# Embeddings of Orlicz-Lorentz spaces into $L_1$

## TL;DR

This paper demonstrates that certain Orlicz-Lorentz spaces can be embedded into L1 space under specific conditions, using combinatorial techniques and new averaging results, expanding understanding of their structure.

## Contribution

It establishes conditions under which Orlicz-Lorentz spaces are uniformly isomorphic to subspaces of L1, including some Lorentz spaces, with a novel approach based on averaging techniques.

## Key findings

- Orlicz-Lorentz spaces embed into L1 under Hardy-type inequalities.
- New averaging results relate Orlicz-Lorentz norms to combinatorial averages.
- Includes embedding results for some Lorentz spaces d^n(a,p).

## Abstract

In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces $d^n(a,p)$. Our approach is based on combinatorial averaging techniques and we prove a new result of independent interest that relates suitable averages with Orlicz-Lorentz norms.