University of MaraghehSahand Communications in Mathematical Analysis2322-580712120181101$L^p$-Conjecture on Hypergroups1211303138610.22130/scma.2018.66851.256ENSeyyed MohammadTabatabaieDepartment of Mathematics, University of Qom, Qom 3716146611, Iran.0000-0003-4392-2577FaranakHaghighifarDepartment of Mathematics, University of Qom, Qom 3716146611, Iran.Journal Article20170624In 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.http://scma.maragheh.ac.ir/article_31386_a8065572b9d5671e77d5b27bb7d5b341.pdf