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.

21 pages. Corrected a technical error in the argument for the Heisenberg group and added previously omitted proofs. Improved the exposition and corrected typos as suggested by referees. To appear in Revista Matematica Iberoamericana