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 \( 1<p<\infty \) 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 \( (0,\infty) \).