Hypercontractivity, Nash inequalities, and subordination for classes of nonlinear semigroups

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.