{
  "id": "1703.08165",
  "version": "v1",
  "published": "2017-03-23T17:51:26.000Z",
  "updated": "2017-03-23T17:51:26.000Z",
  "title": "Weighted Bergman spaces of domains with Levi-flat boundary: geodesic segments on compact Riemann surfaces",
  "authors": [
    "Masanori Adachi"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CV"
  ],
  "abstract": "The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses the case of the space of all the geodesic segments on a hyperbolic compact Riemann surface, which is a typical example of such a domain in the sense that its realization as a holomorphic disk bundle has the best possible Diederich-Fornaess index $1/2$. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the base Riemann surface via optimal $L^2$-jet extension, and, in particular, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for all the order greater than $-1$ beyond $-1/2$, the limiting order until which known $L^2$-estimates for the $\\overline{\\partial}$-equation work. Some applications are also given thanks to the generalized hypergeometric function $_3F_2$ expressing the norm of the optimal $L^2$-jet extension: a proof for the Liouvilleness which does not appeal to the ergodicity of the Levi foliation, and a Forelli-Rudin construction for the disk bundle over the Riemann surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-23T17:51:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32A36",
      "32A05",
      "32A25",
      "32N99",
      "32T27",
      "32W05",
      "33C20"
    ],
    "keywords": [
      "weighted bergman spaces",
      "levi-flat boundary",
      "geodesic segments",
      "hyperbolic compact riemann surface",
      "holomorphic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}