发表机构
University of Pittsburgh; Chulalongkorn University（匹兹堡大学; 朱拉隆功大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究在\(f^\ast \lambda = 0\)条件下，\(F: \mathbb{B}^{n + 1} \to \mathbb{R}^N\)使得\(F^\ast\lambda = 0\)的扩展的最大赫尔德正则性，分析适用于海森堡群，给出光滑水平映射扩展及嵌入的相关结论。
AI 中文摘要
给定\(\mathbb{R}^N\)中的一个形式\(\lambda\)和\(f: \mathbb{S}^{n} \to \mathbb{R}^N\)且\(f^\ast \lambda = 0\)，讨论扩展\(F: \mathbb{B}^{n + 1} \to \mathbb{R}^N\)（使得在分布意义上\(F^\ast\lambda = 0\)）的最大赫尔德正则性。分析适用于海森堡群\(\mathbb{H}_n\)，特别表明对所有\(n \geq 1\)，任何光滑水平映射\(f: \mathbb{S}^{n} \to \mathbb{H}_n\)可扩展为\(C^\alpha\)映射\(F: \mathbb{B}^{n + 1} \to \mathbb{H}_n\)，\(\alpha > 1/2\)。此外，若\(n \geq 3\)，找到\(\alpha > \frac{1}{2}\)的从\(\mathbb{B}^{n + 1}\)到\(\mathbb{H}_n\)的\(C^\alpha\)嵌入。
英文摘要
Given a one form $λ$ in $\mathbb{R}^N$ and $f: \mathbb{S}^{n} \to \mathbb{R}^N$ with $f^\ast λ= 0$ we discuss the maximal Hölder regularity of extensions $F: \mathbb{B}^{n+1} \to \mathbb{R}^N$ such that $F^\astλ= 0$ in distributional sense. Our analysis applies to the Heisenberg groups $\mathbb{H}_n$. It implies in particular that for all $n \geq 1$ any smooth horizontal map $f: \mathbb{S}^{n} \to \mathbb{H}_n$ can be extended to a $C^α$-map $F: \mathbb{B}^{n+1} \to \mathbb{H}_n$ for some $α> 1/2$. Moreover, if $n \geq 3$ we find $C^α$-embeddings from $\mathbb{B}^{n+1}$ into $\mathbb{H}_n$ for some $α> \frac{1}{2}$. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find $C^α$-embeddings from $\mathbb{B}^{2}$ into $\mathbb{H}_1$ for some $α> \frac{1}{2}$, assisted by GPT-6 Astra.
Commentsadded comment about the Gromov conjecture
