scholarly journals Poincaré inequality for abstract spaces

2005 ◽  
Vol 71 (2) ◽  
pp. 193-204 ◽  
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Ricci Curvature ◽  
Poincaré Inequality ◽  
Strong Version ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Poincare Inequality ◽  
Contraction Property ◽  
Measure Spaces ◽  
Measure Contraction Property ◽  
Abstract Spaces

The Poincaré inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincaré inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

scholarly journals The ∞-Poincaré inequality on metric measure spaces

2012 ◽  
Vol 61 (1) ◽  
pp. 63-85 ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Jesús Jaramillo ◽  
Nageswari Shanmugalingam
Keyword(s):  
Poincaré Inequality ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Measure Spaces

scholarly journals Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

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.

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 The Poincaré Inequality Does Not Improve with Blow-Up

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Andrea Schioppa
Keyword(s):  
Blow Up ◽  
Poincaré Inequality ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Measure Spaces

AbstractFor each β > 1 we construct a family Fβ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e. blow-ups), and such that each element of Fβ admits a (1, P)-Poincaré inequality if and only if P > β.

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 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)} .

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