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1、ARTICLEINPRESSJournalofFunctionalAnalysis217(2004)192–220HolomorphicSobolevspacesandthegeneralizedSegal–Bargmanntransforma,1b,BrianC.HallandWicharnLewkeeratiyutkulaDepartmentofMathematics,UniversityofNotreDame,NotreDame,Indiana46556-4618,USAbDepartmentofMathe
2、matics,FacultyofScience,ChulalongkornUniversity,Bangkok10330,ThailandReceived23November2003;accepted12March2004Availableonline1July2004CommunicatedbyL.GrossAbstractWeconsiderthegeneralizedSegal–BargmanntransformCtforacompactgroupK;introducedinHall(J.Funct.Anal
3、.122(1994)103).LetKCdenotethecomplexificationofK:Wegiveanecessary-and-sufficientpointwisegrowthconditionforaholomorphicfunctiononKtobeintheimageunderCofCNðKÞ:WealsocharacterizetheimageunderCofSobolevCttspacesonK:TheproofsmakeuseofaholomorphicversionoftheSoboleve
4、mbeddingtheorem.r2004ElsevierInc.Allrightsreserved.1.IntroductionandstatementofresultsTheSegal–Bargmanntransform,inaformconvenientforthepurposesofthispaper,isthemapC:L2ðRdÞ-HðCdÞgivenbytZd=22CfðzÞ¼ð2ptÞeðzxÞ=2tfðxÞdx;zACd:ð1ÞtRd222dHereðzxÞ¼ðz1x1Þþ?þðzdx
5、dÞandHðCÞdenotesthespaceof(entire)dholomorphicfunctionsonC:Itiseasilyverifiedthattheintegralin(1)isabsolutelyCorrespondingauthor.E-mailaddresses:bhall@nd.edu(B.C.Hall),wicharn.l@chula.ac.th(W.Lewkeeratiyutkul).1SupportedinpartbytheNSFGrantDMS-0200649.0022-1236
6、/$-seefrontmatterr2004ElsevierInc.Allrightsreserved.doi:10.1016/j.jfa.2004.03.018ARTICLEINPRESSB.C.Hall,W.Lewkeeratiyutkul/JournalofFunctionalAnalysis217(2004)192220193dconvergentforallzACandthattheresultisaholomorphicfunctionofz:IfwerestrictattentiontozARd;th
7、enwemayrecognizethefunctiond=22ð2ptÞeðzxÞ=2tð2ÞastheheatkernelforRd;thatis,theintegralkernelforthetime-theatoperator.ThismeansthatCtfmayalternativelybedescribedasCf¼analyticcontinuationofetD=2f:ð3ÞtHeretheanalyticcontinuationisfromRdtoCdwithtfixed,andetD=2is
8、thetime-t(forward)heatoperator.(WetaketheLaplaciantobeanegativeoperatorandfollowtheprobabilists’normalizationoftheheatoperator.)Theorem1(Segal–Bargmann).Foreacht40;themapCisaunitar
