Martingale transforms of the Rademacher sequence in rearrangement invariant spaces

Abstract:

Let $v_k=c_k\chi _{\{\tau \ge k\}}$, where $\tau$ is a stopping time with respect to the Rademacher system $\{r_k\}$ and $c_k\in \mathbb {R}$, $k=1,2,\dots$. Then $\big \|\sum _{k=1}^n v_kr_k\big \|_X\asymp \big \|(\sum _{k=1}^n v_k^2)^{1/2}\big \|_X$ if and only if the rearrangement invariant Banach function space $X$ has nontrivial Boyd indices. If the $v_k$ are the vectors $\sum _{i=0}^{k-1} a_k^ir_i$, $k=1,2,\dots$, the same relation is fulfilled if and only if $X$ contains the closure of $L_\infty$ in the Orlicz space $\exp L_1$. In the second part of the paper, a new unconditionality criterion for the Haar system in a rearrangement invariant space is obtained in terms of a decoupling version of the transforms $f_n=\sum _{k=1}^n v_kr_k$.