The ${\mathcal{C}}^{2,\alpha }$ estimate of complex Monge-Ampere equation

Slawomir Dinew, Xi Zhang, Xiangwen Zhang

We prove that any ${\mathcal{C}}^{1,1}$ solution to complex Monge-Ampère equation $det(u_{i\bar{j}})=f$ with $0<f\in {\mathcal{C}}^{\alpha }$ is in ${\mathcal{C}}^{2,\alpha }$ for $\alpha \in (0,1)$.