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

Abstract

For every finite measure μ on R^n we define a decomposability bundle V(μ, ·) related to the decompositions of μ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on R^n is differentiable at μ-a.e. x with respect to the subspace V(μ, x), and prove that this differentiability result is optimal, in the sense that, following [4], we can construct Lipschitz functions which are not differentiable at μ-a.e. x in any direction which is not in V(μ,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

On the structure of \mathscr A-free measures and applications

“…Let us mention that the last two theorems will also follow by a stronger result announced by Csörnyei and Jones in [23], namely that for every Lebesgue null set E ⊂ R d there exists a Lipschitz map f : R d → R d which is nowhere differentiable in E, see the discussion in the introduction of [4] for a detailed account of these type of results.…”

“…In order to do so, we assume the reader to be familiar with the work of Alberti & Marchese [4] concerning differentiability of Lipschitz functions, with the definition of metric currents given in [12], as well as with the work of Schioppa in [40]. We refer to these papers also for notations and definitions.…”

“…We also prove a structure theorem for the singular part of a finite family of normal currents in the spirit of the rank-one theorem. By relying on the results of Alberti & Marchese [4] and of Schioppa [40], the latter result immediately implies that the Rademacher theorem can hold only for absolutely continuous measures and that every top-dimensional Ambrosio-Kirchheim metric current in R d is a Federer-Fleming flat chain (a part of the so-called "flat chain conjecture", see [12,Section 11]). …”

“…Relying on [4,40], we use Corollary 1.12 to extend these results to every dimension. In particular, Theorem 1.15 below provides a positive answer to the case k = d of the "flat chain conjecture" stated in [12,Section 11], see [40,Theorem 1.6] for the case k = 1.…”

Cone unrectifiable sets and non-differentiability of Lipschitz functions

“…Given a Radon measure µ in R n , these authors assign to µ-a.a. points a linear subspace T (x) of R n that in certain sense represents the directions of curves on which µ is "seen". For [3], the definition of "seen" is exactly the assumption of Corollary 1.12 while [4] bases the definition on a related but different property and shows in [4,Lemma 7.5] that the assumption of Corollary 1.12 is satisfied. It is, however, important to point out that both these references define the linear space T (x) which is in a natural sense smallest, and this allows them to obtain also a counterpart to Corollary 1.12 that every Lipschitz function is µ-a.e.…”