{
  "id": "1012.3977",
  "version": "v1",
  "published": "2010-12-17T20:28:08.000Z",
  "updated": "2010-12-17T20:28:08.000Z",
  "title": "Green functions, the fine topology and restoring coverings",
  "authors": [
    "Tony L. Perkins"
  ],
  "categories": [
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-17T20:28:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B05",
      "31C40"
    ],
    "keywords": [
      "restoring coverings",
      "fine topology",
      "fine interior",
      "jensen measures",
      "equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3977P"
    }
  }
}