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

Abstract

Let G be a locally compact group, and let $1\leqslant p < \infty$. Consider the weighted $L^p$-space $L^p(G,\omega )=\{f:\int |f\omega |^p<\infty \}$, where $\omega :G\rightarrow \mathbb {R}$ 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 $L^p(G,\omega )$? Salas (Trans Am Math Soc 347:993–1004, 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.