  Consider $F$ an element of the second Wiener chaos with variance one. In full
generality, we show that, for every integer $p\ge 1$, there exists $\eta_p>0$
such that if $\kappa_4(F)<\eta_p$ then the Malliavin derivative of $F$ admits a
negative moment of order $p$. This entails that any sequence of random
variables in the second Wiener chaos converging in distribution to a
non--degenerated Gaussian is getting more regular as its distribution is
getting close to the normal law. This substantially generalizes some recent
findings contained in \cite{hu2014convergence,hu2015density,nourdin2016fisher}
where analogous statements were given with additional assumptions which we are
able to remove here. Moreover, we provide a multivariate version of this
Theorem.
  Our main contribution concerns the case of the third Wiener chaos which is
notoriously more delicate as one cannot anymore decompose the random variables
into a linear combination of i.i.d. random variables. We still prove that the
same phenomenon of regularization along central convergence occurs.
Unfortunately, we are not able to provide a statement as strong as the previous
one, but we can show that the usual non--degeneracy estimates of the Malliavin
derivative given by the Carbery-Wright inequality can be improved by a factor
three. Our proof introduces new techniques such that a specific Malliavin
gradient enabling us to encode the distribution of the Malliavin derivative by
the spectrum of some Gaussian matrix. This allows us to revisit the fourth
moment phenomenon in terms of the behavior of its spectral radius.
