{
  "id": "1010.1554",
  "version": "v1",
  "published": "2010-10-07T21:46:44.000Z",
  "updated": "2010-10-07T21:46:44.000Z",
  "title": "A Harnack inequality and Hölder continuity for weak solutions to parabolic operators involving Hörmander vector fields",
  "authors": [
    "Garrett Rea"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with H\\\"ormander vector fields. Adapting the iteration scheme of J\\\"urgen Moser for elliptic and parabolic equations in $\\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \\sum X_i (a_{ij} X_j u)$ we use this to show H\\\"older continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parabolic equations of the form $u_t = -X_i^* A_j(X_j u)$. With this we show both a Harnack inequality and H\\\"older continuity of weak solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-07T21:46:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak solutions",
      "hörmander vector fields",
      "parabolic operator",
      "hölder continuity",
      "iteration scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.1554R"
    }
  }
}