Disjoint strict singularity of inclusions between rearrangement invariant spaces

It is studied when inclusions between rearrangement invariant function spac es on the interval [0, infinity) are disjointly strictly singular operators . In particular suitable criteria, in terms of the fundamental function, fo r the inclusions L-1 boolean AND L-infinity hooked right arrow E and E hook ed right arrow L-1 + L-infinity to be disjointly strictly singular are show n. Applications to the classes of Lorentz and Marcinkiewicz spaces are give n.