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 \hbox{$p$-disjointly} homogeneous rearrangement invariant space on $[0,1]$ with nontrivial Boyd indices which has two rearrangement invariant representations on the half-axis $(0,\infty)$.