Fundamental Solutions to $\Box_b$ on Certain Quadrics

TL;DR

This paper computes explicit fundamental solutions for the Kohn Laplacian on certain quadric submanifolds of complex spaces using Lie group structures and Fourier analysis, expanding known examples in several complex variables.

Contribution

It introduces a method to explicitly solve the $oxb$ equation on new classes of quadric submanifolds by leveraging Lie group symmetries and Hermite functions.

Findings

Explicit fundamental solutions for $oxb$ on quadric hypersurfaces with non-uniform Levi eigenvalues.

Solutions for canonical quadrics in $C^4$ of codimension two.

Application of Fourier transform and Hermite functions to complex analysis problems.

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.