This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure μ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration that transfers the information through the scales. © 2021 Wiley Periodicals LLC.

中文翻译：

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Monge-Ampère 方程的定量线性化结果

本文是关于具有粗略数据的 Monge-Ampère 方程的定量线性化结果。我们开发了一个大规模的正则性理论，并证明如果一个测度μ在所有尺度上都接近 Wasserstein 距离的 Lebesgue 测度，那么宏观最优耦合的位移在所有尺度上在定量上都接近于解的梯度对应的泊松方程。我们使用的主要成分是任意措施之间最优运输计划的谐波近似结果。这用于通过尺度传输信息的 Campanato 迭代。© 2021 威利期刊有限责任公司。