From semigroups to subelliptic estimates for quadratic operators

TL;DR

This paper introduces a new proof technique using FBI transforms to establish global subelliptic estimates for non-selfadjoint quadratic operators, linking derivative loss to algebraic properties of Hamilton maps and providing precise smoothing estimates for the heat semigroup.

Contribution

It offers a simplified proof of subelliptic estimates and connects algebraic properties of Hamilton maps to derivative loss, also deriving Gelfand-Shilov smoothing estimates.

Findings

New proof of subelliptic estimates using FBI transforms

Derivative loss depends on Hamilton map properties

Gelfand-Shilov smoothing estimates for heat semigroup

Abstract

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.