Sobolev Homeomorphisms and Composition Operators

Details Sobolev Homeomorphisms and Composition Operators Sobolev Homeomorphisms and Composition Operators

2009-03-21

arxiv.org/abs/0903.3677v1

Mathematics

Complex Variables

Scientific paper

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

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