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

Abstract

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