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

Abstract

We establish an estimate of the BMO-norm of a divergence-free vector field in $\boldsymbol{\mathbb{R}^{3}}$ in terms of the operator norm of an associated paracommutator. The latter is essentially a $\boldsymbol{\Psi\text{DO}}$ (bounded in $\boldsymbol{L_{2}(\mathbb{R}^{3};\mathbb{C}^{3})}$ ), 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.