Holomorphic L^{p}-functions on Coverings of Strongly Pseudoconvex Manifolds
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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.
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)
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.
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=∞.
5
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
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.
series,asintheclassicalcaseweobtainthathcanbeapproximateduniformlyonKbyHp,ψ(M′)-valuedholomorphicfunctionsonM.Takingrestrictionsofthesefunctionstotheset{(r(y),y):y∈M′},weobtaintherequiredapproximationtheorem.2
References
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[Br1]A.Brudnyi,OnholomorphicL2functionsoncoveringsofstronglypseudocon-vexmanifolds.PublicationsofRIMS,toappear.
[Br2]A.Brudnyi,HartogstypetheoremsforCRL2functionsoncoveringsofstronglypseudoconvexmanifolds.NagoyaMath.J.,toappear.
[Br3]A.Brudnyi,Onholomorphicfunctionsofslowgrowthoncoveringsofstronglypseudoconvexmanifolds.J.Funct.Anal.,249(2007),no.2,354-371.
[Br4]A.Brudnyi,Representationofholomorphicfunctionsoncoveringsofpseu-doconvexdomainsinSteinmanifoldsviaintegralformulasonthesedomains.J.Funct.Anal.231(2006),418-437.
[C]H.Cartan,Surlesfonctionsdeplusieursvariablescomplexes.Lesespacesan-alytiques.Proc.Intern.CongressMathematiciansEdinbourgh1958,CambridgeUniv.Press,1960,33-52.
[GHS]M.Gromov,G.HenkinandM.Shubin,HolomorphicL2-functionsoncover-ingsofpseudoconvexmanifolds.GAFA,Vol.8(1998),552-585.
[L]J.Leiterer,HolomorphicvectorbundlesandtheOka-Grauertprinciple.Sev-eralcomplexvariables.IV.Algebraicaspectsofcomplexanalysis.AtranslationofSovremennyeproblemymatematiki.Fundamental´nyenapravleniya,Tom10,Akad.NaukSSSR,rm.,Moscow1986.Ency-clopaediaofMathematicalSciences,10.Springer-Verlag,Berlin,1990.
[R]R.Remmert,Surlesespacesanalytiquesholomorphiquements´eparablesetholo-morphiquementconvexes.C.R.Acad.Sci.Paris,243(1956),118-121.
[Ru]W.Rudin,Realandcomplexanalysis.Secondedition.McGraw-HillSeriesinHigherMathematics,NewYork,1974.
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