HARMONIC FUNCTION SPACES ON GROUPS

Abstract

A class of harmonic function spaces is introduced and studied, namely the spaces H p σ (G) of σ-harmonic Lp functions on a locally compact group G, for 1⩽ p ⩽ ∞ and a given complex measure σ on G of unit norm. It is shown that there is a contractive projection from Lp(G) onto H p σ (G), for 1< p⩽ ∞, and structural results for H p σ (G) are deduced. Given an adapted probability measure σ on G, a uniqueness result is proved, that the space H p σ (G) contains only constant functions, for 1⩽ p<∞. For any σ, a result on the dimension of H 1 σ (G) is proved.