{
  "id": "1001.1087",
  "version": "v1",
  "published": "2010-01-07T15:20:09.000Z",
  "updated": "2010-01-07T15:20:09.000Z",
  "title": "A Liouville type theorem for Carnot groups",
  "authors": [
    "Alessandro Ottazzi",
    "Ben Warhurst"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "L. Capogna and M. Cowling showed that if $\\phi$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $\\phi$ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that $\\phi$ is in fact $C^\\infty$. As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2. In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-07T15:20:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30L10",
      "20F18"
    ],
    "keywords": [
      "carnot group",
      "liouville type theorem holds",
      "preserves q-harmonic functions",
      "tanaka prolongation theory",
      "open subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}