On the Best Constants in Some Non‐Commutative Martingale Inequalities

By Marius Junge, Quanhua Xu. 2005

70 citations recorded by the source corpus.

The optimal orders are determined for the best constants in the non‐commutative Burkholder–Gundy, Doob and Stein inequalities obtained recently in non‐commutative martingale theory. 2000 Mathematics Subject Classification 46L53, 46L51.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9cde64cea055354894d6f76e35eae7a1b1cad115

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Citations in the corpus

Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, background): “This choice is useful to obtain certain counterexamples, see [16].”
Conditioned square functions for noncommutative martingales (2005, methodology): “p close to p 0. A fundamentalresultinvolving Hardyspacesis thenoncommutative Burkholder– Gundy inequalities which we now state for further use. Theorem2.3(Noncommutative Burkholder–Gundy inequalities [19, 27, 31]). Let 1 < p < ∞ and let x = (xn)∞ n=1 be an L p-martingale. Then x is bounded in ”
User-Friendly Tail Bounds for Sums of Random Matrices (2010, background): “The article [26] contains a reasonably accessible and comprehensive discussion.”
Noncommutative Bennett and Rosenthal inequalities (2011, methodology): “rsion) of (0.7) with several coauthors independently by using a diﬀerent method in a later paper [31]. In fact, Rosenthal inequalities in the noncommutative setting have been successively explored in [23, 24] and [25]. The martingale situation is completely settled due to the work of [40] which show”
A Weak Type Inequality for Non‐Commutative Martingales and Applications (2004, methodology): “This solves a question left open in [26] (see the remark after the main theorem of [26]).”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, methodology): “3) is now obtained by using the lower estimate in the noncommutative Burkholder-Gundy inequality (see [JX2] for the optimal order of the constant): max{‖(dx)‖Lp(M;lc2) , ‖(dx)‖Lp(M;lr2)} ≤ c p ‖x‖p .”