Non-commutative Stein inequality and its applications (1811.00546v1)

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 $1 \leq p < \infty$ are known. In this paper, we verify (i) $(S_{p,\infty})$ for $1 < p \leq \infty$; (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.