On the equivalence of the BMO-norm of divergence-free vector fields and norm of related paracommutators

We establish an estimate of the BMO-norm of a divergence-free vector field in ${\mathbb{R}}^3$ in terms of the operator norm of an associated paracommutator. The latter is essentially a $\mathrm{\Psi }$DO, whose symbol depends linearly on the vector field. Together with the result of P.~Auscher and M.~Taylor concerning the converse estimate, this provides an equivalent norm in the space of divergence-free fields from BMO.