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

Abstract

In this paper, we prove a ${\mathcal{C}}^{2,\alpha }$-estimate for the solution to the complex Monge-Ampère equation $\det (u_{i\bar{j}})=f$ with $0<f\in {\mathcal{C}}^{\alpha }$, under the assumption that $u\in {\mathcal{C}}^{1,\beta }$ for some $\beta <1$ which depends on $n$ and $\alpha$.