  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
$|\operatorname{div}{\bf{v}}|+|\operatorname{curl}{\bf{v}}|\in L^\infty$.
