{
  "id": "1911.13075",
  "version": "v1",
  "published": "2019-11-29T12:10:40.000Z",
  "updated": "2019-11-29T12:10:40.000Z",
  "title": "Sharp Sobolev inequalities via projection averages",
  "authors": [
    "Philipp Kniefacz",
    "Franz E. Schuster"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of $i$-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$ Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them -- the affine $L^p$ Sobolev inequality of Lutwak, Yang, and Zhang. When $p = 1$, the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-29T12:10:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "46E35"
    ],
    "keywords": [
      "sobolev inequality",
      "sharp sobolev inequalities",
      "projection averages",
      "dimensional projections",
      "extremal functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}