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 $L^p(G,\omega )=\{f:\int |f\omega {|}^p<\infty \}$, where $\omega :G\to \mathbb{R}$ is a positive measurable function. Under appropriate conditions on $\omega$, $G$ acts on $L^p(G,\omega )$ 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 $L^p(G,\omega )$? 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.

to appear in Monatshefte für Mathematik

Cited by in corpus (1)

• $\Gamma$-supercyclicity of families of translates in weighted $L^p$-spaces on locally compact groups