Some Remarks on Tangent Martingale Difference Sequences in L-spaces

Let X be a Banach space. Suppose that for all p ∈ (1,∞) a constant Cp,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‖ ≤ Cp,XE‖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. We state several problems related to (∗) and other decoupling inequalities.