{
  "id": "2207.03658",
  "version": "v1",
  "published": "2022-07-08T02:36:00.000Z",
  "updated": "2022-07-08T02:36:00.000Z",
  "title": "Pseudo-Differential Operators, Wigner Transform, and Weyl Transform on the Affine Poincaré Group",
  "authors": [
    "Aparajita Dasgupta",
    "Santosh Kumar Nayak"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we study harmonic analysis on the affine Poincar\\'e group $\\mathcal{P}_{aff}$, which is a non-unimodular group, and obtain pseudo-differential operators with operator valued symbols. More precisely, we study the boundedness properties of pseudo-differential operators on $\\mathcal{P}_{aff}$. We also provide a necessary and sufficient condition on the operator-valued symbols such that the corresponding pseudo-differential operators are in the class of Hilbert--Schmidt operators. Consequently, we obtain a characterization of the trace class pseudo-differential operators on the Poincar\\'e affine group $\\mathcal{P}_{aff}$, and provide a trace formula for these trace class operators. Finally, we study the Wigner transform, and Weyl transform associated with the operator valued symbol on the Poincar\\'e affine group $\\mathcal{P}_{aff}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-08T02:36:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G10",
      "47G30",
      "42C40"
    ],
    "keywords": [
      "wigner transform",
      "weyl transform",
      "poincare affine group",
      "operator valued symbol",
      "trace class pseudo-differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}