Logarithmic Sobolev Inequalities

By Stéphane Boucheron, Gábor Lugosi, Pascal Massart. 2013

763 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6f0937f926799255cc6018486229db00fac3f38e

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Citations in the corpus

Nonlinear flows and rigidity results on compact manifolds (2013, background): “With some exceptions (see for instance [24, 35]), almost all related papers are concerned either with the case of critical exponents and related concentration issues or with the limiting case p = 2, where Inequality (3) has to be replaced by the logarithmic Sobolev inequality (see [38, 41, 56, 52] f”
Discrete Curvature and Abelian Groups (2015, neutral): “, [20],[14]): namely, deûne ρ to be the greatest value so that whenever 1 < r < q <∞ and √ (q − 1)/(r − 1) ≤ eρt , then n−1/q ∥Pt f ∥q ≤ n −1/r ∥ f ∥r .”
GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR A LOGARITHMIC WAVE EQUATION ARISING FROM Q-BALL DYNAMICS (2013, neutral): “See [17].”
Geometry of manifolds with densities (2012, background): “th function u on M satisfying Z M u2 e−f dv = Z M e−f dv. Specializing to the Euclidean space Rn with f(x) = 1 4 |x| 2 , one then recovers the classical logarithmic Sobolev inequality due to L. Gross [11]. More generally, in recentworkofLott and Villani [16], and Sturm [26, 27], a lower bound for th”
Ricci flow on asymptotically Euclidean manifolds (2016, methodology): “nimizer. In the case of Euclidean space, we have the following type of logarithmic Sobolev inequality, which appears in the work by Weissler [17] and is equivalent to the Gross logarithmic inequality [18]. Theorem 3.1. For any function w∈ W1,2(Rn) and R w2 dx= 1, we have (3.4) Z w 2logw2 dx≤ n 2 log”
Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms (2012, neutral): “, [19–21] and the references therein.”
Quantitative logarithmic Sobolev inequalities and stability estimates (2014, background): “Gross [21] for the standard Gaussian measure dγ(x) = dγ(x) = e (2)/2 dx (2π)n/2 on the Borel sets of R (cf.”
An asymmetric affine Pólya–Szegö principle (2012, background): “However, it is well known that inequality (44) is equivalent to the logarithmic Sobolev inequality with respect to Gauss measure due to Stam [64] and Gross [34].”
Monotonicity and its analytic and geometric implications (2012, methodology): “The function b satisfies the following crucial formula: Δðj∇bj2uÞ 1⁄4 1 2 Hessb2 − Δb 2 n g 2 þRicð∇b2;∇b2Þ b−n: [10]”
SHARP HARDY-LITTLEWOOD-SOBOLEV INEQUALITY ON THE UPPER HALF SPACE (2013, background): “The sharp HLS inequality implies Moser-Trudinger-Onofri and Beckner inequalities [2], as well as Gross’s logarithmic Sobolev inequality [24].”
The local entropy along Ricci flow---Part A: the no-local-collapsing theorems (2017, background): “he functional µ(a)(Ω,g,τ) reveals the information of the Riemannian geometry of (Ω,∂Ω,g), by choosing diﬀerent a. If a = 0, then µ(0)(Ω,g,τ) is exactly the classical Logarithmic Sobolev constant(c.f. [20], [12]). In [33], Perelman choose a as the scalar curvature function Rand discovered the monoton”
Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion (2012, background): “iener space; see [3, 27] for the relationship between T2(C) and other properties such as the Poincar´e inequality and Hamilton–Jacobi equations. The logarithmic Sobolev inequality introduced by Gross [17] plays a particular role in this theory since it implies T2(C) (see [3, 27, 32]). With regard to”
Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces (2015, background): “ and has been on relevant and non-trivial examples, such as nding the sharp constant in Sobolev’s inequality [1,45], the Hardy-Littlewood-Sobolev inequality [39] or the logarithmic Sobolev inequality [33] to mention classical examples. In the former two instances, rearrangement inequalities are the ”