Noncommutative Davis type decompositions and applications

TL;DR

This paper establishes a noncommutative Davis decomposition for Hardy spaces, providing new control over norms and extending key inequalities like Burkholder/Rosenthal to noncommutative symmetric spaces and Orlicz functions.

Contribution

It introduces a novel noncommutative Davis decomposition with simultaneous norm control and extends classical inequalities to noncommutative symmetric and Orlicz spaces.

Findings

Proves noncommutative Davis decomposition for all 0<p≤1.

Extends Burkholder/Rosenthal inequality to noncommutative symmetric spaces.

Derives $ ext{Phi}$-moment Burkholder/Rosenthal inequalities for specific Orlicz functions.

Abstract

We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\leq 1$. A new feature of our Davis decomposition is a simultaneous control of $\H_1^c$ and $\H_q^c$ norms for any noncommutative martingale in $\H_1^c \cap \H_q^c$ when $q\geq 2$. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space $E$ that is either an interpolation of the couple $(L_p, L_2)$ for some $1<p<2$ or is an interpolation of the couple $(L_2, L_q)$ for some $2<q<\infty$. We also obtain the corresponding $\Phi$-moment Burkholder/Rosenthal inequality for Orlicz functions that are either $p$-convex and $2$-concave for some $1<p<2$ or are $2$-convex and $q$-concave for some $2<q<\infty$.