For a bounded domain $D$ of ${\mathbb{C}}^n$, we introduce a notion of «$q$-pseudoconvexity» of new type and prove that for a given $\overline{\partial }$-closed $\left(p,r\right)$-form $f$ that is smooth up to the boundary on $D$, and for $r\ge q$, there exists a $\left(p,r-1\right)$-form $u$ smooth up to the boundary on $D$ which is a solution of the equation $\overline{\partial }u=f$