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.