Compactness of the $$\bar{\partial }$$-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 \(\bar{\partial }\)-Neumann operators on \(\Omega _{1}\) and on \(\Omega _{2}\) are compact, then so is the \(\bar{\partial }\)-Neumann operator on \(\Omega _{1} \cap \, \Omega _{2}\). The corresponding result holds for the \(\bar{\partial }\)-Neumann operators on \((0,n-1)\)-forms on domains in \(\mathbb {C}^{n}\).