JournalsrmiVol. 22, No. 3pp. 993–1067

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

  • Franck Barthe

    Université Toulouse III, France
  • Patrick Cattiaux

    Ecole Polytechnique, Palaiseau, France
  • Cyril Roberto

    Université Paris Ouest Nanterre la Défense, France
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry cover
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Abstract

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general FF-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 μα(dx)=(Zα)1e2xαdx\mu_{\alpha}(dx) = (Z_{\alpha})^{-1} e^{-2|x|^{\alpha}} dx, when α(1,2)\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.

Cite this article

Franck Barthe, Patrick Cattiaux, Cyril Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 993–1067

DOI 10.4171/RMI/482