Johnson–Schechtman inequalities for noncommutative martingales

Abstract

We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distributional versions of the noncommutative Stein, dual Doob, martingale transform and Burkholder-Gundy inequalities. Our proof relies upon new and powerful extrapolation theorems. As an application, we obtain some new martingale inequalities in symmetric quasi-Banach operator spaces and some interesting endpoint estimates. Our main approach demonstrates a method to build the noncommutative and classical probabilistic inequalities in an entirely operator theoretic way.

Noncommutative Davis type decompositions and applications

“…We are now in a position of stating our ‐moment version of Burkholder/Rosenthal inequality. It should be compared with a recent version of the Burkholder–Gundy inequality from [, Theorem 7.2]. Our result is much more general than the version obtained in .…”

“…We remark that (ii) was also obtained recently by Jiao et al . in [, Theorem 1.5] under the more restrictive assumption that . It is important to note that through the use of interpolation, it is not difficult to deduce that if and then for .…”

“…Subsequently, Φ-moment analogues of other inequalities were also considered (see for instance, [2,12,14]). Recently, the sharpest result for the Φmoment analogue of the noncommutative Burkholder-Gundy inequalities was obtained by Jiao et al (see [22,Theorem 7.2]). Using our general approach, we extend their result to the Φ-moment analogues of the noncommutative Burkholder inequalities.…”

Square Functions for Noncommutative Differentially Subordinate Martingales

“…In this case, Φ(|x|) ∈ L 0 (M, τ ) and its trace τ (Φ(|x|)) is referred to as the Φ-moment of the operator |x|. For more information and background on moment inequalities involving convex functions, we refer to [2,3,10,17,18,40,41]. Below, the following two basic facts will be used.…”

Modular inequalities in martingale Orlicz–Karamata spaces

“…Inequalities of type (1.1) are precisely the so-called modular martingale inequalities. Subsequently, such kind of inequalities was extensively treated in various situations; see for example [1,3,4,8,[18][19][20]27].…”