The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N).

中文翻译：

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距离函数和应用的拉普拉斯算子的新公式

该论文的目标是在综合意义上满足 Ricci 曲率下界的度量空间框架中证明距离的拉普拉斯算子（更一般地用于任意 1-Lipschitz 函数）的精确表示公式（更准确地说是在本质上是非分支的 MCP(K,N)-空间）。这样的表示公式使经典的上界和一些新的下界以及对奇异部分的精确描述变得明显。1-Lipschitz 函数（特别是距离函数）的拉普拉斯算子的精确表示公式在一般完全黎曼流形中也成立（并且似乎是新的）。我们应用这些结果来证明 CD(K,N) 和带符号距离函数上的维 Bochner 不等式的等价性。