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VOL. 44 | 2006 Some potential theoretic results on an infinite network
A. K. Kalyani

Editor(s) Hiroaki Aikawa, Takashi Kumagai, Yoshihiro Mizuta, Noriaki Suzuki

## Abstract

The greatest harmonic minorant of a superharmonic function is determined as the limit of a sequence of solutions for discrete Dirichlet problems on finite subnetworks. Without using the Green kernel explicitly, a positive superharmonic function is decomposed uniquely as a sum of a potential and a harmonic function. The infimum of a left directed family of harmonic functions is shown to be either $-\infty$ or harmonic. As applications, we study the reduced functions and their properties. We show the existence of the Green kernel with the aid of our reduced function.

## Information

Published: 1 January 2006
First available in Project Euclid: 16 December 2018

zbMATH: 1128.31002
MathSciNet: MR2279768

Digital Object Identifier: 10.2969/aspm/04410353

Subjects:
Primary: 31C20

Keywords: greatest harmonic minorant , Hyperbolic network , Poisson integral , potential , reduced function