Inner functions in the hyperbolic little Bloch class
type: Item
instance of: scholarly article
title: Inner functions in the hyperbolic little Bloch class
zbMATH Open document ID: 0976.30018
author: Wayne Smith
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: The hyperbolic derivative of an analytic self-map \(\varphi\:{\mathcal D}\to{\mathcal D}\) of the unit disk is given by \(|\varphi'|/(1-|\varphi|^2)\). An analytic self-map \(\varphi\) of~\({\mathcal D}\) induces a composition operator \(C_\varphi\:H({\mathcal D})\to H({\mathcal D})\) defined by \(C_\varphi f=f\circ \varphi\). Here \(H({\mathcal D})\) is the space of analytic functions on~\({\mathcal D}\). An analytic function~\(f\) on~\({\mathcal D}\) is said to belong to the Bloch space \({\mathcal B}\) provided that \(\sup_{z\in{\mathcal D}}(1-|z|^2)|f'(z)|<\infty\). Similarly, \(f\in{\mathcal B}_0\), the little Bloch space, if \(\lim_{|z|\to 1}(1-|z|^2)|f'(z)|=0\). The hyperbolic Bloch class \({\mathcal B}^h\) is defined in the following way: \(\varphi\in{\mathcal B}^h\) if \(\varphi\:{\mathcal D}\to{\mathcal D}\) is analytic and \[ \sup_{z\in{\mathcal D}}\tau_{\varphi}(z)= \sup_{z\in{\mathcal D}}\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}<\infty. \] Similarly, \(\varphi\in{\mathcal B}_0^h\), the hyperbolic little Bloch class, if \(\varphi\in{\mathcal B}^h\) and \[ \lim_{|z|\to 1}\tau_{\varphi}(z)=\lim_{|z|\to 1}\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}=0. \] It is well known that \(C_\varphi\:{\mathcal B}\to{\mathcal B}\) is bounded for every analytic self-map of~\({\mathcal D}\), while \(C_\varphi\:{\mathcal B}_0\to{\mathcal B}_0\) is bounded if and only if \(\varphi\in{\mathcal B}_0\). Recently \textit{K. Madigan} and \textit{A. Matheson} [Trans. Am. Math. Soc. 347, No. 7, 2679-2687 (1995; Zbl 0826.47023)] proved that \(C_\varphi\:{\mathcal B}_0\to{\mathcal B}_0\) is compact if and only if \(\varphi\in{\mathcal B}_0^h\). Earlier in [Trans. Am. Math. Soc. 348, No. 6, 2331-2348 (1996; Zbl 0857.47020)] the author showed that if \(\varphi\) is univalent and in \({\mathcal B}_0^h\), then \(C_\varphi\:L_a^p\to H^q\) is compact for all \(0<p<q<\infty\). Here \(L_a^p\) and \(H^q\) are the classical Bergman and Hardy spaces. It is known that \({\mathcal B}_0…
reviewed by: Rauno Aulaskari
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 1548257
zbMATH Keywords: hyperbolic Bloch and little Bloch class
zbMATH Keywords: composition operator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132085
OpenAlex ID: W1980310393
DOI: 10.1307/MMJ/1030132085
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description: scientific article; zbMATH DE number 1548257