Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group
Scott Zimmerman · Revista Matemática Iberoamericana · 2022
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.