The Furstenberg Boundary and Bounded Harmonic Functions

Abstract

Let L denote the Laplace—Beltrami operator on X = G/K. The main purpose of this chapter is to give another, elementary, and self-contained proof of the so-called Poisson formula (see Theorem 12.10) for the integral representation of the bounded harmonic functions, i.e., solutions of the equation Lf = 0 [F3]. This was proved earlier (see Corollary 8.29), using the Martin boundary of X for λ = 0. The key to the proof, presented here, is the fact that (G, K) is a Gelfand pair. As a result it follows, see Corollary 12.9, that a bounded C²-function is harmonic if and only if it satisfies the mean-value property. This is not so easily proved as in Euclidean space because, if the rank of X is greater than one, K is not transitive on the geodesic spheres centered at o.