{
  "id": "2408.04513",
  "version": "v1",
  "published": "2024-08-08T15:14:47.000Z",
  "updated": "2024-08-08T15:14:47.000Z",
  "title": "Extensions of divergence-free fields in $\\mathrm{L}^{1}$-based function spaces",
  "authors": [
    "Franz Gmeineder",
    "Stefan Schiffer"
  ],
  "comment": "48 pages, 8 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish the first extension results for divergence-free (or solenoidal) elements of $\\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying $\\mathrm{L}^{1}$-boundedness. While previous results as in Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] for $\\mathrm{L}^{p}$-based function spaces, $1<p<\\infty$, rely on PDE approaches, basic principles from harmonic analysis rule out such strategies in the $\\mathrm{L}^{1}$-context. By means of a novel method adapted to the divergence-free constraint via differential forms, we establish the existence of such extension operators in the $\\mathrm{L}^{1}$-based situation. This applies both to the case of convex domains, where a global extensions can be achieved, as well as to the Lipschitz case, where a local extension can be achieved. Being applicable to $1<p<\\infty$ too, our method provides a unifying approach to the cases $p\\in\\{1,\\infty\\}$ and $1<p<\\infty$. Specifically, covering the exponents $p\\in\\{1,\\infty\\}$, this answers a borderline case left open by Kato et al. [Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 2000] in the affirmative. By use of explicit examples, the assumptions on the underlying domains are shown to be almost optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-08T15:14:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E40",
      "35D30",
      "53A05",
      "26B20"
    ],
    "keywords": [
      "function spaces",
      "divergence-free fields",
      "divergence-free vector fields",
      "borderline case left open",
      "first extension results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}