Hölder maps under Pfaffian constraints

Abstract

Given a one form $\lambda$ in ${\mathbb{R}}^N$ and $f:{\mathbb{S}}^n\to {\mathbb{R}}^N$ with $f^{\ast }\lambda =0$ we discuss the maximal Hölder regularity of extensions $F:{\mathbb{B}}^{n+1}\to {\mathbb{R}}^N$ such that $F^{\ast }\lambda =0$ in distributional sense. Our analysis applies to the Heisenberg groups ${\mathbb{H}}_n$. It implies in particular that for all $n\ge 1$ any smooth horizontal map $f:{\mathbb{S}}^n\to {\mathbb{H}}_n$ can be extended to a $C^{\alpha }$-map $F:{\mathbb{B}}^{n+1}\to {\mathbb{H}}_n$ for some $\alpha >1/2$. Moreover, if $n\ge 3$ we find $C^{\alpha }$-embeddings from ${\mathbb{B}}^{n+1}$ into ${\mathbb{H}}_n$ for some $\alpha >\frac{1}{2}$. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find $C^{\alpha }$-embeddings from ${\mathbb{B}}^2$ into ${\mathbb{H}}_1$ for some $\alpha >\frac{1}{2}$, assisted by GPT-6 Astra.

added comment about the Gromov conjecture