A \(C^m\) Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

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.