Compensated Compactness, Paracommutators, and Hardy Spaces

✍ Scribed by C. Li; A. McIntosh; K. Zhang; Z. Wu

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 384 KB
• Volume: 150
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3125

✦ Synopsis

1

be bilinear forms which are related as follows: if + and & satisfy B 1 (!, +)=0 and B 2 (!, &)=0 for some !{0, then + { Q&=0. Suppose p &1 +q &1 =1. Coifman, Lions, Meyer and Semmes proved that, if u # L p (R n ) and v # L q (R n ), and the first order systems B 1 (D, u)=0, B 2 (D, v)=0 hold, then u { 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 (!, } ): R N j Ä R m 1 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.

1997 Academic Press

1. Introduction

Recent discoveries tie the weak continuity of various nonlinear quantities in compensated compactness with the theory of harmonic analysis, showing that many of these quantities are in fact in well-known Hardy spaces. We refer readers to the paper [CLMS2] for more details. Further related results can be found in [D], [CG], [JJ], [M] and [Zk].

The problem we are concerned with is set up as follows. Let

article no. FU973125