  We study accretive quadratic operators with zero singular spaces. These
degenerate non-selfadjoint differential operators are known to be hypoelliptic
and to generate contraction semigroups which are smoothing in the Schwartz
space for any positive time. In this work, we study the short-time asymptotics
of the regularizing effect induced by these semigroups. We show that these
short-time asymptotics of the regularizing effect depend on the directions of
the phase space, and that this dependence can be nicely understood through the
structure of the singular space. As a byproduct of these results, we derive
sharp subelliptic estimates for accretive quadratic operators with zero
singular spaces pointing out that the loss of derivatives with respect to the
elliptic case also depends on the phase space directions according to the
structure of the singular space. Some applications of these results are then
given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and
degenerate hypoelliptic Fokker-Planck operators.
