Fundamental Solutions to $\Box_b$ on Certain Quadrics

Abstract

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of $\mathbb C^n\times\mathbb C^m$ and the group Fourier transform to reduce the $\Box_b$ equation to ones that can be solved using modified Hermite functions. We use Mehler's formula and investigate 1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including possibly zero eigenvalues), and 2) the canonical quadrics in $\mathbb C^4$ of codimension two.