{
  "id": "2204.04002",
  "version": "v1",
  "published": "2022-04-08T11:22:22.000Z",
  "updated": "2022-04-08T11:22:22.000Z",
  "title": "Gradient estimates under integral Ricci bounds",
  "authors": [
    "Ludovico Marini",
    "Stefano Pigola",
    "Giona Veronelli"
  ],
  "comment": "9 pages, comments are welcome!",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "In this paper we study $W^{1,p}$ global regularity estimates for solutions of $\\Delta u = f$ on Riemannian manifolds. Under integral (lower) bounds on the Ricci tensor we prove the validity of $L^p$-gradient estimates of the form $|| \\nabla u ||_{L^p} \\le C (|| u ||_{L^p} + || \\Delta u||_{L^p})$. We also construct a counterexample which proves that the previously known constant lower bounds on the Ricci curvature are optimal in the pointwise sense. The relation between $L^p$-gradient estimates and different notions of Sobolev spaces is also investigated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-08T11:22:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "35A23",
      "58J05",
      "35B45",
      "46E35"
    ],
    "keywords": [
      "gradient estimates",
      "integral ricci bounds",
      "global regularity estimates",
      "constant lower bounds",
      "sobolev spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}