A test for membership in Lorentz spaces and some applications
Let G be a locally compact Abelian group with dual group \(\Gamma\). The authors investigate the Lorentz spaces \(L_{p,q}(G)\) for \(p=1=q\), or \(1<p<\infty\) and \(1\leq q<\infty\). The main result of the paper is the Theorem 1: Let \(\{e_{\alpha}\}_{\alpha \in D}\) be an approximate identity in \(L_ 1(G)\) with \(\sup_{\alpha}\| e_{\alpha}\| \leq 1\). (i) If \(1<p,q<\infty\) or \(p=1=q\), and if k is a continuous function on \(\Gamma\) such that each kê\({}_{\alpha}\in L_{p,q}(\Gamma)\) and \(\sup_{\alpha}\| k\hat e_{\alpha}\|_{(p,q)}<\infty\), then \(k\in L_{p,q}(\Gamma)\) and \(\| k\|_{(p,q)}=\sup_{\alpha}\| k\hat e_{\alpha}\|_{(p,q)}\), where in the case \(p=1=q\) it is further assumed that k is bounded. (ii) If \(p=1=q\), \(1<p<2\) and \(1<q<\infty\), or \(1<q\leq p=2\), and if \(\mu\in M(G)\) such that each \({\hat \mu}\)ê\({}_{\alpha}\in L_{p,q}(\Gamma)\) and \(\sup \| {\hat \mu}\hat e_{\alpha}\|_{(p,q)}<\infty\), then \(\mu\) is absolutely continuous. It concerns the multiplier functions defined on \(\Gamma\) for the multiplier operators from \(L^ 1(G)\) into \(L_{p,q}(G)\) like Theorem 3.4 stated in \textit{Y.-K. Chen} and \textit{H.-C. Lai} [ibid. 4, 247-260 (1975; Zbl 0311.43006)]. Using the result of Theorem 1, one can get a different approach to establish the multiplier theorems given by Chen and Lai [loc. cit., Theorem 3.12 and 3.13] which are restated as Theorem 2 in this paper. Some other related results are also established in the paper.
scientific article; zbMATH DE number 4082225