代数Davis分解与非对称Doob不等式
Algebraic Davis decomposition and asymmetric Doob inequalities
摘要：
本文研究了Doob极大不等式的非对称形式.这种不对称性是由不可交换性造成的。设$(\M,τ)$是一个非交换概率空间，它具有von Neumann子代数$(\M_n)_{n \ge 1}$的弱- $*$稠密滤子.令$\E_n$表示相应的条件期望族。作为不对称结果的一个例子，我们证明了对于$1 < p < 2$和$x \in L_p(\M,τ)$，可以找到$a, b \in L_p(\M,τ)$和压缩$u_n, v_n \in \M$，使得$$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$。此外，还证明了$a u_n$和$v_n b$分别收敛于行/列Hardy空间$\H_p^r(\M)$和$\H_p^c(\M)$。特别是，这解决了Defant和Junge在2004年提出的问题。在$p=1$的情况下，我们的结果建立了关于$L_1$中鞅极大值与平方函数之间关系的Davis著名定理的一个非交换形式，其非交换形式在相当长的一段时间内一直是开放的。在给定$1 \le p \le 2$的条件下，我们还给出了新的弱型极大值估计，它们依次暗示了$\E_n(x)$在行/列Hardy空间中的左/右几乎一致收敛性.这改善了迄今为止已知的双边收敛。我们的方法是基于新形式的戴维斯鞅分解，这是独立的利益，代数原子描述所涉及的哈代空间。后者的结果是新的，甚至交换冯诺依曼代数。
英文摘要：
In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,τ)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\M_n)_{n \ge 1}$. Let $\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \in L_p(\M,τ)$ one can find $a, b \in L_p(\M,τ)$ and contractions $u_n, v_n \in \M$ such that $$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, it turns out that $a u_n$ and $v_n b$ converge in the row/column Hardy spaces $\H_p^r(\M)$ and $\H_p^c(\M)$ respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case $p=1$, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in $L_1$, whose noncommutative form has remained open for quite some time. Given $1 \le p \le 2$, we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of $\E_n(x)$ in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.
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