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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Boundary behavior of Green potentials


Author: Daniel H. Luecking
Journal: Proc. Amer. Math. Soc. 96 (1986), 481-488
MSC: Primary 31A15; Secondary 30C85
DOI: https://doi.org/10.1090/S0002-9939-1986-0822445-0
MathSciNet review: 822445
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Abstract: A Green potential on the unit disk $ \{ \left\vert z \right\vert < 1\} $ is a function $ u(z)$ of the form

$\displaystyle u(z) = \int {\log \left\vert {\frac{{1 - \bar wz}}{{z - w}}} \right\vert{\text{ }}} d\alpha (w),$

where $ \alpha $ is a positive measure such that $ \smallint (1 - \left\vert w \right\vert)d\alpha (w)$ is finite. In this note I give a necessary and sufficient condition on a relatively closed subset $ F$ of the unit disk in order that, for all such $ u(z)$,

$\displaystyle \mathop {\lim {\text{ inf}}}\limits_{F \mathrel\backepsilon z \to 1} (1 - \left\vert z \right\vert)u(z) = 0.$

The condition is that the hyperbolic capacity of the portion of $ F$ in arbitrarily small neighborhoods of 1 is bounded away from zero.

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DOI: https://doi.org/10.1090/S0002-9939-1986-0822445-0
Keywords: Green potentials, hyperbolic capacity
Article copyright: © Copyright 1986 American Mathematical Society