The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic non-Hörmander operators

Battaglia, Erika; Biagi, Stefano; Bonfiglioli, Andrea

Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 589-631.

Nous considérons une classe d’opérateurs du second ordre hypoelliptiques $ℒ$ sous la forme de divergence et nous prouvons les principes du maximum fort et faible, et l’inégalité de Harnack. $ℒ$ n’est pas assumé dans la classe de hypoellipticité de Hörmander, ni satisfaisant des estimations sous-elliptique ou de dégénérescence à la Muckenhoupt ; en effet nos résultats sont valables dans le cas infiniment dégénéré et pour des opérateurs qui ne sont pas des sommes de carrés. Nous utilisons un résultat de la théorie du contrôle sur l’hypoellipticité pour récupérer une information géométrique sur la connectivité et la propagation des maximums, en l’absence de la condition de rang maximal. Quand $ℒ$ a coefficients $C^{\omega }$, ce résultat implique également une propriété de prolongement unique pour les fonctions $ℒ$–harmoniques. Le théorème de Harnack est obtenue par une inégalité faible de Harnack au moyen d’un argument de la théorie du potentiel et la solvabilité du problème de Dirichlet.

We consider a class of hypoelliptic second-order operators $ℒ$ in divergence form, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality. The operators are not assumed in the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates or Muckenhoupt-type degeneracy conditions; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily sums of squares. We use a Control Theory result on hypoellipticity to recover a meaningful geometric information on connectivity and maxima propagation, in the absence of any maximal rank condition. For operators $ℒ$ with $C^{\omega }$ coefficients, this control-theoretic result also implies a Unique Continuation property for the $ℒ$–harmonic functions. The Harnack theorem is obtained via a weak Harnack inequality by means of a Potential Theory argument and the solvability of the Dirichlet problem for $ℒ$.

Classification : 35B50, 35B45, 35H20, 35J25, 35J70, 35R03
Keywords: Degenerate-elliptic operators, maximum principles, Harnack inequality, Unique Continuation, divergence form operators
Mot clés : Opérateurs elliptique-dégénérés, principe du maximum, inégalité de Harnack, prolongement unique, opérateurs en forme de divergence

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