A few notes on Lorentz spaces

TL;DR

This paper revisits classical results on Lorentz spaces, emphasizing their properties in $L^p$ spaces, providing detailed proofs and clarifying overlooked aspects in existing literature.

Contribution

It offers new insights and detailed proofs on Lorentz spaces, highlighting their contractive properties in $L^p$ spaces that were previously underexplored.

Findings

Clarification of contractive properties of Lorentz spaces in $L^p$

Provision of detailed proofs missing in literature

Comments on classical results and their implications

Abstract

In the sequel, we recall and comment some classical results on the non-increasing rearrangement and Lorentz spaces. There are papers in the existing literature that seemed to have been bypassed as regards its contractive property in~$L^p$ spaces. Also, we provide detailed proofs and some properties that does not seem to arise in the existing literature.