The Poincaré Inequality Does Not Improve with Blow-Up

Schioppa, Andrea

doi:10.1515/agms-2016-0018

Cited by 6 publications

(7 citation statements)

References 22 publications

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“…By now, several classes of spaces with Poincaré inequalities are known. See for example [53,55,20,45,60,17,24,63,64,48]. The proofs that these examples satisfy Poincaré inequalities are often challenging, and make extensive use of the geometry of the underlying space.…”

Section: Remarkmentioning

confidence: 99%

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Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

Eriksson-Bique¹

2019

Geom. Funct. Anal.

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We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local (1, p)-Poincaré inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new "thickening" construction, which can be used to enlarge subsets into spaces admitting Poincaré inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincaré inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincaré inequalities for the class of M CP (K, n)-spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincaré inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincaré inequalities "self-improve" to classical (1, q)-Poincaré inequalities for some q ∈ [1, ∞), which is related to Keith's and Zhong's theorem on selfimprovement of Poincaré inequalities.

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Section: Remarkmentioning

confidence: 99%

“…Previous characterizations either assumed Ahlfors regularity [43], or presumed knowledge of the exponent p, such as in [48] and [30]. As demonstrated by examples of Schioppa [63], it is possible for this exponent to be arbitrarily large. Thus, applying characterizations from [48] seem difficult in some cases where the exponent is a priori unknown.…”

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confidence: 99%

Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

Eriksson-Bique¹

2019

Geom. Funct. Anal.

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“…If it were to hold for any bounded ρ then a simple scaling argument would show that we in fact have a 1-Poincaré inequality in the tangent. This is impossible by the result of Schioppa [Sch15a] that constructs, for any p ≥ 1, a space whose tangents (and the original space) satsify a p ′ -Poincaré inequality for every p ′ > p but not for p. Remark 6.8. As a corollary we see that for almost every x in a RNP-LDS, every element of Tan(X, µ, x) is also a RNP-LDS.…”

Section: Continuous With Respect To Eachmentioning

confidence: 99%

Differentiability and Poincaré-type inequalities in metric measure spaces

Bate

2018

Advances in Mathematics

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We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect nearby points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger-Kleiner [CK15], we show that our conditions are also sufficient.We also develop another characterization of RNP Lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in [Bat15].

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“…In fact, for a given limiting metric arising from their construction, they construct an uncountable collection of distinct measures which make the limit into a PI space. Schioppa observed that the measures in an uncountable subset of these are in actuality, mutually singular; [62]. For additional examples of PI spaces, see [61], [48].…”

Section: Introductionmentioning

confidence: 99%

Thin Loewner carpets and their quasisymmetric embeddings in $S^2$

Cheeger¹,

Eriksson-Bique²

2019

Preprint

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A carpet is a metric space which is homeomorphic to the standard Sierpinski carpet in R 2 , or equivalently, in S 2 . A carpet is called thin if its Hausdorff dimension is < 2. A metric space is called Q-Loewner if its Q-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a (1, Q)-Poincaré inequality. As we will show, Q-Loewner planar metric spaces are always carpets, and admit quasisymmetric embeddings into the plane.In this paper, for every pair (Q, Q ), with 1 < Q < Q < 2 we construct infinitely many pairwise quasi-symmetrically distinct Q-Loewner carpets X which admit explicit snowflake embeddings, f : X → S 2 , for which the image, f (X), admits an explicit description and is Q -Ahlfors regular. In particular, these f are quasisymmetric embeddings. By a result of Tyson, the Hausdorff dimension of a Loewner space cannot be lowered by a quasisymmetric homeomorphism. By definition, this means that the carpets X and f (X) realize their conformal dimension. Each of images f (X) can be further uniformized via post composition with a quasisymmetric homeomorphism of S 2 , so as to yield a circle carpet and also a square carpet.Our Loewner carpets X are constructed via what we call an admissable quotiented inverse system. This mechanism extends the inverse limit construction for PI spaces given in [25], which however, does not yield carpets. Loewner spaces are a particular subclass of PI spaces. They have strong rigidity properties which which do not hold for PI spaces in general.In many cases the construction of the carpets and their snowflake embeddings, f , can also be described in terms of replacement rules. The statement above concerning (Q, Q ) is already a consequence of these examples. The images of these snowflake embeddings can be de-snowflaked using a deformation by a strong A∞ weight, which multiplies the metric infinitesimally by a conformal factor of the form ω = d(Image(f ), •) α . Consequently, our examples also yield new examples of strong A∞-weights for which the associated metrics admit no bi-Lipschitz embeddings into Banach spaces with the Radon Nikodym Property such as Lp, for 1 < p < ∞ and 1.

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The Poincaré Inequality Does Not Improve with Blow-Up

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

Cited by 6 publications

References 22 publications

Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

Differentiability and Poincaré-type inequalities in metric measure spaces

Thin Loewner carpets and their quasisymmetric embeddings in $S^2$

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