Of Commutators and Jacobians

Abstract

I discuss the prescribed Jacobian equation Ju= det ∇ 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∈ Lp(ℝd) remains open, but recent partial results in this direction will be presented. These are based on a complete characterisation of the Lp-to-Lq 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 ≥ 2 elsewhere, but I will here present a simplified approach to the important special case d = 2, using a framework suggested by S. Lindberg.