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

Abstract

On a smooth, bounded pseudoconvex domain Ω \Omega in C n \mathbb {C}^n , to verify that Catlin’s Property ( P P ) holds for b Ω 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 C ( N ) S_{\mathfrak {C}(\mathcal {N})} , a subset of the infinite type points, and show that Property ( P P ) holds for b Ω b\Omega if and only if it holds for S C ( N ) S_{\mathfrak {C}(\mathcal {N})} . Consequently, if Property ( P P ) holds on S C ( N ) S_{\mathfrak {C}(\mathcal {N})} , then the ∂ ¯ \overline {\partial } -Neumann operator N 1 N_1 is compact on Ω \Omega .