A note on the Jacobian problem of Coifman, Lions, Meyer and Semmes

Coifman, Lions, Meyer and Semmes asked in 1993 whether the Jacobian operator and other compensated compactness quantities map their natural domain of definition onto the real-variable Hardy space ${\mathcal{H}}^1({\mathbb{R}}^n)$. We present an axiomatic, Banach space geometric approach to the problem in the case of quadratic operators. We also make progress on the main open case, the Jacobian equation in the plane.