We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K,N) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.

中文翻译：

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通过拉格朗日流的正则性，RCD(K ,N ) 空间的维数不变

我们证明了 RCD(K,N) 度量空间上 Sobolev 向量场的拉格朗日流的正则性结果，根据从拉普拉斯算子的格林函数构建的新定义的准度量来理解正则性。它的主要应用是 RCD(K,N) 空间具有恒定的维数。通过这种方式，我们将 Colding-Naber 证明的 Ricci 极限空间证明的结果推广到这种抽象框架，引入了即使在平滑设置中也是新的成分。