Noncommutative Davis type decompositions and applications

Abstract

We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\le 1$. 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 $q\ge 2$. 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 $2<q<\infty$.

References in corpus (1)

• Algebraic Davis decomposition and asymmetric Doob inequalities

Cited by in corpus (2)

• Square functions for noncommutative differentially subordinate martingales
• Sum of Free Variables in Fully Symmetric Spaces