THE FUNDAMENTAL SOLUTION TO □_b ON QUADRIC MANIFOLDS WITH NONZERO EIGENVALUES AND NULL VARIABLES

We prove sharp pointwise bounds on the complex Green operator and its derivatives on a class of embedded quadric manifolds of high codimension. In particular, we start with the class of quadrics that we previously analyzed (Trans. Amer. Math. Soc. Ser. B 10 (2023), 507–541) — ones whose directional Levi forms are nondegenerate, and add in null variables. The null variables do not substantially affect the estimates or analysis at the form levels for which □_b is solvable and hypoelliptic. In the nonhypoelliptic degrees, however, the estimates and analysis are substantially different. In the earlier paper, when hypoellipticity of □_b failed, so did solvability. Here, however, we show that if there is at least one null variable, □_b is always solvable, and the estimates are qualitatively different than in the other cases. Namely, the complex Green operator has blow-ups off of the diagonal. We also characterize when a quadric M whose Levi form vanishes on a complex subspace admits a □_b-invariant change of coordinates so that M presents with a null variable.