{
  "id": "1710.05217",
  "version": "v1",
  "published": "2017-10-14T17:55:07.000Z",
  "updated": "2017-10-14T17:55:07.000Z",
  "title": "Modular inequalities for the maximal operator in variable Lebesgue spaces",
  "authors": [
    "David Cruz-Uribe",
    "Giovanni Di Fratta",
    "Alberto Fiorenza"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\\cdot)}(\\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \\[ \\int_\\Omega Mf(x)^{p(x)}\\,dx \\ \\leq c_1 \\int_\\Omega |f(x)|^{q(x)}\\,dx + c_2, \\] where $c_1,\\,c_2$ are non-negative constants and $\\Omega$ is any measurable subset of $\\mathbb{R}^n$. As a corollary we get sufficient conditions for the modular inequality \\[ \\int_\\Omega |Tf(x)|^{p(x)}\\,dx \\ \\leq c_1 \\int_\\Omega |f(x)|^{q(x)}\\,dx + c_2, \\] where $T$ is any operator that is bounded on $L^p(\\Omega)$, $1<p<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-14T17:55:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "46E30",
      "26D15"
    ],
    "keywords": [
      "variable lebesgue spaces",
      "maximal operator",
      "sufficient conditions",
      "hardy-littlewood maximal function",
      "weaker modular inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}