In this article, we study the -properties of powers of positive Rockland operators and define Sobolev spaces on general graded Lie groups. We establish that the defined Sobolev spaces are independent of the choice of a positive Rockland operator, and that they are interpolation spaces. Although this generalises the case of sub-Laplacians on stratified groups studied by G. Folland in [12], many arguments have to be different since Rockland operators are usually of higher degree than two. We also prove results regarding duality and Sobolev embeddings, together with inequalities of Hardy–Littlewood–Sobolev type and of Gagliardo–Nirenberg type.
Dans cet article, nous étudions les propriétés des puissances des opérateurs de Rockland positifs et nous définissons les espaces de Sobolev sur tous les groupes de Lie nilpotents gradués. Nous montrons que les espaces de Sobolev ainsi définis sont indépendants du choix de l’opérateur de Rockland positif et qu’ils sont des espaces d’interpolation. Quoique cela généralise le cas des sous-laplaciens sur les groupes stratifiés étudiés par G. Folland dans [12], plusieurs arguments sont différents car les opérateurs de Rockland sont souvent de degrée plus haut que deux. Nous montrons aussi des résultats concernant la dualité et les injections de Sobolev, ainsi que des inégalités de type Littlewood–Sobolev et de type Gagliardo–Nirenberg.
Revised : 2016-10-03
Accepted : 2016-10-27
Published online : 2017-09-26
DOI : https://doi.org/10.5802/aif.3119
Classification: 13A50, 43A32, 43A85, 43A90
Keywords: Harmonic analysis on nilpotent Lie groups, Sobolev spaces, graded Lie groups, Rockland operators, heat semigroup
@article{AIF_2017__67_4_1671_0,
author = {Fischer, Veronique and Ruzhansky, Michael},
title = {Sobolev spaces on graded Lie groups},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {4},
year = {2017},
pages = {1671-1723},
doi = {10.5802/aif.3119},
language = {en},
url = {http://www.numdam.org/item/AIF_2017__67_4_1671_0}
}
Fischer, Veronique; Ruzhansky, Michael. Sobolev spaces on graded Lie groups. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1671-1723. doi : 10.5802/aif.3119. http://www.numdam.org/item/AIF_2017__67_4_1671_0/
[1] Spectral multipliers on Lie groups of polynomial growth, Proc. Am. Math. Soc., Tome 120 (1994) no. 3, pp. 973-979 | Article | MR 1172944
[2] On positive Rockland operators, Colloq. Math., Tome 67 (1994) no. 2, pp. 197-216 | Article | MR 1305212
[3] Refined inequalities on graded Lie groups, C. R. Math. Acad. Sci. Paris, Tome 350 (2012) no. 7-8, pp. 393-397 | Article | MR 2922090
[4] Paraproduit sur le groupe de Heisenberg et applications, Rev. Mat. Iberoamericana, Tome 17 (2001) no. 1, pp. 69-105 | Article | MR 1846091
[5] Pseudodifferential operators on groups with dilations, Duke Math. J., Tome 68 (1992) no. 1, pp. 31-65 | Article | MR 1185817
[6] Analyse harmonique non-commutative sur certains espaces homogènes, Springer, Lecture Notes in Mathematics, Tome 242 (1971), v+160 pages (Étude de certaines intégrales singulières) | MR 0499948
[7] Sobolev algebras on Lie groups and Riemannian manifolds, Am. J. Math., Tome 123 (2001) no. 2, pp. 283-342 http://muse.jhu.edu/journals/american_journal_of_mathematics/v123/123.2coulhon.pdf | Article | MR 1828225
[8] On semigroups generated by subelliptic operators on homogeneous groups, Colloq. Math., Tome 64 (1993) no. 2, pp. 215-231 | Article | MR 1218485
[9] Note on semigroups generated by positive Rockland operators on graded homogeneous groups, Studia Math., Tome 110 (1994) no. 2, pp. 115-126 | Article | MR 1279987
[10] Spectral estimates for positive Rockland operators, Algebraic groups and Lie groups, Cambridge University Press (Austral. Math. Soc. Lect. Ser.) Tome 9 (1997), pp. 195-213 | MR 1635682
[11] Quantization on nilpotent Lie groups, Birkhäuser/Springer, Progress in Mathematics, Tome 314 (2016), xiii+557 pages | Article | MR 3469687
[12] Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Tome 13 (1975) no. 2, pp. 161-207 | Article | MR 0494315
[13] Estimates for the complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., Tome 27 (1974), pp. 429-522 | Article | MR 0367477
[14] Hardy spaces on homogeneous groups, Princeton University Press; University of Tokyo Press, Mathematical Notes, Tome 28 (1982), xii+285 pages | MR 657581
[15] Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth, Math. Nachr., Tome 279 (2006) no. 9-10, pp. 1028-1040 | Article | MR 2242964
[16] Besov algebras on Lie groups of polynomial growth, Studia Math., Tome 212 (2012) no. 2, pp. 119-139 | Article | MR 3008437
[17] Liouville’s theorem for homogeneous groups, Comm. Partial Differ. Equations, Tome 8 (1983) no. 15, pp. 1665-1677 | Article | MR 729197
[18] Nilpotent Lie groups: structure and applications to analysis, Springer, Lecture Notes in Mathematics, Tome 562 (1976), x+210 pages | MR 0442149
[19] Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differ. Equations, Tome 4 (1979) no. 8, pp. 899-958 | Article | MR 537467
[20] Semi-groups of measures on Lie groups, Trans. Am. Math. Soc., Tome 81 (1956), pp. 264-293 | Article | MR 0079232
[21] The theory of fractional powers of operators, North-Holland Publishing Co., North-Holland Mathematics Studies, Tome 187 (2001), xii+365 pages | MR 1850825
[22] Parametrices for hypoelliptic operators on step two nilpotent Lie groups, Comm. Partial Differ. Equations, Tome 5 (1980) no. 11, pp. 1153-1184 | Article | MR 589618
[23] Besov spaces and Sobolev spaces on a nilpotent Lie group, Tôhoku Math. J., Tome 31 (1979) no. 4, pp. 383-437 | Article | MR 558675
[24] Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton Mathematical Series, Tome 32 (1971), x+297 pages | MR 0304972
[25] Analysis and geometry on groups, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 100 (1992), xii+156 pages | MR 1218884
