Interior Hölder estimate for the linearized complex Monge-Ampere equation

2023/01/04 by Yulun Xu, Xu, Yulun
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2301.01793

openalex publication_date 2023/01/04 · openalex created_date 2023/01/08 · openalex updated_date 2026/07/28

Abstract

Let w₀ be a bounded, C³, strictly plurisubharmonic function defined on B₁⊂ ℂⁿ. Then w₀ has a neighborhood in L^∞(B₁). Suppose that we have a function ϕ in this neighborhood with 1-ε≤ MA(u)≤ 1+ε and there exists a function u solving the linearized complex Monge-Ampere equation: det(ϕ_kl)ϕ^Iju_Ij=0. Then one has an estimate on |u|_C^α(B_(1)/(2)) for some α>0 depending on n, as long as ε is small depending on n. This partially generalizes Caffarelli's estimate for linearized real Monge-Ampere equation to the complex version.

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