{
  "id": "0804.1949",
  "version": "v1",
  "published": "2008-04-11T19:30:19.000Z",
  "updated": "2008-04-11T19:30:19.000Z",
  "title": "The $(L^{1},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages",
  "authors": [
    "Idris Assani",
    "Zoltan Buczolich"
  ],
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Let $(X,\\mathcal{B}, \\mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\\dis R^*:(f, g) \\in L^1\\times L^1 \\to R^*(f, g)(x) = \\sup_{n} \\frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\\in L^1\\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^{p},L^{q})$ functions with $\\frac{1}{p}+\\frac{1}{q}\\leq 1.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-11T19:30:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37A50",
      "28D05"
    ],
    "keywords": [
      "bilinear hardy-littlewood function",
      "furstenberg averages",
      "bilinear hardy-littlewood maximal function fails",
      "non-atomic finite measure space",
      "ergodic dynamical system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.1949A"
    }
  }
}