{
  "id": "2002.03373",
  "version": "v1",
  "published": "2020-02-09T14:31:05.000Z",
  "updated": "2020-02-09T14:31:05.000Z",
  "title": "Globally hypoelliptic triangularizable systems of periodic pseudo-differential operators",
  "authors": [
    "Fernando de Ávila Silva"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This article presents an investigation on the global hypoellipticity problem for systems belonging to the class $P = D_t + Q(t,D_x)$, where $Q(t,D_x)$ is a $m\\times m$ matrix with entries $c_{j,k}(t)Q_{j,k}(D_x)$. The coefficients $c_{j,k}(t)$ are smooth, complex-valued functions on the torus $\\mathbb{T} \\simeq \\mathbb{R}/2\\pi\\mathbb{Z}$ and $Q_{j,k}(D_x)$ are pseudo-differential operators on $ \\mathbb{T}^n$. The approach consists in establishing conditions on the matrix symbol $Q(t,\\xi)$ such that it can be transformed into a suitable triangular form $\\Lambda(t,\\xi) + \\mathcal{N}(t,\\xi)$, where $\\Lambda(t,\\xi)$ is the diagonal matrix $diag(\\lambda_{1}(t,\\xi) \\ldots \\lambda_{m}(t,\\xi))$ and $\\mathcal{N}(t,\\xi)$ is a nilpotent upper triangular matrix. Hence, the global hypoellipticity of $P$ is studied by analyzing the behavior of the eigenvalues $\\lambda_{j}(t,\\xi)$ and its averages $\\lambda_{0,j}(\\xi)$, as $|\\xi| \\to \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-09T14:31:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B10",
      "35B65",
      "35H10",
      "35S05"
    ],
    "keywords": [
      "globally hypoelliptic triangularizable systems",
      "periodic pseudo-differential operators",
      "nilpotent upper triangular matrix",
      "global hypoellipticity problem",
      "diagonal matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}