Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups

Xuan Thinh Duong, Hong-Quan Li, Ji Li, Brett D. Wick

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We provide a study of the Riesz transforms on stratified nilpotent Lie groups, and obtain a certain version of the pointwise lower bound of the Riesz transform kernel. Then we establish the characterisation of the BMO space on stratified nilpotent Lie groups via the boundedness of the commutator of the Riesz transforms and the BMO function. This extends the well-known Coifman, Rochberg, Weiss theorem from Euclidean space to the setting of stratified nilpotent Lie groups. In particular, these results apply to the well-known example of the Heisenberg group. As an application, we also study the curl operator on the Heisenberg group and stratified nilpotent Lie groups, and establish the div–curl lemma with respect to the Hardy space on stratified nilpotent Lie groups.

LanguageEnglish
Pages273-299
Number of pages27
JournalJournal des Mathematiques Pures et Appliquees
Volume124
DOIs
Publication statusPublished - Apr 2019

Fingerprint

Electric commutators
Lie groups
Riesz Transform
Nilpotent Lie Group
Commutator
Lower bound
kernel
Theorem
Curl
Heisenberg Group
BMO Space
Hardy Space
Euclidean space
Boundedness
Lemma
Operator

Keywords

  • BMO space
  • Commutator
  • div–curl lemma
  • Nehari theorem
  • Riesz transforms
  • Stratified nilpotent Lie groups

Cite this

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Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups. / Duong, Xuan Thinh; Li, Hong-Quan; Li, Ji; Wick, Brett D.

In: Journal des Mathematiques Pures et Appliquees, Vol. 124, 04.2019, p. 273-299.

Research output: Contribution to journalArticleResearchpeer-review

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