Diederich-Fornæss index and global regularity of the complex Green operator: domains with comparable Levi eigenvalues

Abstract

Let $\Omega \subset {\mathbb{C}}^n$, with $n\ge 3$, 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 $\min \{q_0,n-1-q_0\}\le q\le \max \{q_0,n-1-q_0\}$.

Corrected the formula for the Hessian of $\phi$ in inequality (6.1), taking into account that the vector fields $L_u^K$ are not invariant under a change of frame when $q>1$. This entailed corresponding changes in various places throughout the paper. None of the results are affected