A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

Abstract

We prove a C m Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is m − 1 times L 1 differentiable almost everywhere coincides with a C m horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if L 1 differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.