{
  "id": "1005.3030",
  "version": "v4",
  "published": "2010-05-17T20:11:30.000Z",
  "updated": "2014-12-28T05:39:49.000Z",
  "title": "On a discrete version of Tanaka's theorem for maximal functions",
  "authors": [
    "Jonathan Bober",
    "Emanuel Carneiro",
    "Kevin Hughes",
    "Lillian B. Pierce"
  ],
  "comment": "V4 - Proof of Lemma 3 updated",
  "journal": "Proc. Amer. Math. Soc. 140 (2012), 1669-1680",
  "doi": "10.1090/S0002-9939-2011-11008-6",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we prove a discrete version of Tanaka's Theorem \\cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\\widetilde{M} $ we prove that, given a function $f: \\mathbb{Z} \\to \\mathbb{R}$ of bounded variation, $$\\textrm{Var}(\\widetilde{M} f) \\leq \\textrm{Var}(f),$$ where $\\textrm{Var}(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \\mathbb{Z} \\to \\mathbb{R}$ such that $f \\in \\ell^1(\\mathbb{Z})$, $$\\textrm{Var}(Mf) \\leq C \\|f\\|_{\\ell^1(\\mathbb{Z})}.$$ This provides a positive solution to a question of Haj{\\l}asz and Onninen \\cite{HO} in the discrete one-dimensional case.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-08-10T18:00:41.000Z",
      "abstract": "In this paper we prove a discrete version of Tanaka's Theorem \\cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\\wM $ we prove that, given a function $f: \\Z \\to \\R$ of bounded variation, $$\\Var(\\wM f) \\leq \\Var(f),$$ where $\\Var(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \\Z \\to \\R$ such that $f \\in \\ell^1(\\Z)$, $$\\Var(Mf) \\leq C \\|f\\|_{\\ell^1(\\Z)}.$$ This provides a positive solution to a question of Haj{\\l}asz and Onninen \\cite{HO} in the discrete one-dimensional case.",
      "comment": "V3 - some typos corrected"
    },
    {
      "version": "v4",
      "updated": "2014-12-28T05:39:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "46E35"
    ],
    "keywords": [
      "discrete version",
      "tanakas theorem",
      "maximal functions",
      "hardy-littlewood maximal operator",
      "discrete non-centered maximal operator"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.3030B"
    }
  }
}