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


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.


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


Published: 2000
First available in Project Euclid: 27 April 2005

zbMATH: 0981.32001
MathSciNet: MR1783562

Primary: ‎32A36‎
Secondary: ‎46E15 , 47B38

Rights: Copyright © 2000 Editorial Board, Nagoya Mathematical Journal

Vol.159 • 2000
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