Theorems on the embedding of Bergman spaces in Lebesgue spaces for domains with nonsmooth boundary
type: Item
instance of: scholarly article
title: Theorems on the embedding of Bergman spaces in Lebesgue spaces for domains with nonsmooth boundary
zbMATH Open document ID: 0873.30030
DOI: 10.1007/BF02509560
author: Juan-Miguel Gracia
publication date: 1997-11-02T00:00:00Z
review text: For an open set \(G\subset\mathbb{C}\), \(G\neq\mathbb{C}\), and a positive Borel measure \(\mu\) on \(G\), let \(d(z)=\text{dist}(z,\partial G)\), \(z\in G\), \(\mu(z)= \mu(B(z,d(z)/2))\), where \(B(z,r)\) is the open disk in \(\mathbb{C}\) with center \(z\in\mathbb{C}\) and radius \(r>0\). Let \(dm_\beta= d(z)^\beta dm\), where \(m\) is the plane Lebesgue measure and \(\beta>-1\). The Bergman space \(L^p_a(G,\mu)\) is defined to be the subspace of \(L^p(G,\mu)\) consisting of analytic functions. The authors give sufficiently conditions for a domain \(G\) that for any \(p>0\), \(q>0\) and a nonnegative Borel measure \(\mu\) on \(G\) the following conditions are equivalent: (i) \(\mu(z)m_\beta(z)^{-1}\in L^{p/(p-q)}(G,m_\beta)\) if \(q<p\), and \(\sup_{z\in G}\mu(z)m_\beta(z)^{-q/p}<\infty\) if \(q\geq p\); (ii) the embedding operator \(I:L^p_a(G,m_\beta)\subset L^q(G,\mu)\) is bounded.
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 1002984
zbMATH Keywords: Bergman space
zbMATH Keywords: Lebesgue space
zbMATH Keywords: domain with nonsmooth boundary
reviewed by: Victor I. Tkachenko
MaRDI profile type: Publication
cites work: Q3999007
cites work: Embedding theorems for Bergman spaces in quasiconformal balls
full work available at URL: https://doi.org/10.1007/bf02509560
OpenAlex ID: W1984348294
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1002984