Removable sets for the poincare inequality on metric spaces

2000 ◽  
Vol 49 (1) ◽  
pp. 0-0 ◽  
Author(s):  
P. Koskela ◽  
N. Shanmugalingam ◽  
H. Tuominen
Keyword(s):  
Metric Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Removable Sets

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 166-181
Author(s):  
Rebekah Jones ◽  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Poincaré Inequality ◽  
Quasiconformal Mappings ◽  
Duality Relation ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
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 Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Sylvester Eriksson-Bique ◽  
Riikka Korte ◽  
Nageswari Shanmugalingam
Keyword(s):  
Bounded Variation ◽  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
The Other ◽  
Functions Of Bounded Variation ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
Measure Spaces

Abstract We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.

The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Boundary Values ◽  
Poincare Inequality ◽  
Compactly Supported ◽  
Bv Functions

AbstractIn the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of{\mathrm{BV}}functions with zero boundary values. In particular, we show that the class is the closure of compactly supported{\mathrm{BV}}functions in the{\mathrm{BV}}norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and{\mathrm{BV}}analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

scholarly journals Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

2021 ◽  
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Irregular Boundary ◽  
Complete Metric Spaces ◽  
Boundary Points ◽  
Local Properties ◽  
Harmonic Measures

AbstractThe trichotomy between regular, semiregular, and strongly irregular boundary points for $$p$$ p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $$p$$ p -harmonic measures, removability, and semibarriers.

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
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 ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS

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

2014 ◽  
Vol 2015 (17) ◽  
pp. 8116-8151
Author(s):  
Christian Houdré ◽  
Ionel Popescu
Keyword(s):  
Poincaré Inequality ◽  
Poincare Inequality ◽  
One Dimensional ◽  
The One

scholarly journals ORLICZ–POINCARÉ INEQUALITIES

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

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