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 $W^{s,p}(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary. The weak forms of the formulae are proved for the range $sp>n-1$, $s>\frac{n-1}{n}$, while the strong versions are proved for the range $sp\ge n$, $s\ge \frac{n}{n+1}$. We also provide a chain rule for distributional Jacobian determinants of H"older functions and point out its relation to two open problems in geometric analysis.