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

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.