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We prove weak and strong versions of the coarea formula and the chain rulefor distributional Jacobian determinants $Ju$ for functions $u$ in fractionalSobolev spaces $W^{s,p}(\Omega)$, where $\Omega$ is a bounded domain in$\mathbb{R}^n$ with smooth boundary. The weak forms of the formulae are provedfor the range $sp>n-1$, $s> \frac{n-1}{n}$, while the strong versions areproved for the range $sp\geq n$, $s\geq \frac{n}{n+1}$. We also provide a chainrule for distributional Jacobian determinants of H\"older functions and pointout its relation to two open problems in geometric analysis.

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