Hölder and \(L^ p\) estimates for \({\bar \partial}_ b\) on weakly pseudo-convex boundaries in \({\mathbb C}^ 2\) (Q1821915)

Let M be the boundary of a weakly pseudoconvex domain of strict finite type in \({\mathbb{C}}^ 2\). We solve the tangential Cauchy-Riemann equation \({\bar \partial}_ b\) on M globally with sharp Hölder and \(L^ p\) estimates for the solutions. The regularity of the solutions reflects the finite type condition, and examples are given to show that these estimates are sharp in the Hölder and \(L^ p\) classes. This result shows that \({\bar \partial}_ b\) has closed range on M globally, even though locally \({\bar \partial}_ b\) may not have closed range in any function space.