Asymmetric Doob inequalities in continuous time

The present paper is devoted to the second part of our project on asymmetric maximal inequalities, where we consider martingales in continuous time. Let $(\mathcal M,\tau)$ be a noncommutative probability space equipped with a continuous filtration of von Neumann subalgebras $(\mathcal M_t)_{0\leq t\leq1}$ whose union is weak-$*$ dense in $\mathcal{M}$. Let $\mathcal E_t$ denote the corresponding family of conditional expectations. As for discrete filtrations, we shall prove that for $1 < p < 2$ and $x \in L_p(\mathcal M,\tau)$ one can find $a, b \in L_p(\mathcal M,\tau)$ and contractions $u_t, v_t \in \mathcal M$ such that $$\mathcal E_t(x) = a u_t + v_t b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, $a u_t$ and $v_t b$ converge in the row/column Hardy spaces $\mathcal H_p^r(\mathcal M)$ and $\mathcal H_p^c(\mathcal M)$ respectively. We also confirm in the continuous setting the validity of related asymmetric maximal inequalities which we recently found for discrete filtrations, including $p=1$. As for other results in noncommutative martingale theory, the passage from discrete to continuous index is quite technical and requires genuinely new methods. Our approach towards asymmetric maximal inequalities is based on certain construction of conditional expectations for a sequence of projective systems of $L_p$-modules. The convergence in $\mathcal H_p^r(\mathcal M)$ and $\mathcal H_p^c(\mathcal M)$ also imposes new algebraic atomic decompositions.