Compensated compactness: continuity in optimal weak topologies

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