Square functions for noncommutative differentially subordinate martingales

Abstract

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: $$\Vert (a_n{)}_{n\ge 1}{\Vert }_{L_{1,\infty }(\mathcal{M}\overline{\otimes }{\ell }_{\infty })}+\Vert (\sum _{n\ge 1}{\mathcal{E}}_{n-1}|b_n{|}^2{)}^{1/2}{\Vert }_{1,\infty }+\Vert (\sum _{n\ge 1}{\mathcal{E}}_{n-1}|c_n^{\ast }{|}^2{)}^{1/2}{\Vert }_{1,\infty }\le C\Vert x{\Vert }_1$$ (resp. $$\Vert (\sum _{n\ge 1}|z_n{|}^2{)}^{1/2}{\Vert }_{1,\infty }+\Vert (\sum _{n\ge 1}|w_n^{\ast }{|}^2{)}^{1/2}{\Vert }_{1,\infty }\le C\Vert x{\Vert }_1).$$ 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 $1\le p<2$, 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 $p\to 1$.