On the equivalence of the BMO-norm of divergence-free vector fields and norm of related paracommutators
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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{\mathbb{R}^{3}}$</tex> in terms of the operator norm of an associated paracommutator. The latter is essentially a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{\Psi\text{DO}}$</tex> (bounded in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{L_{2}(\mathbb{R}^{3};\mathbb{C}^{3})}$</tex> ), whose symbol depends linearly on the vector field. Together with …