Sobolev Homeomorphisms and Composition Operators

Abstract

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 $D^{\prime }$ generates by the composition rule ${\phi }^{\ast }f=f\circ \phi$ a bounded composition operator of Sobolev spaces ${\phi }^{\ast }:L_{\infty }^1(D^{\prime })\to L_p^1(D)$, $p>n-1$, has finite distortion and Luzin $N$-property then its inverse ${\phi }^{-1}$ generates the bounded composition operator from $L_{p^{\prime }}^1(D)$, $p^{\prime }=p/(p-n+1)$, into $L_1^1(D^{\prime })$.