非交换Stein不等式及其应用
Non-commutative Stein inequality and its applications
摘要：
非交换Stein不等式询问是否存在仅依赖于$p, q$的常数$C_{p,q}$，使得对于$L_p(\mathcal{M})$中的（正）序列$(x_n)$，\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*}。已知$(S_{p,2})$对$1 < p < \infty$和$(S_{p,1})$对$1 \leq p < \infty$的有效性。在本文中，我们验证（i）$(S_{p,\infty})$为$1 < p \leq \infty$ ;（ii）$(S_{p,p})$为$1 \leq p < \infty$ ;（iii）$(S_{p,q})$为$1 \leq q \leq 2$和$q<p<\infty$。我们还介绍了一些应用。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
