Pointwise descriptions of nearly incompressible vector fields with bounded curl

Abstract

Among those nearly incompressible vector fields $\mathbf{v}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with $|x| \log |x|$ growth at infinity, we give a pointwise characterization of the ones for which curl $\mathbf{v}=D \mathbf{v}-D^t \mathbf{v}$ belongs to $L^{\infty}$. When $n=2$ we can go further and describe, still in pointwise terms, the vector fields $\mathbf{v}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ for which $|\operatorname{div} \mathbf{v}|+|\operatorname{curl} \mathbf{v}| \in L^{\infty}$.