A characterization of Bloch space and Besov space
type: Item
instance of: scholarly article
title: A characterization of Bloch space and Besov space
zbMATH Open document ID: 1106.30018
author: E. G. Kwon
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: The analytic Besov space \(B_p\), \(1<p<\infty\) on the unit disk \(D\) is defined to be the space of all analytic functions \(f\) on \(D\) such that \(| f'(z)| (1-| z| ^2)\) is \(L^p\)-integrable with respect to the invariant measure. A characterization with derivatives replaced by differences is also well known. That is, it is well known that an analytic function \(f\) is a member of \(B_p\) if and only if the difference function \(f(z)-f(w)\) is \(L^p\)-integrable on \(D\times D\) with respect to the weight given by the square of the absolute value of the Bergman kernel. In this paper the author obtains a new characterization, involving both differences and derivatives, that contains those two characterizations as special cases. A similar characterization that interpolates the two well-known characterizations for the Bloch space is also obtained. The key tool is a new characterization for the Bergman space that also interpolates the very definition and the derivative integrability characterization of the Bergman space. Recently, this new characterization for the Bergman space has been also proved on higher dimensional balls independently by \textit{M. Pavlović} and \textit{K. Zhu} [New characterizations of Bergman spaces (to appear)].
reviewed by: Boo Rim Choe
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5073035
zbMATH Keywords: Besov space
zbMATH Keywords: Bloch space
zbMATH Keywords: Bergman space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.052
OpenAlex ID: W2136841314
cites work: The Bergman space, the Bloch space, and commutators of multiplication operators
cites work: Q3681112
cites work: Q5618011
cites work: Weighted Subspaces of Hardy Spaces
cites work: A New Proof of an Inequality of Littlewood and Paley
cites work: Q4354138
cites work: An Extremal Property of the Bloch Space
cites work: Besov-type characterisations for the Bloch space
cites work: The Bloch space and Besov spaces of analytic functions
cites work: Q3999007
DOI: 10.1016/J.JMAA.2006.01.052
description: scientific article; zbMATH DE number 5073035