A note on commutators of the fractional sub-Laplacian on Carnot groups

In this manuscript, we provide a point-wise estimate for the 3-commutators involving fractional powers of the sub-Laplacian on Carnot groups of homogeneous dimension \begin{document}$Q$\end{document} . This can be seen as a fractional Leibniz rule in the sub-elliptic setting. As a corollary of the point-wise estimate, we provide an \begin{document}$(L^{p}, L^{q})\to L^{r}$\end{document} estimate for the commutator, provided that \begin{document}$\frac{1}{r} = \frac{1}{p}+\frac{1}{q}-\frac{α}{Q}$\end{document} for \begin{document}$α ∈ (0, Q)$\end{document} .