Green functions, the fine topology and restoring coverings

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.

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