Sobolev spaces on graded lie groups

Veronique Fischer, Michael Ruzhansky

Research output: Contribution to journalArticle

6 Citations (Scopus)

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

Fingerprint

Positive Operator
Sobolev Spaces
Sub-Laplacian
Sobolev Embedding
Interpolation Spaces
Duality
Generalise
Operator

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

Cite this

Sobolev spaces on graded lie groups. / Fischer, Veronique; Ruzhansky, Michael.

In: Annales de l'institut Fourier, Vol. 67, No. 4, 01.07.2017, p. 1671-1723.

Research output: Contribution to journalArticle

Fischer, Veronique ; Ruzhansky, Michael. / Sobolev spaces on graded lie groups. In: Annales de l'institut Fourier. 2017 ; Vol. 67, No. 4. pp. 1671-1723.
@article{5dc56ff0a66347cf8e33df2fd6145ce9,
title = "Sobolev spaces on graded lie groups",
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.",
keywords = "Graded Lie groups, Harmonic analysis on nilpotent Lie groups, Heat semigroup, Rockland operators, Sobolev spaces",
author = "Veronique Fischer and Michael Ruzhansky",
year = "2017",
month = "7",
day = "1",
doi = "10.5802/aif.3119",
language = "English",
volume = "67",
pages = "1671--1723",
journal = "Annales de l'institut Fourier",
issn = "0373-0956",
publisher = "Association des Annales de l'Institut Fourier",
number = "4",

}

TY - JOUR

T1 - Sobolev spaces on graded lie groups

AU - Fischer, Veronique

AU - Ruzhansky, Michael

PY - 2017/7/1

Y1 - 2017/7/1

N2 - 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.

AB - 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.

KW - Graded Lie groups

KW - Harmonic analysis on nilpotent Lie groups

KW - Heat semigroup

KW - Rockland operators

KW - Sobolev spaces

UR - http://www.scopus.com/inward/record.url?scp=85017415427&partnerID=8YFLogxK

U2 - 10.5802/aif.3119

DO - 10.5802/aif.3119

M3 - Article

VL - 67

SP - 1671

EP - 1723

JO - Annales de l'institut Fourier

JF - Annales de l'institut Fourier

SN - 0373-0956

IS - 4

ER -