{
  "id": "1205.3606",
  "version": "v4",
  "published": "2012-05-16T09:17:00.000Z",
  "updated": "2015-06-08T14:39:22.000Z",
  "title": "On directional maximal operators in higher dimensions",
  "authors": [
    "Javier Parcet",
    "Keith M. Rogers"
  ],
  "comment": "23 pages; final version to appear in Amer. J. Math",
  "categories": [
    "math.CA"
  ],
  "abstract": "We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously considered by Nagel--Stein--Wainger, Sj\\\"ogren--Sj\\\"olin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in $L_{\\text{loc}}^p(\\mathbb{R}^n)$, replacing balls by tubes which point in these directions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-07-03T14:51:53.000Z",
      "abstract": "We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the (almost disjoint) classes previously considered by Nagel--Stein--Wainger, Sj\\\"ogren--Sj\\\"olin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in $L_{\\text{loc}}^p(\\mathbb{R}^n)$, replacing balls by tubes which point in these directions.",
      "comment": "Description of the proof and references added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-06-08T14:39:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "26B05"
    ],
    "keywords": [
      "higher dimensions",
      "bounds enable lebesgue type differentiation",
      "associated directional maximal operators",
      "bounded maximal operators",
      "finite order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.3606P"
    }
  }
}