{
  "id": "2210.14073",
  "version": "v1",
  "published": "2022-10-25T14:56:14.000Z",
  "updated": "2022-10-25T14:56:14.000Z",
  "title": "Pointwise Multipliers for Besov Spaces $B^{0,b}_{p,\\infty}(\\mathbb{R}^n)$ with Only Logarithmic Smoothness",
  "authors": [
    "Ziwei Li",
    "Winfried Sickel",
    "Dachun Yang",
    "Wen Yuan"
  ],
  "comment": "64 pages; Submitted",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "In this article, we establish a characterization of the set $M(B^{0,b}_{p,\\infty}(\\mathbb{R}^n))$ of all pointwise multipliers of Besov spaces $B^{0,b}_{p,\\infty}(\\mathbb{R}^n)$ with only logarithmic smoothness $b\\in\\mathbb{R}$ in the special cases $p=1$ and $p=\\infty$. As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions $e^{ik\\cdot x}$ with $k\\in\\mathbb{Z}^n$ and $x\\in\\mathbb{R}^n$, are pointwise multipliers of $B^{0,b}_{1,\\infty}(\\mathbb{R}^n)$ and $B^{0,b}_{\\infty,\\infty}(\\mathbb{R}^n)$, respectively; furthermore, we obtain the explicit estimates of $\\|e^{ik \\cdot x}\\|_{M(B^{0,b}_{1,\\infty}(\\mathbb{R}^n))}$ and $\\|e^{ik \\cdot x}\\|_{M(B^{0,b}_{\\infty,\\infty}(\\mathbb{R}^n))}$. In the case that $p\\in(1,\\infty)$, we give some sufficient conditions and some necessary conditions of the pointwise multipliers of $B^{0,b}_{p,\\infty}(\\mathbb{R}^n)$ and a complete characterization of $M(B^{0,b}_{p,\\infty}(\\mathbb{R}^n))$ is still open. However, via a different method, we are still able to accurately calculate $\\|e^{ik \\cdot x}\\|_{M(B^{0,b}_{p,\\infty}(\\mathbb{R}^n))}$, $k\\in\\mathbb{Z}^n$, in this situation. The novelty of this article is that most of the proofs are constructive and these constructions strongly depend on the logarithmic structure of Besov spaces under consideration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-25T14:56:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "42B35"
    ],
    "keywords": [
      "pointwise multipliers",
      "besov spaces",
      "logarithmic smoothness",
      "logarithmic structure",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}