Complex Alexandrov-Bakelman-Pucci estimate and its applications

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 $2^{2n}\mathrm{\mid }\det (u_{i\bar{j}}){\mathrm{\mid }}^2\ge \mathrm{\mid }\det ({\mathrm{\nabla }}^{2}u)\mathrm{\mid }$) 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.