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Composition Operators on Weighted Sobolev Spaces and the Theory of \({{\mathcal{Q}}_{p}}\)-Homeomorphisms


We define the scale \({{\mathcal{Q}}_{p}}\), \(n - 1 < p < \infty \), of homeomorphisms of spatial domains in \({{\mathbb{R}}^{n}}\), a geometric description of which is due to the control of the behavior of the p-capacity of condensers in the image through the weighted p-capacity of the condensers in the preimage. For p = n the class \({{\mathcal{Q}}_{n}}\) of mappings contains the class of so-called Q-homeomorphisms, which have been actively studied over the past 25 years. An equivalent functional and analytic description of these classes \({{\mathcal{Q}}_{p}}\) is obtained. It is based on the problem of the properties of the composition operator of a weighted Sobolev space into a nonweighted one induced by a map inverse to some of the class \({{\mathcal{Q}}_{p}}\).

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  1. Here, \({\text{adj}}A = \{ {{A}_{{ji}}}\} \) is the adjoint of a matrix \(A = \{ {{a}_{{ij}}}\} \), \(i,j = 1, \ldots ,n\); Aji is the cofactor of the element aij of A.


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This work was supported by the Mathematical Center in Akademgorodok with the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2019-1613.

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Correspondence to S. K. Vodopyanov.

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Translated by I. Ruzanova

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Vodopyanov, S.K. Composition Operators on Weighted Sobolev Spaces and the Theory of \({{\mathcal{Q}}_{p}}\)-Homeomorphisms. Dokl. Math. 102, 371–375 (2020).

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  • Sobolev space
  • composition operator
  • quasiconformal analysis
  • capacity estimate