Regular Article
Compensated Compactness, Paracommutators, and Hardy Spaces

https://doi.org/10.1006/jfan.1997.3125Get rights and content
Under an Elsevier user license
open archive

Abstract

LetB1: Rn×RN1Rm1,B2: Rn×RN2Rm2andQ: Rm2Rm1be bilinear forms which are related as follows: ifμandνsatisfyB1(ξ, μ)=0 andB2(ξ, ν)=0 for someξ≠0, thenμτ=0. Supposep−1+q−1=1. Coifman, Lions, Meyer and Semmes proved that, ifuLp(Rn) andvLq(Rn), and the first order systemsB1(D, u)=0,B2(D, v)=0 hold, thenuτQvbelongs to the Hardy spaceH1(Rn), provided that both (i)p=q=2, and (ii) the ranks of the linear mapsBj(ξ, ·) : RNjRm1are 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 whenp=q=2 involves the use of a deep result of Lojasiewicz from singularity theory.

Cited by (0)

The authors were supported by the Australian Government through the Australian Research Council.