On embeddings between classical Lorentz spaces

Abstract

Let p ∈ (0, ∞) , let v be a weight on (0, ∞) and let Λp(v) be the classical Lorentz space, determiined by the norm ∥f∥Λp(v) := (∫∞0(f*(t))pv(t)dt) 1/p. When p ∈ (1, ∞), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if Λp(v) = Γp(v), where ∥f∥Γp(v) := (∫∞0(f**(t))pv(t) t/p. We may thus conclude that, for p ∈ (1, ∞), the space Λp(v) is equivalent to a Banach space if and only if the norm of a function f in it can be expressed in terms of f**. We study the question whether an analogous assertion holds when p = 1. Motivated by this problem, we consider general embeddings between four types of classical and weak Lorentz spaces, namely, Λp(v), Λp,∞(v), Γp(v), Γp,∞(v), where Λp,∞(v) and Γp∞(v) are certain weak analogues of the spaces Λp(v) and Γp(v), respectively. We present a unified approach to these embeddings, based on rearrangement techniques. We survey all the known results and prove new ones. Our main results concern the embedding Γp,∞(v) → Λq(w) which had not been characterized so far. We apply our results to the characterization of associate spaces of classical and weak Lorentz spaces and we give a characterization of fundamental functions for which the endpoint Lorentz space and the endpoint Marcinkiewicz space coincide.