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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

MSC:
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
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[1] Maurice Heins, The minimum modulus of a bounded analytic function, Duke Math. J. 14 (1947), 179 – 215. · Zbl 0030.05001
[2] W. C. Nestlerode and M. Stoll, Radial limits of \?-subharmonic functions in the polydisc, Trans. Amer. Math. Soc. 279 (1983), no. 2, 691 – 703. · Zbl 0552.31004
[3] Joel H. Shapiro and Allen L. Shields, Unusual topological properties of the Nevanlinna class, Amer. J. Math. 97 (1975), no. 4, 915 – 936. · Zbl 0323.30033 · doi:10.2307/2373681 · doi.org
[4] Manfred Stoll, Boundary limits of Green potentials in the unit disc, Arch. Math. (Basel) 44 (1985), no. 5, 451 – 455. · Zbl 0553.31003 · doi:10.1007/BF01229328 · doi.org
[5] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. · Zbl 0087.28401
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