Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ϕ ∈ W_M ¹, we determine the conditions that guarantee the belonging of the inverse of ϕ in some Sobolev–Orlicz space W_F ¹. We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.