De Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the $$\bar \partial - Neumann$$ problem

Abstract

We consider smooth bounded pseudoconvex domains Ω in Cn whose boundary points of infinite type are contained in a smooth submanifoldM (with or without boundary) of the boundary having its (real) tangent space at each point contained in the null space of the Levi form ofbΩ at the point. (In particular, complex submanifolds satisfy this condition.) We consider a certain one-form α onbΩ and show that it represents a De Rham cohomology class on submanifolds of the kind described. We prove that if α represents the trivial cohomology class onM, then the Bergman projection and the $$\bar \partial - Neumann$$ operator on Ω are continuous in Sobolev norms. This happens, in particular, ifM has trivial first De Rham cohomology, for instance, ifM is simply connected.