*Accepted Paper*

**Inserted:** 26 mar 2018

**Last Updated:** 1 nov 2019

**Journal:** Analysis and PDE

**Year:** 2018

**Abstract:**

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 together with lower bounds and a precise description of the singular part. The exact representation formula for the Laplacian of a general 1-Lipschitz function 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).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).

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