Some interior regularity estimates for solutions of complex Monge-Amp\`ere equations on a ball

In this paper, we consider the Dirichlet problem of a complex Monge-Amp\`ere 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.