A remark on the equality \(\text{det} Df=\text{Det} Df\) (Q1308517)

A class of functions mapping an open subset \(\Omega\) of \(\mathbb{R}^ N\) into \(\mathbb{R}^ N\) is introduced, which is larger than \(W^{1,N}(\Omega,\mathbb{R}^ N)\) and smaller than each \(W^{1,p}(\Omega,\mathbb{R}^ N)\), with \(p<N\). For the functions of this class equality between the pointwise Jacobian and the so-called weak determinant introduced by J. Ball is proved.