Of commutators and Jacobians

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.