Mappings of finite signed distortion: Sobolev spaces and composition of mappings

We study the optimal conditions on a homeomorphism f:Ω→Rn which guarantee that the composition u∘f belongs to the Sobolev space W1,p for every u∈W1,q. To prove it we characterize when the inverse mapping f−1 maps sets of measure zero onto sets of measure zero (satisfies the Luzin (N−1) condition).