{
  "id": "2106.14051",
  "version": "v1",
  "published": "2021-06-26T16:02:23.000Z",
  "updated": "2021-06-26T16:02:23.000Z",
  "title": "Limiting weak-type behaviors for singular integrals with rough $L\\log L(\\mathbb{S}^n)$ kernels",
  "authors": [
    "Moyan Qin",
    "Huoxiong Wu",
    "Qingying Xue"
  ],
  "comment": "26 pages. arXiv admin note: text overlap with arXiv:2011.11512",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\Omega$ be a function of homogeneous of degree zero and vanish on the unit sphere $\\mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_\\Omega$ associated with rough kernel $\\Omega$. We show that, if $\\Omega\\in L\\log L(\\mathbb S^{n})$, then $\\lim_{\\lambda\\to0^+}\\lambda|\\{x\\in\\mathbb{R}^n:|T_\\Omega(f)(x)|>\\lambda\\}| = n^{-1}\\|\\Omega\\|_{L^1(\\mathbb {S}^n)}\\|f\\|_{L^1(\\mathbb{R}^n)},\\quad0\\le f\\in L^1(\\mathbb{R}^n).$ Moreover,$(n^{-1}\\|\\Omega\\|_{L^1(\\mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_\\Omega$ when $\\Omega\\in L\\log L(\\mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{\\vec\\Omega}(f_1,f_2) = T_{\\Omega_1}(f_1)\\cdot T_{\\Omega_2}(f_2)$ have also been established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-26T16:02:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "limiting weak-type behavior",
      "rough bilinear singular integral operators",
      "lower bound",
      "weak-type norm",
      "rough kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}