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The explicit solution of the $\bar\partial$-Neumann problem in a non-isotropic Siegel domain

  Published:1997-12-01
 Printed: Dec 1997
  • Jingzhi Tie
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Abstract

In this paper, we solve the $\dbar$-Neumann problem on $(0,q)$ forms, $0\leq q \leq n$, in the strictly pseudoconvex non-isotropic Siegel domain: \[ \cU=\left\{ \begin{array}{clc} &\bz=(z_1,\ldots,z_n) \in \C^{n},\\ (\bz,z_{n+1}):&&\Im (z_{n+1}) > \sum_{j=1}^{n}a_j |z_j|^2 \\ &z_{n+1}\in \C; \end{array} \right\}, \] where $a_j> 0$ for $j=1,2,\ldots, n$. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.
MSC Classifications: 32F15, 32F20, 35N15 show english descriptions unknown classification 32F15
unknown classification 32F20
$\overline\partial$-Neumann problem and generalizations; formal complexes [See also 32W05, 32W10, 58J10]
32F15 - unknown classification 32F15
32F20 - unknown classification 32F20
35N15 - $\overline\partial$-Neumann problem and generalizations; formal complexes [See also 32W05, 32W10, 58J10]
 

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