{
  "id": "0902.0708",
  "version": "v1",
  "published": "2009-02-04T11:51:36.000Z",
  "updated": "2009-02-04T11:51:36.000Z",
  "title": "New results on multiplication in Sobolev spaces",
  "authors": [
    "Carlo Morosi",
    "Livio Pizzocchero"
  ],
  "comment": "LaTex, 47 pages",
  "journal": "Advances in Applied Mathematics 44 (2010) 393-432",
  "doi": "10.1016/j.aam.2009.11.006",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \\leqs K_{ell m n d} || f ||_{m} || g ||_n (f in H^m(R^d, C), g in H^n(R^d, C); 0 <= ell <= m <= n, m + n - ell > d/2); we derive upper and lower bounds K^{+}_{ell m n d}, K^{-}_{ell m n d} for this constant. As examples, we give a table of these bounds for d=1, d=3 and many values of (ell, m, n); here the ratio K^{-}_{ell m n d}/K^{+}_{ell m n d} ranges between 0.75 and 1 (being often near 0.90, or larger), a fact indicating that the bounds are close to the sharp constant. Finally, we discuss the asymptotic behavior of the upper and lower bounds for K_{ell, b ell, c ell, d} when 1 <= b <= c and ell -> + Infinity. As an example, from this analysis we obtain the ell -> + Infinity limiting behavior of the sharp constant K_{ell, 2 ell, 2 ell, d}; a second example concerns the ell -> + Infinity limit for K_{ell, 2 ell, 3 ell, d}. The present work generalizes our previous paper [16], entirely devoted to the constant K_{ell m n d} in the special case ell = m = n; many results given therein can be recovered here for this special case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-04T11:51:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "26D10",
      "47A60"
    ],
    "keywords": [
      "sobolev spaces",
      "infinity limit",
      "lower bounds",
      "multiplication",
      "second example concerns"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.0708M"
    }
  }
}