Rearrangement invariance of Rademacher multiplicator spaces

Let X be a rearrangement invariant function space on [0, 1]. We consider the Rademacher multiplicator space Λ(R,X) of all measurable functions x such that x · h ∈ X for every a.e. converging series h = anrn ∈ X, where (rn) are the Rademacher functions. We study the situation when Λ(R,X) is a rearrangement invariant space different from L∞. Particular attention is given to the case when X is an interpolation space between the Lorentz space Λ(ϕ) and the Marcinkiewicz space M(ϕ). Consequences are derived regarding the behaviour of partial sums and tails of Rademacher series in function spaces. © 2009 Elsevier Inc. All rights reserved.