Curvature and other local inequalities in Markov semigroups

We investigate the relationship between curvature bounds and local functional inequalities. In particular, we show that local inequalities derived through the maximum principle approach of Ivanisvili and Volberg (In 50 Years with Hardy Spaces (2018) 281–305, Birkhäuser/Springer) are equivalent to lower bounds on the Ricci curvature in the sense of Bakry–Emery. Moreover, we will develop this Euclidean space technique to metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in nonsmooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semigroup operators on Rn in the absence of positive curvature, based on a local eigenvalue criteria.