Carleson measures for variable exponent Bergman spaces
type: Item
instance of: scholarly article
title: Carleson measures for variable exponent Bergman spaces
zbMATH Open document ID: 1387.30081
author: Gerardo R. Chacón
author: Humberto Rafeiro
author: Juan Camilo Vallejo
published in: Complex Analysis and Operator Theory
publication date: 2017-12-01T00:00:00Z
review text: Let \(A^p(\mathbb{D})\), \(0< p <\infty\), denote the Bergman space of analytic functions in the unit disk \(\mathbb{D}\) in the complex plane, i.e., the space of holomorphic functions in the unit disk for which the integral \[ \int_{\mathbb{D}}|f(z)|^p dxdy \] is finite. A positive Borel measure \(\mu\) on \(\mathbb{D}\) is said to be a \textit{Carleson measure} for the Bergman space if there exists a constant \(C\) depending only on \(\mu\) such that for every function \(f\in A^p\) we have \[ \int_{\mathbb{D}}|f(z)|^p d\mu (z)\leq C\|f\|^p_{A^p(\mathbb{D})}. \] This notion was first introduced by Lenart Carleson in his proof of the Corona theorem for \(H^\infty(\mathbb{D})\). Around 1960, Carleson provided a complete characterisation of Carleson measures for the Hardy space \(H^p(\mathbb{D})\) where \(1\leq p<\infty\). He proved that a necessary and sufficient condition for \(\mu\) to be a Carleson measure is that \(\mu (S)\leq Ch\) for some constant \(C\) and all \textit{Carleson squares} \[ S=\{ re^{i\theta}: 1-h\leq r<1, |\theta -\theta_0|\leq h\} \] of side-length \(h\) at the boundary of the disk. According to a result of W. W. Hastings, the Carleson measures for the Bergman spaces have the analogous description \(\mu (S)\leq Ch^2\). There is also a description of Carleson measures in terms of the pseudo-hyperbolic metric by D. H. Luecking. In the paper under review, the authors provide a similar characterization for the variable-exponent Bergman spaces. A variable-exponent is a measurable function \(p:\mathbb{D} \to [1, \infty)\) such that its essential supremum on the unit disk is finite. The variable-exponent Bergman space \(A^{p(\cdot)}\) consists of the holomorphic functions in the unit disk for which the integral \[ \int_{\mathbb{D}}|f(z)|^{p(z)} dxdy \] is finite. By definition, a positive Borel measure \(\mu\) is said to be a Carleson measure for \(A^{p(\cdot)}(\mathbb{D})\) if there is a constant \(C\) such that \[ \| f\|_{L^{p(\cdot)}(\mathbb{D},\mu…
reviewed by: A. Abkar
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 6814632
zbMATH Keywords: variable-exponent Bergman spaces
zbMATH Keywords: Carleson measures
MaRDI profile type: Publication
full work available at URL: s11785-016-0573-0
OpenAlex ID: W2464324424
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DOI: 10.1007/S11785-016-0573-0
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description: scientific article; zbMATH DE number 6814632