{
  "id": "1802.08550",
  "version": "v1",
  "published": "2018-02-18T21:51:12.000Z",
  "updated": "2018-02-18T21:51:12.000Z",
  "title": "Morrey spaces related to certain nonnegative potentials and fractional integrals on the Heisenberg groups",
  "authors": [
    "Hua Wang"
  ],
  "comment": "22 pages. arXiv admin note: text overlap with arXiv:1802.02481",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\mathcal L=-\\Delta_{\\mathbb H^n}+V$ be a Schr\\\"odinger operator on the Heisenberg group $\\mathbb H^n$, where $\\Delta_{\\mathbb H^n}$ is the sub-Laplacian on $\\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\\\"older class $RH_s$ with $s\\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\\mathbb H^n$. For given $\\alpha\\in(0,Q)$, the fractional integrals associated to the Schr\\\"odinger operator $\\mathcal L$ is defined by $\\mathcal I_{\\alpha}={\\mathcal L}^{-{\\alpha}/2}$. In this article, we first introduce the Morrey space $L^{p,\\kappa}_{\\rho,\\infty}(\\mathbb H^n)$ and weak Morrey space $WL^{p,\\kappa}_{\\rho,\\infty}(\\mathbb H^n)$ related to the nonnegative potential $V$. Then we establish the boundedness of fractional integrals ${\\mathcal L}^{-{\\alpha}/2}$ on these new spaces. Furthermore, in order to deal with certain extreme cases, we also introduce the spaces $\\mathrm{BMO}_{\\rho,\\infty}(\\mathbb H^n)$ and $\\mathcal{C}^{\\beta}_{\\rho,\\infty}(\\mathbb H^n)$ with exponent $\\beta\\in(0,1]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-18T21:51:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "35J10",
      "22E25",
      "22E30"
    ],
    "keywords": [
      "fractional integrals",
      "nonnegative potential",
      "heisenberg group",
      "weak morrey space",
      "extreme cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}