{
  "id": "1402.3886",
  "version": "v4",
  "published": "2014-02-17T04:06:21.000Z",
  "updated": "2015-05-27T02:19:28.000Z",
  "title": "Bounds for the Hilbert Transform with Matrix $A_2$ Weights",
  "authors": [
    "Kelly Bickel",
    "Stefanie Petermichl",
    "Brett Wick"
  ],
  "comment": "13 pages. v3: Revised to address referee comments and include additional references. v4: Grant information added",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(\\mathbb{R},\\mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(\\mathbb{R},\\mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(\\mathbb{R},W,\\mathbb{C}^d)$ to itself with dependence on on the $A_2$ characteristic at most $[W]_{A_2}^{\\frac{3}{2}} \\log [W]_{A_2}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-11T14:42:27.000Z",
      "comment": "13 pages. Revised to address referee comments and include additional references",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-05-27T02:19:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A50"
    ],
    "keywords": [
      "hilbert transform",
      "deduce square-function type results",
      "haar multipliers map",
      "scalar argument",
      "square function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3886B"
    }
  }
}