Pointwise descriptions of nearly incompressible vector fields with bounded curl

Abstract

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