{
  "id": "1208.6447",
  "version": "v1",
  "published": "2012-08-31T10:08:52.000Z",
  "updated": "2012-08-31T10:08:52.000Z",
  "title": "Nonlocal Hardy type inequalities with optimal constants and remainder terms",
  "authors": [
    "Vitaly Moroz",
    "Jean Van Schaftingen"
  ],
  "comment": "9 pages",
  "journal": "Ann. Univ. Buchar. Math. Ser. 3 (LXI) (2012), no. 2, 187-200",
  "categories": [
    "math.AP"
  ],
  "abstract": "Using a groundstate transformation, we give a new proof of the optimal Stein-Weiss inequality of Herbst [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N,\\alpha, 0}\\int_{\\R^N} \\abs{\\varphi}^2,] and of its combinations with the Hardy inequality by Beckner [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha + s}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha + s}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N, \\alpha, 1} \\int_{\\R^N} \\abs{\\nabla \\varphi}^2,] and with the fractional Hardy inequality [\\int_{\\R^N} \\int_{\\R^N} \\frac{\\varphi (x)}{\\abs{x}^\\frac{\\alpha + s}{2}} I_\\alpha (x - y) \\frac{\\varphi (y)}{\\abs{y}^\\frac{\\alpha + s}{2}}\\dif x \\dif y \\le \\mathcal{C}_{N, \\alpha, s} \\mathcal{D}_{N, s} \\int_{\\R^N} \\int_{\\R^N} \\frac{\\bigabs{\\varphi (x) - \\varphi (y)}^2}{\\abs{x-y}^{N+s}}\\dif x \\dif y] where (I_\\alpha) is the Riesz potential, (0 < \\alpha < N) and (0 < s < \\min(N, 2)). We also prove the optimality of the constants. The method is flexible and yields a sharp expression for the remainder terms in these inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-31T10:08:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal hardy type inequalities",
      "remainder terms",
      "optimal constants",
      "fractional hardy inequality",
      "optimal stein-weiss inequality"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.6447M"
    }
  }
}