Sobolev spaces on graded Lie groups

Fischer, Veronique; Ruzhansky, Michael

Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1671-1723.

Dans cet article, nous étudions les propriétés $L^p$ 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 $L^p$-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.

Classification : 13A50, 43A32, 43A85, 43A90
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

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