1302 5121 y Fq(j)p Fo(p)p Fq(\000)p Fo(k)q Fq(j)p Fo(>\017)p Fq(j)p Fo(k)q Fq(j)1620 5048 y Fv(\(1)18 b(+)g Fr(j)p Fu(p)p Fr(j)h Fv(+)f Fr(j)p Fu(k)s Fr(j)p Fv(\))2109 5014 y Fo(N)2172 5048 y Fr(j)p Fu(I)2231 5060 y Fo(\036)2276 5048 y Fv(\()p Fu(p;)c(k)s(;)g(\014)2517 5060 y Fo(\025)2560 5048 y Fv(\))p Fr(j)23 b Fu(<)g Fr(1)28 b Fu(:)437 b Fv(\(44\))p eop %%Page: 57 57 57 56 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(57)456 425 y Fv(No)n(w)27 b(let)h(\000)23 b Fr(\032)f Fn(R)987 395 y Fo(d)p Fy(+)p Fo(\016)1137 425 y Fv(b)r(e)28 b(a)f(closed)g(cone)g(suc)n(h)g(that) 1227 764 y(sup)1264 834 y Fo(C)1352 764 y Fv(\(1)18 b(+)g Fr(j)p Fu(p)p Fr(j)p Fv(\))1647 730 y Fo(N)1711 764 y Fr(j)1739 740 y Fn([)1734 764 y Fu( )s(t)1821 727 y Fo(\013)1864 735 y FC(1)1821 789 y Fo(\025)1900 764 y Fv(\()p Fu(p)p Fv(\))p Fr(j)p Fu(\025)2077 730 y Fo(!)2149 764 y Fr(!)23 b Fv(0)28 b Fu(;)41 b(\025)24 b Fr(!)f Fv(0)k Fu(;)638 b Fv(\(45\))456 1103 y(for)27 b(all)g(closed)g(cones)g Fu(C)34 b Fv(with)28 b Fu(C)d Fr(\\)18 b Fv(\000)23 b(=)g Fr(;)k Fv(and)h(all)f Fu(N)32 b Fr(2)23 b Fn(N)p Fv(.)555 1302 y(W)-7 b(e)36 b(no)n(w)f(w)n(an)n(t)h(to)f(sho)n(w)g(that)h(the)g (same)f(prop)r(ert)n(y)g(holds)g(for)2705 1276 y Fj(d)2686 1302 y Fu(\036t)2765 1265 y Fo(\013)2808 1273 y FC(2)2765 1327 y Fo(\025)2845 1302 y Fv(.)61 b(So)36 b(let)g Fu(C)42 b Fv(b)r(e)36 b(a)456 1501 y(closed)27 b(cone)g(suc)n(h)g(that)h(the)g (closed)f(cone)1104 1840 y Fu(C)1169 1806 y Fq(0)1216 1840 y Fv(=)c Fr(f)p Fu(p)f Fr(2)h Fn(R)1548 1806 y Fo(d)p Fy(+)p Fo(\016)1670 1840 y Fu(;)14 b Fr(j)p Fu(p)k Fr(\000)h Fu(k)s Fr(j)j(\024)h Fu(\017)p Fr(j)p Fu(k)s Fr(j)k Fv(for)h(some)f Fu(k)e Fr(2)f Fu(C)6 b Fr(g)501 b Fv(\(46\))456 2179 y(do)r(es)29 b(not)g(in)n(tersect)g(\000.)42 b(Then)30 b(w)n(e)f(split)g(the)h(region)e(of)h(in)n(tegration)g(o)n(v)n(er)e Fu(p)i Fv(in)n(to)g(the)h(parts)456 2378 y Fr(j)p Fu(p)19 b Fr(\000)g Fu(k)s Fr(j)26 b(\024)f Fu(\017)p Fr(j)p Fu(k)s Fr(j)k Fv(and)g(the)h(rest.)41 b(In)30 b(the)g(\014rst)f(region) f(w)n(e)g(can)h(estimate)h Fu(k)i Fv(b)n(y)d Fu(p)g Fv(and)g(use)g(the) 456 2577 y(fast)f(deca)n(y)f(of)h(\(1)18 b(+)g Fr(j)p Fu(p)p Fr(j)p Fv(\))1237 2547 y Fo(N)1301 2577 y Fr(j)1329 2554 y Fn([)1324 2577 y Fu( )s(t)1411 2540 y Fo(\013)1454 2548 y FC(1)1411 2602 y Fo(\025)1490 2577 y Fv(\()p Fu(p)p Fv(\))p Fr(j)29 b Fv(within)g Fu(C)1972 2547 y Fq(0)2023 2577 y Fv(and)f(the)g(p)r(olynomial)g(b)r(oundedness)g(of)g Fu(I)3377 2589 y Fo(\036)3421 2577 y Fv(;)456 2776 y(in)23 b(the)h(second)f(region)f(the)h(p)r(olynomial)g(b)r(oundedness)g(of)h (\(1)10 b(+)g Fr(j)p Fu(p)p Fr(j)p Fv(\))2612 2746 y Fo(N)2674 2776 y Fr(j)2702 2753 y Fn([)2697 2776 y Fu( )s(t)2784 2739 y Fo(\013)2827 2747 y FC(1)2784 2801 y Fo(\025)2864 2776 y Fv(\()p Fu(p)p Fv(\))p Fr(j)24 b Fv(and)f(the)g(fast)456 2976 y(deca)n(y)j(of)i Fu(I)817 2988 y Fo(\036)889 2976 y Fv(\(44\).)37 b(This)27 b(pro)n(v)n(es)f(the)i(desired)f(estimate)g (for)2447 2949 y Fj(d)2427 2976 y Fu(\036t)2506 2939 y Fo(\013)2549 2947 y FC(2)2506 3001 y Fo(\025)2587 2976 y Fv(.)770 b Fl(\004)555 3225 y Fv(The)32 b(microlo)r(cal)f(scaling)g (degree)g Fu(\026)p Fv(sd)h(has)f(similar)h(prop)r(erties)f(as)g(the)h (scaling)f(degree,)456 3424 y(as)c(describ)r(ed)g(in)h(Lemma)f(5.1.)36 b(In)28 b(addition,)f(one)h(\014nds)f(the)h(follo)n(wing)f(t)n(w)n(o)g (prop)r(erties:)456 3722 y Fs(Lemma)k(6.6.)40 b Ft(L)l(et)32 b Fu(t)1160 3734 y Fy(1)1197 3722 y Fu(;)14 b(t)1264 3734 y Fy(2)1329 3722 y Fr(2)28 b Fm(D)1481 3692 y Fq(0)1504 3722 y Fv(\()p Fm(M)p Fv(\))k Ft(with)h Fu(\026)p Fv(sd)f Fu(!)2078 3734 y Fy(1)2148 3722 y Ft(r)l(esp.)46 b Fu(!)2416 3734 y Fy(2)2485 3722 y Ft(at)32 b Fm(N)d Fr(\032)f Fm(M)p Ft(,)k(w.r.t.)47 b Fv(\000)3204 3692 y Fy(1)3204 3743 y(0)3274 3722 y Ft(r)l(esp.)456 3921 y Fv(\000)508 3891 y Fy(2)508 3942 y(0)574 3921 y Ft(and)29 b(such)g(that)g Fu(Z)6 b Fv(\()p Fu(N)1262 3891 y Fq(\003)1300 3921 y Fm(N)q Fv(\))33 b Fu(=)-51 b Fr(2)23 b Fv(\(\000)1583 3891 y Fy(1)1583 3942 y(0)1637 3921 y Fv(+)16 b(\000)1770 3891 y Fy(2)1770 3942 y(0)1808 3921 y Fv(\))p Ft(.)38 b(Then)30 b(the)f(p)l(ointwise)h(pr)l(o)l(duct)f Fu(t)2941 3933 y Fy(1)2978 3921 y Fu(t)3008 3933 y Fy(2)3074 3921 y Ft(exists)g(in)g(a)456 4121 y(smal)t(l)k(neighb)l(ourho)l(o)l(d)h(of) f Fm(N)g Ft(and)g(has)g(the)f(micr)l(olo)l(c)l(al)i(sc)l(aling)f(de)l (gr)l(e)l(e)f Fu(!)e Fr(\024)d Fu(!)3014 4133 y Fy(1)3071 4121 y Fv(+)20 b Fu(!)3208 4133 y Fy(2)3277 4121 y Ft(at)32 b Fm(N)456 4320 y Ft(w.r.t.)39 b Fv(\000)740 4332 y Fy(0)800 4320 y Fv(=)23 b(\000)940 4290 y Fy(1)940 4340 y(0)995 4320 y Fr([)c Fv(\000)1121 4290 y Fy(2)1121 4340 y(0)1177 4320 y Fr([)g Fv(\(\000)1335 4290 y Fy(1)1335 4340 y(0)1390 4320 y Fv(+)f(\000)1525 4290 y Fy(2)1525 4340 y(0)1563 4320 y Fv(\))p Ft(.)456 4618 y(Pr)l(o)l(of.)43 b Fv(By)30 b(assumption,)h(the)g(w)n(a)n(v)n(e)e(fron)n(t)h(sets)g(of)h Fu(t)2166 4588 y Fo(\013)2166 4641 y Fy(1)p Fo(;\025)2292 4618 y Fv(and)g Fu(t)2487 4588 y Fo(\013)2487 4641 y Fy(2)p Fo(;\025)2583 4618 y Fv(,)h(some)d Fu(\013)g Fr(2)f Fm(Z)p Fv(,)j(on)f(a)g(suf-)456 4817 y(\014cien)n(tly)h(small)g(neigh)n (b)r(ourho)r(o)r(d)f(of)h Fm(N)h Fv(satisfy)f(the)h(condition)f(\(WF\() p Fu(t)2770 4787 y Fo(\013)2770 4841 y Fy(1)p Fo(;\025)2867 4817 y Fv(\))21 b(+)g(WF\()p Fu(t)3207 4787 y Fo(\013)3207 4841 y Fy(2)p Fo(;\025)3304 4817 y Fv(\)\))g Fr(\\)456 5016 y Fu(Z)6 b Fv(\()p Fu(T)612 4986 y Fq(\003)649 5016 y Fv(\()p Fu(T)730 5028 y Fd(N)789 5016 y Fm(M)p Fv(\)\))39 b(=)g Fr(;)p Fv(,)g(hence)e(their)g(pro)r(duct)g(exists)g(there)g(b)n (y)f Fs(M1)p Fv(.)65 b(Because)36 b(of)i(the)f(se-)456 5216 y(quen)n(tial)25 b(con)n(tin)n(uit)n(y)f(of)i(the)f(pro)r(ducts)g (in)h(the)g(H\177)-42 b(ormander)24 b(top)r(ology)g Fs(M2)p Fv(,)h(the)h(microlo)r(cal)p eop %%Page: 58 58 58 57 bop 456 226 a Fy(58)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(scaling)j(degree)h(is)g (giv)n(en)g(b)n(y)g(the)h(sum)g(w.r.t.)55 b(the)33 b(stated)h(conic)f (region)f(as)h(follo)n(ws)g(from)456 624 y Fs(M1)27 b Fv(and)g(do)r(es)g(not)h(dep)r(end)g(on)g(the)f(c)n(hoice)g(of)h(the)g (map)f Fu(\013)p Fv(.)971 b Fl(\004)555 1777 y Fv(The)26 b(follo)n(wing)e(nice)h(prop)r(ert)n(y)f(follo)n(ws)g(from)h(the)h (sequen)n(tial)e(con)n(tin)n(uit)n(y)h(of)g(the)h(restric-)456 1976 y(tion)h(op)r(erator)f(to)i(submanifolds)f Fs(M3)p Fv(:)456 2998 y Fs(Lemma)k(6.7.)40 b Ft(L)l(et)27 b Fm(N)1189 3010 y Fy(1)1253 2998 y Ft(b)l(e)g(a)g(submanifold)h(of)g Fm(N)q Ft(,)g(and)f(let)g Fu(t)c Fr(2)h Fm(D)2564 2968 y Fq(0)2587 2998 y Fv(\()p Fm(M)p Fv(\))j Ft(have)h(the)f(micr)l(olo)l (c)l(al)456 3197 y(sc)l(aling)g(de)l(gr)l(e)l(e)h Fu(!)h Ft(at)e Fm(N)i Ft(w.r.t.)38 b Fv(\000)1523 3209 y Fy(0)1560 3197 y Ft(.)g(Then)27 b(the)g(micr)l(olo)l(c)l(al)i(sc)l(aling)f(de)l (gr)l(e)l(e)f(of)h Fu(t)f Ft(at)g Fm(N)3186 3209 y Fy(1)3250 3197 y Ft(w.r.t.)456 3397 y(the)j(r)l(estriction)g Fv(\000)1047 3409 y Fy(1)1113 3397 y Ft(of)h Fv(\000)1263 3409 y Fy(0)1330 3397 y Ft(to)e Fm(N)1493 3409 y Fy(1)1561 3397 y Ft(is)h(less)g(or)g(e) l(qual)g(to)g Fu(!)s Ft(,)g Fu(\026)p Fv(sd)2462 3360 y Fy(\000)2503 3368 y FC(1)2462 3422 y Fd(N)2516 3430 y FC(1)2553 3397 y Fv(\()p Fu(t)p Fv(\))24 b Fr(\024)f Fu(\026)p Fv(sd)2888 3360 y Fy(\000)2929 3368 y FC(0)2888 3422 y Fd(N)2965 3397 y Fv(\()p Fu(t)p Fv(\))p Ft(.)555 4419 y Fv(A)k(last)g(w)n(ord)e(is)i(dev)n(oted)f(to)g(a)g(p)r(oin)n (tlik)n(e)h(trivialization)e(of)i(the)g(ab)r(o)n(v)n(e)e(pro)r(cedure.) 36 b(This)456 4618 y(case)31 b(can)h(b)r(e)h(deriv)n(ed)f(straigh)n (tforw)n(ardly)d(b)n(y)j(considering)f Fm(N)i Fr(\021)e(f)p Fu(p)p Fr(g)p Fv(,)h(where)g Fu(p)f Fr(2)g Fm(M)i Fv(is)f(a)456 4817 y(generic)26 b(p)r(oin)n(t)i(and)f(though)n(t)h(of)f(as)g(\(a)h (rather)e(singular)h(case)f(of)6 b(\))29 b(a)e(submanifold.)555 5016 y(The)37 b(translation)f(to)h(this)h(simpler)e(case)h(is)f(done)h (via)g(the)g(follo)n(wing)f(corresp)r(ondence)456 5216 y(b)r(et)n(w)n(een)27 b(geometrical)f(and)i(analytical)e(quan)n (tities:)p eop %%Page: 59 59 59 58 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(59)994 764 y Fu(U)32 b Fr(2)23 b Fu(T)1210 776 y Fd(N)1269 764 y Fm(M)427 b Fr(\000)-14 b(!)189 b Fu(U)2158 776 y Fo(p)2219 764 y Fr(2)23 b Fu(T)2346 776 y Fo(p)2384 764 y Fm(M)k Fu(;)890 1262 y(\013)c Fv(:)g Fu(U)32 b Fr(!)23 b Fm(N)d Fr(\002)e Fm(M)322 b Fr(\000)-14 b(!)189 b Fu(\013)23 b Fv(:)g Fu(U)2280 1274 y Fo(p)2341 1262 y Fr(!)h Fm(M)j Fu(;)97 b Fv(\()p Fu(i)p Fv(\))28 b(and)f(\()p Fu(iv)s Fv(\))h(v)-5 b(alid)p Fu(;)891 1760 y(t)921 1730 y Fo(\013)991 1760 y Fv(=)23 b(\()p Fs(1)18 b Fr(\012)g Fu(t)p Fv(\))h Fr(\016)f Fu(\013)324 b Fr(\000)-14 b(!)189 b Fu(t)2131 1726 y Fo(\013)2201 1760 y Fv(=)23 b Fu(t)18 b Fr(\016)g Fu(\013)28 b(;)650 2258 y Fv(\000)23 b Fr(\032)838 2237 y Fv(_)813 2258 y Fu(T)874 2228 y Fq(\003)911 2258 y Fu(U)36 b(;)97 b Fv(\000)23 b Fr(?)g Fu(T)12 b Fv(\()p Fu(T)g Fm(N)18 b Fr(\\)h Fu(U)9 b Fv(\))83 b Fr(\000)-14 b(!)189 b Fv(\000)23 b Fr(\032)2289 2237 y Fv(_)2263 2258 y Fu(T)2324 2224 y Fq(\003)2362 2258 y Fu(U)2419 2270 y Fo(p)2484 2258 y Fu(;)97 b Fv(\000)23 b Fr(?)g Fu(T)12 b(U)2885 2270 y Fo(p)2950 2258 y Fu(;)983 2756 y Fv(\000)1035 2768 y Fy(0)1095 2756 y Fr(\032)1216 2735 y Fv(_)1183 2756 y Fu(N)1259 2726 y Fq(\003)1297 2756 y Fm(N)417 b Fr(\000)-14 b(!)189 b Fv(\000)2153 2768 y Fo(p)2214 2756 y Fr(\032)2327 2735 y Fv(_)2302 2756 y Fu(T)2363 2722 y Fq(\003)2351 2777 y Fo(p)2400 2756 y Fm(M)27 b Fu(:)456 3095 y Fv(6.3.)46 b Fs(T)-8 b(ransv)m(ersal)40 b(scaling)e(degree.)j Fv(Instead)33 b(of)g(blo)n(wing)g(up)g(distributions)g(on)h Fm(M)f Fv(to)456 3294 y(distributions)e(on)g Fm(N)22 b Fr(\002)f Fm(M)31 b Fv(in)h(the)f(de\014nition)h(of)g(the)f(scaling)g(degree,)g (one)g(could)g(also)g(use)456 3494 y(a)i(\014xed)g(\014bration)g(of)g (a)g(neigh)n(b)r(ourho)r(o)r(d)f(of)i Fm(N)h Fv(in)e Fm(M)g Fv(b)n(y)g(transv)n(ersal)e(surfaces.)53 b(F)-7 b(or)33 b(this)456 3693 y(purp)r(ose)20 b(w)n(e)g(decomp)r(ose)g Fu(T)1336 3705 y Fd(N)1395 3693 y Fm(M)g Fv(in)n(to)g(complemen)n(tary) f(subbundles,)k Fu(T)2729 3705 y Fd(N)2788 3693 y Fm(M)f Fv(=)h Fu(T)12 b Fm(N)5 b Fv(+)t Fu(C)h Fv(.)34 b(The)456 3892 y(map)22 b Fu(\013)688 3904 y Fo(C)767 3892 y Fv(:=)h Fu(\031)925 3904 y Fy(2)970 3892 y Fr(\016)8 b Fu(\013)l Fl(\026)1104 3904 y Fo(C)t Fq(\\)p Fo(V)1280 3892 y Fv(,)23 b(with)g(the)g(pro)5 b(jection)22 b Fu(\031)2085 3904 y Fy(2)2145 3892 y Fv(:)h Fm(N)9 b Fr(\002)f Fm(M)23 b Fr(!)g Fm(M)f Fv(on)n(to)g(the)h(second)e(factor)456 4091 y(and)h(with)h Fu(V)42 b Fv(b)r(eing)23 b(a)f(su\016cien)n(tly)h (small)f(neigh)n(b)r(ourho)r(o)r(d)g(of)g(the)h(zero)f(section)g Fu(Z)6 b Fv(\()p Fu(T)3169 4103 y Fd(N)3228 4091 y Fm(M)p Fv(\),)24 b(is)456 4291 y(then)j(a)g(di\013eomorphism)f(on)n(to)g(some) h(neigh)n(b)r(ourho)r(o)r(d)f(of)g Fm(N)q Fv(.)38 b(The)27 b(images)f(of)g(the)i(\014b)r(ers)f(of)456 4490 y Fu(C)36 b Fv(are)28 b(transv)n(ersal)g(surfaces.)42 b(The)30 b(transv)n(ersally)d(\(w.r.t.)43 b Fu(\013)30 b Fv(and)g Fu(C)6 b Fv(\))30 b(scaled)f(distribution)456 4689 y(is)e(then)h (de\014ned)g(b)n(y)1245 5028 y Fu(t)1275 5040 y Fo(\025;)p Fq(?)1390 5028 y Fv(\()p Fu(x;)14 b(\021)s Fv(\))24 b(=)f Fu(t)18 b Fr(\016)g Fu(\013)1855 5040 y Fo(C)1911 5028 y Fv(\()p Fu(x;)c(\025\021)s Fv(\))29 b Fu(;)42 b Fv(\()p Fu(x;)14 b(\021)s Fv(\))24 b Fr(2)f Fu(C)34 b(:)642 b Fv(\(47\))p eop %%Page: 60 60 60 59 bop 456 226 a Fy(60)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(The)k Ft(tr)l(ansversal)g Fv(microlo)r(cal)f(scaling)g(degree)g(ma)n(y)g(then)h(b)r(e)h (de\014ned)f(as)g(the)g(in\014m)n(um)h(of)456 624 y(all)28 b Fu(!)h Fr(2)d Fn(R)j Fv(suc)n(h)g(that)g(the)h(sequence)e Fu(\025)1731 594 y Fo(!)1780 624 y Fu(t)1810 636 y Fo(\025;)p Fq(?)1954 624 y Fv(con)n(v)n(erges)e(to)j(zero)f(within)i Fm(D)2936 594 y Fq(0)2936 647 y Fy(\000)2977 655 y Fh(C)3030 624 y Fv(\()p Fu(C)6 b Fv(\),)30 b(with)g(a)456 824 y(closed)23 b(conical)g(set)h(\000)1148 836 y Fo(C)1227 824 y Fr(\032)1340 803 y Fv(_)1314 824 y Fu(T)1375 794 y Fq(\003)1412 824 y Fu(C)31 b Fv(with)24 b(\000)1739 836 y Fo(C)1790 824 y Fl(\026)1825 839 y Fo(Z)t Fy(\()p Fo(C)t Fy(\))1982 824 y Fv(=)e Fu(\013)2122 794 y Fq(\003)2122 847 y Fo(C)2178 824 y Fv(\(\000)2262 836 y Fy(0)2300 824 y Fv(\).)36 b(F)-7 b(ortunately)g(,)24 b(it)g(turns)g(out)g(that)456 1023 y(this)j(new)g(concept)h(of)f(a)g(scaling)f(degree)g(at)h(a)g (surface)f(coincides)h(with)h(the)f(old)g(one.)37 b(Th)n(us,)456 1222 y(in)31 b(particular,)h(the)g(transv)n(ersal)d(scaling)h(degree)h (do)r(es)g(not)g(dep)r(end)i(on)e(the)h(c)n(hoice)f(of)g(the)456 1421 y(\014bration.)456 1998 y Fs(Prop)s(osition)f(6.8.)40 b Ft(The)24 b(tr)l(ansversal)f(micr)l(olo)l(c)l(al)h(sc)l(aling)g(de)l (gr)l(e)l(e)f(de\014ne)l(d)g(ab)l(ove)h(c)l(oincides)456 2197 y(with)30 b(the)g(micr)l(olo)l(c)l(al)h(sc)l(aling)g(de)l(gr)l(e)l (e)f(de\014ne)l(d)g(in)g(\(42\).)456 2774 y(Pr)l(o)l(of.)43 b Fv(W)-7 b(e)32 b(ma)n(y)e(restrict)h(ourselv)n(es)f(to)h(a)g (su\016cien)n(tly)h(small)f(neigh)n(b)r(ourho)r(o)r(d)g(of)g(a)g(p)r (oin)n(t)456 2973 y(at)22 b(the)i(surface)e Fm(N)q Fv(.)36 b(In)23 b(suitable)g(co)r(ordinates,)f Fm(M)h Fv(is)f(a)h(subset)g(of)g Fn(R)2602 2943 y Fo(\016)2647 2973 y Fr(\002)9 b Fn(R)2781 2943 y Fo(d)p Fq(\000)p Fo(\016)2927 2973 y Fv(where)22 b(the)i(\014rst)456 3172 y(factor)j(corresp)r(onds)f(to)i Fm(N)i Fv(and)d(the)i(second)e(factor)h(to)g(the)g(transv)n(ersal)e (surfaces.)37 b Fu(T)3220 3184 y Fd(N)3279 3172 y Fm(M)28 b Fv(is)456 3372 y(a)e(subset)h(of)g Fn(R)932 3342 y Fo(\016)986 3372 y Fr(\002)16 b Fn(R)1127 3342 y Fo(\016)1181 3372 y Fr(\002)h Fn(R)1323 3342 y Fo(d)p Fq(\000)p Fo(\016)1473 3372 y Fv(with)27 b(the)h(\014rst)e(factor)g(corresp)r(onding)f(to)i Fm(N)q Fv(,)h(the)f(second)g(to)456 3571 y(the)32 b(tangen)n(t)f (spaces)g(of)h Fm(N)i Fv(and)d(the)i(third)f(one)f(to)h(the)h(\014b)r (ers)e(of)h(the)g(transv)n(ersal)e(bundle)456 3770 y Fu(C)6 b Fv(.)37 b(The)27 b(map)h Fu(\013)g Fv(ma)n(y)f(b)r(e)h(c)n (hosen)f(as)1498 4109 y Fu(\013)p Fv(\()p Fu(x;)14 b(\030)t(;)g(\021)s Fv(\))25 b(=)d(\()p Fu(x;)14 b(x)20 b Fv(+)e Fu(\030)t(;)c(\021)s Fv(\))28 b Fu(:)895 b Fv(\(48\))456 4448 y(Then)21 b Fu(\013)719 4460 y Fo(C)796 4448 y Fv(b)r(ecomes)g(the)g(iden)n(tit)n (y)-7 b(.)35 b(The)21 b(distribution)g Fu(t)g Fv(ma)n(y)f(b)r(e)i (replaced)e(b)n(y)h(a)f(distribution)456 4647 y(with)30 b(compact)f(supp)r(ort,)h(and)f(the)h(factor)f Fs(1)g Fv(in)h(the)g(blo)n(w)f(up)h(of)f Fu(t)h Fv(ma)n(y)f(b)r(e)h(replaced)f (b)n(y)g(a)456 4846 y(test)e(function)i Fu(\037)23 b Fr(2)g Fm(D)p Fv(\()p Fm(N)q Fv(\).)38 b(F)-7 b(or)27 b(the)h(F)-7 b(ourier)26 b(transforms)h(w)n(e)g(then)h(obtain)1179 5170 y Fj(b)1165 5185 y Fu(t)1195 5197 y Fo(\025)1239 5185 y Fv(\()p Fu(p;)14 b(q)s(;)g(k)s Fv(\))23 b(=)f Fu(\025)1663 5151 y Fq(\000)p Fo(d)1762 5185 y Fj(b)-54 b Fu(\037)p Fv(\()p Fu(p)19 b Fr(\000)f Fu(\025)2030 5151 y Fq(\000)p Fy(1)2119 5185 y Fu(q)s Fv(\))2190 5170 y Fj(b)2191 5185 y Fu(t)q Fv(\()p Fu(\025)2302 5151 y Fq(\000)p Fy(1)2392 5185 y Fu(q)s(;)c(\025)2517 5151 y Fq(\000)p Fy(1)2606 5185 y Fu(k)s Fv(\))28 b Fu(;)562 b Fv(\(49\))p eop %%Page: 61 61 61 60 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(61)456 425 y Fv(and)1451 749 y Fj(d)1438 764 y Fu(t)1468 776 y Fo(\025;)p Fq(?)1583 764 y Fv(\()p Fu(p;)14 b(k)s Fv(\))23 b(=)g Fu(\025)1931 730 y Fo(d)p Fq(\000)p Fo(\016)2053 749 y Fj(b)2054 764 y Fu(t)p Fv(\()p Fu(p;)14 b(\025)2243 730 y Fq(\000)p Fy(1)2333 764 y Fu(k)s Fv(\))28 b Fu(:)835 b Fv(\(50\))456 1103 y(F)-7 b(urthermore,)27 b(using)g(a)h(corresp)r(onding)e (trivialization)g(of)i(the)g(resp)r(ectiv)n(e)g(cotangen)n(t)e(bun-)456 1302 y(dles,)33 b(w)n(e)e(ma)n(y)g(iden)n(tify)i(\000)1320 1314 y Fy(0)1387 1302 y Fv(=)d(\000)1534 1314 y Fo(C)1622 1302 y Fv(with)j Fr(f)p Fv(0)p Fr(g)20 b(\002)g Fu(K)6 b Fv(,)33 b(where)f Fu(K)37 b Fv(is)32 b(a)f(closed)h(cone)f(in)h Fn(R)3298 1272 y Fo(d)p Fq(\000)p Fo(\016)3421 1302 y Fv(,)456 1501 y(considered)21 b(as)g(the)i(transv)n(ersal)c(part)j(of)g (the)g(cotangen)n(t)g(space,)g(and)g(\000)g(with)h Fr(f)p Fv(0)p Fr(g)7 b(\002)g(f)p Fv(0)p Fr(g)g(\002)g Fu(K)f Fv(.)456 1700 y(The)26 b(con)n(v)n(ergence)f(of)i Fu(t)1208 1712 y Fo(\025)1278 1700 y Fv(ma)n(y)f(b)r(e)h(discussed)g(in)g(terms)f (of)h(seminorms)f(of)h(the)g(form)3192 1634 y Fj(R)3232 1730 y Fo(V)3289 1700 y Fv(\(1)17 b(+)456 1900 y Fr(j)p Fu(p)p Fr(j)p Fv(\))576 1870 y Fo(N)639 1900 y Fr(j)p Fu(t)692 1912 y Fo(\025)735 1900 y Fr(j)35 b Fv(with,)i(resp)r(ectiv)n (ely)-7 b(,)35 b(conical)f(neigh)n(b)r(ourho)r(o)r(ds)f Fu(V)54 b Fv(of)34 b(\000)h(and)f(some)g Fu(N)43 b Fr(2)35 b Fv(\()p Fr(\000)p Fn(N)p Fv(\))456 2099 y(and)22 b(closed)g(conical)g (sets)g Fu(V)42 b Fv(in)23 b(the)g(complemen)n(t)f(of)h(\000)f(and)h (all)f Fu(N)32 b Fr(2)24 b Fn(N)p Fv(.)35 b(Since)23 b Fu(\037)f Fv(is)h(strongly)456 2298 y(decreasing,)29 b(these)g(seminorms)g(of)h Fu(t)1639 2310 y Fo(\025)1713 2298 y Fv(can)f(b)r(e)i(estimated)f(in)g(terms)f(of)h(the)g(corresp)r (onding)456 2497 y(seminorms)c(of)i Fu(t)992 2509 y Fo(\025;)p Fq(?)1107 2497 y Fv(.)2250 b Fl(\004)456 2746 y Fv(6.4.)46 b Fs(Extension)28 b(of)h(distributions)e(to)i(surfaces.)41 b Fv(W)-7 b(e)25 b(no)n(w)g(w)n(an)n(t)f(to)h(apply)f(these)h(con-)456 2946 y(cepts)31 b(to)h(the)g(extension)g(problem)f(of)h(distributions)f Fu(t)f Fr(2)h Fm(D)2427 2916 y Fq(0)2450 2946 y Fv(\()p Fm(M)5 b Fr(n)g Fm(N)p Fv(\).)50 b(The)32 b(w)n(a)n(v)n(efron)n(t)d (set)456 3145 y(of)36 b(the)h(extension)f(shall)g(b)r(e)g(orthogonal)e (to)j Fu(T)12 b Fm(N)q Fv(,)38 b(hence)e(a)g(necessary)f(condition)h (is)g(that)456 3344 y(this)h(holds)g(true)g(for)f(the)i(closure)d(of)j (WF\()p Fu(t)p Fv(\))g(within)f Fu(T)2318 3314 y Fq(\003)2356 3344 y Fm(M)p Fv(.)65 b(W)-7 b(e)37 b(extend)g(the)h(notion)f(of)456 3544 y(the)32 b(microlo)r(cal)g(scaling)f(degree)g(to)i(suc)n(h)f (distributions)g(in)h(an)f(analogous)e(w)n(a)n(y)i(as)f(in)i(the)456 3743 y(extension)d(problem)h(to)f(a)h(single)f(p)r(oin)n(t.)48 b(Namely)-7 b(,)31 b(for)g(an)g(arbitrary)e(function)i Fu(\037)e Fr(2)g Fm(E)p Fv(\()p Fm(M)p Fv(\))456 3942 y(with)22 b(supp)p Fu(\037)8 b Fr(\\)g Fm(N)24 b Fv(=)f Fr(;)p Fv(,)g(\(1)8 b Fr(\012)g Fu(\037)p Fv(\))g Fr(\016)g Fu(\013)g Fr(\001)g Fu(t)1616 3912 y Fo(\013)1616 3966 y(\025)1683 3942 y Fv(can)21 b(b)r(e)i(considered)e(as)h(a)f (distribution)i(on)e Fu(U)32 b Fr(\032)23 b Fu(T)3281 3954 y Fd(N)3340 3942 y Fm(M)p Fv(.)456 4141 y(The)k(microlo)r(cal)f (scaling)g(degree)g(of)h Fu(t)g Fv(at)g Fm(N)i Fv(is)e(then)h (de\014ned)f(in)h(terms)f(of)g(all)g(sequences)f(so)456 4341 y(obtained.)555 4540 y(W)-7 b(e)24 b(c)n(ho)r(ose)f(a)g (\014bration)g(of)h(a)f(neigh)n(b)r(ourho)r(o)r(d)g(of)g Fm(N)j Fv(b)n(y)d(transv)n(ersal)e(surfaces.)35 b(It)24 b(is)f(easy)456 4739 y(to)34 b(see)g(that)h(if)g Fu(t)g Fv(has)f(a)g(scaling)f(degree)h Fu(!)j Fv(at)e Fm(N)q Fv(,)h(than)f(its)g(restriction)e(to)i(a)f(transv)n(ersal)456 4938 y(surface)h(has)h(a)f(scaling)h(degree)f(at)h(the)g(p)r(oin)n(t)h (of)f(in)n(tersection)f(with)i Fm(N)g Fv(whic)n(h)g(is)f(less)f(or)456 5138 y(equal)27 b(to)g Fu(!)s Fv(.)37 b(W)-7 b(e)28 b(therefore)e (obtain)i(the)g(corresp)r(onding)d(extension)j(theorem.)p eop %%Page: 62 62 62 61 bop 456 226 a Fy(62)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fs(Theorem)i(6.9.)40 b Ft(L)l(et)27 b Fm(N)i Ft(b)l(e)f(a)g(submanifold)i(of)e(the)g (manifold)i Fm(M)p Ft(,)e(and)g(let)g Fu(t)2926 437 y Fy(0)2986 425 y Fr(2)23 b Fm(D)3133 395 y Fq(0)3157 425 y Fv(\()p Fm(M)5 b Fr(n)g Fm(N)p Fv(\))p Ft(.)527 669 y Fv(\()p Fu(i)p Fv(\))41 b Ft(If)28 b Fv(sd)824 681 y Fd(N)884 669 y Fv(\()p Fu(t)946 681 y Fy(0)983 669 y Fv(\))c Fu(<)e Fv(co)r(dim\()p Fm(N)q Fv(\))28 b Ft(ther)l(e)f (exists)g(a)g(unique)f(distribution)i Fu(t)23 b Fr(2)g Fm(D)2906 639 y Fq(0)2930 669 y Fv(\()p Fm(M)p Fv(\))j Ft(extending)661 869 y Fu(t)691 881 y Fy(0)758 869 y Ft(with)31 b Fv(sd)1017 881 y Fd(N)1077 869 y Fv(\()p Fu(t)p Fv(\))23 b(=)g(sd)1361 881 y Fd(N)1420 869 y Fv(\()p Fu(t)1482 881 y Fy(0)1520 869 y Fv(\))498 1068 y(\()p Fu(ii)p Fv(\))41 b Ft(If)33 b Fv(co)r(dim\()p Fm(N)q Fv(\))27 b Fr(\024)g Fv(sd)1297 1080 y Fd(N)1356 1068 y Fv(\()p Fu(t)1418 1080 y Fy(0)1456 1068 y Fv(\))g Fu(<)f Fr(1)32 b Ft(ther)l(e)g(exist)f(extensions)h Fu(t)26 b Fr(2)i Fm(D)2737 1038 y Fq(0)2760 1068 y Fv(\()p Fm(M)p Fv(\))k Ft(with)g Fv(sd)3199 1080 y Fd(N)3258 1068 y Fv(\()p Fu(t)p Fv(\))c(=)661 1267 y(sd)740 1279 y Fd(N)800 1267 y Fv(\()p Fu(t)862 1279 y Fy(0)899 1267 y Fv(\))p Ft(.)39 b(They)30 b(ar)l(e)g(uniquely)f(char)l(acterize)l(d)i(by)f (their)g(values)g(on)f(some)h(close)l(d)g(sub-)661 1466 y(sp)l(ac)l(e)c(of)f Fm(D)p Fv(\()p Fm(M)p Fv(\))g Ft(which)i(is)e(c)l (omplementary)h(to)f(the)g(sp)l(ac)l(e)g(of)h(al)t(l)g(test)e (functions)h(which)661 1666 y(vanish)31 b(on)f Fm(N)h Ft(up)f(to)f(or)l(der)i Fv(sd)1648 1678 y Fd(N)1707 1666 y Fv(\()p Fu(t)1769 1678 y Fy(0)1807 1666 y Fv(\))19 b Fr(\000)f Fv(co)r(dim\()p Fm(N)q Fv(\))p Ft(.)456 2028 y(Pr)l(o)l(of.)43 b Fv(According)24 b(to)h(Theorems)e(5.2)i(and)f(5.3)h (there)f(exist)h(extensions)f(of)h(the)h(restrictions)456 2227 y(of)19 b Fu(t)572 2239 y Fy(0)628 2227 y Fv(to)g(ev)n(ery)f (transv)n(ersal)f(surface)i(with)g(the)h(same)f(scaling)f(degree.)33 b(W)-7 b(e)20 b(ha)n(v)n(e)e(to)h(sho)n(w)f(that)456 2426 y(they)h(are)f(restrictions)g(of)h(a)f(unique)i(distribution)f Fu(t)g Fv(on)g Fm(M)f Fv(with)i(the)f(same)g(microlo)r(cal)e(scaling) 456 2625 y(degree.)38 b(W)-7 b(e)28 b(\014rst)g(\014x)h(some)e(normal)h (\014bration)f(as)h(describ)r(ed)g(ab)r(o)n(v)n(e)f(and)h(consider)g Fu(t)3235 2637 y Fy(0)3300 2625 y Fv(as)g(a)456 2825 y(distribution)19 b(on)h Fu(U)14 b Fr(n)5 b(f)p Fv(0)p Fr(g)16 b Fv(with)k(a)g(neigh)n(b)r(ourho)r(o)r(d)e Fu(U)29 b Fv(of)19 b(the)h(zero)f(section)g(of)g Fu(C)6 b Fv(.)35 b(W)-7 b(e)20 b(p)r(erform)456 3024 y(the)30 b(construction)g(of)g Fu(t)g Fv(at)h(all)f(\014b)r(ers,)g(b)n(y)g(c)n(ho)r(osing)f(a)h(smo)r (oth)g(function)h Fu(#)d Fr(2)g Fm(E)p Fv(\()p Fu(C)6 b Fv(\))32 b(whic)n(h)456 3223 y(is)25 b(equal)f(to)h(1)g(in)g(a)g (neigh)n(b)r(ourho)r(o)r(d)f(of)h(the)g(zero)f(section)h(and)g(whose)f (restrictions)g(to)h(ev)n(ery)456 3422 y(\014b)r(er)19 b(ha)n(v)n(e)g(compact)g(supp)r(ort.)34 b(Moreo)n(v)n(er,)18 b(w)n(e)h(c)n(ho)r(ose)g(smo)r(oth)g(functions)h Fk(w)2934 3434 y Fo(\014)3002 3422 y Fr(2)j Fm(E)p Fv(\()p Fu(U)9 b Fv(\),)23 b(also)456 3622 y(with)32 b(compact)f(supp)r(ort)g(on)h (eac)n(h)f(\014b)r(er,)h(whic)n(h)g(satisfy)f(the)h(condition)g Fu(@)2896 3582 y Fo(\015)2891 3647 y(\030)2938 3622 y Fk(w)3002 3634 y Fo(\014)3047 3622 y Fv(\()p Fu(x;)14 b Fv(0\))30 b(=)g Fu(\016)3402 3582 y Fo(\015)3399 3647 y(\014)456 3821 y Fv(where)f Fu(\030)35 b Fv(denotes)29 b(the)i(v)-5 b(ariable)29 b(in)h(the)h(\014b)r(er)f(o)n(v)n(er)e Fu(x)g Fr(2)g Fm(N)q Fv(.)45 b(W)-7 b(e)30 b(ma)n(y)g(tak)n(e)f Fk(w)3015 3833 y Fo(\014)3087 3821 y Fv(=)e Fk(w)p Fu(\030)3283 3791 y Fo(\014)3328 3821 y Fu(=\014)t Fv(!)456 4020 y(with)c(some)f (function)i Fk(w)f Fv(whic)n(h)g(is)g(iden)n(tical)f(to)h(1)f(in)i(a)e (neigh)n(b)r(ourho)r(o)r(d)g(of)h(the)g(zero)f(section.)555 4219 y(W)-7 b(e)44 b(set)g Fu(\032)50 b Fv(=)g(sd)1146 4231 y Fd(N)1206 4219 y Fv(\()p Fu(t)1268 4231 y Fy(0)1305 4219 y Fv(\))30 b Fr(\000)f Fv(co)r(dim\()p Fm(N)q Fv(\),)49 b Fu(#)1930 4235 y Fo(\025)1969 4218 y Ff(\000)p FC(1)2051 4219 y Fv(\()p Fu(x;)14 b(\030)t Fv(\))51 b(=)e Fu(#)p Fv(\()p Fu(x;)14 b(\025)2617 4189 y Fq(\000)p Fy(1)2708 4219 y Fu(\030)t Fv(\))44 b(and)f Fu(W)12 b(\036)p Fv(\()p Fu(x;)i(\030)t Fv(\))52 b(=)456 4419 y Fu(\036)p Fv(\()p Fu(x;)14 b(\030)t Fv(\))t Fr(\000)766 4356 y Fj(P)855 4443 y Fq(j)p Fo(\014)s Fq(j\024)p Fo(\032)1039 4419 y Fk(w)1103 4431 y Fo(\014)1148 4419 y Fv(\()p Fu(x;)g(\030)t Fv(\))p Fu(@)1385 4379 y Fo(\014)1380 4444 y(\030)1431 4419 y Fu(\036)p Fv(\()p Fu(x;)g Fv(0\).)35 b(Let)21 b(\000)1922 4431 y Fo(C)2001 4419 y Fv(=)p 2088 4346 272 4 v 22 w(WF)q(\()p Fu(t)2290 4431 y Fy(0)2327 4419 y Fv(\))5 b Fr([)t Fu(N)2499 4388 y Fq(\003)2538 4419 y Fm(N)q Fv(.)35 b(W)-7 b(e)20 b(already)f(kno)n(w)h(that)456 4618 y(the)j(sequence)f Fu(t)963 4630 y Fo(n)1031 4618 y Fv(=)h Fu(t)1149 4630 y Fy(0)1186 4618 y Fv(\(1)9 b Fr(\000)g Fu(#)1392 4630 y Fy(2)1425 4614 y Fh(n)1469 4618 y Fv(\))g Fr(\016)g Fu(W)34 b Fv(con)n(v)n(erges)20 b(on)i(ev)n(ery)f(\014b)r(er,)j(and)f(it)g(is)g(easy)e(to)i(see)f(that) 456 4817 y(it)27 b(con)n(v)n(erges)d(w)n(eakly)i(in)h Fm(D)1348 4787 y Fq(0)1371 4817 y Fv(\()p Fu(U)9 b Fv(\).)37 b(W)-7 b(e)27 b(no)n(w)f(w)n(an)n(t)h(to)f(sho)n(w)g(that)h(the)h(w)n (a)n(v)n(e)d(fron)n(t)h(set)h(of)g Fu(t)g Fv(is)456 5016 y(p)r(erp)r(endicular)e(to)h Fu(T)12 b Fm(N)q Fv(.)35 b(F)-7 b(or)25 b(this)h(purp)r(ose)f(w)n(e)h(sho)n(w)f(that)h(the)g(ab) r(o)n(v)n(e)e(sequence)h(con)n(v)n(erges)456 5216 y(ev)n(en)f(in)i Fm(D)806 5185 y Fq(0)806 5238 y Fy(\000)847 5246 y Fh(C)900 5216 y Fv(\()p Fu(U)9 b Fv(\),)26 b(i.e.)36 b(that)26 b(for)e(ev)n(ery)g(pseudo)r(di\013eren)n(tial)h(op)r(erator)f Fu(A)i Fv(whose)e(w)n(a)n(v)n(e)g(fron)n(t)p eop %%Page: 63 63 63 62 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(63)456 425 y Fv(set)26 b(do)r(es)g(not)g(in)n(tersect)g(\000)1302 437 y Fo(C)1384 425 y Fv(\(hence)g Fu(At)1737 437 y Fo(n)1809 425 y Fv(is)g(smo)r (oth\))g(the)h(sequence)e Fu(At)2791 437 y Fo(n)2863 425 y Fv(con)n(v)n(erges)f(in)i(the)456 624 y(sense)33 b(of)h(smo)r(oth)g(functions.)57 b(F)-7 b(or)33 b(a)h(pseudo)r (di\013eren)n(tial)g(op)r(erator)e(with)j(smo)r(oth)e(k)n(ernel)456 824 y(the)g(argumen)n(t)f(is)i(essen)n(tially)e(the)i(same)e(as)h(for)g (the)g(w)n(eak)g(con)n(v)n(ergence,)f(hence)h(w)n(e)g(ma)n(y)456 1023 y(restrict)f(ourselv)n(es)f(to)i(pseudo)r(di\013eren)n(tial)g(op)r (erators)e(whose)h(k)n(ernels)g(ha)n(v)n(e)g(supp)r(ort)h(in)g(a)456 1222 y(su\016cien)n(tly)d(small)g(neigh)n(b)r(ourho)r(o)r(d)g(of)h(the) g(diagonal)e(of)h Fu(U)g Fr(\002)20 b Fu(U)39 b Fv(\(only)31 b(here)f(singularities)456 1421 y(ma)n(y)c(o)r(ccur\).)555 1621 y(According)c(to)h(the)g(discussion)f(of)g(condition)h Fs(M2)p Fv(\(b\),)h(w)n(e)e(ma)n(y)g(equiv)-5 b(alen)n(tly)22 b(lo)r(ok)g(at)h(the)456 1820 y(F)-7 b(ourier)28 b(transform)g(of)i Fu(\037t)1307 1832 y Fy(0)1344 1820 y Fv(\()p Fu(#)1425 1832 y Fy(2)1458 1816 y Fh(m)1537 1820 y Fr(\000)19 b Fu(#)1670 1836 y Fy(2)1703 1819 y Fh(m)p FC(+1)1834 1820 y Fv(\))29 b(where)g Fu(\037)h Fv(is)f(a)g(test)h(function)g(with)f (su\016cien)n(tly)456 2019 y(small)i(supp)r(ort)g(whic)n(h)h(do)r(es)f (not)h(v)-5 b(anish)31 b(at)h(some)f(p)r(oin)n(t)h Fu(x)2419 2031 y Fy(0)2486 2019 y Fr(2)e Fm(N)q Fv(.)50 b(In)n(tro)r(ducing)31 b(suitable)456 2218 y(lo)r(cal)c(co)r(ordinates)f(in)i(a)f(neigh)n(b)r (ourho)r(o)r(d)f(of)i Fu(x)1968 2230 y Fy(0)2006 2218 y Fv(,)f(w)n(e)h(\014nd)644 2557 y([)q Fu(\037t)750 2569 y Fy(0)787 2557 y Fv(\()p Fu(#)868 2569 y Fy(2)901 2553 y Fh(m)979 2557 y Fr(\000)18 b Fu(#)1111 2573 y Fy(2)1144 2556 y Fh(m)p FC(+1)1274 2557 y Fv(\)])1330 2515 y Fq(^)1392 2557 y Fv(\()p Fu(p;)c(k)s Fv(\))24 b(=)e(2)1734 2523 y Fq(\000)p Fo(m)p Fy(\()p Fo(d)p Fq(\000)p Fo(\016)r Fy(\))2033 2490 y Fj(\002)2068 2557 y Fv(\()p Fu(\037t)2182 2569 y Fy(0)2219 2557 y Fv(\))2251 2573 y Fy(2)2284 2556 y Ff(\000)p Fh(m)2385 2573 y Fo(;)p Fq(?)2460 2557 y Fv(\()p Fu(#)d Fr(\000)f Fu(#)2692 2569 y Fy(2)2730 2557 y Fv(\))2762 2490 y Fj(\003)2796 2507 y Fq(^)2859 2557 y Fv(\()p Fu(p;)c Fv(2)3012 2523 y Fq(\000)p Fo(m)3127 2557 y Fu(k)s Fv(\))27 b Fu(;)456 2896 y Fv(where)i(the)h(sym)n(b)r(ol) f([)f Fr(\001)g Fv(])1254 2854 y Fq(^)1333 2896 y Fv(means)h(F)-7 b(ourier)29 b(transform,)g(the)h Fu(p)p Fv('s)f(are)g(the)h(dual)g(co)r (ordinates)456 3095 y(w.r.t.)57 b(p)r(oin)n(ts)35 b Fu(x)f Fr(2)h Fm(N)h Fv(and)f(similarly)e(the)i Fu(k)s Fv('s)f(are)g(dual)g (co)r(ordinates)f(w.r.t.)58 b(the)35 b(p)r(oin)n(ts)456 3294 y Fu(\030)27 b Fr(2)c Fu(C)656 3306 y Fo(x)726 3294 y Fv(of)28 b(the)g(\014b)r(ers)f Fu(C)1247 3306 y Fo(x)1317 3294 y Fv(of)g Fu(C)34 b Fv(and)28 b(\014nally)f Fu(\016)k Fv(is)c(the)h(dimension)g(of)f Fm(N)q Fv(.)555 3494 y(By)i(the)g (assumption)g(on)g(the)g(scaling)f(degree)g(of)h Fu(t)2193 3506 y Fy(0)2260 3494 y Fv(at)f Fm(N)j Fv(w)n(e)e(kno)n(w)f(that)h(for) g(all)g Fu(")c(>)g Fv(0,)456 3693 y Fu(N)31 b Fr(2)24 b Fn(N)j Fv(and)h Fu(!)e(>)c Fv(sd)1126 3705 y Fd(N)1186 3693 y Fv(\()p Fu(t)1248 3705 y Fy(0)1285 3693 y Fv(\))28 b(there)f(exists)h(some)f Fu(c)c(>)f Fv(0)27 b(suc)n(h)h(that)968 4032 y Fr(j)p Fv([\()p Fu(\037t)1128 4044 y Fy(0)1166 4032 y Fv(\))1198 4047 y Fy(2)1231 4031 y Ff(\000)p Fh(m)1331 4047 y Fo(;)p Fq(?)1407 4032 y Fv(\()p Fu(#)19 b Fr(\000)f Fu(#)1639 4044 y Fy(2)1676 4032 y Fv(\)])1731 3997 y Fq(^)1780 4032 y Fv(\()p Fu(p;)c(k)s Fv(\))p Fr(j)24 b(\024)e Fu(c)28 b Fv(2)2209 3997 y Fo(m!)2315 4032 y Fv(\(1)19 b(+)f Fr(j)p Fu(p)p Fr(j)g Fv(+)g Fr(j)p Fu(k)s Fr(j)p Fv(\))2804 3997 y Fo(N)2895 4032 y Fu(;)379 b Fv(\(51\))456 4370 y(for)28 b(all)h(\()p Fu(p;)14 b(k)s Fv(\))29 b(with)h Fr(j)p Fu(p)p Fr(j)25 b Fu(>)g(")p Fr(j)p Fu(k)s Fr(j)k Fv(\(i.e.)42 b(outside)29 b(of)g(a)f(certain)h (conical)f(neigh)n(b)r(ourho)r(o)r(d)g(of)h(the)456 4570 y(normal)d(bundle)i Fr(f)p Fv(\()p Fu(p;)14 b(k)s Fv(\))p Fu(;)g(p)23 b Fv(=)f(0)p Fr(g)p Fv(\).)37 b(Therefore,)26 b(the)i(sequence)1095 4908 y(sup)1041 4982 y Fq(j)p Fo(p)p Fq(j)p Fo(>")p Fq(j)p Fo(k)q Fq(j)1288 4908 y Fr(j)p Fv([)p Fu(\037t)1416 4920 y Fy(0)1453 4908 y Fv(\()p Fu(#)1534 4920 y Fy(2)1567 4904 y Fh(m)1645 4908 y Fr(\000)18 b Fu(#)1777 4924 y Fy(2)1810 4907 y Fh(m)p FC(+1)1940 4908 y Fv(\)])1995 4874 y Fq(^)2045 4908 y Fv(\()p Fu(p;)c(k)s Fv(\))p Fr(j)p Fv(\(1)k(+)g Fr(j)p Fu(p)p Fr(j)g Fv(+)g Fr(j)p Fu(k)s Fr(j)p Fv(\))2745 4874 y Fo(N)2836 4908 y Fu(;)438 b Fv(\(52\))p eop %%Page: 64 64 64 63 bop 456 226 a Fy(64)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(is)23 b(summable)f(for)h Fu(!)j(<)c Fv(\()p Fu(d)9 b Fr(\000)g Fu(\016)s Fv(\).)37 b(If)23 b(sd)1657 437 y Fd(N)1717 425 y Fv(\()p Fu(t)1779 437 y Fy(0)1816 425 y Fv(\))g Fu(<)g Fv(\()p Fu(d)9 b Fr(\000)g Fu(\016)s Fv(\))24 b(suc)n(h)f(an)g Fu(!)j Fv(exists,)d(and)g(w)n(e)g(conclude,)456 624 y(that)k(the)h(sequence)g Fu(t)1153 636 y Fy(0)1190 624 y Fv(\(1)18 b Fr(\000)g Fu(#)1414 636 y Fy(2)1447 620 y Fh(m)1507 624 y Fv(\))27 b(con)n(v)n(erges)e(in)j Fm(D)2104 594 y Fq(0)2104 647 y Fy(\000)2145 655 y Fh(C)2198 624 y Fv(\()p Fu(M)9 b Fv(\).)555 824 y(In)32 b(case)f(of)h(sd)1021 836 y Fd(N)1080 824 y Fv(\()p Fu(t)1142 836 y Fy(0)1180 824 y Fv(\))e Fr(\025)f Fv(\()p Fu(d)22 b Fr(\000)f Fu(\016)s Fv(\))32 b(w)n(e)g(ha)n(v)n(e)e(to)i(apply)f(the)h Fu(W)12 b Fv(-op)r(eration)31 b(de\014ned)h(ab)r(o)n(v)n(e,)456 1023 y(whic)n(h,)27 b(for)f Fu(\037)h Fv(with)h(su\016cien)n(tly)f(small)g(supp)r(ort,)g (reduces)f(to)h(a)g(subtraction)f(of)h(the)g(T)-7 b(a)n(ylor)456 1222 y(series)23 b(up)i(to)f(order)g([)p Fu(\032)p Fv(].)36 b(W)-7 b(e)25 b(obtain)f(from)g(the)h(in)n(tegral)e(form)n(ula)h(for)g (the)h(remainder)e(in)i(the)456 1421 y(T)-7 b(a)n(ylor)25 b(expansion)723 1712 y([)p Fu(\037t)828 1724 y Fy(0)866 1712 y Fv(\()p Fu(#)947 1724 y Fy(2)980 1708 y Fh(m)1058 1712 y Fr(\000)18 b Fu(#)1190 1728 y Fy(2)1223 1711 y Fh(m)p FC(+1)1353 1712 y Fv(\))h Fr(\016)f Fu(W)12 b Fv(])1577 1678 y Fq(^)1626 1712 y Fv(\()p Fu(p;)i(k)s Fv(\))23 b(=)812 1851 y Fj(Z)895 1872 y Fy(1)858 2040 y(0)1030 1886 y Fj(X)946 2067 y Fq(j)p Fo(\014)s Fq(j)p Fy(=[)p Fo(\032)p Fy(]+1)1234 1964 y Fv([)p Fu(\037t)1339 1976 y Fy(0)1376 1964 y Fv(\()p Fu(#)1457 1976 y Fy(2)1490 1960 y Fh(m)1568 1964 y Fr(\000)18 b Fu(#)1700 1980 y Fy(2)1733 1963 y Fh(m)p FC(+1)1863 1964 y Fv(\))p Fu(\030)1935 1930 y Fo(\014)1981 1964 y Fv(])2004 1930 y Fq(^)2053 1964 y Fv(\()p Fu(p;)c(\026k)s Fv(\))p Fu(k)2338 1930 y Fo(\014)2392 1908 y Fv(\(1)k Fr(\000)h Fu(\026)p Fv(\))2650 1878 y Fy([)p Fo(\032)p Fy(])2726 1908 y Fv(\([)p Fu(\032)p Fv(])g(+)f(1\))p 2392 1945 631 4 v 2670 2021 a Fu(\014)t Fv(!)3033 1964 y Fu(d\026)27 b(:)456 2309 y Fv(Since)32 b Fu(\037t)759 2321 y Fy(0)796 2309 y Fu(\030)836 2278 y Fo(\014)913 2309 y Fv(has)f(scaling)h(degree)e(sd)1684 2321 y Fd(N)1744 2309 y Fv(\()p Fu(t)1806 2321 y Fy(0)1843 2309 y Fv(\))22 b Fr(\000)f Fv([)p Fu(\032)p Fv(])g Fr(\000)g Fv(1)30 b Fu(<)g Fv(\()p Fu(d)22 b Fr(\000)f Fu(\016)s Fv(\),)34 b(w)n(e)d(ma)n(y)h(use)g(the)g(same)456 2508 y(estimate)21 b(as)g(b)r(efore)h(and)g(\014nd)g(that)g(the)g(sequence)f Fu(t)2117 2520 y Fy(0)2155 2508 y Fv(\(1)7 b Fr(\000)g Fu(#)2357 2520 y Fy(2)2390 2504 y Fh(m)2448 2508 y Fv(\))g Fr(\016)g Fu(W)33 b Fv(con)n(v)n(erges)20 b(in)i Fm(D)3174 2478 y Fq(0)3174 2531 y Fy(\000)3215 2539 y Fh(C)3267 2508 y Fv(\()p Fu(M)9 b Fv(\).)555 2707 y(It)31 b(remains)e(to)i(pro)n (v)n(e)d(the)j(stabilit)n(y)f(of)g(the)h(scaling)e(degree)h(under)g (the)h(extension)f(pro-)456 2906 y(cedure.)56 b(The)34 b(argumen)n(t)f(is)h(a)g(straigh)n(tforw)n(ard)d(com)n(bination)i(of)h (the)h(tec)n(hniques)f(in)g(the)456 3106 y(corresp)r(onding)17 b(pro)r(ofs)i(in)i(Section)e(5)h(and)f(the)i(argumen)n(ts)d(ab)r(o)n(v) n(e)h(and)g(is)h(therefore)f(omitted.)3380 3305 y Fl(\004)850 3970 y Fv(7.)45 b Fw(Extension)31 b(to)h(the)f(Dia)n(gonal)g(and)g (Renormaliza)-6 b(tion)555 4219 y Fv(W)f(e)41 b(come)f(to)g(the)g(main) g(p)r(oin)n(t,)k(namely)c(to)g(pro)n(v)n(e)e(that)j(the)f(inductiv)n(e) h(analysis)e(of)456 4419 y(Section)30 b(4)g(closes)g(when)g(supplemen)n (ted)h(b)n(y)f(the)h(information)f(ab)r(out)g(the)h(scaling)f(degrees) 456 4618 y(and)23 b(giv)n(es)f(w)n(ell)h(de\014ned)h(op)r(erator-v)-5 b(alued)21 b(distributions)j Fu(T)2380 4630 y Fo(n)2448 4618 y Fv(all)f(o)n(v)n(er)f(the)h(space)g Fu(M)3179 4588 y Fo(n)3224 4618 y Fv(.)36 b(This)456 4817 y(pro)r(cess)26 b(is)i(what)f(it)h(is)g(usually)f(called)g(\\renormalization.")555 5016 y(T)-7 b(o)n(w)n(ard)29 b(this)j(goal)d(w)n(e)i(m)n(ust)g(sho)n(w) f(that)h(the)h(scaling)e(degree)g(for)g(the)i(distributions)f(of)456 5216 y(the)e Fu(n)p Fv(th)h(order)e(of)h(the)h(induction)f(can)g(b)r(e) h(estimated)f(in)h(terms)f(of)g(those)g(of)g(lo)n(w)n(er)f(orders.)p eop %%Page: 65 65 65 64 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(65)456 425 y Fv(As)27 b(explained)h(b)r(efore)f(it)h(is)f(su\016cien)n(t)h(to)g(do)f(it)h (for)f(the)h(n)n(umerical)f(distributions)1301 730 y Fy(0)1338 764 y Fu(t)p Fv(\()p Fu(x)1447 776 y Fy(1)1485 764 y Fu(;)14 b(:)g(:)g(:)g(;)g(x)1717 776 y Fo(n)1762 764 y Fv(\))24 b(=)e Fu(!)s Fv(\()1992 730 y Fy(0)2030 764 y Fu(T)12 b Fv(\()p Fu(x)2170 776 y Fy(1)2207 764 y Fu(;)i(:)g(:)g(:)f(;)h(x)2438 776 y Fo(n)2484 764 y Fv(\)\))28 b Fu(;)698 b Fv(\(53\))456 1103 y(where)31 b Fu(!)j Fv(is)e(our)e(reference)h(Hadamard)g(state.)48 b(According)31 b(to)h(\(31\))2722 1073 y Fy(0)2759 1103 y Fu(t)g Fv(is)f(a)h(\014nite)g(sum)g(of)456 1302 y(terms)1178 1641 y Fu(f)1219 1653 y Fo(I)1257 1641 y Fv(\()p Fu(x)p Fv(\))p Fu(t)1398 1606 y Fo(I)1437 1641 y Fv(\()p Fu(x)1516 1653 y Fo(I)1554 1641 y Fv(\))p Fu(t)1616 1606 y Fo(I)1650 1581 y Fh(c)1686 1641 y Fv(\()p Fu(x)1765 1653 y Fo(I)1799 1636 y Fh(c)1835 1641 y Fv(\))1989 1562 y Fj(Y)1881 1744 y Fy(\()p Fo(i;j)s Fy(\))p Fq(2)p Fo(I)5 b Fq(\002)p Fo(I)2171 1727 y Fh(c)2216 1641 y Fu(!)2268 1653 y Fy(2)2305 1641 y Fv(\()p Fu(x)2384 1653 y Fo(i)2413 1641 y Fu(;)14 b(x)2497 1653 y Fo(j)2532 1641 y Fv(\))2564 1606 y Fo(a)2600 1614 y Fh(ij)2685 1641 y Fu(;)589 b Fv(\(54\))456 1979 y(with)38 b(nonnegativ)n(e)e(in)n(tegers)h Fu(a)1488 1991 y Fo(ij)1584 1979 y Fv(and)h(a)g(smo)r(oth)f(partition)h(of)f (unit)n(y)h(of)g Fu(M)3022 1949 y Fo(n)3072 1979 y Fr(n)5 b Fv(\001)3188 1991 y Fo(n)3270 1979 y Fu(f)3311 1991 y Fo(I)3389 1979 y Fr(2)456 2179 y Fm(E)p Fv(\()p Fu(M)628 2148 y Fo(n)678 2179 y Fr(n)g Fv(\001)794 2191 y Fo(n)838 2179 y Fv(\))21 b(with)g(supp)p Fu(f)1285 2191 y Fo(I)1346 2179 y Fr(\032)i(C)1478 2191 y Fo(I)1515 2179 y Fv(.)35 b Fu(M)1663 2148 y Fo(n)1728 2179 y Fv(inherits)20 b(a)g(natural)g (metric)g(from)g Fu(M)9 b Fv(,)22 b(and)e(all)g(partial)456 2378 y(diagonals)28 b(\001)892 2390 y Fo(I)961 2378 y Fv(are)i(totally)g(geo)r(desic)g(submanifolds.)45 b(The)31 b(map)f Fu(\013)f Fv(:)f Fu(T)12 b(M)2887 2348 y Fo(n)2958 2378 y Fr(!)28 b Fu(M)3159 2348 y Fo(n)3225 2378 y Fr(\002)20 b Fu(M)3400 2348 y Fo(n)456 2577 y Fv(ma)n(y)36 b(therefore)f(b)r(e)j (de\014ned)f(b)n(y)f Fu(\013)p Fv(\()p Fu(x;)14 b(\030)t Fv(\))40 b(=)e(\()p Fu(x;)14 b Fv(exp)2171 2597 y Fo(x)2227 2577 y Fu(\030)t Fv(\).)65 b(Then)37 b(all)f(restrictions)g(of)g Fu(\013)i Fv(to)456 2776 y(partial)31 b(sub)r(diagonals,)h Fu(\013)1302 2788 y Fo(I)1372 2776 y Fv(=)f Fu(\013)t Fl(\026)1560 2788 y Fo(T)1599 2796 y FC(\001)1645 2810 y Fh(I)1681 2788 y Fo(M)1750 2772 y Fh(n)1828 2776 y Fv(satify)i(the)f(conditions)g(b)r(efore)g(Prop)r(osition)f(6.4.)456 2976 y(W)-7 b(e)24 b(can)g(c)n(ho)r(ose)f(the)h(functions)h Fu(f)1539 2988 y Fo(I)1600 2976 y Fv(with)g(smo)r(oth)f(scaling)f (degree)g(0)h(at)g(the)g(small)g(diagonal.)456 3175 y(W)-7 b(e)36 b(then)h(ma)n(y)e(consider)h(all)g(factors)f(in)h(\(54\))g(as)g (distributions)g(on)g Fu(M)2837 3145 y Fo(n)2882 3175 y Fv(.)62 b(According)36 b(to)456 3374 y(Lemma)28 b(6.7,)g(their)g (microlo)r(cal)f(scaling)h(degrees)f(with)i(resp)r(ect)f(to)h(\001)2723 3386 y Fo(n)2796 3374 y Fv(are)f(b)r(ounded)h(from)456 3573 y(ab)r(o)n(v)n(e)38 b(b)n(y)h(their)h(microlo)r(cal)e(scaling)h (degrees)f(with)i(resp)r(ect)g(to)f(the)h(resp)r(ectiv)n(e)f(partial) 456 3773 y(diagonals.)48 b(Moreo)n(v)n(er)29 b(the)j(con)n(v)n(ex)f (com)n(binations)f(of)i(the)h(resp)r(ectiv)n(e)e(conical)g(subsets)h (of)456 3972 y(the)26 b(conormal)e(bundle)i(of)g(the)g(small)f (diagonal)g(do)g(not)h(meet)g(the)g(zero)f(section.)35 b(Hence,)27 b(b)n(y)456 4171 y(Lemma)g(6.6,)g(the)h(scaling)e(degree)h (of)g(the)h(distribution)g(in)g(\(54\))f(is)h(b)r(ounded)g(b)n(y)1162 4510 y Fu(!)e Fv(=)d(sd)1407 4522 y Fy(\001)1462 4530 y Fh(I)1499 4510 y Fv(\()p Fu(t)1561 4522 y Fo(I)1599 4510 y Fv(\))c(+)f(sd)1812 4522 y Fy(\001)1867 4532 y Fh(I)1896 4520 y(c)1935 4510 y Fv(\()p Fu(t)1997 4522 y Fo(I)2031 4506 y Fh(c)2066 4510 y Fv(\))h(+)2200 4431 y Fj(X)2233 4608 y Fo(ij)2334 4510 y Fu(a)2378 4522 y Fo(ij)2436 4510 y Fv(\()p Fu(d)g Fr(\000)f Fv(2\))28 b Fu(:)559 b Fv(\(55\))p eop %%Page: 66 66 66 65 bop 456 226 a Fy(66)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(F)-7 b(rom)27 b(this)h(w)n(e)f(get)g(the)h(form)n(ula)1214 764 y(sd)1293 776 y Fy(\001)1348 784 y Fh(n)1392 764 y Fv(\()p Fu(!)s Fv(\()p Fu(T)12 b Fv(\(:)1669 660 y Fo(n)1637 685 y Fj(Y)1636 862 y Fo(i)p Fy(=1)1758 764 y Fu(')1812 730 y Fo(l)1833 738 y Fh(i)1863 764 y Fv(\()p Fu(x)1942 776 y Fo(i)1971 764 y Fv(\))d(:\)\)\))24 b Fr(\024)2282 660 y Fo(n)2243 685 y Fj(X)2249 862 y Fo(i)p Fy(=1)2377 764 y Fu(l)2402 776 y Fo(i)2439 708 y Fu(d)19 b Fr(\000)f Fv(2)p 2439 745 187 4 v 2511 821 a(2)2663 764 y Fu(:)611 b Fv(\(56\))555 1103 y(W)-7 b(e)28 b(th)n(us)g(obtain)456 1540 y Fs(Theorem)j(7.1)g Fv(\(Main)d(Theorem\))p Fs(.)40 b Ft(A)n(l)t(l)35 b(p)l(olynomial)t(ly) j(inter)l(acting)e(quantum)e(\014eld)h(the)l(o-)456 1739 y(ries)h(b)l(ase)l(d)h(on)f(the)g(the)g(sc)l(alar)g(\014eld)h(on)f Fu(d)e Fr(\025)g Fv(2)h Ft(dimensional)j(glob)l(al)t(ly)g(hyp)l(erb)l (olic)g(sp)l(ac)l(e-)456 1938 y(times)23 b(fol)t(low)i(the)e(same)g (short-distanc)l(e)h(p)l(erturb)l(ative)f(classi\014c)l(ation)i(as)e (on)g(their)h(r)l(esp)l(e)l(ctive)456 2138 y(Minkowskian)31 b(c)l(ases.)555 2575 y Fv(W)-7 b(e)31 b(recall)e(that)i(for)f (simplicit)n(y)g(w)n(e)g(ha)n(v)n(e)g(considered)f(only)h(pure)g (monomials)f(as)h(in)n(ter-)456 2774 y(acting)c(terms,)i(i.e.)36 b(without)28 b(deriv)-5 b(ativ)n(es,)26 b(m)n(ultiple)i(in)n(teracting) f(\014elds)g(and)g(so)g(forth.)36 b(The)456 2973 y(general)31 b(case)g(ho)n(w)n(ev)n(er)f(can)i(b)r(e)h(deriv)n(ed)e(straigh)n(tforw) n(ardly)f(from)i(our)f(construction)h(and)456 3172 y(w)n(e)27 b(lea)n(v)n(e)f(the)i(task)f(to)h(the)g(reader.)555 3372 y(W)-7 b(e)36 b(close)e(this)i(section)f(b)n(y)g(sp)r(ecifying)g(the)h (c)n(hoices)e(whic)n(h)h(ha)n(v)n(e)g(to)g(b)r(e)g(made)h(in)f(the)456 3571 y(abstract)29 b(geometrical)h(setting)h(of)g(Section)g(6.)46 b(First)31 b(w)n(e)f(c)n(ho)r(ose)g(the)h(normal)f(\014bration)h(of)456 3770 y Fu(M)546 3740 y Fo(n)618 3770 y Fv(in)c(a)f(neigh)n(b)r(ourho)r (o)r(d)g(of)h(\001)1512 3782 y Fo(n)1585 3770 y Fv(whic)n(h)g(w)n(as)f (used)h(in)g(the)h(pro)r(of)e(of)h(the)h(Theorem)e(6.9)g(for)456 3969 y(the)e(construction)g(of)g(the)h(extension.)35 b(Let)25 b Fu(N)9 b Fv(\001)1988 3981 y Fo(n)2057 3969 y Fv(b)r(e)25 b(the)f(orthogonal)e(complemen)n(t)j(of)f Fu(T)12 b Fv(\001)3400 3981 y Fo(n)456 4169 y Fv(in)27 b Fu(T)601 4181 y Fy(\001)656 4189 y Fh(n)700 4169 y Fu(M)790 4139 y Fo(n)863 4169 y Fv(w.r.t.)37 b(the)28 b(metric)f(in)h Fu(M)1682 4139 y Fo(n)1727 4169 y Fv(,)1096 4507 y Fu(N)9 b Fv(\001)1241 4519 y Fo(n)1309 4507 y Fv(=)23 b Fr(f)p Fv(\()p Fu(x;)14 b(\030)1591 4519 y Fy(1)1629 4507 y Fu(;)g(:)g(:)g(:)27 b(;)14 b(\030)1863 4519 y Fo(n)1908 4507 y Fv(\))24 b Fr(2)f Fu(T)2091 4519 y Fy(\001)2146 4527 y Fh(n)2190 4507 y Fu(M)2280 4473 y Fo(n)2325 4507 y Fu(;)2362 4429 y Fj(X)2496 4507 y Fu(\030)2532 4519 y Fo(i)2583 4507 y Fv(=)f(0)p Fr(g)27 b Fu(:)493 b Fv(\(57\))456 4846 y(Then)27 b Fu(\031)719 4858 y Fy(2)775 4846 y Fr(\016)18 b Fu(\013)l Fl(\026)919 4858 y Fo(N)6 b Fy(\001)1033 4866 y Fh(n)1105 4846 y Fv(describ)r(es)27 b(the)h(desired)f(\014bration,)1118 5185 y Fu(\031)1165 5197 y Fy(2)1221 5185 y Fr(\016)18 b Fu(\013)p Fv(\()p Fu(x;)c(\030)1486 5197 y Fy(1)1525 5185 y Fu(;)g(:)g(:)g(:)27 b(;)14 b(\030)1759 5197 y Fo(n)1805 5185 y Fv(\))23 b(=)g(\(exp)2107 5205 y Fo(x)2163 5185 y Fu(\030)2199 5197 y Fy(1)2236 5185 y Fu(;)14 b(:)g(:)g(:)28 b(;)14 b Fv(exp)2561 5205 y Fo(x)2617 5185 y Fu(\030)2653 5197 y Fo(n)2699 5185 y Fv(\))27 b Fu(:)516 b Fv(\(58\))p eop %%Page: 67 67 67 66 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(67)456 425 y Fu(x)19 b Fv(ma)n(y)g(b)r(e)g(considered)f(as)h(the)g(cen)n(ter)g(of)g(mass)g (of)g(the)g(p)r(oin)n(ts)g Fu(x)2453 437 y Fo(i)2504 425 y Fv(=)k(exp)2719 445 y Fo(x)2774 425 y Fu(\030)2810 437 y Fo(i)2838 425 y Fv(,)e(and)e(the)h(tangen)n(t)456 624 y(v)n(ectors)30 b Fu(\030)778 636 y Fo(i)839 624 y Fv(with)j(the)g(constrain)n(t)1576 562 y Fj(P)1678 624 y Fu(\030)1714 636 y Fo(i)1773 624 y Fv(=)d(0)i(pla)n(y)g(the)h(r^) -42 b(ole)31 b(of)i(relativ)n(e)e(co)r(ordinates.)50 b(W)-7 b(e)456 824 y(further)23 b(c)n(ho)r(ose)f(a)h(smo)r(oth)h (function)g Fk(w)f Fv(on)h Fu(T)12 b(M)31 b Fv(whic)n(h)23 b(is)h(equal)f(to)g(1)g(on)g(a)h(neigh)n(b)r(ourho)r(o)r(d)456 1023 y(of)j(the)h(zero)f(section)g(and)g(has)g(compact)g(supp)r(ort)h (on)f(eac)n(h)g(\014b)r(er)h(and)f(use)g(the)h(function)1365 1362 y Fk(w)1429 1374 y Fo(n)1474 1362 y Fv(\()p Fu(x;)14 b(\030)1626 1374 y Fy(1)1665 1362 y Fu(;)g(:)g(:)g(:)27 b(;)14 b(\030)1899 1374 y Fo(n)1944 1362 y Fv(\))24 b(=)2087 1283 y Fj(Y)2207 1362 y Fk(w)p Fv(\()p Fu(x;)14 b(\030)2423 1374 y Fo(i)2452 1362 y Fv(\))28 b Fu(;)762 b Fv(\(59\))456 1700 y(in)27 b(the)h(extension)g(to)f(the)h(diagonal)e(\001)1708 1712 y Fo(n)1754 1700 y Fv(.)555 1900 y(By)k(these)f(con)n(v)n(en)n (tions)f(w)n(e)i(get)f(a)h(reference)e(de\014nition)i(of)g (time-ordered)f(pro)r(ducts)g(for)456 2099 y(all)24 b(Wic)n(k)g(pro)r (ducts)g(whic)n(h)g(in)n(v)n(olv)n(es)f(only)h(the)g(c)n(hosen)g (Hadamard)f(state)h(and)g(the)h(function)456 2298 y Fk(w)p Fv(.)35 b(The)24 b(algebra)e(of)h(in)n(teracting)g(\014elds)g(can)h (then)g(b)r(e)g(de\014ned)g(b)n(y)f(c)n(ho)r(osing)f(the)i(co)r (e\016cien)n(ts)456 2497 y(in)j(the)h(Lagrangian.)503 2804 y(8.)45 b Fw(On)32 b(the)f(Definition)h(of)g(the)f(Net)h(of)f (Local)g(Algebras)g(of)h(Obser)-7 b(v)g(ables)555 3053 y Fv(In)31 b(the)g(preceding)f(Section)h(w)n(e)f(\014nished)h(the)h (construction)e(of)g(time-ordered)g(pro)r(ducts)456 3252 y(of)36 b(Wic)n(k)h(p)r(olynomials)e(of)i(the)g(free)f(\014elds.)64 b(W)-7 b(e)36 b(no)n(w)g(w)n(an)n(t)g(to)h(sho)n(w)e(that)i(this)g (already)456 3451 y(giv)n(es)26 b(the)h(full)h(net)g(of)f(lo)r(cal)g (algebras)f(of)h(observ)-5 b(ables)26 b(\(within)i(p)r(erturbation)f (theory\).)36 b(An)456 3651 y(\\adiabatic)18 b(limit,")k(whatev)n(er)c (this)j(migh)n(t)e(mean)h(on)g(a)f(curv)n(ed)g(space-time,)i(is)f(not)f (required.)456 3850 y(Actually)-7 b(,)23 b(these)e(observ)-5 b(ations)20 b(are)g(not)i(completely)f(new.)35 b(In)22 b([52)o(])g(it)g(w)n(as)e(already)g(observ)n(ed)456 4049 y(that)34 b(the)g(lo)r(cal)f Fu(S)5 b Fv(-matrices)32 b(of)i(the)g(St)r(\177)-44 b(uc)n(k)n(elb)r(erg-Bogoliub)r(o)n (v-Epstein-Glaser)28 b(approac)n(h)456 4248 y(giv)n(e)e(a)h(lo)r(cal)g (net)h(of)g(observ)-5 b(ables.)555 4448 y(Let)33 b Fm(W)g Fv(b)r(e)h(the)f(set)g(of)f(Wic)n(k)h(p)r(olynomials)f(with)h(co)r (e\016cien)n(ts)g(from)f Fm(D)p Fv(\()p Fu(M)9 b Fv(\).)53 b(So)33 b(ev)n(ery)456 4647 y Fu(A)23 b Fr(2)g Fm(W)k Fv(is)f(an)g(op)r(erator)e(v)-5 b(alued)26 b(distribution)g(with)h (compact)f(supp)r(ort)g(whic)n(h)g(is)g(relativ)n(ely)456 4846 y(lo)r(cal)h(to)g(the)h(free)f(\014eld.)38 b(Our)26 b(starting)h(relation)g(is)g(the)h(causal)f(factorization)f(\(2\))1113 5185 y Fu(S)5 b Fv(\()p Fu(A)18 b Fv(+)g Fu(B)23 b Fv(+)18 b Fu(C)6 b Fv(\))23 b(=)g Fu(S)5 b Fv(\()p Fu(A)19 b Fv(+)f Fu(B)t Fv(\))p Fu(S)5 b Fv(\()p Fu(B)t Fv(\))2279 5151 y Fq(\000)p Fy(1)2369 5185 y Fu(S)g Fv(\()p Fu(B)22 b Fv(+)c Fu(C)6 b Fv(\))29 b Fu(;)523 b Fv(\(60\))p eop %%Page: 68 68 68 67 bop 456 226 a Fy(68)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)456 425 y Fv(for)e Fu(A;)14 b(B)t(;)g(C)29 b Fr(2)24 b Fm(W)p Fv(,)j(whenev)n(er)g(supp\()p Fu(A)p Fv(\))i(is)e(later)g(than)h(supp\()p Fu(C)6 b Fv(\).)555 624 y(No)n(w)24 b(let)h Fm(L)g Fv(b)r(e)g(our)f(in)n (teraction)g(Lagrangian.)33 b(Then)25 b Fu(g)s Fm(L)e Fr(2)g Fm(W)i Fv(for)f Fu(g)i Fr(2)d Fm(D)p Fv(\()p Fu(M)9 b Fv(\).)36 b(W)-7 b(e)25 b(ma)n(y)456 824 y(de\014ne)i(observ)-5 b(ables)26 b(with)j(resp)r(ect)e(to)g(the)h(in)n(teraction)f Fu(g)s Fm(L)h Fv(b)n(y)f(Bogoliub)r(o)n(v)f(form)n(ula)1387 1162 y Fu(S)1438 1174 y Fo(g)r Fd(L)1527 1162 y Fv(\()p Fu(A)p Fv(\))e(=)f Fu(S)5 b Fv(\()p Fu(g)s Fm(L)p Fv(\))1987 1128 y Fq(\000)p Fy(1)2076 1162 y Fu(S)g Fv(\()p Fu(g)s Fm(L)19 b Fv(+)f Fu(A)p Fv(\))28 b Fu(:)784 b Fv(\(61\))456 1501 y(W)-7 b(e)30 b(no)n(w)e(sho)n(w)h(that)h(the)g(in)n(teracting)e (observ)-5 b(ables)28 b(dep)r(end)j(only)e(lo)r(cally)g(on)g(the)h(in)n (terac-)456 1700 y(tion.)37 b(More)26 b(precisely)-7 b(,)27 b(w)n(e)g(ha)n(v)n(e)g(the)h(follo)n(wing)456 2585 y Fs(Prop)s(osition)i(8.1.)40 b Ft(L)l(et)35 b Fm(O)g Ft(b)l(e)h(a)g(c)l(ausal)t(ly)h(close)l(d)g(r)l(e)l(gion,)h(and)e(let)g (the)f(test)g(functions)h Fu(g)456 2785 y Ft(and)d Fu(g)663 2755 y Fq(0)719 2785 y Ft(c)l(oincide)h(on)f(some)h(neighb)l(ourho)l(o) l(d)h(of)e Fm(O)p Ft(.)48 b(Then)34 b(ther)l(e)f(exists)f(a)i(unitary)f Fu(V)51 b Ft(such)456 2984 y(that)29 b(for)i(al)t(l)g Fu(A)23 b Fr(2)g Fm(W)30 b Ft(with)h Fv(supp\()p Fu(A)p Fv(\))24 b Fr(\032)e Fm(O)1479 3323 y Fu(V)d(S)1597 3335 y Fo(g)r Fd(L)1686 3323 y Fv(\()p Fu(A)p Fv(\))p Fu(V)1879 3288 y Fq(\000)p Fy(1)1992 3323 y Fv(=)j Fu(S)2130 3335 y Fo(g)2164 3319 y Ff(0)2187 3335 y Fd(L)2242 3323 y Fv(\()p Fu(A)p Fv(\))30 b Fu(:)876 b Fv(\(62\))456 3661 y Ft(Pr)l(o)l(of.)43 b Fv(W)-7 b(e)28 b(ma)n(y)f(split)i Fu(g)1268 3631 y Fq(0)1309 3661 y Fr(\000)19 b Fu(g)26 b Fv(=)d Fu(a)c Fv(+)f Fu(b)28 b Fv(where)g Fu(a)g Fv(do)r(es)f(not)h (in)n(tersect)g(the)g(past)g(of)g Fm(O)g Fv(and)g Fu(b)456 3861 y Fv(not)f(the)h(future.)37 b(Then)28 b(supp\()p Fu(a)p Fm(L)p Fv(\))h(is)e(later)g(than)h(supp\()p Fu(A)p Fv(\),)h(hence)e(from)h(\(60\))f(w)n(e)g(\014nd)985 4199 y Fu(S)5 b Fv(\()p Fu(g)1116 4165 y Fq(0)1138 4199 y Fm(L)19 b Fv(+)f Fu(A)p Fv(\))24 b(=)f Fu(S)5 b Fv(\()p Fu(g)1636 4165 y Fq(0)1658 4199 y Fm(L)p Fv(\))p Fu(S)g Fv(\(\()p Fu(g)22 b Fv(+)c Fu(b)p Fv(\))p Fm(L)p Fv(\))2173 4165 y Fq(\000)p Fy(1)2263 4199 y Fu(S)5 b Fv(\(\()p Fu(g)21 b Fv(+)d Fu(b)p Fv(\))p Fm(L)h Fv(+)f Fu(A)p Fv(\))28 b Fu(;)396 b Fv(\(63\))456 4538 y(th)n(us)27 b Fu(S)689 4550 y Fo(g)723 4534 y Ff(0)746 4550 y Fd(L)801 4538 y Fv(\()p Fu(A)p Fv(\))d(=)e Fu(S)1089 4553 y Fy(\()p Fo(g)r Fy(+)p Fo(b)p Fy(\))p Fd(L)1310 4538 y Fv(\()p Fu(A)p Fv(\).)38 b(Moreo)n(v)n(er,)25 b(supp\()p Fu(A)p Fv(\))k(is)e(later)g(than)h(supp\()p Fu(b)p Fm(L)p Fv(\),)g(hence)1015 4877 y Fu(S)5 b Fv(\(\()p Fu(g)21 b Fv(+)d Fu(b)p Fv(\))p Fm(L)h Fv(+)f Fu(A)p Fv(\))24 b(=)e Fu(S)5 b Fv(\()p Fu(g)s Fm(L)19 b Fv(+)f Fu(A)p Fv(\))p Fu(S)5 b Fv(\()p Fu(g)s Fm(L)p Fv(\))2321 4843 y Fq(\000)p Fy(1)2411 4877 y Fu(S)g Fv(\(\()p Fu(g)21 b Fv(+)d Fu(b)p Fv(\))p Fm(L)p Fv(\))28 b Fu(:)412 b Fv(\(64\))456 5216 y(Hence)27 b(w)n(e)h(obtain)f (\(62\))g(with)h Fu(V)42 b Fv(=)23 b Fu(S)1676 5228 y Fo(g)r Fd(L)1765 5216 y Fv(\()p Fu(b)p Fm(L)p Fv(\))1924 5185 y Fq(\000)p Fy(1)2014 5216 y Fv(.)1343 b Fl(\004)p eop %%Page: 69 69 69 68 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(69)555 425 y Fv(W)-7 b(e)26 b(conclude)f(that)g(the)h(algebra)d Fk(A)1701 437 y Fo(g)r Fd(L)1790 425 y Fv(\()p Fm(O)p Fv(\))i(whic)n(h)g(is)g (generated)f(b)n(y)h Fu(S)2794 437 y Fo(g)r Fd(L)2883 425 y Fv(\()p Fu(A)p Fv(\),)i(supp\()p Fu(A)p Fv(\))d Fr(\032)456 624 y Fm(O)j Fv(with)i Fu(g)d Fr(\021)e Fv(1)j(in)i(a)e (neigh)n(b)r(ourho)r(o)r(d)g(of)h Fm(O)p Fv(,)g(is)g(up)g(to)g(unitary) g(equiv)-5 b(alence)27 b(uniquely)h(\014xed)456 824 y(b)n(y)f Fm(L)p Fv(.)555 1023 y(W)-7 b(e)25 b(ma)n(y)e(formalize)g(the)i (construction)e(in)h(the)h(follo)n(wing)e(w)n(a)n(y)-7 b(.)34 b(Let)24 b(\002\()p Fm(O)p Fv(\))g(for)g(a)g(causally)456 1222 y(closed)i(compact)h(region)f Fm(O)h Fv(b)r(e)h(the)g(set)f(of)h (test)f(functions)h(whic)n(h)f(equal)g(unit)n(y)h(on)f(a)g(neigh-)456 1421 y(b)r(ourho)r(o)r(d)35 b(of)h Fm(O)p Fv(.)63 b(Consider)35 b(for)g Fu(g)s(;)14 b(g)1701 1391 y Fq(0)1761 1421 y Fr(2)38 b Fv(\002\()p Fm(O)p Fv(\))e(the)g(set)h Fm(V)2423 1433 y Fo(g)r(g)2491 1417 y Ff(0)2518 1421 y Fv(\()p Fm(O)p Fv(\))g(of)f(unitaries)f Fu(V)56 b Fv(whic)n(h)456 1621 y(satisfy)27 b(the)h(in)n(tert)n(wining)f(relation)1244 1959 y Fu(V)19 b(S)1362 1971 y Fo(g)r Fd(L)1451 1959 y Fv(\()p Fu(A)p Fv(\))24 b(=)f Fu(S)1740 1971 y Fo(g)1774 1955 y Ff(0)1797 1971 y Fd(L)1851 1959 y Fv(\()p Fu(A)p Fv(\))p Fu(V)48 b(;)41 b Fv(supp)q(\()p Fu(A)p Fv(\))23 b Fr(\032)g Fm(O)k Fu(:)642 b Fv(\(65\))456 2298 y(The)26 b(algebra)e(of)i(observ)-5 b(ables)24 b Fk(A)1502 2310 y Fd(L)1556 2298 y Fv(\()p Fm(O)p Fv(\))j(can)e(no)n(w)h(b)r(e)g (de\014ned)h(as)e(the)h(algebra)f(of)h(co)n(v)-5 b(arian)n(tly)456 2497 y(constan)n(t)26 b(sections)h(of)h(the)g(bundle)1629 2757 y Fj([)1559 2939 y Fo(g)r Fq(2)p Fy(\002\()p Fd(O)p Fy(\))1791 2836 y Fr(f)p Fu(g)s Fr(g)17 b(\002)h Fk(A)2078 2848 y Fo(g)r Fd(L)2167 2836 y Fv(\()p Fm(O)p Fv(\))28 b Fu(:)956 b Fv(\(66\))456 3175 y(Here,)27 b(a)g(section)g Fu(A)d Fv(=)e(\()p Fu(A)1288 3187 y Fo(g)1327 3175 y Fv(\))1359 3190 y Fo(g)r Fq(2)p Fy(\002\()p Fd(O)p Fy(\))1624 3175 y Fv(is)27 b(called)g(co)n(v)-5 b(arian)n(tly)26 b(constan)n(t)h(if)1331 3514 y Fu(V)19 b(A)1460 3526 y Fo(g)1522 3514 y Fv(=)j Fu(A)1671 3526 y Fo(g)1705 3509 y Ff(0)1733 3514 y Fu(V)46 b(;)180 b Fr(8)p Fu(V)42 b Fr(2)23 b Fm(V)2299 3526 y Fo(g)r(g)2367 3509 y Ff(0)2395 3514 y Fv(\()p Fm(O)p Fv(\))28 b Fu(:)728 b Fv(\(67\))456 3852 y Fk(A)516 3864 y Fd(L)570 3852 y Fv(\()p Fm(O)p Fv(\))31 b(con)n(tains)g(for)f(example)h(the)g(elemen)n(ts)g Fu(S)2053 3864 y Fd(L)2108 3852 y Fv(\()p Fu(A)p Fv(\))f(=)e(\()p Fu(S)2440 3864 y Fo(g)r Fd(L)2530 3852 y Fv(\()p Fu(A)p Fv(\)\))2688 3867 y Fo(g)r Fq(2)p Fy(\002\()p Fd(O)p Fy(\))2925 3852 y Fv(.)48 b(T)-7 b(o)31 b(complete)456 4052 y(the)22 b(construction)g(of)g(the)g(net)h(of)f(lo)r(cal)g (algebras)e(of)i(observ)-5 b(ables)21 b(w)n(e)g(ha)n(v)n(e)g(to)i (\014x)f(the)g(im)n(b)r(ed-)456 4251 y(dings)28 b Fu(i)703 4263 y Fd(O)753 4271 y FC(2)785 4263 y Fd(O)835 4271 y FC(1)896 4251 y Fv(:)d Fk(A)1004 4263 y Fd(L)1059 4251 y Fv(\()p Fm(O)1150 4263 y Fy(1)1187 4251 y Fv(\))g Fr(!)g Fk(A)1412 4263 y Fd(L)1466 4251 y Fv(\()p Fm(O)1557 4263 y Fy(2)1594 4251 y Fv(\))30 b(for)e Fm(O)1843 4263 y Fy(1)1905 4251 y Fr(\032)c Fm(O)2053 4263 y Fy(2)2090 4251 y Fv(.)40 b(But)29 b(this)g(structure)f(is)h(inherited)g(from)456 4450 y(the)22 b(\014b)r(ers)f(and)h(ma)n(y)f(b)r(e)h(de\014ned)g(b)n(y) f(the)i(restriction)d(of)i(the)g(section)f(from)h(\002\()p Fm(O)3010 4462 y Fy(1)3047 4450 y Fv(\))g(to)f(\002\()p Fm(O)3352 4462 y Fy(2)3389 4450 y Fv(\).)p eop %%Page: 70 70 70 69 bop 456 226 a Fy(70)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)1415 425 y Fv(9.)46 b Fw(Summar)-6 b(y)31 b(and)g(Outlook)555 674 y Fv(W)-7 b(e)39 b(ha)n(v)n(e)d(pro)n(v)n(en)h(that)h(renormalization)d(on)j (curv)n(ed)f(bac)n(kgrounds)f(can)h(b)r(e)i(done)e(in)456 874 y(close)26 b(analogy)f(to)i(renormalization)e(on)i(Mink)n(o)n(wski) f(space,)h(and)g(that)g(the)h(remo)n(v)-5 b(al)26 b(of)h(sin-)456 1073 y(gularities)k(follo)n(ws)h(the)i(w)n(ell)f(kno)n(wn)f(p)r(o)n(w)n (er)g(coun)n(ting)g(rules.)53 b(This)32 b(result)h(w)n(as)f(exp)r (ected)456 1272 y(since)26 b(the)h(ultra)n(violet)e(b)r(eha)n(viour)h (on)g(smo)r(oth)g(manifolds)h(should)f(b)r(e)h(essen)n(tially)f(iden)n (tical)456 1471 y(to)k(that)h(on)f(Mink)n(o)n(wski)f(space.)45 b(But)31 b(w)n(e)f(had)g(to)h(o)n(v)n(ercome)d(t)n(w)n(o)i(ma)5 b(jor)29 b(obstacles:)42 b(On)30 b(a)456 1671 y(generic)d(space-time)h (there)g(is)g(no)g(reason)f(to)h(exp)r(ect)g(a)g(decen)n(t)h(infrared)e (b)r(eha)n(viour,)g(hence)456 1870 y(w)n(e)32 b(had)h(to)g(use)f(a)h (metho)r(d)g(whic)n(h)g(decouples)g(completely)f(the)i(short)e (distance)h(from)f(the)456 2069 y(long)37 b(distance)g(problem;)43 b(the)38 b(other)f(source)g(of)h(problem)f(w)n(as)g(the)h(absence)f(of) h(transla-)456 2268 y(tion)32 b(in)n(v)-5 b(ariance)31 b(whic)n(h,)j(on)e(the)h(tec)n(hnical)g(side)f(mak)n(es)f(obsolete)h (the)h(usual)f(momen)n(tum)456 2468 y(space)g(metho)r(ds,)j(and)e(on)g (the)g(side)h(of)f(ph)n(ysics,)h(forbids)e(to)h(base)g(the)h (construction)e(on)h(a)456 2667 y(distinguished)27 b(v)-5 b(acuum)28 b(state)f(and)h(on)f(the)h(notion)f(of)h(particles.)555 2866 y(W)-7 b(e)37 b(solv)n(ed)e(these)h(problems)f(b)n(y)h(basing)f (the)h(construction)f(on)h(the)g(lo)r(cal)g Fu(S)5 b Fv(-matrices)456 3065 y(of)35 b(St)r(\177)-44 b(uc)n(k)n(elb)r(erg)35 b(and)h(Bogoliub)r(o)n(v,)g(b)n(y)f(in)n(v)n(oking)g(the)h(general)e (ideas)h(of)h(algebraic)e(quan-)456 3265 y(tum)26 b(\014eld)h(theory)e (and)h(b)n(y)g(replacing)f(translation)f(in)n(v)-5 b(ariance)25 b(b)n(y)h(smo)r(othness)f(prop)r(erties,)456 3464 y(making)h(extensiv)n (e)h(use)h(of)f(tec)n(hniques)h(and)f(concepts)g(from)h(microlo)r(cal)e (analysis.)555 3663 y(Besides)i(the)h(solution)f(of)h(the)g(problem)f (to)g(whic)n(h)g(this)h(pap)r(er)f(is)h(addressed,)e(w)n(e)i(solv)n(ed) 456 3862 y(sev)n(eral)39 b(other)h(problems)g(whic)n(h)g(migh)n(t)h(b)r (e)g(of)g(indep)r(enden)n(t)h(in)n(terest.)76 b(First)40 b(w)n(e)h(ha)n(v)n(e)456 4062 y(found)34 b(a)g(new)g(construction)f(of) h(Wic)n(k)g(p)r(olynomials)f(on)g(a)h(domain)g(whic)n(h)f(dep)r(ends)i (only)456 4261 y(on)e(the)g(represen)n(tation)f(asso)r(ciated)g(to)i(a) f(\014xed)g(Hadamard)f(state)h(but)h(not)g(on)f(the)h(state)456 4460 y(itself.)75 b(Since)40 b(according)f(to)h(V)-7 b(erc)n(h)39 b([57)o(],)44 b(all)c(represen)n(tations)e(induced)j(b)n (y)f(Hadamard)456 4659 y(states)22 b(are)g(lo)r(cally)g(quasiequiv)-5 b(alen)n(t)22 b(this)h(amoun)n(ts)f(to)h(an)g(algebraic)e(construction) h(of)h(Wic)n(k)456 4859 y(p)r(olynomials.)p eop %%Page: 71 71 71 70 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(71)555 425 y Fv(Second,)31 b(w)n(e)f(ga)n(v)n(e)f(a)g(p)r(erturbativ)n(e)h(construction)f(of)i (algebraic)d(quan)n(tum)i(\014eld)h(theory)-7 b(.)456 624 y(In)22 b(particular,)h(w)n(e)f(pro)n(v)n(ed)f(that)i(the)g(theory) f(\(in)h(the)g(algebraic)e(sense\),)i(is)g(completely)f(\014xed)456 824 y(if)j(it)g(is)f(kno)n(wn)g(lo)r(cally)-7 b(.)35 b(Actually)25 b(this)g(holds)f(indep)r(enden)n(tly)h(of)g(p)r (erturbation)f(theory)g(and)456 1023 y(migh)n(t)19 b(b)r(e)g(a)g(hin)n (t)h(for)e(a)h(construction)f(\(in)i(the)g(sense)e(of)h(constructiv)n (e)f(quan)n(tum)i(\014eld)f(theory\))456 1222 y(of)h(asymptotically)g (free)g(theories.)33 b(There)20 b(the)h(construction)f(in)h(small)f(v)n (olumes)g(seems)g(to)g(b)r(e)456 1421 y(p)r(ossible)g([46)o(])h(but)g (the)g(infrared)f(problem)g(p)r(oses,)i(at)e(presen)n(t,)i(unsormoun)n (table)d(di\016culties.)456 1621 y(The)28 b(message)f(of)h(this)h(pap)r (er)f(is)g(that)h(the)g(construction)f(of)g(the)h(algebra)d(of)j (observ)-5 b(ables)27 b(is)456 1820 y(nev)n(ertheless)k(p)r(ossible)h (once)g(the)h(algebra)e(of)i(observ)-5 b(ables)31 b(for)h(small)g (space-time)g(regions)456 2019 y(ha)n(v)n(e)f(b)r(een)j(constructed.)52 b(The)33 b(long)f(distance)g(b)r(eha)n(viour)g(of)h(suc)n(h)f(a)h (theory)f(w)n(ould)g(still)456 2218 y(require)25 b(an)h(extra)f(in)n(v) n(estigation,)h(but)g(it)h(w)n(ould)f(b)r(e)h(the)g(b)r(eha)n(viour)e (of)h(an)g(existing)g(theory)-7 b(,)456 2418 y(quite)22 b(similar)g(to)h(the)g(computation)f(of)h(sp)r(ectra)f(of)g (Hamiltonians)g(whic)n(h)h(ha)n(v)n(e)e(b)r(een)i(sho)n(wn)456 2617 y(to)k(b)r(e)h(self-adjoin)n(t)f(op)r(erators.)555 2816 y(On)33 b(the)g(tec)n(hnical)f(side)h(w)n(e)f(had)h(to)f(study)h (the)g(extension)f(problem)g(for)h(distributions)456 3015 y(whic)n(h)41 b(are)f(de\014ned)h(on)g(the)g(complemen)n(t)g(of)h (some)e(submanifold.)77 b(This)41 b(seems)g(to)g(b)r(e)456 3215 y(a)35 b(natural)h(mathematical)g(question,)i(and)e(in)g(a)g (simple)g(case)f(it)i(w)n(as)e(treated)h(in)g([23)o(])h(b)n(y)456 3414 y(similar)27 b(metho)r(ds.)40 b(What)29 b(seems)f(to)g(b)r(e)h (new)f(is)h(our)e(concept)i(of)f(the)h(\(microlo)r(cal\))e(scaling)456 3613 y(degree)34 b(at)i(a)f(surface)g(whic)n(h)h(com)n(bines)f(the)h (condition)g(of)f(smo)r(othness)g(along)g(a)g(surface)456 3812 y(with)28 b(a)f(classi\014cation)f(of)i(the)g(singularit)n(y)e(in) i(a)f(transv)n(ersal)e(direction.)555 4012 y(The)34 b(main)f(op)r(en)g (p)r(oin)n(t)g(in)h(this)g(pap)r(er)e(is)i(the)f(\014xing)g(of)g(the)h (\014nite)g(renormalizations.)456 4211 y(One)27 b(exp)r(ects)h(that)f (they)h(can)f(b)r(e)h(c)n(hosen)f(in)h(terms)f(of)h(lo)r(cal)f (functions)h(of)f(the)h(metric,)g(but)456 4410 y(a)e(precise)f(form)n (ulation)g(meets)i(a)f(lot)g(of)g(problems.)36 b(A)27 b(similar)e(problem)h(w)n(as)g(studied)g(\(and)456 4610 y(partially)e(solv)n(ed\))g(in)h(the)g(de\014nition)h(of)f(the)g(exp)r (ectation)g(v)-5 b(alue)25 b(of)g(a)f(renormalized)g(energy)456 4809 y(momen)n(tum)g(tensor)f(of)g(free)h(\014elds)g(b)n(y)f(R.)h(W)-7 b(ald)24 b([61)o(].)36 b(W)-7 b(e)24 b(hop)r(e)g(to)g(return)f(to)h (this)g(problem)456 5008 y(in)j(a)h(future)g(publication)f([10)o(].)p eop %%Page: 72 72 72 71 bop 456 226 a Fy(72)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)1538 425 y Fw(A)n(ckno)n(wledgements) 555 674 y Fv(W)-7 b(e)32 b(are)f(particularly)f(grateful)h(to)h(Ra)n (ymond)f(Stora)f(who)i(long)f(ago)f(suggested)g(us)i(the)456 874 y(relev)-5 b(ance)25 b(of)h(the)h(Epstein)f(and)g(Glaser)g(pro)r (cedure)f(for)h(renormalization)e(on)i(curv)n(ed)g(bac)n(k-)456 1073 y(grounds.)56 b(In)35 b(an)f(early)f(stage)h(of)g(this)h(w)n(ork,) g(some)e(results,)j(in)f(particular)e(on)h(the)h(w)n(a)n(v)n(e)456 1272 y(fron)n(t)24 b(sets)g(of)g(time-ordered)f(functions,)j(ha)n(v)n (e)d(b)r(een)i(obtained)f(in)g(collab)r(oration)f(with)i(Mar-)456 1471 y(tin)i(K\177)-42 b(ohler)26 b(whic)n(h)h(is)f(gratefully)g(ac)n (kno)n(wledged.)35 b(The)27 b(\014rst)g(named)f(author)h(w)n(as)e (partially)456 1671 y(supp)r(orted)k(b)n(y)g(a)h(gran)n(t)e(of)i(T)-7 b(raining)28 b(and)i(Mobilit)n(y)f(of)h(Researc)n(hers)d(\(TMR\))j (programme)456 1870 y(of)d(Europ)r(ean)f(Comm)n(unit)n(y)-7 b(.)1708 2343 y Fw(References)526 2559 y FB([1])35 b(Ash)n(tek)l(ar,)29 b(A.:)41 b(Mathematical)30 b(problems)e(of)h(non-p)r(erturbativ)n(e)h (quan)n(tum)g(general)f(relativit)n(y)-6 b(.)29 b(In:)636 2725 y(Zinn-Justin)34 b(et)g(al.)f(\(eds.\))h(Les)g(Houc)n(hes)g (Summer)g(Sc)n(ho)r(ol)g(on)g(Gra)n(vitation)h(and)f(Quan)n(tization.) 636 2891 y(North-Holland,)23 b(1994)526 3057 y([2])35 b(Beem,)d(J.)f(K.,)h(Ehlric)n(h,)f(P)-6 b(.)31 b(E.)g(and)i(Easley)-6 b(,)31 b(K.)g(L.:)47 b(Global)32 b(Loren)n(tzian)g(Geometry)-6 b(.)33 b(New)f(Y)-6 b(ork:)636 3223 y(Marcel)23 b(Dekk)n(er)h(Inc.,)g (1996)526 3389 y([3])35 b(Birrel,)24 b(N.)i(D.,)g(and)h(Da)n(vies,)f(P) -6 b(.)26 b(C.)g(W.:)37 b(Quan)n(tum)27 b(Fields)f(in)g(Curv)n(ed)h (Space.)g(Cam)n(bridge:)37 b(Cam-)636 3555 y(bridge)24 b(Univ)n(ersit)n(y)f(Press,)g(1982)526 3721 y([4])35 b(Blanc)n(hard,)c(P)-6 b(.,)32 b(and)g(S)n(\023)-33 b(en)n(\023)g(eor,) 31 b(R.:)46 b(Green's)31 b(functions)h(for)e(theories)h(with)h (massless)e(particles)h(\(in)636 3887 y(p)r(erturbation)25 b(theory\).)f(Ann.)g(Inst.)g(Henry)f(P)n(oincar)n(\023)-33 b(e)25 b Fb(23)p FB(,)d(147)i(\(1975\))526 4053 y([5])35 b(Bogoliub)r(o)n(v,)22 b(N.)f(N.,)f(and)i(Shirk)n(o)n(v,)g(D.)f(V.:)29 b(In)n(tro)r(duction)23 b(to)f(the)h(Theory)f(of)f(Quan)n(tized)i (Fiels.)d(New)636 4219 y(Y)-6 b(ork:)31 b(John)24 b(Wiley)g(and)g (Sons,)g(1976,)g(3rd)g(edition)526 4385 y([6])35 b(Bourbaki,)23 b(N.:)31 b(Alg)n(\022)-33 b(ebre,)23 b(Chap.VI)r(I)r(I.)i(P)n(aris:)30 b(Hermann,)23 b(1970)526 4551 y([7])35 b(Bros,)d(J.,)j(Epstein,)f(H.,)g (and)g(Mosc)n(hella,)f(U.:)50 b(Analyticit)n(y)34 b(prop)r(erties)f (and)h(thermal)f(e\013ects)i(for)636 4717 y(general)g(quan)n(tum)i (\014eld)e(theory)h(on)f(de)g(Sitter)h(space-time.)f(Comm)n(un.)g (Math.)g(Ph)n(ys.)f Fb(186)p FB(,)j(535)636 4883 y(\(1998\))526 5050 y([8])e(Brunetti,)22 b(R.,)f(F)-6 b(redenhagen,)23 b(K.,)e(and)i(K\177)-35 b(ohler,)20 b(M.:)30 b(The)22 b(microlo)r(cal)f(sp)r(ectrum)h(condition)h(and)f(the)636 5216 y(Wic)n(k's)i(p)r(olynomials)f(of)g(free)h(\014elds.)f(Comm)n(un.) h(Math.)f(Ph)n(ys.)h Fb(180)p FB(,)e(633)i(\(1996\))p eop %%Page: 73 73 73 72 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(73)526 425 y FB([9])35 b(Brunetti,)16 b(R.,)h(and)g(F)-6 b(redenhagen,)18 b(K.:)26 b(Microlo)r(cal)16 b(analysis)g(and)g(in)n(teracting)i(quan)n(tum)f (\014eld)g(theory:)636 591 y(Renormalizabilit)n(y)28 b(of)g Fa(')1341 568 y Fp(4)1375 591 y FB(.)g(In:)41 b(Doplic)n(her,)27 b(S.,)i(Longo,)f(R.,)h(Rob)r(erts,)f(J.)g(E.,)g(and) h(Zsido,)e(L)i(\(eds.\))636 757 y(Op)r(erator)17 b(Algebras)g(and)g (Quan)n(tum)h(Field)f(Theory)-6 b(.)17 b(Pro)r(ceedings,)h(Roma)f (1996,)i(In)n(ternational)g(Press)636 923 y(1997)491 1089 y([10])35 b(Brunetti,)24 b(R.,)e(and)j(F)-6 b(redenhagen,)25 b(K.:)30 b(w)n(ork)24 b(in)f(progress)491 1255 y([11])35 b(Brunetti,)28 b(R.,)f(and)h(F)-6 b(redenhagen,)29 b(K.:)38 b(On)28 b(the)g(connection)i(b)r(et)n(w)n(een)f(in)n(teracting)g(quan)n (tum)g(\014eld)636 1421 y(theory)c(and)f(microlo)r(cal)f(analysis.)g(F) -6 b(orthcoming)25 b(review)e(pap)r(er)491 1588 y([12])35 b(Buc)n(hholz,)24 b(D.:)31 b(Curren)n(t)23 b(trends)i(in)e(axiomatic)h (quan)n(tum)h(\014eld)f(theory)-6 b(.)25 b FA(hep-th/9811233)491 1754 y FB([13])35 b(Bunc)n(h,)d(T.)e(S.:)47 b(BPHZ)31 b(renormalization)g(of)g Fa(\025\036)2020 1730 y Fp(4)2086 1754 y FB(\014eld)h(theory)g(in)f(curv)n(ed)h(space-times.)f(Ann.)h(of) 636 1920 y(Ph)n(ys.)23 b Fb(131)p FB(,)g(118)h(\(1981\))491 2086 y([14])35 b(De)29 b(Witt,)g(B.)f(S.,)h(and)g(Brehme,)f(R.)h(W.:)41 b(Radiation)29 b(damping)g(in)f(a)h(gra)n(vitational)h(\014eld.)e(Ann.) h(of)636 2252 y(Ph)n(ys.)23 b Fb(9)p FB(,)g(220)i(\(1965\))491 2418 y([15])35 b(Dimo)r(c)n(k,)21 b(J.:)30 b(Scalar)21 b(quan)n(tum)i(\014eld)f(in)f(an)h(external)g(gra)n(vitational)g (\014eld.)g(J.)f(Math.)g(Ph)n(ys.)h Fb(20)p FB(,)e(2549)636 2584 y(\(1979\))491 2750 y([16])35 b(Dosc)n(h,)25 b(H.)f(G.,)h(and)h(M) r(\177)-37 b(uller,)23 b(V.)h(F.:)33 b(Renormalization)26 b(of)f(quan)n(tum)h(electro)r(dynamics)g(in)f(an)g(arbi-)636 2916 y(trary)e(strong)i(time)e(indep)r(enden)n(t)j(external)f(\014eld.) e(F)-6 b(ort.)24 b(der)f(Ph)n(ysik)h Fb(23)p FB(,)e(661)j(\(1975\))491 3082 y([17])35 b(Duistermaat,)24 b(J.)f(J.,)g(and)i(H\177)-35 b(ormander,)23 b(L.:)31 b(F)-6 b(ourier)23 b(in)n(tegral)h(op)r (erators)g(I)r(I.)g(Acta)h(Math.)e Fb(128)p FB(,)g(183)636 3248 y(\(1973\))491 3414 y([18])35 b(D)r(\177)-37 b(utsc)n(h,)33 b(M.,)g(and)g(F)-6 b(redenhagen,)34 b(K.:)48 b(A)32 b(lo)r(cal)h(\(p)r (erturbativ)n(e\))h(construction)g(of)e(observ)l(ables)i(in)636 3580 y(gauge)25 b(theories:)31 b(the)25 b(example)f(of)g(QED.)f(T)-6 b(o)24 b(app)r(ear)g(on)g(Comm)n(un.)f(Math.)h(Ph)n(ys.)f(\(1999\))491 3746 y([19])35 b(Dyson,)25 b(F.:)34 b(Collected)26 b(w)n(orks.)f (American)g(Mathematical)i(So)r(ciet)n(y)-6 b(.)26 b(Pro)n(vidence)g (RI:)g(In)n(ternational)636 3912 y(Press,)d(1996)491 4078 y([20])35 b(Epstein,)28 b(H.,)g(and)g(Glaser,)g(V.:)39 b(The)28 b(role)g(of)f(lo)r(calit)n(y)i(in)e(p)r(erturbation)i(theory) -6 b(.)29 b(Ann.)f(Inst.)g(Henri)636 4244 y(P)n(oincar)n(\023)-33 b(e-Section)25 b(A,)e(v)n(ol.)g(XIX,)h(n.3,)f(211)h(\(1973\))491 4410 y([21])35 b(Epstein,)g(H.,)h(and)f(Glaser,)f(V.:)52 b(Adiabatic)36 b(limit)d(in)h(p)r(erturbation)i(theory)-6 b(.)35 b(In:)54 b(V)-6 b(elo,)34 b(G.,)j(and)636 4576 y(Wigh)n(tman,)32 b(A.)e(S.)h(\(eds.\))h(Renormalization)f(Theory)-6 b(.)32 b(Pro)r(ceedings,)h(D.)d(Reidel)h(Publishing)g(Co.,)636 4742 y(Do)r(drec)n(h)n(t-Holland,)24 b(1976)491 4908 y([22])35 b(Epstein,)24 b(H.:)30 b(On)24 b(the)g(Borc)n(hers)g(class)f (of)h(a)g(free)f(\014eld.)g(Nuo)n(v)n(o)i(Cimen)n(to)f Fb(27)p FB(,)f(886)h(\(1966\))491 5074 y([23])35 b(Estrada,)21 b(R.:)29 b(Regularization)21 b(of)g(distributions.)f(In)n(ternat.)i(J.) e(Math.)h(&)g(Math.)f(Sci.)h Fb(21)p FB(,)f(625)h(\(1998\))p eop %%Page: 74 74 74 73 bop 456 226 a Fy(74)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)491 425 y FB([24])35 b(F)-6 b(redenhagen,)23 b(K.,)e(and)h(Haag,)f(R.:)30 b(Generally)22 b(co)n(v)l(arian)n(t)g(quan)n(tum)h(\014eld)f(theory)-6 b(.)22 b(Comm)n(un.)f(Math.)636 591 y(Ph)n(ys.)i Fb(108)p FB(,)g(91)h(\(1987\))491 757 y([25])35 b(F)-6 b(ulling,)28 b(S.:)42 b(Asp)r(ects)30 b(of)f(Quan)n(tum)h(Field)e(Theory)i(in)f (Curv)n(ed)g(Space-Time.)h(Cam)n(bridge:)42 b(Cam-)636 923 y(bridge)24 b(Univ)n(ersit)n(y)f(Press,)g(1989)491 1089 y([26])35 b(Glimm,)30 b(J.,)j(and)f(Ja\013e,)g(A.:)46 b(Quan)n(tum)32 b(Ph)n(ysics:)47 b(A)31 b(F)-6 b(unctional)33 b(In)n(tegral)f(P)n(oin)n(t)g(of)f(View.)g(New)636 1255 y(Y)-6 b(ork,)23 b(Berlin,)f(Heidelb)r(erg:)32 b(Springer-V)-6 b(erlag,)22 b(1981)491 1421 y([27])35 b(Guillemin,)18 b(V.,)h(and)h(P)n(ollac)n(k,)f(A.:)28 b(Di\013eren)n(tial)19 b(T)-6 b(op)r(ology)g(.)21 b(Englew)n(o)r(o)r(d-Cli\013s,)e(N.J.:)28 b(Pren)n(tice-Hall,)636 1588 y(Inc.,)c(1974)491 1754 y([28])35 b(Green,)19 b(D.)g(B.,)g(Sc)n(h)n(w)n(arz,)g(J.)g(H.,)g(and)h (Witten,)g(E.:)28 b(Sup)r(erstring)19 b(Theory)-6 b(.)20 b(V)-6 b(oll.)18 b(1)h(and)h(2.)f(Cam)n(bridge:)636 1920 y(Cam)n(bridge)24 b(Univ)n(ersit)n(y)f(Press,)g(1987)491 2086 y([29])35 b(Haag,)23 b(R.:)30 b(Lo)r(cal)24 b(Quan)n(tum)g(Ph)n (ysics:)31 b(Fields,)22 b(P)n(articles)h(and)h(Algebras.)e(Berlin:)30 b(Springer-V)-6 b(erlag,)636 2252 y(2nd)24 b(ed.,)f(1996)491 2418 y([30])35 b(Haag,)c(R.,)i(Narnhofer,)d(H.,)i(and)g(Stein,)f(U.:)46 b(On)31 b(quan)n(tum)i(\014eld)e(theory)i(in)d(gra)n(vitational)j(bac)n (k-)636 2584 y(ground.)24 b(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fb(94)p FB(,)g(219)h(\(1984\))491 2750 y([31])35 b(Halzen,)28 b(F.,)g(and)h(Martin,)e(A.)g(D.:)39 b(Quarks)28 b(and)h(Leptons:)41 b(An)28 b(In)n(tro)r(ductory)i(Course)e(in)f(Mo)r(dern)636 2916 y(P)n(article)d(Ph)n(ysics.)f(New)h(Y)-6 b(ork:)31 b(John)24 b(Wiley)g(and)g(Sons,)g(1984)491 3082 y([32])35 b(Ha)n(wking,)24 b(S.:)30 b(P)n(article)24 b(creation)h(b)n(y)f(blac)n (k)g(holes.)f(Comm)n(un.)h(Math.)f(Ph)n(ys.)h Fb(43)p FB(,)e(199)j(\(1975\))491 3248 y([33])35 b(Ha)n(wking,)24 b(S.:)30 b(The)24 b(c)n(hronology)h(protection)g(conjecture.)g(Ph)n (ys.)f(Rev.)f Fb(D)28 b(46)p FB(,)22 b(603)j(\(1992\))491 3414 y([34])35 b(Hepp,)j(K.:)58 b(Th)n(\023)-33 b(eorie)38 b(de)g(la)f(Renormalisation.)h(Lect.)g(Notes)g(in)f(Ph)n(ys.)h Fb(2)p FB(.)f(Berlin,)i(Heidelb)r(erg:)636 3580 y(Springer-V)-6 b(erlag,)22 b(1969)491 3746 y([35])35 b(H\177)-35 b(ormander,)30 b(L.:)44 b(The)30 b(Analysis)g(of)g(Linear)g(P)n(artial)g(Di\013eren)n (tial)g(Op)r(erators.)g(V)-6 b(oll.)30 b(I-IV.)g(Berlin:)636 3912 y(Springer-V)-6 b(erlag,)22 b(1983-1986)491 4078 y([36])35 b(Iagolnitzer,)30 b(D.:)42 b(Scattering)31 b(in)e(Quan)n(tum)i(Field)e(Theories:)42 b(The)30 b(Axiomatic)g(and)g (Constructiv)n(e)636 4244 y(Approac)n(hes.)24 b(Princeton)h(NJ:)e (Princeton)h(Univ)n(ersit)n(y)g(Press,)f(1993)491 4410 y([37])35 b(Iagolnitzer,)26 b(D.:)34 b(Microlo)r(cal)24 b(analysis)h(and)h(phase)g(space)g(decomp)r(osition.)g(Lett.)g(Math.)f (Ph)n(ys.)g Fb(21)p FB(,)636 4576 y(323)f(\(1991\))491 4742 y([38])35 b(Itzykson,)25 b(C.,)e(and)h(Zub)r(er,)g(J.)f(B.:)30 b(Quan)n(tum)25 b(Field)e(Theory)-6 b(.)24 b(New-Y)-6 b(ork:)31 b(McGra)n(w-Hill,)21 b(1980)491 4908 y([39])35 b(Junk)n(er,)27 b(W.:)38 b(Hadamard)27 b(states,)h(adiabatic)g(v)l (acua)h(and)e(the)h(construction)g(of)f(ph)n(ysical)g(states)h(for)636 5074 y(scalar)23 b(quan)n(tum)i(\014elds)f(on)g(curv)n(ed)h (spacetimes.)f(Rev.)f(Math.)h(Ph)n(ys.)g Fb(8)p FB(,)e(1091)j(\(1996\)) p eop %%Page: 75 75 75 74 bop 1104 226 a Fy(MICR)n(OLOCAL)30 b(ANAL)-5 b(YSIS)28 b(AND)h(RENORMALIZA)-5 b(TION)582 b(75)491 425 y FB([40])35 b(Ka)n(y)-6 b(,)24 b(B.)g(S.,)f(Radzik)n(o)n(wski,)h(M.,)f(and)i(W)-6 b(ald,)24 b(R.)g(M.:)31 b(Quan)n(tum)25 b(\014eld)f(theories)h(on)f (spacetimes)h(with)636 591 y(a)f(compactly)h(generated)g(Cauc)n(h)n(y)g (horizon.)f(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fb(183)p FB(,)g(533)h(\(1997\))491 757 y([41])35 b(Ka)n(y)-6 b(,)27 b(B.)f(S.,)h(and)h(W)-6 b(ald,)27 b(R.)f(M.:)36 b(Theorems)27 b(on)h(the)f(uniqueness)h(and)g(thermal)f(prop)r(erties)g(of)f(sta-)636 923 y(tionary)-6 b(,)23 b(non)h(singular,)e(quasifree)g(states)j(on)e (spacetimes)h(with)f(a)g(bifurcate)g(Killing)f(horizon.)g(Ph)n(ys.)636 1089 y(Rep.)i Fb(207)p FB(,)e(49)i(\(1991\))491 1255 y([42])35 b(Kinoshita,)23 b(T.)g(\(ed.\):)32 b(Quan)n(tum)25 b(Electro)r(dynamics.)f(Singap)r(ore:)32 b(W)-6 b(orld)24 b(Scien)n(ti\014c,)g(1990)491 1421 y([43])35 b(K\177)-35 b(ohler,)20 b(M.:)29 b(The)21 b(stress)f(energy)i(tensor)f(of)g(a)g(lo) r(cally)f(sup)r(ersymmetric)h(quan)n(tum)h(\014eld)f(on)h(a)e(curv)n (ed)636 1588 y(spacetime.)k(Ph.D.)f(thesis,)h(Univ)n(ersit)n(y)f(of)g (Ham)n(burg)h(1995.)g(Desy)g(95-080,)g FA(gr-qc/9505014)491 1754 y FB([44])35 b(Liess,)29 b(O.:)43 b(Conical)30 b(Refractions)g (and)g(Higher)f(Microlo)r(calization.)h(Lect.)g(Notes)h(in)e(Math.)h Fb(1555)p FB(.)636 1920 y(Berlin:)g(Springer-V)-6 b(erlag,)23 b(1993)491 2086 y([45])35 b(L)r(\177)-37 b(usc)n(her,)23 b(M.:)30 b(Dimensional)22 b(regularization)h(in)f(the)i(presence)g(of)e (large)h(bac)n(kground)i(\014elds.)d(Ann.)h(of)636 2252 y(Ph)n(ys.)g Fb(142)p FB(,)g(359)h(\(1982\))491 2418 y([46])35 b(Magnen,)26 b(J.,)g(Riv)l(asseau,)g(V.,)f(and)i(S)n(\023)-33 b(en)n(\023)g(eor,)26 b(R.:)35 b(Construction)27 b(of)f(YM-4)f(with)h (an)g(infrared)f(cuto\013.)636 2584 y(Comm)n(un.)e(Math.)h(Ph)n(ys.)g Fb(155)p FB(,)e(325)i(\(1993\))491 2750 y([47])35 b(Osterw)n(alder,)22 b(K.)f(and)i(Sc)n(hrader,)f(R.:)30 b(Axioms)22 b(for)f(Euclidean)i (Green's)f(functions:)31 b(I,)22 b(I)r(I.)g(Comm)n(un.)636 2916 y(Math.)i(Ph)n(ys.)f Fb(31)p FB(,)g(81)h(\(1973\);)h(ibidem)e Fb(42)p FB(,)g(281)h(\(1975\))491 3082 y([48])35 b(Prange,)24 b(D.:)30 b(Causal)24 b(p)r(erturbation)h(theory)f(and)g(di\013eren)n (tial)g(renormalization.)g FA(hep-th/9710225)491 3248 y FB([49])35 b(Radzik)n(o)n(wski,)22 b(M.:)29 b(Micro-lo)r(cal)21 b(approac)n(h)i(to)f(the)h(Hadamard)f(condition)h(in)e(quan)n(tum)i (\014eld)f(theory)636 3414 y(on)i(curv)n(ed)g(space-time.)g(Comm)n(un.) g(Math.)f(Ph)n(ys.)h Fb(179)p FB(,)e(529)i(\(1996\))491 3580 y([50])35 b(Sc)n(harf,)f(G.:)51 b(Finite)34 b(Quan)n(tum)h (Electro)r(dynamics:)53 b(The)35 b(Causal)f(Approac)n(h.)g(Berlin:)51 b(Springer-)636 3746 y(V)-6 b(erlag,)23 b(1995,)h(2nd)h(edition)491 3912 y([51])35 b(Sc)n(h)n(winger,)22 b(J.)g(\(ed.\):)31 b(Selected)24 b(P)n(ap)r(ers)e(on)g(Quan)n(tum)h(Electro)r(dynamics.)f (New)g(Y)-6 b(ork:)30 b(Do)n(v)n(er,)22 b(1960)491 4078 y([52])35 b(Il'in,)c(V.)h(A.,)h(and)g(Sla)n(vno)n(v,)f(D.)g(A.:)47 b(Observ)l(able)33 b(algebras)f(in)g(the)h Fa(S)t FB(-matrix)e(approac) n(h.)i(Theor.)636 4244 y(Math.)24 b(Ph)n(ys.)f Fb(36)p FB(,)g(32)h(\(1978\))491 4410 y([53])35 b(Steinmann,)27 b(O.:)35 b(P)n(erturbation)28 b(Expansions)e(in)g(Axiomatic)h(Field)f (Theory)-6 b(.)26 b(Lect.)h(Notes)g(in)e(Ph)n(ys.)636 4576 y Fb(11)p FB(.)d(Berlin:)31 b(Springer-V)-6 b(erlag,)22 b(1971)491 4742 y([54])35 b(Stora,)19 b(R.:)28 b(Di\013eren)n(tial)19 b(algebras)g(in)g(Lagrangean)h(\014eld)g(theory)-6 b(.)19 b(ETH)g(Lectures,)h(Jan)n(uary-F)-6 b(ebruary)636 4908 y(1993.)24 b(Man)n(uscript)p eop %%Page: 76 76 76 75 bop 456 226 a Fy(76)803 b(R.)29 b(BR)n(UNETTI)h FC(AND)e Fy(K.)h(FREDENHA)n(GEN)491 425 y FB([55])35 b(Streater,)29 b(R.)f(F.,)g(and)h(Wigh)n(tman,)g(A.)f(S.:)40 b(PCT,)28 b(Spin)56 b(&)f(Statistics)30 b(and)e(all)g(that.)h(New)g(Y) -6 b(ork:)636 591 y(W.A.)23 b(Benjamin,)g(Inc.,)h(1964)491 757 y([56])35 b(St)r(\177)-37 b(uc)n(k)n(elb)r(erg,)27 b(E.)e(C.)g(G.,)h(and)g(P)n(etermann,)h(A.:)34 b(La)26 b(normalisation)g(des)g(constan)n(ts)h(dans)f(la)g(theorie)636 923 y(des)e(quan)n(ta.)h(Helv.)e(Ph)n(ys.)g(Acta)i Fb(26)p FB(,)d(499)j(\(1953\);)g(and)f(earlier)f(references)h(therein)491 1089 y([57])35 b(V)-6 b(erc)n(h,)24 b(R.:)30 b(Lo)r(cal)25 b(de\014niteness,)g(primarit)n(y)e(and)h(quasiequiv)l(alence)i(of)d (quasifree)h(Hadamard)g(quan-)636 1255 y(tum)g(states)h(in)e(curv)n(ed) h(spacetime.)h(Comm)n(un.)e(Math.)h(Ph)n(ys.)f Fb(160)p FB(,)g(507)h(\(1994\))491 1421 y([58])35 b(V)-6 b(erc)n(h,)24 b(R.:)30 b(W)-6 b(a)n(v)n(efron)n(t)25 b(sets)f(in)f(algebraic)h(quan)n (tum)h(\014eld)f(theory)-6 b(.)25 b FA(math-ph/9807022)491 1588 y FB([59])35 b(W)-6 b(ald,)24 b(R.)f(M.:)30 b(General)24 b(Relaitvit)n(y)-6 b(.)24 b(Chicago:)32 b(The)24 b(Univ)n(ersit)n(y)g (of)f(Chicago)i(Press,)d(1984)491 1754 y([60])35 b(W)-6 b(ald,)17 b(R.)f(M.:)27 b(Quan)n(tum)17 b(Field)g(Theory)g(in)f(Curv)n (ed)h(Spacetime)h(and)f(Blac)n(k)g(Hole)g(Thermo)r(dynamics.)636 1920 y(Chicago:)32 b(The)24 b(Univ)n(ersit)n(y)f(of)h(Chicago)g(Press,) f(1994)491 2086 y([61])35 b(W)-6 b(ald,)27 b(R.)f(M.:)37 b(The)27 b(bac)n(k)h(reaction)g(e\013ect)g(in)f(particle)g(creation)h (in)e(curv)n(ed)i(spacetime.)f(Comm)n(un.)636 2252 y(Math.)d(Ph)n(ys.)f Fb(54)p FB(,)g(1)g(\(1977\))491 2418 y([62])35 b(W)-6 b(ein)n(b)r(erg,)27 b(S.:)37 b(The)27 b(Quan)n(tum)h(Theory)f(of)g (Fields.)e(v)n(ol.I-I)r(I.)i(Cam)n(bridge:)38 b(Cam)n(bridge)26 b(Univ)n(ersit)n(y)636 2584 y(Press,)d(1995-1996)491 2750 y([63])35 b(W)-6 b(rezinski,)28 b(W.)f(F.:)39 b(Note)29 b(on)f(the)h(construction)h(of)d(the)i(Bogolyub)r(o)n(v)g(scattering)g (op)r(erator)g(in)e(the)636 2916 y(\(:)20 b Fa(')749 2892 y Fp(4)803 2916 y FB(:\))850 2925 y Fp(2)908 2916 y FB(theory)-6 b(.)24 b(Theor.)g(Math.)f(Ph)n(ys.)h Fb(11)p FB(,)e(331)j(\(1972\))491 3082 y([64])35 b(Zimmermann,)c(W.:)46 b(Con)n(v)n(ergence)34 b(of)d(Bogoliub)r(o)n(v)i(metho)r(d)f(of)g (renormalization)f(in)g(momen)n(tum)636 3248 y(space.)24 b(Comm)n(un.)g(Math.)f(Ph)n(ys.)h Fb(15)p FB(,)e(208)j(\(1969\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9903180412792--