# Lower bound of the Riesz transform kernel on Stratified Lie groups, commutators and applications

It is well-known that the commutator of Riesz transform (Hilbert transform in dimension 1) and a symbol $b$ is bounded on $L^2(\mathbb R^n)$ if and only if $b$ is in the BMO space BMO$(\mathbb R^n)$ (Coifman--Rochberg--Weiss).

Inspired by this result, it is natural to ask whether it holds for commutator of Riesz transform on Heisenberg groups $\mathbb H^n$. Note that in the setting of several complex variables, the Heisenberg group $\mathbb H^n$ is the boundary of the Siegel upper half space, whose roles are holomorphically equivalent to the unit sphere and the unit ball in $\mathbb C^n$ respectively, and hence the role of Riesz transform on $\mathbb H^n$ is similar to that of Hilbert transform on the real line $\mathbb R$.

We answer this question confirmatively in a more general setting: stratified Lie groups $\mathcal G$.
%, which is more general than the Heisenberg group $\mathbb H^n$.
We first obtain a suitable version of lower bound for the kernel of the Riesz transform on $\mathcal G$, and then  establish a characterisation for the boundedness of the Riesz commutator, i.e., the commutator of Riesz transform and a symbol $b$ is bounded on $L^2(\mathcal G)$ if and only if $b$ is in the BMO space BMO$(\mathcal G)$ studied by Folland and Stein.
This yields directly a weak factorisation of
%In the mean time we also establish  characterisations for the endpoint boundedness of Riesz commutators, including the weak type $(1,1)$, $H^1(\mathcal G)\to L^1(\mathcal G)$, and  $L^\infty(\mathcal G)$ to BMO$(\mathcal G)$, where $H^1(\mathcal G)$ is the Hardy space  studied by Folland and Stein.
the Hardy space $H^1(\mathcal G)$ and a div-curl lemma with respect to $H^1(\mathcal G)$ via
a suitable curl operator that we introduced on $\mathcal G$.

Our method here bypassed the use of Fourier transform, and hence, as applications, it can be further developed and adapted to the study of commutator
in other settings, such as the Cauchy type integrals for domains in $\mathbb C^n$ with minimal smoothness (studied by Lanzani--Stein),
which are singular integrals with non-smooth kernels (beyond the standard frame of Calder\'on--Zygmund operators).

The results we provide here are based on recent joint works with Xuan Thinh Duong, Michael Lacey, Hong-Quan Li and Brett D. Wick.