Universal differentiability sets and maximal directional derivatives in Carnot groups

Abstract

Abstract. We show that every Carnot group G of step 2 admits a Hausdorff dimension one 'universal differentiability set' N such that every Lipschitz map f : G → R is Pansu differentiable at some point of N . This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.

A dichotomy of sets via typical differentiability

“…Moreover, by our assumption, we have that < < for any ≥ 1, but we may have < < for some ≥ 1 and < < for others. Let 14) and note for future reference that G ∪ B = N and ∪ = ≥1 = 1 \ ′ . Write 1 = ( , ), and denote by ∈ S −1 the vector such that is -flat around 0 in direction .…”

“…This demonstrates the extent to which the -null criterion from [22] fails in higher dimensions: in dimension one, countable unions of closed null sets are typically non-differentiability sets; but in all higher dimensions, they may actually capture differentiability points of every Lipschitz function. We expect that, in the same spirit as for UDSs, typical differentiability sets will be explored further, in particular providing insight into typical behaviour of Lipschitz functions on non-Euclidean spaces; in this context, one should mention recent research into UDSs in Heisenberg and, more generally, Carnot groups [21,19,14].…”

Structure of porous sets in Carnot groups

“…The following lemma [15] will be used in Section 5. There is no similar statement if R is replaced by R n (n > 1) or H n : these spaces admit measure zero sets containing points of (Pansu) differentiability for every real-valued Lipschitz function [23,25,16]. In Banach spaces which admit a suitable bump function, one can construct a Lipschitz function which is differentiable at no point of any given σ-porous set.…”

Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

“…In the Banach space setting, [21,22] give a version of Rademacher's theorem for Frechét differentiability of Lipschitz mappings on Banach spaces in which porous sets are negligible in a suitable sense. Other results have also been studied in stratified groups [20,29,31,32].…”