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

Let G be a locally compact group, and let 1 <= p < infinity. Consider the weighted L-p-space L-p(G, omega) = {f : integral |f omega|(p) < infinity}, where : omega: G -> 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 = 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 of G are considered.