Hölder maps under Pfaffian constraints

Abstract

Given a one form $\hat{I}»$ in ${\mathbb{R}}^N$ and $f:{\mathbb{S}}^n\to {\mathbb{R}}^N$ with $f^{\ast }\hat{I}»=0$ we discuss the maximal Hölder regularity of extensions $F:{\mathbb{B}}^{n+1}\to {\mathbb{R}}^N$ such that $F^{\ast }\hat{I}»=0$ 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 $C^{\hat{I}}\pm$-map $F:{\mathbb{B}}^{n+1}\to {\mathbb{H}}_n$ for some $\hat{I}\pm >1/2$. Moreover, if $n\ge 3$ we find $C^{\hat{I}}\pm$-embeddings from ${\mathbb{B}}^{n+1}$ into ${\mathbb{H}}_n$ for some $\hat{I}\pm >\frac{1}{2}$.