Distribution-valued Ricci bounds for metric measure spaces, singular time changes, and gradient estimates for Neumann heat flows (Q2216472)

In [Acta Math. 196, No. 1, 133--177 (2006; Zbl 1106.53032)] the author introduced a condition \(CD(K,n)\) on metric measure spaces, which for Riemannian manifolds is satisfied if and only if the dimension is at most \(n\) and the Ricci curvature is at least \(K\) times norm square. This led to the development of a sophisticated theory of metric measure spaces with lower bounded Ricci curvature. The purpose of the paper under review is to extend the scope to metric measure spaces with non-uniform lower Ricci bounds. While the condition \(CD(K,n)\) is equivalent to a Bakry-Émery-condition \(BE_2(K,n)\), the author introduces a notion of distribution-valued Ricci bounds and a version \(BE_1(K,\infty)\) of the Baker-Émery condition which allows him to prove equivalence with sharp estimates for gradient flows of locally semiconvex functions, which is preserved under Lipschitzian ``time change'' (a certain transformation of metric measure spaces corresponding to time change of Brownian motion), and which is hereditary for subspaces \(Y\subset X\) in the sense that a variable synthetic lower Ricci bound on \(X\) and a variable lower bound for the curvature of \(\partial Y\) imply a distribution-valued lower Ricci bound for \(Y\). The latter result builds on a gradient estimate for the Neumann heat flow on not necessarily convex subsets \(Y\subset X\) with a quite involved proof.