Calculus of Variations and Geometric Measure Theory
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A. Pinamonti - G. Speight

A Measure Zero Universal Differentiability Set in the Heisenberg Group

created by pinamonti on 29 May 2015
modified by speight on 16 May 2017


Accepted paper: Math.Ann.

Inserted: 29 may 2015
Last Updated: 16 may 2017

Year: 2015

ArXiv: 1505.07986 PDF


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.


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