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arXiv: math.PR/0407219

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Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

F. Barthe 1, P. Cattiaux 2, Cyril Roberto 3

(2004)

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.

1:	Laboratoire de Statistiques et Probabilités (LSProba)
	CNRS : UMR5583 – Université Paul Sabatier [UPS] - Toulouse III – Institut National des Sciences Appliquées (INSA) - Toulouse
2:	Centre de Mathématiques Appliquées (CMAP)
	CNRS : UMR7641 – Université de Versailles Saint-Quentin-en-Yvelines – Polytechnique - X
3:	Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
	Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout

Subject

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Mathematics/Probability

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Submitted on: Sunday, 20 November 2005 13:34:45

Updated on: Sunday, 20 November 2005 13:34:45