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CARLESON MEASURES AND REPRODUCING KERNEL THESIS IN DIRICHLET TYPE SPACES

20 pages
CARLESON MEASURES AND REPRODUCING KERNEL THESIS IN DIRICHLET-TYPE SPACES GERARDO R. CHACON, EMMANUEL FRICAIN, AND MAHMOOD SHABANKHAH Abstract. In this paper, using a generalization of a Richter and Sundberg representation theorem, we give a new characterization of Carleson measures for the Dirichlet-type space D(µ) when µ is a finite sum of point masses. A reproducing kernel thesis result is also established in this case. 1. Introduction Dirichlet-type spaces (also called local Dirichlet spaces) have been in- troduced by S. Richter [21] when investigating analytic two-isometries. This class of operators appeared for the first time in [1] in connection with the compression of a first-order differential operator to the Hardy space H2 of the unit disc. The study of two-isometries and related operators is also of interest for its relations with the theory of dilations and invariant subspaces of the shift operator on the classical Dirichlet space D [19]. It is an immediate consequence of the norm definitions that Mz, the operator of multiplication by the independant variable z, is an isometry on H2 but not on D (see Section 2 for precise defi- nitions). In fact, one can verify that Mz is an analytic two isometry on D. It is a remarkable result of S. Richter [21] that every analytic two-isometry satisfying dim Ker(T ?) = 1 is unitarily equivalent to Mz on some Dirichlet-type space D(µ).

  • hilbert spaces

  • carleson measures

  • spaces induced

  • thesis result

  • richter-sundberg representation

  • pre- liminary material concerning

  • carleson measure

  • dirichlet space


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CARLESONMEASURESANDREPRODUCINGKERNELTHESISINDIRICHLET-TYPESPACESGERARDOR.CHACO´N,EMMANUELFRICAIN,ANDMAHMOODSHABANKHAHAbstract.Inthispaper,usingageneralizationofaRichterandSundbergrepresentationtheorem,wegiveanewcharacterizationofCarlesonmeasuresfortheDirichlet-typespaceD(µ)whenµisafinitesumofpointmasses.Areproducingkernelthesisresultisalsoestablishedinthiscase.1.IntroductionDirichlet-typespaces(alsocalledlocalDirichletspaces)havebeenin-troducedbyS.Richter[21]wheninvestigatinganalytictwo-isometries.Thisclassofoperatorsappearedforthefirsttimein[1]inconnectionwiththecompressionofafirst-orderdifferentialoperatortotheHardyspaceH2oftheunitdisc.Thestudyoftwo-isometriesandrelatedoperatorsisalsoofinterestforitsrelationswiththetheoryofdilationsandinvariantsubspacesoftheshiftoperatorontheclassicalDirichletspaceD[19].ItisanimmediateconsequenceofthenormdefinitionsthatMz,theoperatorofmultiplicationbytheindependantvariablez,isanisometryonH2butnotonD(seeSection2forprecisedefi-nitions).Infact,onecanverifythatMzisananalytictwoisometryonD.ItisaremarkableresultofS.Richter[21]thateveryanalytictwo-isometrysatisfyingdimKer(T)=1isunitarilyequivalenttoMzonsomeDirichlet-typespaceD(µ).Thesespaceshavebeenstudiedeversincebyseveralauthors,seeforexample[2],[7],[8],[9],[10],[20],[24],[25],[26],[28]and[29].Inparticular,in[7],thefirstauthorintroducesanotionofcapacityadaptedtoDirichlet-typespacesandhegivesacharacterizationofCar-lesonmeasuresforD(µ)intermsofthiscapacity.Carlesonmeasures2010MathematicsSubjectClassification.Primary:46E22;Secondary:30H10,31C25.Keywordsandphrases.Dirichlet-typespaces,Carlesonmeasures,reproducingkernelthesis.ThesecondauthorispartiallysupportedbytheANRFRAB.Thethirdauthor’sresearchissupportedbyANRDYNOPandFQRNT.1
2G.R.CHACO´N,E.FRICAIN,ANDM.SHABANKHAHfortheHardyspacehaveprovedtobeobjectsoffundamentalimpor-tanceinthedevelopmentofmodernfunctiontheory.Inparticular,theyhaveappearedinareasrangingfromthecelebratedCoronaprob-lemanditssolutionbyCarleson[6],tothedevelopmentofboundedmeanoscillation(BMO)functionsbyC.FeffermanandE.Stein[12],P.Jones[15]andmanyothers.Thecharacterizationobtainedin[7]issimilartothosegivenbyD.Stegengain[30]fortheclassicalDirichletspace.Inthepresentpaper,wewillprovideanewcharacterizationoftheCarlesonmeasuresforthespaceD(µ)whenµisafinitesumofpointmasses.Aswewillsee,inthiscase,CarlesonmeasuresforD(µ)aredetermined,inaveryspecificway,fromthoseofH2.ThekeyideaisageneralizationofRichter-Sundbergrepresentationtheorem.Theothernaturalquestionweaddressisthereproducingkernelthe-sisfortheembeddingD(µ),L2(ν),whereνisapositiveBorelmea-sureontheunitdisc.RecallthatanoperatoronareproducingkernelHilbertspaceissaidtosatisfytheReproducingKernelThesis(RKT)ifitsboundednessisdeterminedbyitsbehaviouronthereproducingkernels.Ingeneral,thereisnoreasonwhythisshouldbetruebutitturnsout,aswasprovedbyL.Carleson,thatthisisindeedthecasefortheidentitymapI:H2L2(ν).Moreexplicitly,Iisbounded(compact,respectively)onH2ifandonlyifitactsasabounded(com-pact,respectively)operatorontheset{kz,zD}.ThoughthereweremanyresultsofthistypesinceCarleson’sresult,philosophicallytheideatostudy(RKT)forclassesofoperatorsingeneralreproducingkernelHilbertspacescomesfrom[14](seealso[18]).WewillshowthattheidentitymapI:D(µ)L2(ν)isanotherexampleofopera-torssatisfyingthe(RKT),inthecasewhereµisafinitesumofpointmasses.LetusmentionthatthereisanothernaturalgeneralizationoftheclassicalDirichletspaceD,theso-calledweightedDirichletspaces,wherethe(RKT)fortheembeddinginL2(ν)spaceisnotvalid(seeRemark6.3forfurtherdetails).Theplanofthepaperisthefollowing.Thenextsectioncontainspre-liminarymaterialconcerningDirichlet-typespacesandCarlesonmea-sures.Section3containsarepresentationtheoremforfunctionsinD(µ)spacescorrespondingtothecasewhereµisafinitesumofpointmasses.InSection4,wegiveanewcharacterizationofCarlesonmeasuresinDirichlet-typespacesinducedbyfinetelyatomicmeasures.Insections5and6areproducingkernelthesisfortheembeddingD(µ),L2(ν)isestablished.Finally,insection7,compactCarlesonmeasuresforDirichlet-typespacesinducedbyfinitelyatomicmeasuresarecharac-terizedintermsofthenormalizedreproducingkernelsofthespace.