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On admissible changes of variables for Sobolev functions on (sub)Riemannian manifolds. (English. Russian original) Zbl 1365.46030

Dokl. Math. 93, No. 3, 318-321 (2016); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 468, No. 6, 609-613 (2016).
The article under review is concerned with the following change of variables question: Which mappings \(\phi\) between metric spaces \(X\) and \(Y\) induce isomorphisms between the Sobolev spaces \(W^{1,p}(X)\) and \(W^{1,p}(Y)\) through composition? While this question has been settled in the case that \(X\) and \(Y\) are domains of some \(\mathbb{R}^n\), the author of this article presents a characterization in the case that \(X\) and \(Y\) are domains of some Riemannian \(n\)-manifold.
The main results are Theorem 1 and Theorem 2. In the case \(p=n\), a mapping \(\phi\) has the desired property if and only if \(\phi\) coincides with a quasiconformal homeomorphism almost everywhere. In the case \(p\neq n\), \(\phi\) has the desired property if and only if \(\phi\) coincides with a quasi-isometry (i.e., a homeomorphism that resembles bi-Lipschitz on large scale) almost everywhere.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
58C07 Continuity properties of mappings on manifolds
47B33 Linear composition operators
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