{
  "id": "1304.6199",
  "version": "v2",
  "published": "2013-04-23T08:09:50.000Z",
  "updated": "2013-07-17T10:49:13.000Z",
  "title": "Riesz transforms and multipliers for the Bessel-Grushin operator",
  "authors": [
    "Víctor Almeida",
    "Jorge J. Betancor",
    "Alejandro J. Castro",
    "Kishin Sadarangani"
  ],
  "comment": "33 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We establish that the spectral multiplier $\\frak{M}(G_{\\alpha})$ associated to the differential operator $$ G_{\\alpha}=- \\Delta_x +\\sum_{j=1}^m{{\\alpha_j^2-1/4}\\over{x_j^2}}-|x|^2 \\Delta_y \\; \\text{on} (0,\\infty)^m \\times \\R^n,$$ which we denominate Bessel-Grushin operator, is of weak type $(1,1)$ provided that $\\frak{M}$ is in a suitable local Sobolev space. In order to do this we prove a suitable weighted Plancherel estimate. Also, we study $L^p$-boundedness properties of Riesz transforms associated to $G_{\\alpha}$, in the case $n=1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-17T10:49:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C10",
      "43A90"
    ],
    "keywords": [
      "riesz transforms",
      "denominate bessel-grushin operator",
      "suitable local sobolev space",
      "boundedness properties",
      "differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.6199A"
    }
  }
}