Noncommutative sharp dual Doob inequalities

Abstract: 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 $(\mathcal{M}n){k\geq1}$ 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.