Almost compact embeddings between Orlicz and Lorentz spaces

TL;DR

This paper characterizes when Orlicz spaces are almost compactly embedded into Lorentz spaces using a new method based on Young-type inequalities and level set measures.

Contribution

It provides a new characterization of almost compact embeddings between Orlicz and Lorentz spaces through a novel inequality-based approach.

Findings

Established a balance condition for embeddings involving parameters p, q, and Young functions.

Developed a new method using Young-type inequalities and level set measures.

Characterized almost compact embeddings in terms of these conditions.

Abstract

We characterize when an Orlicz space $L^A$ is almost compactly (uniformly absolutely continuously) embedded into a Lorentz space $L^{p,q}$ in terms of a balance condition involving parameters $p,q\in[1,\infty]$, and a Young function $A$. In the course of the proof, we develop a new method based on an inequality of Young type involving the measure of level sets of a given function.