Lusin Approximation and Horizontal Curves in Carnot Groups

TL;DR

This paper explores the approximation of horizontal curves in Carnot groups, showing that in the Heisenberg group such approximation is possible, but providing a counterexample in the Engel group.

Contribution

It establishes the possibility of $C^1$ horizontal approximation in the Heisenberg group and demonstrates its failure in the Engel group, highlighting differences in geometric structures.

Findings

$C^1$ horizontal approximation exists in the Heisenberg group.

Counterexample of no $C^1$ approximation in the Engel group.

Insights into geometric properties of Carnot groups.

Abstract

We show that, given an absolutely continuous horizontal curve $\gamma$ in the Heisenberg group, there is a $C^1$ horizontal curve $\Gamma$ such that $\Gamma=\gamma$ and $\Gamma'=\gamma'$ outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no $C^1$ horizontal approximation.