A C^k Lusin Approximation Theorem For Real-Valued Functions on Carnot Groups

Abstract

We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C_{\mathbb{G}}^k$ maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.

31 pages, updated with suggestions from referee