Some operators on Lorentz spaces
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.
scientific article; zbMATH DE number 1331852