Some operators on Lorentz spaces
type: Item
instance of: scholarly article
title: Some operators on Lorentz spaces
zbMATH Open document ID: 0942.43002
author: Enji Sato
published in: Hokkaido Mathematical Journal
publication date: 2000-08-16T00:00:00Z
review text: Let \(G\) be a locally compact abelian group with dual group \(\Gamma .\) For \(1\leq p,q\leq \infty \), denote by \(L(p,q)=L(p,q)(G)\) the Lorentz spaces. \textit{Y. K. Chen} and \textit{H. C. Lai} [ibid. 4, 247-260 (1975; Zbl 0311.43006)] have proved that if \(1\leq q\leq p\leq 2\) then \(M(p,q)\cong A(p,q)\), where \(A(p,q)=\left\{ f\in L^{1}|\widehat{f}\in L(p,q)(\Gamma)\right\} \) is a subspace of \(L^{1}(G)\) such that the Fourier transforms are in the Lorentz space \(L(p,q)\) and \(M(p,q)=\left\{ \mu \in M(G)|\widehat{\mu }\in L(p,q)(\Gamma)\right\} \) is a subspace of M(G) such that the Fourier-Stieltjes transforms are in the Lorentz space \(L(p,q)\) for \(1<p<\infty \) and \(1\leq q<\infty \). A question raised in the above mentioned paper by Chen and Lai was whether the condition \(1\leq q\leq p\leq 2\) can be relaxed; for instance, if \(1<p<q\leq 2\), does the identity \(M(p,q)=A(p,q)\) still hold. This paper solves the problem as shown in the following theorem: If \(1<p<2\) and \(1<q<\infty \), then \(M(p,q)=A(p,q)\). It is known that the multipliers of \(L^{1}(G)\) are characterized as the measure algebra \(M(G)\) which is related to the Fourier algebra \(A(G)\). In this paper, some multiplier properties related to the Lorentz space \(L(p,q)\) are also discussed.
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 43A25
zbMATH DE Number: 1331852
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: measure algebra
zbMATH Keywords: approximate identity
reviewed by: Hang-Chin Lai
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351001219
OpenAlex ID: W2019943095
DOI: 10.14492/HOKMJ/1351001219
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description: scientific article; zbMATH DE number 1331852