Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

Semmes, Stephen

doi:10.1007/bf01587936

Cited by 134 publications

(188 citation statements)

References 38 publications

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“…Observe that the proof of quasiconvexity gives only one curve joining any pair of points. This result is also partial converse of the result by Semmes about families of curves implying Poincaré inequality, see [16].…”

Section: Introductionsupporting

confidence: 75%

Geometric Implications of the Poincaré Inequality

Korte¹

2007

Result. Math.

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Section: Introductionsupporting

confidence: 75%

Geometric Implications of the Poincaré Inequality

Korte¹

2007

Result. Math.

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“…Many examples of doubling measures can be found in Stein's book [63]. Theorem 4.6 is standard, see Semmes [59,Lemma C.3] or Heinonen [40,Exercise 8.11]. Theorem 4.5 is due to Volberg and Konyagin [66] in the case of a compact metric space.…”

Section: Historical Notesmentioning

confidence: 99%

Sobolev spaces on metric-measure spaces

Hajłasz¹

2003

Contemporary Mathematics

182

192

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“…It is easy to see [61,Lemma C.3] that if a metric space is Q-regular for some Borel measure µ, then it has Hausdorff dimension equal to Q, in fact, µ is comparable with the Hausdorff measure H Q (cf. Proposition 4.10).…”

Section: Basic Notation and Definitionsmentioning

confidence: 99%

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

Tyson

2001

Conform. Geom. Dyn.

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Abstract. Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu's generalized modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension Q > 1, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality.We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper Q-regular Loewner space, Q > 1, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.

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Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

Cited by 134 publications

References 38 publications

Geometric Implications of the Poincaré Inequality

Geometric Implications of the Poincaré Inequality

Sobolev spaces on metric-measure spaces

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces

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