{
  "id": "2102.10206",
  "version": "v1",
  "published": "2021-02-19T23:30:50.000Z",
  "updated": "2021-02-19T23:30:50.000Z",
  "title": "Continuity of the gradient of the fractional maximal operator on $W^{1,1}(\\mathbb{R}^d)$",
  "authors": [
    "David Beltran",
    "Cristian González-Riquelme",
    "José Madrid",
    "Julian Weigt"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "We establish that the map $f\\mapsto |\\nabla \\mathcal{M}_{\\alpha}f|$ is continuous from $W^{1,1}(\\mathbb{R}^d)$ to $L^{q}(\\mathbb{R}^d)$, where $\\alpha\\in (0,d)$, $q=\\frac{d}{d-\\alpha}$ and $\\mathcal{M}_{\\alpha}$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d >1$ and $\\alpha \\in (0,1)$ in full generality, for which results were only known for radial functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-19T23:30:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "46E35"
    ],
    "keywords": [
      "fractional maximal operator",
      "continuity",
      "non-centered fractional hardy-littlewood maximal operator",
      "full generality",
      "radial functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}