{
  "id": "0903.4536",
  "version": "v2",
  "published": "2009-03-26T09:09:11.000Z",
  "updated": "2009-11-07T04:42:10.000Z",
  "title": "Localized BMO and BLO Spaces on RD-Spaces and Applications to Schrödinger Operators",
  "authors": [
    "Dachun Yang",
    "Dongyong Yang",
    "Yuan Zhou"
  ],
  "comment": "Commun. Pure Appl. Anal. (to appear)",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "An RD-space ${\\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in ${\\mathcal X}$. Let $\\rho$ be an admissible function on RD-space ${\\mathcal X}$. The authors first introduce the localized spaces $\\mathrm{BMO}_\\rho({\\mathcal X})$ and $\\mathrm{BLO}_\\rho({\\mathcal X})$ and establish their basic properties, including the John-Nirenberg inequality for $\\mathrm{BMO}_\\rho({\\mathcal X})$, several equivalent characterizations for $\\mathrm{BLO}_\\rho({\\mathcal X})$, and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to $\\rho$, and the Littlewood-Paley $g$-function associated to $\\rho$, where the Littlewood-Paley $g$-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schr\\\"odinger operator. These results apply in a wide range of settings, for instance, to the Schr\\\"odinger operator or the degenerate Schr\\\"odinger operator on ${{\\mathbb R}}^d$, or the sub-Laplace Schr\\\"odinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-07T04:42:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B35",
      "42B20",
      "42B25",
      "42B30"
    ],
    "keywords": [
      "schrödinger operators",
      "blo spaces",
      "localized bmo",
      "radial maximal functions",
      "maximal operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.4536Y"
    }
  }
}