Sobolev estimates for the complex Green operator on Levi-flat manifolds (Symmetry and Singularity of Geometric Structures and Differential Equations)

Abstract

This note is an announcement of the author's recent work on analysis of the complex Green operator for CR line bundles over compact Levi-flat manifolds. We explain a special but interesting example, the unit tangent bundle of hyperbolic Riemann surface with leafwise pluricanonical bundle, where Sobolev estimates for complex Green operator hold up to certain order, but fail to the order above.