{
  "id": "1809.01256",
  "version": "v1",
  "published": "2018-09-04T22:03:01.000Z",
  "updated": "2018-09-04T22:03:01.000Z",
  "title": "A generalization of the boundedness of certain integral operators in variable Lebesgue spaces",
  "authors": [
    "Lucas Alejandro Vallejos y Marta Susana Urciuolo"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $A_{1},...A_{m}$ be a $n\\times n$ invertible matrices. Let $0 \\leq \\alpha<n$ and $0<\\alpha_{i}<n$ such that $\\alpha_1 + ... + \\alpha_m = n- \\alpha$. We define% \\begin{equation*} T_{\\alpha}f(x)=\\int \\frac{1}{\\left\\vert x-A_{1}y\\right\\vert ^{\\alpha _{1}}...\\left\\vert x-A_{m}y\\right\\vert ^{\\alpha _{m}}}f(y)dy. \\end{equation*}% In \\cite{U-V} we obtained the boundedness of this operator from $L^{p(.)}(% \\mathbb{R}^{n})$ into $L^{q(.)}(\\mathbb{R}^{n})$ for $\\frac{1}{q(.)}=\\frac{1% }{p(.)}-\\frac{\\alpha }{n},$ in the case that $A_{i}$ is a power of certain fixed matrix $A~\\ $and for exponent functions $p$ satisfying log-Holder conditions and $p(Ay)=p(y),$ $y\\in \\mathbb{R}^{n}$ $.$ We will show now that the hypothesis on $p$, in certain cases, is necessary for the boundedness of $T_{\\alpha}$ and we also prove the result for more general matrices $A_{i}.$ \\footnote{Partially supported by CONICET and SECYTUNC} \\footnote{Math. subject classification: 42B25, 42B35.} \\footnote{Key words: Variable Exponents, Fractional Integrals.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-04T22:03:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "variable lebesgue spaces",
      "integral operators",
      "boundedness",
      "generalization",
      "general matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}