Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces

We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants Ju for functions u in fractional Sobolev spaces Ws,p(Ω), where Ω is a bounded domain in Rn with smooth boundary. The weak forms of the formulae are proved for the range sp>n−1, s≥n−1n, while the strong versions are proved for the range sp>n, s≥nn+1. We also provide a chain rule for the distributional Jacobian determinant of Hölder functions and point out its relation to two open problems in geometric analysis.