Multipliers for the weighted Orlicz spaces of a locally compact abelian group
type: Item
instance of: scholarly article
title: Multipliers for the weighted Orlicz spaces of a locally compact abelian group
zbMATH Open document ID: 1481.43001
author: Rüya Üster
published in: Results in Mathematics
publication date: 2021-08-19T00:00:00Z
review text: Throughout this review \(G\) will always denote a locally compact abelian group, and \(\omega\) will always be a weight function on \(G\). Denote by \(L^{\Phi}(G, \omega)\) the weighted Orlicz space with Young function \(\Phi\). All Young functions in this review will satisfy the \(\Delta_2\)-condition. We will use \(\ast\) to indicate convolution of measurable functions on \(G\). With respect to convolution, \(L^{\Phi} (G, \omega)\) is an essential Banach \(L^1(G, \omega)\)-module. The left translation of a function \(f\) is defined to be \(L_af(x) = f(x-a)\), where \(x\) and \(a\) are in \(G\). Suppose that \(\Phi_1\) and \(\Phi_2\) are Young functions on \(G\). A bounded linear operator \(T\) from \(L^{\Phi_2}(G, \omega)\) to \(L^{\Phi_1}(G, \omega)\) is called a left multiplier for the pair \((L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\) if \(T(L_x f) = L_x(Tf)\) for all \(f \in L^{\Phi_2} (G, \omega)\) and \(x \in G\). The set of left multipliers for the pair \((L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\) will be denoted by \(M(L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\). In the paper under review, the author investigates \(M(L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\) and proves several interesting results. For example, it is shown that \(T\) is in \(M(L^{\Phi_2}(g, \omega), L^{\Phi_1}(G, \omega))\) if and only if \(T( h \ast f) = h \ast T(f)\) for all \(h \in L^1(G, \omega)\), and \(f \in L^{\Phi_2}(g, \omega)\). Using this result the author then proves the following: For a bounded linear operator \(T\) from \(L^1(G, \omega)\) to \(L^{\Phi}(G, \omega)\), the following are equivalent: \begin{itemize} \item[1.] \(T \in M(L^1(G, \omega), L^{\Phi}(G, \omega))\). \item[2.] There exists an \(f \in L^{\Phi}(G, \omega)\) such that \(T(h) = h \ast f\) for \(h \in L^1(G, \omega)\). \end{itemize}
reviewed by: Michael J. Puls
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 7383833
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: weighted Orlicz space
zbMATH Keywords: invariant set
zbMATH Keywords: convex set
zbMATH Keywords: Banach module
zbMATH Keywords: compact multiplier
MaRDI profile type: Publication
full work available at URL: s00025-021-01493-4
OpenAlex ID: W3193727636
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DOI: 10.1007/S00025-021-01493-4
description: scientific article; zbMATH DE number 7383833