A remark on irregularity of the dbar-Neumann problem on non-smooth domains

Abstract

It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain D in $C^n$ there exists s>0 such that the dbar-Neumann operator on D maps $W^s_{(0,1)}(D)$ (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 D in $C^2$, smooth except at one point, whose dbar-Neumann operator is not bounded on $W^s_{(0,1)}(D)$ for any s>0.