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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Boundary behavior of Green potentials
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by Daniel H. Luecking PDF
Proc. Amer. Math. Soc. 96 (1986), 481-488 Request permission

Abstract:

A Green potential on the unit disk $\{ \left | z \right | < 1\}$ is a function $u(z)$ of the form \[ u(z) = \int {\log \left | {\frac {{1 - \bar wz}}{{z - w}}} \right |{\text { }}} d\alpha (w),\] where $\alpha$ is a positive measure such that $\smallint (1 - \left | w \right |)d\alpha (w)$ 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 $u(z)$, \[ \liminf _{F \ni z \to 1} (1 - \left | z \right |) u(z) = 0. \] 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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • 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