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

2022/07/28 by Liu, Bingyuan, Straube, Emil J.
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.14197

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 N_q and the Bergman projections P_q-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.

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