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

TL;DR

This paper establishes a connection between approximate differentiability and Lusin approximation by smooth maps for real-valued functions on Carnot groups, extending classical results to this non-Euclidean setting.

Contribution

It proves that k-approximate differentiability and existence of Taylor polynomials are equivalent to Lusin approximation by smooth or Lipschitz maps on Carnot groups.

Findings

k-approximate differentiability implies Lusin approximation by $C^{k}_{ extbf{G}}$ maps

Existence of approximate (k-1)-Taylor polynomial implies Lusin approximation by Lipschitz maps

Results extend classical Euclidean approximation theorems to 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 $C^{k}_{\mathbb{G}}$ 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.