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 $U^n=\{z\in {\mathbf{C}}^n||z_i|<1,i=1,...,n\}$ such that

\[ \|f\|^p_{{\cal L}_{\a}(U^n)}=\int_{U^n}|f(z)|^p\prod_{j=1}^n (1-|z_j|^2)^{\a_j}dm(z_j)<\infty, \]

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

\[ \prod_{j\in S}(1-|z_j|^2)\frac{\pt ^{|S|} f} {\prod_{j\in S}\pt z_j}\big(\chi_S(1)z_1, \chi_S(2)z_2,..., \chi_S(n)z_n\big) \]

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.