  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.
