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

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.