Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

We will study metric measure spaces $(X,d,m)$ beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE$_1(\kappa,\infty)$ $\bullet$ for which we prove the equivalence with sharp gradient estimates, $\bullet$ the class of which will be preserved under time changes with arbitrary $\psi\in{\mathrm Lip}_b(X)$, and $\bullet$ which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $Y\subset X$. In the latter case, the distribution-valued Ricci bound will be given by the signed measure $\kappa= k\, m_Y + \ell\,\sigma_{\partial Y}$ where $k$ denotes a variable synthetic lower bound for the Ricci curvature of $X$ and $\ell$ denotes a lower bound for the "curvature of the boundary" of $Y$, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.