Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory, we clarify which rearrangement invariant (r.i.) spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and any increasing concave function f on [0,1], which is not equivalent neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function f. Moreover, in the class of all interpolation r.i. spaces with respect to the Banach couple of Lorentz and Marcinkiewicz spaces with the same fundamental function, dilation indices of which are non-trivial, for every 1<p< infty, there is only a unique p-disjointly homogeneous space.