{
  "id": "1712.03065",
  "version": "v1",
  "published": "2017-12-08T13:57:27.000Z",
  "updated": "2017-12-08T13:57:27.000Z",
  "title": "A sharp multiplier theorem for a perturbation-invariant class of Grushin operators of arbitrary step",
  "authors": [
    "Gian Maria Dall'Ara",
    "Alessio Martini"
  ],
  "comment": "38 pages",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "We prove a multiplier theorem of Mihlin--H\\\"ormander type for operators of the form $-\\Delta_x - V(x) \\Delta_y$ on $\\mathbb{R}^{d_1}_x \\times \\mathbb{R}^{d_2}_y$, where $V(x) = \\sum_{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \\mapsto |t|^{2\\sigma}$, and $\\sigma \\in (1/2,\\infty)$. The result is sharp whenever $d_1 \\geq (1+\\sigma) d_2$. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schr\\\"odinger operators, which are stable under perturbations of the potential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-08T13:57:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L20",
      "35J70",
      "35H20",
      "42B15"
    ],
    "keywords": [
      "sharp multiplier theorem",
      "grushin operators",
      "perturbation-invariant class",
      "arbitrary step",
      "perturbations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}