Some properties of coarse Lipschitz maps between Banach spaces

TL;DR

This paper investigates the structure of coarse Lipschitz maps between Banach spaces, introducing norm attaining maps, extending known results, and exploring stability of smooth norms under coarse equivalences.

Contribution

It introduces the concept of norm attaining coarse Lipschitz maps and extends classical results to non-separable Banach spaces.

Findings

Introduction of norm attaining coarse Lipschitz maps

Extension of Godefroy's results to coarse Lipschitz equivalences

Stability results for asymptotically uniformly smooth norms

Abstract

We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a result of G. Godefroy on Lipschitz equivalences. This leads us to include the non separable versions of classical results on the stability of the existence of asymptotically uniformly smooth norms under Lipschitz or coarse Lipschitz equivalences.