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 $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) data, we prove an interior $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) estimate for the solution. These estimates are generalized versions of the Bedford-Taylor interior $\mathcal C^{1,1}$ estimate.