Compactness of Riesz transform commutator on stratified Lie groups

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 $\mathcal G$. 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 provide a concrete construction of the "twisted truncated sector" which is related to the pointwise lower bound of the kernel of $R_j$ on $\mathcal G$. Then we obtain the characterisation of compactness of the commutators of $R_j$ with a function $b\in$ VMO$(\mathcal G)$, the space of functions with vanishing mean oscillation on $\mathcal G$.