Holomorphic L^{p}-functions on Coverings of Strongly Pseudoc(3)

In this paper we will show how to construct holomorphic L^{p}-functions on unbranched coverings of strongly pseudoconvex manifolds. Also, we prove some extension and approximation theorems for such functions.

Remark1.2LetdVM′betheRiemannianvolumeformonthecoveringM′obtainedbyaRiemannianmetricpulledbackfromN.Notethateveryf∈Hp,ψ(M′),p1≤p<∞,alsobelongstotheBanachspaceHψ(M′)ofholomorphicfunctionsgonM′withnorm z∈M′|g(z)|ψ(z)dVM′(z)p1/p.

pMoreover,onehasacontinuousembeddingHp,ψ(M′) →Hψ(M′).

Next,weintroducetheBanachspacelp,ψ,x(M′),x∈M,1≤p<∞,offunctionsgonr 1(x)withnorm

|g|p,ψ,x:= y∈r 1(x)|g(y)|pψ(y) 1/p,(1.7)

andtheBanachspacel∞,ψ,x(M′),x∈M,offunctionsgonr 1(x)withnorm

|g|∞,ψ,x:=

y∈r 1(x)sup{|g(y)|ψ(y)}.(1.8)

LetCM MbetheunionofallcompactcomplexsubvarietiesofMofcomplexdimension≥1.ItisknownthatifMisstronglypseudoconvex,thenCMisacompactcomplexsubvarietyofM.

InthesequelforBanachspacesEandF,byB(E,F)wedenotethespaceofalllinearboundedoperatorsE→Fwithnorm||·||.

Ourmainresultisthefollowinginterpolationtheorem.

Theorem1.3Supposethat M\CMisanopenSteinsubsetandK .Thenforanyp∈[1,∞]thereexistsafamily{Lz∈B(lp,ψ,z(M′),Hp,ψ(M′))}z∈ holomorphicinzsuchthat

(Lzh)(x)=h(x)

Moreover,

sup||Lz||<∞.z∈Kforanyh∈lp,ψ,z(M′)andx∈r 1(z).

AsimilarresultforMbeingaboundeddomaininaSteinmanifoldwasprovedin[Br4,Theorem1.3].

1.4.ToformulateapplicationsofTheorem1.3werecallsomede nitionsfrom[Br4].De nition1.4Letr:N′→NbeacoveringandX NbeacomplexsubmanifoldofN.ByHp,ψ(X′),X′:=r 1(X),wedenotetheBanachspaceofholomorphicfunctionsfonX′suchthatf|r 1(x)∈lp,ψ,x(N′)foranyx∈Xwithnorm

x∈Xsup|f|r 1(x)|p,ψ,x.(1.9)

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