{
  "id": "1202.2226",
  "version": "v1",
  "published": "2012-02-10T10:06:16.000Z",
  "updated": "2012-02-10T10:06:16.000Z",
  "title": "The Cauchy Singular Integral Operator on Weighted Variable Lebesgue Spaces",
  "authors": [
    "Alexei Yu. Karlovich",
    "Ilya M. Spitkovsky"
  ],
  "comment": "17 pages",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Let $p:\\R\\to(1,\\infty)$ be a globally log-H\\\"older continuous variable exponent and $w:\\R\\to[0,\\infty]$ be a weight. We prove that the Cauchy singular integral operator $S$ is bounded on the weighted variable Lebesgue space $L^{p(\\cdot)}(\\R,w)=\\{f:fw\\in L^{p(\\cdot)}(\\R)\\}$ if and only if the weight $w$ satisfies \\[ \\sup_{-\\infty<a<b<\\infty} \\frac{1}{b-a}\\|w\\chi_{(a,b)}\\|_{p(\\cdot)}\\|w^{-1}\\chi_{(a,b)}\\|_{p'(\\cdot)}<\\infty \\quad (1/p(x)+1/p'(x)=1). \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-10T10:06:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A50",
      "42B25",
      "46E30"
    ],
    "keywords": [
      "cauchy singular integral operator",
      "weighted variable lebesgue space",
      "continuous variable exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2226K"
    }
  }
}