THE REGULARITY OF INVERSES TO SOBOLEV MAPPINGS AND THE THEORY OF HOMEOMORPHISMS

We prove that each homeomorphism phi : D -> D' of Euclidean domains in R-n, n >= 2, belonging to the Sobolev class W-p,loc(1) (D), where p is an element of [1, infinity), and having finite distortion induces a bounded composition operator from the weighted Sobolev space L-p(1)(D'; omega) into L-p(1)(D) for some weight function omega : D -> (0, infinity). This implies that in the cases p > n-1 and n >= 3 as well as p >= 1 and n >= 2 the inverse phi(-1) : D' -> D belongs to the Sobolev class W-1,loc(1)(D'), has finite distortion, and is differentiable H-n-almost everywhere in D'. We apply this result to Q(q,p)-homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of Q(q,p)-homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes.