Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

By Franck Barthe, Patrick Cattiaux, Cyril Roberto. 2004

151 citations recorded by the source corpus.

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 µa(dx) = (Za)-1 e-2|x|adx, when a I (1,2). As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdre and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2ec8392b949224c649f0ea1901167110f702f4ba

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In Applied Mathematics

Citations in the corpus

Lyapunov conditions for Super Poincaré inequalities (2009, background): “These inequalities are related to the Latala-Oleskiewicz interpolating inequalities [23], see [4] for a complete description.”
Hypercontractive measures, Talagrand's inequality, and influences (2011, background): “Nevertheless, the hypercontractive theorems in Orlicz norms of [5] still indicate that the semigroup (Pt)t≥0 generated by the potential |x| is such”
Applications of Stein's method for concentration inequalities (2009, background): “The papers [2, 14, 20, 28] deal with tails between exponential and Gaussian and [1, 4] deal with subexponential tails.”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “The product probability spaces in R that are taken into account were analyzed in [3,4], and share common features with the Gauss space, namely R endowed with the probability measure dγn(x) = (2π)− n 2 e− |x|2 2 dx.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “We follow a variation on the definition given in [63] (for general q), which was extended by Barthe, Cattiaux and Roberto (with q = 2) in [7] (after being introduced in [10]).”
Poincaré inequalities on intervals – application to sensitivity analysis (2016, unknown): “As explained in [6], it can be interpreted as a reduction from functions to sets (by decomposition functions according to their level sets): a Poincaré inequality is equivalent to a comparison between the measures of sets, and a notion of μ-capacity.”
Poincaré inequalities and dimension free concentration of measure (2010, methodology): “Our approach to compare the inequalities SG(ω, ·) to the super Poincaré inequalities relies on the capacity-measure results of Barthe, Cattiaux and Roberto [5,6].”