Noncommutative sharp dual Doob inequalities

Let $(x_k{)}_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $({\mathcal{E}}_k{)}_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras $({\mathcal{M}}_n{)}_{k\ge 1}$ of the finite von Neumann algebra $\mathcal{M}$. We establish the following sharp noncommutative dual Doob inequalities: \begin{equation*} \Big| \sum_{k=1}^nx_k\Big|{L_p($\mathcal{M}$)}\leq \frac{1}{p} \Big| \sum{k=1}^n$\mathcal{E}$k(x_k)\Big|{L_p($\mathcal{M}$)},\quad 0<p\leq 1, \end{equation*} and \begin{equation*} \Big| \sum_{k=1}^n$\mathcal{E}$k(x_k)\Big|{L_p($\mathcal{M}$)}\leq p\Big| \sum_{k=1}^nx_k\Big|_{L_p($\mathcal{M}$)},\quad 1\leq p\leq 2. \end{equation*} As applications, we obtain several noncommutative martingale inequalities with better constants.