On weaker notions of nonlinear embeddings between Banach spaces

TL;DR

This paper investigates weaker forms of nonlinear embeddings between Banach spaces, demonstrating that many known results for standard embeddings also hold in these weaker contexts, suggesting they may be more powerful than previously thought.

Contribution

It introduces and analyzes weaker notions of coarse and uniform embeddings, showing their robustness and potential strength compared to traditional embeddings.

Findings

Many known results for coarse and uniform embeddings extend to weaker notions.

Weaker embeddings may be more powerful than initially believed.

The study provides evidence supporting the strength of these weaker embedding concepts.

Abstract

In this paper, we study nonlinear embeddings between Banach spaces. More specifically, the goal of this paper is to study weaker versions of coarse and uniform embeddability, and to provide suggestive evidences that those weaker embeddings may be stronger than one would think. We do such by proving that many known results for coarse and uniform embeddings remain valid for those weaker notions of embeddings.