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arXiv · 1908.07624

A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

Marco Capolli·Andrea Pinamonti·Gareth Speight
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Abstract

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.

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Marco Capolli, Andrea Pinamonti, Gareth Speight. 2019-08-20. A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group. https://doi.org/10.1007/s00526-021-01923-9

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