{
  "id": "1510.08604",
  "version": "v1",
  "published": "2015-10-29T08:54:05.000Z",
  "updated": "2015-10-29T08:54:05.000Z",
  "title": "The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian",
  "authors": [
    "Boumediene Abdellaoui",
    "María Medina",
    "Ireneo Peral",
    "Ana Primo"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$ \\left\\{\\begin{array}{rcll} (-\\Delta)^s u-\\lambda \\dfrac{u}{|x|^{2s}}&=&f(x,u) &\\hbox{ in } \\Omega,\\\\ u&=&0 &\\hbox{ in } \\mathbb{R}^N\\setminus\\Omega,\\\\ u&>&0 &\\hbox{ in }\\Omega, \\end{array}\\right. $$ where $(-\\Delta)^s$, $s\\in(0,1)$, is the fractional laplacian operator, $\\Omega\\subset \\mathbb{R}^N$ is a bounded domain with Lipschitz boundary such that $0\\in\\Omega$ and $N>2s$. We will mainly consider the solvability in two cases: 1) The linear problem, that is, $f(x,t)=f(x)$, where according to the summability of the datum $f$ and the parameter $\\lambda$ we give the summability of the solution $u$. 2) The problem with a nonlinear term $f(x,t)=\\frac{h(x)}{t^\\sigma}$ for $t>0$. In this case, existence and regularity will depend on the value of $\\sigma$ and on the summability of $h$. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with \\emph{singular coefficients} that seems to be new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-29T08:54:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy potential",
      "calderón-zygmund properties",
      "summability",
      "nonlocal elliptic problems",
      "weak harnack inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151008604A"
    }
  }
}