Asymmetric Doob's Maximal Inequalities and Algebraic Atomic Decomposition for Noncommutative Martingales

Guixiang Hong   (Wuhan University)

16:00-17:00, Oct 31, 2017   Science Building A1510


Motivated by the research of pointwise principle value of noncommutative singular integrals, we formulate several asymmetric Doob's maximal inequalities for noncommuative martingales with discrete filtration, which incidentally solves two open problems. One of them dates back 1997, when Pisier and Xu established noncommutative Burkholder-Gundy inequalities and Fefferman-Stein duality in their seminal paper. The main tool used in the proof-algebraic atomic decomposition of Hardy space, is new even in the classical martingale theory. For further application to noncommutative harmonic analysis and ergodic theory, we also show similar asymmetric Doob's maximal inequalities for martingales with continuous filtrations. However, there appear new difficulities. The idea we come up with may be helpful to understand noncommutative stochastic calculus. This talk is based on joint works with Marius Junge and Javier Parcet.

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