  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.
