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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Boundary behavior of invariant Green's potentials on the unit ball in $ {\bf C}\sp n$

Authors: K. T. Hahn and David Singman
Journal: Trans. Amer. Math. Soc. 309 (1988), 339-354
MSC: Primary 32A40; Secondary 31B25
MathSciNet review: 957075
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Abstract: Let $ p(z) = \int_B {G(z,\,w)\,d\mu (w)} $ be an invariant Green's potential on the unit ball $ B$ in $ {{\mathbf{C}}^n}\;(n \geqslant 1)$, where $ G$ is the invariant Green's function and $ \mu $ is a positive measure with $ \int_B {{{(1 - \vert w{\vert^2})}^n}\,d\mu (w) < \infty } $.

In this paper, a necessary and sufficient condition on a subset $ E$ of $ B$ such that for every invariant Green's potential $ p$,

$\displaystyle \mathop {\lim }\limits_{z \to e} \,\inf {(1 - \vert z{\vert^2})^n}p(z) = 0,\qquad e = (1,\,0,\, \ldots ,\,0)\; \in \partial B,\;z \in E,$

is given. The condition is that the capacity of the sets $ E \cap \{ z \in B\vert\;\vert z - e\vert < \varepsilon \} $, $ \varepsilon > 0$, is bounded away from 0. The result obtained here generalizes Luecking's result, see [L], on the unit disc in $ {\mathbf{C}}$.

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Keywords: Invariant Green's potential, boundary behavior, Bergman metric, Laplace-Beltrami operator, harmonic, superharmonic, capacity, energy, Brelot space
Article copyright: © Copyright 1988 American Mathematical Society