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We study accretive quadratic operators with zero singular spaces. Thesedegenerate non-selfadjoint differential operators are known to be hypoellipticand to generate contraction semigroups which are smoothing in the Schwartzspace for any positive time. In this work, we study the short-time asymptoticsof the regularizing effect induced by these semigroups. We show that theseshort-time asymptotics of the regularizing effect depend on the directions ofthe phase space, and that this dependence can be nicely understood through thestructure of the singular space. As a byproduct of these results, we derivesharp subelliptic estimates for accretive quadratic operators with zerosingular spaces pointing out that the loss of derivatives with respect to theelliptic case also depends on the phase space directions according to thestructure of the singular space. Some applications of these results are thengiven to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators anddegenerate hypoelliptic Fokker-Planck operators.

Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators