{
  "id": "1605.05176",
  "version": "v1",
  "published": "2016-05-17T14:11:52.000Z",
  "updated": "2016-05-17T14:11:52.000Z",
  "title": "Poincaré inequalities for the maximal function",
  "authors": [
    "Olli Saari"
  ],
  "comment": "19 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study generalized Poincar\\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we get a unified approach to proving that the maximal operator is bounded on Sobolev, Lipschitz and BMO spaces. As another application, we show that the distributional derivatives of the maximal function of $u \\in W^{1,1}(\\mathbb{R}^{n})$ coincide with some functions in $L^{1,\\infty}(\\mathbb{R}^{n})$ outside a closed set of measure zero. The same property holds for $u \\in {\\rm BV}(\\mathbb{R}^{n})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-17T14:11:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "46E35",
      "42B35"
    ],
    "keywords": [
      "study generalized poincare inequalities",
      "hardy-littlewood maximal function",
      "function satisfies",
      "property holds",
      "similar form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}