  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 $p \to 1$.
