Weighted Integrals of Holomorphic Functions on the Polydisc II

Abstract

Let ${\mathcal{L}}_{\alpha }^p(U^n)$ denote the class of all measurable functions defined on the unit polydisc $U^n=\{z\in {\mathbf{C}}^n||z_i|<1,i=1,...,n\}$ such that

where ${\alpha }_j>-1$, $j=1,...,n$, 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 ${\mathcal{A}}_{\alpha }^p(U^n)={\mathcal{L}}_{\alpha }^p(U^n)\cap H(U^n)$ (the weighted Bergman space). In this paper we prove that for $p\in (0,\infty ),$ $f\in {\mathcal{A}}_{\alpha }^p(U^n)$ if and only if the functions

belong to the space ${\mathcal{L}}_{\alpha }^p(U^n)$ 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 ${\prod }_{j\in S}\partial z_j=\partial z_{j_1}\cdots \partial z_{j_{|S|}},$ 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.