{
  "id": "1107.3266",
  "version": "v1",
  "published": "2011-07-17T00:36:38.000Z",
  "updated": "2011-07-17T00:36:38.000Z",
  "title": "Weighted Local Orlicz-Hardy Spaces with Applications to Pseudo-differential Operators",
  "authors": [
    "Dachun Yang",
    "Sibei Yang"
  ],
  "comment": "80 pages, Dissertationes Math. (Rozprawy Mat.) (to appear)",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $\\Phi$ be a concave function on $(0,\\infty)$ of strictly lower type $p_{\\Phi}\\in(0,1]$ and $\\omega\\in A^{\\mathop\\mathrm{loc}}_{\\infty}(\\mathbb{R}^n)$. We introduce the weighted local Orlicz-Hardy space $h^{\\Phi}_{\\omega}(\\mathbb{R}^n)$ via the local grand maximal function. Let $\\rho(t)\\equiv t^{-1}/\\Phi^{-1}(t^{-1})$ for all $t\\in(0,\\infty)$. We also introduce the $\\mathop\\mathrm{BMO}$-type space $\\mathop\\mathrm{bmo}_{\\rho,\\,\\omega}(\\mathbb{R}^n)$ and establish the duality between $h^{\\Phi}_{\\omega}(\\mathbb{R}^n)$ and $\\mathop\\mathrm{bmo}_{\\rho,\\,\\omega}(\\mathbb{R}^n)$. Several real-varaiable characterizations of $h^{\\Phi}_{\\omega}(\\mathbb{R}^n)$ are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h^{\\Phi}_{\\omega}(\\mathbb{R}^n)$. As applications, we show that the local Riesz transforms are bounded on $h^{\\Phi}_{\\omega}(\\mathbb{R}^n)$, the local fractional integrals are bounded from {\\normalsize$h^p_{\\omega^p}(\\mathbb{R}^n)$} to {\\normalsize$L^q_{\\omega^q}(\\mathbb{R}^n)$} when $q>1$ and from {\\normalsize$h^p_{\\omega^p}(\\mathbb{R}^n)$} to {\\normalsize$h^q_{\\omega^q}(\\mathbb{R}^n)$} when $q\\le 1$, and some pseudo-differential operators are also bounded on both $h^{\\Phi}_{\\omega}(\\mathbb{R}^n)$. All results for any general $\\Phi$ even when $\\omega\\equiv 1$ are new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-17T00:36:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "42B35",
      "42B30",
      "42B25",
      "42B20",
      "35S05",
      "47G30",
      "47B06"
    ],
    "keywords": [
      "weighted local orlicz-hardy space",
      "pseudo-differential operators",
      "applications",
      "local grand maximal function",
      "local riesz transforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 80,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.3266Y"
    }
  }
}