{
  "id": "2208.06266",
  "version": "v1",
  "published": "2022-08-12T13:26:04.000Z",
  "updated": "2022-08-12T13:26:04.000Z",
  "title": "Boundedness of Calderón--Zygmund operators on ball Campanato-type function spaces",
  "authors": [
    "Yiqun Chen",
    "Hongchao Jia",
    "Dachun Yang"
  ],
  "comment": "32 pages, Submitted. arXiv admin note: substantial text overlap with arXiv:2206.06551, arXiv:2206.06080",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $X$ be a ball quasi-Banach function space on ${\\mathbb R}^n$ satisfying some mild assumptions. In this article, the authors first find a reasonable version $\\widetilde{T}$ of the Calder\\'on--Zygmund operator $T$ on the ball Campanato-type function space $\\mathcal{L}_{X,q,s,d}(\\mathbb{R}^n)$ with $q\\in[1,\\infty)$, $s\\in\\mathbb{Z}_+^n$, and $d\\in(0,\\infty)$. Then the authors prove that $\\widetilde{T}$ is bounded on $\\mathcal{L}_{X,q,s,d}(\\mathbb{R}^n)$ if and only if, for any $\\gamma\\in\\mathbb{Z}^n_+$ with $|\\gamma|\\leq s$, $T^*(x^{\\gamma})=0$, which is hence sharp. Moreover, $\\widetilde{T}$ is proved to be the adjoint operator of $T$, which further strengthens the rationality of the definition of $\\widetilde{T}$. All these results have a wide range of applications. In particular, even when they are applied, respectively, to weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the properties of the kernel of $T$ under consideration and also on the dual theorem on $\\mathcal{L}_{X,q,s,d}(\\mathbb{R}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-12T13:26:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "42B30",
      "42B35",
      "46E35",
      "47A30"
    ],
    "keywords": [
      "ball campanato-type function space",
      "calderón-zygmund operators",
      "ball quasi-banach function space",
      "boundedness",
      "mixed-norm lebesgue spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}