Extended Cesàro operators on Bergman spaces
type: Item
instance of: scholarly article
title: Extended Cesàro operators on Bergman spaces
zbMATH Open document ID: 1072.47029
author: Zhangjian Hu
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-19T00:00:00Z
review text: Denote by \(H(B)\) the class of all holomorphic functions on the open unit ball \(B\) of \(\mathbb{C}^n\) and, for \(g\in H(B)\), let \(\Re g(z)=\sum_{j=1}^n z_j (\partial g/\partial z_j)\) denote the radial derivative. For \(g\in H(B)\), the extended Cesàro operator \(T_g\) with symbol \(g\) is the operator on \(H(B)\) defined by \[ T_g(f)(z)=\int_0^1 f(tz)\Re g(tz) \frac{dt}{t},\quad f\in H(B),\; z\in B. \] For a positive Lebesgue measurable function \(w\) defined on \(B\) and for \(0<p<\infty\), the weighted Bergman space \(L_{a,w}^p(B)\) is the space of all functions \(f\in H(B)\) such that \[ \int_B | f(z)| ^pw(z)dm(z)<\infty. \] The main result of the paper under review characterizes (for some class of weights \(w\)) all symbols \(g\) for which \(T_g\) is a bounded operator from the Bergman space \(L_{a,w}^p(B)\) to \(L_{a,w}^q(B)\) for \(0<p, q<\infty\). A similar characterization is given for compact operators.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 32A36
zbMATH DE Number: 2092936
zbMATH Keywords: extended Cesàro operator
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: normal weight
reviewed by: Zenon Jan Jabłoński
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.01.045
OpenAlex ID: W2084248791
cites work: An integral operator on \(H^ p\) and Hardy's inequality.
cites work: An Integral Operator on Hp
cites work: Integration operators on Bergman spaces
cites work: A note on weighted Bergman spaces and the Cesaro Operator
cites work: Q5618011
cites work: The dual of an inequality of Hardy and Littlewood and some related inequalities
cites work: Multipliers of Bergman spaces into Lebesgue spaces
cites work: The Cesaro Operator is Bounded on H p for 0 < p < 1
cites work: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation
cites work: Interpolation by functions in Bergman spaces
cites work: Q3958740
cites work: On the rate of growth of the means \(M_ p\) of holomorphic and pluriharmonic functions on bounded symmetric domains of \({\mathbb{C}}^ n\)
cites work: Boundedness of the Cesàro operator on mixed norm spaces
cites work: Composition Semigroups and the Cesàro Operator OnH p
cites work: Bloch functions in several complex variables. II.
cites work: Q4236455
cites work: The Bergman Spaces, The Bloch Space, and Gleason's Problem
DOI: 10.1016/J.JMAA.2004.01.045
description: scientific article; zbMATH DE number 2092936