  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.
