2010/12/17 by Tony L. Perkins, Perkins, Tony L.
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Primary: 31B05 #Secondary: 31C40 #math.CA #msc:31B05 #msc:31C40
paper · pdf · doi:10.48550/arxiv.1012.3977
arxiv created 2010/12/17 · openalex publication_date 2010/12/17 · arxiv updated 2010/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in \mathbb 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 \citeDG74 it was shown that these definitions are equivalent. Using a localization result of \citeBH78 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 \it restoring. Another equivalent definition of H(K) was introduced in \citeP97 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 \citeDG74 and \citeP97. To study these spaces, two notions of Green functions have previously been introduced. One by \citeP97 as the limit of Green functions on domains Dj where the domains Dj are decreasing to K, and alternatively following \citeF72, F75 one has the fine Green function on the fine interior of K. Our Theorem \refT:greenequiv shows that these are equivalent notions. In Section \refS:Jensen a careful study of the set of Jensen measures on K, leads to an interesting extension result (Corollary \refC: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 \citeGL83 and \citeP97 to higher dimensions.