Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space

Abstract

We prove new sharp L p , logarithmic, and weak-type inequalities for martingales under the assumption of differentially subordination. The L p estimates are "Fyenman-Kac" type versions of Burkholder's celebrated martingale transform inequalities. From the martingale L p inequalities we obtain that Riesz transforms on manifolds of nonnegative Bakry-Emery Ricci curvature have exactly the same L p bounds as those known for Riesz transforms in the flat case of R n . From the martingale logarithmic and weak-type inequalities we obtain similar inequalities for Riesz transforms on compact Lie groups and spheres. Combining the estimates for spheres with Poincaré's limiting argument, we deduce the corresponding results for Riesz transforms associated with the Ornstein-Uhlenbeck semigroup, thus providing some extensions of P.A. Meyer's L p inequalities.

Erratum to: Martingale transforms and $$L^p$$ L p -norm estimates of Riesz transforms on complete Riemannian manifolds

“…Similarly, using the correct probabilistic representation formulas of the Riesz transforms and Theorem 2.2 in [2], we can re-derive the original estimates of Theorem 1.4 obtained in the original paper for the case Ric + ∇ 2 φ ≥ a > 0. To save the length of this note, we omit the detail of the proof.…”

“…Therefore the original estimates in Theorem 1.4 and Corollary 1.5 obtained in the original paper remain valid. See also Theorem 1.1 in [2]. More precisely, we have…”

“…Recently, Bañuelos and Osekowski [2] proved a new martingale inequality which improves Theorem 2.6 in [1] and derived that if Ric + ∇ 2 φ ≥ 0, then for all p ∈ (1, ∞), the original estimate stated in the original paper remains valid, i.e., ∇(−L) −1/2 p, p ≤ 2( p * − 1). In particular, ∇(− ) −1/2 p, p ≤ 2( p * − 1) holds on complete Riemannian manifolds with Ric ≥ 0.…”

“…We would like to point out that, based on the correct representation formulas of the Riesz transforms stated below, and using a new martingale subordination inequality due to Bañuelos and Osekowski (Theorem 2.2 in [2]), we can correct the gap in the original paper without major change in the line of our original argument and to arrive at the original estimates obtained in the original paper.…”

“…To see this, let us first recall Theorem 2.2 in Bañuelos and Osekowski [2]: Let X and Y be R n -valued martingales with continuous paths such that Y is differentially subordinate to X . Consider the solution of the matrix equation…”