# Hypercontractivity, Nash inequalities, and subordination for classes of nonlinear semigroups - Mathematics Functional Analysis

Abstract: 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 $u,Hu$. Weuse this fact to prove that, for semigroups generated by operators of$p$-Laplacian-type, hypercontractivity implies ultracontractivity. We thenintroduce the notion of subordinated nonlinear semigroups when thecorresponding Bernstein function is $fx=x^\alpha$, and write down an explicitformula for the associated generator. It is shown that, in certain cases,hypercontractivity still holds for the subordinated semigroup and, hence, thatNash-type inequalities holds as well for the subordinated generator.

Author: Fabio Cipriani, Gabriele Grillo

Source: https://arxiv.org/