Vodopyanov, S. K. 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. Reviewer: Vyron Vellis (Storrs) Cited in 6 Documents 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 Keywords:Sobolev space; change of variables; Riemannian manifold PDFBibTeX XMLCite \textit{S. K. Vodopyanov}, Dokl. Math. 93, No. 3, 318--321 (2016; Zbl 1365.46030); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 468, No. 6, 609--613 (2016) Full Text: DOI References: [1] Vodop’yanov, S. K.; Gol’dshtein, V. M., No article title, Sib. Math. J., 16, 174-189 (1975) · Zbl 0324.46040 · doi:10.1007/BF00967502 [2] Romanov, A. S., On change of variables in Bessel and Riesz potential spaces, 117-133 (1985), Novosibirsk · Zbl 0584.46022 [3] Vodop’yanov, S. K., Potential Lp-theory and quasiconformal mappings on homogeneous groups, 45-89 (1989), Novosibirsk · Zbl 0714.31005 [4] Vodop’yanov, S. K., No article title, Contemp. Math., 382, 327-342 (2005) [5] Vodop’yanov, S. K.; Evseev, N. A., No article title, Sib. Math. J., 55, 817-848 (2014) · Zbl 1322.46025 · doi:10.1134/S0037446614050048 [6] Reshetnyak, Y. G., No article title, Sib. Math. J., 38, 567-582 (1997) · Zbl 0944.46024 · doi:10.1007/BF02683844 [7] G. Federer, Geometric Measure Theory (Springer, Berlin, 1996). · Zbl 0874.49001 · doi:10.1007/978-3-642-62010-2 [8] Vodop’yanov, S. K.; Gol’dshtein, V. M., No article title, Sib. Math. J., 18, 35-50 (1977) · Zbl 0409.46032 · doi:10.1007/BF00966948 [9] Vodop’yanov, S. K.; Ukhlov, A. D., No article title, Sib. Adv. Math., 14, 78-125 (2004) · Zbl 1089.47027 [10] Choquet, G., No article title, Ann. Inst. Fourier (Grenoble), 9, 83-89 (1959) · Zbl 0093.29701 · doi:10.5802/aif.87 [11] Vodop’yanov, S. K.; Evseev, N. A., No article title, Dokl. Math., 92, 232-236 (2015) [12] Vodop’yanov, S. K., No article title, Izv. Math., 74, 663-689 (2010) · Zbl 1203.46025 · doi:10.1070/IM2010v074n04ABEH002502 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.