{
  "id": "1110.0297",
  "version": "v1",
  "published": "2011-10-03T08:31:45.000Z",
  "updated": "2011-10-03T08:31:45.000Z",
  "title": "Pseudodifferential Operators on Variable Lebesgue Spaces",
  "authors": [
    "Alexei Yu. Karlovich",
    "Ilya M. Spitkovsky"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathcal{M}(\\mathbb{R}^n)$ be the class of bounded away from one and infinity functions $p:\\mathbb{R}^n\\to[1,\\infty]$ such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\mathbb{R}^n)$. We show that if $a$ belongs to the H\\\"ormander class $S_{\\rho,\\delta}^{n(\\rho-1)}$ with $0<\\rho\\le 1$, $0\\le\\delta<1$, then the pseudodifferential operator $\\Op(a)$ is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\R^n)$ provided that $p\\in\\cM(\\R^n)$. Let $\\mathcal{M}^*(\\mathbb{R}^n)$ be the class of variable exponents $p\\in\\mathcal{M}(\\mathbb{R}^n)$ represented as $1/p(x)=\\theta/p_0+(1-\\theta)/p_1(x)$ where $p_0\\in(1,\\infty)$, $\\theta\\in(0,1)$, and $p_1\\in\\mathcal{M}(\\mathbb{R}^n)$. We prove that if $a\\in S_{1,0}^0$ slowly oscillates at infinity in the first variable, then the condition \\[ \\lim_{R\\to\\infty}\\inf_{|x|+|\\xi|\\ge R}|a(x,\\xi)|>0 \\] is sufficient for the Fredholmness of $\\Op(a)$ on $L^{p(\\cdot)}(\\R^n)$ whenever $p\\in\\cM^*(\\R^n)$. Both theorems generalize pioneering results by Rabinovich and Samko \\cite{RS08} obtained for globally log-H\\\"older continuous exponents $p$, constituting a proper subset of $\\mathcal{M}^*(\\mathbb{R}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-03T08:31:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G30",
      "42B25",
      "46E30"
    ],
    "keywords": [
      "variable lebesgue space",
      "pseudodifferential operator",
      "hardy-littlewood maximal operator",
      "infinity functions",
      "theorems generalize pioneering results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.0297K"
    }
  }
}