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Riemannian approximation in Carnot groups

Part of: Close-to-elliptic equations and systems Global differential geometry

Published online by Cambridge University Press:  06 September 2021

András Domokos Open the ORCID record for András Domokos [Opens in a new window] ,
Juan J. Manfredi Open the ORCID record for Juan J. Manfredi [Opens in a new window]  and
Diego Ricciotti Open the ORCID record for Diego Ricciotti [Opens in a new window]
András Domokos
Affiliation:
Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, CA 95819, USA (domokos@csus.edu)
Juan J. Manfredi
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (manfredi@pitt.edu)
Diego Ricciotti
Affiliation:
Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, CA 95819, USA (ricciotti@csus.edu)
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Abstract

We present self-contained proofs of the stability of the constants in the volume doubling property and the Poincaré and Sobolev inequalities for Riemannian approximations in Carnot groups. We use an explicit Riemannian approximation based on the Lie algebra structure that is suited for studying nonlinear subelliptic partial differential equations. Our approach is independent of the results obtained in [11].

Keywords

Carnot groupRiemannian approximationPoincaré and Sobolev inequalities

MSC classification

Primary: 53C17: Sub-Riemannian geometry
Secondary: 35H20: Subelliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 5 , October 2022 , pp. 1139 - 1154
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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