Diederich-Fornæss index and global regularity in the ∂-Neumann problem: domains with comparable Levi eigenvalues

Bingyuan Liu, The University of Texas Rio Grande Valley
Emil J. Straube

Abstract

Let Ω be a smooth bounded pseudoconvex domain in Cn. Let 1≤q0≤(n−1). We show that if q0--sums of eigenvalues of the Levi form are comparable, then if the Diederich--Fornæss index of Ω is 1, the ∂¯¯¯--Neumann operators Nq and the Bergman projections Pq−1 are regular in Sobolev norms for q0≤q≤n. In particular, for domains in C2, Diederich--Fornæss index 1 implies global regularity in the ∂¯¯¯--Neumann problem.