Revista Matemática Iberoamericana

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

Franck Barthe , Patrick Cattiaux , and Cyril Roberto

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Abstract

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general $F$-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures $\mu_{\alpha}(dx) = (Z_{\alpha})^{-1} e^{-2|x|^{\alpha}} dx$, when $\alpha\in (1,2)$. As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdré and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems.

Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 993-1067.

Dates
First available in Project Euclid: 22 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1169480039

Mathematical Reviews number (MathSciNet)
MR2320410

Zentralblatt MATH identifier
1118.26014

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60E15: Inequalities; stochastic orderings 60G10: Stationary processes

Keywords
isoperimetry Orlicz spaces hypercontractivity Boltzmann measure Girsanov transform $F$-Sobolev inequalities

Citation

Barthe, Franck; Cattiaux, Patrick; Roberto, Cyril. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22 (2006), no. 3, 993--1067. https://projecteuclid.org/euclid.rmi/1169480039


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