The constant in the Sobolev inequality and the boundedness of subelliptic operators on compact Lie groups

Andrea V. Hurtado-Quiceno; Duván Cardona Sánchez; Andrea V. Hurtado-Quiceno; Duván Cardona Sánchez

doi:10.3934/cam.2025035

Communications in Analysis and Mechanics

2025, Volume 17, Issue 4: 878-897. doi: 10.3934/cam.2025035

Previous Article Next Article

Research article Special Issues

The constant in the Sobolev inequality and the boundedness of subelliptic operators on compact Lie groups

Andrea V. Hurtado-Quiceno ¹,
Duván Cardona Sánchez ^{2
,
,}

1.
Department of Mathematics, RPTU Kaiserslautern-Landau, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany
2.
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, B-9000 Ghent, Belgium

Received: 21 January 2025 Revised: 11 September 2025 Accepted: 16 September 2025 Published: 15 October 2025
26D10, 43A80, 46E35

Given a compact connected Lie group $ G $, we estimated the constant for the Sobolev inequality for an arbitrary Hörmander sub-Laplacian on $ G $. The constant was estimated in terms of quantities depending on the intrinsic geometry of the group and on the Lebesgue parameters. We also provided applications of this result to the estimation of the operator norm for $ L^p $-$ L^q $-bounded subelliptic pseudo-differential operators in subelliptic Hörmander classes.
- compact Lie groups,
- Sobolev inequality,
- subelliptic pseudo-differential operators
Citation: Andrea V. Hurtado-Quiceno, Duván Cardona Sánchez. The constant in the Sobolev inequality and the boundedness of subelliptic operators on compact Lie groups[J]. Communications in Analysis and Mechanics, 2025, 17(4): 878-897. doi: 10.3934/cam.2025035

Related Papers:

Abstract

Given a compact connected Lie group $ G $, we estimated the constant for the Sobolev inequality for an arbitrary Hörmander sub-Laplacian on $ G $. The constant was estimated in terms of quantities depending on the intrinsic geometry of the group and on the Lebesgue parameters. We also provided applications of this result to the estimation of the operator norm for $ L^p $-$ L^q $-bounded subelliptic pseudo-differential operators in subelliptic Hörmander classes.

References

[1]	N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1993. https://doi.org/10.1017/CBO9780511662485
[2]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353–372. https://doi.org/10.1007/BF02418013 doi: 10.1007/BF02418013
[3]	Th. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573–598. https://doi.org/10.4310/jdg/1214433725 doi: 10.4310/jdg/1214433725
[4]	N. Garofalo, D. Vassilev, Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, Math. Ann., 318 (2000), 453–516. https://doi.org/10.1007/s002080000127 doi: 10.1007/s002080000127
[5]	R. Akylzhanov, M. Ruzhansky, $L^p$-$L^q$ multipliers on locally compact groups, J. Funct. Anal., 278 (2020), 108324. https://doi.org/10.1016/j.jfa.2019.108324 doi: 10.1016/j.jfa.2019.108324
[6]	T. Bruno, M. Peloso, A. Tabacco, M. Vallarino, Sobolev spaces on Lie groups: embedding theorems and algebra properties, J. Funct. Anal., 276 (2019), 3014–3050. https://doi.org/10.1016/j.jfa.2018.11.014 doi: 10.1016/j.jfa.2018.11.014
[7]	T. Bruno, M. Peloso, M. Vallarino, The Sobolev embedding constant on Lie groups, Nonlinear Anal, 216 (2022), 112707. https://doi.org/10.1016/j.na.2021.112707 doi: 10.1016/j.na.2021.112707
[8]	V. Fischer, M. Ruzhansky, Sobolev spaces on graded Lie groups, Ann. Inst. Fourier (Grenoble), 67 (2017), 1671–1723. https://doi.org/10.5802/aif.3119 doi: 10.5802/aif.3119
[9]	M. Ruzhansky, N. Yessirkegenov, Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups, 2018, arXiv: 1810.08845. https://doi.org/10.48550/arXiv.1810.08845
[10]	M. Ruzhansky, N. Tokmagambetov, N. Yessirkegenov, Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher-order nonlinear subelliptic equations, Calc. Var. Partial Differential Equations, 59 (2020), 175. https://doi.org/10.1007/s00526-020-01835-0 doi: 10.1007/s00526-020-01835-0
[11]	D. Cardona, M. Ruzhansky, Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups, 2020, arXiv: 2008.09651v3. https://doi.org/10.48550/arXiv.2008.09651
[12]	D. Cardona, J. Delgado, V. Kumar, M. Ruzhansky, $L^p$-$L^q$ estimates for subelliptic pseudo-differential operators on compact Lie groups, 2023, arXiv: 2310.16247. https://doi.org/10.48550/arXiv.2310.16247
[13]	L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147–171. https://doi.org/10.1007/BF02392081 doi: 10.1007/BF02392081
[14]	H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285–346. https://doi.org/10.2140/pjm.1966.19.285 doi: 10.2140/pjm.1966.19.285
[15]	Z. Avetisyan, M. Ruzhansky, A note on the polar decomposition in metric spaces, J. Math. Sci., 280 (2024), 73–82. https://doi.org/10.1007/s10958-023-06674-w doi: 10.1007/s10958-023-06674-w

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)