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On the mean value property of finely harmonic and finely hyperharmonic functions. (English) Zbl 0709.31002

Let U be a bounded finely open set in \({\mathbb{R}}^ 2\) and let \(\tilde U\) denote its fine closure. It is shown that, if f is finely lower semicontinuous on \(\tilde U\) and finely hyperharmonic on U, then f(x)\(\geq \int_{*}f d\epsilon_ x^{CV}\) for any finely open set V with \(\tilde V\subseteq U\). (Here \(\epsilon_ x^{CV}\) denotes fine harmonic measure for V). An example is given to demonstrate that this result fails in \({\mathbb{R}}^ n\) when \(n\geq 3\).
Reviewer: S.J.Gardiner

MSC:

31A99 Two-dimensional potential theory
31C40 Fine potential theory; fine properties of sets and functions
31B99 Higher-dimensional potential theory
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References:

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