Composition operators from Bloch type spaces to Hardy and Besov spaces
type: Item
instance of: scholarly article
title: Composition operators from Bloch type spaces to Hardy and Besov spaces
zbMATH Open document ID: 0930.30031
author: Ruhan Zhao
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-07T00:00:00Z
review text: The space of all analytic functions on the unit disk \(\mathcal D\) is denoted by \(H(\mathcal D)\). Every analytic self-map \(\varphi : \mathcal D\to \mathcal D\) of the unit disk induces through composition a linear composition operator \(C_{\varphi}\) from \(H(\mathcal D)\) to itself and thus \(C_{\varphi}\) is defined by \(C_{\varphi}(f)=f\circ\varphi\) for \(f\in H(\mathcal D)\). An analytic function \(f\) on \(\mathcal D\) is said to be in the \(\alpha\)-Bloch space \(\mathcal B^{\alpha}, 0<\alpha<\infty\), if \(\|f \|_{\mathcal B^{\alpha}}=\sup_{z\in \mathcal D}|f'(z)|(1-|z|^2)^{\alpha}<\infty\). Correspondingly, \(f\) is in the little \(\alpha\)-Bloch space \(\mathcal B_0^{\alpha}\), if \(\lim_{|z|\to 1}|f'(z)|(1-|z|^2)^{\alpha}=0\). For \(0<p<\infty\) and \(-1<q<\infty\) an analytic function \(f\) is in the Besov type space \(B_{p,q}\) if \(\|f\|_{p,q}=(\int_\mathcal D|f'(z)|^p(1-|z|^2)^q dm(z))^{1/p}<\infty\), where \( dm(z)\) denotes the Lebesgue area measure on \(\mathcal D\). The author's main result is the following theorem: Let \(\varphi : \mathcal D\to \mathcal D\) be an analytic self-map, let \(0<p<\infty, -1<q<\infty\), and \(1\leq \beta<\infty\). Then the following statements are equivalent: (i) \(C_{\varphi} : \mathcal B^{1/\beta} \to B_{p\beta,q}\) is bounded (ii) \(C_{\varphi} : \mathcal B^{1/\beta} \to B_{p\beta, q}\) is compact (iii) \(C_{\varphi} : \mathcal B_0^{1/\beta} \to B_{p\beta,q}\) is bounded (iv) \(C_{\varphi} : \mathcal B_0^{1/\beta} \to B_{p\beta,q}\) is compact (v) \(\int_{\mathcal D}|\varphi'(z)|^{p\beta}(1-|z|^2)^q/ (1-|\varphi(z)|^2)^p dm(z)<\infty\). The hyperbolic Besov class \(B_p^h\) is defined as the set of analytic self-maps \(\varphi : \mathcal D\to \mathcal D\) such that \(\int_{\mathcal D}(\varphi^h(z))^p(1-|z|^2)^{p-2} dm(z)<\infty\), where \(\varphi^h(z)=|\varphi'(z)|/(1-|\varphi(z)|^2)\). Noting that \(\mathcal B^1=\mathcal B\) (the Bloch space), \(\mathcal B_0^1=\mathcal B_0\) (the little Bloch space) and \(B_{p,p…
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 1295277
zbMATH Keywords: composition operator
zbMATH Keywords: Bloch type space
zbMATH Keywords: Hardy space
zbMATH Keywords: Besov space
reviewed by: Rauno Aulaskari
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6341
OpenAlex ID: W1968372446
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DOI: 10.1006/JMAA.1999.6341
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description: scientific article; zbMATH DE number 1295277