Of commutators and Jacobians

TL;DR

This paper explores the prescribed Jacobian equation and its link to commutator boundedness, presenting recent partial results and a simplified approach for the two-dimensional case, advancing understanding in nonlinear PDEs and harmonic analysis.

Contribution

It provides a simplified method for analyzing the Jacobian equation in two dimensions and characterizes commutator boundedness in new exponent regimes, extending previous results.

Findings

Partial results on Jacobian equation solvability for general data.

Complete characterization of commutator boundedness in new exponent regimes.

Simplified approach for the two-dimensional case using Lindberg's framework.

Abstract

I discuss the prescribed Jacobian equation $Ju=\det\nabla u=f$ for an unknown vector-function $u$, and the connection of this problem to the boundedness of commutators of multiplication operators with singular integrals in general, and with the Beurling operator in particular. A conjecture of T. Iwaniec regarding the solvability for general datum $f\in L^p(R^d)$ remains open, but recent partial results in this direction will be presented. These are based on a complete characterisation of the $L^p$-to-$L^q$ boundedness of commutators, where the regime of exponents $p>q$, unexplored until recently, plays a key role. These results have been proved in general dimension $d\geq 2$ elsewhere, but I will here present a simplified approach to the important special case $d=2$, using a framework suggested by S. Lindberg.