Sharp dimension free bound for the Bakry-Riesz vector

TL;DR

This paper establishes a sharp, dimensionless weighted bound for the Bakry-Riesz vector on manifolds with non-negative Bakry-Emery curvature, using Bellman functions, applicable to various spaces including Gaussian space.

Contribution

It introduces a novel Bellman function approach to obtain sharp weighted estimates for the Riesz vector in a broad geometric setting.

Findings

Proves a sharp weighted estimate for the Riesz vector

Extends results to non-homogeneous spaces like Gaussian space

Uses explicit Bellman function of six variables

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.