An extension of Lepingle inequality in Von Neumann algebras with finite trace

An inequality of Asmar and Montgomery-Smith states that \begin{eqnarray*} \| (\sum_{n=1}^\infty |\mathbb{E}_{\mathcal{F}_{n-1}}X_n|^q)^{\frac{1}{q}} \|_p \leq C_p \| (\sum_{n=1}^\infty |X_n|^q)^{\frac{1}{q}} \|_p \quad (1 p \infty, 1 \leq q \leq \infty), \end{eqnarray*} where $(X_n)_{n=1}^\infty$ is a stochastic process adapted to the filtration $(\mathcal{F}_n)_{n=0}^\infty$. Recall that a filtration of a von Neumann algebra $\mathcal{M}$ is an increasing sequence $(\mathcal{M}_n)_{n\ge 0}$ of von Neumann subalgebras of $\mathcal{M}$ such that $\bigcup\limits_{n\ge 0} \mathcal{M}_n$ generates $\mathcal{M}$ in the $w^*$-topology. Using the duality argument Lepingle verified this inequality for adapted process with $q=2$. We obtain some versions of Lepingle inequality in the noncommutative setting.