{
  "id": "2403.15758",
  "version": "v1",
  "published": "2024-03-23T08:19:25.000Z",
  "updated": "2024-03-23T08:19:25.000Z",
  "title": "An endpoint estimate for the maximal Calderón commutator with rough kernel",
  "authors": [
    "Guoen Hu",
    "Xudong Lai",
    "Xiangxing Tao",
    "Qingying Xue"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, the authors consider the endpoint estimates for the maximal Calder\\'on commutator defined by $$T_{\\Omega,\\,a}^*f(x)=\\sup_{\\epsilon>0}\\Big|\\int_{|x-y|>\\epsilon}\\frac{\\Omega(x-y)}{|x-y|^{d+1}} \\big(a(x)-a(y)\\big)f(y)dy\\Big|,$$ where $\\Omega$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has vanishing moment of order one, $a$ be a function on $\\mathbb{R}^d$ such that $\\nabla a\\in L^{\\infty}(\\mathbb{R}^d)$. The authors prove that if $\\Omega\\in L\\log L(S^{d-1})$, then $T^*_{\\Omega,\\,a}$ satisfies an endpoint estimate of $L\\log\\log L$ type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-23T08:19:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "endpoint estimate",
      "maximal calderón commutator",
      "rough kernel",
      "maximal calderon commutator",
      "degree zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}