The Regularity of Inverses to Sobolev Mappings and the Theory of -Homeomorphisms.

We prove that each homeomorphism of Euclidean domains in , , belonging to the Sobolev class , where , and having finite distortion induces a bounded composition operator from the weighted Sobolev space into for some weight function . This implies that in the cases and as well as and the inverse belongs to the Sobolev class , has finite distortion, and is differentiable -almost everywhere in . We apply this result to -homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of -homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes. [ABSTRACT FROM AUTHOR]