The classical Davis inequality $\E f^*\simeq \E Sf$, where $(Sf)^2=\sum_{k} \left|f_{k}-f_{k-1}\right|^2$ is the square function and $f^*= \sup_n \left|f_n\right|$ is the maximal function, is true with a universal constant for any martingale $f$ on any filtration. A natural analog in the setting of (F4) doubly indexed filtrations, i.e. $\left(\mathcal{F}_{i,j}\right)_{i,j}$ such that the operators $\mathbb{E}\left(\cdot\mid \mathcal{F}_{i,\infty}\right)$ and $\E\left(\cdot\mid \mathcal{F}_{\infty,j}\right)$ commute, is the inequality \[\mathbb{E}\sup_{n,m} \left|f_{n,m}\right|\simeq\mathbb{E}\left(\sum_{i,j} \left|\underbrace{f_{i,j}-f_{i-1,j}-f_{i,j-1}+f_{i-1,j-1}}_{\Delta f_{i,j}}\right|^2 \right)^\frac{1}{2}.\] It was known to be true only with some highly restrictive additional assumptions, e.g. regularity of the filtration ($g_{n,m}\gtrsim g_{n+1,m},g_{n,m+1}$ for any positive martingale $g$) or $f$ being a strong martingale ($\mathbb{E}\left(\Delta f_{i,j}\mid \mathcal{F}_{i-1,j}\vee \mathcal{F}_{i,j-1}\right)=0$). We prove the inequality $\lesssim $ assuming just some decoupling properties of the underlying filtration, which are true in particular for a purely atomic one.
Meeting Id: 962 0933 2657 Password: 268545