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

  • Scott Zimmerman

    The Ohio State University at Marion, USA
Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group cover
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Abstract

Consider the sub-Riemannian Heisenberg group H\mathbb{H}. In this paper, we answer the following question: given a compact set KRK \subseteq \mathbb{R} and a continuous map f ⁣:KHf\colon K \to \mathbb{H}, when is there a horizontal CmC^m curve F ⁣:RHF\colon \mathbb{R} \to \mathbb{H} such that FK=fF|_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 Rn\mathbb{R}^n. We also prove a finiteness principle for Cm,ωC^{m,\sqrt{\omega}} horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman.

Cite this article

Scott Zimmerman, Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group. Rev. Mat. Iberoam. (2022),

DOI 10.4171/RMI/1339