Weighted composition operators between different Bergman spaces of bounded symmetric domains
type: Item
instance of: scholarly article
title: Weighted composition operators between different Bergman spaces of bounded symmetric domains
zbMATH Open document ID: 1192.32002
author: Xiaofen Lv
author: Xiaomin Tang
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2009-12-09T00:00:00Z
review text: Let \(\Omega\) be a bounded symmetric domain in \(\mathbb C^n\) with Bergman kernel \(K(z,w)\), and consider the measures \(dv_\alpha:=K(z,z)^{1-\alpha} \,dv(z)\), where \(dv\) stands for the Lebesgue volume measure and \(\alpha> \frac{N-1}N\), \(N\)~being the genus of \(\Omega\). The weighted Bergman space \(A^p_\alpha\), \(0<p<\infty\), consists, by definition, of all holomorphic functions in \(L^p(\Omega,dv_\alpha)\). For \(\delta>0\), a measure \(\mu\) on \(\Omega\) is called \(\delta\)-Carleson if the expression \(\mu(E(z,r))/ v(E(z,r))^\delta\) is bounded as \(z\) ranges over~\(\Omega\); if~this expression even tends to zero as \(z\to\partial\Omega\), then one calls \(\mu\) a vanishing \(\delta\)-Carleson measure. Here \(E(z,r)\) is the ball with center \(z\) and radius \(r\) with respect to the Bergman metric. In the paper under review, the authors study the weighted composition operators \[ W_{\phi,\psi} f(z) := \psi(z) \, f(\phi(z)) \] on \(\Omega\), where \(\phi:\Omega\to\Omega\) and \(\psi:\Omega\to\mathbb C\) are holomorphic. The main results are that for \(0<p\leq q<\infty\), \(W_{\phi,\psi}:A^p_\alpha\to A^q_\beta\) is bounded if and only~if \[ \mu_{\phi,\psi,q,\beta}(E):=\int_{\phi^{-1}(E)} |\psi|^q \,dv_\beta \] is an \(\frac{\alpha q}p\)-Carleson measure (and similarly for ``compact'' and ``vanishing Carleson''); while for \(0<q<p<\infty\), \(W_{\phi,\psi}: A^p_\alpha\to A^q_\beta\) is bounded if and only if it is compact if and only if \(\mu_{\phi,\psi,q,\beta}(E(\cdot,r))/v(E(\cdot,r))^\alpha\) belongs to \(L^s(\Omega,dv_\alpha)\), where \(s=\frac p{p-q}\). Applications to multiplication and composition operators are given.
reviewed by: Miroslav Engliš
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32M15
zbMATH DE Number: 5644692
zbMATH Keywords: weighted composition operator
zbMATH Keywords: Bergman space
zbMATH Keywords: bounded symmetric domain
zbMATH Keywords: Carleson measure
MaRDI profile type: Publication
full work available at URL: s12044-009-0050-z
OpenAlex ID: W1976703879
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DOI: 10.1007/S12044-009-0050-Z
description: scientific article; zbMATH DE number 5644692