A note on commutators of the fractional sub-Laplacian on Carnot groups
Ali Maalaoui
Communications on Pure & Applied Analysis 2019, 18(1): 435-453 doi: 10.3934/cpaa.2019022

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 $\frac{1}{r} = \frac{1}{p}+\frac{1}{q}-\frac{α}{Q}$ for $α ∈ (0, Q)$.

keywords: Fractional sub-Laplacian commutator Carnot groups

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