Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects

Paul Alphonse (1) , Joackim Bernier (2, 1)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 MINGUS - Multi-scale numerical geometric schemes

We characterize geometrically the regularizing effects of the semigroups generated by accretive non-selfadjoint quadratic differential operators. As a byproduct, we establish the subelliptic estimates enjoyed by these operators, being expected to be optimal. These results prove conjectures by M. Hitrik, K. Pravda-Starov and J. Viola. The proof relies on a new representation of the polar decomposition of these semigroups. In particular, we identify the selfadjoint part as the evolution operator generated by the Weyl quantization of a time-dependent real-valued nonnegative quadratic form for which we prove a sharp anisotropic lower bound.

Mots clés: Polar decomposition; Quadratic operators; Fourier integral operators; Splitting method; Subelliptic estimates

Domaines: Equations aux dérivées partielles [math.AP]

Paul Alphonse, Joackim Bernier. Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects. Annales Scientifiques de l'École Normale Supérieure, 2023, 56 (2), pp.323-382. ⟨10.24033/asens.2536⟩. ⟨hal-02280971v3⟩