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

Abstract:

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 $s>0$.