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DOI: 10.46698/n0927-3994-6949-u
Mollifications of Contact Mappings of Engel Group
Basalaev, S. G.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 1.
Abstract: The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as Holder continuity, differentiability almost everywhere in the sense of Pansu, Luzin \(\mathcal{N}\)-property.
Keywords: Carnot group, Engel group, quasiconformal mappings, bounded distortion
For citation: Basalaev, S. G. Mollifications of Contact Mappings of Engel Group, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp.5-19. DOI 10.46698/n0927-3994-6949-u
1. Dairbekov, N. S. The Morphism Property for Mappings with Bounded
Distortion on the Heisenberg Group, Siberian Mathematical Journal,
1999, vol. 40, no. 4, pp. 682-649. DOI: 10.1007/BF02675669.
2. Dairbekov, N. S. Mappings with Bounded Distortion of Two-Step Carnot Groups,
Proceedings on Analysis and Geometry, ed. S. K. Vodopyanov, Sobolev Institute Press, Novosibirsk,
2000, pp. 122-155.
3. Reshetnyak, Yu. G. Certain Geometric Properties of Functions
and Mappings with Generalized Derivatives, Siberian Mathematical Journal,
1966, vol. 7, no. 4, pp. 704-732. DOI: 10.1007/BF00973267.
4. Reshetnyak, Yu. G. Space Mappings with Bounded Distortion,
Translation of Mathematical Monographs, vol. 73, American Mathematical Society,
Providence, RI, 1989.
5. Vodopyanov, S. K. Foundations of the Theory of Mappings
with Bounded Distortion on Carnot Groups, Contemporary Mathematics, 2007,
vol. 424, pp. 303-344. DOI: 10.13140/RG.2.1.3112.8480.
6. Kleiner, B., Muller, S. and Xie, X. Pansu Pullback and Exterior Differentiation
for Sobolev Maps on Carnot Groups, 2021, arxiv.org/abs/2007.06694v2.
7. Rotschild, L. P. and Stein, E. M.
Hypoelliptic Differential Operators and Nilpotent Groups,
Acta Mathematica, 1976, vol. 137, pp. 247-320.
DOI: 10.1007/BF02392419.
8. Folland, G. B. and Stein, E. M. Hardy Spaces on Homogeneous Groups,
Princeton Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J., 1982.
9. Bonfiglioli, A., Lanconelli, E. and Uguzonni, F.
Stratified Lie Groups and Potential Theory for their Sub-Laplacians,
Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, 2007.
DOI: 10.1007/978-3-540-71897-0.
10. Vodop'yanov, S. K. and Ukhlov, A. D.-O.
Approximately Differentiable Transformations and
Change of Variables on Nilpotent Groups, Siberian Mathematical Journal,
1996, vol. 37, no. 1, pp. 62-78. DOI: 10.1007/BF02104760.
11. Vodop'yanov, S. K. \(\mathcal{P}\)-Differentiability on Carnot Groups
in Different Topologies and Related Topics, Proceedings on Analysis and Geometry,
ed. S. K. Vodopyanov, Sobolev Institute Press, Novosibirsk, 2000.
12. Franchi, B., Lu, G. and Wheeden, R. L.
A Relationship between Poincare-Type
Inequalities and Representation Formulas In Spaces
of Homogeneous Type, International Mathematics Research Notices, 1996,
vol. 1, pp. 1-14. DOI:10.1155/S1073792896000013.
13. Hajasz, P. Sobolev Spaces on an Arbitrary Metric Space,
Potential Analysis, 1996, vol. 5, pp. 403-415. DOI: 10.1007/BF00275475.
14. Vodop'yanov, S. K. Monotone Functions and Quasiconformal Mappings on Carnot Groups,
Siberian Mathematical Journal, 1996, vol. 37, no. 6, pp. 1113-1136. DOI: 10.1007/BF02106736.
15. Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups,
Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York, 1983. DOI: 10.1007/978-1-4757-1799-0.
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