On a rearrangement-invariant function set that appears in optimal Sobolev embeddings

We give necessary and sufficient conditions for the set of measurable functions Y(A) = {f : t −m/n[f ∗∗ (t) − f ∗ (t)] A <∞, f ∗∗ (∞) = 0} to be a normable linear space.We also give a complete characterization of all spaces B that can be represented in the form B = Y(A) for some space A and of all spaces A that can appear in such representations. © 2008 Elsevier Inc. All rights reserved