The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

中文翻译：

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拉格朗日流在RCD *（K，N）空间上的规律性

本注释的目的是提供Sobolev向量场的规则Lagrangian流在紧凑度量尺度空间上的正则结果，从而验证黎曼曲率维数条件。我们首先借用文献中已经存在的一些思想，证明由带对称对称导数的矢量场产生的流为Lipschitz，从而将标准的Cauchy-Lipschitz定理自然地扩展到了这种情况。然后，我们证明了Sobolev情况下的Lusin型正则结果（在mms为Ahlfors正则的其他假设下），从而扩展了已知的欧几里得结果。