{
  "id": "2011.11512",
  "version": "v1",
  "published": "2020-11-23T16:21:02.000Z",
  "updated": "2020-11-23T16:21:02.000Z",
  "title": "Limiting weak-type behavior for rough bilinear operators",
  "authors": [
    "Moyan Qin",
    "Huoxiong Wu",
    "Qingying Xue"
  ],
  "comment": "29 pages, 2 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\Omega_1,\\Omega_2$ be functions of homogeneous of degree $0$ and $\\vec\\Omega=(\\Omega_1,\\Omega_2)\\in L\\log L(\\mathbb{S}^{n-1})\\times L\\log L(\\mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\\vec\\Omega}$ and bilinear singular integral $T_{\\vec\\Omega}$ associated with rough kernel $\\vec\\Omega$. For all $f,g\\in L^1(\\mathbb{R}^n)$, we show that $$\\lim_{\\lambda\\to 0^+}\\lambda |\\big\\{ x\\in\\mathbb{R}^n:M_{\\vec\\Omega}(f_1,f_2)(x)>\\lambda\\big\\}|^2 = \\frac{\\|\\Omega_1\\Omega_2\\|_{L^{1/2}(\\mathbb{S}^{n-1})}}{\\omega_{n-1}^2}\\prod\\limits_{i=1}^2\\| f_i\\|_{L^1}$$ and $$\\lim_{\\lambda\\to 0^+}\\lambda|\\big\\{ x\\in\\mathbb{R}^n:| T_{\\vec\\Omega}(f_1,f_2)(x)|>\\lambda\\big\\}|^{2} = \\frac{\\|\\Omega_1\\Omega_2\\|_{L^{1/2}(\\mathbb{S}^{n-1})}}{n^2}\\prod\\limits_{i=1}^2\\| f_i\\|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{\\vec\\Omega}$ and $T_{\\vec\\Omega}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-23T16:21:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25"
    ],
    "keywords": [
      "limiting weak-type behavior",
      "rough bilinear operators",
      "rough bilinear fractional maximal function",
      "bilinear maximal function",
      "fractional integral operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}