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
is
solvable and hypoelliptic. In the nonhypoelliptic degrees, however, the estimates and
analysis are substantially different. In the earlier paper, when hypoellipticity of
failed,
so did solvability. Here, however, we show that if there is at least one null variable,
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
whose Levi form vanishes on a complex subspace admits a
-invariant change of
coordinates so that
presents with a null variable.
Keywords
quadric submanifolds, higher codimension, nonzero
eigenvalues, complex Green operator, null variables, zero
eigenvalues
