Primitives of volume forms in Carnot groups

Abstract

In the Euclidean space, it is known that a function $f\in L^2$ of a ball, with vanishing average, is the divergence of a vector field $F\in L^2$ with $\Vert F{\Vert }_{L^2(B)}\le C\Vert f{\Vert }_{L^2(B)}$. In this note, we prove a similar result in any Carnot group $\mathbb{G}$ for a vanishing average $f\in L^p$, $1\le p<Q$, where $Q$ is the so-called homogeneous dimension of $\mathbb{G}$.