Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group

Abstract

Consider the sub-Riemannian Heisenberg group $\mathbb{H}$. In this paper, we answer the following question: given a compact set $K\subseteq \mathbb{R}$ and a continuous map $f:K\to \mathbb{H}$, when is there a horizontal $C^m$ curve $F:\mathbb{R}\to \mathbb{H}$ such that $F{|}_K=f$? Whitney originally answered this question for real valued mappings, and Fefferman provided a complete answer for real valued functions defined on subsets of ${\mathbb{R}}^n$. We also prove a finiteness principle for $C^{m,\sqrt{\omega }}$ horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman.