Algebraic Davis decomposition and asymmetric Doob inequalities

TL;DR

This paper develops noncommutative asymmetric Doob inequalities, providing new decomposition techniques and solving longstanding problems related to martingale maximal functions and Hardy space convergence in noncommutative probability.

Contribution

It introduces novel Davis martingale decompositions and algebraic atomic descriptions for Hardy spaces, addressing open problems in noncommutative martingale inequalities.

Findings

Proves asymmetric Doob inequalities for 1 < p < 2.

Establishes noncommutative Davis theorem for p=1.

Provides weak type maximal estimates and convergence results.

Abstract

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ 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 $x \in L_p(\M,\tau)$ one can find $a, b \in L_p(\M,\tau)$ 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.