The ∞-Poincaré inequality on metric measure spaces

Cartagena, Estibalitz Durand; Jaramillo, Jesús Á.; Shanmugalingam, Nageswari

doi:10.1307/mmj/1331222847

Cited by 15 publications

(29 citation statements)

References 15 publications

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“…The argument also shows that S a does not support an ∞-Poincaré inequality. See [12] for the definition. Note that the ∞-Poincaré inequality is weaker than the p-Poincaré inequality for any finite p.…”

Section: Failure Of the Poincaré Inequalitymentioning

confidence: 99%

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

2013

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Abstract. A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.

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Section: Failure Of the Poincaré Inequalitymentioning

confidence: 99%

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

2013

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“…where B ranges over balls in M of radius r. Recall that (P p ) is weaker and weaker as p increases, that is (P p ) implies (P q ) for q > p, see for instance [36], and the p = ∞ version is trivial in the Riemannian setting (see however interesting developments for more general metric measure spaces in [24] [25,Section 5]. On conical manifolds with compact basis, (P p ) holds at least for p ≥ 2 (see [15]).…”

mentioning

confidence: 99%

Gaussian heat kernel bounds through elliptic Moser iteration

Bernicot

Coulhon

Frey

2016

Journal de Mathématiques Pures et Appliquées

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Abstract. On a doubling metric measure space endowed with a "carré du champ", we consider L p estimates (G p ) of the gradient of the heat semigroup and scale-invariant L p Poincaré inequalities (P p ). We show that the combination of (G p ) and (P p ) for p ≥ 2 always implies two-sided Gaussian heat kernel bounds. The case p = 2 is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in [37]. This relies in particular on a new notion of L p Hölder regularity for a semigroup and on a characterization of (P 2 ) in terms of harmonic functions.

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“…The rst statement can be found in Theorem 17.1 of [6]. The second can be found (in greater generality than we need here) in [9], Theorem 4.7.…”

Section: (B) For Any Bounded Lipschitz Function U On Xmentioning

confidence: 86%

Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

David

2015

Analysis and Geometry in Metric Spaces

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A theorem of Lusin states that every Borel function on R is equal almost everywhere to the derivative of a continuous function. This result was later generalized to R n in works of Alberti and Moonens-Pfe er.In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of di erentiation by a famous theorem of Cheeger.

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The ∞-Poincaré inequality on metric measure spaces

Cited by 15 publications

References 15 publications

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

Gaussian heat kernel bounds through elliptic Moser iteration

Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces

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