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Superharmonic extensions, mean values and Riesz measures. (English) Zbl 0785.31002
Let \(Z\) be a relatively closed polar subset of an open set \(D\) in \(\mathbb{R}^ n\). The classical extension theorem asserts that, if \(u\) is superharmonic on \(D \backslash Z\) and locally bounded below on \(D\), then \(u\) has a superharmonic extension to \(D\).
The starting point for this paper is a generalization of this result in which: (i) the set \(Z\) is no longer required to be closed, and (ii) the lower boundedness condition is replaced by the requirement that there is a positive superharmonic function \(v\) on \(D\) for which \(u/v\) has a nonnegative lower limit at all points of \(Z\). Extension results for harmonic functions are deduced from this. There then follows an extensive collection of related results based on the limiting behaviour of spherical means of superharmonic (or \(\delta\)-subharmonic) functions as the radius tends to 0. Some of these give sufficient conditions for a polar set to be positive for the Riesz measure of a \(\delta\)-subharmonic function. Several of the results involve Hausdorff measure conditions. Complementary extension results have been given independently by the reviewer [J. Lond. Math. Soc., II. Ser. 48, 515-525 (1993)].
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
Full Text: DOI
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