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

TL;DR

This paper proves that the compactness of the $ar{ ext{d}}$-Neumann operator on two intersecting smooth bounded pseudoconvex domains in complex space implies the same property on their intersection, extending to higher dimensions.

Contribution

It establishes a new result linking the compactness of the $ar{ ext{d}}$-Neumann operator on individual domains to their intersection in complex analysis.

Findings

Compactness is preserved under intersection of domains.

Results extend to $(0,n-1)$-forms in $ ext{C}^n$.

Applicable to smooth bounded pseudoconvex 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 $\mathbb{C}^{n}$.