2023/07/20 by Jiaxiang Wang, Bin Zhou, Wang, Jiaxiang +1
Mathematics · #32W20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2307.10530
openalex publication_date 2023/07/20 · openalex created_date 2023/07/22 · openalex updated_date 2026/07/28
Let u be a smooth, strictly k-plurisubharmonic function on a bounded domain Ω∈\mathbb Cn with 2≤ k≤ n. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Ampère and Hessian equations when the complex k-Hessian Hk[u] of u is bounded from above and below. We first establish some estimates of Green's functions associated to the linearized equations. Then we prove a class of new Sobolev inequalities. With these inequalities, we use Moser's iteration to investigate the a priori estimates of Hessian equations and their linearized equations, as well as the Kähler scalar curvature equation. In particular, we obtain the Harnack inequality for the linearized complex Monge-Ampère and Hessian equations under an extra integrability condition on the coefficients. The approach works in both real and complex case.