A Note on Commutators of the Fractional Sub-Laplacian on Carnot Groups

Abstract: 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 $Q$. 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 $(L^{p},L^{q})\to L^{r}$ estimate for the commutator, provided that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-\frac{\alpha}{Q}$.