Sobolev inequalities for probability measures on the real line

By Franck Barthe, Cyril Roberto. 2003

135 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis, Political Science and International Relations

View Paper: https://www.semanticscholar.org/paper/07f33e5e52888ad7026f4365f113a279c787c4bb

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In Applied Mathematics

In Numerical Analysis

In Political Science and International Relations

Citations in the corpus

Poincaré inequalities and dimension free concentration of measure (2010, methodology): “The LO(p,C) inequalities on R were completely characterized by Barthe and Roberto in [7].”
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, methodology): “Our approach is an extension to any dimension of the criteria on the real line recently obtained through Hardy inequalities [15,12].”
Transport Inequalities. A Survey (2010, background): “Poincaré inequalities [55, 19], logarithmic Sobolev inequalities [67, 17], modified logarithmic Sobolev inequalities [19, 21, 47, 10], inf-convolution inequalities [82, 66], Beckner-Lata laOleszkiewicz inequalities [11, 65, 9, 7].”
Isoperimetry between exponential and Gaussian (2006, methodology): “It turns out that the tools of [9] apply to both, see for example Lemma 3 below.”
Asymptotics of the Fast Diffusion Equation via Entropy Estimates (2007, unknown): “The following result is inspired by [3, 8, 19, 38].”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “We can now state the following extension of the tensorization results of Barthe, Cattiaux and Roberto [6, 7], already roughly stated in Section 1.”
Improving (2018, unknown): “Thus we argue that these rainfall forecasts provide skill complementary to that from the hydrological initial conditions.”
Concentration for independent random variables with heavy tails (2005, background): “[10]) yields dimension free Gaussian concentration, whereas recent inequalities devised by Lata la and Oleszkiewicz [9] provide intermediate rates, see also [4, 15, 3].”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, methodology): “It is known that Hardy-type inequalities are useful tool to deal with functional inequalities of Poincaré type in dimension 1 (see [13; 12] for recent contributions on the topic).”
Bilateral Hardy-type inequalities (2012, background): “We mention that there is a large number of publications on the logarithmic Sobolev inequalities, in the one-dimensional case for instance, one may refer to [14], [1], [5; §4.”
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]).”
Orlicz–Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups☆ (2006, methodology): “In [BCR05b], the authors introduce a general tool to obtain a criterium which is based on an appropriate notion of capacity ([Maz85]) initially introduced in [BR03].”
Modified logarithmic Sobolev inequalities and transportation inequalities (2004, methodology): “And in [BR03] the authors present a simple proof of the result of Lata la-Oleszkiewicz and describe the measures on the line which enjoy the same inequality.”
A characterization of dimension free concentration in terms of transportation inequalities (2008, background): “Among the many functional inequalities yielding concentration estimates, let us mention the following: Poincaré inequalities [8, 23], logarithmic-Sobolev inequalities [7, 27], modified logarithmic-Sobolev inequalities [4, 8, 10, 18], transportation-cost inequalities [6, 7, 9, 20, 30, 33–35], inf-con”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, unknown): “[15], [27], [16] for further applications.”
Entropies, convexity, and functional inequalities (2002, result): “which is exactly the inequality studied in [Bec89] for the Gaussian measure and in [LO00], [Wan02], [Bar01], [ BR03 ] and [BCR04] for further developments related to this particular case.”
On quadratic transportation cost inequalities (2006, result): “We refer to [26,23,3,11] and [2] for more refined results in connection with I (α).”