Regularity of mappings inverse to Sobolev mappings

For homeomorphisms $\varphi\colon\Omega\to \Omega'$ on Euclidean domains in $\mathbb R^n$, $n\geqslant2$, necessary and sufficient conditions ensuring that the inverse mapping belongs to a Sobolev class are investigated. The result obtained is used to describe a new two-index scale of homeomorphisms in some Sobolev class such that their inverses also form a two-index scale of mappings, in another Sobolev class. This scale involves quasiconformal mappings and also homeomorphisms in the Sobolev class $W^1_{n-1}$ such that $\operatorname{rank}D\varphi(x)\leqslant n-2$ almost everywhere on the zero set of the Jacobian $\det D\varphi(x)$.
Bibliography: 65 titles.