A quantitative metric differentiation theorem

TL;DR

This paper provides a quantitative version of Kirchheim's metric differentiation theorem, offering estimates on how Lipschitz functions approximate seminorms in metric spaces, with implications for understanding the local structure of Lipschitz maps.

Contribution

It introduces a Carleson-type estimate for the metric differentiation of Lipschitz functions, extending earlier qualitative results to a quantitative framework.

Findings

Quantitative estimates for the pullback metric as a seminorm

A Carleson-type estimate for metric differentiation

Extension of Kirchheim's 1994 metric differentiation result

Abstract

The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson-type estimate.