Mathematical analysis demonstrates a trajectory-based Poincaré inequality for hypoelliptic equations, indicating new pathways to establish De Giorgi-type Hölder regularity.
We propose a systematic approach based on trajectories to prove a Poincaré inequality for weak non-negative sub-solutions to hypoelliptic equations with an arbitrary number of Hörmander commutators, both in the local and in the non-local case. As a consequence, we deduce the weak Harnack inequality and Hölder regularity along the line of the De Giorgi method.
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Anceschi et al. (2026) studied this question.
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