Abstract
There are two interrelated themes to this paper. One is the generalization of recent harmonic and superharmonic extension theorems to the case where the removable set is not relatively closed, with the simultaneous weakening of other hypotheses in the harmonic case. The other is the use of results which are well-known in geometric measure theory, to prove theorems on the relative behaviour of the spherical mean values of a δ-subharmonic and a superharmonic function, and to establish new criteria for harmonic and superharmonic extensions. Some related theorems establish sufficient conditions for a polar set to be positive for the Riesz measure of a δ-subharmonic function, a useful formula for the restriction of such a measure to the infinity set of a superharmonic function, and a condition for such a restriction to be absolutely continuous with respect to an appropriate Hausdorff measure.
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Watson, N.A. Superharmonic extensions, mean values and Riesz measures. Potential Anal 2, 269–294 (1993). https://doi.org/10.1007/BF01048511
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DOI: https://doi.org/10.1007/BF01048511