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 $1\leq q_{0}\leq (n-1)$. 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 $q_{0}\leq q\leq n$. In particular, for domains in $\mathbb {C}^{2}$, Diederich–Fornæss index $1$ implies global regularity in the $\overline {\partial }$–Neumann problem.