Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball
type: Item
instance of: scholarly article
title: Weighted composition operators from the weighted Bergman space to the weighted Hardy space on the unit ball
zbMATH Open document ID: 1197.47040
author: Stevo Stević
author: Sei-Ichiro Ueki
published in: Applied Mathematics and Computation
publication date: 2010-02-09T00:00:00Z
review text: Let \(0<p<\infty\), \(\alpha>-1\), and \(\beta\geq 0\). The weighted Hardy space \(H_\beta^p(\mathbb{B})\) consists of all holomorphic functions on the unit ball \(\mathbb{B}\) of \(\mathbb{C}^n\) such that \(\| f\|^p_{H_\beta^p}=\sup_{0<r<1}(1-r)^\beta\int_{\partial\mathbb{B}}|f(r\zeta)|^p\,d\sigma(\zeta)<\infty\), where \(d\sigma\) is the normalized Lebesgue measure on the boundary \(\partial\mathbb{B}\) of \(\mathbb{B}\). The weighted Bergman space \(A_\alpha^p(\mathbb{B})\) consists of all holomorphic functions on \(\mathbb{B}\) such that \(\| f\|_{A_\alpha^p}^p=\int_{\mathbb{B}}|f(z)|^pc_\alpha(1-|z|^2)^\alpha\,dV(z)<\infty\), where \(dV\) is the normalized Lebesgue measure on \(\mathbb{B}\) and \(c_\alpha\) is a constant chosen such that \(\|1\|_{A_\alpha^p}=1\). For \(\beta=0\), \(H_\beta^p(\mathbb{B})\) coincides with the Hardy space \(H^p(\mathbb{B})\). For \(\alpha\geq 0\), \(H_\alpha^p(\mathbb{B})\) is a closed subspace of \(A_\alpha^p(\mathbb{B})\). The authors establish criteria for boundedness and compactness of a weighted composition operator \(uC_\phi: A_\alpha^p(\mathbb{B})\to H_\beta^q(\mathbb{B})\) defined by \(uC_\phi f=u(f\circ\phi)\), where \(\phi\) is a holomorphic self-map of \(\mathbb{B}\) and \(u\) is a holomorphic function on \(\mathbb{B}\).
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A36
zbMATH DE Number: 5668003
zbMATH Keywords: weighted composition operators
zbMATH Keywords: Bergman spaces
zbMATH Keywords: Hardy spaces
zbMATH Keywords: essential norm
reviewed by: Dmitry Kaliuzhnyi-Verbovetskyi
MaRDI profile type: Publication
full work available at URL: j.amc.2009.10.048
OpenAlex ID: W2005274865
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DOI: 10.1016/J.AMC.2009.10.048
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description: scientific article; zbMATH DE number 5668003