{
  "id": "1603.07988",
  "version": "v1",
  "published": "2016-03-25T19:31:55.000Z",
  "updated": "2016-03-25T19:31:55.000Z",
  "title": "An extension problem related to inverse fractional operators",
  "authors": [
    "Félix del Teso"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP",
    "math.FA",
    "math.NA"
  ],
  "abstract": "It is well known from the work of Caffarelli and Silvestre that the fractional Laplacian $(-\\Delta_x)^{\\frac{\\sigma}{2}}$ for $\\sigma \\in (0,2)$ can be obtained as a Dirichlet-to-Neumann map through an extension problem to the upper half space. In this paper we show that the inverse fractional Laplacian $(-\\Delta_x)^{-\\frac{\\sigma}{2}}$ has a similar property: it can be obtained as a Neumann-to-Dirichlet map via a new extension problem to the upper half space. We also obtain an explicit formula for the solution of the new extension problem. Moreover, we deal with powers of a more general class of second order differential operators defined in open subsets of $\\mathbb{R}^N$. From this characterization we show possible applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-25T19:31:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extension problem",
      "inverse fractional operators",
      "upper half space",
      "second order differential operators",
      "inverse fractional laplacian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160307988D"
    }
  }
}