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

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.

HolomorphicLp-functionsonCoveringsofStrongly

PseudoconvexManifolds

arXiv:0712.4302v1 [math.CV] 28 Dec 2007AlexanderBrudnyi DepartmentofMathematicsandStatisticsUniversityofCalgary,CalgaryCanadaAbstractInthispaperwewillshowhowtoconstructholomorphicLp-functionsonunbranchedcoveringsofstronglypseudoconvexmanifolds.Also,weprovesomeextensionandapproximationtheoremsforsuchfunctions.1.Introduction.1.1.Thepresentpapercontinuesthestudyofholomorphicfunctionsofslowgrowthonunbranchedcoveringsofstronglypseudoconvexmanifoldsstartedin[Br1]-[Br3].Ourworkwasinspiredbytheseminalpaper[GHS]ofGromov,HenkinandShubinonholomorphicL2-functionsoncoveringsofpseudoconvexmanifolds.AparticularinterestinthissubjectisbecauseofitspossibleapplicationstotheShafarevichconjectureonholomorphicconvexityofuniversalcoveringsofcomplexprojectivemanifolds.Theresultsofthispaperdon’timplydirectlyanynewresultsintheareaoftheShafarevichconjecture.However,oneobtainsarichcomplexfunctiontheoryoncoveringsofstronglypseudoconvexmanifoldsthattogetherwithsomeadditional

methodsandideaswouldleadtoaprogressinthisconjecture.

Themainresultof[Br3]dealswithholomorphicL2-functionsonunbranchedcoveringsofstronglypseudoconvexmanifolds.InthepresentpaperweusethisresulttoconstructholomorphicLp-functions(p=2)onsuchcoverings.Also,weprovesomeextensionandapproximationtheoremsforthesefunctions.Inourproofsweexploitsomeideasbasedonin nite-dimensionalversionsofCartan’sAandBtheoremsoriginallyprovedbyBungart[B](seealso[L]andreferencesthereinforsomegeneralizationsofresultsofthecomplexfunctiontheorytothecaseofBanach-valuedholomorphicfunctions).

1.2.Toformulateourresultswe rstrecallsomebasicde nitions.