Compensated compactness: Continuity in optimal weak topologies

For l-homogeneous linear differential operators A of constant rank, we study the implication vj⇀v in X and Avj→Av in W−lY implies F(vj)⇝F(v) in Z, where F is an A-quasiaffine function and ⇝ denotes an appropriate type of weak convergence. Here Z is a local L1-type space, either the space M of measures, or L1, or the Hardy space H1; X,Y are Lp-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 Hp-bounds for the sequence (F(vj))j, for appropriate p<1, and new convergence results in the dual of Hölder spaces when (vj) is A-free and lies in a suitable negative order Sobolev space W−β,s. The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.