A Solution Operator for the ∂ Equation in Sobolev Spaces of Negative Index

2021/11/17 by Ziming Shi, Shi, Ziming, Liding Yao +1
Computer Science · Mathematics · #32A26 (Primary) #32T15 and 46E35 (Secondary) #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2111.09245

openalex publication_date 2021/11/17 · openalex created_date 2021/11/22 · openalex updated_date 2026/07/28

Abstract

Let Ω be a strictly pseudoconvex domain in ℂⁿ with C^k+2 boundary, k ≥ 1. We construct a ∂ solution operator (depending on k) that gains \frac12 derivative in the Sobolev space H^s,p (Ω) for any 1(1)/(p) -k. If the domain is C^∞, then there exists a ∂ solution operator that gains \frac12 derivative in H^s,p(Ω) for all s ∈ ℝ. We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of ``anti-derivative operators'' for distributions defined on bounded Lipschitz domains.

Citations

• Sobolev (1)/(2) estimates for ∂ equations on strictly pseudoconvex domains with C² boundary

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