Lower bound of Riesz transform kernels and Commutator Theorems on stratified nilpotent Lie groups

TL;DR

This paper investigates Riesz transforms on stratified nilpotent Lie groups, establishing kernel bounds, characterizing BMO spaces via commutator boundedness, and extending classical Euclidean results to these groups, including applications to the Heisenberg group.

Contribution

It extends the Coifman-Rochberg-Weiss theorem to stratified nilpotent Lie groups and provides kernel bounds and BMO characterizations in this setting.

Findings

Established pointwise lower bounds for Riesz transform kernels.

Characterized BMO spaces via commutator boundedness on stratified groups.

Extended div-curl lemma and applications to the Heisenberg group.

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.