Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-01T02:08:41.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Riemannian approximation in Carnot groups

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

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 ([email protected])
Juan J. Manfredi
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA ([email protected])
Diego Ricciotti
Affiliation:
Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, CA 95819, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F., Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics (Berlin: Springer, 2007).Google Scholar
Capogna, L. and Citti, G.. Regularity for subelliptic PDE through uniform estimates in multi-scale geometries. Bull. Math. Sci. 6 (2016), 173–230.CrossRefGoogle Scholar
Capogna, L., Citti, G., Donne, E. L. and Ottazzi, A.. Conformality and q- harmonicity in sub-riemannian manifolds. J. Math. Pure Appl. In press, 2018.10.1016/j.matpur.2017.12.006CrossRefGoogle Scholar
Capogna, L., Citti, G. and Manfredini, M.. Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups. Anal. Geom. Metric Spaces 1 (2013), 255–275.Google Scholar
Capogna, L., Citti, G. and Rea, G.. A subelliptic analogue of Aronson-Serrin's Harnack inequality. Math. Ann. 357 (2013), 1175–1198.10.1007/s00208-013-0937-yCrossRefGoogle Scholar
Domokos, A. and Manfredi, J. J.. $C^{1,\alpha}$-subelliptic regularity on ${\rm SU}(3)$ and compact, semi-simple Lie groups. Anal. Math. Phys. 10 (2020), 1–32.CrossRefGoogle Scholar
Hajłasz, P. and Koskela, P.. Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000), 1–101.Google Scholar
Jerison, D. and Sánchez-Calle, A.. Subelliptic, second order differential operators. In Complex analysis, III (College Park, Md., 1985–86), Lecture Notes in Mathematics, vol. 1277, pp. 46–77 (Berlin: Springer, 1987).10.1007/BFb0078245CrossRefGoogle Scholar
Korányi, A.. Geometric aspects of analysis on the Heisenberg group. In Topics in modern harmonic analysis, vol. I, II (Turin/Milan, 1982), Ist. Naz. Alta Mat., pp. 209–258 (Rome: Francesco Severi, 1983).Google Scholar
Monti, R.. Distances, boundaries and surface measures in carnot-carathéodory spaces. Ph.D. thesis. Department of Mathematics, University of Trento, Italy, 2001.Google Scholar
Nagel, A., Stein, E. M. and Wainger, S.. Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155 (1985), 103–147.10.1007/BF02392539CrossRefGoogle Scholar