Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators

We propose a systematic elementary approach based on trajectories to prove Poincaré inequalities for hypoelliptic equations with an arbitrary number of H\"ormander commutators, both in the local and in the non-local case. Our method generalises and simplifies the method introduced in Guerand-Mo...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Anceschi, Francesca, Dietert, Helge, Guerand, Jessica, Loher, Amélie, Mouhot, Clément, Rebucci, Annalaura
Format: Article
Language:English
Subjects:
Commutators
Inequalities
Online Access:Get full text
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Summary:We propose a systematic elementary approach based on trajectories to prove Poincaré inequalities for hypoelliptic equations with an arbitrary number of H\"ormander commutators, both in the local and in the non-local case. Our method generalises and simplifies the method introduced in Guerand-Mouhot (2022) in the local case with one commutator, and extended in Loher (2024) to the non-local case with one commutator. It draws inspiration from the paper Niebel-Zacher (2022), although we use different trajectories. We deduce Harnack inequalities and H\"older regularity along the line of the De Giorgi method. Our results recover those in Anceschi-Rebucci (2022) in the local case, and are new in the non-local case.
ISSN:2331-8422