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

We study the optimal conditions on a homeomorphism f : Ω → R n which guarantee that the composition u ∘ f belongs to the Sobolev space W 1 , p for every u ∈ W 1 , 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).