{
  "id": "1612.02033",
  "version": "v1",
  "published": "2016-12-06T21:26:35.000Z",
  "updated": "2016-12-06T21:26:35.000Z",
  "title": "Global hypoellipticity for a class of pseudo-differential operators on the torus",
  "authors": [
    "Fernando de Ávila Silva",
    "Rafael Borro Gonzalez",
    "Alexandre Kirilov",
    "Cleber de Medeira"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We show that an obstruction of number-theoretical nature appears as a necessary condition for the global hypoellipticity of the pseudo-differential operator $L=D_t+(a+ib)(t)P(D_x)$ on $\\mathbb{T}^1_t\\times\\mathbb{T}_x^{N}$. This condition is also sufficient when the symbol $p(\\xi)$ of $P(D_x)$ has at most logarithmic growth. If $p(\\xi)$ has super-logarithmic growth, we show that the global hypoellipticity of $L$ depends on the change of sign of certain interactions of the coefficients with the symbol $p(\\xi).$ Moreover, the interplay between the order of vanishing of coefficients with the order of growth of $p(\\xi)$ plays a crucial role in the global hypoellipticity of $L$. We also describe completely the global hypoellipticity of $L$ in the case where $P(D_x)$ is positively homogeneous. Additionally, we explore the influence of irrational approximations of a real number in the global hypoellipticity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-06T21:26:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B10",
      "35H10",
      "35S05",
      "58Jxx"
    ],
    "keywords": [
      "global hypoellipticity",
      "pseudo-differential operator",
      "super-logarithmic growth",
      "real number",
      "number-theoretical nature appears"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}