Nagoya Mathematical Journal

A note on weighted Bergman spaces and the Cesáro operator

George Benke and Der-Chen Chang

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Let $B$ denote the unit ball in $\mathbb{C}^n$, and $dV(z)$ normalized Lebesgue measure on $B$. For $\alpha > -1$, define $dV^\alpha (z)=(1-|z|^2)^\alpha dV(z)$. Let ${\cal H}(B)$ denote the space of holomorhic functions on $B$, and for $0 < p <\infty$, let ${\mathcal{A}}^p(dV_\alpha)$ denote $L^p(dV_\alpha)\cap {\cal H}(B)$. In this note we characterize ${\mathcal{A}}^p(dV_\alpha)$ as those functions in ${\cal H}(B)$ whose images under the action of a certain set of differential operators lie in $L^p(dV_\alpha)$. This is valid for $1 \le p <\infty$. We also show that the Ces\`aro operator is bounded on ${\mathcal{A}}^p(dV_\alpha)$ for $0<p<\infty$. Analogous results are given for the polydisc.

Article information

Nagoya Math. J., Volume 159 (2000), 25-43.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A36: Bergman spaces
Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions 47B38: Operators on function spaces (general)


Benke, George; Chang, Der-Chen. A note on weighted Bergman spaces and the Cesáro operator. Nagoya Math. J. 159 (2000), 25--43.

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