{
  "id": "2309.12850",
  "version": "v1",
  "published": "2023-09-22T13:23:03.000Z",
  "updated": "2023-09-22T13:23:03.000Z",
  "title": "Corona theorem for the Dirichlet-type space",
  "authors": [
    "Shuaibing Luo"
  ],
  "journal": "J. Geom. Anal. 32 (2022)",
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper utilizes Cauchy's transform and duality for the Dirichlet-type space $D(\\mu)$ with positive superharmonic weight $U_\\mu$ on the unit disk $\\mathbb{D}$ to establish the corona theorem for the Dirichlet-type multiplier algebra $M\\big(D(\\mu)\\big)$ that: if $$\\{f_1,...,f_n\\}\\subseteq M\\big(D(\\mu)\\big)\\quad\\text{and}\\quad \\inf_{z\\in\\mathbb{D}}\\sum_{j=1}^n|f_j(z)|>0$$ then $$ \\exists\\,\\{g_1,...,g_n\\}\\subseteq M\\big(D(\\mu)\\big)\\quad\\text{such that}\\quad \\sum_{j=1}^nf_jg_j=1, $$ thereby generalizing Carleson's corona theorem for $M(H^2)=H^\\infty$ and Xiao's corona theorem for $M(\\mathscr{D})\\subset H^\\infty$ thanks to $$ D(\\mu)=\\begin{cases} \\text{Hardy space}\\ H^2\\quad &\\text{as}\\quad d\\mu(z)=(1-|z|^2)\\,dA(z)\\ \\ \\forall\\ z\\in\\mathbb{D};\\\\ \\text{Dirichlet space}\\ \\mathscr{D}\\ &\\text{as}\\quad d\\mu(z)=|dz|\\ \\ \\forall\\ z\\in\\mathbb{T}=\\partial{\\mathbb{D}}. \\end{cases} $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-22T13:23:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet-type space",
      "paper utilizes cauchys transform",
      "xiaos corona theorem",
      "generalizing carlesons corona theorem",
      "dirichlet-type multiplier algebra"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}