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

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.

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.

LetM NbeadomainwithsmoothboundarybMinann-dimensionalcomplexmanifoldN,speci cally,

M={z∈N:ρ(z)<0}(1.1)

whereρisareal-valuedfunctionofclassC2( )inaneighbourhood ofthecompactset

zj(z)wj=0}.(1.3)

TheLeviformofρatz∈bMisahermitianformonTzc(bM)de nedinthelocalcoordinatesbytheformula

Lz(w, zj wk.(1.4)

ThemanifoldMiscalledstronglypseudoconvexifLz(w,

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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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

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.

2 ′Remark2.2ThefactsthatRzmapsHψ(M)intol2,ψ,z(M′)andiscontinuous

easilyfollowfromtheuniformcontinuityoflogψandthemeanvaluepropertyforplurisubharmonicfunctions.Similarlyoneobtainsthattherestrictionoperator

2 ′(M)→H2,ψ(M′),g→g|M′,iscontinuous.RM′:Hψ

Weset

Tψ,z:=RM′ Sψ,z.

ThenTψ,zistherequiredinterpolationoperatorforp=2.Letusprovetheresultforp=2.

Wewillnaturallyidentifyr 1(z)with{z}×SwhereSisthe breofr.Let{es}s∈S,es(z,t)=0fort=sandes(z,s)=(ψ(z,s)) 1/2,betheorthonormalbasisofl2,ψ,z(M′).Weset

hs,z:=Tψ,z(es)∈H2,ψ(M′).

Thenforasequencea={as}s∈S∈l2(S)wehave

ha:=

s∈S ashs,z∈H2,ψ(M′)and|ha|2,ψ≤c||a||l2(S).(2.1)

Wede neFs,z∈H1,ψ(M′)bytheformula

Fs,z(w):=ψ(z,s)h2s,z(w),

Then(2.1)yields w∈M′.(2.2)|Fs,z(w)|

s∈S

ψ(z,s)

Also,

(Tψ,za)(z,t):= ψ(w) ≤c2|a|∞,ψ,z.

asFs,z(z,t):=atψ(z,t)e2t(z,t)=at:=a(z,t).

s∈S

ThusTψ,zistherequiredinterpolationoperatorforp=∞.

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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.

Similarlyforp=1wehavefrom(2.1)withha:=hs,z

sup

w∈M|(Tψ,z(a))(y)|ψ(y):=sup y∈r 1(w) w∈M y∈r 1(w)s∈S

s∈S |as|ψ(z,s)· supsupw∈M s∈S |Fs,z(y)| asFs,z(y) ψ(y) ≤

y∈r 1(w)

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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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.

series,asintheclassicalcaseweobtainthathcanbeapproximateduniformlyonKbyHp,ψ(M′)-valuedholomorphicfunctionsonM.Takingrestrictionsofthesefunctionstotheset{(r(y),y):y∈M′},weobtaintherequiredapproximationtheorem.2

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