Pointwise descriptions of nearly incompressible vector fields with bounded curl

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