{
  "id": "1907.09398",
  "version": "v1",
  "published": "2019-07-16T05:26:02.000Z",
  "updated": "2019-07-16T05:26:02.000Z",
  "title": "Morrey spaces for Schrödinger operators with certain nonnegative potentials, Littlewood-Paley and Lusin functions on the Heisenberg groups",
  "authors": [
    "Hua Wang"
  ],
  "comment": "28 pages. arXiv admin note: text overlap with arXiv:1802.08550, arXiv:1907.03573",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "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 sublaplacian on $\\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\\\"older class $RH_q$ with $q\\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\\mathbb H^n$. Assume that $\\{e^{-s\\mathcal L}\\}_{s>0}$ is the heat semigroup generated by $\\mathcal L$. The Littlewood-Paley function $\\mathfrak{g}_{\\mathcal L}$ and the Lusin area integral $\\mathcal{S}_{\\mathcal L}$ associated with the Schr\\\"odinger operator $\\mathcal L$ are defined, respectively, by \\begin{equation*} \\mathfrak{g}_{\\mathcal L}(f)(u) := \\bigg(\\int_0^{\\infty}\\bigg|s\\frac{d}{ds} e^{-s\\mathcal L}f(u) \\bigg|^2\\frac{ds}{s}\\bigg)^{1/2} \\end{equation*} and \\begin{equation*} \\mathcal{S}_{\\mathcal L}(f)(u) := \\bigg(\\iint_{\\Gamma(u)} \\bigg|s\\frac{d}{ds} e^{-s\\mathcal L}f(v) \\bigg|^2 \\frac{dvds}{s^{Q/2+1}}\\bigg)^{1/2}, \\end{equation*} where \\begin{equation*} \\Gamma(u) := \\big\\{(v,s)\\in\\mathbb H^n\\times(0,\\infty): |u^{-1}v| < \\sqrt{s\\,}\\big\\}. \\end{equation*} In this paper the author first introduces a class of Morrey spaces associated with the Schr\\\"odinger operator $\\mathcal L$ on $\\mathbb H^n$. Then by using some pointwise estimates of the kernels related to the nonnegative potential $V$, the author establishes the boundedness properties of these two operators $\\mathfrak{g}_{\\mathcal L}$ and $\\mathcal{S}_{\\mathcal L}$ acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators $\\mathfrak{g}_{\\sqrt{\\mathcal L}}$ and $\\mathcal{S}_{\\sqrt{\\mathcal L}}$ with respect to the Poisson semigroup $\\{e^{-s\\sqrt{\\mathcal L}}\\}_{s>0}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-16T05:26:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "35J10",
      "22E25",
      "22E30"
    ],
    "keywords": [
      "morrey spaces",
      "nonnegative potential",
      "heisenberg group",
      "schrödinger operators",
      "lusin functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}