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\( BV \)-Spaces and the Bounded Composition Operators of \( BV \)-Functions on Carnot Groups

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Abstract

Under study are the homeomorphisms that induce the bounded composition operators of \( BV \)-functions on Carnot groups. We characterize continuous \( BV_{\operatorname{loc}} \)-mappings on Carnot groups in terms of the variation on integral lines and estimate the variation of the \( BV \)-derivative of the composition of a \( C^{1} \)-function and a continuous \( BV_{\operatorname{loc}} \)-mapping.

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Notes

  1. The precise description of the \( BV \)-derivative of composition is available for Euclidean spaces; see [9, 14].

  2. The finiteness of distortion is introduced for mappings of Sobolev class: \( \varphi\in W^{1}_{p,\operatorname{loc}}(𝔾,𝔾) \) has finite distortion whenever on the zero set \( Z \) of the Jacobian we have \( D\varphi(x)=0 \) for almost all \( x\in Z \). For more detail see [1], for instance.

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Acknowledgment

The author is grateful to S.K. Vodopyanov for his attention during the preparation of this article.

Funding

The work was supported by the Mathematical Center in Akademgorodok under Agreement no. 07–15–2022–281 with the Ministry of Science and Higher Education of the Russian Federation.

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    D. A. Sboev

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  1. D. A. Sboev
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Correspondence to D. A. Sboev.

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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1304–1326. https://doi.org/10.33048/smzh.2023.64.614

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Sboev, D.A. \( BV \)-Spaces and the Bounded Composition Operators of \( BV \)-Functions on Carnot Groups. Sib Math J 64, 1420–1438 (2023). https://doi.org/10.1134/S0037446623060149

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  • DOI: https://doi.org/10.1134/S0037446623060149

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