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 $f\colon K \to \mathbb{H}$, when is there a horizontal $C^m$ curve $F\colon \mathbb{R} \to \mathbb{H}$ 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.