Non-tangential limits, fine limits and the Dirichlet integral

Gardiner, Stephen

doi:10.1090/S0002-9939-01-05952-4

Non-tangential limits, fine limits and the Dirichlet integral
HTML articles powered by AMS MathViewer

by Stephen J. Gardiner PDF

Proc. Amer. Math. Soc. 129 (2001), 3379-3387 Request permission

Abstract:

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.

References

D. H. Armitage and M. Goldstein, Tangential harmonic approximation on relatively closed sets, Proc. London Math. Soc. (3) 68 (1994), no. 1, 112–126. MR 1243837, DOI 10.1112/plms/s3-68.1.112
F. F. Bonsall, Domination of the supremum of a bounded harmonic function by its supremum over a countable subset, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 3, 471–477. MR 908454, DOI 10.1017/S0013091500026869
M. Brelot and J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) 13 (1963), no. fasc. 2, 395–415 (French). MR 196107
Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Stephen J. Gardiner, Sets of determination for harmonic functions, Trans. Amer. Math. Soc. 338 (1993), no. 1, 233–243. MR 1100694, DOI 10.1090/S0002-9947-1993-1100694-2
Stephen J. Gardiner, Harmonic approximation, London Mathematical Society Lecture Note Series, vol. 221, Cambridge University Press, Cambridge, 1995. MR 1342298, DOI 10.1017/CBO9780511526220
W. K. Hayman and T. J. Lyons, Bases for positive continuous functions, J. London Math. Soc. (2) 42 (1990), no. 2, 292–308. MR 1083447, DOI 10.1112/jlms/s2-42.2.292
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
Y. Mizuta, On the behaviour of harmonic functions near a hyperplane, Analysis 2 (1982), 203-218.
A. Stray, Simultaneous approximation in the Dirichlet space, in: Advances in Multivariate Approximation, ed. W. Haussmann et al., Wiley, Berlin, 1999, pp. 307-319.
Hans Wallin, On the existence of boundary values of a class of Beppo Levi functions, Trans. Amer. Math. Soc. 120 (1965), 510–525. MR 188473, DOI 10.1090/S0002-9947-1965-0188473-7