Abstract Lorentz spaces and K\"othe duality

TL;DR

This paper explores the K"othe duality of generalized Lorentz spaces built from symmetric Banach function spaces and weights, extending known results for Orlicz-Lorentz spaces and introducing new duality characterizations.

Contribution

It develops a comprehensive duality theory for generalized Lorentz spaces, including the introduction of the class M_{E,w} and the space Q_{E,w}, and applies these to Orlicz-Lorentz spaces.

Findings

K"othe dual of M_{E,w} is Λ_{E',w}

K"othe dual of Λ_{E,w} is Q_{E',w}

Q_{E,w} characterized via Halperin's level functions

Abstract

Given a fully symmetric Banach function space $E$ and a decreasing positive weight $w$ on $I = (0, a)$, $0 < a \le \infty $, the generalized Lorentz space ${\Lambda}_{E,w}$ is defined as the symmetrization of the canonical copy $E_w$ of $E$ on the measure space associated with the weight. If $E$ is an Orlicz space then ${\Lambda}_{E,w}$ is an Orlicz-Lorentz space. An investigation of the K\"othe duality of these classes is developed that is parallel to preceding works on Orlicz-Lorentz spaces. First a class of functions $M_{E,w}$, which does not need to be even a linear space, is similarly defined as the symmetrization of the space $w.E_w$. Let also $Q_{E,w}$ be the smallest fully symmetric Banach function space containing $M_{E,w}$. Then the K\"othe dual of the class $M_{E,w}$ is identified as the Lorentz space ${\Lambda}_{E',w}$, while the K\"othe dual of ${\Lambda}_{E,w}$ is $Q_{E',w}$. The space $Q_{E,w}$ is also characterized in terms of Halperin's level functions. These results are applied to concrete Banach function spaces. In particular the K\"othe duality of Orlicz-Lorentz spaces is revisited at the light of the new results.