Compactification of Symmetric Spaces pp 186-194 | Cite as

# 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 C^{2}-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*.

## Keywords

Harmonic Function Compact Subgroup Harmonic Measure Weyl Chamber Geodesic Sphere## Preview

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