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Using an approach based on the techniques of FBI transforms, we give a newsimple proof of the global subelliptic estimates for non-selfadjointnon-elliptic quadratic differential operators, under a natural averagingcondition on the Weyl symbols of the operators, established by the secondauthor. The loss of the derivatives in the subelliptic estimates dependsdirectly on algebraic properties of the Hamilton maps of the quadratic symbols.Using the FBI point of view, we also give accurate smoothing estimates ofGelfand-Shilov type for the associated heat semigroup in the limit of smalltimes.

From semigroups to subelliptic estimates for quadratic operators