Modifications of the Levi core

TL;DR

This paper introduces a family of modified Levi cores in complex analysis, linking their properties to Catlin's Property (P) and revealing new insights into the structure of pseudoconvex domains.

Contribution

It constructs modified Levi cores indexed by closed distributions, analyzes their properties, and relates them to boundary regularity conditions like Property (P).

Findings

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

In a72, all modified Levi cores coincide.

For certain Hartogs domains, the modified Levi core is trivial, unlike the classical Levi core.

Abstract

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 $T^{1, 0}b\Omega$ of a smooth bounded pseudoconvex domain $\Omega$. We show that Catlin's Property ($P$) holds on $b\Omega$ 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.