Compactness of Riesz transform commutator on stratified Lie groups

Let $\mathcal{G}$ be a stratified Lie group and $\{{\text{X}}_j{\}}_{1\le j\le n}$ a basis for the left-invariant vector fields of degree one on $\mathcal{G}$. Let $\mathrm{\Delta }={\sum }_{j=1}^n{\text{X}}_j^2$ be the sub-Laplacian on $\mathcal{G}$. The $j^{\mathrm{t}\mathrm{h}}$ 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 $\mathcal{G}$.