{
  "id": "0711.4117",
  "version": "v1",
  "published": "2007-11-26T21:03:40.000Z",
  "updated": "2007-11-26T21:03:40.000Z",
  "title": "A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group",
  "authors": [
    "Albert Boggess",
    "Andrew Raich"
  ],
  "comment": "8 pages",
  "journal": "Proc. Amer. Math. Soc. 137 (2009), no. 3, 937--944",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $L = -1/4 (\\sum_{j=1}^n(X_j^2+Y_j^2)+i\\gamma T)$ where $\\gamma$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \\times R^n \\times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\\partial_s\\rho = -L\\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operator in $C^n$ with weight $\\exp(-\\tau P(z_1,...,z_n))$ where $P(z_1,...,z_n) = 1/2(x_1^2 + >... x_n^2)$, $z_j=x_j+iy_j$, and $\\tau\\in R$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-26T21:03:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32W30",
      "33C45",
      "42C10"
    ],
    "keywords": [
      "heat equation",
      "fundamental solution",
      "simplified calculation",
      "left invariant vector fields",
      "fourier transform"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.4117B"
    }
  }
}