Poincar\'e inequality and quantitative De Giorgi method for hypoelliptic operators

Abstract

We propose a systematic approach based on trajectories to prove a Poincar\'e inequality for weak non-negative sub-solutions to hypoelliptic equations with an arbitrary number of H\"ormander commutators, both in the local and in the non-local case. As a consequence, we deduce the weak Harnack inequality and H\"older regularity along the line of the De Giorgi method.