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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      C n ×  C m \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 N       and the projection onto the nullspace of      ◻ b \Box _b      . The main application of our formulas is the critical case of codimension two quadrics in      C 4 \mathbb {C}^4       where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] 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.

The fundamental solution to \Box_{𝑏} on quadric manifolds – Part 1. General formulas