Weak derivatives and metric differentiability almost everywhere
Nikita Evseev
TL;DR
This work extends metric differentiability beyond targets that are duals of separable spaces by introducing weak weak* derivatives. It constructs a linear differential ∇°f with values in the dual Lipschitz space (Lip_{z_0}(X))^* and a seminorm ρ so that the metric differential md(f,x)(ν) is obtained as md(f,x)(ν) = ρ(∇°f(x)·ν) a.e., aligning metric and linearized descriptions even when X lacks a separable dual structure. The framework ensures measurability of derivative norms and connects to Sobolev theory, showing that Sobolev maps W^{1,p}(Ω; X) are metrically differentiable in the W^{1,p} topology, and it provides explicit representations in the linear-target case via ∇^{**}f and ∇^*f. Together, these results yield a robust calculus for metric-valued maps, enabling Rademacher-type differentiability and gradient-control arguments in broad settings beyond classical dual targets.
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.