Characterizations of the Upper Bound of Bakry-Emery Curvature

In this paper, we will present characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on the path space. Moreover, characterizations for general lower and upper bounds of Ricci curvature are also given, which extends the recent results derived by Naber (Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces, arXiv:1306.6512v4) and Wang-Wu (Sci China Math 61:1407-1420, 2018). A crucial point of the present study is to use a symmetrization argument for the lower and upper bounds of Ricci curvature, and a localization technique.