On curve-flat Lipschitz functions and their linearizations

TL;DR

This paper explores the properties of Lipschitz functions and their linearizations, revealing conditions under which various operator ideals coincide and extending the concept of curve-flatness.

Contribution

It establishes equivalences among operator ideals for Lipschitz map linearizations and introduces a metric property extending curve-flatness.

Findings

Linearizations of Lipschitz maps are Dunford-Pettis iff they are Radon-Nikodým.

Such linearizations do not fix any copy of L1.

A new metric property extending curve-flatness is identified.

Abstract

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\widehat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it is Radon-Nikod\'ym if and only if it does not fix any copy of $L_1$. We also identify and study the corresponding metric property of $f$, which is a natural extension of the curve-flatness.