Gains of integrability and local smoothing effects for quadratic\n evolution equations

2021/11/22 by Paul Alphonse, Alphonse, Paul, Joackim Berniér +1
Mathematics · #Nonlinear Differential Equations Analysis #Spectral Theory in Mathematical Physics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2111.11254

Abstract

We characterize geometrically the semigroups generated by non-selfadjoint\nquadratic differential operators (e^-tqw)_{t\≥ 0} enjoying local\nsmoothing effects and providing gains of integrability. More precisely, we\nprove that the evolution operators e^-tqw map L^ mathfrakp on\nL^ mathfrakq \∩ C^\∞, for all 1\≤ mathfrakp \≤ mathfrakq\n\≤ \∞, if and only if the singular space of the quadratic operator q^w\nis included in the graph of a linear map. We also provide quantitative\nestimates for the associated operator norms in the short-time asymptotics 0<t\n\≪ 1.\n

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