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

TL;DR

This paper investigates the mapping properties of the Jacobian operator and similar quantities, providing an axiomatic approach for quadratic operators and advancing understanding of the Jacobian equation in the plane.

Contribution

It introduces a Banach space geometric framework for the Jacobian problem and makes progress on the open case in two dimensions.

Findings

Axiomatic approach to quadratic operators

Progress on the Jacobian equation in the plane

Insights into the Jacobian operator's mapping properties

Abstract

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.