A Measure Zero Universal Differentiability Set in the Heisenberg Group

TL;DR

This paper constructs a measure zero set in the Heisenberg group where every Lipschitz function is Pansu differentiable, extending the concept of universal differentiability sets from Euclidean spaces to sub-Riemannian geometry.

Contribution

It introduces the first measure zero universal differentiability set in the Heisenberg group, adapting Euclidean techniques to a sub-Riemannian setting.

Findings

Existence of measure zero set where all Lipschitz functions are Pansu differentiable

Extension of Euclidean universal differentiability set concepts to Heisenberg group

Identification of points with almost locally maximal horizontal directional derivatives

Abstract

We show that the Heisenberg group $\mathbb{H}^n$ contains a measure zero set $N$ such that every Lipschitz function $f\colon \mathbb{H}^n \to \mathbb{R}$ is Pansu differentiable at a point of $N$. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of $N$ and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.