Geodesic
Multipliers on the Set of Rademacher Series in Symmetric Spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 3, pp. 87-90.

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Let $E$ be a symmetric space on $[0,1]$. Let $\Lambda(\mathcal{R},E)$ be the space of measurable functions $f$ such that $fg\in E$ for every almost everywhere convergent series $g=\sum b_nr_n\in E$, where $(r_n)$ are the Rademacher functions. In [G. P. Curbera, Proc. Edinb. Math. Soc., 40, No. 1, 119–126 (1997)] it was shown that, for a broad class of spaces $E$, the space $\Lambda(\mathcal{R},E)$ is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on $E$ for $\Lambda(\mathcal{R},E)$ to be order isomorphic to $L_\infty$. This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which $\Lambda(\mathcal{R},E)$ is order isomorphic to a symmetric space that differs from $L_\infty$. The answer is positive for the Orlicz spaces $E=L_{\Phi_q}$ with $\Phi_q(t)=\exp|t|^q-1$ and $0$.
Keywords: Rademacher series in symmetric spaces, Orlicz and Marcinkiewicz spaces, multiplier for Rademacher series. 
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G. P. Curbera; V. A. Rodin. Multipliers on the Set of Rademacher Series in Symmetric Spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 3, pp. 87-90. http://geodesic.mathdoc.fr/item/FAA_2002_36_3_a13/

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