  A suitable notion of hypercontractivity for a nonlinear semigroup $\{T_t\}$
is shown to imply Gagliardo--Nirenberg inequalities for its generator $H$,
provided a subhomogeneity property holds for the energy functional $(u,Hu)$. We
use this fact to prove that, for semigroups generated by operators of
$p$--Laplacian--type, hypercontractivity implies ultracontractivity. We then
introduce the notion of subordinated nonlinear semigroups when the
corresponding Bernstein function is $f(x)=x^\alpha$, and write down an explicit
formula for the associated generator. It is shown that, in certain cases,
hypercontractivity still holds for the subordinated semigroup and, hence, that
Nash--type inequalities holds as well for the subordinated generator.
