Non-Commutative Martingale Transforms

By Narcisse Randrianantoanina. 2001

103 citations recorded by the source corpus.

Abstract We prove that non-commutative martingale transforms are of weak type (1,1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and ( M n ) n =1 ∞ is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x =( x n ) n =1 ∞ in L 1 ( M ), adapted to ( M n ) n =1 ∞ , and any sequence of signs (e n ) n =1 ∞ , ∥e 1 x 1 + ∑ n=2 N e n (x n −x n−1 )∥ 1,∞ ⩽C∥x N ∥ 1 for every N ⩾2. This generalizes a result of Burkholder from classical martingale theory to non-commutative setting and answers positively a question of Pisier and Xu. As applications, we get the optimal order of the unconditional Martingale differences (UMD)-constants of the Schatten class S p when p→∞. Similarly, we prove that the UMD-constant of the finite-dimensional Schatten class S n 1 is of order log( n +1). We also discuss the Pisier–Xu non-commutative Burkholder–Gundy inequalities.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b61f15ba0bd60b32d95ac0364482fb286f66636

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Citations in the corpus

$H^{\infty }$ functional calculus and square functions on noncommutative $L^p$-spaces (2006, background): “See in particular [66, 33, 70, 37], and also the recent survey [81] and the references therein.”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, background): “We mention only [JX2, PaR, R2 , R3, R4], where, among many other results, the optimal orders of the best constants in most noncommutative martingale inequalities are determined.”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “In view of [29], we must impose certain commuting condition of the coefficients ξkm, we refer to [24, Section 6] for more details on this topic.”
Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory (2007, methodology): “A noncommutative analogue of Doob’s maximal function [24], the noncommutative John-Nirenberg theorem [26], extensions of Burkholder inequalities for conditioned square functions [30] and related weak type inequalities [50, 51, 52]; see [44] for a simpler approach to some of them.”
On the Best Constants in Some Non‐Commutative Martingale Inequalities (2005, background): “Randrianantoanina’s proof for (6) depends heavily on a non-commutative version of the classical Doob weak type (1, 1) maximal inequality, obtained by Cuculescu [4].”
Gundy's Decomposition for Non‐Commutative Martingales and Applications (2004, methodology): “We illustrate this by reproving the main results in [27] and [28].”
Noncommutative maximal ergodic theorems (2005, background): “These include Burkholder inequalities on conditioned square function [JX2], Burkholder-Gundy inequalities on square function [PX1], Doob maximal inequality [Ju], Rosenthal inequalities on independent random variables [JX4] and boundedness of martingale transforms [ Ra ].”
A Weak Type Inequality for Non‐Commutative Martingales and Applications (2004, methodology): “In order to achieve the decomposition into two martingales, our method of proof (although it follows closely those used in [43] and [41]) requires substantial adjustments.”
Rosenthal inequalities in noncommutative symmetric spaces (2011, methodology): “As a result we obtain a new proof of Rosenthal’s theorem for (Haagerup) Lp-spaces.”
A reduction method for noncommutative L_p-spaces and applications (2008, methodology): “We also emphasize that Randianantoanina’s theorem provides a key to the problem of finding the optimal orders of the best constants in various noncommutative martingale inequalities.”