Embeddings and Extensions in Analysis

Wells, James; Williams, Lynn R.

doi:10.1007/978-3-642-66037-5

Cited by 193 publications

(121 citation statements)

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“…Since for p ∈ [1, 2] the metric space L p , x − y p/2 2 embeds isometrically into L 2 (see [42]), it follows that L p has Markov type p with constant 1. For p > 2 it was shown in [28] that L p has Markov type 2 with constants O √ p .…”

Section: Edge Markov Type Need Not Imply Enflo Typementioning

confidence: 99%

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Embeddings of Discrete Groups and the Speed of Random Walks

Naor

Peres

2008

International Mathematics Research Notices

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Let G be a group generated by a finite set S and equipped with the associated left-invariant word metric d G . For a Banach space X let α * X (G) (respectively α # X (G)) be the supremum over all α ≥ 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f : G → X and c > 0 such that for all x, y ∈ G we have f (We show that if X has modulus of smoothness of power type p,Here β * (G) is the largest β ≥ 0 for which there exists a set of generators S of G and c > 0 such that for all t ∈ N we have E d G (W t , e) ≥ ct β , where {W t } ∞ t=0 is the canonical simple random walk on the Cayley graph of G determined by S , starting at the identity element. This result is sharp when X = L p , generalizes a theorem of Guentner and Kaminker [20], and answers a question posed by Tessera [37]. We also show that if. This improves the previous bound due to Stalder and Valette [36]. We deduce that if we write2−2 1−k , and use this result to answer a question posed by Tessera in [37] on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C 2 ≀ C n embed into L 1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres in [26]. Finally, we use these results to show that edge Markov type need not imply Enflo type.

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Section: Edge Markov Type Need Not Imply Enflo Typementioning

confidence: 99%

“…Since the metric space L 1 , x − y 1 embeds isometrically into L 2 (see [42]), a positive solution to Question 7.5 would imply a positive solution to Question 7.3.…”

Section: Second Embedding Ofmentioning

confidence: 99%

Embeddings of Discrete Groups and the Speed of Random Walks

Naor

Peres

2008

International Mathematics Research Notices

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“…Let us point out that if Ω satisfies a uniform cone property, as described above, then also 25) at least if the height of Γ is sufficiently small relative to r (appearing in (2.24)). Indeed, the existence of a point y ∈ (x 0 − R(Γ)) ∩ Ω would entail x 0 ∈ y + R(Γ).…”

Section: Definition 22mentioning

confidence: 99%

“…Indeed, Kirszbraun's Theorem asserts that any Lipschitz function defined on a subset of a metric space can be extended to a Lipschitz function on the entire space with the same Lipschitz constant (see, e.g., [25]; for a more elementary result which will, nonetheless, do in the current context see Theorem 5.1 on p. 29 in [23]). This shows that Ω is a strongly Lipschitz domain near x 0 , hence concluding the proof of the theorem.…”

Section: Definition 22mentioning

confidence: 99%

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Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

2007

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In the first part of this paper we give intrinsic characterizations of the classes of Lipschitz and C 1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C 1 . In the second part of the paper, we study the invariance of various classes of domains of locally finite perimeter under biLipschitz and C 1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C 1 diffeomorphisms. A number of other applications are also presented.

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Vector‐valued optimal Lipschitz extensions

Sheffield⋆

Smart

2011

Comm Pure Appl Math

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Abstract.Consider a bounded open set U ⊂ R n and a Lipschitz function g : ∂U → R m . Does this function always have a canonical optimal Lipschitz extension to all of U ? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.

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Embeddings and Extensions in Analysis

Cited by 193 publications

References 0 publications

Embeddings of Discrete Groups and the Speed of Random Walks

Embeddings of Discrete Groups and the Speed of Random Walks

Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

Vector‐valued optimal Lipschitz extensions

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