{
  "id": "1204.1159",
  "version": "v1",
  "published": "2012-04-05T09:27:49.000Z",
  "updated": "2012-04-05T09:27:49.000Z",
  "title": "Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators",
  "authors": [
    "Alessio Martini",
    "Adam Sikora"
  ],
  "journal": "Mathematical Research Letters, 19 no. 5 (2012), p. 1075-1088",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Grushin operators acting on $\\R^{d_1}_{x'}\\times \\R^{d_2}_{x\"}$ and defined by the formula \\[ L=-\\sum_{\\jone=1}^{d_1}\\partial_{x'_\\jone}^2 - (\\sum_{\\jone=1}^{d_1}|x'_\\jone|^2) \\sum_{\\jtwo=1}^{d_2}\\partial_{x\"_\\jtwo}^2. \\] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These multiplier results are sharp if $d_1 \\ge d_2$. We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$. The described spectral multiplier theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by D. M\\\"uller and E.M. Stein and by W. Hebisch.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-05T09:27:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted plancherel estimates",
      "sharp spectral multipliers",
      "grushin operators",
      "spectral multiplier theorem",
      "spectral multiplier results"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.1159M"
    }
  }
}