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

Abstract

For every finite measure ${\mu}$ on ${{\mathbb{R}}^n}$ we define a decomposability bundle ${V(\mu,\,\cdot)}$ related to the decompositions of ${\mu}$ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on ${{\mathbb{R}}^n}$ is differentiable at ${\mu}$-a.e. ${x}$ with respect to the subspace ${V(\mu,\,x)}$, and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at ${\mu}$-a.e. ${x}$ in any direction which is not in ${V(\mu,\,x)}$. 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 basic formulas involving normal currents and maps of class ${C^1}$ to Lipschitz maps.