{
  "id": "math/0305331",
  "version": "v3",
  "published": "2003-05-23T13:53:22.000Z",
  "updated": "2004-10-25T16:32:39.000Z",
  "title": "Quantitative functional calculus in Sobolev spaces",
  "authors": [
    "Carlo Morosi",
    "Livio Pizzocchero"
  ],
  "comment": "LaTex, 37 pages. Final version, differing only by minor typographical changes from the versions of May 23, 2003 and March 8, 2004",
  "journal": "Journal of Function Spaces and Applications (JFSA) 2 (2004), 279-321",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the framework of Sobolev (Bessel potential) spaces $H^n(\\reali^d, \\reali {or} \\complessi)$, we consider the nonlinear Nemytskij operator sending a function $x \\in \\reali^d \\mapsto f(x)$ into a composite function $x \\in \\reali^d \\mapsto G(f(x), x)$. Assuming sufficient smoothness for $G$, we give a \"tame\" bound on the $H^n$ norm of this composite function in terms of a linear function of the $H^n$ norm of $f$, with a coefficient depending on $G$ and on the $H^a$ norm of $f$, for all integers $n, a, d$ with $a > d/2$. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the $H^n$ norm of the function $x \\mapsto G(f(x),x)$. When applied to the case $G(f(x), x) = f^2(x)$, this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2004-10-25T16:32:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "26D10",
      "47A60"
    ],
    "keywords": [
      "quantitative functional calculus",
      "sobolev spaces",
      "composite function",
      "nonlinear nemytskij operator sending",
      "assuming sufficient smoothness"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5331M"
    }
  }
}