Complex Alexandrov–Bakelman–Pucci Estimate and Its Applications
Abstract
Abstract We prove an Alexandrov–Bakelman–Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality $$2^{2n}|\det (u_{i\bar{j}})|^2\ge |\det (\nabla ^2u)|$$ 2 2 n | det ( u i j ¯ ) | 2 ≥ | det ( ∇ 2 u ) | ) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge–Ampère equations. The approach is based on the De Giorgi iteration method developed by Guo–Phong–Tong for equations of complex Monge–Ampère type.
Fields of Study & Subjects
Advanced Numerical Analysis Techniques
Advanced Optimization Algorithms Research
Publication details
- Journal / source
- Journal of Geometric Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 36
- Issue
- 6
- Publication date
- 5 May 2026
- Language
- EN