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

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