Authors’ abstract: The purpose of this article is to discuss compactness estimates for the \(\overline \partial\)-Neumann problem at a boundary with mixed Levi signature. We consider a domain \(D\subset \subset \mathbb C^n\) which is \(q\)-pseudoconvex and introduce a ‘\((q-P)\) 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 the \((q-P)\) property implies compactness estimates. Next, in Section 2, we introduce the notion of ‘weak \(q\) regularity’ of \(\partial D\), the natural variant of the classical ‘weak regularity’ by Catlin and prove that it implies the \((q-P)\) 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.