Sobolev Homeomorphisms and Composition Operators

TL;DR

This paper investigates the invertibility of bounded composition operators on Sobolev spaces generated by homeomorphisms with finite distortion, establishing conditions under which the inverse also induces a bounded composition operator.

Contribution

It provides new criteria linking the properties of Sobolev homeomorphisms with finite distortion to the boundedness of their inverse composition operators.

Findings

Invertibility of composition operators is characterized for Sobolev homeomorphisms.

Conditions involving finite distortion and Luzin N-property are crucial.

The inverse homeomorphism induces a bounded composition operator under specified integrability conditions.

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 $\varphi$ 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 $L^1_{1}(D')$.