Maximal function estimates and self-improvement results for Poincaré inequalities

Kinnunen, Juha; Lehrbäck, Juha; Vähäkangas, Antti V.; Zhong, Xiao

doi:10.1007/s00229-018-1016-1

Cited by 5 publications

(35 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the scale of Lipschitz functions, the proof of [16] also works in non-complete geodesic spaces. Recently, new proofs and extensions for the Keith-Zhong result have been given in [6,8,17]. In many respects this paper is a continuation of the work initiated in [17].…”

Section: Introductionmentioning

confidence: 90%

“…This fact explains why the balance condition does not play a visible role in the Keith-Zhong self-improvement results for one measure Poincaré inequalities; cf. [16,8,17].…”

Section: Balance Conditionsmentioning

confidence: 99%

“…holds for p-almost every curve γ : [0, ℓ γ ] → X; that is, there exists a non-negative Borel function ρ ∈ L p loc (X; dµ) such that γ ρ ds = ∞ whenever inequality (17) does not hold or is not defined. We refer to [1,14,15] for further information on p-weak upper gradients.…”

Section: Poincaré Inequalitiesmentioning

confidence: 99%

“…Another new feature in our approach is the introduction of the so-called maximal Poincaré inequalities, in which the left-hand side of (1) is replaced with an integral of a sharp maximal function. In the one-measure case, the corresponding maximal Poincaré inequalities are essentially equivalent to the usual Poincaré inequalities; in fact, the maximal Poincaré inequalities have been used as a tool in the proofs of the self-improvement results, for instance, in [16,17]. However, there is a difference between the usual and maximal Poincaré inequalities in the two-measure case.…”

Section: Introductionmentioning

confidence: 99%

“…First, in Section 7, we establish Theorem 7.1, which plays an important role in the proof of Theorem 6.1. This part is based on two-measure adaptations of the ideas from [17], but due to the subtle modifications that are needed we present most of the details. Finally, in Sections 8 and 9 we conclude the proof of Theorem 6.4.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A maximal Function Approach to Two-Measure Poincaré Inequalities

et al. 2018

Self Cite

View full text Add to dashboard Cite

This paper extends the self-improvement result of Keith and Zhong in [16] to the two-measure case. Our main result shows that a two-measure (p, p)-Poincaré inequality for 1 < p < ∞ improves to a (p, p − ε)-Poincaré inequality for some ε > 0 under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.2010 Mathematics Subject Classification. 31E05, 35A23, 46E35.

show abstract

Section: Introductionmentioning

confidence: 90%

“…This fact explains why the balance condition does not play a visible role in the Keith-Zhong self-improvement results for one measure Poincaré inequalities; cf. [16,8,17].…”

Section: Balance Conditionsmentioning

confidence: 99%

Section: Poincaré Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A maximal Function Approach to Two-Measure Poincaré Inequalities

et al. 2018

Self Cite

View full text Add to dashboard Cite

show abstract

The Hajłasz Capacity Density Condition is Self-improving

Canto

Vähäkangas

2022

J Geom Anal

Self Cite

View full text Add to dashboard Cite

We prove a self-improvement property of a capacity density condition for a nonlocal Hajłasz gradient in complete geodesic spaces with a doubling measure. The proof relates the capacity density condition with boundary Poincaré inequalities, adapts Keith–Zhong techniques for establishing local Hardy inequalities and applies Koskela–Zhong arguments for proving self-improvement properties of local Hardy inequalities. This leads to a characterization of the Hajłasz capacity density condition in terms of a strict upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Hajłasz capacity density condition.

show abstract

Local and semilocal Poincaré inequalities on metric spaces

Björn

2018

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

We consider several local versions of the doubling condition and Poincaré inequalities on metric measure spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball.We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincaré inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward.We also conclude that many qualitative, as well as quantitative, properties of pharmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions.Résumé. Nous considérons plusieurs versions locales des conditions de doublement et des inégalités de Poincaré dans des espaces métriques mesurés. Notre premier résultat stipule que dans un espace propre connexe, les hypothèses locales les plus faibles s'améliorent en semi-locales, c.à.d. elles sont valables dans chaque boule.Nousétudions ensuite certaines conséquences géométriques et analytiques de telles hypothèses locales tel que la quasi-convexité locale, l'auto amélioration des inégalités de Poincaré, l'existence des points Lebesgue, la densité des fonctions Lipschitz et la quasicontinuité des fonctions Sobolev. Il s'avère que les versions locales de ces propriétés restent valables sous les hypothèses locales même si elles ne sont pas toujours immédiates.Nous concluonségalement que sous telles hypothéses locales, plusieurs propriétés qualitatives, ainsi que quantitatives, des fonctions p-harmoniques sur des espaces métriques peuventêtre prouvées sous diverses formes, l'exception principaleétant le théorème de Liouville quiéchoue sans hypothéses globales.

show abstract

Maximal function estimates and self-improvement results for Poincaré inequalities

Cited by 5 publications

References 14 publications

A maximal Function Approach to Two-Measure Poincaré Inequalities

A maximal Function Approach to Two-Measure Poincaré Inequalities

The Hajłasz Capacity Density Condition is Self-improving

Local and semilocal Poincaré inequalities on metric spaces

Contact Info

Product

Resources

About