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

Given a homeomorphism $\varphi\in W^1_M$, we determine the conditions that guarantee the belonging of the inverse of $\varphi$ in some Sobolev–Orlicz space $W^1_F$. 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.