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Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. (English) Zbl 1118.26014
The authors study interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Section 2 contains the required elements on Orlicz spaces. The notions of hyperboundedness and hypercontractivity are introduced. Section 3 presents a criterion for a semigroup to be Orlicz-hyperbounded. In order to get contraction results and explicit bounds the authors build in Section 4 the full analogue of Gross theory. The equivalence between the homogeneous \(F\)-Sobolev inequality and the Orlicz hypercontractivity is one of the main results of this section. Section 5 provides a thorough study of Sobolev type inequalities. The various implications between the inequalities studied in this section are summarized in a figure at the end of the section. Section 6 deals with the consequences of generalized Beckner inequalities for the contraction of the measure for which sharp criteria are obtained. Section 7 illustrates all the previous results in the case of \(x^{\alpha}\) Boltzmann measures. In this concrete situation the authors explain how to deal with the technical conditions involved. A perturbation argument is also developed. The final Section 8 deduces isoperimetric inequalities from semigroup hyperboundedness properties.

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
47D07 Markov semigroups and applications to diffusion processes
60E15 Inequalities; stochastic orderings
60G10 Stationary stochastic processes
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