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

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.

arXiv commentManuscript as accepted by publisher. To appear in Potential Analysis