For a domain Ω of ${\mathbb C}^N$ we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the $\bar \partial$-complex for forms with $C^\infty (\bar \Omega)$-coefficients in degree $\geq q+1$. All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the $L^2$-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-$L^2$ and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.
