On Embeddings between Classical Lorentz Spaces

Let p 2 (0; 1), let v be a weight on (0; 1) and let p (v) be the classical Lorentz space, determined by the norm kfk p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. When p 2 (1; 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 p (v) = ? p (v), where kfk ? p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. We may thus conclude that, for p 2 (1; 1), the space p (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, p (v), (v) are certain weak analogues of the spaces p (v) and ? p (v), respectively. We present a uniied approach to these embeddings, based on rearrangement techniques. We survey all the known results and prove new ones. Our main results concern the embedding ? p;1 (v) , ! q (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.