Solvability for\(\bar \partial \) with regularity at the boundary of a domain Ω ⊂⊂ ℂ n for forms of any degreek≥1 was characterized by pseudoconvexity of ϖΩ in [16]. It is proved here thatq-pseudoconvexity suffices to guarantee solvability of forms of degreek≥q+1. The method relies on theL 2 estimates in [13], and on their Sobolev version in [15] and [16].