Abstract
We study the behavior of Sobolev functions and mappings on the Carnot groups with the left invariant sub-Riemannian metric. We obtain some sufficient conditions for a Sobolev function to be locally Hölder continuous (in the Carnot–Carathéodory metric) on almost every hypersurface of a given foliation. As an application of these results we show that a quasimonotone contact mapping of class \( W^{1,\nu} \) of Carnot groups is continuous, \( \mathcal{P} \)-differentiable almost everywhere, and has the \( \mathcal{N} \)-Luzin property.
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Notes
In view of the properties of capacity zero sets (see [17, Proposition 5]) we have that \( S(x,r)=\{y\in\Omega:\rho(x,y)=r\}=\{y\in\widetilde{\Omega}:\rho(x,y)=r\} \) for all \( x\in\Omega \) and for almost all \( r\in(0,\operatorname{dist}(x,\partial\Omega)) \).
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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. 4, pp. 700–719. https://doi.org/10.33048/smzh.2023.64.404
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Basalaev, S.G., Vodopyanov, S.K. Hölder Continuity of the Traces of Sobolev Functions to Hypersurfaces in Carnot Groups and the \( \mathcal{P} \)-Differentiability of Sobolev Mappings. Sib Math J 64, 819–835 (2023). https://doi.org/10.1134/S0037446623040043
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DOI: https://doi.org/10.1134/S0037446623040043