A Remark On C^{2,\alpha}-Regularity of the Complex Monge-Amp\`ere Equation

Abstract

We prove the $C^{2,\alpha}$-regularity of the solution $u$ of the equation [\det(u_{\bar{k} j}) = f, \quad f^{1/n} \in C^{\alpha}, \quad f \geq \lambda] under the assumption in upper bound of $\Delta u$. Our result settles down the regularity problem related to the paper \cite{Tian} (also see \cite{Zhang}).