Johnson-Schechtman inequalities for noncommutative martingales

TL;DR

This paper establishes Johnson-Schechtman inequalities for noncommutative martingales in symmetric operator spaces, revealing one-sided inequalities except in the $L_2$ case, and extends classical inequalities with new sharp $ ext{Phi}$-moment results.

Contribution

It proves Johnson-Schechtman inequalities for noncommutative martingales without Boyd index restrictions and extends noncommutative Burkholder-Gundy inequalities to symmetric and $ ext{Phi}$-moment spaces.

Findings

Inequalities are one-sided except for $E=L_2(0,1)$.

Partial resolution of a problem by Randrianantoanina and Wu.

Sharp $ ext{Phi}$-moment inequalities for specific Orlicz functions.

Abstract

In this paper we study Johnson-Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space $E(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ without making any assumption on the Boyd indices of $E$. We show that we can obtain Johnson-Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when $E=L_2(0,1)$. As an application, we partly resolve a problem stated by Randrianantoanina and Wu in \cite{NW}. We also show that we can obtain sharp $\Phi$-moment analogues for Orlicz functions satisfying $p$-convexity and $q$-concavity for $1 \leq p \leq 2$, $q=2$ and $p=2$, $2< q < \infty$. This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder-Gundy inequalities in symmetric spaces and in the $\Phi$-moment case.