This is a multi-part message in MIME format. ---------------0603150838937 Content-Type: text/plain; name="06-69.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-69.keywords" Carnot group, Hardy inequality, Uncertainty principle inequality, Rellich inequality ---------------0603150838937 Content-Type: application/postscript; name="kombecarnotfin.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="kombecarnotfin.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: kombecarnotfin.dvi %%CreationDate: Tue Mar 14 21:56:25 2006 %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR10 CMTI10 CMCSC10 CMMI10 CMSY10 CMR12 CMMI12 %%+ CMEX10 MSBM7 CMMI6 CMR8 CMSY8 MSBM10 CMMI8 CMR6 CMR9 CMTI12 CMSY6 %%+ MSAM10 CMBX10 CMMI7 CMTT10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 -p1 -l13 kombecarnotfin %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2006.03.14:2155 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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94e48368d6767c78cfc6bd814e70d65fe71fd1ad7faaef6da2b14ef4743bb839 b773a61f7c07803441bc23e370e74efab92cc7600952fd2f150d4aa5a10df547 ab1e09a9f3c51e8e6d98393a587121e92b283ea2a42a29a91dda727b64d2da19 d6d9da63f6a7f24fbc1de60b2b8efcffa89cebed4a0751c9827e488fa6c95acd 1780217b46d694ef39a48dea0244878e94051c2b6b553f8be481e9a5 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 112 /p put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d e9db1830272ffbf2f4996ccc2d6fe2ae272798989525ef3b67b0d09bffcef749 a805e5f76578222b9c4a8a09b13189a16ab746ddef7875f1ecf83e568f493d82 ff729baa1e0834dbec30a35d3c49c9b10c5e1d90c6e3c8fb737df5ceb3299d96 0fb79632f91f257753b4d2e34e3f54a26c7b950981fb7fe4dee3315db63e75b0 24b4d318baec8aac9cde186f6d65767df1dba35124287c2e805bc660a3a22772 0ae4fa097f1d75107441c0a0fbb9eec17d5516fa03b98756b7df7ac7eff9c7a2 364b9f691ed5cb692dfff9dd80c761e9c4d5d6061782b327302c053ce1cd09ba 0605ebf761cf9b730524e1994e7dfaa956799d522ec53d5acba27a35c4d28279 617771938ece58c5ab664aa00c1bc4efba2d15504247a68e3af0f9bd59a387b3 80db88cca083b06f5af70d768d1699826fb12ce12efc2b4c70923e052dbb964f 5fa957c3ce225c0fda64d9b5321ee1ade207c1cc93105096cc73118f4082594c 8ecddeb0d7bfde6f70bc62f10407c26a476ed88be7999d64a15912493829a9de 62c71f5a34fbf5a843e311873cd98817dc9f8a4aeb7e0738b5de10043f5cfea7 147f31c39019afacba581f8351d51c8da6e4e2994265299ec90628a607d505cc db61b2a6486bfa8088064b7004b6dc5673fe202614b9d7507a41eef30cb16f99 9d60b1506df407632bc293b8aefa0e8b96d00091c7652031f9a84e86c8dfd25e 6185cae38dd51eeaee311120b233a876447c89243006de85d695330cdbe7c098 03b2313198eb830f7efaaaf880fa59ea3256136798969aa7010b86dd4cac4fdc 78e1df2b0bfa1b6c8b87e1b5513dfb8613dcc29c6ee7deb09a34cf4208e96b4d b06d21a148dbad9abb2c15739433b39404540498acf77be0a648df9def1f1c32 18b2d2c1c65596a029c3a0016f3e1b8570fb5e63cd7d88d5c0c1b5f1320d0312 c351a064e1ebe501e971af472a805239685549f03a21105c1bc9f286aa9ac0ba bfa7e6024648afb6148061fe1ffb98776f578888de867408cc2f62a82585bbd1 0e6c014761285e08e2409df64c1d64e73752af863fe5f4631c44f1a335e1e531 d32dfce6883ecd756336ac011c5a55b7849528bd311f7021f3a4c32096 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 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 readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 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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 readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 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 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e7a4a832f89e4f9cc 36f6027a64831992e98b1707ddc56e562de9ad0278d89f3416dfab55ea0e5b25 9cfdc4787442890a5a8986bd394cfd87942f66b3712745aa18e52c6bd1d44d48 3f75ca3a5d39391f2bb9441314fbb25ed2b6813e66bbdec80f12a88c2f94f601 fdb6b48613c26c3ac738dcd4ab05a16c34aac07eb789901b095230281fc71f04 5ac83d60c6b82255c6276398134634a8135902a31a52e84b55aece4be46e9d93 72e8c2738a3ae29ce61298d5c2e792bc098a143f3f08614489 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 67 /C put dup 70 /F put dup 72 /H put dup 76 /L put dup 77 /M put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 59cfdc5d7f526fca5416d741073284f5dd2471e09f5916f4bff1a90eea4d1283 7731f6a36e5ea695f4efc3bf5669b7f60ab606df5819149c0c8a4817f06dc172 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen 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 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 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 readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 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0f190132c35e5ff58ebff67abb69ae71f6ae431cdd80a6ee07241d3627db061b 67003969cd6c3758de622a54b86c1eda01cc8cfd1520314b9c2a5e138f04da46 38ef9c5d377155b46bd7c90e98d17b401a4f0c36 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 49 /one put dup 50 /two put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 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 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d8c1e5c02115408a 277a0db93878d14fcdfef599913c0cd614d43dc6a08bb4db21730e0c454740fb 6efc542a3a65c0080b19eb0787edb1b37a32e81102ec944c67e6712ba229a7b0 f9e92271d62dad21f274948d3b192dac77c79729bf891b3baa4f1d47ba7b900e e71b90feb7bf7fcfe223a9a69bc9bd354545708209e94c9684039464f9c3260a 5317fb1e6c96e982272909867a7c5b7bf00186e1baeff1a4ced4785b56712833 f194bbcd40a7cc64fdc98d6640c0a6aa6c204d6539c74a67431f44edd1087f62 c3b119d72d6fe805137c77578c33857bde012f38e42dc5a83d4961e6f56c6faf f04fc4dbe862f8a3fae8fae508c701b101126b2ce75829c7c9c97d04fc490762 3cf90c0580ded9c7868d0c9662061602e3e79036abb61f28947c545e13dd65de 9c76d43cd585fe89a1b7d35197c0459045303b0a4fc7bb63a23448b719a06ff1 574ba437bc11c5f5b2373ceb7b9c53b14a46f5bba90e25ab72a87734a16b9e17 a8a0f8386a46c180d03691b79747716e2ccf0b73072d3ee6774b5a04af8996c1 463146ae0fda57802440957d719f1799b1edaaca85fb27ca6cacefc645621727 3e166d2ea5d8d1d055dfe81bc203ef1d3eecfd45d8ed4b8c139b6391c24e380e 0a3f14f4b4454bc23ea67e3b0c2b30f1e20c63875c402e9b46dde773239ccba2 6df2b17edaaa439359ff7137d6e75ef19d1007b705123e5370a9ddcd49f52e4c 1e7e008b96d8f5f4fa5855e28f8755aa7f8dc6dde1ade71958b03975365b34d0 734f387b3d6b4456c11c05338b7cb47aafc4ee312082fca752c0af79e6c1af7a 01fbe29ddfed452bb806f0f1d16acb3097d5b12c7046e9e2cc85a4933e0994de 82311a4d3552daa9e7745fc96e8a9a956520d67e9928365f83cacb72539eb165 4899ab87ec0dbb3b080b23f008171f50 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 15 /epsilon1 put dup 21 /lambda put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 81 /Q put dup 97 /a put dup 98 /b put dup 99 /c put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 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0bf810351418a5e81bee07bdba6873989f8e77e27fe7e9d470f0c7964b97be4b 4e496b20b24d179f8d139a616caddc8535147f92cc65419f7b6099283aed30ff 35372447751c85e271236adec7a359d24a6e9a34e1df36c5d4a8ab7be732c301 d7d87209501811db95afcea137180bd2649d561f5a7bc55090a10d9efe0c9257 fb10937bedfa1028b284c453ac1046427601864dfd6fb811384a477f852e81b7 a375c82e3746c7aae1155d96483beb68c4bb9542abf9f2aeaab3f9bb1fe22098 ff6c3ac76bfa5edb60f0bd06ef9d3b4860979144478d3dab1e8d16f2c601b14f 1d16a8cf19f4d41134b5807a3b49cf44fa65cb3c733af7209526960b6781afa3 ac7ebcd82f9462b9e41f9876d9ba70d3f245e2a6f944872394e619f102d4d7a3 63900d7ef0d9f72820d73f869d28bda5c2122be8c92105f844ff54e636ba248d fff1642b0d964c11be9bd26f1171b576b892e8f535bca3569d346634d03e9455 97ee4f0f81e94a83efda306edfa90023c80e15aca27eb5270e1d586cc98429f3 a62ae005ea6efe6dc395f574113b840ecac0f7611d857b76ac5252aa70772945 9a60e9265a50bfde527f7d520d33b418635ac1b0d6e3cbf86030f0901def5ea2 f989eee2ff69ff08577df224013a5bd4aecb88fd7dd90bc46a515e529db5510b 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2c83733ce9700ad8e2320806f9424482e26432b8b843785f655b5430add68fe5 749ee8f6d99b62a897bca4bc675fa47d25fd8109a91c4d438f6589963398ac07 be6ee77c7768c5c15801aabec61157d5bccfc1e9962007ee25e305b0384e56fc e52053d598b89ed55e1867f9977172a3a79daa605f819aba41a4652f536f15de 5bfdcc5626450429da92ed920ccc4081f83187ee07a576d5028cfe611680605c e38ac9c2c0b15192ccfb0fe585819db5af5663192d3aec935756d7ea117a7502 7404d6168ffedca16cba3405063248275ae68cf55e66d22108fc0f9f1905cd15 58130a6a50be086cc03ef0dd0f2cc97cdb0213293c829fb75812d2e9c491488b 6967647800d8b3a9dc2bed8227cd7f47062d562f3caa109d3dd7aea3b02e772d f47b48838015cd970e208dea4f0547117330575554bdd59deaddce0ae844891c b9fad638d0c2dab28d8d28ae78f5c67ac0613ec84cb88c909de8bac4e7fe0359 73bc5139f7abd85838113b1ae895878c9baea670d4e7d0c086a97492cb6344e1 d090acb60e132f9350e93e8670f3d708c526aab9abbc8d31829b52d0ed212475 f4637d96af75b28fafc0b6afeefbb4acf24d561843e02d67f04d7fc938503378 b0cea1dc86397310a41324d2aba07011050161a232902629aeaabf71bdab08e5 da6736ab92e1cce67d50972cc0e1c61ce1faf5a53d802e7a7959d5002e224826 b5f8bca9884420d35aba569b806e4dca057103a3a345fe66070ed67292dff50d eaf35876fb23bf6fc454c31332b46b93ca6566bb511fb8477800e9b3db4639ca 01aefb66dcf6e0c908409bb869051b7f7326 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 67 /C put dup 71 /G put dup 72 /H put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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2fdc4287a101b69d11823493daf2ae2fb1e88eb7b5569e462be2c1cf5e925d52 a474c2c65bb91e83b1e9fc435d74fa09ab7e6ff6068d9571d6986d14d7ee4114 e350fa0ec1a1e4d46d725c52b30f220454ed54eea308f44b9dc421ba237d67b6 62ae54709fb79614818f639dd104f0bd728b86bd52b7990335bdbbbc57a06773 9d9b9dcd4734865637249cc050f60886114b8e5ec4208c216fecd37d37788b71 1045e7da204061b8f9996d21b673379978154145a4aadbae4cc8d2f4a3ed0bd6 46018090b03bc9ddf7769aaf52150def86d7ccd6acf0c07e39b6bbf28c757d32 cda39c8edcd98252913593947135d0d2f7dedb1ff630b5ec4ff94cd78471b62c 9f18f0a4bed34f9fbe6506ac00aed83898b61f1c85ad8d77a9df74d0906259ca 865d52564e369eecba79916280b0c3efef7a68399228c921a9465785ca73b188 3b4406279c4182d2977edf1aea32d6579e3633b687b01b4ffefb5917e2520a03 04ff052a96b006ee8d599c2fab96fd8d5299c2e4f834d0ce61e4d5ee0e06c64c 8e1ce8d92064f55b0cb2ffe524bd575d63b5a7952aba9c227b2cb1fb50d826a0 884b3f8cd8520a1c1aa205af2d1bef9a9d4fef46730fc9d329ebf1f1317e442d d33693e4180bca75ac4410189ff636515f3cfdcc6520ee86ae5a855e9e5e0378 73c973c7f025c8c3cd6e9412f5012af6c95fe59cef7649198b17cd093a5c6339 ff26aebf0e2d04ad2bd7c8440de16043088c08b23294df079432fbdd86cad686 0a41 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put dup 49 /infinity put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a25ad4f81f70a4699b36ad0292e61eb54422c13eebe63d1c463d38c6 e0302853d139f2b956a282191dc1a7503384b993f2d884c577935afc33194e74 1dde8f77d5a7f74bc0cba3ee49b9eb85a278362859cc74e1a4491a9fe383d8c9 e2509d7ba32cb13af3699251e5a94c7c570e780f8ba5801064e12521e1341eab 5591697937f75abd0ec38f14bb840b887df80ce1a4511add0572dc6613c4c93b cdb6e112379c2547f6ad6abcd08b77a63d920104bbe4dbc14fa754f5448eed87 66ce388d97a9a20ad3d24a7fb3980c9a88db421ae721125e8cc779e73662a63a 1b487db6c6f0e996d29f367453e797324c975c109d7f4e7e7a4dae93a5c59b57 6973f79ddf7a496f1bc659e3c4c0d11cda2d4b412e80fd5846cf16e5e1bdf1a6 675ea50d094d71068429472ae26da3821b284b5f6de3d4d987887db4d7f56839 e3e9c6bf82c21b54d59ecfda120b8eaa304fed7061a649140473bd5b3d64fee6 4a07bedf959f9d89516d82a464f23764751d1499923a699908f190f8c439f2a9 896e80063af3b274d446d4134e75d103f271a9c2d117344bba372cd5bd2b3767 5651be57fd178b4f9334ac03ea3cdf4f25b582546e003fd548e9fa3fbd5e120f 7935073a1e777c9a6d27ef4b2abf1fbbd036374f7c0205c17760d61bf9c7c273 2b5e4c67a3f10d8f363ce75d4c933206f5dacea428ced3f637571d901091cc0f fced260bda9f2249e4c9fd19ccdc71d3a9e9e68de84c6b9f02051e53059819a2 1782c87cbe9bd4b4f02e064f8857d1fc041ecccc1dcac43f875fb8cae598c138 5b4f2ea06dd157cb03b577bd76319e14d2391646233428b69f1120a96d09d1d3 75725c7532be02709bcf83945ccf4cea53b9e2b60524c73637410c049e241450 96393bd511c96dcbe0a2d1e3e474da99f032b6c26cf8c8669eda192a050f6053 6190a8ae4464e6c2138e3156727246e7262d62950000fb4317f2e055684dcc88 f715dd62e391c9119308a89f33546ed6f37ea0c737a3f19d71fcc2edc987003b 3462ec83c588f75ba5f4742dd6c4313c9f15f6d921f0cab8bf475c070e6bf2c0 397fd059cdf8684869d28fed64a15b02da1267722b58341c7723efe9ff29486d 9ed5f7981529147f1630001cf9fdb322e9ea274efbee9c3b835546b0d0edbc0f bf5c4854bf3aee0964f848e4da6be587a50989a324dacb73e37fa402f4549097 bb01995d1f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 54 /six put dup 61 /equal put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e50e42374ab0a611be 7cf8003531f4120ca0b8629854d3033ffd5e6b2232bc0767dd4f3a1a3f31f00c d55efb3a27f72351e66b79ef3ce4f39c27263c499aeff1d856729f4ffc7f52bf 4cf61c74eeacf2c3a51b0add4f3da9af46f0ed1d4ded887ac31b6a7b7e1c0418 864064e60a6ecd6d6a1fd7066b7d114bb9acba2b030dbdd67943353a5b72235d 8116af0db46d774c2f3218e0482affe8557623024b975f296c1bbfcf6fe00823 3ffa26ab68a96b979bfd50b7331bee7287b1a8e599cf3d2dcc3094847e3cae16 2b01866597942893e2fbb5ee5d59e6caf9cc47300fd85398de6d8b0096b81715 55c2b5b4b3891b6fe883829e42a9007c0fed3b213dad09ff49a59423ce250164 dac1a5eb38a260e787cde28e5b85f82480ec99a0004f139de827ffa58cc8958e fbcd46e953583095442977c487f5198f60a9b7daa4b941f1b46f1dd6bef5e445 d73deafbe4a2721b1804e0411a49d2bcecd9a2dd6283ab17be3b8b1162ab2cf6 a56f7ab4e91fc6a49c0ab532fe888e616719ac02e5cd3c14c48016de799ea264 e4226b6f67312283cf62e83d1a291807dca5d4ad22018517c1ba4bc0287b5095 f59b99c8c7d8ccd93c30ecb3831192522283b549686906645bd2f14c2e53d3c1 8e650f12298edb2ad1809fae4b40c8ed936c9f8714e1c3abd797236288d0d72f 5eaab99015c1d55ea9dde9c2e07b3295217fdd03c7ccb272625f23266159a095 fe9c6ceef8a3ecde3df3eb521bae64f49cc3e7e873475fb183b6c2ac60d9c4ad d1bdab96751530bebf6168a00f4376010ba6917f17a8f012acbeb21882bf196c 9b30f7749437f56b4e5dbe2a6c50796c7f4ef8a98d251f4efe90b70a28365171 2c9a0a0c3b43033f1b1e3a8489ca9f693e69cacd85c108136e548ddbc08a8ca8 0efdd027d0aa42194df1bc1fc1cadce1194c67fb8df5a7b7f464a6fdf1878086 5cd9c8359c0c823257ce5aed58f8060865e97af9fa4dc4f4343a70c0b54c0b4b e62d6bf196f6cb60aed148318c7a9943f344c8c55a70d410dfa2f079109792ab b2de61c5ec66f933ebf2a20b81d62be2dc19810aa7f241ad29639eac6a24ba18 ba342ffa580a31950608c5acbbc062b651535277db10e8603da186448cfab303 b2df4ffcc4ffde1cd81a17e87f512343d3c6c820fe05bee9cd05115344dd6b5c 64041bfbedc54772bd18542ae1ac36ffe9d1a84518c8e323162b577d427b2ab3 578dfe42849eacf684e7c81e783a6c8e943f491af72616a8ad3a1247367a5dd9 5e5a9d0aa1ec6451d4697cb39e6c5b14f4621fa82f0da3d18dfc8c22f02c6f62 981a5fa8889e8307215419e662ff6ef001f73d1edd615c31af73a1f6280c7695 333890a47712855acf97259ac738ffffd5eaf8d026ef382cc46e77d6ba28691d 73ea9e6def109a768a2536179174f29ad6d74e69d816a662f3168678f3168fd2 33d595da5fb8d7fed0403d66b5321b0a9332f9368495b06205bd04fdaa3af5fe 9ca44aacb4c7294cd3d3dcd86ec780b054412fb3f759e593284dbfdf33955626 7094c4d031490611fb520372bf81697458c82c19d1972c23932e8b8749d1bb65 e6063358ea60f27ab171ab4bc16045ab04d3e23d5428f44e499ee210676d6ce8 bf339018c8511f558ef60b741f5998c1e80b70d8454bfa1289ab5352df6b31d5 092d2f8af54c9b6ce143b370c3c99c415b9218d8afe39a8ae99b1ae6accaec6d d871cbf145ffdff91d0029a3a90e2e6df65a0e8053e0363eee319048b563ea8c 6750ff0d53de1db9df01f76a578b29db0abb76a095abd75ad0d8a48f94101ddb a694a983b87e2bd1fb7b2942dfba65760edf9ca4bad7680a749c114c9de94fb5 9e7e297c42b64a3b2388aed0dbf9a7f8f21a9e700e6d53507e97ea46ed7d7e87 ca0ecacebc39517c777fe5de0e6b11ec81e7c8de58126f5340adfde40096c113 e7f6c45dfca08b78f002b3a2b2320595b9fb4da9a7fbaf22558dbe13025cc450 f86dda683ea48bdafb89b87c244cdfed67599af2daf7d8d1bb84a2d90293f95c f802c12bfc68c202f63bcdb5ae997184e8e0364fe4bbccbfb940080423144ba6 63248dc28dadde5d311b3cd4dcf801a88fd8ef4cd6c1cf861e12e68ebe5bf1da d7b8310e3bd3c87f235dcb8490d4ac57bc8613d29688065d4480c0a33e859564 aec1cf814aa51b709c97f8114b93549be32021968e3453c256d2bd3a794716f8 24ddef1946f64f19ff 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 81 /Q put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put dup 112 /p put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f 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da31f293753245245d5756fff2f6d17a17f514844f7262e790ea2ddcf626eb12 ae2a847d3e3a861e9a4200665e11629b653a7450a5bd2f54617de436c0c6a3b8 c600689f7c4312a4fd4fa1e98aaded9416376a54298f7add02e12339df695f44 f5e7782d89a6b47dc06f4ca3e8580a45c17765874ff8eb9ad584b2b511304b90 0126335de922a8ae9cad3c1ba9e702e36cf77661eb8c0344cd069cb7abde50f0 009c3184ea1260751c4c153b2d3e314143156bb5c86cffd84dc9e9c1ddac6cd2 20296952520873e5a309284f13d259d2234ce124311441f8cf6849c2050fe26c f1ff3c56f024ad6ea9be1a3fcea37fc698e2b31198a2b0ecf4c378b5b402f0d2 22460b191365469fdd2890b2bc617cf9cd327cb3528d971ea6734be00992c78e 9c5b58bbe4d3cb22e351e15c4277714ed668c480aeda8cdc015644446e7327bc 699664c00f312578dfa2426716d39244b74ba42037801798260a9b3852d07292 df863d5c1810c900cc8117cca4ff789e625ee945ac824825cfde0e3ac5c92dc5 c40a87adf561d02efd20d2e6b5b3c33b2e8ee32b2e87266d166312a65bd98bdd 26d0c3e173bfd6d7e41f44735a5ec4857791b2f1789c7c52fda15ff886fafdee 452a12a7d73f05ea1c228435bba541e2f54decf45cd1ce3b8bc8d8b3e969eb48 d46aea1a7718d2913d03e1b31345b5b9907a9756b2719a6333878e863baa3158 e1314e2c07694ca6c8730fec90df14345017857fb77337b5dd6c31d7baf62083 e9ad61d2d091ef59661ed2f2ab9813836f244d9769af9d8a270aa6e79e4149d0 f9f9a28d94d62cada6332f832daedce50436582a8754c34661a16ad1168a7d69 c37ae1a9270371786af2c9226d3b7fdcf2d23551a01277915960a09a200acd1c 74616955198adc883b1f5b2de44124b4d2b2d72beeeb93aa214b9dcf4e6d51b5 03af86d04dddf813b6d947a6f433a2838ac9edf2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /G put dup 72 /H put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 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 e0b94b4dcc94213911ed25a52ec3a8ab35244b9d054a1b2bb41caac27c039234 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41614d20620a30807f4db8eb98310f8cd5cb153b89670fe8c8384f6fc4cf47c8 aa037eb9020c7831bff3d56acc54e8125bd53cf4a4dfd01b2530200ba24629cc bde7b07c12a280c3d577f239067be6b8dd81f0b1c5ca17c632acb467ab46095c a0f95c59a44820e1be638aa1963ebdf4bf2f306ebdfefb76048462174d20f74e 6174c7cb0b108a669fa0c0826e75430d87813efe59250e5eb573a577acbb8911 633539f162b139b50be9876f7573ae7eeb48d9113912ae2983bd9e3369a5da3e 5c5efb56b5a786aba8f82501a6f1f70adc2adb4de8ee0deb390b82721ac59ee8 36e6350742dc644de3a868efc1fa10621839ffc9cf05b6d397a401faeda34b47 04f0665280e3275277c03dbf940a990b4922cc4dc43ccabd7086ff54c3f891e1 c992bfcba09dde69bb6e618d4e38d4acacb9664b73031a5ec5810fecde6b9fea 6925c66680fe93e4c3e7ca65b5e6aca6e01637b445832a7c17887350c3cae7c0 37faf4215c43b28d79036b3d9985b7d0594b804f38e01538029311956a628f3a 24a0ebeaafa362407de7ebb1b1139d19efb6dd5d1021884a5dd5b63e8b327a15 07d38ff452fb156fe1cafa58119d0cc77def24d511581d8d6793b33ad2093855 522b0d5b1c28fc17184304846f6d0b1b67382f5bcef4175d8544a5bf13938c18 b467582fd061e83670e555e0bd293c42b56190c58e78b3d2e42e0473c94cec4c 42e4e1df32867054f253b87803e2410d6afffb26a1f6df2bb70a0d487943a767 1a329ddf75174765330455a44f3eac0248794869422a97be8e6f116dcc8f404d 93068a5d129726f246ac77f151a46fcf086fa190d8ec7459c2fe1c9d1f235f81 418dd6504440982ab5bd18210371740b815e218ecb5244d2b51bcbc6b1ac56f6 331ec9251e2dad77d6f84d1bf5b6ceb8df4c57f534909661d1bd5448cb6aceca ad057aaf2b80dcf0f48e1390cd9cbd3b098caaedf7598efc3d28e809908ed27e 2fa9528328c1e6e96c1c1dc27d836cc959af5e72551815fbb3cfdea81c05d886 cc34d7135e3f9d3940f0fb1143931df266ed9f525029b16c9afb4ff0b3e5ebaa e9012488f828fc2696dcefe0f595f4b2183d7bc5ae6c2eeab715d14bb4c13d60 cd06 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 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2d8c88b36666ffa7b1e51f11fafa867273da9be2637f2a7aef042be98c4914f8 b12dad7d8904b6d1651dec2060c972a56558c6b10ca782ba4f2be7008a9bc0be 57239601d6851d6c7a5dc5aa955670ed8fdb4a7ada76e2fd57e74e9d82f77754 a051585c7efc84e2f674e24a898118153888487ce01ed63d65efc0a5760d9379 f6694bd08fe43ecfde0a18ee5ca40b1417ea0ca82f6ac91dbc97bfb7258c2ebe e22dbdf54430ab364c1a7b0ae132395d736c4cdb381f17c86707a95d848b3523 9adedcad602ccb46d51da52272c216df618cc60c5b8bcee90ae72591c7efe865 6a790cb5b16e86553f0b191712d2e3d57edb86c6173e0865f6452940c52c0e0f 218f74c0dc98690f6e30f145c0c50866c4d2cdca7a5aeb83711ad3d684ee6914 27938f611586b158965ac248ca178d655d05beb974f0b6400de7acdb80682db5 2f0ebeb495d9a786be82652eae2a89b81b711bb095c8a54f8606392949d8090c d1762d5a367cb7e1ae7304855b9faf5a4a782bb2134ab5b909ccd0e095623fe4 f7a644cecb5af29d348ac4278c909ae467d4cb7cdff8b8a7ecb0798d63560deb e97e20db737620367f365810274911545a854355f250bf68938e07267073e473 32dbba24039900e647fd653dd40b6b68a6a3c2c6622d095f5dffe2d12240e5cf 59a3f0cef953dcc24c42a7764ab67b0ec42d0c394cfcd585b70abcb4b3f6c543 ff8b54df7171269e2f094cd1e7d36650f461b0533eadb9cf7ce33fef8cc57a35 b4de16a9c5fdc878a089710e35d906051ce86252886e593f62c6e38bf06e51dd 3b263e184b0961b4d8baf6ba4282faf106a86a9d460903b1ad9831f2e9165826 5faa3b218a9f11d6d1e3bd925e279f3ac7d0fa93a5a4dd0258aa066bf556b9c9 a364fa6929e4d818acc7ba12e5f75586e242175616a46afcf933cbd963dbb4db f9965b10458abb194c22938ae8a24680369efa0aaad968695cf58c584a4830f9 d9590800a310ebcb6d539ebab1d103ad05fec4911dca0e7e558ab0b1e37dd403 e33ce89a2cdca08544b8d6b4188ec265763a8c9a2a1991dc0aee65ccc5d48cce 2525378d6532c993756a8c30df91f9f753ff8ea450422940dde170dfb9d7fcc9 7e3e9524a8e421220d70566cfeca38e0b66a5d2fa148992e396fce9d42329464 05bf713f466bf3ccc5c5299769d6c607de261723daa461752996b9d382747981 a9f5e221a7aa7398a404df38a7a6805b31fa5693984122e86b77b730b3f1dbc9 a2cecb580b3714c5b8133675d7454abaa91a5506c09a8c8114c5b184c76f21ac b749ffb99ac301884784df5b57f4ba4318c77bb0e794fd325346e924de113cb5 5f17d37aa128e49089d0bc7005eea39a649cf301f783f3a79e6944f6c25d3970 8c553b117b05c53fa39ef76b2eb82813c8d18231bcaa29629ef9e81b0dfc6f84 ce4886f48588f399e21a1ec805dab3 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 8 /circleplus put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 71 /G put dup 76 /L put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 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6093b7e8717f29c9183246e8d679ffd5cad4434cbd1aacfc914ee686c160ff3a b9fcc35dc7ca229bc8b23062703eee33c65eae74a2c188b28e40092657b0f8e7 b153b41c7eadc176817dedf039bc1c7e547ea837040c69c97cf8decbb41fe5ae fbc526aa4900d058009f9097860ee4272ab79e2cf3047372dec82461bfaebf84 bbc086f51d77c19a98f0b1cbe769708871f4d373b4795d719e5346a71a2c5190 f53853032a6dc3a3f83150bd45caa87a188a46086abc636bae18851171684236 20ecd620e870e5dd1f98b9c500ff595b3900c8d68261f260db0979f8241ffc72 8798a97fabc2c3e230b2df2782b5b1372c25142cfafecc99da896ecff1f91b6f 828ad7090d60d85a17ef87cf389bd75c4d9e81e8680061ec9a84d39f28bbca59 59c0928c333fb3fb9132c60f5bab0ac5ef9b9b4da0dd62b606c74bf163aab734 948a25b103367da021c459a7cce674ab46092cb9f0e108bccba8c6fa90699c58 a2d4dc1f85b53b7ad62e48e623fef2fa31cbe60cbf49aa1b47887f37505cbbbe a93bfbddc9f25163212472ce3b91a32fa81b0319fe4e674dd41046a71f12943f a1e98644b3a4be780723a815588184276a408e87add22529a3b4c59e00efb03a 2c93866fe6937813a5c92f5f289e703c57e4669e2b15c0fc7b2e20407f7f7636 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3145c0ffb8386f56d26ac2b93ebc5f3f15f9139f5e7a0f9fbac32eb95df85526 22c8b8741d053d049fcc30c7f810b02ec1d5ccdad7b7f11cbe8c167286004393 cb5f5d042306ea01c1da707eb554976e5801572018796dbfccc32d28fa899d63 5befbb82ad6cdc95e9276a032eb0f0dd103b9091fe140e32ccb97546f7cf5d8f d31001dc1d0ed328095555f5848b413576c200e5bfff8a486becd9fa12549299 8d3ea6de251beb84722815af7b0627e7fc2a96d845782cd77630b1a5c3bcf770 a0be40b01e78b9476d08f407b8b846c55b3676ff45a7061b29c3319835717ab5 f32a9811765471cb8d6b04127f2636f94778f9cb2f05769214463c10ce797b61 027801038fb23181b2e1faf35688023e65d922178ffb2de19a1550f5ceb7a8a7 3f06e944e6c3b715dc5028e634d79f59a323b8792f1a9be11dc204651b278118 6c0038c103d7f0e720091a344e81ad5e15c71c424c641b8a9e2012a928e7971c a52a1b81e3031b80ac740bf28f6be02a737b36bd5f3bb9b681003d01d4622dc7 2045a8210746d551d1a279e5881c27f2566e51fdaa412ceccda2eaf8e46b1055 367d3affcbe6fcb7c04de5eee74c1d47fc5e2abc0e83724b456a9661b56624b5 ee3c379a61c0f5e365a00484c8badca519a574357073680481725d6446079c9c 440ebdfe245ed919d8125e5c40a671c93ed7049360ba4bfa233107e687c3bb7b e73036ab67701c642d5bbc44c9a339108fbbe94cc78300430bf8c904f2b625c3 f92cabe99ad09c69bbd025a83ea34e606cb6c019084816bac06b9909079b61aa ce11b6866d9810e57a69dc1d6e01607483b1612eefbeabb95f575291a0e798cd f0d3705e0ea051a8b2d22297e45bd5af48885be746b8dd5e454c261d1eb40ea6 17685b791e785416c50ce2ad34f7ee7878132df5e38328c03623ab409a761f1e 9c96f24b5a159a65496724827c1cfed40c005ff538eb6285633df4aeb4518972 bb4cbda916862a7e88b7d62d0576685435a3aa780bc759b313e2c7cf750d521c 823e90e10f054cc74d3d8518b1575a1846892aa782bb1a287bccb2dd4c591c54 2cabf722980cf1b026e204635e166695e560c405b75ff249acef1fe2f933fb67 4db31263683731bffd91c0fea0c021cf695aea4593344b6fa642d6398c800378 cf7fe709907222581b07e9a68af29919cfd1b5a43b131c8a955dc1134f8e40f3 ebf9683512ab3197a3593f109cca32f9ec2cdb7e84467222e5b1130e80d282bb bec87f13b0a61f5619108939f96d344f104d5a62926959e3d987c3e0d110175a 5e8f128c9977696021e999c19fc935c1382179c98760e03a2b9cae9c80ad1b97 20dbad2c79555c0743f8895f3a4d08c4f507b7c96313bdc9de9060d3989c5a47 2db629c66dd55f466105305420f85b1dd304fa4455e7c558d606defc89e18647 4361fc707bcf8a11a9956291d41f00550b975590c733257786b137ef1dd8612e 53219650f78ec9bdcbbcfbccffda1ad8292964574665c1cdf3f1403e2ac37413 91c7a6d7beb834683bfeb638ddcdc2173cab058a4675434602e65219fb8b207e 1b871ba38c4f677878129144d614ce3ffbf750f714b03631570ecf4d27289f1a e01cddafff1fbe9d6ce292061c9c5b8c50fef639ae17766f5e33ab85c7a88f02 a91e1a7ee85a99f7099090d4a7151b38b695c1db4b07990863bda9d9d404c37f b262cde63735f06603ce05c442f8f4d4d6ff4b1edbc980bf9658cd7dbf5d4f33 f48222e2b07ada6471a44fd6c30400215ec3475079f8de79f434bd9a1fa7f37c ca9fb576d169c9033c412f689c2e078867cb6760e5921afe42e8c3fb7e8778ac a8f8e2e5c4e7fcff06f81afb7a5ae7460b78855e9b65a0375dd4755cf9128959 a837d63162f1184b5919bd43384aa5887458a847983bf6d4a80d113fc79b3297 d68ca443b0323c4cb6b9cb011813721bb2318180f176c8b8a519ff4c84c54144 fdf6396023f67cd4e705c619a1234a209ad7e099388fbeb65878be41b2329a43 4d27737a032742bc82e75e64e0b2321f0a44ffbad64b7c7dda0fa3624d147459 8ee0f7d1243677553b5b6304e13ccc3fc2687823a96d525ce3654f8594840ee1 8894047075f72aeb1269bcd0bae1d33e9aa6a20cdc46677e47a2867115c7f737 4254fb5b1c611433e6c2def8948b436570a6795d38d1d04452e6eb97a6b89115 8539e973ad908c14253c56e1924282973269f788f25a5d0d4096cb299d03b272 9fa7f27d5f9a8e6ac7daac29bcf7b383f83941819759a6c4b4678f615ae89e21 b62f1c343ac2ba6fdf40311ba05a1e15a82e401a2fc7e9fd033a6465f476c826 166a007b07a7f93d2aa4df400c51586090b70f61a88737be91cf14fd42ab08a4 d8e064d27a6c25931f96938938b7db7e476bea1099a93d442430cc19ea89790d eac21368520eb593276f9d60738c91633ea139d24e6d936d5bd4e3999ef86bad e6c37f6a61d5208c4ff10dbb8235ea2c7990ce6bb827e46dfaffd82f8b75db81 b508e94d7f08926f7beac0f873a7680e7a0f8f55333f832c38fd29ddbc83e99d 8ce8fad3a0bd3fb307fbebb468e4c3130152b22780a922588114bb68919ecc6f a2dc0322d5f5fbbd131bf7d6b3900ee592c835704e97f5cf9534b1fc97ffc412 a46c35fd0665ddc76dcd2abfcead5284210ead64686dfdb2a324111d6d04f593 7cd9b7449dab62ad57388c32906ed766ef45d29b489bdbb595675e0f262846c5 9a958d8c28e0c318c2f2b1b30d488a60ed71037042116dd7c611a31354408413 3035c469007ca93eb0262b5c28976ca49e7de17fa8a492332db3a2af5bf9fc28 1bc73452679334d20e4352c917838124127a91b07b674fe6516bc77062eb81a5 feb0e7d610b25dff1d6a460d214495a8aba01ea00815b941d8192eb569e2161c 5209de3e3d1bd88a1759c26620b738c3ac984563ca41f7b60dee8beadfaa8f92 e06a5094f5e85670b24775c87b0265a8037dd648c5bfc66120aa67105535b38a 64dca825b241429079d864c64d4bcd0c741d76ead7fdaec855e51622c8f15e88 7ca475c4b34a119838060600bd71d25de551cc0253fba03be551637e04e56000 f3bb7a74d3fad2381f60f4ddda32c3c6a955bb96d9edf09f2d404dbcf762c79c b2db02e137f705694643416c4cb98b280900d1f350e647ad4996e7dd97149cf6 ea6bcb1386 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 25 /pi put dup 26 /rho put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 86 /V put dup 88 /X put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k 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7ef558582b5257029bd49d616892a2530fa183ff184ec1a0fe9b8157739a 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 38 /ampersand put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf 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0fb9f3f766f6bf233b78d62223fc93db630130e35911647a3eb2c5ce2396a6ef c1b0c41afc2a360e73a4845027b914e3c1285c1c2053ab5f527409edec365e39 82920e197e13e1427627b6c0d624e5d7992ee0c976c3cd7feb877a0abe8f66ba 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 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 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{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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5e474a458624653a3e5db5c2e6e842e0d210f02b 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 76 /L put dup 81 /Q put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd45416d f3459365434123a76ccf883a4973bab19807e1f0c4bab03a45f7ed69ff2660ff 3963e4def11001eec4238c368a39d874fd30b3c14f4186ec7700fbce22abe468 eb32baf7b0164f8f21a9b5e706b91411faaa44467cb2180ab03de375c1ab93e9 76e11ab92fdd4a7280a6d1a8dc65c4c89d04c8ed6988045a2a3de9a4c7b1efd3 75b6d05a48a30c24d13b30ca0bd0790ff38c7bf80a753bb77af8cc8581fefbc5 9a66cd02b21afc75f114fda3797b288355236025ab0238d8dc343616e6fae99a 2445990bffd5daaca7a6ccc36f067dec1eb92c0b9feb7cbca20a8b3905579732 5bb230f3a1182192757f86c309820879eda5244737e57afccb1a3fe2084c25e1 35ef5d376d04fdaf74dbc81a12ede3c84f9be0ed73b49787f153223c4c1e6ae8 bf3450e74927a70a15e4f7184b49f04359d0aaad6285939196959f5e0a309147 86ea6bfadbbb433eb734c5c718163ef4e2fa5de4f948146dbf0bf10d0168a3dd 41771f59e95dda53027257d1d7eacb8689e9980f7bcb011ea04ec97c98841b36 e26c185c18f6a786884dbf6f9947b545b6f761656a9d8dffb7abd74331f1f0bf 00ff000a07405e3c21ebdd3bb2b8689c82ffcde7f1ccfa938113f1919da7a981 2c2101b3a6de6b269096ed33ad459a52091c8454a116d9b2de2f5ea1e7850495 c0e91b6363423cd086e23188d6ff228409713c6834281116986cab6507e5316e 5d6e94a818dc9df9bfca636c4d7128eee88993737ac890802a461b2da4184763 95c0c015e5d0a1321d54034f68f18701ab12c0f5cf72dd6cf2fcb97fa2366fa3 9bf065ce5bfaa98fa152d46feffad7ee12a44362a16a4aee3f059287bfeeb601 094b015beaf216e7356830652e22c52cc0a7c9042b1458ca1201a360699dc3fe 489510eecb607190775520a4dd12f1a6f975e2681b9dc6dbc037792a66b84d0d cbaaf7cd795cabf649e5ef47d5950a228e06c3ac5cfb5430dd86e3ddf27a204c 3fe4beeee57fdc 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 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y 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 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 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6196363d73b578a285959516e066665be856d9e9f343c56cff27513ff726ebbf 097678aa2723de0a5a027e6d06f00444c0d89e738ebe6099370bcb5332509251 e79ef187d9a8fe40c1d3bb86ef2124d9705f891e8e7d6dbab614ec0311f56442 1db65b8d86704ec659332d5a7a2d4df76ce5dd35862af903aaa1037a49787df0 a57957d6b0801cf315bfab7864487e58b2d9cd927d15a3c60092152fb1c9ea6b 751fe9e87c0e77d16467bfcb5032993e5942bc3b4a4cd648eb45c9fd910b4bdb 5aa2e9d10a0f3d8b2dab8efe29b76569a4e047fd7d63f20e5838da3539598acf d51bb6e6f5dcaf7e21acfba3252ae066d31c5671fe3ca1c8f311e388f4bb7038 bf84f4bed943bcc57134e51c5c1afdf500437313bea90c4da9599c38d1bdd9ef 3a484e04014dd1b8f43163983f61eb6835f23a48c76dbaff319e404dfae24b5e 152ef1706ee91e9842a613a9ba4a5955a1b72ac6e2f3972865443eb5c3d6cfb9 3880b15db959f1cd4739e5ed58fcb0715ff07ec67d668a76f2a3b8a07169b4c7 5a86e09947deef70e83150a1e9d02c04a6e3b959fe0995c7ecdd288b42b57d21 9d28aa38512bea5eb1349aad61be3caeca9b8bb0e5f611f96de1a07f023f0310 fb25830a77ca034ec65a3f36f10fd7380fa6a71407872204736f1f2e782aad68 2851685beaf543f54a26802c585f3e2195dcf4d6c6adb0f39a7cb78999ebda9a f8463c7304bbe5513a76664ffab95dc9fde2a6ced88712ba8bd2f5ed17b051d5 a3570f2d6fd3861d138767746e69649cb0506080003aea2fc154c5ffcb041885 dac40e5026dc9b49dfcc1a8b936078f7f6ed9091e78173e003c23e7f039c87f1 ebca8ddaed2b7302d15bb49b256647713783cf317ac83bd92ff1fa7921ffdffd a2b6a2332779a82a2577915994e6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 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 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 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%%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea a29b69d1b622b1f0cffc58291248759607d91150cb0651120970dc9f743bebef 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b(The)335 b(main)f(p)36 b(oin)-36 b(t)334 b(is)g(that)g(the)g(fundamen)-36 b(tal)334 b(solution)g(of)-880 48450 y(the)342 b(sub-Laplacian)h(on)g(the)f(Heisen)-36 b(b)36 b(erg)343 b(group)f Fj(H)24534 47968 y Fi(n)25504 48450 y Fs(and)g Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)342 b(groups)h(are)g(kno)-36 b(wn)343 b(explicitly)i(\(see)-880 49999 y(Section)405 b(3\))h(but)f(not)g(for)h(general)g(Carnot)g (groups.)569 b(The)406 b(pro)36 b(of)406 b(of)g(our)g(theorem)f(on)h (general)g(Carnot)-880 51549 y(groups)353 b(di\013ers)g(sligh)-36 b(tly)354 b(in)f(some)h(steps)f(from)g(the)g(Heisen)-36 b(b)36 b(erg)353 b(group)g Fj(H)35638 51067 y Fi(n)36618 51549 y Fs(and)g Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)352 b(group)h(cases.)-880 53099 y(The)501 b(metho)36 b(d)501 b(w)-36 b(e)502 b(apply)f(here,)518 b(inspired)501 b(b)-36 b(y)501 b(the)g(w)-36 b(ork)502 b(of)g(Allegretto)g([1)q(],)519 b(can)501 b(b)36 b(e)501 b(applied)g(to)g(the)-880 54649 y(Baouendi-Grushin)431 b(t)-36 b(yp)36 b(e)434 b(v)-36 b(ector)434 b(\014elds)f(in)h(that)f(they)g(do)h(not)f(arise)h(from)g (an)-36 b(y)434 b(Carnot)g(group.)448 56198 y(In)580 b(a)g(forthcoming)h(pap)36 b(er,)616 b(w)-36 b(e)580 b(shall)h(presen)-36 b(t)579 b(some)h(impro)-36 b(v)g(ed)580 b(Hardy-t)-36 b(yp)36 b(e)579 b(inequalities)i(and)-880 57748 y(their)433 b(applications)h(in)g(Carnot)g(groups.)448 60107 y(In)417 b(order)g(to)g(state)g(our)g(theorems)g(on)f(Carnot)i (group)e Fj(G)p Fs(,)421 b(w)-36 b(e)418 b(\014rst)e(recall)i(the)e (basic)i(prop)36 b(erties)416 b(of)-880 61657 y(Carnot)434 b(group)f Fj(G)h Fs(and)f(some)h(w)-36 b(ell)434 b(kno)-36 b(wn)434 b(results)f(that)g(will)i(b)36 b(e)433 b(used)g(in)h(the)f (sequel.)19076 65053 y(2.)651 b Fr(Carnot)501 b(gr)-30 b(oup)448 67377 y Fs(A)301 b(Carnot)h(group)f(\(see)g([2],)329 b([3],)g([13)q(],)f([15)q(],)g([16)q(],)g([23)q(])302 b(and)e([28)q(]\))i(is)f(a)h(connected,)327 b(simply)301 b(connected,)-880 68927 y(nilp)36 b(oten)-36 b(t)319 b(Lie)h(group)f Fj(G)h Fs(whose)g(Lie)g(algebra)h Fp(G)399 b Fs(admits)319 b(a)h(strati\014cation.)541 b(That)320 b(is,)343 b(there)319 b(exist)i(linear)-880 70477 y(subspaces)433 b Fq(V)5825 70676 y Fl(1)6351 70477 y Fq(;)221 b(:::;)g(V)9356 70676 y Fi(k)10361 70477 y Fs(of)434 b Fp(G)513 b Fs(suc)-36 b(h)432 b(that)-880 72835 y(\(2.1\))2101 b Fp(G)448 b Fs(=)369 b Fq(V)7271 73034 y Fl(1)8092 72835 y Fp(\010)295 b Fq(:::)i Fp(\010)e Fq(V)12886 73034 y Fi(k)13455 72835 y Fq(;)1522 b Fs([)p Fq(V)16457 73034 y Fl(1)16984 72835 y Fq(;)221 b(V)18324 73034 y Fi(i)18700 72835 y Fs(])370 b(=)e Fq(V)21569 73034 y Fi(i)p Fl(+1)23148 72835 y Fq(;)1522 b Fs(for)1301 b Fq(i)369 b Fs(=)f(1)p Fq(;)221 b Fs(2)p Fq(;)g(:::;)g(k)344 b Fp(\000)296 b Fs(1)1301 b(and)f([)p Fq(V)43022 73034 y Fl(1)43548 72835 y Fq(;)221 b(V)44888 73034 y Fi(k)45458 72835 y Fs(])369 b(=)g(0)p eop %%Page: 3 3 3 2 bop -880 -714 a Fg(SHARP)443 b(WEIGHTED)f(HARD)-28 b(Y-TYPE)442 b(INEQUALITIES)f(AND)i(UNCER)-85 b(T)g(AINTY)440 b(PRINCIPLE)h(INEQUALITY)h(ON)h(CARNOT)g(GR)-28 b(OUPS)-512 b(3)-880 1942 y Fs(where)592 b([)p Fq(V)4155 2141 y Fl(1)4681 1942 y Fq(;)221 b(V)6021 2141 y Fi(i)6398 1942 y Fs(])592 b(is)g(the)f(subspace)g(of)h Fp(G)671 b Fs(generated)591 b(b)-36 b(y)592 b(the)f(elemen)-36 b(ts)592 b([)p Fq(X)32 b(;)221 b(Y)290 b Fs(])592 b(with)g Fq(X)743 b Fp(2)638 b Fq(V)47107 2141 y Fl(1)48224 1942 y Fs(and)-880 3492 y Fq(Y)658 b Fp(2)368 b Fq(V)2545 3691 y Fi(i)2921 3492 y Fs(.)579 b(This)433 b(de\014nes)g(a)h Fq(k)45 b Fs(-step)432 b(Carnot)i(group)f(and)h(in)-36 b(teger)433 b Fq(k)414 b Fp(\025)369 b Fs(1)434 b(is)g(called)g(the)f(step)g(of)i Fj(G)p Fs(.)448 5042 y(Via)407 b(the)e(exp)36 b(onen)-36 b(tial)406 b(map,)411 b(it)406 b(is)g(p)36 b(ossible)406 b(to)f(induce)g(on)h Fj(G)f Fs(a)h(family)i(of)e(automorphisms)f(of)h (the)-880 6592 y(group,)434 b(called)g(dilations,)g Fq(\016)13071 6791 y Fi(\025)14044 6592 y Fs(:)370 b Fj(R)15734 6109 y Fi(n)16729 6592 y Fp(\000)-221 b(!)369 b Fj(R)20197 6109 y Fi(n)20824 6592 y Fs(\()p Fq(\025)g(>)f Fs(0\))434 b(suc)-36 b(h)433 b(that)15432 8847 y Fq(\016)16008 9046 y Fi(\025)16612 8847 y Fs(\()p Fq(x)17857 9046 y Fl(1)18383 8847 y Fq(;)221 b(:::;)g(x)21369 9046 y Fi(n)21997 8847 y Fs(\))369 b(=)g(\()p Fq(\025)25518 8298 y Fi(\013)26122 8421 y Fh(1)26638 8847 y Fq(x)27377 9046 y Fl(1)27903 8847 y Fq(;)221 b(:::;)g(\025)30909 8298 y Fi(\013)31513 8409 y Fm(n)32137 8847 y Fq(x)32876 9046 y Fi(n)33503 8847 y Fs(\))-880 11102 y(where)433 b(1)370 b(=)e Fq(\013)6104 11301 y Fl(1)6999 11102 y Fs(=)h Fq(:::)h Fs(=)f Fq(\013)12041 11301 y Fi(m)13297 11102 y Fq(<)g(\013)15505 11301 y Fi(m)p Fl(+1)17964 11102 y Fp(\024)g Fq(:::)h Fp(\024)f Fq(\013)23048 11301 y Fi(n)24109 11102 y Fs(are)433 b(in)-36 b(tegers)434 b(and)f Fq(m)369 b Fs(=)f(dim\()p Fq(V)39937 11301 y Fl(1)40463 11102 y Fs(\).)448 12652 y(The)434 b(group)f(la)-36 b(w)435 b(can)e(b)36 b(e)433 b(written)h(in)f(the)g (follo)-36 b(wing)436 b(form)-880 14907 y(\(2.2\))12984 b Fq(x)296 b Fp(\001)f Fq(y)417 b Fs(=)368 b Fq(x)296 b Fs(+)e Fq(y)343 b Fs(+)295 b Fq(P)181 b Fs(\()p Fq(x;)221 b(y)48 b Fs(\))p Fq(;)1523 b(x;)221 b(y)417 b Fp(2)369 b Fj(R)34036 14359 y Fi(n)-880 17163 y Fs(where)446 b Fq(P)572 b Fs(:)391 b Fj(R)6011 16681 y Fi(n)6942 17163 y Fp(\002)304 b Fj(R)9238 16681 y Fi(n)10255 17163 y Fp(\000)-221 b(!)391 b Fj(R)13745 16681 y Fi(n)14818 17163 y Fs(has)446 b(p)36 b(olynomial)448 b(comp)36 b(onen)-36 b(ts)446 b(and)f Fq(P)34436 17362 y Fl(1)35353 17163 y Fs(=)391 b Fq(:::)g Fs(=)g Fq(P)40471 17362 y Fi(m)41749 17163 y Fs(=)g(0.)617 b(Note)446 b(that)-880 18713 y(the)433 b(in)-36 b(v)g(erse)434 b Fq(x)6406 18231 y Fk(\000)p Fl(1)8097 18713 y Fs(of)g(an)g(elemen)-36 b(t)433 b Fq(x)369 b Fp(2)g Fj(G)433 b Fs(has)h(the)f(form)h Fq(x)28389 18231 y Fk(\000)p Fl(1)30016 18713 y Fs(=)368 b Fp(\000)p Fq(x)i Fs(=)e(\()p Fp(\000)p Fq(x)37196 18912 y Fl(1)37722 18713 y Fq(;)221 b(:::;)g Fp(\000)p Fq(x)41741 18912 y Fi(n)42370 18713 y Fs(\).)448 20262 y(Let)377 b Fq(X)3801 20461 y Fl(1)4327 20262 y Fq(;)221 b(:::;)g(X)7653 20461 y Fi(m)8921 20262 y Fs(b)36 b(e)377 b(a)g(family)i(of)f(left)g(in)-36 b(v)-72 b(arian)-36 b(t)377 b(v)-36 b(ector)377 b(\014elds)g(whic)-36 b(h)377 b(form)g(an)g(orthonormal)h(basis)-880 21812 y(of)441 b Fq(V)1366 22011 y Fl(1)2274 21812 y Fp(\021)381 b Fj(R)4647 21330 y Fi(m)5976 21812 y Fs(at)441 b(the)f(origin,)j(that) e(is,)i Fq(X)19458 22011 y Fl(1)19984 21812 y Fs(\(0\))381 b(=)g Fq(@)24107 22011 y Fi(x)24637 22134 y Fh(1)25153 21812 y Fq(;)221 b(:::;)g(X)28479 22011 y Fi(m)29369 21812 y Fs(\(0\))381 b(=)g Fq(@)33492 22011 y Fi(x)34022 22122 y Fm(m)34859 21812 y Fs(.)599 b(The)441 b(v)-36 b(ector)441 b(\014elds)g Fq(X)46827 22011 y Fi(j)47755 21812 y Fs(ha)-36 b(v)g(e)-880 23362 y(p)36 b(olynomial)435 b(co)36 b(e\016cien)-36 b(ts)434 b(and)f(can)h(b)36 b(e)433 b(assumed)g(to)h(b)36 b(e)433 b(of)h(the)f(form)9504 26611 y Fq(X)10583 26810 y Fi(j)11071 26611 y Fs(\()p Fq(x)p Fs(\))368 b(=)h Fq(@)15258 26810 y Fi(j)16040 26611 y Fs(+)18404 24951 y Fi(n)17730 25349 y Fo(X)17347 28149 y Fi(i)p Fl(=)p Fi(j)51 b Fl(+1)20254 26611 y Fq(a)20937 26810 y Fi(ij)21744 26611 y Fs(\()p Fq(x)p Fs(\))p Fq(@)24182 26810 y Fi(i)24557 26611 y Fq(;)1522 b(X)27519 26810 y Fi(j)28007 26611 y Fs(\(0\))369 b(=)f Fq(@)32105 26810 y Fi(j)32592 26611 y Fq(;)221 b(j)444 b Fs(=)369 b(1)p Fq(;)221 b(:::;)g(m;)-880 30192 y Fs(where)340 b(eac)-36 b(h)341 b(p)36 b(olynomial)342 b Fq(a)12928 30391 y Fi(ij)14075 30192 y Fs(is)f(homogeneous)g(with)f(resp)36 b(ect)341 b(to)f(the)g(dilations)h(of)h(the)d(group,)360 b(that)340 b(is)-880 31742 y Fq(a)-197 31941 y Fi(ij)610 31742 y Fs(\()p Fq(\016)1692 31941 y Fi(\025)2296 31742 y Fs(\()p Fq(x)p Fs(\)\))369 b(=)f Fq(\025)7061 31260 y Fi(\013)7665 31395 y Fm(i)8017 31260 y Fk(\000)p Fi(\013)9353 31395 y Fm(j)9842 31742 y Fq(a)10525 31941 y Fi(ij)11332 31742 y Fs(\()p Fq(x)p Fs(\).)577 b(The)429 b(horizon)-36 b(tal)430 b(gradien)-36 b(t)429 b(on)h(Carnot)g(group)f Fj(G)h 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Fq(;)22255 62391 y Fs(0)4700 b(if)1301 b Fq(x)369 b Fs(=)g(0)p Fq(:)-880 64841 y Fs(W)-108 b(e)338 b(recall)g(that)f(a)h(homogeneous)g (norm)f(on)h Fj(G)g Fs(is)g(a)g(con)-36 b(tin)g(uous)337 b(function)g Fq(N)507 b Fs(:)369 b Fj(G)h Fp(\000)-222 b(!)370 b Fs([0)p Fq(;)221 b Fp(1)p Fs(\))339 b(smo)36 b(oth)-880 66390 y(a)-36 b(w)g(a)g(y)480 b(from)f(the)f(origin)h(whic) -36 b(h)478 b(satis\014es)h(the)f(conditions)h(:)668 b Fq(N)139 b Fs(\()p Fq(\016)31994 66589 y Fi(\025)32597 66390 y Fs(\()p Fq(x)p Fs(\)\))445 b(=)h Fq(\025N)139 b Fs(\()p Fq(x)p Fs(\),)488 b Fq(N)139 b Fs(\()p Fq(x)43723 65908 y Fk(\000)p Fl(1)44980 66390 y Fs(\))446 b(=)f Fq(N)139 b Fs(\()p Fq(x)p Fs(\))-880 67940 y(and)433 b Fq(N)139 b Fs(\()p Fq(x)p Fs(\))368 b(=)h(0)434 b(i\013)f Fq(x)369 b Fs(=)g(0.)448 69490 y(The)501 b(curv)-36 b(e)500 b Fq(\015)554 b Fs(:)483 b([)p Fq(a;)221 b(b)p Fs(])484 b Fp(\032)f Fj(R)g Fp(\000)-221 b(!)482 b Fj(G)501 b Fs(is)g(called)g(horizon)-36 b(tal)500 b(if)h(its)g(tangen)-36 b(ts)499 b(lie)i(in)g Fq(V)42830 69689 y Fl(1)43356 69490 y Fs(,)517 b(i.e,)h Fq(\015)47160 69008 y Fk(0)47470 69490 y Fs(\()p Fq(t)p Fs(\))482 b Fp(2)-880 71040 y Ff(sp)-66 b(an)99 b Fp(f)p Fq(X)3486 71239 y Fl(1)4013 71040 y Fq(;)221 b(:::;)g(X)7339 71239 y Fi(m)8229 71040 y Fp(g)446 b Fs(for)g(all)h Fq(t)p Fs(.)615 b(Then,)448 b(the)e(Carnot-Car)-36 b(\023)-614 b(ethedory)445 b(distance)h Fq(d)38280 71239 y Fi(C)63 b(C)39801 71040 y Fs(\()p Fq(x;)221 b(y)48 b Fs(\))446 b(b)36 b(et)-36 b(w)g(een)446 b(t)-36 b(w)g(o)-880 72817 y(p)36 b(oin)-36 b(ts)510 b Fq(x;)221 b(y)549 b Fp(2)499 b Fj(G)511 b Fs(is)f(de\014ned)f(to)i(b) 36 b(e)510 b(the)g(in\014m)-36 b(um)509 b(of)i(all)h(horizon)-36 b(tal)511 b(lengths)39671 71747 y Fo(R)40556 72097 y Fi(b)40298 73278 y(a)41014 72817 y Fp(h)p Fq(\015)42278 72335 y Fk(0)42588 72817 y Fs(\()p Fq(t)p Fs(\))p Fq(;)221 b(\015)45399 72335 y Fk(0)45710 72817 y Fs(\()p Fq(t)p Fs(\))p Fp(i)47709 72335 y Fl(1)p Fi(=)p Fl(2)49174 72817 y Fq(dt)p eop %%Page: 4 4 4 3 bop -880 -714 a Fg(4)20957 b(ISMAIL)341 b(K)-28 b(OMBE)-880 1942 y Fs(o)-36 b(v)g(er)509 b(all)h(horizon)-36 b(tal)509 b(curv)-36 b(es)508 b Fq(\015)569 b Fs(:)497 b([)p Fq(a;)221 b(b)p Fs(])498 b Fp(\000)-221 b(!)497 b Fj(G)508 b Fs(suc)-36 b(h)508 b(that)g Fq(\015)72 b Fs(\()p Fq(a)p Fs(\))497 b(=)f Fq(x)509 b Fs(and)f Fq(\015)72 b Fs(\()p Fq(y)48 b Fs(\))496 b(=)g Fq(b)p Fs(.)804 b(Notice)509 b(that)-880 3492 y Fq(d)-204 3691 y Fi(C)63 b(C)1751 3492 y Fs(is)434 b(a)g(homogeneous)g(norm)f(and)g(satis\014es)h(the)f(in)-36 b(v)-72 b(ariance)434 b(prop)36 b(ert)-36 b(y)12454 6982 y Fq(d)13130 7181 y Fi(C)63 b(C)14652 6982 y Fs(\()p Fq(z)354 b Fp(\001)295 b Fq(x;)221 b(z)356 b Fp(\001)295 b Fq(y)48 b Fs(\))369 b(=)f Fq(d)23338 7181 y Fi(C)63 b(C)24860 6982 y Fs(\()p Fq(x;)221 b(y)48 b Fs(\))p Fq(;)1523 b Fp(8)221 b Fq(x;)g(y)48 b(;)221 b(z)430 b Fp(2)368 b Fj(G)p Fq(;)-880 9194 y Fs(and)433 b(is)h(homogeneous)g(of)g(degree)g (one)f(with)h(resp)36 b(ect)433 b(to)h(the)f(dilation)h Fq(\016)35414 9393 y Fi(\025)36018 9194 y Fs(,)g(i.e.)9155 11676 y Fq(d)9831 11875 y Fi(C)63 b(C)11353 11676 y Fs(\()p Fq(\016)12435 11875 y Fi(\025)13039 11676 y Fs(\()p Fq(x)p Fs(\))p Fq(;)221 b(\016)15948 11875 y Fi(\025)16553 11676 y Fs(\()p Fq(y)48 b Fs(\)\))368 b(=)h Fq(\025d)21937 11875 y Fi(C)63 b(C)23458 11676 y Fs(\()p Fq(x;)221 b(y)48 b Fs(\))p Fq(;)1523 b Fp(8)221 b Fq(x;)g(y)48 b(;)221 b(z)430 b Fp(2)368 b Fj(G)p Fq(;)221 b Fp(8)g Fq(\025)370 b(>)f Fs(0)p Fq(:)448 14159 y Fs(The)355 b(Carnot-Careth)-36 b(\023)-614 b(edory)354 b(balls)h(are)g(de\014ned)e(b)-36 b(y)354 b Fq(B)67 b Fs(\()p Fq(x;)221 b(R)11 b Fs(\))370 b(=)f Fp(f)p Fq(y)417 b Fp(2)368 b Fj(G)p Fp(j)p Fq(d)37217 14358 y Fi(C)63 b(C)38739 14159 y Fs(\()p Fq(x;)221 b(y)48 b Fs(\))369 b Fq(<)g(R)11 b Fp(g)p Fs(.)552 b(By)355 b(left-)-880 15709 y(translation)420 b(and)g(dilation,)j(it)d(is)h (easy)g(to)f(see)g(that)f(the)h(Haar)g(measure)g(of)h Fq(B)67 b Fs(\()p Fq(x;)221 b(R)11 b Fs(\))420 b(is)h(prop)36 b(ortional)-880 17258 y(b)-36 b(y)433 b Fq(R)1927 16776 y Fi(Q)2723 17258 y Fs(.)578 b(More)434 b(precisely)13878 19741 y Fp(j)p Fq(B)67 b Fs(\()p Fq(x;)221 b(R)11 b Fs(\))p Fp(j)369 b Fs(=)g Fq(R)21756 19193 y Fi(Q)22551 19741 y Fp(j)p Fq(B)67 b Fs(\()p Fq(x;)221 b Fs(1\))p Fp(j)370 b Fs(=)f Fq(R)30079 19193 y Fi(Q)30874 19741 y Fp(j)p Fq(B)67 b Fs(\(0)p Fq(;)221 b Fs(1\))p Fp(j)-880 22224 y Fs(where)19741 25133 y Fq(Q)369 b Fs(=)23224 23473 y Fi(k)22521 23871 y Fo(X)22664 26671 y Fi(j)51 b Fl(=1)24661 25133 y Fq(j)75 b Fs(\(dim)p Fq(V)28706 25332 y Fi(j)29193 25133 y Fs(\))-880 28670 y(is)434 b(the)f(homogeneous)h(dimension)f(of) h Fj(G)p Fs(.)448 30220 y(It)328 b(is)f(w)-36 b(ell)328 b(kno)-36 b(wn)327 b(that)g(Sob)36 b(olev)328 b(inequalities)g(are)g (imp)36 b(ortan)-36 b(t)326 b(in)h(the)g(study)g(of)h(partial)f (di\013eren)-36 b(tial)-880 31770 y(equations,)350 b(esp)36 b(ecially)330 b(in)f(the)f(study)g(of)h(those)f(arising)i(from)f (geometry)g(and)f(ph)-36 b(ysics.)543 b(The)329 b(follo)-36 b(wing)-880 33320 y(Sob)36 b(olev)434 b(inequalit)-36 b(y)435 b(holds)e(on)h Fj(G)-880 36663 y Fs(\(2.4\))11206 35188 y Fo(\020)12221 34855 y(Z)12959 37862 y Fn(G)13924 36663 y Fp(jr)15400 36862 y Fn(G)16144 36663 y Fq(\036)p Fs(\()p Fq(x)p Fs(\))p Fp(j)19034 36114 y Fl(2)19560 36663 y Fq(dx)20975 35188 y Fo(\021)21768 35486 y Fl(1)p Fi(=)p Fl(2)23604 36663 y Fp(\025)369 b Fq(C)26032 35188 y Fo(\020)27047 34855 y(Z)27785 37862 y Fn(G)28750 36663 y Fp(j)p Fq(\036)p Fs(\()p Fq(x)p Fs(\))p Fp(j)32461 35633 y Fh(2)p Fm(Q)p 32142 35842 1688 40 v 32142 36391 a(Q)p Fe(\000)p Fh(2)34017 36663 y Fq(dx)35432 35188 y Fo(\021)36358 35075 y Fm(Q)p Fe(\000)p Fh(2)p 36358 35284 V 36678 35833 a(2)p Fm(Q)-880 39902 y Fs(\()444 b(see)g([13)q(],)k([30)q(])c(\).)610 b(It)444 b(is)h(a)f(more)g (di\016cult)g(problem)g(to)g(determine)f(the)h(sharp)g(constan)-36 b(t)443 b Fq(C)539 b Fs(in)444 b(\(2.4\))-880 41452 y(on)500 b(general)g(Carnot)h(groups.)777 b(The)500 b(only)h(results)f(that)f (ha)-36 b(v)g(e)501 b(so)f(far)h(b)36 b(een)499 b(pro)-36 b(v)g(en)500 b(are)g(in)g(the)f(case)-880 43001 y(of)528 b(Heisen)-36 b(b)36 b(erg)528 b(group)f Fj(H)12220 42519 y Fi(n)13374 43001 y Fs(b)-36 b(y)528 b(Jerison)f(and)g(Lee)h([21)q(])g (and)f(Iw)-36 b(asa)g(w)g(a-t)g(yp)36 b(e)528 b(groups)g(\()f(a)h (particular)-880 44551 y(sub-class)543 b(of)i Ff(H)o Fs(-t)-36 b(yp)36 b(e)544 b(groups\))f(b)-36 b(y)544 b(Garofalo)h(and)e(V)-108 b(assilev)545 b([18)q(].)910 b(W)-108 b(e)544 b(should)f(men)-36 b(tion)543 b(that)g(the)-880 46101 y(sharp)433 b(constan)-36 b(ts)433 b(in)g([21)q(])h(and)f([18)q (])h(lead)g(us)f(to)h(obtain)f(explicit)i(constan)-36 b(t)433 b(in)g(Corollary)j(4.2.)5882 49851 y(3.)651 b Fr(Hard)-30 b(y-type)502 b(inequalities)h(on)e(Carnot)h(gr)-30 b(oups)502 b(of)h(step)e(2)448 52176 y Fs(Among)431 b(Carnot)g(groups)g (of)g(step)f(t)-36 b(w)g(o,)432 b(the)f(Heisen)-36 b(b)36 b(erg)430 b(group)h(and)f(Heisen)-36 b(b)36 b(erg)431 b(t)-36 b(yp)36 b(e)430 b(\()p Ff(H)209 b Fs(-t)-36 b(yp)36 b(e\))-880 53726 y(groups)419 b(are)g(of)h(particular)f (signi\014cance.)573 b(These)420 b(groups)e(app)36 b(ear)419 b(naturally)h(in)f(analysis,)k(geometry)-108 b(,)-880 55275 y(represen)-36 b(tation)383 b(theory)g(and)g(mathematical)h(ph) -36 b(ysics.)562 b(In)383 b(this)g(section,)394 b(w)-36 b(e)384 b(\014rst)f(pro)-36 b(v)g(e)383 b(Hardy-t)-36 b(yp)36 b(e)-880 56825 y(inequalities)435 b(on)e(the)g(Heisen)-36 b(b)36 b(erg)434 b(group)f(and)g(w)-36 b(e)434 b(extend)f(this)g (result)g(to)h(the)f Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)433 b(group.)-880 58375 y Fy(Heisen)-42 b(b)42 b(erg)408 b(group)p Fs(.)552 b(The)354 b(Heisen)-36 b(b)36 b(erg)353 b(group)g Fj(H)25210 57893 y Fi(n)26190 58375 y Fs(is)g(an)h(example)g (of)g(a)g(noncomm)-36 b(utativ)g(e)353 b(Carnot)-880 59925 y(group.)760 b(Denoting)495 b(p)36 b(oin)-36 b(ts)493 b(in)h Fj(H)15752 59443 y Fi(n)16873 59925 y Fs(b)-36 b(y)494 b(\()p Fq(z)59 b(;)221 b(t)p Fs(\))495 b(with)f Fq(z)532 b Fs(=)471 b(\()p Fq(z)28714 60124 y Fl(1)29240 59925 y Fq(;)221 b(:::;)g(z)32091 60124 y Fi(n)32720 59925 y Fs(\))472 b Fp(2)g Fj(C)36015 59443 y Fi(n)37136 59925 y Fs(and)493 b Fq(t)472 b Fp(2)g Fj(R)495 b Fs(w)-36 b(e)494 b(ha)-36 b(v)g(e)494 b(the)-880 61474 y(group)433 b(la)-36 b(w)435 b(giv)-36 b(en)434 b(as)11525 64951 y(\()p Fq(z)59 b(;)221 b(t)p Fs(\))296 b Fp(\016)f Fs(\()p Fq(z)16676 64402 y Fk(0)16987 64951 y Fq(;)221 b(t)18039 64402 y Fk(0)18350 64951 y Fs(\))369 b(=)f(\()p Fq(z)355 b Fs(+)295 b Fq(z)24040 64402 y Fk(0)24351 64951 y Fq(;)221 b(t)295 b Fs(+)g Fq(t)27475 64402 y Fk(0)28080 64951 y Fs(+)g(2)30933 63290 y Fi(n)30259 63689 y Fo(X)30401 66488 y Fi(j)51 b Fl(=1)32399 64951 y Fq(I)104 b(m)p Fs(\()p Fq(z)35325 65150 y Fi(j)35891 64951 y Fs(\026)-729 b Fq(z)36475 64402 y Fk(0)36416 65279 y Fi(j)36903 64951 y Fs(\)\))-880 68759 y(With)434 b(the)f(notation)g Fq(z)10526 68958 y Fi(j)11382 68759 y Fs(=)369 b Fq(x)13502 68958 y Fi(j)14284 68759 y Fs(+)295 b Fq(iy)16669 68958 y Fi(j)17156 68759 y Fs(,)434 b(the)f(horizon)-36 b(tal)434 b(space)f Fq(V)30530 68958 y Fl(1)31490 68759 y Fs(is)h(spanned)e(b)-36 b(y)434 b(the)f(basis)11908 71995 y Fq(X)12987 72194 y Fi(j)13844 71995 y Fs(=)15970 71097 y Fq(@)p 15357 71690 1985 54 v 15357 72907 a(@)72 b(x)16855 73106 y Fi(j)17770 71995 y Fs(+)295 b(2)p Fq(y)20361 72194 y Fi(j)21216 71097 y Fq(@)p 20981 71690 1229 54 v 20981 72907 a(@)72 b(t)23643 71995 y Fs(and)1300 b Fq(Y)27794 72194 y Fi(j)28650 71995 y Fs(=)30724 71097 y Fq(@)p 30163 71690 1880 54 v 30163 72907 a(@)72 b(y)31556 73106 y Fi(j)32471 71995 y Fp(\000)296 b Fs(2)p Fq(x)35189 72194 y Fi(j)36044 71097 y Fq(@)p 35809 71690 1229 54 v 35809 72907 a(@)72 b(t)37170 71995 y(:)p eop %%Page: 5 5 5 4 bop -880 -714 a Fg(SHARP)443 b(WEIGHTED)f(HARD)-28 b(Y-TYPE)442 b(INEQUALITIES)f(AND)i(UNCER)-85 b(T)g(AINTY)440 b(PRINCIPLE)h(INEQUALITY)h(ON)h(CARNOT)g(GR)-28 b(OUPS)-512 b(5)-880 1973 y Fs(The)397 b(one)g(dimensional)h(cen)-36 b(ter)396 b Fq(V)15817 2172 y Fl(2)16740 1973 y Fs(is)i(spanned)e(b)-36 b(y)397 b(the)f(v)-36 b(ector)398 b(\014eld)f Fq(T)549 b Fs(=)36657 1450 y Fi(@)p 36487 1668 888 54 v 36487 2431 a(@)52 b(t)37507 1973 y Fs(.)567 b(W)-108 b(e)397 b(ha)-36 b(v)g(e)397 b(the)g(comm)-36 b(u-)-880 3538 y(tator)449 b(relations)h([)p Fq(X)9125 3737 y Fi(j)9613 3538 y Fq(;)221 b(Y)10950 3737 y Fi(j)11437 3538 y Fs(])396 b(=)f Fp(\000)p Fs(4)p Fq(T)181 b Fs(,)453 b(and)c(all)h(other)f(brac) -36 b(k)g(ets)449 b(of)h Fp(f)p Fq(X)33177 3737 y Fl(1)33703 3538 y Fq(;)221 b(Y)35040 3737 y Fl(1)35566 3538 y Fq(;)g(:::;)g(X) 38892 3737 y Fi(n)39521 3538 y Fq(;)g(Y)40858 3737 y Fi(n)41485 3538 y Fp(g)449 b Fs(are)h(zero.)625 b(The)-880 5088 y(sub-elliptic)433 b(gradien)-36 b(t)433 b(is)h(the)f(2)p Fq(n)i Fs(dimensional)e(v)-36 b(ector)434 b(\014eld)g(giv)-36 b(en)434 b(b)-36 b(y)16604 7340 y Fp(r)17711 7539 y Fn(H)18400 7287 y Fm(n)19390 7340 y Fs(=)369 b(\()p Fq(X)22356 7539 y Fl(1)22882 7340 y Fq(;)221 b(:::;)g(X)26208 7539 y Fi(n)26837 7340 y Fq(;)g(Y)28174 7539 y Fl(1)28700 7340 y Fq(;)g(:::;)g(Y)31702 7539 y Fi(n)32330 7340 y Fs(\))-880 9591 y(and)433 b(the)g(Kohn)g(Laplacian)h(on)g Fj(H)16212 9109 y Fi(n)17272 9591 y Fs(is)g(the)f(op)36 b(erator)18561 12837 y(\001)19645 13036 y Fn(H)20334 12784 y Fm(n)21325 12837 y Fs(=)23380 11177 y Fi(n)22706 11575 y Fo(X)22848 14374 y Fi(j)51 b Fl(=1)24624 12837 y Fs(\()p Fq(X)26313 12289 y Fl(2)26209 13165 y Fi(j)27135 12837 y Fs(+)295 b Fq(Y)29486 12289 y Fl(2)29197 13165 y Fi(j)30011 12837 y Fs(\))p Fq(:)-880 16415 y Fs(A)434 b(homogeneous)f(norm)g(on)h Fj(H)14720 15933 y Fi(n)15780 16415 y Fs(is)g(giv)-36 b(en)434 b(b)-36 b(y)17401 18698 y Fq(\032)369 b Fs(=)f Fp(j)p Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))p Fp(j)370 b Fs(=)e(\()p Fp(j)p Fq(z)59 b Fp(j)26943 18150 y Fl(4)27764 18698 y Fs(+)295 b Fq(t)29541 18150 y Fl(2)30067 18698 y Fs(\))30573 18150 y Fl(1)p Fi(=)p Fl(4)-880 20950 y Fs(and)433 b(the)g(homogeneous)h(dimension)f(of)h Fj(H)20508 20468 y Fi(n)21568 20950 y Fs(is)g Fq(Q)369 b Fs(=)g(2)p Fq(n)296 b Fs(+)f(2.)448 22500 y(A)505 b(remark)-72 b(able)506 b(analogy)h(b)36 b(et)-36 b(w)g(een)504 b(Kohn)h(Laplacian)h(and)f(the) f(classical)j(Laplace)f(op)36 b(erator)505 b(has)-880 24049 y(b)36 b(een)402 b(obtained)g(b)-36 b(y)403 b(F)-108 b(olland)402 b([12)q(].)568 b(He)403 b(found)f(that)g(the)g(fundamen) -36 b(tal)402 b(solution)h(of)h Fp(\000)p Fs(\001)43327 24248 y Fn(H)44016 23996 y Fm(n)45040 24049 y Fs(with)f(p)36 b(ole)-880 25599 y(at)434 b(zero)g(is)f(giv)-36 b(en)435 b(b)-36 b(y)8173 28715 y(\011\()p Fq(z)59 b(;)221 b(t)p Fs(\))369 b(=)15815 27816 y Fq(c)16375 28015 y Fi(Q)p 13794 28409 5396 54 v 13794 29626 a Fq(\032)p Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))17192 29242 y Fi(Q)p Fk(\000)p Fl(2)20624 28715 y Fs(where)1300 b Fq(c)25808 28914 y Fi(Q)26972 28715 y Fs(=)28486 27816 y(2)29136 27334 y Fl(\()p Fi(Q)p Fk(\000)p Fl(2\))p Fi(=)p Fl(2)32807 27816 y Fs(\000\(\()p Fq(Q)294 b Fp(\000)i Fs(2\))p Fq(=)p Fs(4\))40247 27334 y Fl(2)p 28486 28409 12288 54 v 33369 29626 a Fq(\031)34155 29242 y Fi(Q=)p Fl(2)40906 28715 y Fq(:)448 31693 y Fs(W)-108 b(e)364 b(no)-36 b(w)363 b(pro)-36 b(v)g(e)363 b(the)g(follo)-36 b(wing)366 b(theorem)c(on)i (the)e(Heisen)-36 b(b)36 b(erg)364 b(group)e Fj(H)36372 31211 y Fi(n)36999 31693 y Fs(.)555 b(In)363 b(the)g(v)-72 b(arious)364 b(in)-36 b(tegral)-880 33243 y(inequalities)535 b(b)36 b(elo)-36 b(w)535 b(\(Section)f(3)h(and)f(Section)g(4\),)560 b(w)-36 b(e)534 b(allo)-36 b(w)536 b(the)e(v)-72 b(alues)535 b(of)g(the)e(in)-36 b(tegrals)535 b(on)f(the)-880 34793 y(left-hand)605 b(sides)i(to)f(b)36 b(e)606 b(+)p Fp(1)p Fs(.)1096 b(Before)607 b(w)-36 b(e)607 b(pro)36 b(ceed,)649 b(w)-36 b(e)606 b(should)g(emphasize)g(that)g(the)f(constan)-36 b(t)-880 36351 y Fq(C)95 b Fs(\()p Fq(Q;)221 b(\013)8 b Fs(\))427 b(=)e(\()6108 35767 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 6108 36045 3279 54 v 7512 36809 a(2)9519 36351 y Fs(\))10025 35869 y Fl(2)11018 36351 y Fs(obtained)466 b(in)h(Section)f(3)h(and)g(Section)f(4)i(is)f(sharp)f(in)h(the)f(sense) h(that)f(if)i(it)f(is)-880 37901 y(replaced)433 b(b)-36 b(y)434 b(an)f(grater)h(n)-36 b(um)g(b)36 b(er)432 b(the)h(inequalit) -36 b(y)435 b(fails.)-880 40316 y Fy(Theorem)499 b(3.1.)554 b Ff(L)-66 b(et)464 b Fq(\013)378 b Fp(2)368 b Fj(R)466 b Ff(and)f Fq(\036)368 b Fp(2)h Fq(C)20402 39834 y Fk(1)20307 40645 y Fl(0)21398 40316 y Fs(\()p Fj(H)22937 39834 y Fi(n)23859 40316 y Fp(n)295 b(f)p Fs(0)p Fp(g)p Fs(\))p Ff(.)598 b(Then)464 b(we)h(have)g(:)9609 41604 y Fo(Z)10347 44612 y Fn(H)11036 44360 y Fm(n)11879 43413 y Fq(\032)12550 42864 y Fi(\013)13209 43413 y Fp(jr)14685 43612 y Fn(H)15374 43360 y Fm(n)15996 43413 y Fq(\036)p Fp(j)17135 42864 y Fl(2)17660 43413 y Fq(dz)59 b(dt)369 b Fp(\025)21916 41938 y Fo(\020)22842 42514 y Fq(Q)296 b Fs(+)f Fq(\013)304 b Fp(\000)295 b Fs(2)p 22842 43107 5742 54 v 25388 44324 a(2)28717 41938 y Fo(\021)29510 42235 y Fl(2)30258 41604 y Fo(Z)30996 44612 y Fn(H)31685 44360 y Fm(n)32528 43413 y Fq(\032)33199 42864 y Fi(\013)33991 42514 y Fp(j)p Fq(z)59 b Fp(j)35392 42032 y Fl(2)p 33991 43107 1928 54 v 34356 44324 a Fq(\032)35027 43940 y Fl(4)36051 43413 y Fq(\036)36821 42864 y Fl(2)37347 43413 y Fq(dz)g(dt)-880 46613 y Ff(wher)-66 b(e)452 b Fq(\032)369 b Fs(=)g(\()p Fp(j)p Fq(z)59 b Fp(j)7105 46131 y Fl(4)7898 46613 y Fs(+)267 b Fq(l)9593 46131 y Fl(2)10119 46613 y Fs(\))10625 46131 y Fl(1)p Fi(=)p Fl(4)12544 46613 y Ff(is)453 b(the)f(homo)-66 b(gene)g(ous)452 b(norm)f(on)i Fj(H)30203 46131 y Fi(n)30829 46613 y Ff(.)593 b(Mor)-66 b(e)g(over,)453 b(the)f(c)-66 b(onstant)451 b Fs(\()45877 46030 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 45877 46308 3279 54 v 47281 47072 a(2)49288 46613 y Fs(\))49794 46131 y Fl(2)-880 48163 y Ff(is)465 b(sharp)h(pr)-66 b(ovide)g(d)463 b Fq(Q)296 b Fs(+)e Fq(\013)304 b Fp(\000)296 b Fs(2)369 b Fq(>)g Fs(0)p Ff(.)-880 50579 y(Pr)-66 b(o)g(of.)552 b Fs(Let)435 b Fq(\036)373 b Fs(=)f Fq(\032)8625 50097 y Fi(\014)9254 50579 y Fq( )483 b Fs(where)435 b Fq(\014)447 b Fp(2)372 b Fj(R)297 b Fp(n)g(f)p Fs(0)p Fp(g)436 b Fs(and)f Fq( )420 b Fp(2)372 b Fq(C)27493 50097 y Fk(1)27398 50907 y Fl(0)28489 50579 y Fs(\()p Fj(H)30028 50097 y Fi(n)30951 50579 y Fp(n)297 b(f)p Fs(0)p Fp(g)p Fs(\).)585 b(A)436 b(direct)f(calculation) i(sho)-36 b(ws)-880 52129 y(that)-880 54380 y(\(3.1\))2442 b Fq(\032)4906 53832 y Fi(\013)5566 54380 y Fp(jr)7042 54579 y Fn(H)7731 54328 y Fm(n)8352 54380 y Fq(\036)p Fp(j)9491 53832 y Fl(2)10386 54380 y Fs(=)368 b Fq(\014)12574 53832 y Fl(2)13100 54380 y Fq(\032)13771 53832 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(2)17409 54380 y Fp(jr)18885 54579 y Fn(H)19574 54328 y Fm(n)20195 54380 y Fq(\032)p Fp(j)21235 53832 y Fl(2)21761 54380 y Fq( )22655 53832 y Fl(2)23475 54380 y Fs(+)295 b(2)p Fq(\014)74 b(\032)26911 53832 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(1)30549 54380 y Fq( )i Fp(r)32550 54579 y Fn(H)33239 54328 y Fm(n)33860 54380 y Fq(\032)295 b Fp(\001)g(r)36597 54579 y Fn(H)37286 54328 y Fm(n)37908 54380 y Fq( )342 b Fs(+)295 b Fq(\032)41074 53832 y Fi(\013)p Fl(+2)p Fi(\014)43509 54380 y Fp(jr)44985 54579 y Fn(H)45729 54380 y Fq( )48 b Fp(j)46992 53832 y Fl(2)47517 54380 y Fq(:)-880 56632 y Fs(It)434 b(is)f(easy)i(to)f(see)f(that)20573 59112 y Fp(jr)22049 59311 y Fn(H)22738 59059 y Fm(n)23359 59112 y Fq(\032)p Fp(j)24399 58564 y Fl(2)25294 59112 y Fs(=)26807 58213 y Fp(j)p Fq(z)59 b Fp(j)28208 57731 y Fl(2)p 26807 58807 1928 54 v 27173 60023 a Fq(\032)27844 59640 y Fl(2)-880 61819 y Fs(and)433 b(in)-36 b(tegrating)434 b(\(3.1\))g(o)-36 b(v)g(er)434 b Fj(H)15183 61337 y Fi(n)15809 61819 y Fs(,)g(w)-36 b(e)434 b(get)-880 66552 y(\(3.2\))2965 62964 y Fo(Z)3703 65971 y Fn(H)4392 65719 y Fm(n)5235 64772 y Fq(\032)5906 64224 y Fi(\013)6565 64772 y Fp(jr)8041 64971 y Fn(H)8730 64719 y Fm(n)9352 64772 y Fq(\036)p Fp(j)10491 64224 y Fl(2)11017 64772 y Fq(dz)59 b(dt)368 b Fs(=)15251 62964 y Fo(Z)15989 65971 y Fn(H)16678 65719 y Fm(n)17521 64772 y Fq(\014)18329 64224 y Fl(2)18855 64772 y Fq(\032)19526 64224 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(4)23163 64772 y Fp(j)p Fq(z)59 b Fp(j)24564 64224 y Fl(2)25091 64772 y Fq( )25985 64224 y Fl(2)26510 64772 y Fq(dz)g(dt)295 b Fs(+)30596 62964 y Fo(Z)31334 65971 y Fn(H)32023 65719 y Fm(n)32866 64772 y Fs(2)p Fq(\014)74 b(\032)34995 64224 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(1)38633 64772 y Fq( )i Fp(r)40634 64971 y Fn(H)41323 64719 y Fm(n)41944 64772 y Fq(\032)295 b Fp(\001)g(r)44681 64971 y Fn(H)45370 64719 y Fm(n)45992 64772 y Fq( )48 b(dz)59 b(dt)13797 68277 y Fs(+)15103 66469 y Fo(Z)15841 69477 y Fn(H)16530 69225 y Fm(n)17373 68277 y Fq(\032)18044 67729 y Fi(\013)p Fl(+2)p Fi(\014)20480 68277 y Fp(jr)21956 68476 y Fn(H)22645 68225 y Fm(n)23266 68277 y Fq( )48 b Fp(j)24529 67729 y Fl(2)25054 68277 y Fq(dz)59 b(dt)-880 71286 y Fs(Applying)540 b(in)-36 b(tegration)539 b(b)-36 b(y)539 b(parts)g(to)h(the)f(middle)g(in)-36 b(tegral)540 b(on)f(the)g(righ)-36 b(t-hand)537 b(side)j(of)g(\(3.2\),) 567 b(w)-36 b(e)-880 72835 y(obtain)p eop %%Page: 6 6 6 5 bop -880 -714 a Fg(6)20957 b(ISMAIL)341 b(K)-28 b(OMBE)-880 6321 y Fs(\(3.3\))3806 2739 y Fo(Z)4544 5746 y Fn(H)5233 5494 y Fm(n)6076 4547 y Fq(\032)6747 3999 y Fi(\013)7407 4547 y Fp(jr)8883 4746 y Fn(H)9572 4494 y Fm(n)10193 4547 y Fq(\036)p Fp(j)11332 3999 y Fl(2)11858 4547 y Fq(dz)59 b(dt)369 b Fs(=)16092 2739 y Fo(Z)16830 5746 y Fn(H)17519 5494 y Fm(n)18362 4547 y Fq(\014)19170 3999 y Fl(2)19696 4547 y Fq(\032)20367 3999 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(4)24005 4547 y Fp(j)p Fq(z)59 b Fp(j)25406 3999 y Fl(2)25932 4547 y Fq( )26826 3999 y Fl(2)27351 4547 y Fq(dz)g(dt)295 b Fp(\000)33136 3649 y Fq(\014)p 31592 4242 3896 54 v 31592 5458 a(\013)304 b Fs(+)295 b(2)p Fq(\014)35842 2739 y Fo(Z)36580 5746 y Fn(H)37269 5494 y Fm(n)38112 4547 y Fs(\001)39196 4746 y Fn(H)39885 4494 y Fm(n)40507 4547 y Fs(\()p Fq(\032)41684 3999 y Fi(\013)p Fl(+2)p Fi(\014)44119 4547 y Fs(\))p Fq( )45519 3999 y Fl(2)46044 4547 y Fq(dz)59 b(dt)14638 8053 y Fs(+)15945 6245 y Fo(Z)16683 9252 y Fn(H)17372 9000 y Fm(n)18215 8053 y Fq(\032)18886 7504 y Fi(\013)p Fl(+2)p Fi(\014)21321 8053 y Fp(jr)22797 8252 y Fn(H)23486 8000 y Fm(n)24108 8053 y Fq( )48 b Fp(j)25371 7504 y Fl(2)25896 8053 y Fq(dz)59 b(dt:)-880 10958 y Fs(One)433 b(can)g(sho)-36 b(w)434 b(that)-880 13181 y(\(3.4\))8137 b(\001)11014 13380 y Fn(H)11703 13128 y Fm(n)12324 13181 y Fs(\()p Fq(\032)13501 12632 y Fi(\013)p Fl(+2)p Fi(\014)15936 13181 y Fs(\))369 b(=)g Fp(j)p Fq(z)59 b Fp(j)19593 12632 y Fl(2)20119 13181 y Fq(\032)20790 12632 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(4)24428 13181 y Fs(\()p Fq(\013)303 b Fs(+)295 b(2)p Fq(\014)74 b Fs(\)\()p Fq(\013)304 b Fs(+)295 b(2)p Fq(\014)369 b Fs(+)295 b Fq(Q)g Fp(\000)h Fs(2\))p Fq(:)-880 15404 y Fs(Substituting)432 b(\(3.4\))i(in)-36 b(to)434 b(\(3.3\))g(giv)-36 b(es)434 b(the)f(follo)-36 b(wing)-659 17737 y Fo(Z)79 20745 y Fn(H)768 20492 y Fm(n)1611 19545 y Fq(\032)2282 18997 y Fi(\013)2942 19545 y Fp(jr)4418 19744 y Fn(H)5107 19492 y Fm(n)5729 19545 y Fq(\036)p Fp(j)6868 18997 y Fl(2)7393 19545 y Fq(dz)59 b(dt)369 b Fs(=)g(\()p Fq(\014)12942 18997 y Fl(2)13762 19545 y Fp(\000)296 b Fq(\014)74 b Fs(\()p Fq(\013)303 b Fs(+)295 b(2)p Fq(\014)370 b Fs(+)294 b Fq(Q)i Fp(\000)f Fs(2\)\))26439 17737 y Fo(Z)27177 20745 y Fn(H)27866 20492 y Fm(n)28709 19545 y Fq(\032)29380 18997 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(4)33018 19545 y Fp(j)p Fq(z)59 b Fp(j)34419 18997 y Fl(2)34945 19545 y Fq( )35839 18997 y Fl(2)36364 19545 y Fq(dz)g(dt)295 b Fs(+)40451 17737 y Fo(Z)41189 20745 y Fn(H)41878 20492 y Fm(n)42721 19545 y Fq(\032)43392 18997 y Fi(\013)p Fl(+2)p Fi(\014)45827 19545 y Fp(jr)47303 19744 y Fn(H)47992 19492 y Fm(n)48614 19545 y Fq( )48 b Fp(j)49877 18997 y Fl(2)50402 19545 y Fq(dz)59 b(dt)10247 23051 y Fp(\025)369 b Fs(\()p Fp(\000)p Fq(\014)13996 22502 y Fl(2)14817 23051 y Fp(\000)295 b Fq(\014)74 b Fs(\()p Fq(\013)304 b Fs(+)295 b Fq(Q)g Fp(\000)g Fs(2\)\))24433 21243 y Fo(Z)25172 24250 y Fn(H)25861 23998 y Fm(n)26704 23051 y Fq(\032)27375 22502 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(4)31012 23051 y Fp(j)p Fq(z)59 b Fp(j)32413 22502 y Fl(2)32940 23051 y Fq( )33834 22502 y Fl(2)34359 23051 y Fq(dz)g(dt:)-880 26223 y Fs(Note)491 b(that)f(the)f(function)i Fq(\014)539 b Fp(\000)-221 b(!)466 b(\000)p Fq(\014)18363 25741 y Fl(2)19223 26223 y Fp(\000)334 b Fq(\014)74 b Fs(\()p Fq(\013)342 b Fs(+)334 b Fq(Q)g Fp(\000)g Fs(2\))490 b(attains)h(the)f(maxim)-36 b(um)490 b(for)h Fq(\014)540 b Fs(=)46547 25639 y Fl(2)p Fk(\000)p Fi(\013)p Fk(\000)p Fi(Q)p 46547 25917 3279 54 v 47951 26681 a Fl(2)49959 26223 y Fs(,)-880 27981 y(and)433 b(this)g(maxim)-36 b(um)434 b(is)g(equal)g(to)g(\()17224 27397 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 17224 27675 V 18628 28439 a(2)20635 27981 y Fs(\))21141 27499 y Fl(2)21667 27981 y Fs(.)578 b(Therefore)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)436 b(inequalit)-36 b(y)-880 31174 y(\(3.5\))9428 29366 y Fo(Z)10166 32373 y Fn(H)10855 32121 y Fm(n)11698 31174 y Fq(\032)12369 30626 y Fi(\013)13028 31174 y Fp(jr)14504 31373 y Fn(H)15193 31121 y Fm(n)15815 31174 y Fq(\036)p Fp(j)16954 30626 y Fl(2)17480 31174 y Fq(dz)59 b(dt)368 b Fp(\025)21736 29700 y Fo(\020)22662 30275 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 22662 30869 5742 54 v 25207 32085 a(2)28536 29700 y Fo(\021)29330 29997 y Fl(2)30077 29366 y Fo(Z)30815 32373 y Fn(H)31504 32121 y Fm(n)32347 31174 y Fq(\032)33018 30626 y Fi(\013)33810 30275 y Fp(j)p Fq(z)59 b Fp(j)35211 29793 y Fl(2)p 33810 30869 1928 54 v 34176 32085 a Fq(\032)34847 31701 y Fl(4)35870 31174 y Fq(\036)36640 30626 y Fl(2)37166 31174 y Fq(dz)g(dt:)448 34346 y Fs(It)464 b(only)h(remains)f(to)h(sho)-36 b(w)464 b(that)g(the)f(constan)-36 b(t)464 b(\()25560 33762 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 25560 34041 3279 54 v 26963 34804 a(2)28971 34346 y Fs(\))29477 33864 y Fl(2)30466 34346 y Fs(is)h(sharp.)669 b(Let)464 b Fq(\036)39081 34545 y Fi(\017)39516 34346 y Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))465 b(b)36 b(e)464 b(the)f(family)-880 35896 y(of)434 b(functions)g(de\014ned)e(b)-36 b(y)-880 39263 y(\(3.6\))11318 b Fq(\036)13881 39462 y Fi(\017)14316 39263 y Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))370 b(=)18793 36991 y Fo(\()19863 38346 y Fs(1)8362 b(if)1302 b Fq(\032)368 b Fp(2)h Fs([0)p Fq(;)221 b Fs(1])p Fq(;)19863 40259 y(\032)20534 39768 y Fk(\000)p Fl(\()21765 39358 y Fm(Q)p Fh(+)p Fm(\013)p Fe(\000)p Fh(2)p 21765 39567 2843 40 v 22983 40116 a(2)24740 39768 y Fl(+)p Fi(\017)p Fl(\))28875 40259 y Fs(if)1302 b Fq(\032)368 b(>)h Fs(1)p Fq(;)-880 42853 y Fs(where)433 b Fq(\017)370 b(>)e Fs(0)434 b(and)f Fq(\032)369 b Fs(=)g Fp(j)p Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))p Fp(j)369 b Fs(=)g(\()p Fp(j)p Fq(z)59 b Fp(j)18308 42371 y Fl(4)19129 42853 y Fs(+)295 b Fq(t)20906 42371 y Fl(2)21431 42853 y Fs(\))21937 42371 y Fl(1)p Fi(=)p Fl(4)23404 42853 y Fs(.)578 b(It)434 b(follo)-36 b(ws)435 b(that)10248 45775 y Fp(j)p Fq(\032)11288 45227 y Fi(\013)11947 45775 y Fp(r)13054 45974 y Fn(H)13743 45722 y Fm(n)14365 45775 y Fq(\036)15135 45974 y Fi(\017)15570 45775 y Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))p Fp(j)18666 45227 y Fl(2)19561 45775 y Fs(=)20942 44301 y Fo(\020)21868 44877 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)296 b Fs(2)p 21868 45470 5742 54 v 24414 46687 a(2)28038 45775 y(+)f Fq(\017)29870 44301 y Fo(\021)30663 44598 y Fl(2)31189 45775 y Fp(j)p Fq(z)59 b Fp(j)32590 45227 y Fl(2)33116 45775 y Fq(\032)33787 45227 y Fk(\000)p Fl(\()p Fi(Q)p Fl(+2+2)p Fi(\017)p Fl(\))38831 45775 y Fq(:)-880 48467 y Fs(In)530 b(the)f(sequel)h(w)-36 b(e)530 b(indicate)g Fq(B)15156 48666 y Fl(1)16215 48467 y Fs(=)i Fp(f)p Fs(\()p Fq(z)59 b(;)221 b(t)p Fs(\))534 b(:)f Fq(\032)f Fp(\024)h Fs(1)p Fp(g)e Fq(\032)p Fs(-ball)e(cen)-36 b(tered)529 b(at)h(the)f(origin)i(in)f Fj(H)46635 47985 y Fi(n)47791 48467 y Fs(with)-880 50016 y(radius)433 b(1.)448 51566 y(By)h(direct)g(computation)f(w)-36 b(e)434 b(get)-880 58304 y(\(3.7\))4160 52752 y Fo(Z)4898 55759 y Fn(H)5587 55507 y Fm(n)6430 54560 y Fq(\032)7101 54011 y Fi(\013)7894 53661 y Fp(j)p Fq(z)59 b Fp(j)9295 53179 y Fl(2)p 7894 54254 1928 54 v 8259 55471 a Fq(\032)8930 55087 y Fl(4)9954 54560 y Fq(\036)10724 54011 y Fl(2)10724 54888 y Fi(\017)11249 54560 y Fq(dz)g(dt)369 b Fs(=)15484 52752 y Fo(Z)16221 55759 y Fi(B)16931 55882 y Fh(1)17669 54560 y Fq(\032)18340 54011 y Fi(\013)p Fk(\000)p Fl(4)20202 54560 y Fp(j)p Fq(z)59 b Fp(j)21603 54011 y Fl(2)22129 54560 y Fq(\036)22899 54011 y Fl(2)22899 54888 y Fi(\017)23425 54560 y Fq(dz)g(dt)295 b Fs(+)27512 52752 y Fo(Z)28250 55759 y Fn(H)28939 55507 y Fm(n)29505 55759 y Fk(n)p Fi(B)30685 55882 y Fh(1)31423 54560 y Fq(\032)32094 54011 y Fi(\013)p Fk(\000)p Fl(4)33956 54560 y Fp(j)p Fq(z)59 b Fp(j)35357 54011 y Fl(2)35883 54560 y Fq(\036)36653 54011 y Fl(2)36653 54888 y Fi(\017)37179 54560 y Fq(dz)g(dt)14103 58286 y Fs(=)15484 56478 y Fo(Z)16221 59486 y Fi(B)16931 59609 y Fh(1)17669 58286 y Fq(\032)18340 57738 y Fi(\013)p Fk(\000)p Fl(4)20202 58286 y Fp(j)p Fq(z)g Fp(j)21603 57738 y Fl(2)22129 58286 y Fq(dz)g(dt)295 b Fs(+)26216 56478 y Fo(Z)26954 59486 y Fn(H)27643 59233 y Fm(n)28209 59486 y Fk(n)p Fi(B)29389 59609 y Fh(1)30127 58286 y Fp(j)p Fq(z)59 b Fp(j)31528 57738 y Fl(2)32055 58286 y Fq(\032)32726 57738 y Fk(\000)p Fl(\()p Fi(Q)p Fl(+2+2)p Fi(\017)p Fl(\))37769 58286 y Fq(dz)g(dt)14103 62013 y Fs(=)15484 60205 y Fo(Z)16221 63212 y Fi(B)16931 63335 y Fh(1)17669 62013 y Fq(\032)18340 61465 y Fi(\013)p Fk(\000)p Fl(4)20202 62013 y Fp(j)p Fq(z)g Fp(j)21603 61465 y Fl(2)22129 62013 y Fq(dz)g(dt)295 b Fs(+)g(\()26855 61115 y Fq(Q)g Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 26855 61708 5742 54 v 29400 62924 a(2)33024 62013 y(+)g Fq(\017)p Fs(\))35362 61465 y Fk(\000)p Fl(2)36841 60205 y Fo(Z)37579 63212 y Fn(H)38268 62960 y Fm(n)39111 62013 y Fq(\032)39782 61465 y Fi(\013)40442 62013 y Fp(jr)41918 62212 y Fn(H)42607 61960 y Fm(n)43228 62013 y Fq(\036)43998 62212 y Fi(\017)44434 62013 y Fp(j)44803 61465 y Fl(2)45329 62013 y Fq(dz)59 b(dt:)-880 65115 y Fs(Since)413 b Fq(Q)255 b Fs(+)f Fq(\013)263 b Fp(\000)254 b Fs(2)370 b Fq(>)e Fs(0)414 b(then)f(the)g(\014rst)f(in)-36 b(tegral)414 b(on)g(the)f(righ)-36 b(t)413 b(hand)g(side)g(of)i(\(3.7\))f(is)g(in) -36 b(tegrable)413 b(and)-880 66665 y(w)-36 b(e)434 b(conclude)f(b)-36 b(y)434 b Fq(\017)369 b Fp(\000)-222 b(!)370 b Fs(0.)49287 68215 y Fd(\003)448 71262 y Fs(The)451 b(follo)-36 b(wing)453 b(theorem)e(sho)-36 b(ws)451 b(that)f(the)h(w)-36 b(eigh)g(t)451 b(function)g Fq(\032)32489 70780 y Fi(\013)33599 71262 y Fs(has)g(a)g(signi\014can)-36 b(t)451 b(e\013ect)g(on)f(the)-880 72820 y(sharp)433 b(constan)-36 b(t)433 b(\()8553 72236 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 8553 72515 3279 54 v 9957 73278 a(2)11964 72820 y Fs(\))12470 72338 y Fl(2)13429 72820 y Fs(whereas)h(the)f(new)h(w)-36 b(eigh)g(t)433 b(function)h Fp(jr)33953 73019 y Fn(H)34642 72767 y Fm(n)35263 72820 y Fq(\032)p Fp(j)36303 72338 y Fi(\015)37329 72820 y Fs(has)f(no)h(e\013ect.)p eop %%Page: 7 7 7 6 bop -880 -714 a Fg(SHARP)443 b(WEIGHTED)f(HARD)-28 b(Y-TYPE)442 b(INEQUALITIES)f(AND)i(UNCER)-85 b(T)g(AINTY)440 b(PRINCIPLE)h(INEQUALITY)h(ON)h(CARNOT)g(GR)-28 b(OUPS)-512 b(7)-880 1942 y Fy(Theorem)499 b(3.2.)554 b Ff(L)-66 b(et)464 b Fq(\013)8 b(;)221 b(\015)443 b Fp(2)368 b Fj(R)465 b Ff(and)g Fq(\036)369 b Fp(2)g Fq(C)21732 1460 y Fk(1)21637 2271 y Fl(0)22728 1942 y Fs(\()p Fj(H)24267 1460 y Fi(n)25188 1942 y Fp(n)296 b(f)p Fs(0)p Fp(g)p Fs(\))p Ff(.)597 b(Then)465 b(we)g(have)g(:)5372 3088 y Fo(Z)6109 6095 y Fn(H)6798 5843 y Fm(n)7642 4896 y Fq(\032)8313 4348 y Fi(\013)8972 4896 y Fp(jr)10448 5095 y Fn(H)11137 4843 y Fm(n)11759 4896 y Fq(\032)p Fp(j)12799 4348 y Fi(\015)13391 4896 y Fp(jr)14867 5095 y Fn(H)15556 4843 y Fm(n)16177 4896 y Fq(\036)p Fp(j)17316 4348 y Fl(2)17842 4896 y Fq(dz)59 b(dt)369 b Fp(\025)g Fs(\()22737 3998 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 22737 4591 5742 54 v 25282 5807 a(2)28611 4896 y(\))29117 4348 y Fl(2)29864 3088 y Fo(Z)30602 6095 y Fn(H)31291 5843 y Fm(n)32134 4896 y Fq(\032)32805 4348 y Fi(\013)p Fk(\000)p Fl(2)34667 4896 y Fp(jr)36143 5095 y Fn(H)36832 4843 y Fm(n)37454 4896 y Fq(\032)p Fp(j)38494 4348 y Fi(\015)52 b Fl(+2)40288 4896 y Fq(\036)41058 4348 y Fl(2)41584 4896 y Fq(dz)59 b(dt)-880 8098 y Ff(wher)-66 b(e)452 b Fq(\032)369 b Fs(=)g(\()p Fp(j)p Fq(z)59 b Fp(j)7105 7616 y Fl(4)7898 8098 y Fs(+)267 b Fq(l)9593 7616 y Fl(2)10119 8098 y Fs(\))10625 7616 y Fl(1)p Fi(=)p Fl(4)12544 8098 y Ff(is)453 b(the)f(homo)-66 b(gene)g(ous)452 b(norm)f(on)i Fj(H)30203 7616 y Fi(n)30829 8098 y Ff(.)593 b(Mor)-66 b(e)g(over,)453 b(the)f(c)-66 b(onstant)451 b Fs(\()45877 7514 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 45877 7793 3279 54 v 47281 8556 a(2)49288 8098 y Fs(\))49794 7616 y Fl(2)-880 9648 y Ff(is)465 b(sharp)h(pr)-66 b(ovide)g(d)463 b Fq(Q)296 b Fs(+)e Fq(\013)304 b Fp(\000)296 b Fs(2)369 b Fq(>)g Fs(0)p Ff(.)-880 12064 y(Pr)-66 b(o)g(of.)552 b Fs(The)434 b(pro)36 b(of)434 b(is)g(similar)g(to)g(the)f(pro)36 b(of)434 b(of)h(Theorem)e(3.1.)580 b(W)-108 b(e)433 b(only)h(need)f(to) h(note)f(that)17391 14317 y Fp(r)18498 14516 y Fn(H)19187 14264 y Fm(n)19809 14317 y Fq(\032)295 b Fp(\001)g(r)22546 14516 y Fn(H)23235 14264 y Fm(n)23857 14317 y Fs(\()p Fp(jr)25839 14516 y Fn(H)26528 14264 y Fm(n)27149 14317 y Fq(\032)p Fp(j)28189 13769 y Fi(\015)28781 14317 y Fs(\))369 b(=)g(0)p Fq(:)49287 16570 y Fd(\003)-880 18981 y Fy(Heisen)-42 b(b)42 b(erg-t)-42 b(yp)42 b(e)659 b(group)p Fs(.)995 b(Another)571 b(imp)36 b(ortan)-36 b(t)572 b(mo)36 b(del)572 b(of)h(Carnot)f(groups)g(are)g(the)g Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)-880 20530 y(\(Heisen)-36 b(b)36 b(erg)371 b(t)-36 b(yp)36 b(e\))371 b(groups)g(whic)-36 b(h)371 b(w)-36 b(ere)372 b(in)-36 b(tro)36 b(duced)370 b(b)-36 b(y)371 b(Kaplan)h([22)q(])g(as)f(direct)h(generalizations)g (of)-880 22080 y(the)409 b(Heisen)-36 b(b)36 b(erg)410 b(group)g Fj(H)12626 21598 y Fi(n)13252 22080 y Fs(.)571 b(An)409 b Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)409 b(group)h(is)g(a)g (Carnot)g(group)g(with)g(a)g(t)-36 b(w)g(o-step)409 b(Lie)h(algebra) -880 23630 y Fp(G)448 b Fs(=)369 b Fq(V)2497 23829 y Fl(1)3318 23630 y Fp(\010)295 b Fq(V)5404 23829 y Fl(2)6364 23630 y Fs(and)433 b(an)g(inner)g(pro)36 b(duct)433 b Fp(h)p Fq(;)221 b Fp(i)433 b Fs(in)h Fp(G)512 b Fs(suc)-36 b(h)433 b(that)g(the)g(linear)h(map)19681 25883 y Fq(J)491 b Fs(:)370 b Fq(V)22380 26082 y Fl(2)23275 25883 y Fp(\000)-221 b(!)369 b Fs(End)o Fq(V)28872 26082 y Fl(1)29398 25883 y Fq(;)-880 28135 y Fs(de\014ned)432 b(b)-36 b(y)434 b(the)f(condition)13226 30388 y Fp(h)p Fq(J)14462 30587 y Fi(z)14992 30388 y Fs(\()p Fq(u)p Fs(\))p Fq(;)221 b(v)48 b Fp(i)368 b Fs(=)h Fp(h)p Fq(z)59 b(;)221 b Fs([)p Fq(u;)g(v)48 b Fs(])p Fp(i)p Fq(;)1523 b(u;)221 b(v)418 b Fp(2)368 b Fq(V)31534 30587 y Fl(1)32061 30388 y Fq(;)221 b(z)429 b Fp(2)368 b Fq(V)35688 30587 y Fl(2)-880 32641 y Fs(satis\014es)20619 34351 y Fq(J)21461 33803 y Fl(2)21338 34680 y 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Fs(:)369 b Fj(G)g Fp(\000)-221 b(!)369 b Fq(V)28225 47576 y Fl(1)29185 47377 y Fs(and)433 b Fq(z)c Fs(:)369 b Fj(G)g Fp(\000)-221 b(!)369 b Fq(V)38146 47576 y Fl(2)39106 47377 y Fs(b)-36 b(y)18755 49630 y Fq(x)369 b Fs(=)f(exp\()p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))295 b(+)f Fq(z)59 b Fs(\()p Fq(x)p Fs(\)\))-880 51882 y(for)500 b(ev)-36 b(ery)500 b Fq(x)482 b Fp(2)e Fj(G)p Fs(.)777 b(In)499 b([22)q(])h(Kaplan)f(pro) -36 b(v)g(ed)499 b(that)g(there)g(exists)i(a)e(constan)-36 b(t)499 b Fq(c)481 b(>)g Fs(0)500 b(suc)-36 b(h)498 b(that)h(the)-880 53432 y(function)15287 55745 y(\010\()p Fq(x)p Fs(\))369 b(=)g Fq(c)20287 54270 y Fo(\020)21080 55745 y Fp(j)p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))p Fp(j)24246 55196 y Fl(4)25066 55745 y Fs(+)295 b(16)p Fp(j)p Fq(z)59 b Fs(\()p Fq(x)p Fs(\))p Fp(j)30825 55196 y Fl(2)31351 54270 y Fo(\021)32277 54157 y Fh(2)p Fe(\000)p Fm(Q)p 32277 54366 1688 40 v 32918 54915 a Fh(4)-880 58073 y Fs(is)434 b(a)g(fundamen)-36 b(tal)433 b(solution)h(for)g(the)f(op)36 b(erator)434 b Fp(\000)p Fs(\001)25531 58272 y Fn(G)26275 58073 y Fs(.)578 b(W)-108 b(e)434 b(note)f(that)-880 61112 y(\(3.9\))14283 b Fq(K)95 b Fs(\()p Fq(x)p Fs(\))369 b(=)20779 59638 y Fo(\020)21572 61112 y Fp(j)p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))p Fp(j)24738 60564 y Fl(4)25558 61112 y Fs(+)295 b(16)p Fp(j)p Fq(z)59 b Fs(\()p Fq(x)p Fs(\))p Fp(j)31317 60564 y Fl(2)31843 59638 y Fo(\021)32770 59573 y Fh(1)p 32770 59734 406 40 v 32770 60283 a(4)-880 63712 y Fs(de\014nes)475 b(a)i(homogeneous)f(norm)g(and)g Fq(Q)441 b Fs(=)h Fq(m)324 b Fs(+)f(2)p Fq(k)522 b Fs(is)477 b(the)e(homogeneous)i(dimension)f(of) h Fj(G)f Fs(where)-880 65262 y Fq(m)448 b Fs(=dim)p Fq(V)4644 65461 y Fl(1)5650 65262 y Fs(and)479 b Fq(k)494 b Fs(=dim)p Fq(V)13333 65461 y Fl(2)13858 65262 y Fs(.)718 b(This)481 b(result)f(generalized)g(F)-108 b(olland's)481 b(fundamen)-36 b(tal)479 b(solution)i(for)f(the)-880 66812 y(Heisen)-36 b(b)36 b(erg)478 b(group)h Fj(H)10547 66329 y Fi(n)11652 66812 y Fs([12)q(].)713 b(It)479 b(is)g(remark)-72 b(able)479 b(that)f(the)g(homogeneous)h(norm)f Fq(K)95 b Fs(\()p Fq(x)p Fs(\))478 b(is)h(in)-36 b(v)g(olv)g(ed)-880 68361 y(also)395 b(in)f(the)g(expression)h(of)g(the)f(fundamen)-36 b(tal)393 b(solution)i(of)g(the)f(follo)-36 b(wing)396 b Fq(p)p Fp(\000)p Fs(sub-Laplace)e(op)36 b(erator)-880 71608 y(\(3.10\))10524 b Fp(L)13883 71807 y Fi(p)14412 71608 y Fq(u)369 b Fs(=)17445 69947 y Fi(m)16902 70346 y Fo(X)17100 73145 y Fi(i)p Fl(=1)19042 71608 y Fq(X)20121 71807 y Fi(i)20497 71608 y Fs(\()p Fp(j)p Fq(X)104 b(u)p Fp(j)23664 71059 y Fi(p)p Fk(\000)p Fl(2)25396 71608 y Fq(X)26475 71807 y Fi(i)26852 71608 y Fq(u)p Fs(\))p Fq(;)1522 b Fs(1)369 b Fq(<)g(p)g(<)f Fp(1)p Fq(:)p eop %%Page: 8 8 8 7 bop -880 -714 a Fg(8)20957 b(ISMAIL)341 b(K)-28 b(OMBE)-880 1942 y Fs(More)436 b(precisely)-108 b(,)438 b(Cap)36 b(ogna,)438 b(Danielli)g(and)e(Garofalo)i([9)q(])e(pro)-36 b(v)g(ed)436 b(that)g(for)h(ev)-36 b(ery)437 b(1)374 b Fq(<)g(p)f(<)h Fp(1)437 b Fs(there)-880 3492 y(exists)d 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b(e)433 b(a)h(radial)g(function,)g(i.e.,)h Fq(u)p Fs(\()p Fq(x)p Fs(\))369 b(=)f Fq(f)142 b Fs(\()p Fq(K)95 b Fs(\()p Fq(x)p Fs(\)\))433 b(where)h Fq(f)511 b Fp(2)368 b Fq(C)95 b Fs(\()p Fj(R)p Fs(\))434 b(then)18349 17549 y Fp(jr)19825 17748 y Fn(G)20569 17549 y Fq(u)p Fp(j)21678 17000 y Fl(2)22573 17549 y Fs(=)24086 16650 y Fp(j)p Fq(v)48 b Fp(j)25501 16168 y Fl(2)p 24086 17243 1941 54 v 24192 18460 a Fq(K)25394 18076 y Fl(2)26159 17549 y Fp(j)p Fq(f)27311 17000 y Fk(0)27622 17549 y Fs(\()p Fq(K)95 b Fs(\))p Fp(j)30205 17000 y Fl(2)30730 17549 y Fq(:)-880 20123 y Fs(Moreo)-36 b(v)g(er)434 b(if)g Fq(u)p Fs(\()p Fq(x)p Fs(\))369 b(=)g Fq(f)142 b Fs(\()p Fq(K)95 b Fs(\()p Fq(x)p Fs(\)\))433 b(and)g Fq(f)511 b Fp(2)368 b Fq(C)21403 19641 y Fl(2)21929 20123 y Fs(\()p Fj(R)p Fs(\))434 b(then)-880 24708 y(\(3.12\))12990 23143 y(\001)14074 23342 y Fn(G)14818 23143 y Fq(u)369 b Fs(=)g Fp(jr)18784 23342 y Fn(G)19528 23143 y Fq(K)95 b Fs(\()p Fq(x)p Fs(\))p Fp(j)22850 22595 y Fl(2)23376 21669 y 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b(satis\014es)g(the)f(so-called)h Fp(1)p Fs(-sub-Laplace)e(equation)i(:)14147 33812 y(\001)15231 34011 y Fn(G)p Fi(;)p Fk(1)17177 33812 y Fq(K)464 b Fs(=)20262 32913 y(1)p 20262 33506 651 54 v 20262 34723 a(2)21045 33812 y Fp(hr)22669 34011 y Fn(G)23412 33812 y Fs(\()p Fp(jr)25394 34011 y Fn(G)26138 33812 y Fq(K)95 b Fp(j)27709 33263 y Fl(2)28235 33812 y Fs(\))p Fq(;)221 b Fp(r)30430 34011 y Fn(G)31175 33812 y Fq(K)95 b Fp(i)368 b Fs(=)h(0)-880 36601 y(at)434 b(ev)-36 b(ery)434 b(p)36 b(oin)-36 b(t)433 b Fq(x)369 b Fp(2)g Fj(G)295 b Fp(n)h(f)p Fs(0)p Fp(g)p Fs(.)579 b(\(See)433 b([22)q(])h(and)f([9])h(for)g(further)f (information)h(on)g Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)432 b(groups\))448 38922 y(W)-108 b(e)434 b(no)-36 b(w)434 b(ha)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)436 b(theorem)d(on)h(the) f Ff(H)209 b Fs(-t)-36 b(yp)36 b(e)432 b(group)i(:)-880 41377 y Fy(Theorem)557 b(3.3.)580 b Ff(L)-66 b(et)510 b Fj(G)h Ff(b)-66 b(e)510 b(an)h(H-typ)-66 b(e)510 b(gr)-66 b(oup)511 b(with)g(homo)-66 b(gene)g(ous)510 b(dimension)g Fq(Q)454 b Fs(=)g Fq(m)329 b Fs(+)g(2)p Fq(k)557 b Ff(and)-880 42927 y(let)464 b Fq(\013)8 b(;)221 b(\015)443 b Fp(2)368 b Fj(R)466 b Ff(and)f Fq(\036)368 b Fp(2)h Fq(C)12103 42445 y Fk(1)12008 43256 y Fl(0)13099 42927 y Fs(\()p Fj(G)295 b Fp(n)h(f)p Fs(0)p Fp(g)p Fs(\))p Ff(.)597 b(Then)465 b(the)f(fol)66 b(lowing)465 b(ine)-66 b(quality)463 b(is)i(valid)f(:)-880 45949 y Fs(\(3.13\))6795 44141 y Fo(Z)7533 47148 y Fn(G)8499 45949 y Fq(K)9701 45401 y Fi(\013)10361 45949 y Fp(jr)11837 46148 y Fn(G)12581 45949 y Fq(K)95 b Fp(j)14152 45401 y Fi(\015)14744 45949 y Fp(jr)16220 46148 y Fn(G)16964 45949 y Fq(\036)p Fp(j)18103 45401 y Fl(2)18629 45949 y Fq(dx)368 b Fp(\025)i Fs(\()22454 45051 y Fq(Q)295 b Fs(+)f Fq(\013)304 b Fp(\000)296 b Fs(2)p 22454 45644 5742 54 v 24999 46860 a(2)28328 45949 y(\))28834 45401 y Fl(2)29581 44141 y Fo(Z)30319 47148 y Fn(G)31284 45949 y Fq(K)32486 45401 y Fi(\013)p Fk(\000)p Fl(2)34348 45949 y Fp(jr)35824 46148 y Fn(G)36568 45949 y Fq(K)95 b Fp(j)38139 45401 y Fi(\015)52 b Fl(+2)39934 45949 y Fq(\036)40704 45401 y Fl(2)41229 45949 y Fq(dx)-880 49219 y Ff(wher)-66 b(e)496 b Fq(K)95 b Fs(\()p Fq(x)p Fs(\))427 b(=)g(\()p Fp(j)p Fq(v)48 b Fs(\()p Fq(x)p Fs(\))p Fp(j)11312 48737 y Fl(4)12155 49219 y Fs(+)318 b(16)p Fp(j)p Fq(z)59 b Fs(\()p Fq(x)p Fs(\))p Fp(j)17937 48737 y Fl(2)18464 49219 y Fs(\))18970 48737 y Fl(1)p Fi(=)p Fl(4)20436 49219 y Ff(.)692 b(Mor)-66 b(e)g(over,)502 b(the)496 b(c)-66 b(onstant)496 b Fs(\()35721 48635 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 35721 48914 3279 54 v 37124 49677 a(2)39132 49219 y Fs(\))39638 48737 y Fl(2)40660 49219 y Ff(is)g(sharp)h(pr)-66 b(ovide)g(d)-880 50769 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)370 b Fq(>)e Fs(0)p Ff(.)-880 53224 y(Pr)-66 b(o)g(of.)552 b Fs(The)434 b(pro)36 b(of)434 b(is)g(iden)-36 b(tical)434 b(to)f(the)g(Heisen)-36 b(b)36 b(erg)434 b(group)f(case.)16762 b Fd(\003)2893 56513 y Fs(4.)651 b Fr(Hard)-30 b(y-type)502 b(inequalities)g(on)g(Carnot)g(gr)-30 b(oups)502 b(of)g(arbitrar)-89 b(y)503 b(step)448 58837 y Fs(In)466 b(this)g(section,)475 b(w)-36 b(e)467 b(consider)f(Carnot)h(group)f Fj(G)g Fs(of)h(an)-36 b(y)467 b(step)e Fq(k)512 b Fs(with)466 b(the)g(homogeneous)g(norm)-880 60387 y Fq(N)648 b Fs(=)510 b Fq(u)3072 59905 y Fl(1)p Fi(=)p Fl(\(2)p Fk(\000)p Fi(Q)p Fl(\))7259 60387 y Fs(asso)36 b(ciated)517 b(to)g(F)-108 b(olland's)516 b(solution)h Fq(u)g Fs(for)g(the)e(sub-Laplacian)h(\001) 40961 60586 y Fn(G)42221 60387 y Fs([13)q(].)827 b(W)-108 b(e)517 b(ha)-36 b(v)g(e)-880 61937 y(the)433 b(follo)-36 b(wing)436 b(theorem:)-880 64392 y Fy(Theorem)715 b(4.1.)650 b Ff(L)-66 b(et)636 b Fj(G)i Ff(b)-66 b(e)636 b(a)i(Carnot)f(gr)-66 b(oup)637 b(with)h(homo)-66 b(gene)g(ous)636 b(dimension)g Fq(Q)689 b Fp(\025)f Fs(3)638 b Ff(and)f(let)-880 65942 y Fq(\036)369 b Fp(2)f Fq(C)2539 65460 y Fk(1)2444 66270 y Fl(0)3536 65942 y Fs(\()p Fj(G)295 b Fp(n)g(f)p Fs(0)p Fp(g)p Fs(\))p Ff(,)465 b Fq(\013)378 b Fp(2)368 b Fj(R)p Ff(,)465 b Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)296 b Fs(2)369 b Fq(>)f Fs(0)p Ff(.)598 b(Then)464 b(the)h(fol)66 b(lowing)465 b(ine)-66 b(quality)463 b(is)h(valid)-880 69107 y Fs(\(4.1\))9654 67299 y Fo(Z)10392 70306 y Fn(G)11358 69107 y Fq(N)12539 68559 y Fi(\013)13198 69107 y Fp(jr)14674 69306 y Fn(G)15418 69107 y Fq(\036)p Fp(j)16557 68559 y Fl(2)17082 69107 y Fq(dx)369 b Fp(\025)20268 67633 y Fo(\020)21195 68208 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 21195 68802 5742 54 v 23740 70018 a(2)27069 67633 y Fo(\021)27862 67930 y Fl(2)28610 67299 y Fo(Z)29348 70306 y Fn(G)30313 69107 y Fq(N)31494 68559 y Fi(\013)32286 68208 y Fp(jr)33762 68407 y Fn(G)34506 68208 y Fq(N)139 b Fp(j)36056 67726 y Fl(2)p 32286 68802 4296 54 v 33580 70018 a Fq(N)34761 69635 y Fl(2)36714 69107 y Fq(\036)37484 68559 y Fl(2)38010 69107 y Fq(dx:)-880 72377 y Ff(F)-100 b(urthermor)-66 b(e,)464 b(the)h(c)-66 b(onstant)463 b Fq(C)95 b Fs(\()p Fq(Q;)221 b(\013)8 b Fs(\))370 b(=)f(\()21239 71793 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 21239 72072 3279 54 v 22643 72835 a(2)24650 72377 y Fs(\))25156 71895 y Fl(2)26147 72377 y Ff(is)464 b(sharp.)p eop %%Page: 9 9 9 8 bop -880 -714 a Fg(SHARP)443 b(WEIGHTED)f(HARD)-28 b(Y-TYPE)442 b(INEQUALITIES)f(AND)i(UNCER)-85 b(T)g(AINTY)440 b(PRINCIPLE)h(INEQUALITY)h(ON)h(CARNOT)g(GR)-28 b(OUPS)-512 b(9)-880 1942 y Ff(Pr)-66 b(o)g(of.)552 b Fs(Let)440 b Fq(\036)381 b Fs(=)f Fq(N)9156 1460 y Fi(\014)9785 1942 y Fq( )487 b Fs(where)441 b Fq( )428 b Fp(2)380 b Fq(C)18449 1460 y Fk(1)18354 2271 y Fl(0)19446 1942 y Fs(\()p Fj(G)300 b Fp(n)g(f)p Fs(0)p Fp(g)p Fs(\))441 b(and)f Fq(\014)455 b Fp(2)380 b Fj(R)300 b Fp(n)g(f)p Fs(0)p Fp(g)p Fs(.)600 b(A)441 b(direct)f(calculation)i(sho)-36 b(ws)-880 3492 y(that)-880 5707 y(\(4.2\))3499 b Fp(jr)6768 5906 y Fn(G)7512 5707 y Fs(\()p Fq(N)9199 5159 y Fi(\014)9827 5707 y Fq( )48 b Fs(\))p Fp(j)11596 5159 y Fl(2)12490 5707 y Fs(=)368 b Fq(\014)14678 5159 y Fl(2)15204 5707 y Fq(N)16385 5159 y Fl(2)p Fi(\014)46 b Fk(\000)p Fl(2)18686 5707 y Fp(jr)20162 5906 y Fn(G)20906 5707 y Fq(N)139 b Fp(j)22456 5159 y Fl(2)22981 5707 y Fq( )23875 5159 y Fl(2)24695 5707 y Fs(+)295 b(2)p Fq(\014)74 b(N)28641 5159 y Fl(2)p Fi(\014)46 b Fk(\000)p Fl(1)30942 5707 y Fq( )i Fp(r)32943 5906 y Fn(G)33687 5707 y Fq(N)433 b Fp(\001)295 b(r)36933 5906 y Fn(G)37677 5707 y Fq( )343 b Fs(+)295 b Fq(N)41354 5159 y Fl(2)p Fi(\014)42452 5707 y Fp(jr)43928 5906 y Fn(G)44672 5707 y Fq( )48 b Fp(j)45935 5159 y Fl(2)46460 5707 y Fq(:)-880 7923 y Fs(Multiplying)434 b(b)36 b(oth)433 b(sides)h(of)g(\(4.2\))g(b)-36 b(y)433 b(the)h Fq(N)22210 7441 y Fi(\013)23302 7923 y Fs(and)f(applying)h(in) -36 b(tegration)434 b(b)-36 b(y)433 b(parts)g(o)-36 b(v)g(er)434 b Fj(G)g Fs(giv)-36 b(es)-880 15603 y(\(4.3\))4577 10262 y Fo(Z)5315 13269 y Fn(G)6281 12070 y Fq(N)7462 11522 y Fi(\013)8120 12070 y Fp(jr)9596 12269 y Fn(G)10341 12070 y Fq(\036)p Fp(j)11480 11522 y Fl(2)12005 12070 y Fq(dx)369 b Fs(=)g Fq(\014)15978 11522 y Fl(2)16725 10262 y Fo(Z)17463 13269 y Fn(G)18428 12070 y Fq(N)19609 11522 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(2)23246 12070 y Fp(jr)24722 12269 y Fn(G)25466 12070 y Fq(N)139 b Fp(j)27016 11522 y Fl(2)27542 12070 y Fq( )28436 11522 y Fl(2)28961 12070 y Fq(dx)295 b Fp(\000)33676 11172 y Fq(\014)p 32132 11765 3896 54 v 32132 12981 a(\013)304 b Fs(+)295 b(2)p Fq(\014)36382 10262 y Fo(Z)37120 13269 y Fn(G)38086 12070 y Fs(\001)39170 12269 y Fn(G)39914 12070 y Fs(\()p Fq(N)41601 11522 y Fi(\013)p Fl(+2)p Fi(\014)44035 12070 y Fs(\))p Fq( )45435 11522 y Fl(2)45960 12070 y Fq(dx)13715 15576 y Fs(+)15022 13768 y Fo(Z)15760 16775 y Fn(G)16726 15576 y Fq(N)17907 15027 y Fi(\013)p Fl(+2)p Fi(\014)20341 15576 y Fp(jr)21817 15775 y Fn(G)22561 15576 y Fq( )48 b Fp(j)23824 15027 y Fl(2)24349 15576 y Fq(dx)13789 19094 y Fp(\025)369 b Fq(\014)15999 18546 y Fl(2)16746 17286 y Fo(Z)17484 20293 y Fn(G)18450 19094 y Fq(N)19631 18546 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(2)23268 19094 y Fp(jr)24744 19293 y Fn(G)25488 19094 y Fq(N)139 b Fp(j)27038 18546 y Fl(2)27563 19094 y Fq( )28457 18546 y Fl(2)28982 19094 y Fq(dx)295 b Fp(\000)33698 18195 y Fq(\014)p 32153 18789 V 32153 20005 a(\013)304 b Fs(+)295 b(2)p Fq(\014)36404 17286 y Fo(Z)37142 20293 y Fn(G)38107 19094 y Fs(\001)39191 19293 y Fn(G)39935 19094 y Fs(\()p Fq(N)41622 18546 y Fi(\013)p Fl(+2)p Fi(\014)44057 19094 y Fs(\))p Fq( )45457 18546 y Fl(2)45982 19094 y Fq(dx:)-880 21992 y Fs(A)434 b(straigh)-36 b(tforw)g(ard)434 b(calculation)g(sho)-36 b(ws)434 b(that)-880 26139 y(\(4.4\))665 b Fp(\000)5168 25241 y Fq(\014)p 3624 25834 V 3624 27051 a(\013)304 b Fs(+)295 b(2)p Fq(\014)7653 26139 y Fs(\001)8737 26338 y Fn(G)9481 26139 y Fs(\()p Fq(N)11168 25591 y Fi(\013)p Fl(+2)p Fi(\014)13602 26139 y Fs(\))369 b(=)g Fp(\000)p Fq(\014)74 b Fs(\()p Fq(\013)131 b Fs(+)122 b(2)p Fq(\014)196 b Fs(+)122 b Fq(Q)g Fp(\000)g Fs(2\))p Fq(N)27655 25591 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(2)31293 26139 y Fp(jr)32769 26338 y Fn(G)33514 26139 y Fq(N)139 b Fp(j)35064 25591 y Fl(2)35711 26139 y Fp(\000)38247 25241 y Fq(\014)p 36999 25834 3305 54 v 36999 27051 a Fs(2)296 b Fp(\000)f Fq(Q)40436 26139 y(N)41617 25591 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fl(+)p Fi(Q)p Fk(\000)p Fl(2)46726 26139 y Fs(\001)47810 26338 y Fn(G)48554 26139 y Fq(u:)-880 29182 y Fs(Substituting)432 b(\(4.4\))i(in)-36 b(to)434 b(\(4.3\))g(and)f(using)g(the)g(fact)h(that)f Fq( )29871 28700 y Fl(2)30766 29182 y Fs(=)368 b Fq(N)33327 28700 y Fk(\000)p Fl(2)p Fi(\014)35158 29182 y Fq(\036)35928 28700 y Fl(2)36453 29182 y Fs(,)434 b(w)-36 b(e)434 b(get)g(the)f (follo)-36 b(wing)435 b(:)-880 31652 y Fo(Z)-142 34659 y Fn(G)823 33460 y Fq(N)2004 32911 y Fi(\013)2663 33460 y Fp(jr)4139 33659 y Fn(G)4884 33460 y Fq(\036)p Fp(j)6023 32911 y Fl(2)6548 33460 y Fq(dx)369 b Fp(\025)g Fs(\()p Fp(\000)p Fq(\014)12081 32911 y Fl(2)12831 33460 y Fp(\000)224 b Fq(\014)74 b Fs(\()p Fq(\013)232 b Fs(+)224 b Fq(Q)g Fp(\000)g Fs(2\))21585 31652 y Fo(Z)22323 34659 y Fn(G)23289 33460 y Fq(N)24470 32911 y Fi(\013)25261 32561 y Fp(jr)26737 32760 y Fn(G)27481 32561 y Fq(N)139 b Fp(j)29031 32079 y Fl(2)p 25261 33154 4296 54 v 26556 34371 a Fq(N)27737 33987 y Fl(2)29689 33460 y Fq(\036)30459 32911 y Fl(2)30985 33460 y Fq(dx)224 b Fp(\000)35262 32561 y Fq(\014)p 34014 33154 3305 54 v 34014 34371 a Fs(2)295 b Fp(\000)h Fq(Q)37672 31652 y Fo(Z)38410 34659 y Fn(G)39154 33460 y Fs(\(\001)40744 33659 y Fn(G)41488 33460 y Fq(u)p Fs(\))p Fq(N)43915 32911 y Fi(\013)p Fl(+)p Fi(Q)p Fk(\000)p Fl(2)47248 33460 y Fq(\036)48018 32911 y Fl(2)48544 33460 y Fq(dx:)-880 36358 y Fs(Since)433 b Fq(u)h Fs(is)g(the)f(fundamen)-36 b(tal)433 b(solution)h(of)g(sub-Laplacian)f(\001)30574 36557 y Fn(G)31751 36358 y Fs(on)h(Carnot)g(group)f Fj(G)p Fs(,)h(w)-36 b(e)434 b(get)11127 39201 y Fp(\000)12381 37393 y Fo(Z)13120 40400 y Fn(G)13864 39201 y Fs(\(\001)15454 39400 y Fn(G)16198 39201 y Fq(u)p Fs(\))p Fq(N)18625 38652 y Fi(\013)p Fl(+)p Fi(Q)p Fk(\000)p Fl(2)21958 39201 y Fq(\036)22728 38652 y Fl(2)23253 39201 y Fq(dx)369 b Fs(=)g Fq(N)27599 38652 y Fi(\013)p Fl(+)p Fi(Q)p Fk(\000)p Fl(2)30932 39201 y Fs(\(0\))p Fq(\036)33364 38652 y Fl(2)33890 39201 y Fs(\(0\))g(=)f(0)p Fq(:)-880 42099 y Fs(W)-108 b(e)434 b(no)-36 b(w)433 b(obtain)7685 44026 y Fo(Z)8423 47033 y Fn(G)9389 45834 y Fq(N)10570 45285 y Fi(\013)11229 45834 y Fp(jr)12705 46033 y Fn(G)13449 45834 y Fq(\036)p Fp(j)14588 45285 y Fl(2)15113 45834 y Fq(dx)369 b Fp(\025)h Fs(\()p Fp(\000)p Fq(\014)20647 45285 y Fl(2)21467 45834 y Fp(\000)296 b Fq(\014)74 b Fs(\()p Fq(\013)303 b Fs(+)295 b Fq(Q)h Fp(\000)f Fs(2\))30578 44026 y Fo(Z)31316 47033 y Fn(G)32282 45834 y Fq(N)33463 45285 y Fi(\013)34255 44935 y Fp(jr)35731 45134 y Fn(G)36475 44935 y Fq(N)139 b Fp(j)38025 44453 y Fl(2)p 34255 45528 4296 54 v 35549 46745 a Fq(N)36730 46361 y Fl(2)38683 45834 y Fq(\036)39453 45285 y Fl(2)39978 45834 y Fq(dx:)-880 48461 y Fs(Cho)36 b(osing)20427 50759 y Fq(\014)443 b Fs(=)23117 49861 y(2)296 b Fp(\000)f Fq(Q)g Fp(\000)h Fq(\013)p 23117 50454 5764 54 v 25674 51671 a Fs(2)-880 53098 y(giv)-36 b(es)435 b(the)e(follo)-36 b(wing)435 b(sharp)e(inequalit)-36 b(y)-880 56084 y(\(4.5\))9654 54276 y Fo(Z)10392 57283 y Fn(G)11358 56084 y Fq(N)12539 55536 y Fi(\013)13198 56084 y Fp(jr)14674 56283 y Fn(G)15418 56084 y Fq(\036)p Fp(j)16557 55536 y Fl(2)17082 56084 y Fq(dx)369 b Fp(\025)20268 54610 y Fo(\020)21195 55186 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 21195 55779 5742 54 v 23740 56995 a(2)27069 54610 y Fo(\021)27862 54907 y Fl(2)28610 54276 y Fo(Z)29348 57283 y Fn(G)30313 56084 y Fq(N)31494 55536 y Fi(\013)32286 55186 y Fp(jr)33762 55385 y Fn(G)34506 55186 y Fq(N)139 b Fp(j)36056 54704 y Fl(2)p 32286 55779 4296 54 v 33580 56995 a Fq(N)34761 56612 y Fl(2)36714 56084 y Fq(\036)37484 55536 y Fl(2)38010 56084 y Fq(dx:)49287 58957 y Fd(\003)-880 61321 y Fy(Remark)654 b(4.1.)985 b Fs(In)569 b(the)g(ab)36 b(elian)570 b(case,)603 b(when)569 b Fj(G)600 b Fs(=)f Fj(R)28057 60839 y Fi(n)29253 61321 y Fs(with)569 b(the)g(ordinary)g(dilations,)604 b(one)569 b(has)-880 62871 y Fp(G)448 b Fs(=)369 b Fq(V)2497 63070 y Fl(1)3392 62871 y Fs(=)f Fj(R)5731 62389 y Fi(n)6718 62871 y Fs(so)360 b(that)f Fq(Q)369 b Fs(=)g Fq(n)p Fs(.)554 b(No)-36 b(w)360 b(it)g(is)g(clear)h(that)e(the)g(inequalit)-36 b(y)361 b(\(4.1\))g(with)f(the)f(homogeneous)-880 64421 y(norm)433 b Fq(N)139 b Fs(\()p Fq(x)p Fs(\))368 b(=)h Fp(j)p Fq(x)p Fp(j)433 b Fs(and)h Fq(\013)377 b Fs(=)369 b(0)434 b(reco)-36 b(v)g(ers)434 b(the)f(Hardy)h(inequalit)-36 b(y)434 b(\(1.1\).)-880 66636 y Fy(Uncertain)-42 b(t)g(y)622 b(Principle)e(Inequalit)-42 b(y)-125 b(.)894 b Fs(The)539 b(classical)h(uncertain)-36 b(t)g(y)538 b(principle)h(w)-36 b(as)539 b(dev)-36 b(elop)36 b(ed)-880 68186 y(in)558 b(the)g(con)-36 b(text)558 b(of)g(quan)-36 b(tum)558 b(mec)-36 b(hanics)558 b(b)-36 b(y)558 b(Heisen)-36 b(b)36 b(erg)558 b([20)q(].)952 b(It)558 b(sa)-36 b(ys)559 b(that)e(the)h(p)36 b(osition)558 b(and)-880 69736 y(momen)-36 b(tum)444 b(of)j(a)e(particle)h(cannot)f(b)36 b(e)445 b(determined)f(exactly)j (at)f(the)f(same)h(time)f(but)f(only)j(with)e(an)-880 71286 y(\\uncertain)-36 b(t)g(y".)540 b(The)319 b(harmonic)f(analysis)i (v)-36 b(ersion)319 b(of)g(uncertain)-36 b(t)g(y)318 b(principle)g(states)h(that)f(a)h(function)-880 72835 y(on)452 b(the)f(real)i(line)f(and)f(its)h(F)-108 b(ourier)452 b(transform)g(can)f(not)h(b)36 b(e)452 b(sim)-36 b(ultaneously)452 b(w)-36 b(ell)453 b(lo)36 b(calized.)635 b(It)452 b(has)p eop %%Page: 10 10 10 9 bop -880 -714 a Fg(10)20445 b(ISMAIL)341 b(K)-28 b(OMBE)-880 1942 y Fs(b)36 b(een)406 b(widely)h(studied)f(in)g(quan)-36 b(tum)405 b(mec)-36 b(hanics)407 b(and)f(signal)h(analysis.)570 b(There)407 b(are)f(v)-72 b(arious)407 b(forms)g(of)-880 3492 y(the)433 b(uncertain)-36 b(t)g(y)433 b(principle.)578 b(F)-108 b(or)433 b(an)g(o)-36 b(v)g(erview)436 b(w)-36 b(e)433 b(refer)h(to)g(F)-108 b(olland's)434 b(and)f(Sitaram's)h(pap)36 b(er)433 b([14)q(].)448 5042 y(The)399 b(uncertain)-36 b(t)g(y)398 b(principle)h(on)g(the)f(Euclidean)h(space)g Fj(R)29624 4560 y Fi(n)30649 5042 y Fs(can)g(b)36 b(e)399 b(stated)g(in)f(the)h(follo)-36 b(wing)401 b(w)-36 b(a)g(y:)-880 8009 y(\(4.6\))7371 6534 y Fo(\020)8385 6201 y(Z)9123 9208 y Fn(R)9763 8956 y Fm(n)10606 8009 y Fp(j)p Fq(x)p Fp(j)12083 7460 y Fl(2)12609 8009 y Fp(j)p Fq(f)142 b Fs(\()p Fq(x)p Fs(\))p Fp(j)15881 7460 y Fl(2)16407 8009 y Fq(dx)17822 6534 y Fo(\021\020)19630 6201 y(Z)20368 9208 y Fn(R)21008 8956 y Fm(n)21850 8009 y Fp(jr)p Fq(f)g Fs(\()p Fq(x)p Fs(\))p Fp(j)26229 7460 y Fl(2)26755 8009 y Fq(dx)28170 6534 y Fo(\021)29332 8009 y Fp(\025)30867 7110 y Fq(n)31643 6628 y Fl(2)p 30867 7703 1303 54 v 31193 8920 a Fs(4)32302 6534 y Fo(\020)33317 6201 y(Z)34055 9208 y Fn(R)34695 8956 y Fm(n)35538 8009 y Fp(j)p Fq(f)g Fs(\()p Fq(x)p Fs(\))p Fp(j)38810 7460 y Fl(2)39335 8009 y Fq(dx)40750 6534 y Fo(\021)41543 6832 y Fl(2)-880 11018 y Fs(for)535 b(all)g Fq(f)683 b Fp(2)540 b Fq(L)6750 10536 y Fl(2)7276 11018 y Fs(\()p Fj(R)8741 10536 y Fi(n)9368 11018 y Fs(\).)881 b(An)534 b(analogue)h(of)g(the)f(ab)36 b(o)-36 b(v)g(e)535 b(inequalit)-36 b(y)536 b(\(4.6\))f(for)g(the)f (Heisen)-36 b(b)36 b(erg)534 b(group)-880 12568 y Fj(H)153 12086 y Fi(n)1233 12568 y Fs(w)-36 b(as)454 b(established)f(b)-36 b(y)453 b(Garofalo)i(and)e(Lanconelli)g([17)q(].)638 b(W)-108 b(e)454 b(should)f(men)-36 b(tion)452 b(that)h(Thanga)-36 b(v)g(elu)-880 14117 y([29)q(],)379 b(and)364 b(Sitaram,)379 b(Sundari)364 b(and)g(Thanga)-36 b(v)g(elu)366 b([27)q(])f(ha)-36 b(v)g(e)365 b(also)g(obtained)g(related)g(but)e(inequiv)-72 b(alen)-36 b(t)-880 15667 y(analogues)434 b(of)h(Heisen)-36 b(b)36 b(erg's)434 b(inequalit)-36 b(y)434 b(for)g(functions)g(on)f (the)g(Heisen)-36 b(b)36 b(erg)434 b(group)f Fj(H)43038 15185 y Fi(n)43664 15667 y Fs(.)448 17217 y(W)-108 b(e)359 b(no)-36 b(w)358 b(giv)-36 b(e)360 b(an)e(analogue)h(of)g(\(4.6\))g(on) f(general)h(Carnot)g(groups)f(whic)-36 b(h)358 b(is)g(an)h(immediate)f (conse-)-880 18767 y(quence)433 b(of)i(the)e(Hardy-t)-36 b(yp)36 b(e)433 b(inequalit)-36 b(y)435 b(\(4.1\))f(with)f Fq(\013)378 b Fs(=)369 b(0)434 b(and)f(the)g(Cauc)-36 b(h)g(y-Sc)g(h)g(w)g(arz)433 b(inequalit)-36 b(y)-108 b(.)-880 21151 y Fy(Corollary)758 b(4.1.)660 b Ff(\(Unc)-66 b(ertainty)653 b(ine)-66 b(quality\).)1166 b(L)-66 b(et)654 b Fj(G)h Ff(b)-66 b(e)655 b(a)g(Carnot)g(gr)-66 b(oup)655 b(with)g(homo)-66 b(gene)g(ous)-880 22700 y(dimension)463 b Fq(Q)369 b Fp(\025)h Fs(3)p Ff(.)597 b(Then)465 b(for)f(every)g Fq(\036)369 b Fp(2)g Fq(C)22131 22218 y Fk(1)22036 23029 y Fl(0)23127 22700 y Fs(\()p Fj(G)295 b Fp(n)g(f)p Fs(0)p Fp(g)p Fs(\))-880 25598 y(\(4.7\))4774 24123 y Fo(\020)5789 23790 y(Z)6527 26797 y Fn(G)7492 25598 y Fq(N)8673 25049 y Fl(2)9198 25598 y Fp(jr)10674 25797 y Fn(G)11418 25598 y Fq(N)139 b Fp(j)12968 25049 y Fl(2)13493 25598 y Fq(\036)14263 25049 y Fl(2)14789 25598 y Fq(dx)16204 24123 y Fo(\021\020)18012 23790 y(Z)18750 26797 y Fn(G)19715 25598 y Fp(jr)21191 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Fy(Corollary)668 b(4.2.)620 b Ff(\(Har)-66 b(dy-Sob)g(olev\))581 b(L)-66 b(et)582 b Fj(G)h Ff(b)-66 b(e)582 b(a)h(Carnot)f(gr)-66 b(oup)583 b(with)g(homo)-66 b(gene)g(ous)582 b(dimension)-880 36344 y Fq(Q)369 b Fp(\025)g Fs(3)455 b Ff(and)f Fs(0)370 b Fp(\024)f Fq(s)f Fp(\024)i Fs(2)p Ff(.)594 b(Then)453 b(for)h(every)g Fq(\036)369 b Fp(2)f Fq(C)24275 35862 y Fk(1)24180 36673 y Fl(0)25271 36344 y Fs(\()p Fj(G)272 b Fp(n)g(f)p Fs(0)p Fp(g)p Fs(\))455 b Ff(then)e(ther)-66 b(e)454 b(exists)g(a)g(c)-66 b(onstant)453 b Fq(C)464 b(>)369 b Fs(0)-880 37894 y Ff(such)466 b(that)10757 38342 y Fo(Z)11494 41349 y Fn(G)12460 40150 y Fp(jr)13936 40349 y Fn(G)14680 40150 y Fq(\036)p Fp(j)15819 39602 y Fl(2)16345 40150 y Fq(dx)369 b Fp(\025)g Fq(C)20557 38676 y Fo(\020)21571 38342 y(Z)22309 41349 y Fn(G)23275 38676 y Fo(\020)24201 39251 y Fp(jr)25677 39450 y Fn(G)26421 39251 y Fq(N)139 b Fp(j)p 24201 39845 3770 54 v 25496 41061 a Fq(N)28103 38676 y Fo(\021)28897 38973 y Fi(s)29387 40150 y Fp(j)p 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Fl(2)19640 51627 y Fq(dx)f Fp(\025)22959 50728 y Fq(n)23735 50246 y Fl(2)24261 50728 y Fs(\()p Fq(n)296 b Fp(\000)f Fs(4\))28323 50246 y Fl(2)p 22959 51322 5890 54 v 25254 52538 a Fs(16)29203 49819 y Fo(Z)29941 52826 y Fn(R)30581 52574 y Fm(n)31557 50728 y Fp(j)p Fq(\036)p Fs(\()p Fq(x)p Fs(\))p Fp(j)34816 50246 y Fl(2)p 31557 51322 3785 54 v 32448 52538 a Fp(j)p Fq(x)p Fp(j)33925 52155 y Fl(4)35474 51627 y Fq(dx)-880 55001 y Fs(for)460 b(all)h Fq(\036)413 b Fp(2)f Fq(C)6473 54519 y Fk(1)6378 55330 y Fi(c)7470 55001 y Fs(\()p Fj(R)8935 54519 y Fi(n)9874 55001 y Fp(n)313 b(f)p Fs(0)p Fp(g)p Fs(\))460 b(and)f(the)g(constan) -36 b(t)24021 54368 y Fi(n)24592 54056 y Fh(2)25053 54368 y Fl(\()p Fi(n)p Fk(\000)p Fl(4\))27558 54056 y Fh(2)p 24021 54696 3999 54 v 25550 55459 a Fl(16)28612 55001 y Fs(is)459 b(sharp.)656 b(The)460 b(Rellic)-36 b(h)460 b(inequalit)-36 b(y)461 b(is)e(the)-880 56551 y(\014rst)434 b(generalization)i(of)f(Hardy's)g(inequalit)-36 b(y)435 b(to)g(higher-order)f(deriv)-72 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b(the)g(ordinary)g(dilations,)604 b(one)569 b(has)-880 71286 y Fp(G)448 b Fs(=)369 b Fq(V)2497 71485 y Fl(1)3392 71286 y Fs(=)f Fj(R)5731 70804 y Fi(n)6718 71286 y Fs(so)360 b(that)f Fq(Q)369 b Fs(=)g Fq(n)p Fs(.)554 b(No)-36 b(w)360 b(it)g(is)g(clear)h(that)e(the)g(inequalit)-36 b(y)361 b(\(4.9\))g(with)f(the)f(homogeneous)-880 72835 y(norm)433 b Fq(N)139 b Fs(\()p Fq(x)p Fs(\))368 b(=)h Fp(j)p Fq(x)p Fp(j)433 b Fs(and)h Fq(\013)377 b Fs(=)369 b(0)434 b(reco)-36 b(v)g(ers)434 b(the)f(Rellic)-36 b(h)434 b(inequalit)-36 b(y)435 b(\(4.8\).)p eop %%Page: 11 11 11 10 bop -880 -714 a Fg(SHARP)443 b(WEIGHTED)f(HARD)-28 b(Y-TYPE)442 b(INEQUALITIES)f(AND)i(UNCER)-85 b(T)g(AINTY)440 b(PRINCIPLE)h(INEQUALITY)h(ON)h(CARNOT)g(GR)-28 b(OUPS)-1025 b(11)-880 1942 y Fy(P)-42 b(olarizable)347 b(Carnot)f(group.)535 b Fs(The)299 b(second)h(main)g(result)f(of)i(this)e(section)h(is)g(to)g (establish)g(w)-36 b(eigh)g(ted)-880 3492 y(Hardy-t)g(yp)36 b(e)610 b(inequalit)-36 b(y)611 b(including)e(the)h(w)-36 b(eigh)g(t)610 b(function)1219 b Fp(jr)32287 3691 y Fn(G)33031 3492 y Fq(N)139 b Fp(j)34581 3010 y Fi(\015)35783 3492 y Fs(as)610 b(in)g(the)f(Section)h(3.)1108 b(W)-108 b(e)-880 5042 y(no)-36 b(w)519 b(establish)g(suc)-36 b(h)519 b(a)g(inequalit)-36 b(y)520 b(on)f(p)36 b(olarizable)521 b(Carnot)e(groups.)835 b(This)519 b(class)h(of)g(groups)f(w)-36 b(ere)-880 6592 y(in)g(tro)36 b(duced)432 b(b)-36 b(y)434 b(Balogh)g(and)f(T)-36 b(yson)435 b([2])f(and)f(admit)h(the)f(analogue)h(of)g(p)36 b(olar)434 b(co)36 b(ordinates.)448 8181 y(A)584 b(Carnot)f(group)g Fj(G)h Fs(is)f(said)h(to)f(b)36 b(e)583 b(p)36 b(olarizable)585 b(if)f(the)f(homogeneous)g(norm)g Fq(N)762 b Fs(=)624 b Fq(u)46288 7698 y Fl(1)p Fi(=)p Fl(\(2)p Fk(\000)p Fi(Q)p Fl(\))49959 8181 y Fs(,)-880 9730 y(asso)36 b(ciated)625 b(to)g(F)-108 b(olland's)624 b(solution)h Fq(u)g Fs(for)f(the)g (sub-Laplacian)g(\001)33686 9929 y Fn(G)34430 9730 y Fs(,)672 b(satis\014es)625 b(the)f(follo)-36 b(wing)626 b Fp(1)p Fs(-)-880 11280 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Fq(N)139 b Fp(j)33425 64520 y Fi(\015)52 b Fl(+2)35219 65068 y Fq( )36113 64520 y Fl(2)36638 65068 y Fq(dx)20124 68573 y Fs(+)295 b(2)p Fq(\014)23111 66765 y Fo(Z)23849 69773 y Fn(G)24814 68573 y Fq(N)25995 68025 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(1)29632 68573 y Fp(jr)31108 68772 y Fn(G)31852 68573 y Fq(N)139 b Fp(j)33402 68025 y Fi(\015)33994 68573 y Fq( )48 b Fp(r)35995 68772 y Fn(G)36738 68573 y Fq(N)434 b Fp(\001)295 b(r)39985 68772 y Fn(G)40729 68573 y Fq( )48 b(dx)20124 72079 y Fs(+)21431 70271 y Fo(Z)22169 73278 y Fn(G)23134 72079 y Fq(N)24315 71530 y Fi(\013)p Fl(+2)p Fi(\014)26750 72079 y Fp(jr)28226 72278 y Fn(G)28970 72079 y Fq(N)139 b Fp(j)30520 71530 y Fi(\015)31112 72079 y Fp(jr)32588 72278 y Fn(G)33332 72079 y Fq( )48 b Fp(j)34595 71530 y Fl(2)35120 72079 y Fq(dx:)p eop %%Page: 12 12 12 11 bop -880 -714 a Fg(12)20445 b(ISMAIL)341 b(K)-28 b(OMBE)-880 1942 y Fs(Applying)434 b(in)-36 b(tegration)433 b(b)-36 b(y)434 b(parts)f(to)h(the)f(middle)g(term:)-880 9706 y(\(4.15\))7034 4365 y Fo(Z)7772 7373 y Fn(G)8737 6173 y Fq(N)9918 5625 y Fi(\013)10577 6173 y Fp(jr)12053 6372 y Fn(G)12797 6173 y Fq(N)139 b Fp(j)14347 5625 y Fi(\015)14939 6173 y Fp(jr)16415 6372 y Fn(G)17159 6173 y Fq(\036)p Fp(j)18298 5625 y Fl(2)18823 6173 y Fq(dx)369 b Fs(=)g Fq(\014)22796 5625 y Fl(2)23543 4365 y Fo(Z)24281 7373 y Fn(G)25247 6173 y Fq(N)26428 5625 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(2)30065 6173 y Fp(jr)31541 6372 y Fn(G)32285 6173 y Fq(N)139 b Fp(j)33835 5625 y Fi(\015)52 b Fl(+2)35628 6173 y Fq( )36522 5625 y Fl(2)37048 6173 y Fq(dx)20534 9679 y Fp(\000)295 b Fq(\014)22891 7871 y Fo(Z)23629 10878 y Fn(G)24595 9679 y Fq( )25489 9130 y Fl(2)26014 9679 y Fs(div)27784 8603 y Fo(\000)28393 9679 y Fq(N)29574 9130 y Fi(\013)p Fl(+2)p Fi(\014)46 b Fk(\000)p Fl(1)33211 9679 y Fp(jr)34687 9878 y Fn(G)35431 9679 y Fq(N)139 b Fp(j)36981 9130 y Fi(\015)37573 9679 y Fp(r)38680 9878 y Fn(G)39424 9679 y Fq(N)40605 8603 y Fo(\001)41213 9679 y Fq(dx)20534 13184 y Fs(+)21840 11376 y Fo(Z)22578 14383 y Fn(G)23544 13184 y Fq(N)24725 12636 y Fi(\013)p Fl(+2)p Fi(\014)27160 13184 y Fp(jr)28636 13383 y Fn(G)29380 13184 y Fq(N)g Fp(j)30930 12636 y Fi(\015)31521 13184 y Fp(jr)32997 13383 y Fn(G)33741 13184 y Fq( )48 b Fp(j)35004 12636 y Fl(2)35529 13184 y Fq(dx:)-880 16179 y Fs(W)-108 b(e)434 b(no)-36 b(w)433 b(c)-36 b(ho)36 b(ose)434 b Fq(\015)441 b Fs(=)369 b Fq(p)295 b Fp(\000)g Fs(2)369 b Fq(>)g Fs(1)434 b(and)f Fq(\013)304 b Fs(+)295 b(2)p Fq(\014)369 b Fp(\000)296 b Fs(1)369 b(=)f(1)296 b Fp(\000)f Fq(Q)p Fs(,)434 b(w)-36 b(e)434 b(get)-880 17729 y(\(4.16\))912 18172 y Fo(Z)1650 21179 y Fn(G)2615 19980 y Fq(N)3796 19431 y Fi(\013)4455 19980 y Fp(jr)5931 20179 y Fn(G)6675 19980 y Fq(N)139 b Fp(j)8225 19431 y Fi(\015)8817 19980 y Fp(jr)10293 20179 y Fn(G)11037 19980 y Fq(\036)p Fp(j)12176 19431 y Fl(2)12702 19980 y Fq(dx)368 b Fs(=)15866 18505 y Fo(\020)16792 19081 y Fq(Q)296 b Fs(+)e Fq(\013)304 b Fp(\000)296 b Fs(2)p 16792 19674 5742 54 v 19338 20891 a(2)22667 18505 y Fo(\021)23460 18803 y Fl(2)24207 18172 y Fo(Z)24945 21179 y Fn(G)25911 19980 y Fq(N)27092 19431 y Fi(\013)p Fk(\000)p Fl(2)28953 19980 y Fp(jr)30429 20179 y Fn(G)31173 19980 y Fq(N)139 b Fp(j)32723 19431 y Fi(\015)52 b Fl(+2)34517 19980 y Fq(\036)35287 19431 y Fl(2)35813 19980 y Fq(dx)14412 23485 y Fp(\000)295 b Fq(\014)74 b(c)17108 22937 y Fl(1)p Fk(\000)p Fi(p)17108 23814 y(p)19060 21677 y Fo(Z)19798 24684 y Fn(G)20543 23485 y Fs(\(\001)22133 23684 y Fn(G)p Fi(;p)23611 23485 y Fs(\()p Fq(u)24857 23684 y Fi(p)25386 23485 y Fs(\)\))p Fq(N)27579 22937 y Fk(\000)p Fl(2)p Fi(\014)29409 23485 y Fq(\036)30179 22937 y Fl(2)30705 23485 y Fq(dx)295 b Fs(+)33722 21677 y Fo(Z)34460 24684 y Fn(G)35426 23485 y Fq(N)36607 22937 y Fl(2)p Fk(\000)p Fi(Q)38604 23485 y Fp(jr)40080 23684 y Fn(G)40824 23485 y Fq(N)139 b Fp(j)42374 22937 y Fi(\015)42965 23485 y Fp(jr)44441 23684 y Fn(G)45185 23485 y Fq( )48 b Fp(j)46448 22937 y Fl(2)46973 23485 y Fq(dx:)-880 26480 y Fs(Since)433 b Fq(u)3256 26679 y Fi(p)4219 26480 y Fs(is)h(the)f(fundamen)-36 b(tal)433 b(solution)h(of)g(sub-p-Laplacian)e Fp(\000)p Fs(\001)33292 26679 y Fn(G)p Fi(;p)34771 26480 y Fs(,)i(w)-36 b(e)434 b(get)9989 29533 y Fp(\000)11243 27725 y Fo(Z)11982 30732 y Fn(G)12726 29533 y Fs(\(\001)14316 29732 y Fn(G)p Fi(;p)15795 29533 y Fs(\()p Fq(u)17041 29732 y Fi(p)17570 29533 y Fs(\)\))p Fq(N)19763 28984 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)23096 29533 y Fq(\036)23866 28984 y Fl(2)24391 29533 y Fq(dx)369 b Fs(=)g Fq(N)28737 28984 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)32070 29533 y Fs(\(0\))p Fq(\036)34502 28984 y Fl(2)35028 29533 y Fs(\(0\))f(=)h(0)p Fq(:)-880 32528 y Fs(W)-108 b(e)434 b(no)-36 b(w)433 b(obtain)h(the)f(desired)g(inequalit)-36 b(y)-880 35467 y(\(4.17\))6371 33659 y Fo(Z)7109 36667 y Fn(G)8074 35467 y Fq(N)9255 34919 y Fi(\013)9914 35467 y Fp(jr)11390 35666 y Fn(G)12134 35467 y Fq(N)139 b Fp(j)13684 34919 y Fi(\015)14276 35467 y Fp(jr)15752 35666 y Fn(G)16496 35467 y Fq(\036)p Fp(j)17635 34919 y Fl(2)18160 35467 y Fq(dx)369 b Fp(\025)21346 33993 y Fo(\020)22273 34569 y Fq(Q)295 b Fs(+)g Fq(\013)304 b Fp(\000)295 b Fs(2)p 22273 35162 V 24818 36379 a(2)28147 33993 y Fo(\021)28941 34290 y Fl(2)29688 33659 y Fo(Z)30426 36667 y Fn(G)31391 35467 y Fq(N)32572 34919 y Fi(\013)p Fk(\000)p Fl(2)34433 35467 y Fp(jr)35909 35666 y Fn(G)36654 35467 y Fq(N)139 b Fp(j)38204 34919 y Fi(\015)52 b Fl(+2)39997 35467 y Fq(\036)40767 34919 y Fl(2)41293 35467 y Fq(dx:)-880 39288 y Fs(T)-108 b(o)434 b(sho)-36 b(w)434 b(that)f(the)g(constan)-36 b(t)14562 37813 y Fo(\020)15488 38704 y Fi(Q)p Fl(+)p Fi(\013)p Fk(\000)p Fl(2)p 15488 38982 3279 54 v 16892 39746 a(2)18899 37813 y Fo(\021)19693 38111 y Fl(2)20652 39288 y Fs(is)434 b(sharp,)f(w)-36 b(e)434 b(use)f(the)g(follo)-36 b(wing)436 b(family)f(of)g(functions)-880 43349 y(\(4.18\))9771 b Fq(\036)12984 43548 y Fi(\017)13420 43349 y Fs(\()p Fq(x)p Fs(\))369 b(=)16920 41078 y Fo(\()17990 42433 y Fs(1)8872 b(if)1301 b Fq(N)139 b Fs(\()p Fq(x)p Fs(\))368 b Fp(2)h Fs([0)p Fq(;)221 b Fs(1])p Fq(;)17990 44345 y(N)19171 43855 y Fk(\000)p Fl(\()20402 43444 y Fm(Q)p Fh(+)p Fm(\013)p Fe(\000)p Fh(2)p 20401 43653 2843 40 v 21620 44202 a(2)23377 43855 y Fl(+)p Fi(\017)p Fl(\))27512 44345 y Fs(if)1301 b Fq(N)139 b Fs(\()p Fq(x)p Fs(\))368 b Fq(>)h Fs(1)p Fq(;)-880 46879 y Fs(and)523 b(pass)h(to)f(the)g(limit) h(as)g Fq(\017)f Fp(\000)-222 b(!)523 b Fs(0.)848 b(Here)524 b(w)-36 b(e)524 b(notice)f(that)g Fp(jr)32593 47078 y Fn(G)33338 46879 y Fq(N)139 b Fp(j)522 b Fs(is)i(uniformly)g(b)36 b(ounded)522 b(and)-880 48429 y(p)36 b(olar)434 b(co)36 b(ordinate)434 b(in)-36 b(tegration)434 b(form)-36 b(ula)434 b(holds)f(on)h Fj(G)f Fs(\(see)h([2)q(],)g([15)q(]\).)-880 49979 y Fy(Ac)-42 b(kno)g(wledgemen)g(t.)813 b Fs(I)511 b(w)-36 b(ould)510 b(lik)-36 b(e)512 b(to)f(thank)f(Nicola)i(Garofalo)h (and)d(Jerem)-36 b(y)511 b(T)-36 b(yson)511 b(for)g(their)-880 51528 y(v)-72 b(aluable)434 b(commen)-36 b(ts.)20847 54792 y Fr(References)-880 56895 y Fx([1])832 b(W.)385 b(Allegretto,)393 b Fw(Nonoscil)57 b(lation)411 b(the)-57 b(ory)411 b(of)h(el)57 b(liptic)411 b(e)-57 b(quations)412 b(of)f(or)-57 b(der)411 b(2n)p Fx(,)391 b(P)-31 b(aci\014c)387 b(journal)f(of)h(mathematics,)1119 58223 y Fc(64)370 b Fx(\(1976\),)i(no.)e(1,)g(1-16.)-880 59552 y([2])832 b(Z.)370 b(Balogh)h(and)e(J.)g(T)-31 b(yson,)767 b Fw(Polar)395 b(c)-57 b(o)g(or)g(dinates)397 b(on)f(Carnot)g(gr)-57 b(oups)p Fx(,)369 b(Math.)h(Z.)g Fc(241)g Fx(\(2002\),)j(no.)d(4,)g (697-730.)-880 60880 y([3])832 b(Z.)457 b(Balogh,)480 b(I.)456 b(Holopainen)j(and)d(J.)g(T)-31 b(yson,)480 b Fw(Singular)d(solutions,)495 b(homo)-57 b(gene)g(ous)477 b(norms)f(and)g(quasic)-57 b(onformal)1119 62209 y(mappings)396 b(in)h(Carnot)e(gr)-57 b(oups)p Fx(,)369 b(Math.)i(Ann.)e Fc(324)i Fx(\(2002\),)h(no.)e(1,)g(159-186.)-880 63537 y([4])832 b(P)-92 b(.)434 b(Baras)g(and)g(J.)f(A.)i(Goldstein,)451 b Fw(The)456 b(he)-57 b(at)455 b(e)-57 b(quation)456 b(with)f(a)h(singular)f(p)-57 b(otential)p Fx(,)450 b(T)-92 b(rans.)434 b(Amer.)h(Math.)f(So)31 b(c.)1119 64865 y Fc(284)370 b Fx(\(1984\),)j(121{139)-880 66194 y([5])832 b(G.)420 b(Barbatis,)434 b(S.)420 b(Filippas)h(and)f(A.)h(T)-92 b(ertik)-61 b(as)863 b Fw(A)442 b(uni\014e)-57 b(d)445 b(appr)-57 b(o)g(ach)442 b(to)g(impr)-57 b(ove)g(d)444 b Fu(L)38194 65792 y Fb(p)39149 66194 y Fw(Har)-57 b(dy)443 b(ine)-57 b(qualities)444 b(with)1119 67522 y(b)-57 b(est)396 b(c)-57 b(onstants)p Fx(,)369 b(T)-92 b(rans.)369 b(Amer.)h(Math.)g(So) 31 b(c.)370 b Fc(356)g Fx(\(2004\),)j(2169-2196.)-880 68850 y([6])832 b(A.)271 b(Bon\014glioli,)294 b(E.)271 b(Lanconelli)h(and)f(F.)g(Uguzzoni,)292 b Fw(Uniform)306 b(Gaussian)f(estimates)g(of)h(the)f(fundamental)h(solutions)1119 70179 y(for)396 b(he)-57 b(at)396 b(op)-57 b(er)g(ators)395 b(on)h(Carnot)g(gr)-57 b(oups)p Fx(,)369 b(Adv.)h(Di\013eren)-31 b(tial)370 b(Equations)h Fc(7)f Fx(\(2002\),)i(1153-1192.)-880 71507 y([7])832 b(H.)399 b(Brezis)f(and)h(J.)g(L.)g(V\023)-553 b(azquez,)407 b Fw(Blow-up)423 b(solutions)f(of)h(some)h(nonline)-57 b(ar)424 b(el)57 b(liptic)424 b(pr)-57 b(oblems)p Fx(,)406 b(Rev.)399 b(Mat.)h(Univ.)1119 72835 y(Complutense)371 b(Madrid)e Fc(10)h Fx(\(1997\),)i(443-469.)p eop %%Page: 13 13 13 12 bop -880 -714 a Fg(SHARP)443 b(WEIGHTED)f(HARD)-28 b(Y-TYPE)442 b(INEQUALITIES)f(AND)i(UNCER)-85 b(T)g(AINTY)440 b(PRINCIPLE)h(INEQUALITY)h(ON)h(CARNOT)g(GR)-28 b(OUPS)-1025 b(13)-880 1942 y Fx([8])832 b(X.)322 b(Cabr)-31 b(\023)-523 b(e)322 b(and)g(Y.)f(Martel,)332 b Fw(Existenc)-57 b(e)353 b(versus)e(explosion)h(instantane)g(p)-57 b(our)352 b(des)g(quations)g (de)g(la)f(chaleur)h(linair)-57 b(es)1119 3271 y(ave)g(c)397 b(p)-57 b(otentiel)396 b(singulier)489 b Fx(,)370 b(C.)g(R.)f(Acad.)h (Sci.)g(P)-31 b(aris)370 b(Sr.)f(I.)g(Math.,)i Fc(329)f Fx(\(1999\),)j(973-978.)-880 4599 y([9])832 b(L.)381 b(Cap)31 b(ogna,)386 b(D.)380 b(Danielli)j(and)e(N.)g(Garofalo,)796 b Fw(Cap)-57 b(acitary)407 b(estimates)f(and)h(the)f(lo)-57 b(c)g(al)407 b(b)-57 b(ehavior)408 b(of)f(solutions)f(of)1119 5927 y(nonline)-57 b(ar)397 b(sub)-57 b(el)57 b(liptic)397 b(e)-57 b(quations)p Fx(,)370 b(Amer.)f(J.)h(Math.)g Fc(118)g Fx(\(1996\),)j(no.)d(6,)g(1153-1196.)-880 7256 y([10])555 b(L.)458 b(D'Am)-31 b(brosio,)481 b Fw(Some)c(Har)-57 b(dy)477 b(ine)-57 b(qualities)479 b(on)e(the)g(Heisenb)-57 b(er)g(g)478 b(gr)-57 b(oup)p Fx(,)480 b(Di\013eren)-31 b(tial)458 b(Equations)h Fc(40)f Fx(\(2004\))1119 8584 y(no.)370 b(4,)g(552-564.)-880 9912 y([11])555 b(D.)450 b(Danielli,)473 b(N.)450 b(Garofalo)i(and)e(Duy-Minh)g(Nhieu,)471 b Fw(On)g(the)f(b)-57 b(est)470 b(p)-57 b(ossible)471 b(char)-57 b(acter)470 b(of)g(the)g(norm)h(in)g(some)1119 11241 y(a)437 b(priori)i(estimates)e(for)h(non-diver)-57 b(genc)g(e)441 b(form)d(e)-57 b(quations)439 b(in)g(Carnot)f(gr)-57 b(oups)p Fx(,)426 b(Pro)31 b(c.)415 b(Amer.)g(Math.)h(So)31 b(c.)415 b Fc(131)1119 12569 y Fx(\(2003\),)372 b(3487-3498.)-880 13897 y([12])555 b(G.)414 b(B.)h(F)-92 b(olland,)427 b Fw(A)437 b(fundamental)h(solution)f(for)h(a)f(sub)-57 b(el)57 b(liptic)438 b(op)-57 b(er)g(ator)p Fx(,)425 b(Bull.)415 b(Amer.)g(Math.)g(So)31 b(c.)415 b Fc(79)g Fx(\(1973\),)1119 15226 y(373-376.)-880 16554 y([13])555 b(G.)371 b(B.)f(F)-92 b(olland,)372 b Fw(Sub)-57 b(el)57 b(liptic)399 b(estimates)d(and)h(function)i(sp)-57 b(ac)g(es)397 b(on)g(nilp)-57 b(otent)398 b(Lie)g(gr)-57 b(oups)p Fx(,)371 b(Arkiv.)g(f)31 b(\177)-584 b(ur)370 b(Math.)h Fc(13)1119 17883 y Fx(\(1975\),)h(161-207.)-880 19211 y([14])555 b(G.)406 b(B.)f(F)-92 b(olland)407 b(and)f(A.)f(Sitaram,)417 b Fw(The)429 b(Unc)-57 b(ertainty)431 b(Principle:)577 b(A)430 b(Mathematic)-57 b(al)428 b(Survey)p Fx(,)416 b(J.)406 b(F)-92 b(ourier)404 b(Anal.)1119 20539 y(Appl.)370 b Fc(3)f Fx(\(1997\),)k(207-238.)-880 21868 y([15])555 b(G.)428 b(B.)f(F)-92 b(olland)429 b(and)e(E.)h(Stein,)443 b(\\)428 b(Hardy)f(Spaces)g(on)h(Homogeneous)h(Groups",)442 b(Princeton)428 b(Univ)-31 b(ersit)g(y)429 b(Press,)1119 23196 y(Princeton,)370 b(NJ.)-880 24524 y([16])555 b(B.)547 b(F)-92 b(ranc)-31 b(hi,)591 b(R.)546 b(Serapioni)i(and)e(F.)g(Serra)g (Cassano,)1191 b Fw(R)-57 b(e)g(gular)560 b(Hyp)-57 b(ersurfac)g(es,) 601 b(Intrinsic)560 b(Perimeter)f(and)1119 25853 y(Implicit)396 b(F)-85 b(unction)397 b(The)-57 b(or)g(em)397 b(in)g(Carnot)e(Gr)-57 b(oups)p Fx(,)370 b(Comm.)h(Anal.)g(Geom.)f Fc(11)g Fx(5)f(\(2003\),)k (909-944.)-880 27181 y([17])555 b(N.)327 b(Garofalo)i(and)d(E.)h (Lanconelli,)338 b Fw(F)-85 b(r)-57 b(e)g(quency)359 b(functions)e(on)g(the)g(Heisenb)-57 b(er)g(g)358 b(gr)-57 b(oup,)365 b(the)357 b(unc)-57 b(ertainty)358 b(principle)1119 28509 y(and)396 b(unique)h(c)-57 b(ontinuation)p Fx(,)371 b(Ann.)f(Inst.)f(F)-92 b(ourier\(Grenoble\))369 b Fc(40)h Fx(\(1990\),)j(313-356.)-880 29838 y([18])555 b(N.)291 b(Garofalo)g(and)f(D.)g(V)-92 b(assilev,)306 b Fw(Symmetry)325 b(pr)-57 b(op)g(erties)323 b(of)g(p)-57 b(ositive)324 b(entir)-57 b(e)324 b(solutions)e(of)h(Y)-85 b(amab)-57 b(e-typ)g(e)324 b(e)-57 b(quations)1119 31166 y(on)396 b(gr)-57 b(oups)396 b(of)g(Heisenb)-57 b(er)g(g)398 b(typ)-57 b(e)p Fx(,)370 b(Duk)-31 b(e)369 b(Math.)h(J.)f Fc(106)p Fx(,)i(3)f(\(2001\),)i(411-448.)-880 32494 y([19])555 b(J.)526 b(Heinonen)g(and)f(I.)g(Holopainen,)567 b Fw(Quasir)-57 b(e)g(gular)540 b(maps)f(on)g(Carnot)g(gr)-57 b(oups)p Fx(,)565 b(J.)525 b(Geom.)h(Anal.)g Fc(7)1130 b Fx(\(1997\),)1119 33823 y(109-148.)-880 35151 y([20])555 b(W.)363 b(Heisen)-31 b(b)31 b(erg,)9340 34871 y Fw(\177)9142 35151 y(Ub)-57 b(er)390 b(den)g(anschaulichen)h(Inhalt)e(der)h(quantenthe)-57 b(or)g(etischen)391 b(Kinematik)h(und)e(Me)-57 b(chanik)p Fx(,)365 b(Z.)1119 36479 y(Ph)-31 b(ysik)370 b Fw(43)518 b Fx(\(1927\),)373 b(172198.)-880 37808 y([21])555 b(D.)426 b(Jerison)g(and)g(J.)g(M.)g(Lee,)440 b Fw(Extr)-57 b(emals)447 b(for)i(the)f(Sob)-57 b(olev)449 b(ine)-57 b(quality)450 b(on)e(the)g(Heisenb)-57 b(er)g(g)450 b(gr)-57 b(oup)448 b(and)g(the)g(CR)1119 39136 y(Y)-85 b(amab)-57 b(e)396 b(pr)-57 b(oblem)p Fx(,)370 b(J.)f(Amer.)h(Math.)g(So)31 b(c.)370 b Fc(1)f Fx(\(1988\),)k(1-13.)-880 40465 y([22])555 b(A.)262 b(Kaplan,)283 b Fw(F)-85 b(undamental)297 b(solutions)e(for)h (a)g(class)f(of)i(hyp)-57 b(o)g(el)57 b(liptic)297 b(PDE)e(gener)-57 b(ate)g(d)297 b(by)g(c)-57 b(omp)g(osition)297 b(of)g(quadr)-57 b(atic)1119 41793 y(forms)p Fx(,)369 b(T)-92 b(rans.)369 b(Amer.)h(Math.)g(So)31 b(c.)370 b Fc(258)p Fx(,)h(1)e(\(1980\),)k (147-153.)-880 43121 y([23])555 b(R.)332 b(Mon)-31 b(ti)334 b(and)e(F.)g(Serra)f(Cassano,)709 b Fw(Surfac)-57 b(es)363 b(me)-57 b(asur)g(es)362 b(in)h(Carnot-Car)-57 b(eth)-28 b(\023)-537 b(eo)-57 b(dory)360 b(sp)-57 b(ac)g(es)p Fx(,)340 b(Calc.)333 b(V)-92 b(ar.)332 b(P)-31 b(artial)1119 44450 y(Di\013er.)369 b(Equations)i Fc(13)f Fx(\(2001\),)i(339-376.) -880 45778 y([24])555 b(P)-92 b(.)447 b(Niu,)467 b(H.)447 b(Zhang)h(and)f(Y.)g(W)-92 b(ang,)952 b Fw(Har)-57 b(dy)467 b(typ)-57 b(e)468 b(and)f(R)-57 b(el)57 b(lich)468 b(typ)-57 b(e)468 b(ine)-57 b(qualities)469 b(on)e(the)g(Heisenb)-57 b(er)g(g)469 b(gr)-57 b(oup)p Fx(,)1119 47106 y(Pro)31 b(c.)369 b(Amer.)h(Math.)g(So)31 b(c.)370 b Fc(129)g Fx(\(2001\),)j(3623-3630.)-880 48435 y([25])555 b(I.)428 b(P)-31 b(eral)429 b(and)f(J.)g(L.)g(V\023)-553 b(azquez,)907 b Fw(On)450 b(the)g(stability)g(or)g(instability)h(of)f(the)g(singular) g(solution)g(of)g(the)g(semiline)-57 b(ar)1119 49763 y(he)g(at)396 b(e)-57 b(quation)397 b(with)e(exp)-57 b(onential)397 b(r)-57 b(e)g(action)397 b(term)p Fx(,)369 b(Arc)-31 b(h.)370 b(Rational)i(Mec)-31 b(h.)369 b(Anal.)i Fc(129)f Fx(\(1995\),)i(201-224.)-880 51091 y([26])555 b(F.)369 b(Rellic)-31 b(h,)371 b(\\P)-31 b(erturbation)372 b(theory)e(of)f(eigen)-31 b(v)-61 b(alue)371 b(problems",)g(Gordon)e (and)g(Breac)-31 b(h,)370 b(New)g(Y)-92 b(ork,)370 b(1969.)-880 52420 y([27])555 b(A.)458 b(Sitaram,)482 b(M.Sundari)457 b(and)h(S.Thanga)-31 b(v)g(elu,)483 b Fw(Unc)-57 b(ertainty)478 b(principles)g(on)f(c)-57 b(ertain)478 b(Lie)g(gr)-57 b(oups)p Fx(,)479 b(Pro)31 b(c.)458 b(Ind.)1119 53748 y(Acad.)370 b(Sci.)g(\(Math.)g(Sci.\))h Fc(105)f Fx(\(1995\),)j (133-151.)-880 55076 y([28])555 b(E.)627 b(Stein,)692 b(\\Harmonic)628 b(Analysis,)692 b(Real-V)-92 b(ariable)628 b(Metho)31 b(ds,)691 b(Orthgonalit)-31 b(y)-92 b(,)694 b(and)627 b(Oscillatory)h(In)-31 b(tegrals",)1119 56405 y(Princeton)370 b(Univ)-31 b(ersit)g(y)371 b(Press,)d(Princeton,)j(NJ.) -880 57733 y([29])555 b(S.Thanga)-31 b(v)g(elu,)373 b Fw(Some)397 b(unc)-57 b(ertainty)398 b(ine)-57 b(qualities)p Fx(,)371 b(Pro)31 b(c.)370 b(Ind.)f(Acad.)h(Sci.)g Fc(100)g Fx(\(1990\),)j(137-145.)-880 59061 y([30])555 b(N.)370 b(V)-92 b(arop)31 b(oulos,)767 b Fw(A)-28 b(nalysis)396 b(on)g(nilp)-57 b(otent)397 b(gr)-57 b(oups)p Fx(,)369 b(J.)g(F)-92 b(unct.)369 b(Anal.)i Fc(66)f Fx(\(1986\),)i(406-431.)-880 60390 y([31])555 b(J.)377 b(L.)g(V\023)-553 b(azquez)377 b(and)g(E.)g(Zuazua,)j Fw(The)403 b(Har)-57 b(dy)403 b(c)-57 b(onstant)402 b(and)h(the)g(asymptotic)f(b)-57 b(ehaviour)404 b(of)f(the)f(he)-57 b(at)403 b(e)-57 b(quation)1119 61718 y(with)395 b(an)h(inverse-squar)-57 b(e)397 b(p)-57 b(otential)484 b Fx(,)369 b(J.)h(F)-92 b(unct.)369 b(Anal.)i Fc(173)f Fx(\(2000\),)i(103-153.)448 64209 y Fv(Ismail)384 b(K)-32 b(ombe,)390 b(Ma)-74 b(thema)g(tics)385 b(Dep)-74 b(ar)g(tment,)392 b(D)-32 b(a)-98 b(wson-Loeffler)383 b(Science)j(&Ma)-74 b(thema)g(tics)385 b(Bldg,)-880 65537 y(Oklahoma)417 b(City)h(University,)e(2501)i(N.)g(Bla)-25 b(ckwelder,)419 b(Oklahoma)e(City,)h(OK)g(73106-1493)448 66865 y Fw(E-mail)397 b(addr)-57 b(ess)91 b Fx(:)491 b Fa(ikombe@okcu.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0603150838937--