{
  "id": "2104.03497",
  "version": "v1",
  "published": "2021-04-08T04:15:44.000Z",
  "updated": "2021-04-08T04:15:44.000Z",
  "title": "The limiting weak type behaviors and The lower bound for a new weak $L\\log L$ type norm of strong maximal operators",
  "authors": [
    "Moyan Qin",
    "Huoxiong Wu",
    "Qingying Xue"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "It is well known that the weak ($1,1$) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak $L\\log L$ type norm inequality. Let $\\Phi_n(t)=t(1+(\\log^+t)^{n-1})$ and the space $L_{\\Phi_n}({\\mathbb R^{n}})$ be the set of all measurable functions on ${\\mathbb R^{n}}$ such that $\\|f\\|_{L_{\\Phi_n}({\\mathbb R^{n}})} :=\\|\\Phi_n(|f|)\\|_{L^1({\\mathbb R^{n}})}<\\infty$. In this paper, we introduce a new weak norm space $L_{\\Phi_n}^{1,\\infty}({\\mathbb R^{n}})$, which is more larger than $L^{1,\\infty}({\\mathbb R^{n}})$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $ \\max\\{{2^n}{((n-1)!)^{-1}},1\\}$ is a lower bound for the best constant of the $L_{\\Phi_n}\\to L_{\\Phi_n}^{1,\\infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-08T04:15:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25"
    ],
    "keywords": [
      "lower bound",
      "correspondng limiting weak type behaviors",
      "multilinear strong maximal operators",
      "type norm inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}