Vector-Valued Functions .pdf

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1、-88-ChapterVVector-ValuedFunctionsI.InthischapterweshallexploreaspectsofthetheoryofHardyclassesofvector-valuedanalytLcfunctions.Itisnotamereroutineparaphraseoftheclassicaltheory.Foronething,evensuchstandardphenomenaasthep.p.existenceofradiallimitsarenolongerassured.Nevertheless,theb

2、oundarystudiesofCh.IVmaybeextendedtothevector-valuedcasewhenonereplacestheboundaryfunctionbyasuitablyrestrictedvector-valuedRadonmeasureontheboundary.(Byavector-valuedRadonmeasureontheboundaryweunderstandaboundedlinearmapofthespaceofcomplex-valuedcontinuousfunctionsontheboundaryinto

3、thefixedBanachspaceunderconsideration.)2.Basicdefinitions.WeassumethatXisafixedBanachspaceoverC.ThenorminXwillbedenotedby,IILetfmappinganeighborhoodofapointaofaRiemannsurfaceintoXbegiven.Thenotionsoftheanalyticityoffataandtheharmonicityoffataarereferredtotheplanecasebymeansofalocalu

4、niformizerFortheplanecaseanalyticitywillbedefinedinacustomaryway-viatheexistenceofthederivate(inthestrongsense)ateachpointofaneighborhood.Theclassicalapparatusextendswithouttheintoductionofanyessentiallynewideas.Continuingintheplanecaseweintroducethenotionofharmonicityata~Cbyanyoneo

5、fthefollowingequivalentmeans:(I)theexistenceofalocalrepresentationoftheform~(z)+~(z)where~isanalyticat~and~at~,(2)thevalidityoftheLaplaceequationu-=Oinaneighborhoodofa,(3)thevalidityoftheGauss-Koebemean-zzvalueproperty,i.e.jforeachzsufficientlynear~thefunctionuundercon-sideration(as

6、sumedcontinuousinaneighborhoodof~)satisfiesu(z)=~Iu(z+reiS)deforsufficientlysmallpositiver.Weassumethattheusualprelimariesconcerningthesemattersareatourdisposal.Theirtreatmentisroutine.Using(3)weconcludethatiffisharmonic(inparticularanalytic)onan-89-opensetOofaRiemannsurface,thenilf

7、liissubharmoniconO.Whenfisana-lyticon0wehavethestrongerconclusionthatlog]lfltissubharmoniconO.Toseethis,itsufficestoconsidertheplanecasewherefisanalyticat0andtoshowthatthemean-valuepropertyholdsrelativetoO.WeintroducealinearfunctionalonXofnormIwhichsatisfiesA[f(O)]=llf(O)it.Usingthe

8、subharmonicityoflog