{
  "id": "1503.04008",
  "version": "v1",
  "published": "2015-03-13T10:20:24.000Z",
  "updated": "2015-03-13T10:20:24.000Z",
  "title": "The $L(\\log L)^ε$ endpoint estimate for maximal singular integral operators",
  "authors": [
    "Tuomas Hytönen",
    "Carlos Pérez"
  ],
  "comment": "21 pages, final version, accepted for publication in J. Math. Anal. Appl",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \\|T^*f\\|_{L^{1,\\infty}(w)} \\lesssim \\frac{1}{\\epsilon} \\int_{\\mathbb{R}^n} |f(x)| M_{L(\\log L)^{\\epsilon}} (w)(x)dx$, for $w\\geq 0, 0<\\epsilon \\leq 1.$ This follows from the sharp $L^p$ estimate $ \\|T^*f \\|_{ L^{p}(w) } \\lesssim p' (\\frac{1}{\\delta})^{1/p'} \\|f \\|_{L^{p}(M_{ L(\\log L)^{p-1+\\delta}} (w))}$, for $1<p<\\infty, w\\geq 0, 0<\\delta \\leq 1. $ As as a consequence we deduce that $ \\|T^*f\\|_{L^{1,\\infty}(w)} \\lesssim [w]_{A_{1}} \\log(e+ [w]_{A_{\\infty}}) \\int_{\\mathbb{R}^n} |f| w dx, $ extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-13T10:20:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "46E30"
    ],
    "keywords": [
      "maximal singular integral operators",
      "endpoint estimate",
      "weight bump estimate",
      "maximal operator",
      "consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}