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

TL;DR

This paper establishes that for certain smooth pseudoconvex domains with comparable Levi eigenvalues, a Diederich--Forn ext{ae}ss index of 1 guarantees regularity of the $ar{ ext{d}}$-Neumann operators and Bergman projections in Sobolev spaces, especially in $ ext{C}^2$.

Contribution

It proves that domains with comparable Levi eigenvalues and Diederich--Forn ext{ae}ss index 1 exhibit Sobolev regularity for $ar{ ext{d}}$-Neumann operators, extending known results.

Findings

Regularity of $ar{ ext{d}}$-Neumann operators in Sobolev norms for domains with comparable Levi eigenvalues.

Diederich--Forn ext{ae}ss index 1 implies global regularity in $ ext{C}^2$ domains.

Conditions on Levi eigenvalues are key to regularity results.

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--Forn\ae ss 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--Forn\ae ss index $1$ implies global regularity in the $\overline{\partial}$--Neumann problem.