COMPACTNESS ESTIMATE FOR THE ∂-NEUMANN PROBLEM ON A Q-PSEUDOCONVEX DOMAIN IN A STEIN MANIFOLD

Abstract

Abstract. We consider a smoothly bounded q-pseudoconvex domain Ω in an ndimensional Stein manifold X and suppose that the boundary bΩ of Ω satisfies (q − P) property, which is the natural variant of the classical P property. Then, one prove the compactness estimate for the ∂-Neumann operator Nr,s in the Sobolev kspace. Applications to the boundary global regularity for the ∂-Neumann operator Nr,s in the Sobolev k-space are given. Moreover, we prove the boundary global regularity of the ∂-operator on Ω.