# 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

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topBarthe, 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 -

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