{
  "id": "2101.07407",
  "version": "v1",
  "published": "2021-01-19T01:44:03.000Z",
  "updated": "2021-01-19T01:44:03.000Z",
  "title": "Compactness Characterizations of Commutators on Ball Banach Function Spaces",
  "authors": [
    "Jin Tao",
    "Dachun Yang",
    "Wen Yuan",
    "Yangyang Zhang"
  ],
  "comment": "36 pages, Submitted. arXiv admin note: text overlap with arXiv:1911.04953, arXiv:1906.03653",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $X$ be a ball Banach function space on ${\\mathbb R}^n$. Let $\\Omega$ be a Lipschitz function on the unit sphere of ${\\mathbb R}^n$,which is homogeneous of degree zero and has mean value zero, and let $T_\\Omega$ be the convolutional singular integral operator with kernel $\\Omega(\\cdot)/|\\cdot|^n$. In this article, under the assumption that the Hardy--Littlewood maximal operator $\\mathcal{M}$ is bounded on both $X$ and its associated space, the authors prove that the commutator $[b,T_\\Omega]$ is compact on $X$ if and only if $b\\in{\\rm CMO}({\\mathbb R}^n)$. To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm in $X$ of the commutators and the characteristic functions of some measurable subset,which are implied by the assumed boundedness of ${\\mathcal M}$ on $X$ and its associated space as well as the geometry of $\\mathbb R^n$; the complete John--Nirenberg inequality in $X$ obtained by Y. Sawano et al.; the generalized Fr\\'{e}chet--Kolmogorov theorem on $X$ also established in this article. All these results have a wide range of applications. Particularly, even when $X:=L^{p(\\cdot)}({\\mathbb R}^n)$ (the variable Lebesgue space), $X:=L^{\\vec{p}}({\\mathbb R}^n)$ (the mixed-norm Lebesgue space), $X:=L^\\Phi({\\mathbb R}^n)$ (the Orlicz space), and $X:=(E_\\Phi^q)_t({\\mathbb R}^n)$ (the Orlicz-slice space or the generalized amalgam space), all these results are new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-19T01:44:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47",
      "42B20",
      "42B25",
      "42B30",
      "42B35",
      "46E30"
    ],
    "keywords": [
      "ball banach function space",
      "compactness characterizations",
      "commutator",
      "convolutional singular integral operator",
      "complete john-nirenberg inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}