[Espaces de Sobolev sur les groupes de Lie gradués]
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.
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.
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Keywords: Harmonic analysis on nilpotent Lie groups, Sobolev spaces, graded Lie groups, Rockland operators, heat semigroup
Mot clés : analyse harmonique sur les groupes de Lie nilpotents, espaces de Sobolev, groupes de Lie gradués, opérateus de Rockland, semi-groupe de la chaleur
Fischer, Veronique 1 ; Ruzhansky, Michael 2
CC-BY-ND 4.0
@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},
pages = {1671--1723},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {67},
number = {4},
year = {2017},
doi = {10.5802/aif.3119},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3119/}
}
TY - JOUR AU - Fischer, Veronique AU - Ruzhansky, Michael TI - Sobolev spaces on graded Lie groups JO - Annales de l'Institut Fourier PY - 2017 SP - 1671 EP - 1723 VL - 67 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3119/ DO - 10.5802/aif.3119 LA - en ID - AIF_2017__67_4_1671_0 ER -
%0 Journal Article %A Fischer, Veronique %A Ruzhansky, Michael %T Sobolev spaces on graded Lie groups %J Annales de l'Institut Fourier %D 2017 %P 1671-1723 %V 67 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3119/ %R 10.5802/aif.3119 %G en %F AIF_2017__67_4_1671_0
Fischer, Veronique; Ruzhansky, Michael. Sobolev spaces on graded Lie groups. Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1671-1723. doi : 10.5802/aif.3119. https://aif.centre-mersenne.org/articles/10.5802/aif.3119/
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