Bellman function and linear dimension-free estimates in a theorem of Bakry

TL;DR

This paper employs an explicit Bellman function to establish a bilinear embedding theorem for the Laplacian on weighted Riemannian manifolds with curvature bounds, leading to dimension-free estimates of Riesz transforms.

Contribution

It introduces a new analytic approach using Bellman functions to obtain dimension-free $L^p$ estimates for Riesz transforms on weighted manifolds with curvature bounds.

Findings

Dimension-free $L^p$ estimates for Riesz transforms

Bilinear embedding theorem for weighted Laplacian

Analytic proofs using Bellman functions

Abstract

By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,\mu_\phi)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,\mu_\phi)$ and $L^q(T^*M,\mu_\phi)$, $1/p+1/q=1$, involves estimates which are independent of the dimension of the manifold and linear in $p$. As a consequence we obtain linear dimension-free estimates of the $L^p$ norms of the corresponding shifted Riesz transform. All our proofs are analytic.