{
  "id": "1405.0368",
  "version": "v2",
  "published": "2014-05-02T10:05:06.000Z",
  "updated": "2014-10-01T09:20:55.000Z",
  "title": "Fredholmness and Index of Simplest Weighted Singular Integral Operators with Two Slowly Oscillating Shifts",
  "authors": [
    "Alexei Yu. Karlovich"
  ],
  "comment": "To appear in the Banach Journal of Mathematical Analysis, 19 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\alpha$ and $\\beta$ be orientation-preserving diffeomorphisms (shifts) of $\\mathbb{R}_+=(0,\\infty)$ onto itself with the only fixed points 0 and $\\infty$, where the derivatives $\\alpha'$ and $\\beta'$ may have discontinuities of slowly oscillating type at 0 and $\\infty$. For $p\\in(1,\\infty)$, we consider the weighted shift operators $U_\\alpha$ and $U_\\beta$ given on the Lebesgue space $L^p(\\mathbb{R}_+)$ by $U_\\alpha f=(\\alpha')^{1/p}(f\\circ\\alpha)$ and $U_\\beta f= (\\beta')^{1/p}(f\\circ\\beta)$. For $i,j\\in\\mathbb{Z}$ we study the simplest weighted singular integral operators with two shifts $A_{ij}=U_\\alpha^i P_\\gamma^++U_\\beta^j P_\\gamma^-$ on $L^p(\\mathbb{R}_+)$, where $P_\\gamma^\\pm=(I\\pm S_\\gamma)/2$ are operators associated to the weighted Cauchy singular integral operator $$ (S_\\gamma f)(t)=\\frac{1}{\\pi i}\\int_{\\mathbb{R}_+} (\\frac{t}{\\tau})^\\gamma\\frac{f(\\tau)}{\\tau-t}d\\tau $$ with $\\gamma\\in\\mathbb{C}$ satisfying $0<1/p+\\Re\\gamma<1$. We prove that all $A_{ij}$ are Fredholm operators on $L^p(\\mathbb{R}_+)$ and have zero indices. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for $\\gamma=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-02T10:05:06.000Z",
      "abstract": "Let $\\alpha$ and $\\beta$ be orientation-preserving diffeomorphisms (shifts) of $\\mathbb{R}_+=(0,\\infty)$ onto itself with the only fixed points $0$ and $\\infty$, where the derivatives $\\alpha'$ and $\\beta'$ may have discontinuities of slowly oscillating type at $0$ and $\\infty$. For $p\\in(1,\\infty)$, we consider the weighted shift operators $U_\\alpha$ and $U_\\beta$ given on the Lebesgue space $L^p(\\mathbb{R}_+)$ by $U_\\alpha f=(\\alpha')^{1/p}(f\\circ\\alpha)$ and $U_\\beta f= (\\beta')^{1/p}(f\\circ\\beta)$. For $i,j\\in\\mathbb{Z}$ we study the simplest weighted singular integral operators with two shifts $A_{ij}=U_\\alpha^i P_\\gamma^++U_\\beta^j P_\\gamma^-$ on $L^p(\\mathbb{R}_+)$, where $P_\\gamma^\\pm=(I\\pm S_\\gamma)/2$ are operators associated to the weighted Cauchy singular integral operator $$ (S_\\gamma f)(t)=\\frac{1}{\\pi i}\\int_{\\mathbb{R}_+} \\left(\\frac{t}{\\tau}\\right)^\\gamma\\frac{f(\\tau)}{\\tau-t}d\\tau $$ with $\\gamma\\in\\mathbb{C}$ satisfying $0<1/p+\\Re\\gamma<1$. We prove that all $A_{ij}$ are Fredholm operators on $L^p(\\mathbb{R}_+)$ and have zero indices. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for $\\gamma=0$.",
      "comment": "17 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-01T09:20:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "45E05",
      "47A53",
      "47G10",
      "47G30"
    ],
    "keywords": [
      "simplest weighted singular integral operators",
      "slowly oscillating shifts",
      "cauchy singular integral operator",
      "fredholmness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0368K"
    }
  }
}