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

Abstract

By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,{\mu }_{\varphi })$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,{\mu }_{\varphi })$ and $L^q(T^{\ast }M,{\mu }_{\varphi })$, $1/p+1/q=1$, 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.

16 pages, to appear in Journal of Functional Analysis; identical to v2 except that the first-named author's address was updated