{
  "id": "1612.09502",
  "version": "v1",
  "published": "2016-12-30T14:59:57.000Z",
  "updated": "2016-12-30T14:59:57.000Z",
  "title": "Vector-valued local approximation spaces",
  "authors": [
    "Tuomas Hytönen",
    "Jori Merikoski"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove that for every Banach space $Y,$ the Besov space $B^{s}_{pq}( \\mathbb{R}^n; Y)$ of functions $f: \\mathbb{R}^n \\to Y$ agrees with a suitable local approximation space $A^{s}_{pq}( \\mathbb{R}^n; Y)$ with equivalent norms. In adddition, we prove that the Sobolev space $W^{k,q}( \\mathbb{R}^n; Y)$ is continuously embedded in the Besov space $B^k_{qq}( \\mathbb{R}^n; Y)$ if and only if $Y$ has martingale cotype $q.$ We interpret this as an extension of earlier results of Xu (1998), and Mart\\'inez, Torrea and Xu (2006), which correspond to the case $k=0.$ These two results combined give the satisfactory characterization that $Y$ admits an equivalent norm with modulus of convexity of power type q if and only if weakly differentiable functions have good local approximations with polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-30T14:59:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector-valued local approximation spaces",
      "besov space",
      "equivalent norm",
      "suitable local approximation space",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}