We characterize geometrically the semigroups generated by non-selfadjoint quadratic differential operators $(e^{ -tq^w})_{ t\geq 0}$ enjoying local smoothing effects and providing gains of integrability. More precisely, we prove that the evolution operators $e^{ -tq^w}$ map $L^{ \mathfrak{ p}}$ on $L^{ \mathfrak{ q}} \cap C^\infty$, for all $1\leq \mathfrak{ p} \leq \mathfrak{ q} \leq \infty$, if and only if the singular space of the quadratic operator $q^w$ is included in the graph of a linear map. We also provide quantitative estimates for the associated operator norms in the short-time asymptotics $0<t \ll 1$.
Categories: math.AP math.FA