Abstract
We prove that a mapping of finite distortion \( f:\Omega\to 𝔾 \) in a domain \( \Omega \) of an \( H \)-type Carnot group \( 𝔾 \) is continuous, open, and discrete provided that the distortion function \( K(x) \) of \( f \) belongs to \( L_{p,\operatorname{loc}}(\Omega) \) for some \( p>\nu-1 \). In fact, the proof is suitable for each Carnot group provided it has a \( \nu \)-harmonic function of the form \( \log\rho \), where the homogeneous norm \( \rho \) is \( C^{2} \)-smooth.
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Notes
Here and henceforth all functions involved into integration over \( x \) are assumed depending on \( x \).
Otherwise \( B(z,\beta)=f^{-1}(f(S(z,\beta))) \) and \( \det Df(x)=0 \) a.e. in \( B(z,\beta) \) by [19, Proposition 20, formula 18]. Since \( f \) has finite distortion, it follows that \( f=\operatorname{const} \) on \( B(z,\beta) \), which contradicts the assumption.
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Funding
The work was supported by the Mathematical Center in Akademgorodok under the agreement no. 075–15–2022–281 with the Ministry of Science and Higher Education of the Russian Federation.
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1151–1159. https://doi.org/10.33048/smzh.2023.64.604
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Basalaev, S.G., Vodopyanov, S.K. Openness and Discreteness of Mappings of Finite Distortion on Carnot Groups. Sib Math J 64, 1289–1296 (2023). https://doi.org/10.1134/S0037446623060046
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DOI: https://doi.org/10.1134/S0037446623060046