Non-commutative martingale inequalities

Abstract

We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an $L^p$-martingale via its integrand, and then extend the Ito-Clifford integral theory in $L^2$, developed by Barnett, Streater and Wilde, to $L^p$ for all $1<p<\infty$. We include an appendix on the non-commutative analogue of the classical Fefferman duality between $H^1$ and $BMO$.

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• Harmonic analysis on quantum tori
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• BMO spaces associated with semigroups of operators
• Calderón-Zygmund operators associated to matrix-valued kernels
• Algebraic Davis decomposition and asymmetric Doob inequalities
• Square functions for noncommutative differentially subordinate martingales
• Noncommutative Khintchine inequalities in interpolation spaces of $L_p$-spaces
• Noncommutative Davis type decompositions and applications
• Average-case Speedup for Product Formulas
• Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori
• Noncommutative martingale concentration inequalities
• Large BMO spaces vs interpolation
• $L^p$-solutions of QSDE driven by Fermion fields with nonlocal conditions under non-Lipschitz coefficients
• Quantum $S{L}_2$, Infinite curvature and Pitman's 2M-X theorem
• K-closedness results in noncommutative Lebesgue spaces with filtrations
• A martingale approach to noncommutative stochastic calculus
• Non-commutative Stein inequality and its applications