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EVALUATION OF SUPERHARMONIC FUNCTIONS USING LIMITS ALONG LINES

Published online by Cambridge University Press:  01 March 2000

HIROAKI AIKAWA  and
STEPHEN J. GARDINER
HIROAKI AIKAWA
Affiliation:
Department of Mathematics, Shimane University, Matsue 690-8504, Japan
STEPHEN J. GARDINER
Affiliation:
Department of Mathematics, University College Dublin, Dublin 4, Ireland
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Abstract

If u is a superharmonic function on ℝ2, then

formula here

for all (x, y) ∈ ℝ2. This follows from the fact that a line segment in ℝ2 is non-thin at each of its constituent points. (See Doob [1, 1.XI] or Helms [7, Chapter 10] for an account of thin sets and the fine topology.) The situation is different in higher dimensions. For example, if u is the Newtonian potential on ℝ3 defined by

formula here

then

formula here

Corollary 2 below will show that, nevertheless, for nearly every vertical line L, the value of a superharmonic function at any point X of L is determined by its lower limit along L at X.

Throughout this paper, we let n [ges ] 3. A typical point of ℝn will be denoted by X or (X′, x), where X′ ∈ ℝn−1 and x ∈ ℝ. Given any function f[ratio ]ℝn → [−∞, +∞] and any point X, we define the vertical cluster set of f at X by

formula here

and the fine cluster set of f at X by

formula here

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 32 , Issue 2 , March 2000 , pp. 209 - 213
Copyright
© The London Mathematical Society 2000

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Footnotes

This work was supported in part by Grant-in-Aid for Scientific Research (B) No. 09440062, Japanese Ministry of Education, Science and Culture. The second author is grateful for the hospitality of the Department of Mathematics at Shimane University.