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

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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