Differentiability of Lipschitz Functions, Structure of Null Sets, and Other Problems

Abstract

The research presented here developed from rather mysterious observations, originally made by the authors independently and in different circumstances, that Lebesgue null sets may have uniquely defined tangent directions that are still seen even if the set is much enlarged (but still kept Lebesgue null). This phenomenon appeared, for example, in the rank-one property of derivatives of BV functions and, perhaps in its most striking form, in attempts to decide whether Rademacher's theorem on differentiability of Lipschitz functions may be strengthened or not.We describe the non-differentiability sets of Lipschitz functions on R n and use this description to explain the development of the ideas and various approaches to the definition of the tangent fields to null sets. We also indicate connections to other current results, including results related to the study of structure of sets of small measure, and present some of the main remaining open problems. (2000). Primary 26B05; Secondary 28A75. Mathematics Sub ject Classification

The Lip-lip equality is stable under blow-up

“…This result is a functional-analytic interpretation of constructions in [ACP05, ACP10,Bat12], that was obtained in [Sch14b,Sch13]; to obtain the optimal constants we were greatly helped by Andrea Marchese's PhD thesis.…”

“…In connection with embeddings into RNP-Banach spaces, Cheeger and Kleiner [CK09] showed that if (X, µ) is a PI-space the fibres of T X are spanned by "tangent vectors" to Lipschitz curves. Putting T X and T * X on a complete equal footing has required substantial effort: Bate's beautiful work [Bat12,Bat15] on Alberti representations in differentiability spaces, which was partly motivated by a deep structure theory for measures and sets in R n developed by Alberti, Csörnyei and Preiss [ACP05,ACP10], and the formulation of metric differentiation for differentiability spaces [CKS15], which was partly motivated by unpublished results of Cheeger and Kleiner on metric differentiation in PI-spaces, and unpublished results of mine on prescribing the norms on T X and T * X.…”

Cone unrectifiable sets and non-differentiability of Lipschitz functions

“…We follow this trend in our main notion, introduced in Definition 1.6. Before coming to it, we recall the main starting motivation behind what we do, namely the following definition from [2] and a special case of their result (proved in [3]) which is most relevant for us.…”

“…Obviously, if E is a k-dimensional embedded C 1 submanifold of R n , its tangent field τ (x) satisfies (1.3). However, the following theorem proved in [2,3] shows that many non-smooth sets admit a k-dimensional tangent field. Before stating it, we notice that Definition 1.2 uses the value α in two different meanings and so it is sensitive on the choice of the notion of width.…”

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

“…Alberti representations were introduced in [1] to prove the so-called rank-one property for BV functions; they were later applied to study the di erentiability properties of Lipschitz functions f : R N → R [2,3] and have recently been used to obtain a description of measures in di erentiability spaces [10]. We rst give an informal de nition.…”