Rate of decay of concentration functions for spread out measures

Abstract

Let $G$ be a locally compact unimodular group and $\mu$ an adapted spread out probability measure on $G$. We relate the rate of decay of the concentration functions associated with $\mu$ to the growth of a certain subgroup $N_{\mu}$ of $G$. In particular, we show that when $\mu$ is strictly aperiodic (i.e., when $N_{\mu}=G$) and $G$ satisfies the growth condition $V_G(m)\geq Cm^D$, then for any compact neighborhood $K\subset G$ we have $\sup_{g\in G}\mu^{*n}(gK) \leq C'n^{-D/2}$. This extends recent results of Retzlaff \cite{Retzlaff} on discrete groups for adapted probability measures.