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 $dy=a +b +c$ (resp. $dy=z +w$) where $a$, $b$, and $c$ are adapted sequences (resp. $z$ and $w$ are martingale difference sequences) such that: $$\Big\| (a_n)_{n\geq 1}\Big\|_{L_{1,\infty}({\mathcal M}\overline{\otimes}\ell_\infty)} +\Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|b_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|c_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1$$ (resp. $$\Big\| \Big(\sum_{n\geq1} |z_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} |w_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1).$$ We also prove strong-type $(p,p)$ versions of the above weak-type results for $1<p<2$. In order to provide more insights into the interactions between noncommutative differential subordinations and martingale Hardy spaces when $1\leq p<2$, we also provide several martingale inequalities with sharp constants which are new and of independent interest. As a byproduct of our approach, we obtain new and constructive proofs of both the noncommutative Burkholder-Gundy inequalities and the noncommutative Burkholder/Rosenthal inequalities for $1<p<2$ with the optimal order of the constants when $p \to 1$.