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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

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.

Keywords

compact Lie groups / Sobolev inequality / subelliptic pseudo-differential operators

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

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

[3] Th. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598. https://doi.org/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

[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

[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

[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

[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

[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

[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

[14] H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. https://doi.org/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