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,μφ)(M,μφ) having the Bakry–Emery curvature bounded from below. The embedding, acting on the Cartesian product of Lp(M,μφ)Lp(M,μφ) and Lq(T⁎M,μφ)Lq(T⁎M,μφ), 1/p+1/q=11/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 LpLp norms of the corresponding shifted Riesz transform. All our proofs are analytic.