Addendum to the paper "A note on weighted Bergman spaces and the Cesàro operator"

Abstract

Let $H({\bf D}_{n})$ be the space of holomorphic functions on the unit polydisk ${\bf D}_{n}$, and let ${\cal L}^{p, q}_{\alpha}({\bf D}_{n})$, where $p, q > 0$, $\alpha = (\alpha_{1}, \dots, \alpha_{n})$ with $\alpha_{j} > -1$, $j = 1, \dots, n$, be the class of all measurable functions $f$ defined on ${\bf D}_{n}$ such that

$\int_{[0, 1)^{n}} M^{q}_{p}(f, r) \prod_{j=1}^{n} (1-r_{j})^{\alpha_{j}} dr_{j} < \infty$,

where $M_{p}(f, r)$ denote the $p$-integral means of the function $f$. Denote the weighted Bergman space on ${\bf D}_{n}$ by ${\cal A}^{p, q}_{\alpha}({\bf D}_{n}) = {\cal L}^{p, q}_{\alpha}({\bf D}_{n}) \cap H({\bf D}_{n})$. We provide a characterization for a function $f$ being in ${\cal A}^{p, q}_{\alpha}({\bf D}_{n})$. Using the characterization we prove the following result: Let $p > 1$, then the Cesàro operator is bounded on the space ${\cal A}^{p, p}_{\alpha}({\bf D}_{n})$.