{
  "id": "1503.02508",
  "version": "v1",
  "published": "2015-03-09T15:10:06.000Z",
  "updated": "2015-03-09T15:10:06.000Z",
  "title": "Riesz transforms through reverse Hölder and Poincaré inequalities",
  "authors": [
    "Frédéric Bernicot",
    "Dorothee Frey"
  ],
  "comment": "36 pages",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CA"
  ],
  "abstract": "We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $\\omega$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse H\\\"older inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \\in [2,p)$. This improves results of \\cite{ACDH} and \\cite{AC}, where the stronger assumption of a Poincar\\'e inequality and the assumption $e^{-tL}(1)=1$ were made. The Poincar\\'e inequality assumption is also weakened in the setting of a sectorial operator $L$. In the last section, we study elliptic perturbations of Riesz transforms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-09T15:10:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J35",
      "42B20"
    ],
    "keywords": [
      "riesz transform",
      "reverse hölder",
      "study elliptic perturbations",
      "poincare inequality assumption",
      "gradient operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}