Weak derivatives and metric differentiability almost everywhere

TL;DR

This paper explores the differentiability properties of Lipschitz maps into metric spaces, introducing a new framework based on weak weak* derivatives to represent metric differentials linearly.

Contribution

It introduces a novel approach using weak weak* derivatives to linearize metric differentials for Lipschitz maps into arbitrary metric or Banach spaces.

Findings

Established a framework for weak weak* derivatives

Provided a linear representation for metric differentials

Extended differentiability concepts to broader classes of metric spaces

Abstract

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear counterpart -- weak* differential. However, for an arbitrary metric or Banach space, a Lipschitz map is not necessarily weak* differentiable. This paper introduces an approach based on a concept of weak weak* derivatives. This framework yields a linear representation for the metric differential, allowing for its calculation as the norm of an associated linear operator.