  Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of
$\mathbb{C}^{n}$ of hypersurface type. Our main result says that when a certain
$1$-form on $M$ is exact on the null space of the Levi form, then the complex
Green operator on $M$ satisfies Sobolev estimates. This happens in particular
when $M$ admits a set of plurisubharmonic defining functions or when $M$ is
strictly pseudoconvex except for the points on a simply connected complex
submanifold.

Complex Variables