Holomorphic L^{p}-functions on Coverings of Strongly Pseudoc(4)

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.

AsanapplicationofTheorem1.3weprovearesultonextensionofholomorphicfunctionsfromcomplexsubmanifolds.

LetUbearelativelycompactopensubsetofaholomorphicallyconvexdomainV NcontainingCMandY V\CMbeaclosedcomplexsubmanifoldofV.WesetX:=Y∩U.Consideracoveringr:N′→N.

Theorem1.5Foreveryf∈Hp,ψ(Y′),thereisafunctionF∈Hp,ψ(U′)suchthatF=fonX′.

Remark1.6LetM Nbeastronglypseudoconvexmanifold.Asbefore,weassumethatπ1(M)=π1(N)andNisstronglypseudoconvex,aswell.ThenthereexistanormalSteinspaceXN,aproperholomorphicsurjectivemapp:N→XNwithconnected bresandpointsx1,...,xl∈XNsuchthat

p:N\

1≤i≤l p 1(xi)→XN\1≤i≤l {xi}

isbiholomorphic,see[C],[R].Byde nition,thedomainXM:=p(M) XNisstronglypseudoconvex(soitisStein).Withoutlossofgeneralitywewillassumethatx1,...,xl∈XMsothat∪1≤i≤lp 1(xi)=CM.Next,XV:=p(V)isaSteinsubdomainofXN.Now,asYwetakethepreimageunderpofaclosedcomplexsubmanifoldofXVthatdoesnotcontainpointsx1,...,xl.

AnotherapplicationofTheorem1.3isthefollowingapproximationresult.Theorem1.7LetK M\CMbeacompactholomorphicallyconvexsubsetandO M\CMbeaneighbourhoodofK.Theneveryfunctionf∈Hp,ψ(O′)canbeuniformlyapproximatedonK′inthenormofHp,ψ(K′)byholomorphicfunctionsfromHp,ψ(M′).

InthecaseofcoveringsofSteinmanifoldstheresultssimilartoTheorems1.5and1.7areprovedin[Br4,Theorems1.8,1.10].

2.ProofofTheorem1.3.

2.1.Webegintheproofwiththefollowingauxiliaryresult.

Proposition2.1Foreveryz∈M\CMandp∈[1,∞]thereisalinearoperatorTψ,z∈B(lp,ψ,z(M′),Hp,ψ(M′))suchthat

(Tψ,zh)(x)=h(x)foranyh∈lp,ψ,z(M′)andx∈r 1(z).

(InthesequelwecallsuchTψ,zalinearinterpolationoperator.)

NbeastronglypseudoconvexmanifoldcontainingProof.LetM