{
  "id": "2007.12827",
  "version": "v1",
  "published": "2020-07-25T01:55:54.000Z",
  "updated": "2020-07-25T01:55:54.000Z",
  "title": "Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings",
  "authors": [
    "Jian-Feng Zhu"
  ],
  "comment": "17 pages",
  "journal": "The Journal of Geometric Analysis, 2020",
  "doi": "10.1007/s12220-020-00488-x",
  "categories": [
    "math.CV"
  ],
  "abstract": "Suppose $p\\geq1$, $w=P[F]$ is a harmonic mapping of the unit disk $\\mathbb{D}$ satisfying $F$ is absolutely continuous and $\\dot{F}\\in L^p(0, 2\\pi)$, where $\\dot{F}(e^{it})=\\frac{\\mathrm{d}}{\\mathrm{d}t}F(e^{it})$. In this paper, we obtain Bergman norm estimates of the partial derivatives for $w$, i.e., $\\|w_z\\|_{L^p}$ and $\\|\\overline{w_{\\bar{z}}}\\|_{L^p}$, where $1\\leq p<2$. Furthermore, if $w$ is a harmonic quasiregular mapping of $\\mathbb{D}$, then we show that $w_z$ and $\\overline{w_{\\bar{z}}}$ are in the Hardy space $H^p$, where $1\\leq p\\leq\\infty$. The corresponding Hardy norm estimates, $\\|w_z\\|_{p}$ and $\\|\\overline{w_{\\bar{z}}}\\|_{p}$, are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-25T01:55:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C55",
      "30C62",
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "harmonic quasiregular mapping",
      "partial derivatives",
      "harmonic mapping",
      "corresponding hardy norm estimates",
      "bergman norm estimates"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}