Sharp differentiability results for lip

TL;DR

This paper establishes precise conditions under which Lipschitz mappings from metric spaces with Poincaré inequalities to Banach spaces are differentiable almost everywhere, and uses these results to prove a non-embedding theorem.

Contribution

It provides sharp differentiability criteria for Lipschitz maps into Banach spaces with the Radon-Nikodym property, advancing understanding of metric space embeddings.

Findings

Derived a sharp lower local Lipschitz constant condition for differentiability

Proved differentiability almost everywhere under these conditions

Established a non-embedding theorem for certain metric space mappings

Abstract

We give a sharp condition on the lower local Lipschitz constant of a mapping from a metric space supporting a Poincar\'e inequality to a Banach space with the Radon-Nikodym property that guarantees differentiability at almost every point. We apply these results to obtain a non-embedding theorem for a corresponding class of mappings.