{
  "id": "2305.00840",
  "version": "v1",
  "published": "2023-04-28T08:29:44.000Z",
  "updated": "2023-04-28T08:29:44.000Z",
  "title": "Endpoint Sobolev inequalities for vector fields and cancelling operators",
  "authors": [
    "Jean Van Schaftingen"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "The injectively elliptic vector differential operators $A (\\mathrm{D})$ from $V$ to $E$ on $\\mathbb{R}^n$ such that the estimate \\[ \\Vert D^\\ell u\\Vert_{L^{n/(n - \\ell)} (\\mathbb{R}^n)} \\le \\Vert A (\\mathrm{D}) u\\Vert_{L^1 (\\mathbb{R}^n)} \\] holds can be characterized as the operators satisfying a cancellation condition \\[ \\bigcap_{\\xi \\in \\mathbb{R}^n \\setminus \\{0\\}} A (\\xi)[V] = \\{0\\}\\;. \\] These estimates unify existing endpoint Sobolev inequalities for the gradient of scalar functions (Gagliardo and Nirenberg), the deformation operator (Korn-Sobolev inequality by M.J. Strauss) and the Hodge complex (Bourgain and Brezis). Their proof is based on the fact that $A (\\mathrm{D}) u$ lies in the kernel of a cocancelling differential operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-28T08:29:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A23",
      "26D15",
      "35E05",
      "42B30",
      "42B35",
      "46E35"
    ],
    "keywords": [
      "vector fields",
      "cancelling operators",
      "existing endpoint sobolev inequalities",
      "inequality",
      "injectively elliptic vector differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}