Differential of metric valued Sobolev maps

TL;DR

This paper introduces a new notion of differential for Sobolev maps between metric spaces, aligning with existing concepts in Euclidean and real-valued cases, advancing the understanding of calculus in metric measure spaces.

Contribution

It develops a consistent differential framework for Sobolev maps in metric spaces using tangent and cotangent modules, bridging gaps with classical differentials.

Findings

Differential aligns with Kirchheim's metric differential in Euclidean spaces

Differential matches the abstract differential when the target is real numbers

Provides a unified approach to Sobolev map differentials in metric spaces

Abstract

We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is $\mathbb{R}$.