Lp -Conjecture on Hypergroups

In this paper, we study Lp -conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space Lp(K,w) to be a convolution Banach algebra, where 1<p<∞, 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 f∗g exists for all f,g∈Lp(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 Lp(K) exists.