Strictly singular embeddings in L 1 +L ∞

Abstract

The paper deals with the strict singularity and the disjoint strict singularity of the canonical embedding between couples of rearrangement invariant spaces E and F. Let E c F. This embedding is called strictly singular (disjointly strictly singular) if for any infinite-dimensional subspace (any infinite-dimensional subspace generated by a sequence of disjoint functions) B of E the norms of E and F are non-equivalent on B. Denote by Lp,1 (1 < p < 1) the space of all measurable functions x(t) on (0,1) such that kxkLp,1 = sup s>0 s1/p−1 Z s 0 x(t) dt <1.