Convolution operators on Banach-Orlicz algebra
For a locally compact group \(G\), let \(\mathcal{L}^{\Phi} (G)\) and \(\mathcal{L}_{\omega}^{\Phi} (G)\) denote the Orlicz and weighted Orlicz spaces, respectively, where \(\Phi\) is a Young function and \(\omega\) is a weight on \(G\). The authors study harmonic and convolution operators on Orlicz and weighted Orlicz spaces. In Section 4, they consider the harmonic operators on \(\mathcal{L}^{\Phi} (G)\) and study their compactness. In Proposition 4.4, they prove that, for a compact group \(G\) with normalized Haar measure \(\lambda\), and \(\mu\in M(G)\) and \(\Phi\in \Delta'\) satisfying a certain extra hypotheses, then a harmonic operator \(T_{\mu}\) on \(\mathcal{L}^{\Phi} (G)\) is weakly compact if and only if it is compact if and only if \(\mu\) is absolutely continuous. Compact harmonic operators on \(\mathcal{L}_{\omega}^{\Phi} (G)\) are characterized in Theorem 4.14. The last section is devoted to characterizing the convolution operators on Orlicz and weighted Orlicz spaces.
scientific article; zbMATH DE number 7254982
scientific article