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.

中文翻译：

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球上复杂Monge-Ampère方程解的一些内部正则估计

在本文中，我们考虑了\（{\ mathbb {C}} ^ n \）中球上复Monge-Ampère方程的Dirichlet问题。使用\（{\ mathcal {C}} ^ {1，\ alpha} \）（分别为\（{\ mathcal {C}} ^ {0，\ alpha} \））数据，我们证明了内部\（{ \ mathcal {C}} ^ {1，\ alpha} \）（分别为\（{\ mathcal {C}} ^ {0，\ alpha} \）估计值。这些估计是Bedford-Taylor内部\（{{数学{C}} ^ {1,1} \）估计的广义版本。