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

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

2007-04-26

arxiv.org/abs/0704.3527v2

Mathematics

Functional Analysis

19 pages

Scientific paper

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.

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