Compensated compactness, paracommutators, and hardy spaces

Let B-1:R-n x R-N1 --> R-m1, B-2:R-n x R-N2 --> R-m2 and Q:R-m2 --> R-m1 be bilinear forms which are related as follows: if mu and nu satisfy B-1(xi, mu) = 0 and B-2(xi, nu) = 0 for some xi not equal 0, then mu(tau)Q nu = 0. Suppose p(-1) + q(-1) = 1. Coifman, Lions, Meyer and Semmes proved that, if u is an element of L-p(R-n) and v is an element of L-q(R-n), and the first order systems B-1(D, u) = 0, B-2(D, v) = 0 hold, then u(tau)Qv belongs to the Hardy space H-1(R-n), provided that both (i) p = q = 2, and (ii) the ranks of the linear maps B-j(xi,.):R-Nj --> R-m1 are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition when p = q = 2 involves the use of a deep result of Lojasiewicz from singularity theory. (C) 1997 Academic Press.