CARLESON MEASURES AND REPRODUCING KERNEL THESIS IN DIRICHLET TYPE SPACES

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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

Sujets

• Hilbert space
• Carleson measure

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 13
• Langue: English

Extrait

CARLESONMEASURESANDREPRODUCING KERNELTHESISINDIRICHLET-TYPESPACES GERARDOR.CHACO ´ N,EMMANUELFRICAIN, ANDMAHMOODSHABANKHAH Abstract. Inthispaper,usingageneralizationofaRichterand Sundbergrepresentationtheorem,wegiveanewcharacterization ofCarlesonmeasuresfortheDirichlet-typespace D (µ)whenµis afinitesumofpointmasses.Areproducingkernelthesisresultis alsoestablishedinthiscase. 1. Introduction Dirichlet-typespaces(alsocalledlocalDirichletspaces)havebeenin- troducedbyS.Richter[21]wheninvestigatinganalytictwo-isometries. Thisclassofoperatorsappearedforthefirsttimein[1]inconnection withthecompressionofafirst-orderdifferentialoperatortotheHardy spaceH 2 oftheunitdisc.Thestudyoftwo-isometriesandrelated operatorsisalsoofinterestforitsrelationswiththetheoryofdilations andinvariantsubspacesoftheshiftoperatorontheclassicalDirichlet space D [19].Itisanimmediateconsequenceofthenormdefinitions thatM z ,theoperatorofmultiplicationbytheindependantvariable z,isanisometryonH 2 butnoton D (seeSection2forprecisedefi- nitions).Infact,onecanverifythatM z isananalytictwoisometry on D .ItisaremarkableresultofS.Richter[21]thateveryanalytic two-isometrysatisfyingdimKer(T ∗ )=1isunitarilyequivalenttoM z onsomeDirichlet-typespace D (µ).Thesespaceshavebeenstudied eversincebyseveralauthors,seeforexample[2],[7],[8],[9],[10],[20], [24],[25],[26],[28]and[29]. Inparticular,in[7],thefirstauthorintroducesanotionofcapacity adaptedtoDirichlet-typespacesandhegivesacharacterizationofCar- lesonmeasuresfor D (µ)intermsofthiscapacity.Carlesonmeasures 2010MathematicsSubjectClassification.Primary:46E22;Secondary:30H10, 31C25. Keywordsandphrases.Dirichlet-typespaces,Carlesonmeasures,reproducing kernelthesis. ThesecondauthorispartiallysupportedbytheANRFRAB.Thethirdauthor’s researchissupportedbyANRDYNOPandFQRNT. 1

2G.R.CHACO ´ N,E.FRICAIN,ANDM.SHABANKHAH fortheHardyspacehaveprovedtobeobjectsoffundamentalimpor- tanceinthedevelopmentofmodernfunctiontheory.Inparticular, theyhaveappearedinareasrangingfromthecelebratedCoronaprob- lemanditssolutionbyCarleson[6],tothedevelopmentofbounded meanoscillation(BMO)functionsbyC.FeffermanandE.Stein[12], P.Jones[15]andmanyothers.Thecharacterizationobtainedin[7]is similartothosegivenbyD.Stegengain[30]fortheclassicalDirichlet space.Inthepresentpaper,wewillprovideanewcharacterizationof theCarlesonmeasuresforthespace D (µ)whenµisafinitesumof pointmasses.Aswewillsee,inthiscase,Carlesonmeasuresfor D (µ) aredetermined,inaveryspecificway,fromthoseofH 2 .Thekeyidea isageneralizationofRichter-Sundbergrepresentationtheorem. Theothernaturalquestionweaddressisthereproducingkernelthe- sisfortheembedding D (µ), → L 2 (ν),whereνisapositiveBorelmea- sureontheunitdisc.Recallthatanoperatoronareproducingkernel HilbertspaceissaidtosatisfytheReproducingKernelThesis(RKT) ifitsboundednessisdeterminedbyitsbehaviouronthereproducing kernels.Ingeneral,thereisnoreasonwhythisshouldbetruebutit turnsout,aswasprovedbyL.Carleson,thatthisisindeedthecase fortheidentitymapI:H 2 → L 2 (ν).Moreexplicitly,Iisbounded (compact,respectively)onH 2 ifandonlyifitactsasabounded(com- pact,respectively)operatorontheset { k z ,z ∈ D } .Thoughtherewere manyresultsofthistypesinceCarleson’sresult,philosophicallythe ideatostudy(RKT)forclassesofoperatorsingeneralreproducing kernelHilbertspacescomesfrom[14](seealso[18]).Wewillshow thattheidentitymapI: D (µ) → L 2 (ν)isanotherexampleofopera- torssatisfyingthe(RKT),inthecasewhereµisafinitesumofpoint masses.Letusmentionthatthereisanothernaturalgeneralizationof theclassicalDirichletspace D ,theso-calledweightedDirichletspaces, wherethe(RKT)fortheembeddinginL 2 (ν)spaceisnotvalid(see Remark6.3forfurtherdetails). Theplanofthepaperisthefollowing.Thenextsectioncontainspre- liminarymaterialconcerningDirichlet-typespacesandCarlesonmea- sures.Section3containsarepresentationtheoremforfunctionsin D (µ) spacescorrespondingtothecasewhereµisafinitesumofpointmasses. InSection4,wegiveanewcharacterizationofCarlesonmeasuresin Dirichlet-typespacesinducedbyfinetelyatomicmeasures.Insections 5and6areproducingkernelthesisfortheembedding D (µ), → L 2 (ν) isestablished.Finally,insection7,compactCarlesonmeasuresfor Dirichlet-typespacesinducedbyfinitelyatomicmeasuresarecharac- terizedintermsofthenormalizedreproducingkernelsofthespace.