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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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  • 1. D. H. Armitage and M. Goldstein, Tangential harmonic approximation on relatively closed sets, Proc. London Math. Soc. (3), 68 (1994), 112-126. MR 94i:31005
  • 2. F. F. Bonsall, Domination of the supremum of a bounded harmonic function by its supremum over a countable subset, Proc. Edinburgh Math. Soc., 30 (1987), 471-477. MR 88k:31001
  • 3. M. Brelot and J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier Grenoble, 13 (1963), 395-415. MR 33:4299
  • 4. L. Carleson, Selected problems on exceptional sets, Van Nostrand, Princeton, 1967. MR 37:1576
  • 5. E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, Cambridge, 1966. MR 38:325
  • 6. J. Deny, Les potentiels d'énergie finie, Acta Math. 82 (1950), 107-183. MR 12:98e
  • 7. J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer, New York, 1983. MR 85k:31001
  • 8. S. J. Gardiner, Sets of determination for harmonic functions, Trans. Amer. Math. Soc. 338 (1993), 233-243. MR 93j:31005
  • 9. S. J. Gardiner, Harmonic approximation, London Math. Soc. Lecture Note Series 221, Cambridge Univ. Press, Cambridge, 1995. MR 96j:31001
  • 10. W. K. Hayman and T. J. Lyons, Bases for positive continuous functions, J. London Math. Soc. (2) 42 (1990), 292-308. MR 92a:31002
  • 11. N. S. Landkof, Foundations of modern potential theory , Springer, Berlin, 1972. MR 50:2520
  • 12. Y. Mizuta, On the behaviour of harmonic functions near a hyperplane, Analysis 2 (1982), 203-218.
  • 13. A. Stray, Simultaneous approximation in the Dirichlet space, in: Advances in Multivariate Approximation, ed. W. Haussmann et al., Wiley, Berlin, 1999, pp. 307-319.
  • 14. H. Wallin, On the existence of boundary values of a class of Beppo Levi functions, Trans. Amer. Math. Soc. 120 (1965), 510-525. MR 32:5911

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Additional Information

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

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

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