Lusin approximation and horizontal curves in 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 ${\Gamma }^{\prime }={\gamma }^{\prime }$ outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no $C^1$ horizontal approximation.