${\cal H}^1$-estimates of Jacobians by subdeterminants

Abstract.

Let $f:\Omega \rightarrow{\Bbb R}^n$ be a mapping in the Sobolev space $W^{1,n-1}_{loc}(\Omega,{\Bbb R}^n), n\geq 2$. We assume that the cofactors of the differential matrix Df(x) belong to $L^\frac{n}{n-1}(\Omega)$. Then, among other things, we prove that the Jacobian determinant detDf lies in the Hardy space ${\cal H}^1(\Omega)$.