Arkiv för Matematik

, Volume 29, Issue 1–2, pp 107–126 | Cite as

Representations of bounded harmonic functions

  • T. S. Mountford
  • S. C. Port


An open subsetD ofR d ,d≧2, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a GreenianD to be Poissonian in terms of its Martin boundary.


Brownian Motion Harmonic Function Open Ball Harmonic Measure Geometric Boundary 
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  1. 1.
    Ancona, A., Sur un conjecture concernant la capacite et l'effilement,Seminar on harmonic analysis 1983–84, Publ. Math. Orsay 85-2. 56–91.Google Scholar
  2. 2.
    Bishop, C. J., Carleson, L., Garnett, J. B. andJones, P. W., Harmonic measures supporte curves,Pacific J. Math. 138 (1989), 233–236.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bishop, C. J., A characterization of Poissonian domains,Ark. Mat. 29 (1991), 1–24.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Port, S. C. andStone, C. J.,Brownian motion and classical potential theory, Academic Press, New York, 1978.zbMATHGoogle Scholar

Copyright information

© Institut Mittag-Leffler 1991

Authors and Affiliations

  • T. S. Mountford
    • 1
  • S. C. Port
    • 1
  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesU.S.A.

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