Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds

Let (M, g) be a complete Riemannian manifold, L = ∆ −∇φ · ∇ be a Markovian symmetric diffusion operator with an invariant measure dμ(x) = e−φ(x)dν(x), where φ ∈ C2(M), ν is the Riemannian volume measure on (M, g). A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform Ra(L) = ∇(a − L)−1/2 is bounded in Lp(μ) for all 1 < p < ∞ and for certain a ≥ 0. An affirmative answer to this problem has many important applications in elliptic or parabolic PDEs, potential theory, probability theory, the Lp-Hodge decomposition theory and in the study of Navier-Stokes equations and boundary value problems. Using some new interplays between harmonic analysis, differential geometry and probability theory, we prove that the Riesz transform Ra(L) = ∇(a − L)−1/2 is bounded in Lp(μ) for all a > 0 and p ≥ 2 provided that L generates a ultracontractive Markovian semigroup Pt = etL in the sense that Pt1 = 1 for all t ≥ 0, ‖Pt‖1,∞ < Ct−n/2 for all t ∈ (0, 1] for some constants C > 0 and n > 1, and satisfies (K + c)− ∈ Ln2 + (M, μ) for some constants c ≥ 0 and > 0, where K(x) denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature Ric(L) = Ric + ∇2φ on TxM , i.e., K(x) = inf{Ric(L)(v, v) : v ∈ TxM, ‖v‖ = 1}, ∀ x ∈ M. 2000 Mathematics Subject Classification: 31C12, 53C20, 58J65, 60H30.