$L^2$-Sobolev theory for the complex Green operator

Abstract

These notes are concerned with the $L^2$-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed CR--submanifolds of ${\mathbb{C}}^n$ of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR--geometry of general CR--submanifolds and then specialize to this class. Next, we review the basic $L^2$-theory of the tangential Cauchy-Riemann operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

arXiv commentThis revision incorporates suggestions from the referee's report for International Journal of Mathematics