Weighted \(L^p\)-spaces on locally compact nilpotent groups
Let \(G\) be a locally compact Hausdorff group with left Haar measure \(\mu\). A measurable function \(\omega \colon G \rightarrow \mathbb{R}_+\) that is submultiplicative is known as a weight. Denote by \(L^p_{\omega}(G)$, $1 < p < \infty\), the Banach space of functions \(f \colon G \rightarrow \mathbb{C}\) that satisfy \[ \int_G \vert f \cdot \omega \vert^p d\mu < \infty. \] A weight is called diagonally bounded if there exists \(M > 0\) such that \(\omega(x)\omega(x^{-1}) < M\) for all \(x \in G\). It was shown by \textit{S. Saeki} [Ill. J. Math. 34, No. 3, 614--627 (1990; Zbl 0701.22003)] that if \(1 < p < \infty\) and \(L^p(G)\) is closed under convolution, then \(G\) is compact. \par In the paper under review the author investigates this result with \(L^p(G)\) replaced by \(L^p_{\omega} (G)\). In particular, the following result is obtained: \par Let \(G\) be a nilpotent, locally compact group and let \(\omega\) be a diagonally bounded weight. If \(L^p_{\omega}(G)\) is closed under convolution, then \(G\) is compact.
scientific article; zbMATH DE number 7364767
scientific article