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

We prove new sharp Lp, logarithmic, and weak-type inequalities for martingales under the assumption of differential subordination. The Lp estimates are “Feynman–Kac” type versions of Burkholder's celebrated martingale transform inequalities. From the martingale Lp inequalities we obtain that Ri...

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Bibliographic Details
Published in:Journal of functional analysis 2015-09, Vol.269 (6), p.1652-1713
Main Authors: Bañuelos, Rodrigo, Osȩkowski, Adam
Format: Article
Language:English
Subjects:
Lie group
Martingale
Riesz transform
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Summary:We prove new sharp Lp, logarithmic, and weak-type inequalities for martingales under the assumption of differential subordination. The Lp estimates are “Feynman–Kac” type versions of Burkholder's celebrated martingale transform inequalities. From the martingale Lp inequalities we obtain that Riesz transforms on manifolds of nonnegative Bakry–Emery Ricci curvature have exactly the same Lp bounds as those known for Riesz transforms in the flat case of Rd. 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 Lp inequalities.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2015.06.015