Sufficient condition for compactness of the $\overline{\partial}$-Neumann operator using the Levi core

TL;DR

This paper establishes a new criterion for the compactness of the $ar{ abla}$-Neumann operator on pseudoconvex domains, linking it to the Levi core support and Property (P) on boundary points.

Contribution

It shows that Property (P) on the Levi core support is equivalent to Property (P) on the entire boundary, leading to a new sufficient condition for the compactness of the $ar{ abla}$-Neumann operator.

Findings

Property (P) holds on the Levi core support if and only if it holds on the boundary.

Compactness of the $ar{ abla}$-Neumann operator follows from Property (P) on the Levi core support.

The Levi core support is a key subset of infinite type points determining boundary regularity.

Abstract

On a smooth, bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^n$, to verify that Catlin's Property ($P$) holds for $b\Omega$, it suffices to check that it holds on the set of D'Angelo infinite type boundary points. In this note, we consider the support of the Levi core, $S_{\mathfrak{C}(\mathcal{N})}$, a subset of the infinite type points, and show that Property ($P$) holds for $b\Omega$ if and only if it holds for $S_{\mathfrak{C}(\mathcal{N})}$. Consequently, if Property ($P$) holds on $S_{\mathfrak{C}(\mathcal{N})}$, then the $\overline{\partial}$-Neumann operator $N_1$ is compact on $\Omega$.