Rademacher type and Enflo type coincide

Ivanisvili, Paata; Handel, Ramon van; Volberg, Alexander

doi:10.4007/annals.2020.192.2.8

Cited by 14 publications

(33 citation statements)

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“…Remark added in proofs. After the submission of this paper, Ivanisvili, van Handel and Volberg circulated a preprint [IvHV20] showing that a Banach space satisfies for every (equivalently, for some) if and only if X has finite cotype.…”

Section: Discussionmentioning

confidence: 99%

On Pisier’s inequality for UMD targets

Eskenazis

2020

Can. Math. Bull.

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Section: Discussionmentioning

confidence: 99%

On Pisier’s inequality for UMD targets

Eskenazis

2020

Can. Math. Bull.

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“…In the proof of Theorem 2, the role of L q (µ) as the target space is only through the validity of (47). For this, it suffices for the target to have Enflo type q (using terminology of [BMW86]), and by the remarkable work [IvHV20] this is equivalent to the target having type q in the linear sense (recall (17)).…”

Section: Approximate Smoothing Of Powers Of the Crinkled Arcmentioning

confidence: 99%

Impossibility of almost extension

Naor

2020

Preprint

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Let (X, • X ),(Y, • Y ) be normed spaces with dim(X) = n. Bourgain's almost extension theorem asserts that for any ε > 0, if N is an ε-net of the unit sphere of X and f :We prove that this is optimal up to lower order factors, i.e., sometimes max a∈N F (athen the approximation in the almost extension theorem can be improved to max a∈N F (a) − f (a) Y nε. We prove that this is sharp, i.e., sometimes max a∈N F (a) − f (a) Y nε for every O(1)-Lipschitz F : ℓ n 2 → Y.

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“…Vector-valued influence inequalities. In view of Enflo's problem [Enf78] and its recent solution in [IvHV20], it would be most natural to try and understand for which Banach spaces (E, • E ) there exists a constant C = C(E) ∈ (0, ∞) such that for every n ∈ N, every function f :…”

Section: Asymptotic Notationmentioning

confidence: 99%

“…The proof of Theorem 1 builds upon a novel idea exploited in [IvHV20], which in turn is reminiscent of a trick due to Maurey [Pis86]. It remains unclear whether one can deduce from this idea a vector-valued extension of Talagrand's inequality (6) for spaces of Rademacher type 2 and whether the doubly logarithmic error term σ(f ) on the right hand side of (9) is needed.…”

Section: Asymptotic Notationmentioning

confidence: 99%

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Talagrand's influence inequality revisited

Cordero–Erausquin¹,

Eskenazis²

2020

Preprint

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Let Cn = {−1, 1} n be the discrete hypercube equipped with the uniform probability measure σn. Talagrand's influence inequality (1994), also known as the L1 − L2 inequality, asserts that there exists C ∈ (0, ∞) such that for every n ∈ N, every function f :.In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach spacevalued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. Moreover, our proof implies vector-valued versions of a general family of L1 − Lp inequalities, each refining the dimension independent Lp-Poincaré inequality on (Cn, σn). We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman ( 2002) on the action of negative powers of the hypercube Laplacian on functions f : Cn → E, whose target space (E, • E ) has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube Cn equipped with the Hamming metric, thus deriving new nonembeddability results for these finite metrics. Our proofs make use of Banach space-valued Itô calculus, Riesz transform inequalities, Littlewood-Paley-Stein theory and hypercontractivity.

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Rademacher type and Enflo type coincide

Cited by 14 publications

References 0 publications

On Pisier’s inequality for UMD targets

On Pisier’s inequality for UMD targets

Impossibility of almost extension

Talagrand's influence inequality revisited

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