A ${\mathcal{C}}^{2,\alpha }$ estimate of the complex Monge-Amp\`ere equation

In this paper, we prove a ${\mathcal{C}}^{2,\alpha }$-estimate for the solution to the complex Monge-Amp`ere equation $\det (u_{i\bar{j}})=f$ with $0< f\in \mathcal C^{\alpha}$,undertheassumptionthat$u\in \mathcal C^{1,\beta }$forsome$\beta <1$whichdependson$n$and$\alpha $.