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

We study the optimal conditions on a homeomorphism f : Omega -> R(n) which guarantee that the composition u o f belongs to the Sobolev space W(1,p) for every u is an element of 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). (C) 2011 Elsevier Inc. All rights reserved.