From semigroups to subelliptic estimates for quadratic operators

Michael Hitrik (1) , Karel Pravda-Starov (2) , Joe Viola (3)

1 Department of Mathematics [UCLA]
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LMJL - Laboratoire de Mathématiques Jean Leray

Using an approach based on the techniques of FBI transforms, we give a new simple proof of the global subelliptic estimates for non-selfadjoint non-elliptic quadratic differential operators, under a natural averaging condition on the Weyl symbols of the operators, established by the second author. The loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the Hamilton maps of the quadratic symbols. Using the FBI point of view, we also give accurate smoothing estimates of Gelfand-Shilov type for the associated heat semigroup in the limit of small times.

Mots clés: subelliptic estimate; quadratic differential operator; FBI transform; heat semigroup; Non-selfadjoint operator

Domaines: Equations aux dérivées partielles [math.AP]

Michael Hitrik, Karel Pravda-Starov, Joe Viola. From semigroups to subelliptic estimates for quadratic operators. Transactions of the American Mathematical Society, 2018, 370 (10), pp.7391-7415. ⟨10.1090/tran/7251⟩. ⟨hal-01219682⟩