Characterization of the Orlicz spaces whose convergence is equivalent to convergence in measure on reflexive subspaces
type: Item
instance of: scholarly article
title: Characterization of the Orlicz spaces whose convergence is equivalent to convergence in measure on reflexive subspaces
zbMATH Open document ID: 1469.46026
DOI: 10.1134/S0037446619040141
author: S. I. Strakhov
published in: Siberian Mathematical Journal
publication date: 2019-11-27T00:00:00Z
review text: Consider a (closed) subspace \(Y\) of a rearrangement invariant space of integrable functions \(X\) on \([0, 1]\). The subspace \(Y\) is said to be strongly embedded in \(X\) if the norm convergence of \(Y \) is equivalent to convergence in measure. \(F\) is an increasing convex function on \([0, \infty)\), $F(0) = 0$, and \(L_F\) is the corresponding Orlicz space on the interval \([0, 1]\). As was shown by \textit{E. Lavergne} [Colloq. Math. 113, No. 2, 333--340 (2008; Zbl 1152.46018)], the condition \[ \lim_{t \to \infty} \frac{\tilde{F}(ct)}{\tilde{F}(t)} = \infty \tag{1} \] for some \(c > 1\), where \(\tilde{F}\) is the complementary function to \(F, \) implies that every reflexive subspace of \(L_F\) is strongly embedded in \(L_F\). The goal of this article is to find necessary and sufficient conditions on \(F\) under which every reflexive subspace of \(L_F\) is strongly embedded in \(L_F\). Condition (1) on \(F\) guarantees ``proximity'' of the Orlicz space \(L_F\) to \(L_1\). Theorem~1, as the principal result of this article, demonstrates that all reflexive subspaces of an Orlicz space which are sufficiently ``close'' to \(L_\infty\) are also strongly embedded. Other interesting related results are also shown.
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 7137341
zbMATH Keywords: rearrangement invariant space
zbMATH Keywords: reflexive subspace
zbMATH Keywords: Orlicz space
zbMATH Keywords: Orlicz-Matuszewska indices
zbMATH Keywords: convergence in measure
MaRDI profile type: Publication
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full work available at URL: s0037446619040141
OpenAlex ID: W2968207237
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description: scientific article; zbMATH DE number 7137341