Diederich-Forn\ae ss index and global regularity of the complex Green operator: domains with comparable Levi eigenvalues

TL;DR

This paper establishes a link between the Diederich-Forn ae ss index and the global regularity of the complex Green operator on certain pseudoconvex domains with comparable Levi eigenvalues.

Contribution

It proves that a Diederich-Forn ae ss index of one implies global regularity of the complex Green operator under symmetric eigenvalue conditions.

Findings

Global regularity holds for a range of q values.

Diederich-Forn ae ss index of one is sufficient.

Results apply to domains with comparable Levi eigenvalues.

Abstract

Let $\Omega\subset \mathbb{C}^{n}$, with $n \geq 3$, be a smooth bounded pseudoconvex domain satisfying the symmetric eigenvalue comparability condition $D(q_0)$ for some $1\le q_0\le n-2$. We show that if the Diederich-Fornaess-index of $\Omega$ is one, then the complex Green operator $G_q$, associated with $\Omega$, is globally regular for $q$ in the range $\min\{q_0,\, n - 1 - q_0\} \leq q \leq \max\{q_0,\, n - 1 - q_0\}$.