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

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.