Composition operators on the polydisc induced by smooth symbols
type: Item
instance of: scholarly article
title: Composition operators on the polydisc induced by smooth symbols
zbMATH Open document ID: 1147.47022
author: Hyungwoon Koo
author: Michael Stessin
author: Kehe Zhu
published in: Journal of Functional Analysis
publication date: 2008-06-09T00:00:00Z
review text: Let \(\Phi\) be a holomorphic map from \(\mathbb{D}^n\) into itself, then it induces a composition operator \(C_{\Phi}\) on \(H(\mathbb{D}^n)\) defined by \(C_{\Phi}f=f\circ\Phi\). In this paper, the authors prove that for \(\Phi\in C^2(\mathbb{D}^n)\), \(C_{v}\) is bounded on weighted Bergman space \(A_\alpha^p(\mathbb{D}^n)\) if and only if the Jacobian of \(\Phi\) does not vanish on those points \(\zeta\) on the distinguished boundary \(\mathbb{T}^n\) such that \(\Phi(\zeta)\in\mathbb{T}^n\). Also, they show that there is a ``jump phenomenon'': if \(\Phi\) is smooth enough and if \(C_{\Phi}\) does not map \(A_\alpha^p(\mathbb{D}^n)\) into itself, then it does not map \(A_\alpha^p(\mathbb{D}^n)\) into the larger space \(A_\beta^p(\mathbb{D}^n)\), where \(\alpha<\beta<\alpha+\frac{m}{2n}\), and \(m\) is a positive integer such that \(n-m\) is the rank of the Jacobian matrix of \(\Phi\).
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 5285682
zbMATH Keywords: composition operator
zbMATH Keywords: polydisc
zbMATH Keywords: distinguished boundary
zbMATH Keywords: Jacobian
reviewed by: Pengyan Hu
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.03.003
OpenAlex ID: W2042630216
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DOI: 10.1016/J.JFA.2008.03.003
description: scientific article; zbMATH DE number 5285682