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

2011/05/31 by Carbonaro, Andrea, Dragičević, Oliver
#Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1105.6330

Abstract

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.

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