Compactness estimate for the partial derivative-Neumann problem on a Q-pseudoconvex domain

The purpose of this article is to discuss compactness estimate for the partial derivative-Neumann problem at a boundary with mixed Levi signature. We consider a domain D subset of subset of C-n which is q-pseudoconvex and introduce the '(q - P) property' which is the natural variant of the classical 'P property' by Catlin adapted to the new class of domains. In Section 1, we prove that (q - P) property implies compactness estimate. Next, in Section 2, we introduce the notion of ` weak q regularity' of partial derivative D, the natural variant of the classical ` weak regularity' by Catlin and prove that it implies (q - P) property. In Section 3, we recall how compactness yields Sobolev estimates. In Section 4, we give a criterion for weak q regularity of a real-analytic boundary and finally, in Section 5, we exhibit a class of weakly q regular domains.