{
  "id": "1901.08318",
  "version": "v1",
  "published": "2019-01-24T10:02:31.000Z",
  "updated": "2019-01-24T10:02:31.000Z",
  "title": "The fundamental solution of a class of ultra-hyperbolic operators on Pseudo $H$-type groups",
  "authors": [
    "Wolfram Bauer",
    "André Froehly",
    "Irina Markina"
  ],
  "comment": "40 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Pseudo $H$-type Lie groups $G_{r,s}$ of signature $(r,s)$ are defined via a module action of the Clifford algebra $C\\ell_{r,s}$ on a vector space $V \\cong \\mathbb{R}^{2n}$. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let $\\mathcal{N}_{r,s}$ denote the Lie algebra corresponding to $G_{r,s}$. A choice of left-invariant vector fields $[X_1, \\ldots, X_{2n}]$ which generate a complement of the center of $\\mathcal{N}_{r,s}$ gives rise to a second order operator \\begin{equation*} \\Delta_{r,s}:= \\big{(}X_1^2+ \\ldots + X_n^2\\big{)}- \\big{(}X_{n+1}^2+ \\ldots + X_{2n}^2 \\big{)}, \\end{equation*} which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of $\\Delta_{r,s}$ in the case $r=0$, $s>0$ and study their properties. In the case of $r>0$ we prove that $\\Delta_{r,s}$ admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of $\\Delta_{r,s}$ and the existence of a fundamental solution in the space of Schwartz distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-24T10:02:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65M80",
      "22E25"
    ],
    "keywords": [
      "fundamental solution",
      "type groups",
      "ultra-hyperbolic operators",
      "nilpotent lie groups",
      "second order operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}