This paper describes – in terms of potential-theoretic properties – the necessary and sufficient conditions required for a bounded domain satisfying the capacity density condition to be, respectively: a John domain, a uniform domain and an inner uniform domain.

Let $D$ be a bounded domain in ${\bb R}^n$ . For a function $f$ on the boundary $\partial D$ , the Dirichlet solution of $f$ over $D$ is denoted by $H_D f$ , provided that such a solution exists. Conditions on $D$ for $H_D$ to transform a Hölder continuous function on $\partial D$ to a Hölder continuous function on $D$ with the same Hölder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Hölder continuity with exponent 1. It is also also proved that a bounded regular domain $D$ preserves the Hölder continuity with some exponent $\alpha$ , $0 < \alpha < 1$ , if and only if $\partial D$ satisfies the capacity density condition, which is equivalent to the uniform perfectness of $\partial D$ if $n = 2$ .

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

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