Functional inequalities for a class of nonlocal hypoelliptic equations of Hörmander type

Abstract

We consider a class of second-order partial differential operators $\mathcal{A}$ of Hörmander 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 Poincaré inequalities adapted to the underlying geometry.

The paper is going to appear in Nonlinear Anal., Special Issue on "Fractional and Nonlocal equations". The content of Section 5 had been originally announced in a first preliminary version of the ArXiv preprint 1811.02968