Abstract
We prove the continuity of the mappings with finite distortion of the Sobolev class \( W^{1}_{\nu,\operatorname{loc}} \) on Carnot groups and establish that these mappings are \( \mathcal{P} \)-differentiable almost everywhere and have the Luzin \( \mathcal{N} \)-property.
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Notes
If \( dx \) is the volume form on \( 𝔾 \) of degree \( N \), then \( i(X_{1j}) \) is a degree \( N-1 \) form which on smooth vector fields \( Y_{1},Y_{2},\dots,Y_{N-1} \) defined on \( 𝔾 \) takes the value \( i(X_{1j})(Y_{1},Y_{2},\dots,Y_{N-1})=dx(X_{1j},Y_{1},Y_{2},\dots,Y_{N-1}) \).
Henceforth we denote the Riemannian differential of \( \varphi:\Omega\to 𝔾 \) at \( x \) by \( {\mathcal{D}}\varphi(x) \).
Here we should appreciate Lemma 1 and Property 2 (claims (3) and (4)) in order for the restriction \( \varphi|_{\partial U} \) to be continuous, and for the convergence \( \varphi_{l}|_{\partial U}\to\varphi|_{\partial U} \) as \( l\to\infty \) to be uniform, see Property 2 (claim (6)), as well as the details in [11].
According to the properties of capacity zero sets (see [11, Proposition 5]) we have \( S(x,r)=\{y\in U:\rho(x,y)=r\}=\{y\in\widetilde{U}:\rho(x,y)=r\} \) for all \( x\in U \) and almost all \( r\in\bigl{(}0,\frac{1}{2}\operatorname{dist}(x,\partial U)\bigr{)} \).
The mapping \( \varphi\in W_{\nu}^{1}(\Omega) \) has the Luzin \( N^{-1} \)-property whenever the preimage of each negligible set is negligible. This property is obviously equivalent to the following: The image of each set of nonzero measure is a set of nonzero measure.
At that time this class of mappings was nameless. Its modern name was proposed in [21].
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Funding
This research was carried out in the framework of the State Task of the Ministry of Science and Higher Education of the Russian Federation for the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Project FWNF–2022–0006).
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 5, pp. 912–934. https://doi.org/10.33048/smzh.2023.64.503
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Vodopyanov, S.K. Continuity of the Mappings with Finite Distortion of the Sobolev Class \( W^{1}_{\nu,\operatorname{loc}} \) on Carnot Groups. Sib Math J 64, 1091–1109 (2023). https://doi.org/10.1134/S0037446623050038
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DOI: https://doi.org/10.1134/S0037446623050038