{
  "id": "2209.09751",
  "version": "v1",
  "published": "2022-09-20T14:30:14.000Z",
  "updated": "2022-09-20T14:30:14.000Z",
  "title": "Global pseudo-differential operators on the Lie group $G= (-1,1)^n$",
  "authors": [
    "Duván Cardona",
    "Roland Duduchava",
    "Arne Hendrickx",
    "Michael Ruzhansky"
  ],
  "comment": "34 Pages; 1 Figure",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "In this work we characterise the H\\\"ormander classes $\\symbClassOn{m}{\\rho}{\\delta}{\\group,\\textnormal{H\\\"or}}$ on the open manifold $\\group = (-1,1)^n$. We show that by endowing the open manifold $\\group = (-1,1)^n$ with a group structure, the corresponding global Fourier analysis on the group allows one to define a global notion of symbol on the phase space $\\group \\times \\R^n$. Then, the class of pseudo-differential operators associated to the global H\\\"ormander classes $\\symbClassOn{m}{\\rho}{\\delta}{\\group \\times \\R^n}$ recovers the H\\\"ormander classes $\\symbClassOn{m}{\\rho}{\\delta}{\\group,\\textnormal{loc}}$ defined by local coordinate systems. The analytic and qualitative properties of the classes $\\symbClassOn{m}{\\rho}{\\delta}{\\group \\times \\R^n}$ are presented in terms of the corresponding global symbols. In particular, $L^p$-Fefferman type estimates and Calder\\'on-Vaillancourt theorems are analysed, as well as the spectral properties of the operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-20T14:30:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global pseudo-differential operators",
      "lie group",
      "open manifold",
      "fefferman type estimates",
      "corresponding global fourier analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}