# Three examples of Sharp Commutator Estimates via Harmonic Extensions

## TL;DR

This paper reviews a harmonic extension approach to derive sharp commutator estimates, illustrating its application through three key examples in harmonic analysis and PDEs.

## Contribution

It introduces a unified method using harmonic extensions to obtain sharp commutator estimates, demonstrated through three classical examples.

## Key findings

- Unified harmonic extension approach simplifies derivation of sharp estimates
- Provides new insights into classical commutator inequalities
- Enhances understanding of trace space techniques in harmonic analysis

## Abstract

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.