Martin boundary of a fine domain and a Fatou-Naim-Doob theorem for finely superharmonic functions

TL;DR

This paper constructs the Martin boundary for fine domains in Euclidean space, develops an integral representation for non-negative finely superharmonic functions using the Riesz-Martin kernel, and proves a Fatou-Naim-Doob theorem in this context.

Contribution

It introduces the Martin compactification for fine domains and establishes a new Fatou-Naim-Doob theorem for finely superharmonic functions.

Findings

Martin boundary constructed for fine domains

Integral representation of finely superharmonic functions obtained

Fatou-Naim-Doob theorem proved in the fine potential theory setting

Abstract

We construct the Martin compactification ${\bar U}$ of a fine domain $U$ in $R^n$, $n\ge 2$, and the Riesz-Martin kernel $K$ on $U \times{\bar U}$. We obtain the integral representation of finely superharmonic fonctions $\ge 0$ on $U$ in terms of $K$ and establish the Fatou-Naim-Doob theorem in this setting.