{
  "id": "1507.08854",
  "version": "v1",
  "published": "2015-07-31T12:29:50.000Z",
  "updated": "2015-07-31T12:29:50.000Z",
  "title": "Some $s$-numbers of an integral operator of Hardy type in Banach function spaces",
  "authors": [
    "David Edmunds",
    "Amiran Gogatishvili",
    "Tengiz Kopaliani",
    "Nino Samashvili"
  ],
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $s_{n}(T)$ denote the $n$th approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator $T$ given by $$ Tf(x)=v(x)\\int_{a}^{x}u(t)f(t)dt,\\,\\,\\,x\\in(a,b)\\,\\,(-\\infty<a<b<+\\infty) $$ and mapping a Banach function space $E$ to itself. We investigate some geometrical properties of $E$ for which $$ C_{1}\\int_{a}^{b}u(x)v(x)dx \\leq\\limsup\\limits_{n\\rightarrow\\infty}ns_{n}(T) \\leq \\limsup\\limits_{n\\rightarrow\\infty}ns_{n}(T)\\leq C_{2}\\int_{a}^{b}u(x)v(x)dx $$ under appropriate conditions on $u$ and $v.$ The constants $C_{1},C_{2}>0$ depend only on the space $E.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-31T12:29:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P30",
      "46E30",
      "46E35",
      "47A75",
      "47B06",
      "47B10",
      "47B40",
      "47G10"
    ],
    "keywords": [
      "banach function space",
      "hardy type",
      "hardy-type integral operator",
      "bernstein number",
      "th approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150708854E"
    }
  }
}