Rearrangement-invariant norms commuting with dilations

Abstract

We study rearrangement-invariant spaces X over [0, ∞) for which there exists a function h : (0, ∞) → (0, ∞) such that ∥Drf∥X = h(r)∥f∥X for all f ∈ X and all r > 0, where Dr is the dilation operator. It is shown that this may hold only if h(r) = r− 1 p for all r > 0, in which case the norm ∥·∥X is called p-homogeneous. We investigate which types of r.i. spaces satisfy this condition and show some important embedding properties.