Diederich--Fornæss index and global regularity in the $\overline{\partial}$--Neumann problem: domains with comparable Levi eigenvalues
Bingyuan Liu, Emil J. Straube
TL;DR
This work links the global regularity of the $ar\partial$-Neumann problem to the Diederich–Fornæss index under a Levi eigenvalue comparability condition. By recasting the index in terms of D'Angelo forms $\alpha_\eta$ and proving an integrated $L^2$-estimate that avoids pointwise Hessian bounds, the authors obtain maximal estimates sufficient to deduce Sobolev regularity of $N_q$ and $P_{q-1}$ for all $q$ with $q_0\le q\le n$ when $DF(\Omega)=1$. In particular, for domains in $\mathbb{C}^2$, index one implies global regularity for $P_0$ and $N_1$. The method broadens the scope of index-one results to domains with comparable Levi eigenvalues and does not require the delicate geometric control of infinite-type boundary sets, providing a robust, self-contained approach via $\alpha_\eta$-based estimates and commutator analysis. These results enhance the understanding of how geometric boundary data governs analytical regularity in several complex variables.
Abstract
Let $Ω$ 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 $Ω$ 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.