{
  "id": "2206.06551",
  "version": "v1",
  "published": "2022-06-14T01:58:08.000Z",
  "updated": "2022-06-14T01:58:08.000Z",
  "title": "Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces",
  "authors": [
    "Yiqun Chen",
    "Hongchao Jia",
    "Dachun Yang"
  ],
  "comment": "47 pages, Submitted. arXiv admin note: text overlap with arXiv:2203.15165, 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 and let $\\alpha\\in(0,n)$ and $\\beta\\in(1,\\infty)$. In this article, when $\\alpha\\in(0,1)$, the authors first find a reasonable version $\\widetilde{I}_{\\alpha}$ of the fractional integral $I_{\\alpha}$ 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 $\\widetilde{I}_{\\alpha}$ is bounded from $\\mathcal{L}_{X^{\\beta},q,s,d}(\\mathbb{R}^n)$ to $\\mathcal{L}_{X,q,s,d}(\\mathbb{R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\\subset \\mathbb{R}^n$, $|B|^{\\frac{\\alpha}{n}}\\leq C \\|\\mathbf{1}_B\\|_X^{\\frac{\\beta-1}{\\beta}}$, where $X^{\\beta}$ denotes the $\\beta$-convexification of $X$. Furthermore, the authors extend the range $\\alpha\\in(0,1)$ in $\\widetilde{I}_{\\alpha}$ to the range $\\alpha\\in(0,n)$ and also obtain the corresponding boundedness in this case. Moreover, $\\widetilde{I}_{\\alpha}$ is proved to be the adjoint operator of $I_\\alpha$. All these results have a wide range of applications. Particularly, even when they are applied, respectively, to 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 dual theorem on $\\mathcal{L}_{X,q,s,d}(\\mathbb{R}^n)$ and also on the special atomic decomposition of molecules of $H_X(\\mathbb{R}^n)$ (the Hardy-type space associated with $X$) which proves the predual space of $\\mathcal{L}_{X,q,s,d}(\\mathbb{R}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-14T01:58:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G40",
      "42B20",
      "47A30",
      "42B30",
      "46E35",
      "42B25",
      "42B35"
    ],
    "keywords": [
      "ball campanato-type function space",
      "fractional integral",
      "boundedness",
      "ball quasi-banach function space",
      "mixed-norm herz spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}