A remark on irregularity of the $\overline{\partial}$-Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^n$ there exists $s>0$ such that the $\overline{\partial}$-Neumann operator on $\Omega$ maps $W^s_{(0,1)}(\Omega)$ (the space of $(0,1)$-forms with coefficient functions in $L^2$-Sobolev space of order $s$) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^{2}$, smooth except at one point, whose $\overline{\partial}$-Neumann operator is not bounded on $W^s_{(0,1)}(\Omega)$ for any $s>0$.