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In this paper, we consider the Dirichlet problem of a complex Monge-Amp\`ereequation 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 aregeneralized versions of the Bedford-Taylor interior $\mathcal C^{1,1}$estimate.

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