{
  "id": "1605.01251",
  "version": "v1",
  "published": "2016-05-04T12:44:52.000Z",
  "updated": "2016-05-04T12:44:52.000Z",
  "title": "Oscillation and variation for Riesz transform associated with Bessel operators",
  "authors": [
    "Huoxiong Wu",
    "Dongyong Yang",
    "Jing Zhang"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\lambda>0$ and $\\triangle_\\lambda:=-\\frac{d^2}{dx^2}-\\frac{2\\lambda}{x} \\frac d{dx}$ be the Bessel operator on $\\mathbb R_+:=(0,\\infty)$. We show that the oscillation operator $\\mathcal{O}(R_{\\Delta_{\\lambda},\\ast})$ and variation operator $\\mathcal{V}_{\\rho}(R_{\\Delta_{\\lambda},\\ast})$ of the Riesz transform $R_{\\Delta_{\\lambda}}$ associated with $\\Delta_\\lambda$ are both bounded on $L^p(\\mathbb R_+, dm_{\\lambda})$ for $p\\in(1,\\,\\infty)$, from $L^1(\\mathbb{R}_{+},dm_{\\lambda})$ to $L^{1,\\,\\infty}(\\mathbb{R}_{+},dm_{\\lambda})$, and from $L^{\\infty}(\\mathbb{R}_{+},dm_{\\lambda})$ to $BMO(\\mathbb{R}_{+},dm_{\\lambda})$, where $\\rho\\in (2,\\infty)$ and $dm_{\\lambda}(x):=x^{2\\lambda}dx$. As an application, we give the corresponding $L^p$-estimates for $\\beta$-jump operators and the number of up-crossing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-04T12:44:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B35"
    ],
    "keywords": [
      "riesz transform",
      "bessel operator",
      "oscillation operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}