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 $\C^n$. 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 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 H\"ormander and Kohn and the tangential $\bar\partial$-Neumann operator by Kohn. As for the notion of $q$ pseudoconvexity we follow closely Zampieri. The main technical result is a version for "perturbed" $q$-pseudoconvex domains of a similar result by Ahn who generalizes in turn Chen-Shaw.