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Transactions of the American Mathematical Society

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A conformal inequality related to the conditional gauge theorem


Author: Terry R. McConnell
Journal: Trans. Amer. Math. Soc. 318 (1990), 721-733
MSC: Primary 60J45; Secondary 31A05, 60J65
DOI: https://doi.org/10.1090/S0002-9947-1990-0957083-8
MathSciNet review: 957083
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Abstract: We prove the inequality $ h{(x)^{ - 1}}G(x,y)h(y) \leqslant cG(x,y) + c$, where $ G$ is the Green function of a plane domain $ D,\;h$ is positive and harmonic on $ D$, and $ c$ is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains $ c$ may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant $ c$ in the above inequality is finite.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0957083-8
Keywords: Green functions, conditioned Brownian motion, conditional gauge theorems, Schrödinger operators, positive harmonic functions
Article copyright: © Copyright 1990 American Mathematical Society

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