Coarse and uniform embeddings between Orlicz sequence spaces

TL;DR

This paper characterizes when Orlicz sequence spaces can be coarsely or uniformly embedded into each other, mainly based on their Matuszewska-Orlicz indices, but also presents exceptions to this rule.

Contribution

It provides a nearly complete classification of embeddability between Orlicz sequence spaces, highlighting the role of Matuszewska-Orlicz indices and presenting counterexamples.

Findings

Embeddability mostly determined by upper Matuszewska-Orlicz indices.

Counterexamples show embeddability not always dictated by these indices.

Provides a comprehensive description of coarse and uniform embeddings for these spaces.

Abstract

We give an almost complete description of the coarse and uniform embeddability between Orlicz sequence spaces. We show that the embeddability between two Orlicz sequence spaces is in most cases determined only by the values of their upper Matuszewska-Orlicz indices. On the other hand, we present examples which show that sometimes the embeddability is not determined by the values of these indices.