{
  "id": "1702.04790",
  "version": "v1",
  "published": "2017-02-15T21:45:35.000Z",
  "updated": "2017-02-15T21:45:35.000Z",
  "title": "Weighted Estimates for Rough Bilinear Singular Integrals via Sparse Domination",
  "authors": [
    "Alexander Barron"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove weighted estimates for rough bilinear singular integral operators with kernel $$K(y_1, y_2) = \\frac{\\Omega((y_1,y_2)/|(y_1,y_2)|)}{|(y_1, y_2)|^{2d}},$$ where $y_i \\in \\mathbb{R}^{d}$ and $\\Omega \\in L^{\\infty}(S^{2d-1})$ with $\\int_{S^{2d-1}}\\Omega d\\sigma = 0.$ The argument is by sparse domination of rough bilinear operators, via an abstract theorem that is a multilinear generalization of recent work by Conde-Alonso, Culiuc, Di Plinio and Ou. We also use recent results due to Grafakos, He, and Honz\\'{\\i}k for the application to rough bilinear operators. In particular, since the weighted estimates are proved via sparse domination, we obtain some quantitative estimates in terms of the $A_{p}$ characteristics of the weights. The abstract theorem is also shown to apply to multilinear Calder\\'{o}n-Zygmund operators with a standard smoothness assumption. Due to the generality of the sparse domination theorem, future applications not considered in this paper are expected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-15T21:45:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "weighted estimates",
      "rough bilinear operators",
      "rough bilinear singular integral operators",
      "abstract theorem",
      "sparse domination theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}