Abstract
A Carnot group \(\mathbb {G}\) admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve \(\gamma \) in \(\mathbb {G}\) and \(\varepsilon >0\), there is a \(C^1\) horizontal curve \(\Gamma \) such that \(\Gamma =\gamma \) and \(\Gamma '=\gamma '\) outside a set of measure at most \(\varepsilon \). We verify this property for free Carnot groups of step 2 and show that it is preserved by images of Lie group homomorphisms preserving the horizontal layer. Consequently, all step 2 Carnot groups admit Lusin approximation for horizontal curves.
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Acknowledgments
Part of this work was carried out when G.S. was visiting the University of Jyvaskyla. He thanks the mathematics department at the University of Jyvaskyla for its kind hospitality. E.L.D. acknowledges the support of the Academy of Finland project no. 288501. The authors also thank a kind referee for suggesting improvements to the paper.
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Communicated by L. Ambrosio.
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Le Donne, E., Speight, G. Lusin approximation for horizontal curves in step 2 Carnot groups. Calc. Var. 55, 111 (2016). https://doi.org/10.1007/s00526-016-1054-z
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DOI: https://doi.org/10.1007/s00526-016-1054-z