Jacobians and Hardy spaces (Q1315159)

We wish to illustrate here the links between various nonlinear quantities identified by the compensated-compactness method and the Hardy spaces. A systematic study of this matter can be found in \textit{R. R. Coifman}, \textit{P. L. Lions}, \textit{Y. Meyer} and \textit{S. Semmes} [C. R. Acad. Sci., Paris, Sér. I 309, No. 18, 945-949 (1989; Zbl 0684.46044)], and we want to illustrate it on the particular example of the Jacobian. More precisely, we consider \(J(u)= \text{det} (\nabla u)\) when \(u\in W^{1,p} (\mathbb{R}^N)^N\) for some \(p\in [1,\infty ]\). This quantity clearly makes sense in \(L^1 (\mathbb{R}^N)\) if \(p= N\); however, in that case, there are various reasons to guess that \(J(u)\) might be slightly better than \(L^1\). First of all, it is a classical fact that \(J\) is weakly sequentially continuous on \(W^{1, N} (\mathbb{R}^N )^N\). This fact is one of the key ingredients in J. Ball's theory of polyconvex functionals in nonlinear elasticity. Next, Several results by H. Wente, L. Tartar indicate that \(L^1\) is not optimal. Finally, the last piece of evidence is a striking recent result due to \textit{S. Müller} [Bull. Am. Math. Soc., New Ser. 21, No. 2, 245-248 (1989; Zbl 0689.49006)]\ showing that if \(u\in W^{1,N}_{\text{loc}} (\mathbb{R}^N )^N\) and \(J(u)\geq 0\) a.e. then \[ J(u) \log(1+ J(u))\in L^1_{\text{loc}} (\mathbb{R}^N). \] We will show below in section 2 how all this results can be recovered from the following statement: \(J(u)\in {\mathcal H}^1 (\mathbb{R}^N)\) if \(u\in W^{1,N} (\mathbb{R}^N)^N\).