ON ISOMORPHIC EMBEDDINGS IN THE CLASS OF DISJOINTLY HOMOGENEOUS REARRANGEMENT INVARIANT SPACES

The equivalence of the Haar system in a rearrangement invariant space X on [0, 1] and a sequence of pairwise disjoint functions in some Lorentz space is known to imply that X = L-2[0, 1] up to the equivalence of norms. We show that the same holds for the class of uniform disjointly homogeneous rearrangement invariant spaces and obtain a few consequences for the properties of isomorphic embeddings of such spaces. In particular, the L-p[0, 1] space with 1 < p < infinity is the only uniform p-disjointly homogeneous rearrangement invariant space on [0, 1] with nontrivial Boyd indices which has two rearrangement invariant representations on the half-axis (0, infinity).