Density of translates in weighted $L^p$ spaces on locally compact groups

Abstract

Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,ω)=\{f:\int|fω|^p<\infty\}$, where $ω:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on $ω$, $G$ acts on $L^p(G,ω)$ by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in $L^p(G,ω)$? H. Salas (1995) gave a criterion of hypercyclicity in the case $G=\mathbb Z$ . Under mild assumptions, we present a corresponding characterization for a general locally compact group $G$. Our results are obtained in a more general setting when the translations only by a subset $S\subset G$ are considered.