- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:44
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
22
2006
3
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
Franck
Barthe
Université Toulouse III, TOULOUSE CEDEX, FRANCE
Patrick
Cattiaux
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
Cyril
Roberto
Université Paris Ouest Nanterre la Défense, NANTERRE, FRANCE
Isoperimetry, Orlicz spaces, hypercontractivity, Boltzmann measure, Girsanov Transform, F-Sobolev inequalities
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.
Real functions
Operator theory
Probability theory and stochastic processes
General
993
1067
10.4171/RMI/482
http://www.ems-ph.org/doi/10.4171/RMI/482