{
  "id": "2203.01582",
  "version": "v1",
  "published": "2022-03-03T09:18:54.000Z",
  "updated": "2022-03-03T09:18:54.000Z",
  "title": "On solid cores and hulls of weighted Bergman spaces $A_μ^1$",
  "authors": [
    "José Bonet",
    "Wolfgang Lusky",
    "Jari Taskinen"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider weighted Bergman spaces $A_\\mu^1$ on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces we characterize the solid core of $A_\\mu^1$. Also, as a consequence of a characterization of solid $A_\\mu^1$-spaces we show that, in the case of entire functions, there indeed exist solid $A_\\mu^1$-spaces. The second part of the paper is restricted to the case of the unit disc and it contains a characterization of the solid hull of $A_\\mu^1$, when $\\mu $ equals the weighted Lebesgue measure with weight $v$. The results are based on a duality relation of weighted $A^1$- and $H^\\infty$-spaces, the validity of which requires the assumption that $- \\log v$ belongs to the class $\\mathcal{W}_0$, studied in a number of publications; moreover, $v$ has to satisfy condition $(b)$, introduced by the authors. The exponentially decreasing weight $v(z) = \\exp( -1 /(1-|z|)$ provides an example satisfying both assumptions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-03T09:18:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted bergman spaces",
      "solid core",
      "unit disc",
      "entire functions",
      "non-atomic borel measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}