The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the $k$-th order momenta with respect to the heat kernel.