  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.

Functional Analysis;Classical Analysis and ODEs