Holomorphic L^{p}-functions on Coverings of Strongly Pseudoc(7)

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.

moreover,thereexistsalinearcontinuousmapCzofthe breEp,φ,zofEp,φ(M)overztothe breE0,z(M)ofE0(M)overzsuchthatRz Cz=id.Repeatingliterallytheargumentsof[Br4,section3.2]weobtainfromherethatforeveryz∈M\CMthereisaneighbourhoodUz M\CMofzsuchthatKerR|UzisbiholomorphictoUz×KerRzandthisbiholomorphismislinearoneveryKerRxandmapsthisspaceontox×KerRz,x∈Vz.

ThelattershowsthatthebundleEp,φ(M)islocallycomplementedinE0(M)overM\CM.Now,if M\CMisanopenSteinmanifold,thenbytheBungarttheorem

[B]basedonthepreviousstatementoneobtainsthatEp,φ(M)iscomplementedinE0(M)over ,see[Br4,section3.2]fordetails.Thismeansthatthereisa(holomorphic)homomorphismofbundlesF:Ep,φ(M)| →E0(M)| suchthatR F=id.Moreover,bythede nitionF|KisboundedoverK .

Finally,weset

Lz:=F(z),z∈M.

Thenbyde nition,everyLzisalinearcontinuousmapoflp,ψ,z(M′)intoHp,ψ(M′),thefamily{Lz}isholomorphicinz∈M,Rz Lz=id,andsupz∈K||Lz||<∞.Thiscompletestheproofofthetheorem.2

3.Proofs.

3.1.ProofofTheorem1.5.Letf∈Hp,ψ(Y′)beafunctionwhereY′satis esassumptionsofthetheorem.Theseassumptionsimplythatthereisastrongly NsuchthatV M .WeapplyTheorem1.3withpseudoconvexmanifoldM ′.ThenweconsiderthefunctionM′substitutedforM

h(z):=Lz(f|r 1(z)),z∈Y.

By[Br4,Proposition2.4]andbythepropertiesof{Lz}weobtainthathisa ′)-valuedholomorphicfunctiononY.(Itcanbewrittenasthescalarfunc-Hp,ψ(M

′.)Thusitsu cestoprovetheextensiontionofthevariables(z,w)∈Y×M

theoremfortheBanach-valuedholomorphicfunctionhonYextendingittoV.EvaluatingtheextendedBanach-valuedfunctionatthepoints(r(y),y),y∈U′,wegettherequiredfunctionF(cf.argumentsin[Br4,section4]).Now,theaboveBanach-valuedextensiontheoremfollowsdirectlyfromtheBanach-valuedversionoftheclassicalCartanBtheoremforSteinmanifoldsduetoBungart[B].2

3.2.ProofofTheorem1.7.WeretainthenotationofRemark1.6.BytheconditionsofthetheoremweobtainthatXK:=p(K)isaholomorphicallyconvexcompactsubsetofXMthatdoesnotcontainpointsxi,1≤i≤l.Thenthereisanon-degenerateanalyticpolyhedronP XOcontainingKandformedbyholomorphicfunctionsonXM.Now,forf∈Hp,ψ(O′)weconsiderthefunction

h(z):=Lz(f|r 1(z)),z∈O,

with{Lz}asinTheorem1.3.ThenhisaHp,ψ(M′)-valuedholomorphicfunctiononO.Next,weapplytoh|p 1(P)theWeilintegralformula(alsovalidforBanach-valuedholomorphicfunctions).Expandingthekernelinthisformulainananalytic

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