Compactness of the $\overline{\partial }$-Neumann operator on the intersection of two domains

Abstract

Assume that ${\Omega }_1$ and ${\Omega }_2$ are two smooth bounded pseudoconvex domains in ${\mathbb{C}}^2$ that intersect (real) transversely, and that ${\Omega }_1\cap {\Omega }_2$ is a domain (i.e. is connected). If the $\overline{\partial }$-Neumann operators on ${\Omega }_1$ and on ${\Omega }_2$ are compact, then so is the $\overline{\partial }$-Neumann operator on ${\Omega }_1\cap {\Omega }_2$. The corresponding result holds for the $\overline{\partial }$-Neumann operators on $(0,n-1)$-forms on domains in ${\mathbb{C}}^n$.