Sergi Baena-Miret, Amiran Gogatishvili, Zdeněk Mihula, Luboš Pick
We study interpolation properties of operators (not necessarily linear) which satisfy a specific $K$-inequality corresponding to endpoints defined in terms of Orlicz--Karamata spaces modeled upon the example of the Gaussian--Sobolev embedding. We prove a reduction principle for a fairly wide class of such operators.
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In this note we bound the deficit in the logarithmic Sobolev Inequality and in the Talagrand transport-entropy Inequality for the Gaussian measure, in any dimension, by mean of a distance introduced by Bucur and Fragalà.
A Gordon-Chevet type Inequality
证明了高斯过程的一个新的不等式，该不等式蕴涵了Gordon-Chevet不等式.给出了Dvoretzky定理的高斯证明的一些注记。
A sharp Poincaré-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure.
A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural semigroup associated to the Gaussian $\partial$-Laplacian.
We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free improvement for a related result given in [Eldan '18].
A refinement of the Šidák-Khatri inequality and a strong Gaussian correlation conjecture
本文证明了Tehranchi强高斯相关性猜想的两个特例，证明了其渐近成立则任意维度均成立，并以此改进了Šidák-Khatri不等式。
The Gaussian Conjugate Rogers-Shephard Inequality
本文推导了一个关于高斯测度的共轭罗杰斯-谢普德不等式，并扩展了相关高斯不等式。
We establish dual equivalent forms involving relative entropy, Fisher information and optimal transport costs of inverse Santal{ó} inequalities. We show in particular that the Mahler conjecture is equivalent to some dimensional lower bound on the deficit in the Gaussian logarithmic Sobolev inequality. We also derive from existing results on inverse Santal{ó} inequalities some sharp lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality.
The symmetric property tau for the Gaussian measure
基于Poincaré不等式给出了高斯测度的对称性τ的一个证明。这个性质被证明等价于布拉施克-桑塔洛不等式的某种函数形式，正如Artstein，Klartag和Milman在一篇论文中所解释的那样。
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
本文基于马尔可夫半群与Γ-微分算子，推导加权高斯测度下的广义贝克纳不等式及皮卡德不等式稳定性，进一步研究其梯度和L^2稳定性，并应用至海森堡不确定性原理分析及对数Sobolev不等式构建。
We establish a dimension-free improvement of Talagrand's Gaussian transport-entropy inequality, under the assumption that the measures satisfy a Poincaré inequality. We also study stability of the inequality, in terms of relative entropy, when restricted to measures whose covariance matrix trace is smaller than the ambient dimension. In case the covariance matrix is strictly smaller than the identity, we give dimension-free estimates which depend on its eigenvalues. To complement our results, we show that our conditions cannot be relaxed, and that there exist measures with covariance larger th…
Stability of Gaussian Poincaré inequalities and Heisenberg Uncertainty Principle with monimial weights
该研究利用Bakry-Émery准则与Γ-演算建立单项式高斯测度庞加莱不等式，通过对偶方法研究其改进与梯度稳定性，构建尺度依赖不等式并用于推导带单项式权重的海森堡不确定性原理的稳定性，还得到经典情形下的相关稳定性结果。
Let $Z$ be an $n$-dimensional Gaussian vector and let $f: \mathbb R^n \to \mathbb R$ be a convex function. We show that: $$\mathbb P \left( f(Z) \leq \mathbb E f(Z) -t\sqrt{ {\rm Var} f(Z)} \right) \leq \exp(-ct^2),$$ for all $t>1$, where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.
$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $\lat \subseteq \R^n$ and vectors $\vec{x}, \vec{y} \in \R^n$, \[ ρ(\lat + \vec{x})^2 ρ(\lat + \vec{y})^2 \leq ρ(\lat)^2 ρ(\lat + \vec{x} + \vec{y}) ρ(\lat + \vec{x} - \vec{y}) \; , \] where $ρ$ is the Gaussian measure $ρ(A) := \sum_{\vec{w} \in A} \exp(-π\| \vec{w} \|^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive corr…
Gaussian Correlation via Inverse Brascamp-Lieb
本研究借助推广的Nakamura-Tsuji对称逆Brascamp-Lieb不等式，给出Royen高斯相关性不等式的简洁替代证明，明确该逆不等式是正向不等式的对偶对应，且对数凹性假设不可省略。
This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.
We provide explicit lower bounds for the deficit in the Gaussian logarithmic Sobolev inequality in terms of differential operators that are naturally associated with the so-called Stein characterization of the Gaussian distribution. The techniques are based on a crucial use of the representation of the relative Fisher information, along the Ornstein-Uhlenbeck semigroup, in terms of the Minimal Mean-Square Error from information theory.
Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality
本文建立涉及不同协方差矩阵的异方差高斯Brunn-Minkowski不等式，融合两类已有高斯不等式，推导双线性版本及高斯混合相关不等式，证明基于前期的高斯正反Brascamp-Lieb不等式。
Stability for the logarithmic Sobolev inequality
该论文聚焦高斯对数Sobolev不等式的稳定性问题，推导得出了显式的稳定性常数，为相关不等式的稳定性分析提供了量化结果。
The Gaussian correlation inequality for centered convex sets and the case of equality
本文受Milman启发，证明高斯相关不等式对同重心凸集（尤其是中心凸集）成立，回答了Szarek和Werner的问题，刻画等式情形，并给出同重心凸集下高斯向量事件及范数随机变量独立的充要条件。
Grünbaum's inequality for Gaussian and convex probability measures
本文在特定凹性假设下建立了一系列精确的Grünbaum型概率测度不等式并给出等号刻画，推导出高斯测度的Ehrhard-Grünbaum不等式，简化了$s$-凹Radon测度的证明并补全了等号刻画。
The de cit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances.
Stability of the Poincaré-Korn inequality
本文在矩约束下解决了 Carrapatoso 等人关于 Poincaré-Korn 不等式锐常数高斯最优性的问题，并证明接近最优常数的测度在定量意义上接近标准高斯分布。
Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for gaussian measure
令$μ$为高斯测度（例如，在${\bf R}^n$上），且令$K, L \subset {\bf R}^n$使得K为凸的，$L$为“层”（即，对于某些$a$、$b \in {\bf R}$和$u \in {\bf R}^n$，为$L = \{x : a \leq < x,u > \leq b \}$），且$K$和$L$的质心（相对于$μ$）重合。然后是$μ(K \cap L) \geq μ(K) \cdot μ(L)$。这是由著名的对称集的“正相关猜想”和Sidak关于多元正态分布的置信区域的相关不等式所激发的。证明使用了一个显然是迄今为止未知的（标准）高斯累积分布函数的估计：$Φ(x) > 1 - \frac{(8/π)^{1/2}}{3x + (x^2 +8)^{1/2}} e^{-x^2/2}$（对$x > -1$有效）。
We study the $L^p$ boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on $L^p$ for $1 \textless{} p \textless{} 2$, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this.In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for $1 \textless{} p \textless{} 2$. This yields a full picture of the ranges of $p\in (1,+\infty)$ for whi…
Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $λ\in(0,1)$, $$γ_n(λK+(1-λ)L)^{\frac{1}{n}} \geq λγ_n(K)^{\frac{1}{n}}+(1-λ)γ_n(L)^{\frac{1}{n}},$$ thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure. We also show that, for a fixed $λ\in(0,1)$, equality is attained if and only if $K=L$.
In this paper we adapt the well-estabilished $Γ$-calculus techniques to the context of $RCD(K,\infty)$ spaces, proving Bobkov's local isoperimetric inequality and, when $K$ is positive, the Gaussian isoperimetric inequality in this class of spaces. The proof relies on the measure-valued $Γ_2$ operator introduced by Savaré.
An approach to sub-Gaussian heat kernel estimates via analysis on metric spaces
本文通过引入截断能量条件，将截断索博列夫不等式转化为庞加莱型不等式，并利用度量空间分析技术，为强局部正则狄利克雷形式的次高斯热核估计建立了新刻画。
A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$
通过模函数参数化和范数1分裂、ADE根计数及影子正性，证明秩不超过32的整幺模格的高斯质量在稳定格中由整数格最大化，并得到秩40偶幺模格的严格不等式。
The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on Rd. We especially focus on getting good dependence of the constants on the dimension. We prove that the Poincar{é} inequality holds with a dimension-free bound. For the logarithmic Sobolev inequality, we improve the best known results (Zimmermann, JFA 2013) by getting a bound that grows linearly with the dimension. We also establish transport-entropy inequalities for various transport costs.
A proof of the strong Gaussian product inequality conjecture
本文证明了强高斯乘积不等式猜想，即中心化高斯向量绝对值的任意正幂次乘积期望不小于各分量期望之积，并由此解决了实线性极化常数猜想，推导出多项相关不等式与刻画。
On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices
本文将 GPI 猜想相关结果推广至 Wishart 矩阵不相交主子式，建立乘积型与定量不等式，并利用 MTP₂ 证明特征值情形的扩展 GPI。
Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert, we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo--Nirenberg--Sobolev inequalities due to Del Pino and Dolbeault \cite{DelPino}.
Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality
设$\M$是光滑连通流形，具有光滑测度$μ$和关于$μ$对称的光滑局部次椭圆扩散算子$L$.我们假设$L$满足Baudoin-Garofalo \cite{BG1}引入的一个广义曲率维数不等式.我们的目标是讨论$μ$的函数不等式，如Poincaré不等式，log-Sobolev不等式或高斯对数等周不等式.
A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $e^{-|t|^p}$ and symmetric $p$-stable random variables, where $p\in(0,2]$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian…
A generalized Legendre duality relation and Gaussian saturation
本研究推广了勒让德对偶关系，基于勒让德对偶与逆Brascamp–Lieb不等式的关联，证明了退化Brascamp–Lieb数据下偶函数与对数凹函数输入的中心化高斯饱和性，解决了凸几何及信息论、最优输运领域的两个相关猜想。
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