On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces

Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates

“…Though they can be applied to more general situation than in the setting of Theorem 2.1, we exclude it from the main theorem for simplicity of presentation since the statements of them look more complicated. We also prove another space-time W 2 -control (2.8) studied in [20] directly from (1.2) (Theorem 2.5). In Section 4, we will prove Theorem 2.6, which concerns with L p -estimates, on a complete Riemannian manifold satisfying (1.7) for a (possibly nonsymmetric) diffusion generator L .…”

“…Since those new conditions are stable under geometric operations such as the measured Gromov-Hausdorff limit, the same stability holds for (1.7) once we prove the equivalence between them. This equivalence is finally established by Ambrosio, Gigli, Savaré, Mondino and Rajala [1,2,3,4] when N = ∞ and by Erbar, Sturm and the author [20] when N < ∞. For connecting Bakry-Èmery theory based on (1.7) with optimal transport approach, the estimate (1.3) or (1.4) works as a bridge, though what we actually used when N < ∞ is (2.8) below.…”

“…Actually, an L p -type estimate of (1.4) (see (2.10)) holds on complete Riemannian manifolds, and we can obtain the L p * -analog of (1.2) by our result. It is not yet known whether we can obtain the same estimate on metric measure spaces satisfying the Riemannian curvature-dimension condition in [20], where the L 2 -estimate holds. Note that, in the case N = ∞, the L p -version of (1.3) follows from (1.3) itself (see [37]).…”

From volume cone to metric cone in the nonsmooth setting

“…Other relevant geometric properties, both on their own and for the purposes of the current paper, are the splitting theorem, proved in [12] by the first author, and the maximal diameter theorem, proved by Ketterer in [19]. Another remarkable result has been established by Erbar-Kuwada-Sturm in [10]: they proved that, in a suitable sense, RCD * (K, N ) spaces can be characterized as those spaces where the Bochner inequality with parameters K, N holds (the case N = ∞ was already known by [16] and [2], see also [3]).…”

“…Finally, on RCD * (0, N ) spaces the Bochner inequality holds ( [10]) in the sense that for f ∈ Test(X) the function |Df | 2 has a measure valued Laplacian and…”