Harmonic functions on nilpotent groups

Abstract

For a probability measure σ on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL∞(G) is called σ-harmonic if ∫F(st)dσ(t)=F(s), for almost alls inG. Constant functions are σ-harmonic and it is known that for abelianG all σ-harmonic functions are constant. For other groups it is known that non constant σ-harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant σ-harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.