{
  "id": "1105.0407",
  "version": "v2",
  "published": "2011-05-02T19:09:32.000Z",
  "updated": "2011-06-03T08:47:55.000Z",
  "title": "On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces",
  "authors": [
    "Alexei Yu. Karlovich",
    "Ilya M. Spitkovsky"
  ],
  "comment": "23 pages. An inaccuracy in Lemma 3.11 is corrected. The proof of the main result is corrected accordingly",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $a$ be a semi-almost periodic matrix function with the almost periodic representatives $a_l$ and $a_r$ at $-\\infty$ and $+\\infty$, respectively. Suppose $p:\\mathbb{R}\\to(1,\\infty)$ is a slowly oscillating exponent such that the Cauchy singular integral operator $S$ is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\mathbb{R})$. We prove that if the operator $aP+Q$ with $P=(I+S)/2$ and $Q=(I-S)/2$ is Fredholm on the variable Lebesgue space $L_N^{p(\\cdot)}(\\mathbb{R})$, then the operators $a_lP+Q$ and $a_rP+Q$ are invertible on standard Lebesgue spaces $L_N^{q_l}(\\mathbb{R})$ and $L_N^{q_r}(\\mathbb{R})$ with some exponents $q_l$ and $q_r$ lying in the segments between the lower and the upper limits of $p$ at $-\\infty$ and $+\\infty$, respectively.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-03T08:47:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35"
    ],
    "keywords": [
      "variable lebesgue space",
      "semi-almost periodic coefficients",
      "cauchy singular integral operator",
      "semi-almost periodic matrix function",
      "standard lebesgue spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.0407K"
    }
  }
}