{
  "id": "1810.05069",
  "version": "v1",
  "published": "2018-10-11T15:06:24.000Z",
  "updated": "2018-10-11T15:06:24.000Z",
  "title": "Surjectivity of the $\\overline{\\partial}$-operator between spaces of weighted smooth vector-valued functions",
  "authors": [
    "Karsten Kruse"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\\overline{\\partial}$ between spaces of weighted smooth Fr\\'echet-valued functions. This is done by establishing an analog of H\\\"ormander's theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth $\\mathbb{C}$-valued functions whose topologyis given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag-Leffler procedure. Using tensor products, we deduce the corresponding result on the solvability of the inhomogeneous Cauchy-Riemann equation for Fr\\'echet-valued functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-11T15:06:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "32W05",
      "46A32",
      "46E40"
    ],
    "keywords": [
      "weighted smooth vector-valued functions",
      "surjectivity",
      "inhomogeneous cauchy-riemann equation",
      "derive sufficient conditions",
      "proof relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}