{
  "id": "1501.07746",
  "version": "v1",
  "published": "2015-01-30T12:11:50.000Z",
  "updated": "2015-01-30T12:11:50.000Z",
  "title": "Symbolic calculus and convolution semigroups of measures on the Heisenberg group",
  "authors": [
    "Krystian Bekała"
  ],
  "categories": [
    "math.RT",
    "math.FA"
  ],
  "abstract": "Let $P$ be a symmetric generalised laplacian on $R^{2n+1}$. It is known that $P$ generates semigroups of measures $\\mu_{t}$ on the Heisenberg group $H^{n}$ and $\\nu_{t}$ on the Abelian group $R^{2n+1}$. Recall that the underlying manifold of the Heisenberg group is $R^{2n+1}$. Suppose that the negative defined function $\\psi(\\xi)=-\\hat{P}(\\xi)$ satisfies some weight conditions and $|D^{\\alpha}\\psi(\\xi)| \\leq c_{\\alpha}\\psi(\\xi)(1+\\|\\xi\\|)^{-|\\alpha|}, \\xi \\in R^{2n+1}.$ We show that the semigroup $\\mu_{t}$ is a kind of perturbation of the semigroup $\\nu_{t}$. More precisely, we give pointwise estimates for the difference between the densities of $\\mu_{t}$ and $\\nu_{t}$ and we show that it is small with respect to $t$ and $x$. As a consequence we get a description of the asymptotic behaviour at origin of densities of a semigroup of measures which is analogon of the symmetrized gamma (gamma-variance) semigroup on the Heisenberg group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-30T12:11:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heisenberg group",
      "convolution semigroups",
      "symbolic calculus",
      "abelian group",
      "asymptotic behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150107746B"
    }
  }
}