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.
"Three Examples of Sharp Commutator Estimates via Harmonic Extensions." Taiwanese J. Math. 23 (6) 1365 - 1388, December, 2019. https://doi.org/10.11650/tjm/190204