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We characterize geometrically the regularizing effects of the semigroupsgenerated by accretive non-selfadjoint quadratic differential operators. As abyproduct, 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 thepolar decomposition of these semigroups. In particular, we identify theselfadjoint part as the evolution operator generated by the Weyl quantizationof a time-dependent real-valued nonnegative quadratic form for which we prove asharp anisotropic lower bound.

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