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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)].
MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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References:
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