{
  "id": "2411.13538",
  "version": "v1",
  "published": "2024-11-20T18:41:24.000Z",
  "updated": "2024-11-20T18:41:24.000Z",
  "title": "An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces",
  "authors": [
    "Gonzalo Flores"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For a domain $\\Omega$ in a finite-dimensional space $E$, we consider the space $M=(\\Omega,d)$ where $d$ is the intrinsic distance in $\\Omega$. We obtain an isometric representation of the space $\\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\\infty}(\\Omega;E^{*})$ and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space $\\mathcal{F}(M)$ as a quotient of the space $L^{1}(\\Omega;E)$. We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as a corollaries of our result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-20T18:41:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B10",
      "46F10"
    ],
    "keywords": [
      "lipschitz-free space",
      "finite-dimensional space",
      "length spaces",
      "intrinsic distance",
      "corresponding isometric representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}