scholarly journals Poincaré-Type Inequalities for the Composite Operator inL𝒜-Averaging Domains

10.1155/2014/675464 â—˝  
2014 â—˝  
Vol 2014 â—˝  
pp. 1-7 â—˝  
Author(s):  
Guannan Shi â—˝  
Yuming Xing â—˝  
Baiqing Sun
Keyword(s):  
Maximal Operator â—˝  
Poincaré Inequality â—˝  
Composite Operator â—˝  
Potential Operator â—˝  
Poincare Inequality â—˝  
Definition Of â—˝  
Harmonic Equation â—˝  
Poincaré Type Inequalities

We first establish the local Poincaré inequality withL𝒜-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenousA-harmonic equation. Then, according to the definition ofL𝒜-averaging domains and relative properties, we demonstrate the global Poincaré inequality withL𝒜-averaging domains. Finally, we give some illustrations for these theorems.

scholarly journals A new approach to Poincaré-type inequalities on the Wiener space

2015 â—˝  
Vol 16 (01) â—˝  
pp. 1650002 â—˝  
Author(s):  
Alberto Lanconelli
Keyword(s):  
Wide Class â—˝  
Gaussian Measure â—˝  
Poincaré Inequality â—˝  
Probability Measures â—˝  
Poincare Inequality â—˝  
Absolutely Continuous â—˝  
New Approach â—˝  
Abstract Wiener Spaces â—˝  
New Type â—˝  
Poincaré Type Inequalities

We prove a new type of Poincaré inequality on abstract Wiener spaces for a family of probability measures that are absolutely continuous with respect to the reference Gaussian measure. This class of probability measures is characterized by the strong positivity (a notion introduced by Nualart and Zakai in [22]) of their Radon–Nikodym densities. In general, measures of this type do not belong to the class of log-concave measures, which are a wide class of measures satisfying the Poincaré inequality (Brascamp and Lieb [2]). Our approach is based on a pointwise identity relating Wick and ordinary products and on the notion of strong positivity which is connected to the non-negativity of Wick powers. Our technique also leads to a partial generalization of the Houdré and Kagan [11] and Houdré and Pérez-Abreu [12] Poincaré-type inequalities.

scholarly journals Functional Inequalities for Two-Level Concentration

Potential Analysis â—˝  
2021 â—˝  
Author(s):  
Franck Barthe â—˝  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality â—˝  
Poincaré Inequality â—˝  
Logarithmic Sobolev Inequality â—˝  
Probability Measures â—˝  
Exponential Rate â—˝  
Functional Inequalities â—˝  
Concentration Inequality â—˝  
Poincare Inequality â—˝  
Free Concentration â—˝  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 â—˝  
Vol 8 (1) â—˝  
pp. 166-181
Author(s):  
Rebekah Jones â—˝  
Panu Lahti
Keyword(s):  
Metric Space â—˝  
Metric Spaces â—˝  
Complete Metric Space â—˝  
Poincaré Inequality â—˝  
Quasiconformal Mappings â—˝  
Duality Relation â—˝  
Doubling Measure â—˝  
Poincare Inequality â—˝  
Family Of Curves â—˝  
The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

scholarly journals ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS

Econometric Theory â—˝  
2021 â—˝  
pp. 1-37
Author(s):  
Florian F. Gunsilius
Keyword(s):  
Convergence Rate â—˝  
Poincaré Inequality â—˝  
Optimal Transportation â—˝  
Matching Theory â—˝  
Economic Research â—˝  
Poincare Inequality â—˝  
Optimal Transportation Problem â—˝  
Minimax Rate â—˝  
The Cost â—˝  
Kantorovich Problem

The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

scholarly journals Refinements of the One Dimensional Free Poincaré Inequality

10.1093/imrn/rnu189 â—˝  
2014 â—˝  
Vol 2015 (17) â—˝  
pp. 8116-8151
Author(s):  
Christian Houdré â—˝  
Ionel Popescu
Keyword(s):  
Poincaré Inequality â—˝  
Poincare Inequality â—˝  
One Dimensional â—˝  
The One

scholarly journals ORLICZ–POINCARÉ INEQUALITIES

2008 â—˝  
Vol 51 (2) â—˝  
pp. 529-543 â—˝  
Author(s):  
Feng-Yu Wang
Keyword(s):  
Porous Media â—˝  
Probability Measure â—˝  
Poincaré Inequality â—˝  
Dirichlet Forms â—˝  
Sobolev Inequalities â—˝  
Young Function â—˝  
Poincare Inequality â—˝  
Poincaré Inequalities â—˝  
Poincare Inequalities â—˝  
Porous Media Equations

AbstractCorresponding to known results on Orlicz–Sobolev inequalities which are stronger than the Poincaré inequality, this paper studies the weaker Orlicz–Poincaré inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz–Poincaré inequality$$ \|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0 $$is studied by using the well-developed weak Poincaré inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.

$p$-harmonic functions on complete manifolds with a weighted Poincaré inequality

2017 â—˝  
Vol 40 (2) â—˝  
pp. 343-357
Author(s):  
Bui Van Binh â—˝  
Nguyen Thac Dung â—˝  
Nguyen Thi Le Hai
Keyword(s):  
Harmonic Functions â—˝  
Poincaré Inequality â—˝  
Poincare Inequality â—˝  
Weighted Poincaré Inequality â—˝  
Complete Manifolds

A Remark on the Steklov–Poincaré Inequality

Mathematical Notes â—˝  
2021 â—˝  
Vol 110 (1-2) â—˝  
pp. 221-225
Author(s):  
Sh. M. Nasibov
Keyword(s):  
Poincaré Inequality â—˝  
Poincare Inequality
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