Lower $N$-weighted Ricci curvature bound with $\epsilon$-range and displacement convexity of entropies

Abstract

In the present article, we provide a characterization of a lower $N$-weighted Ricci curvature bound for $N\in ]-\infty ,1]\cup [n,+\infty ]$ with $\epsilon$-range introduced by Lu-Minguzzi-Ohta in terms of a convexity of entropies over Wasserstein space. We further derive various interpolation inequalities and functional inequalities.

25 pages