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

## TL;DR

This paper investigates nonlocal hypoelliptic operators of H"ormander type, establishing sharp inequalities and estimates for their associated semigroups, and introduces related interpolation spaces to understand their properties.

## Contribution

It introduces new interpolation spaces and sharp Harnack and Poincaré inequalities for nonlocal operators of H"ormander type, extending classical results to fractional and nonlocal settings.

## Key findings

- Sharp pointwise Harnack estimates for the semigroup
- Global and local Poincaré inequalities adapted to the geometry
- Introduction of interpolation spaces related to fractional operators

## Abstract

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 $\mathscr A$, 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.