首页/论文/2011.05207

arXiv 2011.052072020-11math.CAmath.PR

On the variational interpretation of local logarithmic Sobolev inequalities

Gauthier Clerc, Giovanni Conforti, Ivan Gentil

英文摘要：

The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $Γ$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{ö}dinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly unlock novel class of local inequalities.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Trace inequalities for Sobolev martingales

We study limiting trace inequalities in the style of Maz'ya and Meyers--Ziemer for Sobolev martingales. We develop the Bellman function approach to such estimates, which allows to provide sufficient and almost necessary conditions on the martingale space and the martingale transform under which the trace inequalities hold true

2022-11-24math.APmath.CAmath.PR

Fractional Sobolev-Poincaré and fractional Hardy inequalities in unbounded John domains

We prove fractional Sobolev-Poincaré inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.

2013-11-12math.CA

Improved Poincaré inequalities in fractional Sobolev spaces

We obtain improved fractional Poincaré and Sobolev Poincaré inequalities including powers of the distance to the boundary in John, $s$-John domains and Hölder-$α$ domains, and discuss their optimality.

2017-05-11math.CA

致敬Guido Weiss及其对Saint Louis团队的领导：奇异积分的交换子与Sobolev不等式

A homage to Guido Weiss and his leadership of the Saint Louis team: Commutators of Singular Integrals and Sobolev inequalities

本文扩展了线性与非线性交换子的经典Sobolev型不等式。

2023-07-28math.CA

粗糙算子的逐点估计及其在Sobolev不等式中的应用

Pointwise estimates for rough operators with applications to Sobolev inequalities

本文研究粗糙算子的Sobolev不等式，证明其满足被Riesz势作用于梯度的逐点界，并由此推导出左侧带算子的新Sobolev型不等式。

2024-01-03math.CA

Markov-type inequalities and duality in weighted Sobolev spaces

The aim of this paper is to provide Markov-type inequalities in the setting of weighted Sobolev spaces when the considered weights are generalized classical weights. Also, as results of independent interest, some basic facts about Sobolev spaces with respect to certain vector measures are stated.

2015-01-25math.CA

更多相关论文收起更多论文

Gagliardo-Nirenberg type inequalities using fractional Sobolev spaces and Besov spaces

Our main purpose is to establish Gagliardo-Nirenberg type inequalities using fractional homogeneous Sobolev spaces, and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in [1, 2, 3, 7, 16, 21].

2022-12-10math.CA

Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces

In this article we generalize the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces instead the classical Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality. We consider for simplicity only the case of whole space.

2010-02-25math.CAmath.FA

Logarithmic Sobolev inequality for diffusion semigroups

Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequality.

2010-09-17math.APmath.CAmath.DG

On improved fractional Sobolev-Poincaré inequalities

We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a `separation property', is a John domain.

2013-12-18math.CA

Relative Nash-type and $L^2$-Sobolev inequalities for Dunkl operators and applications

We investigate local variants of Nash inequalities in the context of Dunkl operators. Pseudo-Poincaré inequalities are first established using pointwise gradient estimates of the Dunkl heat kernel. These inequalities allow to obtain relative Nash-type inequalities which are used to derive mean value inequalities for subsolutions of the heat equation on orbits of balls not necessarily centered on the origin.

2021-02-09math.CA

In between the inequalities of Sobolev and Hardy

We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.

2015-02-04math.APmath.CA

Sobolev Trace Inequalities

The existence of extremal functions for the Sobolev trace inequalities is studied using the concentration compactness theorem. The conjectured extremal, the function of conformal factor, is considered and is proved to be an actual extremal function with extra symmetry condition on functions. One of the limiting cases of the Sobolev trace inequalities is investigated and the best constant for this case is computed.

2001-07-09math.APmath.CA

Higher-order multilinear Poincaré and Sobolev inequalities in Carnot groups

The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.

2013-05-15math.APmath.CA

New Proofs of Davies-Simon's Theorems about Ultracontractivity and Logarithmic Sobolev Inequalities related to Nash Type Inequalities

We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the semigroup. In our proof, we use neither the $L^p$ version of the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We also present new results. In particular, we show that Nash type inequality and LSIWP are equivalent.

2006-09-05math.CAmath.FA

Limiting Sobolev and Hardy inequalities on stratified homogeneous groups

We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.

2022-06-02math.APmath.CAmath.CV

An Endpoint Version of Uniform Sobolev Inequalities

We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [2]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.

2018-07-29math.APmath.CA

Hardy-Sobolev inequalities and weighted capacities in metric spaces

Let $Ω$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $Ω$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $Ω$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.

2021-06-10math.APmath.CA

Sharp Hardy-Littlewood-Sobolev Inequalities on Quaternionic Heisenberg Groups

In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invaria…

2014-07-12math.APmath.CAmath.FA

Strengthened Fractional Sobolev Type Inequalities in Besov Spaces

The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of th…

2022-02-20math.APmath.CAmath.FA