{
  "id": "2005.06347",
  "version": "v1",
  "published": "2020-05-13T14:42:08.000Z",
  "updated": "2020-05-13T14:42:08.000Z",
  "title": "Dimension Free Estimates for the Riesz Transforms Associated with some Fractional Operators",
  "authors": [
    "Benjamin Arras",
    "Christian Houdré"
  ],
  "comment": "35 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "In these notes, boundedness properties of the Riesz transforms associated with symmetric $\\alpha$-stable probability measures, $\\alpha \\in (1,2)$, are investigated on appropriate $L^p$ spaces, for $p \\in (1,+\\infty)$. Our approach is based on Bismut-type formulae in order to obtain useful representations for the different Riesz transforms. In the Euclidean setting, the method of transference and one-dimensional multiplier theory combined with fine properties of stable distributions imply dimension free estimate for the fractional Laplacian. A limiting argument as $\\alpha$ tends to $2^-$ then recovers the well-known result for the standard Laplacian. In the non-Euclidean setting, a regularization phenomenon, specific to the non-Gaussian stable case, provides the boundedness result as well as a dimension free estimate when the reference measure is the rotationally invariant $\\alpha$-stable probability measure on $\\mathbb{R}^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-13T14:42:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "60E07",
      "42B15",
      "26A33"
    ],
    "keywords": [
      "riesz transforms",
      "fractional operators",
      "stable probability measure",
      "distributions imply dimension free estimate",
      "one-dimensional multiplier theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}