{
  "id": "2106.13363",
  "version": "v1",
  "published": "2021-06-25T00:21:26.000Z",
  "updated": "2021-06-25T00:21:26.000Z",
  "title": "Hardy's inequality and (almost) the Landau equation",
  "authors": [
    "Maria Gualdani",
    "Nestor Guillen"
  ],
  "comment": "18 pages, no figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "In this manuscript we establish an $L^\\infty$ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent $\\gamma$ in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincar\\'e inequality, which in turn implies the propagation of some $L^p$ norms of solutions. From here, the $L^\\infty$ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-25T00:21:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "47G20",
      "35Q20"
    ],
    "keywords": [
      "hardys inequality",
      "interaction exponent",
      "soft potentials",
      "main observation",
      "classical weighted hardy inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}