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Boundary behavior of Green potentials. (English) Zbl 0594.31009
Let F be a relatively closed subset of the unit disc U. It is shown that the equation \(\lim \inf_{z\to 1,z\in F}(1-| z|)u(z)=0\) holds for every Green potential u in U if and only if \(\lim_{\epsilon \to 0} \eta (F\cap \{z:\quad | z-1| <\epsilon \})>0,\) where \(\eta\) (E) is the hyperbolic capacity of a subset E of U. (It turns out that the above limit is always either 0 or 1.) The result generalizes a recent theorem of M. Stoll.
The paper concludes with a discussion of the possibility of extending the work to other domains; as an example, it is mentioned that an unbounded closed set F in \(R^ 3\) has the property that \(\lim \inf_{x\to \infty,x\in F} | x|^{-1} u(x)=0\) for every Newtonian potential u if and only if F has infinite capacity.
Reviewer: D.Armitage

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
Full Text: DOI
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