Geometric Implications of the Poincaré Inequality

Korte, Riikka

doi:10.1007/s00025-006-0237-x

Cited by 47 publications

(45 citation statements)

References 14 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since X is an Ahlfors Q-regular complete metric space that satisfies a weak (1, p)-Poincaré inequality with 1 ≤ p < Q, there exists (see [Kor07,Theorem 4.2]) a constant C depending only on p and on the data of X such that…”

Section: ş Costeamentioning

confidence: 99%

Besov capacity and Hausdorff measures in metric measure spaces

Costea¹

2009

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

Section: ş Costeamentioning

confidence: 99%

Besov capacity and Hausdorff measures in metric measure spaces

Costea¹

2009

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…Note that the LLCcondition is not a serious restriction in our case, since it follows from the (1, Q)-Poincaré inequality, see for example [11]. Theorem 3.17.…”

Section: Resultsmentioning

confidence: 89%

“…As X is quasiconvex (which follows from the Poincaré inequality, see for example [11]) and hence path-connected, and as X \ B(x 0 , 2 k+1 r 0 ) is nonempty, there is a point…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Equivalence and self-improvement of p-fatness and Hardy’s inequality, and association with uniform perfectness

Korte

Shanmugalingam

2008

Math. Z.

Self Cite

View full text Add to dashboard Cite

Abstract. We present an easy proof that p-Hardy's inequality implies uniform p-fatness of the boundary when p = n. The proof works also in metric space setting and demonstrates the self-improving phenomenon of the p-fatness. We also explore the relationship between p-fatness, pHardy inequality, and the uniform perfectness for all p ≥ 1, and demonstrate that in the Ahlfors Q-regular metric measure space setting with p = Q, these three properties are equivalent. When p = 2, our results are new even in the Euclidean setting.

show abstract

“…One can consider for example the half line X = [0, ∞) endowed with the euclidean metric which is annularly quasiconvex only with respect to a = 0. Annular quasiconvexity was introduced in [21] and has been further used for example in [8], [19] and [20].…”

Section: Basic Conceptsmentioning

confidence: 99%

Preservation of bounded geometry under sphericalization and flattening: quasiconvexity and ∞-Poincaré inequality

Durand-Cartagena¹,

Li²

2017

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

Abstract. In this work we explore the preservation of quasiconvexity and ∞-Poincaré inequality under sphericalization and flattening in the metric setting. The results developed in [22] show the preservation of Ahlfors regularity, doubling property and the p-Poincaré inequality for 1 ≤ p < ∞ under the sphericalization and flattening transformations provided the underlying metric space is annularly quasicovex. In this work, we propose a weaker assumption to still preserve quasiconvexity and ∞-Poincaré inequality, called radially star-like quasiconvexity (corresponding to sphericalization) and meridian-like quasiconvexity (corresponding to flattening) extending in particular a result in [8] to a wider class of metric spaces and covering the case p = ∞ in [22].

show abstract

Geometric Implications of the Poincaré Inequality

Cited by 47 publications

References 14 publications

Besov capacity and Hausdorff measures in metric measure spaces

Besov capacity and Hausdorff measures in metric measure spaces

Equivalence and self-improvement of p-fatness and Hardy’s inequality, and association with uniform perfectness

Preservation of bounded geometry under sphericalization and flattening: quasiconvexity and ∞-Poincaré inequality

Contact Info

Product

Resources

About