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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Non-tangential limits, fine limits and the Dirichlet integral

Author: Stephen J. Gardiner
Journal: Proc. Amer. Math. Soc. 129 (2001), 3379-3387
MSC (2000): Primary 31B25
Published electronically: April 25, 2001
MathSciNet review: 1845016
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Let $B$ denote the unit ball in $\mathbb{R}^{n}.$ This paper characterizes the subsets $E$ of $B$ with the property that $ \sup_{E}h=\sup_{B}h$ for all harmonic functions $h$ on $B$ which have finite Dirichlet integral. It also examines, in the spirit of a celebrated paper of Brelot and Doob, the associated question of the connection between non-tangential and fine cluster sets of functions on $B$ at points of the boundary.

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Additional Information

Stephen J. Gardiner
Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland

PII: S 0002-9939(01)05952-4
Received by editor(s): December 17, 1999
Received by editor(s) in revised form: April 3, 2000
Published electronically: April 25, 2001
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2001 American Mathematical Society