A $$C^m$$ C m Lusin approximation theorem for horizontal curves in the Heisenberg group

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

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