The Furstenberg Boundary and Bounded Harmonic Functions

  • Yves Guivarc’h
  • Lizhen Ji
  • J. C. Taylor
Part of the Progress in Mathematics book series (PM, volume 156)

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.

Keywords

Convolution Radon 

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Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Yves Guivarc’h
    • 1
  • Lizhen Ji
    • 2
  • J. C. Taylor
    • 3
  1. 1.IRMAR UFR MathématiquesUniversité de Rennes-IRennesFrance
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Department of Mathematics and StatisticsMcGill UniversityQuebecCanada

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