A compact null set containing a differentiability point of every Lipschitz function

Abstract

ABSTRACT. We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

“…Remarkably, the ℓ 2 summability condition on the defining sequence a has recently arisen in a different (although related) context. Doré and Maleva [11] show that when a ∈ c 0 \ ℓ 2 , the compact set S a is a universal differentiability set, i.e., it contains a differentiability point for every real-valued Lipschitz function on R 2 . Question 8.3.…”

A measure zero universal differentiability set in the Heisenberg group

“…Finally one must show that the directional derivative Ef (x) exists and is almost maximal; to prove this it is crucial that at each stage we maximize over a constrainted set of points and directions. Our argument follows very closely that of [10], modified to use horizontal directions, H-linear maps and Hölder equivalence of the Carnot-Carathéodory and Euclidean distance. Finally we observe that combining Theorem 5.6 and Proposition 6.1 gives Theorem 2.12.…”

“…We now prove that the limit directional derivative E * f (x * ) is almost locally maximal in horizontal directions. This is our adaptation of [10,Lemma 3.5].…”

Universal differentiability sets and maximal directional derivatives in Carnot groups

“…The curve h 1 (t) defines h(t) for t ∈ [s/2, s] with the desired properties. The construction of h(t) for t ∈ [−s, −s/2] is essentially identical.Using the constructed curves g and h, together with Lemma 5.1, the rest of the proof is identical to the Heisenberg group case treated in[31, Theorem 5.6].Proposition 5.8 below is a generalization of [31, Proposition 6.1] in the Heisenberg group, itself based on[14, Theorem 3.1] in Euclidean spaces, to free Carnot groups of step 2. We omit the proof since it is identical to[31, Proposition 6.1], replacing an application of[31, Lemma 5.1] in the Heisenberg group with Lemma 5.1 in the step 2 free Carnot group.…”