Heisenberg群中曲线的Whitney扩张定理和有限性原理
Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group
摘要：
考虑次黎曼海森堡群$\mathbb{H}$。本文回答了如下问题：给定紧集$K \subseteq \mathbb{R}$和连续映射$f:K \to \mathbb{H}$，何时存在水平$C^m$曲线$F:\mathbb{R} \to \mathbb{H}$使得$F|_K = f$？惠特尼最初回答了这个问题的实值映射，和Febriman提供了一个完整的答案实值函数定义的子集$\mathbb{R}^n$。在Brudnyi和Shvartsman的意义下，证明了海森堡群中$C^{m,\sqrtω}$水平曲线的有限性原理.
英文摘要：
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 $C^{m,\sqrtω}$ horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman.
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