{
  "id": "2409.01292",
  "version": "v1",
  "published": "2024-09-02T14:34:06.000Z",
  "updated": "2024-09-02T14:34:06.000Z",
  "title": "Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces",
  "authors": [
    "Takashi Kumagai",
    "Nageswari Shanmugalingam",
    "Ryosuke Shimizu"
  ],
  "comment": "24 pages, 3 figures",
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "In the context of a metric measure space $(X,d,\\mu)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^\\theta_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $\\theta$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents $\\theta_p(X)$ and $\\theta_p^{\\ast}(X)$ for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces $X$ formed by glueing copies of $n$-dimensional cubes, the Sierpi\\'{n}ski gaskets, and of the Sierpi\\'{n}ski carpet.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-02T14:34:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31E05",
      "28A80",
      "46E36",
      "31C25"
    ],
    "keywords": [
      "metric spaces",
      "finite dimensionality",
      "potential-theoretic decomposition",
      "critical exponents",
      "finite-dimensional besov space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}