Some interior regularity estimates for solutions of complex Monge-Ampère equations on a ball

Abstract

In this paper, we consider the Dirichlet problem of a complex Monge-Ampère equation on a ball in ${\mathbb{C}}^n$. With ${\mathcal{C}}^{1,\alpha }$ (resp. ${\mathcal{C}}^{0,\alpha }$) data, we prove an interior ${\mathcal{C}}^{1,\alpha }$ (resp. ${\mathcal{C}}^{0,\alpha }$) estimate for the solution. These estimates are generalized versions of the Bedford-Taylor interior ${\mathcal{C}}^{1,1}$ estimate.