Multiplication operators on the space of Rademacher series in rearrangement invariant spaces

Abstract

Let E be a rearrangement invariant (r.i.) function space on [0; 1]. We consider the space (R;E) of measurable functions f such that fg2E for every a.e. converging seriesg = P anrn2 E, where (rn) are the Rademacher functions. Curbera [4] showed that, for a broad class of spaces E, the space (R;E) is not order-isomorphic to a r.i. space. We study cases when (R;E) is order-isomorphic to a r.i. space. We give conditions on E so that (R;E) is order-isomorphic to L1. This includes certain classes of Lorentz and Marcinkiewicz spaces. We study further when (R;E) is orderisomorphic to a r.i. space different fromL1. This occurs for the Orlicz spacesE = Lq withq(t) asymptotically equivalent to expjtj q 1 and 0 <q< 2.