Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.

Franck Barthe; Patrick Cattiaux; Cyril Roberto

Revista Matemática Iberoamericana (2006)

  • Volume: 22, Issue: 3, page 993-1067
  • ISSN: 0213-2230

Abstract

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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 measuresμα(dx) = (Zα)-1 e-2|x|αdx, when α ∈ (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.

How to cite

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Barthe, Franck, Cattiaux, Patrick, and Roberto, Cyril. "Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.." Revista Matemática Iberoamericana 22.3 (2006): 993-1067. <http://eudml.org/doc/42001>.

@article{Barthe2006,
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 measuresμα(dx) = (Zα)-1 e-2|x|αdx, when α ∈ (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.},
author = {Barthe, Franck, Cattiaux, Patrick, Roberto, Cyril},
journal = {Revista Matemática Iberoamericana},
keywords = {Espacio de Orlicz; Semigrupo de operadores; Procesos estocásticos; Procesos estacionarios; Isoperimetría; Desigualdades de Sobolev; isoperimetry; Orlicz spaces; Boltzmann measure; Girsanov Transform; -Sobolev inequality},
language = {eng},
number = {3},
pages = {993-1067},
title = {Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.},
url = {http://eudml.org/doc/42001},
volume = {22},
year = {2006},
}

TY - JOUR
AU - Barthe, Franck
AU - Cattiaux, Patrick
AU - Roberto, Cyril
TI - Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry.
JO - Revista Matemática Iberoamericana
PY - 2006
VL - 22
IS - 3
SP - 993
EP - 1067
AB - 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 measuresμα(dx) = (Zα)-1 e-2|x|αdx, when α ∈ (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.
LA - eng
KW - Espacio de Orlicz; Semigrupo de operadores; Procesos estocásticos; Procesos estacionarios; Isoperimetría; Desigualdades de Sobolev; isoperimetry; Orlicz spaces; Boltzmann measure; Girsanov Transform; -Sobolev inequality
UR - http://eudml.org/doc/42001
ER -

Citations in EuDML Documents

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  1. Pierre-André Zitt, Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré
  2. Patrick Cattiaux, Arnaud Guillin, Trends to equilibrium in total variation distance
  3. Patrick Cattiaux, Arnaud Guillin, Deviation bounds for additive functionals of Markov processes
  4. Sasha Sodin, An isoperimetric inequality on the ℓp balls
  5. Patrick Cattiaux, Arnaud Guillin, Pierre André Zitt, Poincaré inequalities and hitting times
  6. Djalil Chafaï, Florent Malrieu, On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities
  7. Patrick Cattiaux, Arnaud Guillin, deviation bounds for additive functionals of markov processes
  8. Nathael Gozlan, Poincaré inequalities and dimension free concentration of measure

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