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 $(M,\mu_\phi)$ 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. 
Andrea Carbonaro and Oliver Dragi\v{c}evi\'c
math.FA math.DG