Lusin approximation and horizontal curves in Carnot groups
Summary: 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.
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