A convolution theorem for probability measures on finite groups (Q762402)

Let G be a finite group; P(G) be the probability measures on G. For \(\mu\in P(G)\) let G(\(\mu)\) be the subgroup of G generated by the collection of \(i^{-1}j\) where i and j vary over the support of the measure \(\mu\). The main result of this paper is the following. Theorem 1. (a) If \(\mu\in P(G)\) then there is a \(p<2\), dependent on \(\mu\), such that \(\| \mu * x\|_ 2\leq \| x\|_ p\) for every x in \(L_ p(G)\) if and only if \(G(\mu)=G\). (b) In addition, if C is a compact subset of P(G) with every \(\mu\) in C satisfying \(G(\mu)=G\), then there is a \(p<2\), dependent on C, such that \(\| \mu * x\| \leq \| x\|_ p\) for every \(\mu\in C\) and every x in \(L^ p(G)\). Similar results have been obtained by W. Beckner, S. Janson, and D. Jerison in a joint work entitled ''Convolution inequalities on the circle''.