Compensated compactness: Continuity in optimal weak topologies

For $l$-homogeneous linear differential operators $\mathcal{A}$ of constant rank, we study the implication $v_j\rightharpoonup v$ in $X$ and $\mathcal{A} v_j\rightarrow \mathcal{A} $W^{-l}Y$ implies $F(v_j)\rightsquigarrow F(v)$ $Z$, where $F$ is an $\mathcal{A}$-quasiaffine function $\rightsquigarrow$ denotes appropriate type weak convergence. Here $Z$ a local $L^1$-type space, either space $\mathscr{M}$ measures, or $L^1$, Hardy $\mathscr{H}^1$; $X,\, Y$ are $L^p$-type spaces, by which mean Lebesgue Zygmund spaces. Our conditions for each choice $X,\,Y,\,Z$ sharp. Analogous statements also given case when $F(v)$ not locally integrable it instead defined as distribution. In this case, prove $\mathscr{H}^p$-bounds sequence $(F(v_j))_j$, $p<1$, new convergence results dual H\"older spaces $(v_j)$ $\mathcal{A}$-free lies suitable negative order Sobolev $W^{-\beta,s}$. The these sharp, shown construction explicit counterexamples. Some even distributional Jacobians.