{
  "id": "2004.12947",
  "version": "v1",
  "published": "2020-04-27T17:04:46.000Z",
  "updated": "2020-04-27T17:04:46.000Z",
  "title": "Subexponential decay and regularity estimates for eigenfunctions of localization operators",
  "authors": [
    "Federico Bastianoni",
    "Nenad Teofanov"
  ],
  "comment": "29 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider time-frequency localization operators $A_a^{\\varphi_1,\\varphi_2}$ with symbols $a$ in the wide weighted modulation space $ M^\\infty_{w}(\\mathbb{R}^{2d})$, and windows $ \\varphi_1, \\varphi_2 $ in the Gelfand-Shilov space $\\mathcal{S}^{\\left(1\\right)}(\\mathbb{R}^{d})$. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $A_a^{\\varphi_1,\\varphi_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $ \\mathcal{S}^{(\\gamma)} (\\mathbb{R}^{2d}) $, where the parameter $\\gamma \\geq 1 $ is related to the growth of the considered weight. An important role is played by $\\tau$-pseudodifferential operators $\\mathrm{Op}_\\tau(\\sigma)$. In that direction we show convenient continuity properties of $\\mathrm{Op}_\\tau(\\sigma)$ when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $\\mathrm{Op}_\\tau(\\sigma)$ when the symbol $\\sigma$ belongs to a modulation space with appropriately chosen weight functions. As a tool we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-27T17:04:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G30",
      "47B10",
      "46F05",
      "35S05"
    ],
    "keywords": [
      "regularity estimates",
      "eigenfunctions",
      "wide weighted modulation space",
      "banach weighted modulation spaces",
      "appropriately chosen weight functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}