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 $\phi$ of Euclidean domains $D$ and $D^{\mathrm{\prime }}$ generates by the composition rule ${\phi }^{\ast }f=f\circ \phi$ a bounded composition operator of Sobolev spaces $\varphi^{\ast}: L^1_{\infty}(D’)\to L^1_p(D)$,$p>n-1$,hasfinitedistortionandLuzin$N$-property then its inverse ${\phi }^{-1}$ generates the bounded composition operator from $L_{p^{\mathrm{\prime }}}^1(D)$, $p^{\mathrm{\prime }}=p\mathrm{/}(p-n+1)$, into $L_1^1(D^{\mathrm{\prime }})$.