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

TL;DR

This paper proves that the BMO-norm of divergence-free vector fields in three dimensions is equivalent to the operator norm of a related paracommutator, linking harmonic analysis and PDEs.

Contribution

It establishes the equivalence of BMO-norms and paracommutator operator norms for divergence-free vector fields, extending previous results to a new setting.

Findings

BMO-norm of divergence-free fields can be estimated via associated paracommutators.

The paracommutator is essentially a pseudodifferential operator with a symbol depending linearly on the vector field.

The result provides an equivalent norm characterization in the space of divergence-free fields.

Abstract

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 $\Psi$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.