Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 822445
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] M. Heins, The minimum modulus of a bounded analytic function, Duke Math. J. 14 (1947), 179-215. MR 0020639 (8:575e)
  • [2] W. C. Nestlerode and M. Stoll, Radial limits of $ n$-superharmonic functions in the polydisc, Trans. Amer. Math. Soc. 279 (1983), 691-703. MR 709577 (85h:32002)
  • [3] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna class, Amer. J. Math. 97 (1976), 915-936. MR 0390227 (52:11053)
  • [4] M. Stoll, Boundary limits of Green potential in the unit disc, preprint. MR 792369 (86g:31003)
  • [5] M. Tsuji, Potential theory in modern function theory, Chelsea, New York, 1959. MR 0114894 (22:5712)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 31A15, 30C85

Retrieve articles in all journals with MSC: 31A15, 30C85

Additional Information

Keywords: Green potentials, hyperbolic capacity
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society