q-Pseudoconvexity and regularity at the boundary for solutions of the (partial derivative)over-bar-problem

For a domain Omega of bb C-N we introduce a fairly general and intrinsic co ndition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the <(partial derivative)over bar>-complex for forms with C-infinity (<(Ome ga)over bar>)-coefficients in degree greater than or equal to q+1. All domains whose boundary have a constant number of negative Levi eigenval ues are easily recognized to fulfill our condition of q-pseudoconvexity; th us we regain the result of Michel (with a simplified proof). Our method deeply relies on the L-2-estimates by Hormander (with some varia nts). 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 exploi t the technique by Dufresnoy for regularity up to the boundary.