Revista Matemática Iberoamericana


Full-Text PDF (533 KB) | Metadata | Table of Contents | RMI summary
Volume 22, Issue 3, 2006, pp. 993–1067
DOI: 10.4171/RMI/482

Published online: 2006-12-31

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

Franck Barthe[1], Patrick Cattiaux[2] and Cyril Roberto[3]

(1) Université Toulouse III, France
(2) Ecole Polytechnique, Palaiseau, France
(3) Université Paris Ouest Nanterre la Défense, France

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.

Keywords: Isoperimetry, Orlicz spaces, hypercontractivity, Boltzmann measure, Girsanov Transform, F-Sobolev inequalities

Barthe Franck, Cattiaux Patrick, Roberto Cyril: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22 (2006), 993-1067. doi: 10.4171/RMI/482