{
  "id": "0806.3073",
  "version": "v3",
  "published": "2008-06-18T19:45:34.000Z",
  "updated": "2010-09-17T01:33:33.000Z",
  "title": "Graphs of bounded degree and the $p$-harmonic boundary",
  "authors": [
    "Michael J. Puls"
  ],
  "comment": "Give a new proof for theorem 4.7. Change the style of the text in the first two sections",
  "categories": [
    "math.FA",
    "math.DG"
  ],
  "abstract": "Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous function on the $p$-harmonic boundary of $G$ can be extended to a function that is $p$-harmonic on $G$. Some properties of this boundary that are preserved under rough-isometries are also given. Now let $\\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced $\\ell^p$-cohomology of $\\Gamma$ in terms of the cardinality of its $p$-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $\\Gamma$, the $p$-harmonic boundary of $\\Gamma$ with the first reduced $\\ell^p$-cohomology of $\\Gamma$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-09-17T01:33:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J50",
      "31C20",
      "43A15"
    ],
    "keywords": [
      "harmonic boundary",
      "bounded degree",
      "translation invariant linear functionals",
      "real number greater",
      "constant functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.3073P"
    }
  }
}