Every absolutely continuous horizontal curve in the Heisenberg group whose horizontal velocity is (m-1)-times L1 differentiable almost everywhere coincides with a C^m horizontal curve outside a set of small measure, and this condition is optimal.
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A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group
Every absolutely continuous horizontal curve in the Heisenberg group whose horizontal velocity is (m-1)-times L1 differentiable almost everywhere coincides with a C^m horizontal curve outside a set of small measure, and this condition is optimal.