Sharp dimension free bound for the Bakry-Riesz vector

Abstract

We prove a sharp dimensionless weighted estimate of the Riesz vector on a Riemannian manifold with non-negative Bakry-Emery curvature. The proof is by the method of Bellman functions, where the explicit expression of a Bellman function of six variables is essential. Notice that our estimate is terms of the Poisson characteristic of the weight includes the case of the Gauss space as well as other spaces that are not necessarily of homogeneous type.