  In this paper we give different estimates between Lebesgue norms of quadratic
time-frequency representations. We show that, in some cases, it is not possible
to have such bounds in classical $L^p$ spaces, but the Lebesgue norm needs to
be suitably weighted. This leads to consider weights of polynomial type, and,
more generally, of ultradifferentiable type, and this, in turn, gives rise to
use as functional setting the ultradifferentiable classes. As applications of
such estimates we deduce uncertainty principles both of Donoho-Stark type and
of local type for representations.
