A conformal inequality related to the conditional gauge theorem

McConnell, Terry R.

doi:10.1090/S0002-9947-1990-0957083-8

A conformal inequality related to the conditional gauge theorem
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by Terry R. McConnell PDF

Trans. Amer. Math. Soc. 318 (1990), 721-733 Request permission

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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