{
  "id": "1803.04570",
  "version": "v1",
  "published": "2018-03-12T23:17:00.000Z",
  "updated": "2018-03-12T23:17:00.000Z",
  "title": "Sharp inequalities for linear combinations of orthogonal martingales",
  "authors": [
    "Yong Ding",
    "Loukas Grafakos",
    "Kai Zhu"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CA",
    "math.PR"
  ],
  "abstract": "For any two real-valued continuous-path martingales $X=\\{X_t\\}_{t\\geq 0}$ and $Y=\\{Y_t\\}_{t\\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of the form $aX+bY$ with $a, b$ real numbers. The best $L^p$ constant is equal to the norm of the operator $aI+bH$ from $L^p$ to $L^p$, where $H$ is the Hilbert transform on the circle or real line. The values of these norms were found by Hollenbeck, Kalton and Verbitsky \\cite{HKV}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-12T23:17:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G44",
      "42A45"
    ],
    "keywords": [
      "linear combinations",
      "orthogonal martingales",
      "sharp inequalities",
      "real line",
      "real-valued continuous-path martingales"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}