Non-commutative Stein inequality and its applications

Abstract

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 $1<p<\infty$ and $(S_{p,1})$ for $1\le p<\infty$ are known. In this paper, we verify (i) $(S_{p,\infty })$ for $1<p\le \infty$; (ii) $(S_{p,p})$ for $1\le p<\infty$; (iii) $(S_{p,q})$ for $1\le q\le 2$ and $q<p<\infty$. We also present some applications.

11 pages, to appear in Comm. Statist. Theory Methods