L²-Sobolev theory for the complex Green operator

2016/06/02 by Biard, Séverine, Straube, Emil J. · 1 citation
#32V20 #32W10 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.00728

Abstract

These notes are concerned with the L²-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed CR--submanifolds of ℂⁿ 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 L²-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).

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• Diederich-Fornæss index and global regularity of the complex Green operator: domains with comparable Levi eigenvalues

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