Noncommutative Davis type decompositions and applications

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 $2<q<\mathrm{\infty }$. We also obtain the corresponding $\mathrm{\Phi }$-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 $2<q<\mathrm{\infty }$.