Sharp Beckner-type inequalities for Cauchy and spherical distributions

Discrete Bochner inequalities via the Bochner-Bakry-Emery approach for Markov chains

Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincaré inequality. Their proof is based on the Bakry-Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra, and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli-Laplace mod…

An extension of the Beckner's type Poincaré inequality to convolution measures on abstract Wiener spaces

We generalize the Beckner's type Poincaré inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $μ\starν$, where $μ$ is the reference Gaussian measure and $ν$ is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and in…

Complete Sobolev Type Inequalities

We establish Sobolev type inequalities in the noncommutative settings by generalizing monotone metrics in the space of quantum states, such as matrix-valued Beckner inequalities. We also discuss examples such as random transpositions and Bernoulli-Laplace models.

广义柯西测度的对数Sobolev不等式

Logarithmic Sobolev inequalities for generalised Cauchy measures

本文通过曲率-维数准则，推导出广义柯西测度的对数Sobolev不等式，并给出最优权重和显式常数，尤其在一维情况下常数最优，进而得到集中性结果，解决病态二维情况。

广义柯西测度与凸测度的庞加莱不等式与积分曲率-维数准则

Poincar{é} inequalities and integrated curvature-dimension criterion for generalised Cauchy and convex measures

本文在黎曼流形上建立一般测度的尖锐加权庞加莱不等式，并由此统一证明广义柯西测度在全部参数范围内的加权庞加莱不等式，同时给出最优谱间隙、误差项与极值函数。

Optimal Weighted Poincaré and Log-Sobolev Inequalities for Cauchy Measures

In this paper, We establish the weighted Poincaré inequalities and Log-Sobolev inequalities for Cauchy distributions with optimal weight functions.

Weighted Poincaré-type inequalities for Cauchy and other convex measures

Brascamp--Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $κ$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

A sharp interpolation between the Hölder and Gaussian Young inequalities

We prove a very general sharp inequality of the Hölder--Young--type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the point--wise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong--Zakai--type approximation theorems, and plays a key role in some generalizations of the Beckner--type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a gener…

Regularization inequalites for one-dimensional Cauchy-type measures

In the paper we investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous properties for one-dimensional sections of multidimensional isotropic Cauchy measure. The paper is a continuation of our previous investigations \cite{BZ}, where we found, among intervals with fixed measure, the ones with the extremal measure of the boundary. Here for the above mentioned measures we investigate inequalities that are analogous to those found for Gaussian measures by Borell in \cite{B} and by Landau and Shepp in \cite{LS}.

论Gilles Pisier的高斯集中、等周性与庞加莱型不等式研究方法

On Gilles Pisier's approach to gaussian concentration, isoperimetry and Poincaré-type inequalities

本研究拓展了Gilles Pisier的研究方法，可在欧氏空间旋转不变概率分布（含多维柯西测度）中探索高斯测度相关集中、等周与庞加莱型不等式结果的对应形式。

Fokker-Planck equations and one-dimensional functional inequalities for heavy tailed densities

We study one-dimensional functional inequalities of the type of Poincaré, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain sharp inequalities satisfied by inverse Gamma densities, taking values on $R_+$, and Cauchy-type densities, taking values on $R$. In this last case, we improve the result obtained by Bobkov and Ledoux in 2009 by introducing a better weight function in the logarithmic Sobolev inequality. The results are obtained by resorting to Fokker-Planck type equations which possess these densities as steady sta…

Some New Gaussian Product Inequalities

The Gaussian product inequality is a long-standing conjecture. In this paper, we investigate the three-dimensional inequality $E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for any centered Gaussian random vector $(X_1,X_2,X_3)$ and $m_2,m_3\in\mathbb{N}$. First, we show that this inequality is implied by a combinatorial inequality. The combinatorial inequality can be verified directly for small values of $m_2$ and arbitrary $m_3$. Hence the corresponding cases of the three-dimensional inequality are proved. Second, we show that the three-dimensional inequality is equ…