Abstract: In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$, where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5].

Key words: Cauchy-Riemann equations, pseudoconcave domains, $\bar{\partial}$-Neumann operator, Bergman spaces