A new class of restricted type spaces

Abstract

We find new properties for the space R(X), introduced by Soria in the study of the best constant for the Hardy operator minus the identity. In particular, we characterize when R(X) coincides with the minimal Lorentz space Λ(X). The condition that R(X) ≠ 0 is also described in terms of the embedding (L1,∞ L∞) X. Finally, we also show the existence of a minimal rearrangement-invariant Banach function space (RIBFS) X among those for which R(X) ≠ 0 (which is the RIBFS envelope of the quasi-Banach space L1,∞ L∞). © 2011 Edinburgh Mathematical Society.