{
  "id": "1010.5336",
  "version": "v1",
  "published": "2010-10-26T08:48:36.000Z",
  "updated": "2010-10-26T08:48:36.000Z",
  "title": "Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data",
  "authors": [
    "Alexei Yu. Karlovich",
    "Yuri I. Karlovich",
    "Amarino B. Lebre"
  ],
  "comment": "26 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Suppose $\\alpha$ is an orientation-preserving diffeomorphism (shift) of $\\mR_+=(0,\\infty)$ onto itself with the only fixed points $0$ and $\\infty$. In \\cite{KKLsufficiency} we found sufficient conditions for the Fredholmness of the singular integral operator with shift \\[ (aI-bW_\\alpha)P_++(cI-dW_\\alpha)P_- \\] acting on $L^p(\\mR_+)$ with $1<p<\\infty$, where $P_\\pm=(I\\pm S)/2$, $S$ is the Cauchy singular integral operator, and $W_\\alpha f=f\\circ\\alpha$ is the shift operator, under the assumptions that the coefficients $a,b,c,d$ and the derivative $\\alpha'$ of the shift are bounded and continuous on $\\mR_+$ and may admit discontinuities of slowly oscillating type at $0$ and $\\infty$. Now we prove that those conditions are also necessary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-26T08:48:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "45E05",
      "47A53",
      "47B35",
      "47G10",
      "47G30"
    ],
    "keywords": [
      "slowly oscillating data",
      "necessary conditions",
      "fredholmness",
      "cauchy singular integral operator",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.5336K"
    }
  }
}