o-O structure of some rearrangement invariant Banach function spaces

摘要

In this paper we extend the results obtained in Angrisani et al. (Ricerche di Matematica 1-17, 2019) to a more general setting, showing that the o-O structure developed in Perfekt (Arkiv Matematik 51(2):345-361, 2013) also includes the pair of spaces (E-0, E), where E = L-phi,L-infinity is a Marcinkiewicz-type rearrangement invariant space corresponding to the function phi and E-0 = E-b = L-phi,L-infinity is the closure of bounded functions in E, thus inheriting the properties of this structure. To show such a result we make use of some properties concerning the inclusion of Marcinkiewicz spaces. We then exploit these properties to prove a dichotomy result for the closed linear subspaces of E-0, generalizing the approach of Kadec and Pelczynski (Stud Math 21(2):161-176, 1962) for the L-p spaces, and extending to those spaces the results obtained in Leibov (J Math Sci 48(5):536-538, 1990) for the (VMO, BMO) pair.