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Talagrand's influence inequality revisited

Dario Cordero-Erausquin and Alexandros Eskenazis

Vol. 16 (2023), No. 2, 571–612

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 , every function f : 𝒞n satisfies

Var σn(f) C i=1n ifL2(σn)2 1 + log (ifL2(σn)ifL1(σn)).

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 space-valued 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.

Hamming cube, Talagrand's inequality, Rademacher type, martingale type, Itô calculus, Riesz transforms, Littlewood–Paley–Stein theory, hypercontractivity, CAT(0) space, bi-Lipschitz embedding
Mathematical Subject Classification
Primary: 42C10
Secondary: 30L15, 46B07, 60G46
Received: 7 January 2021
Revised: 9 July 2021
Accepted: 23 August 2021
Published: 3 May 2023
Dario Cordero-Erausquin
Institut de Mathématiques de Jussieu
Sorbonne Université
Alexandros Eskenazis
Trinity College and Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom

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