Square functions for noncommutative differentially subordinate martingales

We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if $x$ is a self-adjoint noncommutative martingale and $y$ is weakly differentially subordinate to $x$ then $y$ admits a decomposition $y=z +w$ where $z$ and $w$ are two martingales such that:

 $$ \|S_c(z)\|_{1,\infty}+ \|S_r(w)\|_{1,\infty} \leq c\|x\|_1.$$ 

 We also prove strong-type $(p,p)$  version of the above weak-type result for $1<p<2$.  As a byproduct of our approach, we obtain new and constructive proof of the noncommutative Burkholder-Gundy inequalities for $1<p<2$ with the optimal order of the constants as $p \to 1$.