\def\a{\alpha} \def\pt{\partial} Let L\ap(Un) denote the class of all measurable functions defined on the unit polydisc Un={z∈Cn∣∣∣zi∣<1,i=1,...,n} such that
where \aj>−1,j=1,...,n, and dm(zj) is the normalized area measure on the unit disk U,H(Un) the class of all holomorphic functions on Un, and let A\ap(Un)=L\ap(Un)∩H(Un) (the weighted Bergman space). In this paper we prove that for p∈(0,∞),f∈A\ap(Un) if and only if the functions
belong to the space L\ap(Un) for every S⊆{1,2,...,n}, where χS(⋅) is the characteristic function of S,∣S∣ is the cardinal number of S, and ∏j∈S\ptzj=\ptzj1⋯\ptzj∣S∣, where jk∈S,k=1,...,∣S∣. This result extends Theorem 22 of Kehe Zhu in Trans. Amer. Math. Soc. 309 (1988) (1), 253 -- 268, when p∈(0,1). Also in the case p∈[1,∞), we present a new proof.
Cite this article
Stevo Stevic, Weighted Integrals of Holomorphic Functions on the Polydisc II. Z. Anal. Anwend. 23 (2004), no. 4, pp. 775–782