Convergence in variable Lebesgue spaces
type: Item
instance of: scholarly article
title: Convergence in variable Lebesgue spaces
zbMATH Open document ID: 1202.42036
DOI: 10.5565/PUBLMAT_54210_08
author: A. Fiorenza
author: D. Cruz-Uribe
published in: Publicacions Matemàtiques
publication date: 2010-08-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pm/1277731541
review text: Given a measurable set \(\Omega\subset\mathbb R^n\), an exponent function is a measurable function \(p(\cdot): \Omega\to [1,\infty] \). Let \(p_-= \text{ess\,inf}_{x \in \Omega}\, p(x)\) and \(p_+= \text{ess\,sup}_{x \in\Omega}\, p(x)\). We define the modular functional \[ \rho(f)= \int_\Omega |f(x)|^{p(x)}\,dx, \] and define the norm \[ \|f\|_{p(\cdot)}= \inf \{ \lambda >0: \rho(f/\lambda) leq 1 \}. \] Given a sequence \(\{ f_k \} \subset L^{p(\cdot)}(\Omega)\), we say that it converges in norm to \(f\in L^{p(\cdot)}(\Omega)\) if \(\|f - f_k\|_{p(\cdot)} \to 0\) as \(k \to \infty\). We say that \(f_k \to f\) in modular if there exists \(\beta >0\) such that \(\rho( \beta (f - f_k))\to 0\) as \(k \to \infty\). When \(p_{+} < \infty\), \textit{X. Fan} and \textit{D. Zhao} [J. Math. Anal. Appl. 263, No.~2, 424--446 (2001; Zbl 1028.46041)] proved that the following are equivalent: (1) \(f_k \to f\) in norm, (2) \(f_k \to f\) in modular, (3) \(f_k \to f \) in measure and for some \(\gamma >0\), \(\rho(\gamma f_k) \to \rho(\gamma f)\). When \(p_{+} = \infty\) the situation is much more complicated, and these three conditions are not equivalent in general. The authors examine each of these equivalences including the case \(p_{+}= \infty\). For this purpose they define \[ \rho (f) = \int_{\Omega \setminus \Omega_{\infty}}|f(x)|^{p(x)}\,dx+\|f \|_{L^{\infty}(\Omega_{\infty})}, \] where \(\Omega_{\infty} = \{ x \in \Omega : p(x) = \infty \}\).
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5770008
zbMATH Keywords: variable Lebesgue space
zbMATH Keywords: norm convergence
zbMATH Keywords: modular convergence
zbMATH Keywords: convergence in measure
reviewed by: Yasuo Komori-Furuya
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5770008