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Functions Harmonic on a Ball
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Applying approaches assume average ball Borel subsets boundary function bounded called capacity chapter closed compact compact closure compact set component connected Consider constant containing continuous function converges Corollary defined definition denote derivative Dirichlet problem do(z easily seen equal equation example exists extended fact finite fixed follows function f Green function h is harmonic harmonic function harmonic minorant identically inequality integral Lebesgue Lemma lim inf limit mapping measure Moreover neighborhood non-negative Note obtain open set open subset PI(u polar set polar subset positive possibly potential preceding principle Proof properties prove region regular boundary point relative representation resolutive respectively satisfies sequence side signed measure solution space sufficiently superharmonic function Suppose surface term Theorem theory thin transform true uniformly zero