Weighted \(L^p\)-spaces on locally compact nilpotent groups
type: Item
instance of: scholarly article
title: Weighted \(L^p\)-spaces on locally compact nilpotent groups
zbMATH Open document ID: 1474.43004
author: Mateusz Krukowski
published in: Periodica Mathematica Hungarica
publication date: 2021-06-29T00:00:00Z
review text: 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.
reviewed by: Michael J. Puls
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 7364767
zbMATH Keywords: abstract harmonic analysis
zbMATH Keywords: locally compact groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: weights
MaRDI profile type: Publication
full work available at URL: s10998-020-00340-3
OpenAlex ID: W3016262477
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DOI: 10.1007/S10998-020-00340-3
description: scientific article; zbMATH DE number 7364767