Functional inequalities for a class of nonlocal hypoelliptic equations of H\"ormander type

We consider a class of second-order partial differential operators $\mathscr A$ of H"ormander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the '30s. We analyze some properties of the nonlocal operators driven by the fractional powers of $\mathcal{A}$, and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincar'e inequalities adapted to the underlying geometry.