{
  "id": "2207.08545",
  "version": "v1",
  "published": "2022-07-18T12:10:30.000Z",
  "updated": "2022-07-18T12:10:30.000Z",
  "title": "$L^{p}$ gradient estimates and Calderón--Zygmund inequalities under Ricci lower bounds",
  "authors": [
    "Ludovico Marini",
    "Stefano Meda",
    "Stefano Pigola",
    "Giona Veronelli"
  ],
  "comment": "22 pages, comments welcome! arXiv:2204.04002 was merged into this paper",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "In this paper we investigate the validity of first and second order $L^{p}$ estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present $L^{p}$ estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain $L^p$ estimates for the second order Riesz transform (or, equivalently, the validity of $L^{p}$ Calder\\'on--Zygmund inequalities) on the whole scale $1<p<+\\infty$ by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded or non-negative in a global integral sense. When $1<p \\leq 2$, analogous $L^p$ bounds on even higher order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order. In the same range of values of $p$, for manifolds with lower Ricci bounds and positive bottom of the spectrum, we show that the $L^{p}$ norm of the Laplacian controls the whole $W^{2,p}$-norm on compactly supported functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-18T12:10:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05",
      "53C21",
      "35B45",
      "42B20"
    ],
    "keywords": [
      "ricci lower bounds",
      "gradient estimates",
      "calderón-zygmund inequalities",
      "second order riesz transform",
      "ricci tensor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}