{
  "id": "1812.03193",
  "version": "v1",
  "published": "2018-12-07T19:34:29.000Z",
  "updated": "2018-12-07T19:34:29.000Z",
  "title": "Weighted unipolar Hardy inequality with optimal constant",
  "authors": [
    "Anna Canale",
    "Francesco Pappalardo",
    "Ciro Tarantino"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "\\begin{abstract} We state the following weighted Hardy inequality \\begin{equation*} c_\\mu\\int_{{\\R}^N}\\frac{\\varphi^2 }{|x|^2}\\, d\\mu\\le \\int_{{\\R}^N} |\\nabla\\varphi|^2 \\, d\\mu + K \\int_{\\R^N}\\varphi^2 \\, d\\mu \\quad \\forall\\, \\varphi \\in H_\\mu^1 %\\qquad c\\le c_\\mu, \\end{equation*} in the context of the study of the Kolmogorov operators \\begin{equation*} Lu=\\Delta u+\\frac{\\nabla \\mu}{\\mu}\\cdot\\nabla u, \\end{equation*} with $\\mu$ probability density in $\\R^N$, and of the related evolution problems. We prove the optimality of the constant $c_\\mu$ and state existence and nonexistence results following the Cabr\\'e-Martel's approach \\cite{CabreMartel} and using results stated in \\cite{GGR, CGRT}. \\end{abstract}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-07T19:34:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K15",
      "35K65",
      "35B25",
      "34G10",
      "47D03"
    ],
    "keywords": [
      "weighted unipolar hardy inequality",
      "optimal constant",
      "weighted hardy inequality",
      "kolmogorov operators",
      "cabre-martels approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}