We consider a smoothly bounded $q$-pseudoconvex domain $\Omega$ in an $n$-dimensional Stein manifold $X$ and suppose that the boundary $b\Omega$ of $\Omega$ satisfies $(q-P)$ property, which is the natural variant of the classical $P$ property. Then, one prove the compactness estimate for the $\bar\partial$-Neumann operator $N_{r,s}$ in the Sobolev $k$-space. Applications to the boundary global regularity for the $\bar\partial$-Neumann operator $N_{r,s}$ in the Sobolev $k$-space are given. Moreover, we prove the boundary global regularity of the $\overline{\partial}$-operator on $\Omega$.