Green functions, the fine topology and restoring coverings

TL;DR

This paper explores different definitions of harmonic functions on compact sets, showing their equivalence, and extends related results to higher dimensions by analyzing Green functions, Jensen measures, and restoring coverings.

Contribution

It reconciles various definitions of harmonic functions and Green functions, and extends existing results to higher dimensions using Jensen measures and fine topology.

Findings

Green functions defined via decreasing domains and fine Green functions are equivalent.

Restoring coverings for harmonic functions are shown to be the same.

Extensions of previous results to higher dimensions are achieved.

Abstract

There are several equivalent ways to define continuous harmonic functions $H(K)$ on a compact set $K$ in $\mathbb R^n$. One may let $H(K)$ be the unform closures of all functions in $C(K)$ which are restrictions of harmonic functions on a neighborhood of $K$, or take $H(K)$ as the subspace of $C(K)$ consisting of functions which are finely harmonic on the fine interior of $K$. In \cite{DG74} it was shown that these definitions are equivalent. Using a localization result of \cite{BH78} one sees that a function $h\in H(K)$ if and only if it is continuous and finely harmonic on on every fine connected component of the fine interior of $K$. Such collection of sets are usually called {\it restoring}. Another equivalent definition of $H(K)$ was introduced in \cite{P97} using the notion of Jensen measures which leads another restoring collection of sets. The main goal of this paper is to reconcile the results in \cite{DG74} and \cite{P97}. To study these spaces, two notions of Green functions have previously been introduced. One by \cite{P97} as the limit of Green functions on domains $D_j$ where the domains $D_j$ are decreasing to $K$, and alternatively following \cite{F72, F75} one has the fine Green function on the fine interior of $K$. Our Theorem \ref{T:green_equiv} shows that these are equivalent notions. In Section \ref{S:Jensen} a careful study of the set of Jensen measures on $K$, leads to an interesting extension result (Corollary \ref{C:extend}) for superharmonic functions. This has a number of applications. In particular we show that the two restoring coverings are the same. We are also able to extend some results of \cite{GL83} and \cite{P97} to higher dimensions.