Green functions, the fine topology and restoring coverings
Abstract
There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in Rn. 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 DG74 it was shown that these definitions are equivalent. Using a localization result of BH78 one sees that a function h∈ 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 restoring. Another equivalent definition of H(K) was introduced in 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 DG74 and P97. To study these spaces, two notions of Green functions have previously been introduced. One by P97 as the limit of Green functions on domains Dj where the domains Dj are decreasing to K, and alternatively following F72, F75 one has the fine Green function on the fine interior of K. Our Theorem T:greenequiv shows that these are equivalent notions. In Section S:Jensen a careful study of the set of Jensen measures on K, leads to an interesting extension result (Corollary 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 GL83 and P97 to higher dimensions.