  Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant
$C_{p,X}$ depending only on X and p exists such that for any two X-valued
martingales f and g with tangent martingale difference sequences one has
\[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent to the
UMD condition. In fact, it is still equivalent to the UMD condition if in
addition one demands that either f or g satisfy the so-called (CI) condition.
However, for some applications it suffices to assume that (*) holds whenever g
satisfies the (CI) condition. We show that the class of Banach spaces for which
(*) holds whenever only g satisfies the (CI) condition is more general than the
class of UMD spaces, in particular it includes the space L^1. We state several
problems related to (*) and other decoupling inequalities.
