Abstract
In this article, we study the Lp-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.
Original language | English |
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Pages (from-to) | 1671-1723 |
Number of pages | 53 |
Journal | Annales de l'institut Fourier |
Volume | 67 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Jul 2017 |
Funding
Keywords: Harmonic analysis on nilpotent Lie groups, Sobolev spaces, graded Lie groups, Rockland operators, heat semigroup. Math. classification: 13A50, 43A32, 43A85, 43A90. (*) The first author acknowledges the support of the London Mathematical Society via the Grace Chisholm Fellowship held at King’s College London in 2011 and of the University of Padua (Italy). The second author was supported in part by the EPSRC Leadership Fellowship EP/G007233/1 and by EPSRC Grant EP/K039407/1. No new data was collected or generated during research.
Keywords
- Graded Lie groups
- Harmonic analysis on nilpotent Lie groups
- Heat semigroup
- Rockland operators
- Sobolev spaces
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology