We consider Monge-Ampère equations with the right hand side function close to a positive constant and from a function class that is larger than any Hölder class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish explicit bounds regarding the shape of the sequence of shrinking sections, hence revealing the nature of such exponential dependence.

中文翻译：

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右手边接近正常数的Monge-Ampère方程解的内部先验估计

我们考虑Monge-Ampère方程，其右手边函数接近于正常数，且其函数类大于任何Hölder类，并且小于Dini-连续类。我们为解决方案的二阶导数的连续模数确定了一个上限。此范围与Dini半范数相似但大于Dini半范数的数量成指数关系。我们为收缩部分序列的形状建立了明确的界限，从而揭示了这种指数依赖性的本质。