The structure of subspaces in Orlicz spaces lying between L1 and L2

Abstract

A subspace H of a rearrangement invariant space X on [0, 1] is strongly embedded in X if, in H, convergence in the X-norm is equivalent to convergence in measure. We obtain necessary and sufficient conditions on an Orlicz function M, under which the unit ball of an arbitrary strongly embedded subspace in the Orlicz space L<inf>M</inf> has equi-absolutely continuous norms in L<inf>M</inf>. In particular, this extends a well-known Rosenthal’s characterization of Λ (p) -spaces, 1 < p< 2 , to the realm of Orlicz spaces. © 2023 Elsevier B.V., All rights reserved.