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

*Corresponding author for this work

Research output: Contribution to journalArticle

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)273-299
Number of pages27
JournalJournal des Mathematiques Pures et Appliquees
Publication statusPublished - Apr 2019



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

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