The range of the tangential Cauchy-Riemann operator (Q1086711)

Let \(\Omega\) be a bounded pseudoconvex domain with a smooth boundary. This boundary is a CR manifold. We can also define ''abstract'' CR manifold. The author shows that the range of \({\bar \partial}_ b\) in \(L_ 2\) is closed. This property is very important with respect to existence and regularity results. The main tools are \({\bar \partial}\)- Neumann with weights, a priori estimates, microlocalization. Let's quote a typical result: If \(M\subset {\mathbb{C}}^ n\) is a compact pseudo-convex CR manifold which bounds an analytic variety V in the \({\mathcal C}^{\infty}\) sense, \({\bar \partial}_ b\) has closes range in \(L_ 2(M)\).