Bochner's inequality is one of the most fundamental estimates in geometric analysis on Riemannian manifolds.
It states that
\[
\frac12\Delta|\nabla u|^2-\langle\nabla u, \nabla\Delta u\rangle\ge K \cdot |\nabla u|^2+\frac1N \cdot |\Delta u|^2
\]
for each smooth function $u$ on a Riemannian manifold provided $K$ is a lower bound for the Ricci curvature on and $N$ is an upper bound for the dimension. The main result I present in this talk is a Bochner inequality on metric measure spaces with linear heat flow and satisfying the (reduced) curvature-dimension condition. Indeed, also the converse is true: if the heat flow on a metric measure space is linear then an appropriate version of the Bochner inequality (for the canonical gradient and Laplacian) will imply the reduced curvature-dimension condition. Besides that, I will present also new, sharp Wasserstein-contraction results for the heat flow as well as Bakry-Ledoux type gradient estimates each of which is equivalent to the curvature-dimension condition.http://cvgmt.sns.it/seminar/346/
