{
  "id": "1605.01256",
  "version": "v1",
  "published": "2016-05-04T12:51:44.000Z",
  "updated": "2016-05-04T12:51:44.000Z",
  "title": "Oscillation and variation for semigroups 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 ${\\mathcal O(P^{[\\lambda]}_\\ast)}$ and variation operator ${\\mathcal V}_\\rho(P^{[\\lambda]}_\\ast)$ of the Poisson semigroup $\\{P^{[\\lambda]}_t\\}_{t>0}$ associated with $\\Delta_\\lambda$ are both bounded on $L^p(\\mathbb R_+, dm_\\lambda)$ for $p\\in(1, \\infty)$, $BMO({{\\mathbb R}_+},dm_\\lambda)$, from $L^1({{\\mathbb R}_+},dm_\\lambda)$ to $L^{1,\\,\\infty}({{\\mathbb R}_+},dm_\\lambda)$, and from $H^1({{\\mathbb R}_+},dm_\\lambda)$ to $L^1({{\\mathbb R}_+},dm_\\lambda)$, where $\\rho\\in(2, \\infty)$ and $dm_\\lambda(x):=x^{2\\lambda}\\,dx$. As an application, an equivalent characterization of $H^1({{\\mathbb R}_+},dm_\\lambda)$ in terms of ${\\mathcal V}_\\rho(P^{[\\lambda]}_\\ast)$ is also established. All these results hold if $\\{P^{[\\lambda]}_t\\}_{t>0}$ is replaced by the heat semigroup $\\{W^{[\\lambda]}_t\\}_{t>0}$. }",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-04T12:51:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B35",
      "42B30"
    ],
    "keywords": [
      "bessel operator",
      "variation operator",
      "results hold",
      "poisson semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}