Disjoint strict singularity of inclusions between rearrangement invariant spaces

Abstract

Abstract.It is studied when inclusions between rearrangement invariant function spaces on the interval [0, ∞) are disjointly strictly singular operators. In particular suitable criteria, in terms of the fundamental function, for the inclusions

Strictly singular inclusions into $$L^1 + L^\infty$$

“…It is clear that each ω m belongs to the convex hull of {z * n : n ∈ N}. Hence, by (12), we get ω m E ≤ sup n z n E ≤ C for every m ∈ N. And from (14) and…”

“…Suitable criteria for the disjoint strict singularity of embeddings in classes of Orlicz spaces and Lorentz spaces on [0, ∞) have been given in [8,12]:…”

“…It is well known that L 1 ∩ L ∞ → E → L 1 + L ∞ for any r.i. space E on [0, ∞). The singularity of left extreme embeddings L 1 ∩ L ∞ → E has been studied by the authors in [12] finding a suitable characterization of strict singularity in terms of the associated fundamental function φ E . Thus for the operator L 1 ∩ L ∞ → E the statements of strict singularity, disjoint strict singularity and weak compactness coincide and are equivalent to the conditions…”

“…This arises from the non-disjoint strict singularity of the embedding of the order continuous weak L p -spaces L p,∞ 0 into L 1 + L ∞ , which was shown in [12]. One of the main goals of Sect.…”

“…For the special class of r.i. spaces E with submultiplicative fundamental functions, a stronger formulation of Theorem 3.6 can be given: If the embedding E → L 1 + L ∞ is not DSS, then E = W p for 1 ≤ p ≤ ∞, or E = L p,∞ for 1 < p < ∞, or E = L ∞ (see[12, Theorem 4.5]).…”