Sampling measures for Bergman spaces on the unit disk
type: Item
instance of: scholarly article
title: Sampling measures for Bergman spaces on the unit disk
zbMATH Open document ID: 0971.46014
author: Daniel H. Luecking
published in: Mathematische Annalen
publication date: 2001-11-04T00:00:00Z
review text: For the open unit disc \(\mathbb{D}\) in the complex plane and \(\alpha>-1\), define \(dA_\alpha(z)= (1-|z|^2)^\alpha dA\), where \(dA\) is integration with respect to Lebesgue area measure on \(\mathbb{D}\). The weighted Bergman space \(A^p(\alpha)\) is defined to be the collection of all analytic functions \(f\) such that \(\|f\|_{p,\alpha}= [\int_{\mathbb{D}}|f|^p dA_\alpha]^{1/p}< \infty\). A measure \(\tau\) is called a Carleson measure for \(A^p(\alpha)\) if there exists a constant \(c_1>0\) such that \(\int_{\mathbb{D}}|f|^p d\tau\leq c_1\|f\|^p_{p,\alpha}\) for all \(f\in A^p(\alpha)\). Let \(C\) denote the set of all measures \(\mu\) such that \(\tau= (1-|z|^2)^{2+\alpha}\mu\) is a Carleson measure for \(A^p(\alpha)\). Finally, a measure \(\mu\) will be called a sampling measure for \(A^p(\alpha)\) if there is a constant \(\Lambda> 0\) such that \[ {1\over\Lambda}\|f\|^p_{p,\alpha}\leq \int_{\mathbb{D}}|f(z)|^p(1-|z|^2)^{2+\alpha} d\mu(z)\leq \Lambda\|f\|^p_{p,\alpha} \] for all \(f\in A^p(\alpha)\). (The notion of a sampling measure is a continuous analogue of sets of sampling discussed by K. Seip and A. Schuster.) This paper provides a characterization of sampling measures for \(A^p(\alpha)\). The characterization is in terms of weak\(^*\) limits of Moebius transforms of the measure and reads as follows: Let \(M\) be the set of Moebius transforms \(\mathbb{D}\), and, for a fixed measure \(\mu\), \(\mu(M)= \{\mu_\phi: \phi\in M\}\) where \(\mu_\phi(E)= \mu(\phi^{-1}(E))\), \(E\) a measurable subset of \(\mathbb{D}\). Finally, let \(W_\mu\) be the set of all weak limits of measures in \(\mu(M)\), where weak limit is defined in the standard way. Then: Theorem: Let \(\mu\in C\). The following are equivalent (a) \(\mu\) is a sampling measure for \(A^p(\alpha)\). (b) There exists a \(q<p\) such that the support of every measure in \(W_\mu\) is a set of uniqueness for \(A^q(\alpha)\) [that is, a set which fails to be the zero set for any \(f\in A^q(\alpha)\)]. (…
reviewed by: Alan Gluchoff
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 1463208
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: Carleson measure
zbMATH Keywords: sampling measure
zbMATH Keywords: weak\(^*\) limits of Moebius transforms
MaRDI profile type: Publication
full work available at URL: s002080050348
OpenAlex ID: W1987827467
DOI: 10.1007/S002080050348
description: scientific article; zbMATH DE number 1463208