Compensated compactness: Continuity in optimal weak topologies

For l-homogeneous linear differential operators A of constant rank, we study the implicationvj⇀vinXAvj→AvinW−lY}⇒F(vj)⇝F(v)inZ, 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