Non-subelliptic estimates for the tangential Cauchy-Riemann system

We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly "q-pseudoconvex" higher codimensional submanifold M of ℂⁿ. Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of (Trans AMS 181:273-292,1973) in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted L ² estimates by Hörmander (Acta Math 113:89-152,1965) and Kohn (Trans AMS 181:273-292,1973) of Sect. 2 and the tangential ∂̄-Neumann operator by Kohn of Sect 4; for this latter we also refer to the book (Adv Math AMS Int Press 19,2001). As for the notion of q pseudoconvexity we follow closely Zampieri (Compositio Math 121:155-162,2000). The main technical result, Theorem 2.1, is a version for "perturbed" q-pseudoconvex domains of a similar result by Ahn (Global boundary regularity of the ∂̄-equation on q-pseudoconvex domains, Preprint, 2003) who generalizes in turn Chen-Shaw (Adv Math AMS Int Press 19, 2001).