The space of Fourier–Haar multipliers

The Haar system constitutes an unconditional basis for a separable rearrangement invariant (symmetric) space $E$ if and only if the multiplier determined by the sequence $\lambda_{nk}=(-1)^n$, $k=0,1$, for $n=0$ and $k=0,1,\dots,2^n$ for $n\geqslant1$, is bounded in $E$. If the Lorentz space $\Lambda(\varphi)$ differs from $L_1$ and $L_\infty$ then there is a multiplier with respect to the Haar system which is bounded in $\Lambda(\varphi)$ and unbounded in $L_\infty$ and $L_1$.