A Weak Type Inequality for Non‐Commutative Martingales and Applications

By Narcisse Randrianantoanina. 2004

47 citations recorded by the source corpus.

We prove a weak‐type (1,1) inequality for square functions of non‐commutative martingales that are simultaneously bounded in L2 and L1. More precisely, the following non‐commutative analogue of a classical result of Burkholder holds: there exists an absolute constant K > 0 such that if M is a semi‐finite von Neumann algebra and (Mn)n=1∞ is an increasing filtration of von Neumann subalgebras of M then for any given martingale x=(xn)n=1∞ that is bounded in L2 (M) ∩ L1 (M), adapted to (Mn)n=1∞ , there exist two martingale difference sequences, a=(an)n=1∞ and b=(bn)n=1∞ , with dxn = an + bn for every n ⩾ 1, ∥(∑n=1∞an∗an)1/2∥2+∥(∑n=1∞bnbn∗)1/2∥2⩽2∥x∥2,

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ab76df5abbf294dbebb1a93739033508e54311aa

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

Citations in the corpus

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.”
Gundy's Decomposition for Non‐Commutative Martingales and Applications (2004, methodology): “We begin by recalling the construction of the martingales y and z from [28].”
A reduction method for noncommutative L_p-spaces and applications (2008, background): “The second inequality in (BGp) remains valid for p = 1, while the first one has a weak type (1, 1) substitute for p = 1 in the tracial case (see [R2]).”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “In the framework of Theorems A1 and A2, the arguments in [27, 29, 30] are no longer valid.”
Conditioned square functions for noncommutative martingales (2005, background): “t;0 tµ t(x)=sup λ>0 (2.1) λτ(χ (λ,∞)(|x|)) is ﬁnite. The following quasi-triangle inequality on elements of L1,∞(M,τ) holds and will be used repeatedly in the sequel. A short proof can be found in [31], Lemma 1.2. Lemma 2.1. For any x 1,x 2 in L1,∞(M,τ) and λ>0, λτ(χ (λ,∞)(|x 1 +x 2|))≤2λτ(χ (”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, methodology): “Note that cp ≤ c p according to [R3], which improves the original estimate cp ≤ c p(2) from [JX1].”
Rosenthal Type inequalities for free chaos. (2005, methodology): “More precisely, according to the Khintchine and Rosenthal inequalities for free random variables, it is natural to ask whether or not the noncommutative Burkholder-Gundy inequality [34] holds in the free setting for p = ∞, see also [17] for the Burkholder-Gundy inequality over non semifinite von Neu”