Sobolev spaces on graded lie groups

Veronique Fischer, Michael Ruzhansky

Research output: Contribution to journalArticlepeer-review

33 Citations (SciVal)

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 languageEnglish
Pages (from-to)1671-1723
Number of pages53
JournalAnnales de l'institut Fourier
Volume67
Issue number4
DOIs
Publication statusPublished - 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

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