# Lower bound of Riesz transform kernels revisited and commutators on stratified Lie groups

## TL;DR

This paper revisits lower bounds of Riesz transform kernels on stratified Lie groups, establishes new two-weight estimates and endpoint characterizations for commutators with BMO functions, and analyzes the Heisenberg group case.

## Contribution

It provides a new lower bound for Riesz transform kernels and extends endpoint estimates and characterizations for their commutators on stratified Lie groups.

## Key findings

- New lower bound for Riesz transform kernels.
- Bloom-type two weight estimates for commutators.
- Weak type (1,1) characterizations for Riesz commutators.

## Abstract

Let $\mathcal G$ be a stratified Lie group and $\{\X_j\}_{1 \leq j \leq n}$ a basis for the left-invariant vector fields of degree one on $\mathcal G$. Let $\Delta = \sum_{j = 1}^n \X_j^2 $ be the sub-Laplacian on $\mathcal G$ and the $j^{\mathrm{th}}$ Riesz transform on $\mathcal G$ is defined by $R_j:= \X_j (-\Delta)^{-\frac{1}{2}}$, $1 \leq j \leq n$. In this paper we give a new version of the lower bound of the kernels of Riesz transform $R_j$ and then establish the Bloom-type two weight estimates as well as a number of endpoint characterisations for the commutators of the Riesz transforms and BMO functions, including the $L\log^+L(\mathcal G)$ to weak $L^1(\mathcal G)$, $H^1(\mathcal G)$ to $L^1(\mathcal G)$ and $L^\infty(\mathcal G)$ to BMO$(\mathcal G)$. Moreover, we also study the behaviour of the Riesz transform kernel on a special case of stratified Lie group: the Heisenberg group, and then we obtain the weak type $(1,1)$ characterisations for the Riesz commutators.