Amenability and harmonic $L^p$-functions on hypergroups

Abstract

Let $K$ be a locally compact hypergroup with a left invariant Haar measure. We show that the Liouville property and amenability are equivalent for $K$ when it is second countable. Suppose that $\sigma$ is a non-degenerate probability measure on $K$, we show that there is no non-trivial $\sigma$-harmonic function which is continuous and vanishing at infinity. Using this, we prove that the space $H_{\sigma }^p(K)$ of all $\sigma$-harmonic $L^p$-functions, is trivial for all $1\le p<\infty$. Further, it is shown that $H_{\sigma }^{\infty }(K)$ contains only constant functions if and only if it is a subalgebra of $L^{\infty }(K)$. In the case where $\sigma$ is adapted and $K$ is compact, we show that $H_{\sigma }^p(K)=\mathbb{C}1$ for all $1\le p\le \infty$.