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 σ is a non-degenerate probability measure on K, we show that there is no non-trivial σ-harmonic function which is continuous and vanishing at infinity. Using this, we prove that the space H p σ (K) of all σ-harmonic L p-functions is trivial for all 1 ≤ p < ∞. Further, it is shown that H ∞ σ (K) contains only constant functions if and only if it is a subalgebra of L ∞ (K). In the case where σ is adapted and K is compact, we show that H p σ (K) = C1 for all 1 ≤ p ≤ ∞.