Representations of bounded harmonic functions

T. S. Mountford; S. C. Port

doi:10.1007/BF02384334

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Home > Journals > Ark. Mat. > Volume 29 > Issue 1-2 > Article

1991 Representations of bounded harmonic functions

T. S. Mountford, S. C. Port

Author Affiliations +

T. S. Mountford,¹ S. C. Port¹
¹Department of Mathematics, University of California, Los Angeles

Ark. Mat. 29(1-2): 107-126 (1991). DOI: 10.1007/BF02384334

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Abstract

An open subset D of R^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 Greenian D to be Poissonian in terms of its Martin boundary.