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

TL;DR

This paper establishes point-wise estimates and boundedness results for commutators involving fractional sub-Laplacians on Carnot groups, extending fractional Leibniz rules to sub-elliptic settings.

Contribution

It provides the first point-wise estimates for 3-commutators of fractional sub-Laplacians on Carnot groups, generalizing fractional Leibniz rules in sub-elliptic contexts.

Findings

Derived point-wise estimates for 3-commutators.

Established $(L^{p},L^{q}) o L^{r}$ boundedness under specific conditions.

Extended fractional Leibniz rules to Carnot group settings.

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 $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-\frac{\alpha}{Q}$.