[Inégalité de Poincaré et méthode quantitative de De Giorgi pour les opérateurs hypoelliptiques]
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.
Nous proposons une approche systématique fondée sur les trajectoires pour démontrer une inégalité de Poincaré pour les sous-solutions faibles positives d’équations hypoelliptiques comportant un nombre arbitraire de commutateurs de Hörmander, tant dans le cas local que dans le cas non local. Nous en déduisons l’inégalité faible de Harnack et la régularité de Hölder, dans la lignée de la méthode de De Giorgi.
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Keywords: Poincaré inequality, kinetic partial differential equations, hypoellipticity, a priori estimates
Mots-clés : Inégalité de Poincaré, équations cinétiques aux dérivées partielles, hypoellipticité, estimées a priori
Francesca Anceschi 1 ; Helge Dietert 2 ; Jessica Guerand 3 ; Amélie Loher 4 ; Clément Mouhot 4 ; Annalaura Rebucci 5
CC-BY 4.0
Francesca Anceschi; Helge Dietert; Jessica Guerand; Amélie Loher; Clément Mouhot; Annalaura Rebucci. Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1393-1417. doi: 10.5802/jep.350
@article{JEP_2026__13__1393_0,
author = {Francesca Anceschi and Helge Dietert and Jessica Guerand and Am\'elie Loher and Cl\'ement Mouhot and Annalaura Rebucci},
title = {Poincar\'e inequality and quantitative {De~Giorgi} method for hypoelliptic operators},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1393--1417},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.350},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.350/}
}
TY - JOUR AU - Francesca Anceschi AU - Helge Dietert AU - Jessica Guerand AU - Amélie Loher AU - Clément Mouhot AU - Annalaura Rebucci TI - Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1393 EP - 1417 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.350/ DO - 10.5802/jep.350 LA - en ID - JEP_2026__13__1393_0 ER -
%0 Journal Article %A Francesca Anceschi %A Helge Dietert %A Jessica Guerand %A Amélie Loher %A Clément Mouhot %A Annalaura Rebucci %T Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 1393-1417 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.350/ %R 10.5802/jep.350 %G en %F JEP_2026__13__1393_0
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