Rearrangement invariance of Rademacher multiplicator spaces (Q1028321)

Let \(X\) be a rearrangement-invariant (r.i.)\ space on \([0,1]\) and let \({\mathcal R}\) be the set of all functions of the form \(\sum a_n r_n\) where \(r_n\) are the Rademacher functions \(x\) and \(a_n\in{\mathbb R}\). The Rademacher multiplicator space of \(X\) is the space \(\Lambda({\mathcal R},X)\) of all measurable functions on \([0,1]\) such that \(x\sum a_n r_n\in X\) for every \(\sum a_n r_n\in{\mathcal R}\cap X\). The authors proved in their earlier paper [\textit{S.\,Astashkin} and \textit{G.\,Curbera}, J.~Funct.\ Anal.\ 226, No.\,1, 173--192 (2005; Zbl 1083.46015)] that, if the lower Zippin index of the space \(X\) is positive, then \(\Lambda({\mathcal R},X)\) is not an r.i.\ space. Further, they showed in [\textit{S.\,Astashkin} and \textit{G.\,Curbera}, Proc.\ Am.\ Math.\ Soc.\ 136, No.\,10, 3493--3501 (2008; Zbl 1160.46021)] that \(\Lambda({\mathcal R},X)\) coincides with \(L^\infty\) if and only if \(\log^{1/2}(2/t)\) does not belong to the closure of \(X\) in \(L^\infty\). The paper under review addresses the situation when \(\Lambda({\mathcal R},X)\) is an r.i.\ space different from \(L^\infty\). Some conditions (separately, necessary and sufficient) for \(\Lambda({\mathcal R},X)\) being an r.i.\ space are proved. Special attention is paid to the case when \(X\) is an interpolation space between the Lorentz space \(\Lambda(\varphi)\) and the Marcinkiewicz space \(M(\varphi)\).