A Measure Zero Universal Differentiability Set in the Heisenberg Group

Abstract

We show that the Heisenberg group ${\mathbb{H}}^n$ contains a measure zero set $N$ such that every Lipschitz function $f:{\mathbb{H}}^n\to \mathbb{R}$ 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.

35 pages; revised version includes improvements suggested by referees; to appear in Mathematische Annalen