Modifications of the Levi core

We construct a family of subdistributions of the Levi core $\mathfrak{C}(\mathcal{N})$ called modified Levi cores $\{\mathcal{M}{\mathfrak{C}}_{\mathcal{A}}{\}}_{\mathcal{A}}$ indexed over closed distributions $\mathcal{A}$ that contain the Levi null distribution $\mathcal{N}$ and are contained in the complex tangent bundle $T^{1,0}b\mathrm{\Omega }$ of a smooth bounded pseudoconvex domain $\mathrm{\Omega }$. We show that Catlin’s Property ($P$) holds on $b\mathrm{\Omega }$ if and only if Property ($P$) holds on the support of one, and hence all, of the modified Levi cores. In ${\mathbb{C}}^2$, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in ${\mathbb{C}}^2$ that satisfies Property ($P$), we show that its modified Levi core is trivial. This contrasts with $\mathfrak{C}(\mathcal{N})$, which can be nontrivial for such domains.