scholarly journals A remark on Poincaré inequalities on metric measure spaces

2004 ◽  
Vol 95 (2) ◽  
pp. 299 ◽  
Author(s):  
Stephen Keith ◽  
Kai Rajala
Keyword(s):  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Regular Measure ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Lipschitz Constants

We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].

scholarly journals Resistance conditions, Poincaré inequalities, the Lip-lip condition and Hardy’s inequalities

2016 ◽  
Vol 49 (1) ◽  
Author(s):  
Juha Kinnunen ◽  
Pilar Silvestre
Keyword(s):  
Poincaré Inequality ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Hardy Inequalities ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Lipschitz Constants ◽  
Hardy’S Inequalities ◽  
Weaker Conditions

AbstractThis note investigates weaker conditions than a Poincaré inequality in analysis on metric measure spaces. We discuss two resistance conditions which are stated in terms of capacities. We show that these conditions can be characterized by versions of Sobolev–Poincaré inequalities. As a consequence, we obtain so-called Lip-lip condition related to pointwise Lipschitz constants. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent under an appropriate resistance condition.

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

scholarly journals BV and Sobolev homeomorphisms between metric measure spaces and the plane

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Camillo Brena ◽  
Daniel Campbell
Keyword(s):  
Open Subset ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Measure Spaces

Abstract We show that, given a homeomorphism f : G → Ω {f:G\rightarrow\Omega} where G is an open subset of ℝ 2 {\mathbb{R}^{2}} and Ω is an open subset of a 2-Ahlfors regular metric measure space supporting a weak ( 1 , 1 ) {(1,1)} -Poincaré inequality, it holds f ∈ BV loc ⁡ ( G , Ω ) {f\in{\operatorname{BV_{\mathrm{loc}}}}(G,\Omega)} if and only if f - 1 ∈ BV loc ⁡ ( Ω , G ) {f^{-1}\in{\operatorname{BV_{\mathrm{loc}}}}(\Omega,G)} . Further, if f satisfies the Luzin N and N - 1 {{}^{-1}} conditions, then f ∈ W loc 1 , 1 ⁡ ( G , Ω ) {f\in\operatorname{W_{\mathrm{loc}}^{1,1}}(G,\Omega)} if and only if f - 1 ∈ W loc 1 , 1 ⁡ ( Ω , G ) {f^{-1}\in\operatorname{W_{\mathrm{loc}}^{1,1}}(\Omega,G)} .

Orlicz-Poincaré inequalities, maximal functions and AΦ-conditions

2010 ◽  
Vol 140 (1) ◽  
pp. 31-48 ◽  
Author(s):  
Jana Björn
Keyword(s):  
Measure Space ◽  
General Setting ◽  
Maximal Functions ◽  
The Other ◽  
Metric Measure Spaces ◽  
Young Function ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Measure Spaces ◽  
Littlewood Maximal Operator

For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

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.

The Poincaré Inequality and Reverse Doubling Weights

2004 ◽  
Vol 47 (2) ◽  
pp. 206-214 ◽  
Author(s):  
Ritva Hurri-Syrjänen
Keyword(s):  
Large Class ◽  
Poincaré Inequality ◽  
Irregular Domains ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Doubling Weights ◽  
Poincare Inequalities ◽  
John Domains

AbstractWe show that Poincaré inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

ESTIMATES OF BEST CONSTANTS FOR WEIGHTED POINCARÉ INEQUALITIES ON CONVEX DOMAINS

2006 ◽  
Vol 93 (1) ◽  
pp. 197-226 ◽  
Author(s):  
SENG-KEE CHUA ◽  
RICHARD L. WHEEDEN
Keyword(s):  
Poincaré Inequality ◽  
Concave Function ◽  
Continuous Functions ◽  
Directional Derivatives ◽  
Convex Domains ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Best Constant ◽  
Lipschitz Continuous ◽  
Poincare Inequalities

Let $1 \le q \le p <\infty$ and let $\mathcal{C}$ be the class of all bounded convex domains $\Omega$ in $\mathbb{R}^n$. Following the approach in `An optimal Poincaré inequality in $L^1$ for convex domains', by G. Acosta and R. G. Durán (Proc. Amer. Math. Soc. 132 (2003) 195–202), we show that the best constant $C$ in the weighted Poincaré inequality$$ \| f - f_{av} \|_{L^q_w (\Omega)} \le C w(\Omega)^{\frac{1}{q} - \frac{1}{p}} \mbox{diam}(\Omega) \| \nabla f \|_{L^p_w(\Omega)} $$for all $\Omega \in \mathcal{C}$, all Lipschitz continuous functions $f$ on $\Omega$, and all weights $w$ which are any positive power of a non-negative concave function on $\Omega$ is the same as the best constant for the corresponding one-dimensional situation, where $\mathcal{C}$ reduces to the class of bounded intervals. Using facts from `Sharp conditions for weighted 1-dimensional Poincaré inequalities', by S.-K. Chua and R. L. Wheeden (Indiana Math. J. 49 (2000) 143–175), we estimate the best constant. In the case $q = 1$ and $1 <\infty$, our estimate is between the best constant and twice the best constant. Furthermore, when $p = q = 1$ or $p = q = 2$, the estimate is sharp. Finally, in the case where the domains in $\mathbb{R}^n$ are further restricted to be parallelepipeds, we obtain a slightly different form of Poincaré's inequality which is better adapted to directional derivatives and the sidelengths of the parallelepipeds. We also show that this estimate is sharp for a fixed rectangle.

scholarly journals The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Marcello Lucia ◽  
Michael J. Puls
Keyword(s):  
Real Number ◽  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Type Problem ◽  
Locally Compact ◽  
Dirichlet Type ◽  
Algebra Of Functions ◽  
Measure Spaces ◽  
Dirichlet Type Problem

Abstract Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

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