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HYPERCONTRACTIVITY, NASH INEQUALITIES AND SUBORDINATION FOR CLASSES OF NONLINEAR SEMIGROUPS

By Fabio Cipriani and Gabriele Grillo

Abstract

Abstract. A suitable notion of hypercontractivity for a nonlinear semigroup {Tt} is shown to imply Nash–type 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. Then we introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is f(x) = x α, and write an explicit formula for the associated generator. It is shown that hypercontractivity still holds for the subordinated semigroup and, hence, that Nash-type inequalities hold as well for the subordinated generator. 1

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