Compensated compactness and Hardy spaces. (English) Zbl 0864.42009

The following sample results are, probably, not the most important (and, surely, not the most general or difficult ones) in this very interesting paper. They are choosen only to allow the reader to feel the flavor of the topic.
(a) If \(u\in W^{1,N} (\mathbb{R}^N)^N\) (i.e., \(u\) is a vector-valued function with first derivatives in \(L^1)\), then \(\text{det} (\nabla u)\in {\mathcal H}^1 (\mathbb{R}^N)\).
(b) If \(u\in L^p(\mathbb{R}^N)^N\), \(v\in L^q(\mathbb{R}^N)^N\) \((p^{-1} +q^{-1}=1)\) and \(\text{div} u=0\), \(\text{curl} v=0\), then \(\langle u,v \rangle \in {\mathcal H}^1(\mathbb{R}^N)\).
(c) Any function \(f\in{\mathcal H}^1(\mathbb{R}^N)\) can be written in the form \(f=\sum\lambda_k\langle u_k, v_k \rangle\), where \(u_k\) and \(v_k\) are as \(u\) and \(v\) in (b) and of uniformly bounded norms (moreover, \(p=q=2)\), and \(\sum|\lambda_k|<\infty\).
In general, the authors deal with various nonlinear expressions arising in the so-called compensated compactness theory and show that these quantities belong to the real-variable Hardy classes \({\mathcal H}^p (\mathbb{R}^N)\) with certain \(p\leq 1\). Related weak convergence questions are presented. The emphasis is made on the cancellation properties lying beyond the phenomena in question.


42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
35R99 Miscellaneous topics in partial differential equations