DERHAM COHOMOLOGY OF MANIFOLDS CONTAINING THE POINTS OF INFINITE TYPE, AND SOBOLEV ESTIMATES FOR THE PARTIAL-DERIVATIVE-NEUMANN PROBLEM

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