Hölder maps under Pfaffian constraints

Abstract

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 $α> \frac{1}{2}$.

Disclosure

“rganization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).” Discussions with Behnam Esmayli are gratefully acknowledged. Part of the work leading to this article is assisted by chatgpt. All mathematical validation, final proof decisions, and final wording remain the sole responsibility of the human author. 2. Preliminaries, Constant Rank theorem and Proofs of Propositions 1.2, 1.3, 1.5”

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2026-08-24T00:28:51.963Z

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