On the Hardy space theory of compensated compactness quantities

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}$^1(\mathbb R^n)$forany$n \ge 2$\mathrm{.}Therelatedquestionaboutsurjectivityof$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.