Non-commutative Stein inequality and its applications

The non-commutative Stein inequality asks whether there exists a constant $C_{p,q}$ depending only on $p,q$ such that \begin{equation*} \left| \left(\sum_{n} |$\mathcal{E}${n} (x_n) |^{q}\right)^{\frac{1}{q}} \right|p \leq C{p,q} \left| \left(\sum{n} | x_n |^q \right)^{\frac{1}{q}}\right |p\qquad \qquad (S{p,q}), \end{equation*} for (positive) sequences $(x_n)$ in $L_p(\mathcal{M})$. The validity of $(S_{p,2})$ for $1<p<\mathrm{\infty }$ and $(S_{p,1})$ for $1\le p<\mathrm{\infty }$ are known. In this paper, we verify (i) $(S_{p,\mathrm{\infty }})$ for $1<p\le \mathrm{\infty }$; (ii) $(S_{p,p})$ for $1 \leq p < \infty$;(iii)$(S_{p,q})$for$1 \leq q \leq 2$and$q<p<\infty$. We also present some applications.