Continuous asymmetric Doob inequalities in noncommutative symmetric spaces

Let $(M,\tau )$ be a noncommutative probability space equipped with a filtration ${\left(M_t\right)}_{t\in [0,1]}$ whose union is $w^{⁎}$-dense in $M$, and let ${\left(E_t\right)}_{t\in [0,1]}$ be the associated conditional expectations. We prove in the present paper that if the symmetric space $E\in \mathrm{Int}[L_p,L_q]$ with $1<p\le q<2$ and E is $2(1-\theta )$-convex and w-concave with $p<w<2$, then the following holds:${\Vert {\left(E_t\right(x\left)\right)}_{t\in [0,1]}\Vert }_{E(M;{\ell }_{\infty }^{\theta })}\le C_{E,\theta }{\Vert x\Vert }_{H_E^c},x\in H_E^c\left(M\right)$ provided $1-p/2<\theta <1$. Similar result holds for $x\in H_E^r\left(M\right)$. Moreover, if $E\in \mathrm{Int}[L_p,L_q]$ with $1<p\le q<2$ and E is w-concave with $2<w<2p/(2-p)$, then for each $x\in E\left(M\right)$ there exist y, $z\in E\left(M\right)$ such that $x=y+z$ and${\Vert {\left(E_t\right(y\left)\right)}_{t\in [0,1]}\Vert }_{E(M;{\ell }_{\infty }^c)}+{\Vert {\left(E_t\right(z\left)\right)}_{t\in [0,1]}\Vert }_{E(M;{\ell }_{\infty }^r)}\le c_E{\Vert x\Vert }_{E\left(M\right)}.$ These results can be considered as continuous analogues of those due to Randrianantoanina et al. [33]. One of the key ingredients in our proof is a new decomposition theorem of $E\left(M\right)$-modules for general symmetric space E, which extends the known result of Junge and Sherman.