Rearrangement invariance of Rademacher multiplicator spaces

Abstract

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 = ∑ a n r n ∈ X , where ( r n ) 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.