On the expansiveness of coarse maps between Banach spaces and geometry preservation

TL;DR

This paper introduces a new embeddability concept between Banach spaces, compares it with classical notions, and explores its implications for geometric properties and embeddings into $\, ext{ extlbrackdbl}\, ext{ extrtaildbl}\, ext{ extlbrackdbl}$ spaces.

Contribution

It defines a novel embeddability notion, analyzes its relation to classical and nonlinear invariants, and advances understanding of embeddings between Banach spaces, especially $\, ext{ extlbrackdbl}\, ext{ extrtaildbl}\, ext{ extlbrackdbl}$ spaces.

Findings

The new embeddability notion is strictly weaker than coarse embeddability.

Results on cotype preservation under the new embeddability.

Insights into the embeddability of $\, ext{ extlbrackdbl}\, ext{ extrtaildbl}\, ext{ extlbrackdbl}$ spaces into $\, ext{ extlbrackdbl}\, ext{ extrtaildbl}\, ext{ extlbrackdbl}$ spaces.

Abstract

We introduce a new notion of embeddability between Banach spaces. By studying the classical Mazur map, we show that it is strictly weaker than the notion of coarse embeddability. We use the techniques from metric cotype introduced by M. Mendel and A. Naor to prove results about cotype preservation and complete our study of embeddability between $\ell_p$ spaces. We confront our notion with nonlinear invariants introduced by N. Kalton, which are defined in terms of concentration properties for Lipschitz maps defined on countably branching Hamming or interlaced graphs. Finally, we address the problem of the embeddability into $\ell_\infty$.