Higher order Whitney extension and Lusin approximation for horizontal curves in the Heisenberg group (Q6573745)

The paper contains four main results about ``smooth extensions or approximations'' of curves in the Heisenberg group \(\mathbb{H}\) in the Whitney-Lusin spirit. For each of the four results, we will explain below what kind of extensions and approximations we are referring to. Recall that in their classical formulations:\N\begin{itemize}\N\item Whitney's theorem characterizes those continuous functions \((f_k)_{k\le m}\) defined on a compact set of \(K\subset\mathbb{R}\) for which there exists a \(C^m\) extension \(F:\mathbb{R}\to \mathbb{R}\) with \(D^kF_{|K}=f_k\) for indices \(k\le m\).\N\item Lusin's theorem asserts that given a measurable map \(f:\mathbb{R}\to \mathbb{R}\) and an \(\varepsilon>0\), there exists a continuous function \(F:\mathbb{R}\to \mathbb{R}\) such that the Lebesgue measure of the set \(\{x\in \mathbb{R} \ : \ f(x)\neq F(x)\}\) is less than \(\varepsilon\).\N\end{itemize}\N\NSeveral generalizations of these results are available in the literature, which usually consider different spaces as domain and codomain of the functions and/or require a different regularity on \(f\) and \(F\).\N\NThe present paper deals with functions from a subset of the real line to the Heisenberg group \(\mathbb{H}\) (only the first Heisenberg group is considered, but analogues of the results are expected to hold for the Heisenberg group of any dimension). A particular role is played by the horizontal curves, namely the absolutely continuous curves in the Heisenberg group which are almost everywhere tangent to the first layer of the stratification of the associated algebra of left invariant vector fields.\N\NThe four main results of the paper are:\N\begin{enumerate}\N\item[(1)] An answer to the following question: Given a compact set \(K\subset\mathbb{R}\) and a continuous map \(\gamma : K \to \mathbb{H}\), when must there be a horizontal \(C^{m,\omega}\) curve \(\Gamma\) with \(\Gamma_{|K} = \gamma\)? We recall that a \(C^{m,\omega}\) curve is an \(m\)-times differentiable curve with the \(m\)-th order partial derivative uniformly continuous with modulus of continuity \(\omega\).\N\item[(2)] An \((m, \omega)\)-Lusin property for horizontal curves in \(\mathbb{H}\), namely the authors show that every suitable horizontal curve is nearly a \(C^{m,\omega}\) curve in the sense of measure.\N\item[(3)] A Whitney extension theorem for \(C^{\infty}\) horizontal curves in the Heisenberg group.\N\item[(4)] A Lusin approximation result giving conditions under which a horizontal curve in \(\mathbb{H}\) can be approximated by a \(C^{\infty}\) horizontal curve in the sense of measure.\N\end{enumerate}