Weighted Integrals of Holomorphic Functions on the Polydisc II

Abstract

\def\a{\alpha} \def\pt{\partial} Let ${\cal L}^p_{\a}(U^n)$ denote the class of all measurable functions defined on the unit polydisc $U^n=\{z\in {\bf C}^n\, \big| \;|z_i|<1,\ i=1,...,n\}$ such that

where $\a_j>-1$, $j=1,...,n$, and $dm(z_j)$ is the normalized area measure on the unit disk $U$, $H(U^n)$ the class of all holomorphic functions on $U^n$, and let ${\cal A}^p_{\a}(U^n)={\cal L}^p_{\a}(U^n) \cap H(U^n)$ (the weighted Bergman space). In this paper we prove that for $p\in (0,\infty),$ $f\in {\cal A}^p_{\a}(U^n)$ if and only if the functions

belong to the space ${\cal L}^p_{\a}(U^n)$ for every $S\subseteq \{1,2,...,n\},$ where $\chi_S(\cdot)$ is the characteristic function of $S,$ $|S|$ is the cardinal number of $S,$ and $\prod_{j\in S}\pt z_j=\pt z_{j_1}\cdots\pt z_{j_{|S|}},$ where $j_k\in S, \, k=1,...,|S|.$ This result extends Theorem 22 of Kehe Zhu in Trans. Amer. Math. Soc. 309 (1988) (1), 253 -- 268, when $p\in (0,1).$ Also in the case $p\in [1,\infty)$, we present a new proof.