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

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 $Δ= \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 $R_j:= \X_j (-Δ)^{-\frac{1}{2}}$, $1\le j\le n$. 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.