Diederich--Fornæss index and global regularity in the $\overline{\partial }$--Neumann problem: domains with comparable Levi eigenvalues

Abstract

Let $\Omega$ be a smooth bounded pseudoconvex domain in ${\mathbb{C}}^n$. Let $1\le q_0\le (n-1)$. We show that if $q_0$--sums of eigenvalues of the Levi form are comparable, then if the Diederich--Fornæss index of $\Omega$ is $1$, the $\overline{\partial }$--Neumann operators $N_q$ and the Bergman projections $P_{q-1}$ are regular in Sobolev norms for $q_0\le q\le n$. In particular, for domains in ${\mathbb{C}}^2$, Diederich--Fornæss index $1$ implies global regularity in the $\overline{\partial }$--Neumann problem.

17 pages