Rademacher multiplicator spaces equal to $L^\infty$

Let $X$ be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space $\Lambda(\mathcal{R},X)$ of measurable functions $x$ such that $x\cdot h\in X$ for every a.e. converging series $h=\sum a_nr_n\in X$, where $(r_n)$ are the Rademacher functions. We characterize the situation when $\Lambda(\mathcal{R},X)= L^\infty$. We also discuss the behaviour of partial sums and tails of Rademacher series in function spaces.