EVALUATION OF SUPERHARMONIC FUNCTIONS USING LIMITS ALONG LINES

Abstract

If u is a superharmonic function on ℝ², then

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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

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then

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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

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and the fine cluster set of f at X by

formula here