MARTINGALE TRANSFORMS OF THE RADEMACHER SEQUENCE IN REARRANGEMENT INVARIANT SPACES

Let v(k) = c(k chi{tau >= k}), where tau is a stopping time with respect to the Rademacher system {r(k)} and c(k) is an element of R, k = 1, 2,. ... Then parallel to Sigma(n)(k=1) v(k)r(k)parallel to(X) asymptotic to parallel to Sigma(n)(k=1) v(k)(2))(1/2)parallel to(X) if and only if the rearrangement invariant Banach function space X has nontrivial Boyd indices. If the vk are the vectors Sigma(k-1)(i=0)a(k)(i)r(i), k = 1, 2, ... , the same relation is fulfilled if and only if X contains the closure of L-infinity 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) = Sigma(n)(k=1) v(k)r(k).