$L^p$-Conjecture on Hypergroups

Abstract

In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$. Among the other things, we also show that if $K$ is a locally compact hypergroup and $p$ is greater than 2, $K$ is compact if and only if $m(K)$ is finite and $fast g$ exists for all $f,gin L^p(K)$, where $m$ is a left Haar measure for $K$, and in particular, if $K$ is discrete, $K$ is finite if and only if the convolution of any two elements of $L^p(K)$ exists.