The Fundamental Solution to $\Box_b$ on Quadric Manifolds -- Part 1. General Formulas

TL;DR

This paper derives explicit formulas for the complex Green operator and nullspace projection of the Kohn Laplacian on quadric manifolds, with applications to codimension two quadrics and classical examples like the Heisenberg group.

Contribution

It provides a streamlined integral formula for the Green operator and nullspace projection on general quadric submanifolds, advancing understanding of their geometric and analytic properties.

Findings

Explicit formulas for the Green operator and nullspace projection on quadric manifolds.

Application to codimension two quadrics in $\

Examples include calculations on the Heisenberg group and its Cartesian products.

Abstract

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator $N$ and the projection onto the nullspace of $\Box_b$. The main application of our formulas is the critical case of codimension two quadrics in $\mathbb{C}^4$ where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci \cite{PeRi03}. We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.