  Recently, Lenzmann and the author observed how to obtain a large class of
sharp commutator estimates by a combination of an integration by parts, an
harmonic extension, and trace space estimates. In this survey we review this
approach in three concrete examples: the Jacobian estimate by
Coifman-Lions-Meyer-Semmes, the Coifman-Rochberg-Weiss commutator estimate for
Riesz transforms, and a Kato-Ponce-Vega-type inequality.
