The Fundamental Solution to $\Box_b$ on Quadric Manifolds -- Part 4. Nonzero Eigenvalues

TL;DR

This paper analyzes the complex Green operator on quadric manifolds with nonzero eigenvalues, establishing bounds and mapping properties, and exploring geometric conditions and examples in the context of the Kohn Laplacian.

Contribution

It provides new bounds and mapping properties for the complex Green operator on quadric manifolds with nonvanishing eigenvalues, expanding understanding of the Kohn Laplacian's inverse.

Findings

Optimal pointwise upper bounds on the Green operator and derivatives

Characterization of geometric conditions imposed by eigenvalues

Examples illustrating the theoretical results

Abstract

This paper is the fourth of a multi-part series in which we study the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We 1) investigate the geometric conditions on $M$ which the eigenvalue condition forces, 2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, 3) explore the $L^p$ and $L^p$-Sobolev mapping properties of the associated kernels, and 4) provide examples.