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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

Harmonic Function Compact Subgroup Harmonic Measure Weyl Chamber Geodesic Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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