On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

TL;DR

This paper extends Rademacher's theorem to arbitrary measures in Euclidean space, linking Lipschitz function differentiability to measure decompositions and normal currents, broadening classical differentiability results.

Contribution

It proves a differentiability theorem for Lipschitz functions relative to arbitrary measures, connecting differentiability to measure decompositions and normal currents.

Findings

Differentiability of Lipschitz functions relates to measure decompositions.

Established differentiability results for measures associated with normal currents.

Extended formulas involving normal currents to Lipschitz maps.

Abstract

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type, where the Lebesgue measure is replaced by an arbitrary measure $\mu$. In particular we show that the differentiability properties of Lipschitz functions at $\mu$-almost every point are related to the decompositions of $\mu$ in terms of rectifiable one-dimensional measures. As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.