A NOTE ON COMMUTATORS OF THE FRACTIONAL SUB-LAPLACIAN ON CARNOT GROUPS

摘要

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) - L-r estimate for the commutator, provided that 1/r = 1/p + 1/q - alpha/Q for alpha is an element of (0, Q).