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

TL;DR

This paper develops an $L^{2}$-Sobolev framework for the complex Green operator on pseudoconvex CR submanifolds, extending boundary domain theories and analyzing compactness and regularity properties.

Contribution

It introduces an $L^{2}$-Sobolev theory for the complex Green operator on a broad class of CR submanifolds, generalizing boundary cases and exploring compactness and regularity results.

Findings

Established Sobolev regularity of the complex Green operator.

Proved compactness results for the Green operator.

Analyzed regularity properties in Sobolev spaces.

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).