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

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 $\Omega$.