{
  "id": "2302.01201",
  "version": "v1",
  "published": "2023-02-02T16:30:15.000Z",
  "updated": "2023-02-02T16:30:15.000Z",
  "title": "Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields",
  "authors": [
    "Jean Van Schaftingen"
  ],
  "comment": "46 pages, lecture notes for the CIME summer school \"Geometric and analytic aspects of functional variational principles'', June 27 - July 1, 2022",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "Although Ornstein's nonestimate entails the impossibility to control in general all the $L^1$-norm of derivatives of a function by the $L^1$-norm of a constant coefficient homogeneous vector differential operator, the corresponding endpoint Sobolev inequality has been known to hold in many cases: 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). The class of differential operators for which estimates holds can be characterized by a cancelling condition. The proof of the estimates rely on a duality estimate for $L^1$-vector fields lying in the kernel of a cocancelling differential operator, combined with classical linear algebra and harmonic analysis techniques. This characterization unifies classes of known Sobolev inequalities and extends to fractional Sobolev and Hardy inequalities. A similar weaker condition introduced by Rai\\c{t}\\u{a} characterizes the operators for which there is an $L^\\infty$-estimate on lower-order derivatives.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-02T16:30:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A23",
      "26D15",
      "35E05",
      "42B30",
      "42B35",
      "46E35"
    ],
    "keywords": [
      "endpoint gagliardo-nirenberg-sobolev inequalities",
      "vector fields",
      "injective ellipticity",
      "cancelling operators",
      "constant coefficient homogeneous vector differential"
    ],
    "tags": [
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}