Sobolev Homeomorphisms and Composition Operators

We study invertibility of bounded composition operators of Sobolev spaces. The problem is closely connected with the theory of mappings of finite distortion. If a homeomorphism $\varphi$ of Euclidean domains $D$ and $D'$ generates by the composition rule $\varphi^{\ast}f=f\circ\varphi$ a bounded composition operator of Sobolev spaces $\varphi^{\ast}: L^1_{\infty}(D')\to L^1_p(D)$, $p>n-1$, has finite distortion and Luzin $N$-property then its inverse $\varphi^{-1}$ generates the bounded composition operator from $L^1_{p'}(D)$, $p'=p/(p-n+1)$, into $L^1_{1}(D')$.