Gains of integrability and local smoothing effects for quadratic evolution equations

Abstract

We characterize geometrically the semigroups generated by non-selfadjoint quadratic differential operators $(e^{-t{q}^w}{)}_{t\ge 0}$ enjoying local smoothing effects and providing gains of integrability. More precisely, we prove that the evolution operators $e^{-t{q}^w}$ map $L^{\mathfrak{p}}$ on $L^{\mathfrak{q}}\cap C^{\infty }$, for all $1\le \mathfrak{p}\le \mathfrak{q}\le \infty$, 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 $0<t\ll 1$.