{
  "id": "2008.11553",
  "version": "v1",
  "published": "2020-08-26T13:34:47.000Z",
  "updated": "2020-08-26T13:34:47.000Z",
  "title": "Norm estimates of the partial derivatives for harmonic and harmonic elliptic mappings",
  "authors": [
    "Sh. Chen",
    "S. Ponnusamy",
    "X. Wang"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CV"
  ],
  "abstract": "Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\\mathbb{T}$ and $\\dot{F}\\in L_p(0, 2\\pi)$, where $\\dot{F}(e^{it})=\\frac{d}{dt} F(e^{it})$ and $p\\geq 1$. Recently, the author in \\cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\\leq p< 2$, then $f_{z}$ and $\\overline{f_{\\overline{z}}}\\in \\mathcal{B}^{p}(\\mathbb{D}),$ the classical Bergman spaces of $\\mathbb{D}$ \\cite[Theorem 1.2]{Zhu}; $(2)$ if $f$ is a harmonic quasiregular mapping and $1\\leq p\\leq \\infty$, then $f_{z},$ $\\overline{f_{\\overline{z}}}\\in \\mathcal{H}^{p}(\\mathbb{D}),$ the classical Hardy spaces of $\\mathbb{D}$ \\cite[Theorem 1.3]{Zhu}. These are the main results in \\cite{Zhu}. The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, \\cite[Theorem 1.2]{Zhu} is true when $1\\leq p< \\infty$. Also, we show that \\cite[Theorem 1.2]{Zhu} is not true when $p=\\infty$. Second, we demonstrate that \\cite[Theorem 1.3]{Zhu} still holds true when the assumption $f$ being a harmonic quasiregular mapping is replaced by the weaker one $f$ being a harmonic elliptic mapping.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-26T13:34:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C62",
      "31A05",
      "30H10",
      "30H20"
    ],
    "keywords": [
      "harmonic elliptic mapping",
      "partial derivatives",
      "norm estimates",
      "harmonic quasiregular mapping",
      "classical hardy spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}