Gains of integrability and local smoothing effects for quadratic evolution equations

TL;DR

This paper characterizes when quadratic differential operators generate semigroups that improve regularity and integrability, linking these properties to geometric conditions on the operators' singular spaces and providing short-time estimates.

Contribution

It offers a geometric characterization of semigroups with smoothing effects for quadratic operators and provides quantitative short-time norm estimates.

Findings

Semigroups map $L^{rak{p}}$ to $L^{rak{q}}$ under geometric conditions.

The singular space inclusion determines smoothing properties.

Explicit short-time operator norm estimates are derived.

Abstract

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 $0<t \ll 1$.