On the Hardy space theory of compensated compactness quantities

TL;DR

This paper advances the understanding of the Hardy space theory related to the Jacobian operator, showing it does not map certain Sobolev spaces onto the Hardy space, and explores the regularity of compensated compactness quantities and nonlinear PDE operators.

Contribution

It proves the non-surjectivity of the Jacobian operator from Sobolev spaces to Hardy spaces and links $ ext{H}^1$ regularity of compensated compactness quantities to simple arithmetic, also analyzing nonlinear PDE operators.

Findings

Jacobian operator does not map $W^{1,n}$ onto $ ext{H}^1$ for $n \\ge 2$

Reduced $ ext{H}^1$ regularity of compactness quantities to basic arithmetic

Identified nonlinear PDE operators with stronger weak continuity than $ ext{H}^1$ regularity

Abstract

We make progress on a problem of R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes from 1993 by showing that the Jacobian operator $J$ does not map $W^{1,n}(\mathbb R^n,\mathbb R^n)$ onto the Hardy space $\mathcal{H}^1(\mathbb R^n)$ for any $n \ge 2$. The related question about surjectivity of $J \colon \dot{W}^{1,n}(\mathbb R^n,\mathbb R^n) \to \mathcal{H}^1(\mathbb R^n)$ is still open. The second main result and its variants reduce the proof of $\mathcal{H}^1$ regularity of a large class of compensated compactness quantities to an integration by parts or easy arithmetic, and applications are presented. Furthermore, we exhibit a class of nonlinear partial differential operators in which weak sequential continuity is a strictly stronger condition than $\mathcal{H}^1$ regularity, shedding light on another problem of Coifman, Lions, Meyer, and Semmes.