{
  "id": "1209.6104",
  "version": "v3",
  "published": "2012-09-27T01:01:55.000Z",
  "updated": "2015-01-28T02:51:12.000Z",
  "title": "Fractional Laplacian on the torus",
  "authors": [
    "L. Roncal",
    "P. R. Stinga"
  ],
  "comment": "18 pages, 2 figures. To appear in Communications in Contemporary Mathematics",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "We study the fractional Laplacian $(-\\Delta)^{\\sigma/2}$ on the $n$-dimensional torus $\\mathbb{T}^n$, $n\\geq1$. First, we present a general extension problem that describes \\textit{any} fractional power $L^\\gamma$, $\\gamma>0$, where $L$ is a general nonnegative selfadjoint operator defined in an $L^2$-space. This generalizes to all $\\gamma>0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for $(-\\Delta)^{\\sigma/2}$, when $0<\\sigma<2$. We deduce regularity estimates on H\\\"older, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-16T23:56:14.000Z",
      "abstract": "In this paper we study the fractional Laplacian $(-\\Delta)^{\\sigma/2}$ on the $n$-dimensional torus $\\Toron$, $n\\geq1$. Our approach is based on the semigroup language. We obtain a pointwise integro-differential formula for $(-\\Delta)^{\\sigma/2}f(x)$, $0<\\sigma<2$, $x\\in\\Toron$, that is derived via the heat kernel on $\\Toron$. The limits as $\\sigma\\to0^+$ and $\\sigma\\to2$ are computed. Regularity estimates on H\\\"older, Lipschitz and Zygmund spaces are deduced. We also present a general extension problem to characterize \\textit{any} fractional power of an operator $L^\\gamma$, $\\gamma>0$. Here $L$ is a general nonnegative selfadjoint operator defined in an $L^2$-space. In particular, $L$ can be taken to be the Laplace--Beltrami operator on a Riemannian manifold or a divergence form elliptic operator with measurable coefficients on a bounded domain. This generalizes to all $\\gamma>0$ and a large class of operators $L$ previous known results by Caffarelli--Silvestre. This extension result is of independent interest. The extension problem and the theory of degenerate elliptic equations is applied to prove interior and boundary Harnack's inequalities for $(-\\Delta)^{\\sigma/2}$.",
      "comment": "25 pages, 2 figures. Revised version. Results now include the case of the multidimensional torus. The general extension problem for any positive fractional power has been moved to Section 6",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-01-28T02:51:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "divergence form elliptic operator",
      "general extension problem",
      "general nonnegative selfadjoint operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.6104R"
    }
  }
}