The fundamental solution to \Box_{????} 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 Kohn Laplacian its inverse on general quadric submanifolds C n × m \mathbb {C}^n\times \mathbb {C}^m . In this paper, present streamlined calculation for integral formula complex Green operator alttext="upper N"> N encoding="application/x-tex">N projection onto nullspace alttext="white medium square Subscript b Baseline"> ? b encoding="application/x-tex">\Box _b The main application our formulas critical case codimension two quadrics 4"> 4 {C}^4 where discuss known solvability hypoellipticity criteria Peloso Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that yield explicit calculations some well-known cases: Heisenberg group Cartesian product groups.