Talagrand’s influence inequality revisited

. Let C n = {− 1 , 1 } n be the discrete hypercube equipped with the uniform probability measure σ n . Talagrand’s inﬂuence inequality (1994), also known as the L 1 − L 2 inequality, asserts that there exists C ∈ (0 , ∞ ) such that for every n ∈ N , every function f : C n → C satisﬁes


Introduction
Let C n = {−1, 1} n be the discrete hypercube equipped with the uniform probability measure σ n .If (E, • E ) is a complex Banach space, we will denote the vector-valued L p (σ n )-norm of a function f : C n → E by and f L∞(σn;E) def = max ε∈Cn f (ε) E .When E = C, we will abbreviate f Lp(σn;C) simply as f Lp(σn) .We will also denote by E σn f the expectation of f with respect to σ n .The i-th partial derivative of a function f : C n → E is given by The discrete Poincaré inequality asserts that every function f : Extensions and refinements of (3) have been a central object of study in the probability and analysis literature for decades.A natural problem, first raised by Enflo [Enf78], is to understand for which target spaces E, every function f : C n → E satisfies (3) up to a universal multiplicative factor depending only on the geometry of E but not n or the choice of f .Recall that a Banach space (E, • E ) has Rademacher type s with constant T ∈ (0, ∞) if for every n ∈ N and x 1 , . . ., x n ∈ E, ˆCn It is evident that if a Banach space E is such that every function f : then E has Rademacher type 2 with constant C, since this condition coincides with (5) for functions of the form f (ε) = n i=1 ε i x i , where x 1 , . . ., x n ∈ E. The reverse implication, i.e. the fact that Rademacher type 2 implies the vector-valued Poincaré inequality (5), was proven in the recent breakthrough [IvHV20] of Ivanisvili, van Handel and Volberg.
In a different direction, an important refinement of the scalar-valued discrete Poincaré inequality (3) was obtained by Talagrand in the celebrated work [Tal94].Talagrand's influence inequality, also known as the L 1 − L 2 inequality, asserts that there exists a universal constant C ∈ (0, ∞) such that for every n ∈ N, every function f : Observe that (6) is a strengthening of the discrete Poincaré inequality (3) up to the value of the universal constant C, which becomes substantial for functions satisfying Since its conception, Talagrand's inequality has played a major role in Boolean analysis [KKL88, FK96, Ros06, FS07, O'D14], percolation [Rus82, BKS03, BR08, Cha14, GS15] and geometric functional analysis [PVZ17, PV18, Tik18, PTV19].In particular, applying (6) to a Boolean function f : C n → {0, 1}, one readily recovers the celebrated theorem of Kahn, Kalai and Linial [KKL88], quantifying the fact that in any (essentially) unbiased voting scheme, there exists a voter with disproportionately large influence over the outcome of the vote.We refer to the above references and [CEL12,Led19] for further bibliographical information on Talagrand's inequality.
The main purpose of the present paper is to investigate vector-valued versions of Talagrand's inequality (6) and other refinements and extensions of (3).These new vector-valued inequalities motivate the definition of a new bi-Lipschitz invariant for metric spaces called Talagrand type (Definition 10), which captures new KKL-type phenomena in embedding theory (see Theorem 13 and the ensuing discussion).We shall now present a summary of these results, which rely on a range of stochastic and harmonic analytic tools such as Banach space-valued Itô calculus, Riesz transforms and Littlewood-Paley-Stein theory, along with standard uses of hypercontractivity.

Asymptotic notation.
In what follows we use the convention that for a, b ∈ [0, ∞] the notation a b (respectively a b) means that there exists a universal constant c ∈ (0, ∞) such that a cb (respectively a cb).Moreover, a b stands for (a b) ∧ (a b).The notations ξ , χ and ψ mean that the implicit constant c depends on ξ, χ and ψ respectively.
1.1.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 : Evidently, as ( 7) is a strengthening of (5), if a space (E, • E ) satisfies ( 7) then E has Rademacher type 2. Conversely, we shall prove the following theorem.
Theorem 1 (Vector-valued influence inequality for spaces with Rademacher type 2).Let (E, • E ) be a Banach space with Rademacher type 2.Then, there exists C = C(E) ∈ (0, ∞) such that for every ε ∈ (0, 1) and n ∈ N, every function f : In particular, if σ(f ) def = max i∈{1,...,n} log log e + ∂ i f L 2 (σn;E) / ∂ i f L 1 (σn;E) , then 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.Let us mention that, even in the scalar-valued case, the argument of Maurey or the one of Ivanisvili, van Handel and Volberg are slightly different than standard semigroup approaches to functional inequalities, in particular to the semigroup proof of (6) from [CEL12].On the other hand, we will see that a slightly stronger condition on the Banach space allows for different approaches, relying on more intricate connections between the space and the semigroup, which will lead to the desired optimal vectorvalued L 1 − L 2 inequality.Recall first that a Banach space (E, • E ) has martingale type s with constant M ∈ (0, ∞) if for every n ∈ N, every probability space (Ω, F, µ) and every filtration {F i } n i=0 of sub-σ-algebras of F, every E-valued martingale {M i : Ω → E} n i=0 adapted to {F i } n i=0 satisfies Martingale type, which is a strengthening of Rademacher type, was introduced by Pisier in [Pis75], who proved the fundamental fact that for every s ∈ (1, 2], a Banach space E has martingale type s if and only if E admits an equivalent s-uniformly smooth norm (see [Pis75,Pis16] for further information on these important notions).
Theorem 2 (Vector-valued influence inequality for spaces with martingale type 2).Let (E, • E ) be a Banach space with martingale type 2.Then, there exists C = C(E) ∈ (0, ∞) such that for every n ∈ N, every function f : Theorem 2 establishes the optimal vector-valued influence inequality for spaces of martingale type 2. We will present two proofs of Theorem 2. The first one uses a clever stochastic process on the cube which was recently constructed by Eldan and Gross [EG19], while the second relies on Xu's vector-valued Littlewood-Paley-Stein inequalities for superreflexive targets (see [Xu20]).
There exist examples of exotic Banach spaces [Jam78,PX87] which have Rademacher type 2 yet fail to have martingale type 2, thus Theorem 2 does not exhaust the list of potential target spaces satisfying (7).Nevertheless, a combination of classical results of Maurey [Mau74], Pisier [Pis75] and Figiel [Fig76] imply that every Banach lattice of Rademacher type 2 has martingale type 2.
The influence inequalities of Theorems 1 and 2 have analogues for spaces of Rademacher and martingale type s which will be presented in Section 9.1 for the sake of simplicity of exposition.
the L p norm of the gradient of f .It has already been pointed out that Talagrand's influence inequality ( 6) is a refinement of the discrete Poincaré inequality (3).It is therefore worth investigating whether similar strengthenings of the L p discrete Poincaré inequality hold true for other values of p.The fact that for every p ∈ [1, ∞) there exists a constant C p ∈ (0, ∞) such that (13) holds true for every n ∈ N and f : C n → C was established by Talagrand in [Tal93].
In the vector-valued setting which is of interest here, the most common substitute of (12) for the norm of the gradient of a function f : Observe that when E = C, for every p ∈ [1, ∞), we have ∇f Lp(σn;C) p ∇f Lp(σn) by Khintchine's inequality [Khi23].With this definition, the vector-valued extension of ( 13) is called Pisier's inequality, since Pisier established in [Pis86] the validity of ( 14) for every Banach space E and p ∈ [1, ∞) with C p (n) = 2e log n.Understanding for which Banach spaces E and p ∈ [1, ∞) the constant C p (n) in Pisier's inequality could be replaced by a constant C p (E), independent of the dimension n, was a long-standing open problem settled in the recent work [IvHV20].
We will recall below, in (97), the definition of Rademacher cotype; let us simply say here that a Banach space E has finite cotype if E does not isomorphically contain the family { n ∞ } ∞ n=1 with uniformly bounded distortion (see [MP76,Pis16]).In [IvHV20], Ivanisvili, van Handel and Volberg proved that a Banach space E with finite cotype satisfies (14) with C p (n) replaced by a universal constant C p (E), thus complementing a result of Talagrand [Tal93] who proved that if a space does not have finite cotype, then C p (n) p log n.
Theorem 3 (Vector-valued L 1 −L p inequality for spaces of finite cotype).Let (E, • E ) be a Banach space with finite Rademacher cotype and p ∈ (1, ∞).Then, there exists C p = C p (E) ∈ (0, ∞) and α p = α p (E) ∈ 0, 1 2 such that for every n ∈ N, every function f : The proof of Theorem 3 builds upon the technique of [IvHV20].A stronger inequality for functions on the Gauss space will be presented in Theorem 27.This approach seems insufficient to yield the optimal α p = 1 2 exponent for E = C and all p > 1, yet we derive the following result using Lust-Piquard's Riesz transform inequalities [LP98,BELP08].
Theorem 4 (Scalar-valued L 1 − L p inequality).For every p ∈ (1, ∞), there exists C p ∈ (0, ∞) such that for every n ∈ N, every function f : The formula is also true if g has values in the dual E * and the product is the duality bracket.The operator ∆ is the (positive) infinitesimal generator of the discrete heat semigroup {P t } t 0 on C n , that is, P t = e −t∆ (see, e.g., [O'D14]).Let us mention that functional calculus involving ∆ can be easily expressed using the Walsh basis.This is the case for all Fourier multipliers appearing below which are defined by formula (106).All available proofs of Talagrand's inequality (6) make crucial use of the hypercontractivity of {P t } t 0 (first proven by Bonami in [Bon70]) along with some version of "orthogonality" [Tal94] or semigroup identites [BKS03,CEL12] specific to the scalar case.In particular, Talagrand [Tal94] used Parseval's identity for the Walsh basis to express the variance of a function f : and thus reduced the problem to obtaining effective estimates for ∆ −1/2 h L 2 (σn) .One tool which allows us to circumvent algebraic representations such as (18) (see the proof of Theorem 4 below) are one-sided Riesz transform inequalities, which can combined with certain new vector-valued estimates on negative powers of the generator of the semigroup {P t } t 0 .Let α 0. We say that a Banach space E has nontrivial Rademacher type if E has Rademacher type s for some s ∈ (1, 2].It has been proven by Naor and Schechtman [NS02] that if a Banach space (E, • E ) has nontrivial Rademacher type, then for every p ∈ (1, ∞) and α ∈ (0, ∞), there exists K p (α) = K p (α, E) ∈ (0, ∞) such that for every n ∈ N and f : C n → E, we have Conversely, if (19) holds true for some p and α, then E has nontrivial Rademacher type.The proof of Theorem 4 relies on the following strengthening of Naor and Schechtman's inequality (19).
Theorem 5. Let (E, • E ) be a Banach space of nontrivial Rademacher type.Then, for every p ∈ (1, ∞) and α ∈ (0, ∞), there exists K p (α) = K p (α, E) ∈ (0, ∞) such that for every n ∈ N and f : C n → E, we have We note in passing that when E = C, α = 1 2 and p = 2, Theorem 5 had been proven in [Tal94, Proposition 2.3].However, Talagrand's argument heavily uses orthogonality via Parseval's identity for the Walsh basis and is unlikely to work in the vector-valued setting which is of interest here.
1.4.Vector-valued multipliers and inequalities involving Orlicz norms.In Talagrand's original work [Tal94], he observed that (6) admits a strengthening in terms of Orlicz norms (see [RR91]).Recall that if ψ : [0, ∞) → [0, ∞) is a Young function, i.e. a convex function satisfying and (E, • E ) is a Banach space, then the ψ-Orlicz norm of a function f : C n → E is given by It is evident that for ψ(t) = t p , we have • L ψ (σn;E) = • Lp(σn;E) .More generally, for p ∈ (1, ∞) and r ∈ R we will denote by • Lp(log L) r (σn;E) the Orlicz norm correspoding to a Young function ψ p,r with ψ p,r (x) = x p log r (e + x) for x large enough (to ensure convexity of ψ p,r when r < 0).In [Tal94, Theorem 1.6], Talagrand showed that (6) can be strengthened as follows.There exists a universal constant C ∈ (0, ∞) such that for every n ∈ N, every function f : It is in fact true (see [Tal94, Lemma 2.5] or Lemma 17 below) that (23) formally implies (6).In this direction we can prove the following strengthening of Theorem 1.
Theorem 6.Let (E, • E ) be a Banach space with Rademacher type 2.Then, there exists C = C(E) ∈ (0, ∞) such that for every ε ∈ (0, 1) and n ∈ N, every function f : Furthermore, the proofs of Theorem 2 in fact yield the following improvement of (11), which extends (23) to spaces of martingale type 2.
Theorem 7. Let (E, • E ) be a Banach space with martingale type 2.Then, there exists C = C(E) ∈ (0, ∞) such that for every n ∈ N, every function f : We now turn to Orlicz space strengthenings of Theorem 5.The scalar-valued analogue of this problem had first been studied by Feissner [Fei75] and was later completely settled by Bakry and Meyer [BM82], who showed the following.For every p ∈ (1, ∞) and α ∈ (0, ∞) there exists K p (α) ∈ (0, ∞) such that for every n ∈ N and f : In [BM82] It will be shown in Lemma 17 below, that Theorem 8 is indeed a strengthening of Theorem 5.In view of the result of [NS02], it is evident that the assumption that the target space E has nontrivial type is both necessary and sufficient in Theorem 8.While the ingredients used in the proof of [BM82, Théorème 6] cannot be applied in the vector-valued setting of Theorem 8, (27) will be proven as a consequence of the scalar inequality (26) using the following vector-valued multiplier theorem.
Theorem 9. Let (E, • E ) be a Banach space of nontrivial Rademacher type and consider a holomorphic function h : D r → C where D r = {z ∈ C : |z| < r}, where r ∈ (0, ∞).Then, for every α ∈ (0, ∞) and p ∈ (1, ∞), there exists a constant C h (α, p) = C h (α, p, E) ∈ (0, ∞) such that for every n ∈ N, every function f : When E = C, Theorem 9 is a classical result of Meyer [Mey84, Thèoréme 3].The vector-valued extension presented here crucially relies on the bounds on the action of negative powers of ∆ on vector-valued tail spaces obtained by Mendel and Naor in [MN14].
Definition 10 (Talagrand type).Let ψ : [0, ∞) → [0, ∞) be a Young function and p ∈ (0, ∞).We say that a metric space (M, d M ) has Talagrand type (p, ψ) with constant τ ∈ (0, ∞) if for every n ∈ N, every function f : where It can be easily seen that if a Banach space E has the property that for every n ∈ N, every f : for some τ * ∈ (0, ∞), then E also has Talagrand type (p, ψ).Indeed, applying (31) to the function Theorem 11.There exists τ ∈ (0, ∞) such that every ε ∈ (0, 1) the following holds.Every Gromov hyperbolic group G equipped with the shortest path metric on the Cayley graph with respect to a finite generating set The proof of Theorem 11 relies on a result of Ostrovskii [Ost14], according to which the Cayley graph of every Gromov hyperbolic groups admits a bi-Lipschitz embeddng in an arbitrary nonsuperreflexive Banach space, combined with a classical construction of James [Jam78].
We say that a Riemannian manifold has pinched negative curvature if its sectional curvature takes values in the interval [−R, −r] for some r, R ∈ (0, ∞) with r < R.After the proof of Theorem 11 in Section 7, we also prove the following result.
Theorems 11 and 12 describe two classes of nonpositively curved spaces which satisfy a Talagrandtype inequality that strengthens Enflo type 2. It remains an intriguing open problem to understand whether every CAT(0) space has this property (see also Section 9).1.6.Embeddings of nonlinear quotients of the cube and Talagrand type.
We will denote by c N (M) the infimal bi-Lipschitz distortion of a function f : M → N. When N = L p (R), we will abbreviate c Lp(R) (M) as c p (M).Consider the hypercube C n endowed with the Hamming metric ρ(ε, δ) = ε − δ 1 .The geometric significance of Enflo type stems (partially) from the fact (see [NS02]) that if a metric space M has Enflo type p with constant T ∈ (0, ∞), then In this section, we will establish a more delicate bi-Lipschitz nonembeddability property which is a consequence of the Talagrand type inequality (29).Let R ⊆ C n ×C n be an arbitrary equivalence relation and denote by C n /R the set of all equivalence classes of R equipped with the quotient metric, which is given by here the minimum is taken over all k 1 and η 1 , . . ., ] for every j ∈ {1, . . ., k − 1}.We shall now present an implication of Talagrand type on embeddings of nonlinear quotients 1 of the cube which strengthens the corresponding bounds that one can deduce from Enflo type.We will denote by ∂ i R the boundary of R in the direction i, that is and by a p (R) the quantity Theorem 13.Fix p ∈ (0, ∞) and a Young function It is worth noting that in the setting of Theorem 13, if M has Talagrand type (p, t → t p ) with constant τ (a property which is very closely related to Enflo type p, see Remark 38), then This estimate, which generalizes (33), is substantially weaker than (37) when ψ(t) < < t p for large values of t.In particular, this is the case for Banach spaces of Rademacher or martingale type p (see Theorems 40 and 41).It is also worth mentioning that, in view of Theorem 42 below, Theorem 13 provides nontrivial distortion lower bounds even for bi-Lipschitz embeddings into L 1 (µ) spaces.Theorem 13 is reminiscent of the celebrated theorem of Kahn, Kalai and Linial [KKL88], which asserts that there exists a constant c ∈ (0, ∞) such that for every Boolean function f : where p = E σn f .Viewing f as a voting scheme, (39) asserts that if all influences ∂ i f 2 L 2 (σn) are small, then f is necessarily an unfair system in the sense that its expectation is very close to either 0 or 1. Inequality (37) puts forth a similar phenomenon in embedding theory: if all geometric influences σ n (∂ i R) of the partition are small, then the quotient C n /R is incompatible with the geometry of the target space M.Moreover, the quantitative improvement (37) of ( 38) is in direct analogy with the improvement that the KKL inequality (39) offers to the weaker estimate max i∈{1,...,n} which follows readily from the Poincaré inequality (3) for any function f : C n → C. 1 The term "nonlinear" here is meant to emphasize the distinction between quotients of the hypercube with respect to an arbitrary equivalence relation and quotients by linear codes (see [MS77] and Remark 39 below).Recall that if we identify Cn with F n 2 , where F2 is the field with two elements, a linear code is an F2-subspace C ⊆ Cn and the corresponding quotient is the F2-vector space F n 2 /C endowed with the quotient metric.
Organization of the paper.In Section 2, we will present some elementary inequalities and properties of Orlicz norms which we shall use in the sequel.Section 3 contains the proof of Theorems 1 and 6 and Section 4 contains two proofs of Theorems 2 and 7, one using stochastic calculus and one Fourier analytic.In Section 5, we prove Theorems 3 and 4 and their analogue in Gauss space, Theorem 27, by a combination of semigroup methods and Riesz transforms.Section 6 contains the proof of Theorem 9 and the derivation of Theorems 5 and 8.In Section 7 we present the proof of Theorems 11 and 12 and in Section 8 we present the proof of the nonembeddability result of Theorem 13.Finally, Section 9 contains some concluding remarks and open problems.

Some preliminary calculus lemmas
In this section, we present a few elementary facts related to Orlicz norms which we shall repeatedly use in the sequel.While these results are central for our proofs, they are mostly technical and therefore can be skipped on first reading.We gather them here in order to avoid digressions in the main part of the text.
Lemma 14.Let (E, • E ) be a Banach space and (Ω, µ) a probability space.For every r Proof.Since both sides only depend on the norm of h, we can assume that E = C and h 0.Moreover, without loss of generality η 1 = γ.Suppose, by homogeneity, that the right hand side satisfies h Lr(log L) −1+ε (µ) 1, which implies that Moreover, observe that where the inequality follows from the monotonicity of L s (µ) norms and the equivalence by the change of variables ν = 1 + (r − 1)e −ηt .
The right hand side then satisfies , where the second inequality follows from Jensen's inequality for the convex function t → t r/ν with weights (41).Now, by multiplying k by r, one can easily see that where the second equivalence follows by the change of variables u = ν/r and a further change of variables in k.For k 0 and ε ∈ (0, 1), write and notice that which implies that Finally, to bound R k , we integrate by parts which, after rearranging, readily implies that R k r 1 and the proof is complete.
Using Hölder's inequality, we can easily deduce the following variant of Lemma 14 which we will need to prove Theorems 29 and 30 below.
The following lemma will be used to prove Theorems 3 and 5.
Lemma 16.Let (E, • E ) be a Banach space and (Ω, µ) a probability space.For every r Proof.Without loss of generality, we will again assume that E = C, h 0 and η 1 = γ.As in the proof of Lemma 14, a change of variables shows that and the conclusion follows from (45) and (46).
The following lemma shows that the Orlicz norm statements of Theorems 6 and 7 indeed strengthen Theorems 1 and 2 respectively.In the special case r = 2 and s = 1, this has been proven by Talagrand in [Tal94, Lemma 2.5] and the general case treated here is similar.
Lemma 17.Let (E, • E ) be a Banach space and (Ω, µ) a probability space.For every r ∈ (1, ∞) and s ∈ (0, ∞), there exists D = D(r, s) ∈ (0, ∞) such that every function h : Ω → E satisfies Proof.Without loss of generality, we will again assume that E = C and h 0. We will prove that ˆΩ h r log s (e + h) Let a ∈ (0, ∞).We will distinguish two cases.
Then, ˆΩ h r dµ log s (e + a) Case 2. Suppose that Notice that on {h < a}, we have h r / log s (e + h) a r−1 h, which implies that Now choose a = e h Lr(µ) / h L 1 (µ) 1/r so that b = r log a.In Case 1, (48) then implies that h r Lr(µ) On the other hand since b = r log a, in Case 2, (49) gives since x s 1 + log s x for every s, x ∈ (0, ∞).This completes the proof of the lemma.

Influence inequalities under Rademacher type
In this section we shall present the proofs of Theorems 1 and 6 which rely on the novel approach introduced in the recent work [IvHV20] of Ivanisvili, van Handel and Volberg.For t ∈ (0, ∞), let ξ(t) = ξ 1 (t), . . ., ξ n (t) be a random vector on C n whose coordinates are independent and identically distributed with distribution given by for i ∈ {1, . . ., n}.Moreover, consider the normalized vector δ(t) = (δ 1 (t), . . ., δ n (t)) with In the following statements, we will denote by ε a random vector independent of ξ(t), uniformly distributed on C n .We will need the following (straightforward) refinement of [IvHV20, Theorem 1.4].
Proposition 18.For every Banach space (E, where the expectation on the right hand side is with respect to ε and δ(t).
Let us mention here that we will apply the previous proposition to P t f instead of f , and use the semigroup property P 2t f = P t (P t f ).This is more easily done after reformulating (52) with ∆P t in place of ∂ ∂t P t .So, keeping the notation of Proposition 18, we have that Proof of Proposition 18.The crucial observation of Ivanisvili, van Handel and Volberg is that one can write, for x ∈ C n , where xξ(t) denotes the point (x 1 ξ 1 (t), . . ., x n ξ n (t)).This formula can be proved by writing where, for ξ ∈ C n , ω t (ξ) = 2 −n n i=1 1 + e −t ξ i ; then we note that, with some abuse of notation (denoting Hence, using the integration by parts formula (17) together with the fact that and this concludes the proof of (54).Alternatively, it suffices to readily check the validity of formula (54) in the case of the scalar-valued Walsh basis w J (x) = j∈J x j , where J ⊆ {1, . . ., n}, for which P t w J (x) = e −t|J| w J (x) and ∂ i w J (x) = 1 i∈J w J (x).Therefore, using Jensen's inequality and (54) we have We conclude by noting that the couple (εξ(t), ξ(t)) has the same law as the couple (ε, ξ(t)).This can be seen as a proxy of the rotational invariance of the Gaussian measure (compare with the proof of Proposition 28 below).
Theorems 1 and 6 are consequences of the following lemma.
Lemma 19.Let (E, • E ) be a Banach space with Rademacher type 2. Then there exists a constant K = K(E) ∈ (0, ∞) such that for every ε ∈ (0, 1) and n ∈ N, every f : Proof.We will apply Proposition 18 to P t f instead of f .We have that Suppose now that E has Rademacher type 2 with constant T .Then for ε ∈ (0, 1), by (56) and the Rademacher type condition for centered random variables [LT91, Proposition 9.11], we have where in the second line we used the Cauchy-Schwarz inequality.Therefore, since the integral ´∞ 0 dt (e 2t −1) 1−ε 1 ε as ε → 0 + , we deduce that there exists a universal constant C ∈ (0, ∞) with dt (e 2t − 1) ε , and the conclusion follows readily since e 2t − 1 te t for every t 0.
Remark 20.It was pointed out to us by an anonymous referee that plugging in the standard application (46) of Hölder's inequality along with hypercontractivity to bound the middle term of (57) cannot remove the dependence on ε in inequality (8).Indeed, by hypercontractivity and Hölder's inequality, we have ˆ∞ 0 where . Suppose, for contradiction, that for every n 1 and every 0 a i b i where i ∈ {1, . . ., n}, we have where . The parameters n 1, x i 0 and the weights p i are all arbitrary, thus we conclude from (59) that for every positive random variable X, the inequality holds true.To reach a contradiction, consider a discrete random variable X 0 such that and notice that where in the last inequality we bounded the inner sum by the k = term.This contradicts (60).
Remark 21.A combination of Proposition 18 and Lemma 16 implies a different Talagrand-type strengthening of the vector-valued discrete Poincaré inequality (5) for spaces of Rademacher type 2, which is weaker than (7) (see also [Cha14,Theorem 5.4] for a similar scalar-valued inequality).For a function f : C n → E, we will use the notation Df : C n → E n for the gradient vector Then, the first inequality in (57) can be rewritten as Now, by the hypercontractivity of {P t } t 0 , we have Df L 1+e −2t (σn; n 2 (E)) .Therefore, combining the last two inequalities, we get dt √ t and Lemma 16 then implies that The argument above shows that spaces of Rademacher type 2 satisfy (62) and the reverse implication is clear by choosing a function of the form f (ε) = n i=1 ε i x i .When E = C, this coincides with (16) where p = 2 (see also Remark 32 below for comparison with (6)).

Influence inequalities under martingale type
In this section, we shall present two proofs of Theorems 2 and 7, one probabilistic and one Fourier analytic.As a warmup, we present a simple proof of Talagrand's inequality in Gauss space for functions with values in a space of martingale type 2 using a classical stochastic representation for the variance.The scalar-valued case of this inequality was shown in [CEL12] via semigroup methods which do not seem to be adaptable to the case of vector-valued functions (see Section 4.3 for a harmonic analytic variant).We will denote by γ n the standard Gaussian measure on R n , i.e. the measure dγ n (x) = dx, where • 2 denotes the usual Euclidean norm on R n .

4.1.
A simple stochastic proof in Gauss space.We will denote by {U t } t 0 the Ornstein-Uhlenbeck semigroup on R n , whose action on an integrable function f : R n → E, where (E, • E ) is a Banach space, is given by the Mehler formula Let {X t } t 0 be an Ornstein-Uhlenbeck process, i.e. a stochastic process of the form X t = e −t X 0 + e −t B e 2t −1 , where {B t } t 0 is a standard Brownian motion and X 0 is a standard Gaussian random vector, independent of {B t } t 0 .We will use the following well-known consequence of the Clark-Ocone formula (see [CHL97] for a proof and further applications in functional inequalities).
Lemma 22.Let (E, • E ) be a Banach space.For every smooth function f : R n → E, we have We will also need the following one-sided version of the Itô isometry for 2-smooth spaces, which is essentially due to Dettweiler [Det91].We include the crux of the (simple) proof for completeness.
Proposition 23.Let (E, • E ) be a Banach space of martingale type 2.Then, there exists M ∈ (0, ∞) such that for every n ∈ N, if {B t } t 0 is a standard Brownian motion on R n and {Y t } t 0 is an E n -valued square integrable stochastic process adapted to the filtration {F t } t 0 of {B t } t 0 , then where G = (G(1), . . ., G(n)) is a standard Gaussian random vector on R n , independent of {F t } t 0 .
Proof.We shall assume that {Y t } t 0 is a simple process of the form where 0 = t 1 < t 2 < . . .< t N +1 and each α t k (i) is an F t k −measurable random variable.The general case will follow by standard approximation arguments.By definition, and Now, for a fixed k, B t k+1 (i)−B t k (i) n i=1 conditioned of F t k is equidistributed to a Gaussian random vector with covariance matrix (t k+1 − t k ) • Id n .Therefore, where G = (G(1), . . ., G(n)) is a standard Gaussian random vector, independent of {F t } t 0 .Hence, after taking expectation in (67) and summing over k, (66) becomes thus completing the proof of this simple fact.
We are now well-equipped to prove the following result.
Theorem 24.Let (E, • E ) be a Banach space with martingale type 2.Then, there exists C = C(E) ∈ (0, ∞) such that for every n ∈ N, every smooth function f : Proof.If E has martingale type 2 with constant M , then Lemma 22 and Proposition 23 imply that Thus, applying the Rademacher type 2 condition for Gaussian variables, we deduce that where T is the Rademacher type 2 constant of E. Now, integrating (69) with respect to the standard Gaussian random vector X 0 and using the stationarity of the Ornstein-Uhlenbeck process {X t } t 0 along with Nelson's hypercontractive inequalities [Nel66,Nel73], we derive where the equality follows from the standard commutation relation Since for every i ∈ {1, . . ., n} the correlation EX 0 (i)X s (i) = e −s , taking s → ∞ in (70) we get and the conclusion follows by Lemma 14.

4.2.
A proof of Theorems 2 and 7 via the Eldan-Gross process.In a recent paper, Eldan and Gross [EG19] constructed a clever stochastic process on the cube which resembles the behavior of Brownian motion on R n and used it to prove several important inequalities relating the variance and influences of Boolean functions.We shall briefly describe their construction.Let {B t } t 0 = B t (1), . . ., B t (n) t 0 be a standard Brownian motion on R n .For every i ∈ {1, . . ., n} and t 0, consider the stopping time τ t (i) given by and then, let X t (i) def = B τt(i) (i).Then, the jump process {X t } t 0 def = X t (1), . . ., X t (n) t 0 satisfies the following properties (see [EG19, Section 3] for detailed proofs): 1.For every t 0 and i ∈ {1, . . ., n}, X t (i) = t almost surely and in fact X t ∼ Unif{−t, t} n .2. The process {X t } t 0 is a martingale.3.For every coordinate i ∈ {1, . . ., n}, the jump probabilities of {X t (i)} t 0 are Proof of Theorems 2 and 7. Fix a function f : C n → E and recall (see, e.g., [O'D14]) that there exists a unique multilinear polynomial on R n , which coincides with f on C n , i.e. we can write for some coefficients f (A) ∈ E. By abuse of notation, we will also denote by f that unique multilinear extension on R n .Since f is a multilinear polynomial and {X t } t 0 is a martingale with independent coordinates, it follows that the process {f (X t )} t 0 is itself a martingale.Fix some large N ∈ N and for k ∈ {0, 1, . . ., N }, let t k = k N and M k = f (X t k ).Since E has martingale type 2, there exists where ∂f ∂x i are the usual partial derivatives of f on R n and the remainder R k (f ) satisfies However, since f is a multilinear polynomial, all second derivatives of the form ∂ 2 f /∂x 2 i vanish and (75) implies that for some K(f ) ∈ (0, ∞), so that The fact that only i = j enters the sum will be crucial below to ensure that the error tends to zero as N → +∞ after summing over k.Now, by (71), we have sign( with probability 2k−1 2k , so the conditional second moment of the increments is By the tower property of conditional expectation, the estimate (77) can finally be written as and thus (74) implies that i=1 is a sequence of independent centered random variables, when conditioned on {X s } s t k−1 .Therefore, applying the Rademacher type condition for centered random variables [LT91, Proposition 9.11] and (78), we deduce that where T is the type 2 constant of E. By the tower property of conditional expectation, (80) combined with (81) gives Now, summing over k ∈ {1, . . ., N } and using (73), we get Since X t is uniformly distributed on {−t, t} n , the random variable ∂f ∂x i (X t ) satisfies where ∼ denotes equality in distribution, ε is uniformly distributed on C n and the last equality follows, e.g., by [O'D14, Proposition 2.47].Therefore, by (85) and the change of variables u = log(1/t), we can rewrite (84) as In the scalar-valued case, formula (84) is then an equality with M 2 T 2 = 1 and appears in [EG19].However, in this case, its equivalent form (86) can also be proved by elementary semigroup arguments as in [CEL12] which we can follow to conclude the proof.Using hypercontractivity [Bon70] and (86), we get du.
The conclusions of Theorems 2 and 7 now follow from (86) combined with Lemmas 14 and 17 since for every i ∈ {1, . . ., n}, we have

4.3.
A proof of Theorems 2 and 7 by Littlewood-Paley-Stein theory.We shall now present a second, more analytic proof of Theorems 2 and 7.The main tool for this proof, is a deep vector-valued Littlewood-Paley-Stein inequality (see [Ste70]) due to Xu [Xu20], which is the culmination of the series of works [Xu98,MTX06] (see also [Hyt07] for some similar inequalities for UMD targets).We will need the following statement which is a special case of [Xu20, Theorem 2].
Theorem 25 (Xu).Let (E, • E ) be a Banach space of martingale type 2.Then, there exists a constant C = C(E) ∈ (0, ∞) such that for symmetric diffusion semigroup {T t } t>0 on a probability space (Ω, µ), every function Second proof of Theorems 2 and 7. Since E has martingale type 2, there exists T ∈ (0, ∞) such that E also has Rademacher type 2 with constant T .Then, applying Proposition 18 to P t f and using the Rademacher type condition for centered random variables [LT91, Proposition 9.11], we deduce that Plugging ( 88) in (87) for {T t } t 0 = {P t } t 0 and doing a change of variables, we get As before, the conclusion now follows from hypercontractivity [Bon70] along with Lemmas 14 and 17.
Remark 26.A careful inspection of the proof of [Xu98, Theorem 3.1] shows that if we denote by X 2 (E) the least constant C in Xu's inequality (87), then X 2 (E) M 2 (E), where M 2 (E) is the martingale type 2 constant of E. On the other hand, in [Xu20] it is shown that and the fact that sup t 0 t∂ t T t L 2 (µ;E)→L 2 (µ;E) < ∞ is proven as a consequence of the uniform convexity of E * .Specifically for the case of the heat semigroup {P t } t 0 on C n , a different proof of this statement which only relies on Pisier's K-convexity theorem [Pis82] is presented in [EI20, Lemma 37].In the particular case of E = p , where p 2, an optimization of the argument of [EI20, Lemma 37] using the recent proof of Weissler's conjecture on the domain of contractivity of the complex heat flow by Ivanisvili and Nazarov [IN19], reveals that Therefore, since the Rademacher and martingale type 2 constants of p are both of the order of √ p, the probabilistic proof of Theorem 2 presented in Section 4.2 shows that for every n ∈ N, every function f : C n → p , where p 2, satisfies whereas the proof via Xu's inequality (87) implies a weaker O(p 3 ) bound because of the current best known bounds (90) and ( 91).We refer to [Xu21a,Xu21b] for recent updates on the optimal order of the constant X 2 (E).

Vector-valued L 1 − L p inequalities
In this section, we will prove Theorems 3 and 4. We start by presenting a joint strengthening of the two for functions from the Gauss space instead of the discrete hypercube.
In this section, we will prove the following Talagrand-type strengthening of (92).
Theorem 27.For every p ∈ (1, ∞), there exists C p ∈ (0, ∞) such that the following holds.For every Banach space (E, • E ) and n ∈ N, every smooth function f : R n → E satisfies We will denote by L the (negative) generator of the Ornstein-Uhlenbeck semigroup {U t } t 0 , whose action on a smooth function f : R n → E is given by We will need the following (classical) Gaussian analogue of Proposition 18.
Proposition 28.Let (E, • E ) be a Banach space and p ∈ [1, ∞).Then, for every n ∈ N, every smooth function f : R n → E satisfies Proof.Here we can follow Maurey's trick [Pis86], setting for given independent standard Gaussian vectors X, Y ∈ R n .Then, we have and we conclude using Jensen's inequality together with the fact that (X t , Y t ) has the same distribution as (X, Y ) for every t 0.
Proof of Theorem 27.Arguing as in (56) and using (94) for U t f instead of f , we can write Now, by Nelson's hypercontractive inequalities [Nel66, Nel73] and Kahane's inequality [Kah64] for Gaussian variables, we have dγ n (y) dγ n (y) and the conclusion follows from (95), (96) and Lemma 16.

Proof of Theorem
The discrete vector-valued L 1 − L p inequality of Theorem 3 can be proven along the same lines as Theorem 27 using Proposition 18 instead of Proposition 28.
Remark 32.We note in passing that for p = 2, ( 16) is a consequence of Talagrand's influence inequality (6).To see this, note that it has been observed in [Cha14,Theorem 5.4] that Talagrand's inequality (6) along with an application of Jensen's inequality imply that for every n ∈ N, every f : ) and C ∈ (0, ∞) is a universal constant.Then, (16) for p = 2 follows by Minkowski's integral inequality, since .
Using the vector-valued Bakry-Meyer inequality of Theorem 8 instead of Theorem 5, one obtains the following Orlicz space strengthening of Theorem 4.
Theorem 33.For every p ∈ (1, ∞), there exists C p ∈ (0, ∞) such that for every n ∈ N, every f : 6. Holomorphic multipliers and the vector-valued Bakry-Meyer theorem In this section, we will present the proofs of Theorems 5, 8 and 9.In the proof of Theorem 9, we will need some preliminary terminology from discrete Fourier analysis.Recall that for every Banach space (E, • E ) and every n ∈ N, all functions f : C n → E admit a unique expansion of the form Suppose now that r ∈ (0, ∞) and that h : (0, r) → C is a function.Then, for every α ∈ (0, ∞), the operator h(∆ −α ) is defined spectrally by Finally, for a function f : C n → E and k ∈ {0, 1, . . ., n} we will define the k-th level Rademacher projection of f to be the function with Walsh expansion Pisier's deep K-convexity theorem [Pis82]  Proof of Theorem 5. Since P t = e −t∆ , we can express the action of ∆ −α on functions with expectation equal to 0 as Hence, every function f : If E has nontrivial type, it is a standard consequence of Pisier's K-convexity theorem [Pis82] that there exist K p = K p (E) ∈ (0, ∞) and η p = η p ∈ 0, 1 2 , independent of n and f , such that Combining ( 108) and ( 109), we deduce that and the conclusion follows by hypercontractivity [Bon70] and Lemma 16. 6.2.Proof of Theorem 9.The proof of Theorem 9 relies on the following result of Mendel and Naor from [MN14] (see also [EI20] for a different proof and further results in this direction).
Proof of Theorem 9. Let d p (α) = (2K p (α)/r) for every such function f .Now, let f : C n → E be an arbitrary function and write . By Pisier's K-convexity theorem [Pis82], we have for some M p = M p (E) ∈ (0, ∞).To bound the action of h(∆ −α ) on f 2 , consider the power series expansion h(z) = 0 c z of h around 0, which converges absolutely and uniformly on D r/2 .Then, the triangle inequality implies that Finally, observe that, again by Pisier's K-convexity theorem, for some M p = M p (E) ∈ (0, ∞).The conclusion follows readily from ( 113), ( 114) and (115).
6.3.Proof of Theorem 8. Equipped with Theorem 9, we can now deduce Theorem 8 from (26).We will also need the following simple lemma.
Lemma 35.For every Banach space (E, • E ), every function f : C n → E and every α ∈ (0, ∞), Proof.A change of variables shows that so that for every ε ∈ C n , we have where the second inequality follows from Jensen's inequality because P t is an averaging operator.

Influence inequalities in nonpositive curvature
Theorems 11 and 12 will be proven by combining Theorem 6 with results from geometry and Banach space theory.We first prove Theorem 11.
Proof of Theorem 11.It immediately follows from definition (29) that if a metric space M has Talagrand type (p, ψ) with constant τ ∈ (0, ∞) and another metric space N embeds bi-Lipscitzly in M with distortion D ∈ [1, ∞), then N has Talagrand type (p, ψ) with constant τD.Let G be a Gromov hyperbolic group equipped with the shortest path metric d G associated to the Cayley graph of any (finite) generating set S. Then, by a theorem of Ostrovskii [Ost14], (G, d G ) admits a bi-Lipschitz embedding of bounded distortion into any nonsuperreflexive Banach space.In particular, (G, d G ) embeds bi-Lipschitzly in the classical exotic Banach space (J, • J ) of James [Jam78], which has Rademacher type 2 yet is not superreflexive.By Theorem 6, there exists a universal constant C ∈ (0, ∞) such that for every ε ∈ (0, 1), (J, • J ) has Talagrand type (2, ψ 2,1−ε ) with constant C/ √ ε and thus the same holds true for the group (G, d G ).
The binary R-tree of depth d is the geodesic metric space which is obtained by replacing every edge of the combinatorial binary tree of depth d by the interval [0, 1].In order to prove Theorem 12, we will need the following structural result for Riemannian manifolds of pinched negative curvature which is essentially due to Naor, Peres, Schramm and Sheffield [NPSS06].
Theorem 36.Fix n ∈ N and r, R ∈ (0, ∞) with r < R.Then, there exists N ∈ N and D ∈ (0, ∞) such that any n-dimensional complete simply connected Riemannian manifold (M, g) with sectional curvature in [−R, −r] embeds bi-Lipschitzly with distortion at most D in a product of N binary R-trees of infinite depth.
In [NPSS06, Corollary 6.5], the authors proved an analogue of Theorem 36, in which binary Rtrees are replaced by R-trees of infinite degree.In order to prove the (stronger) theorem presented here, one needs to repeat the argument of [NPSS06] verbatim, replacing the use of [BS05] with a more recent result of Dranishnikov and Schroeder [DS05], who showed that the hyperbolic space H m admits a quasi-isometric in a finite product of binary R-trees of infinite depth.
We shall also need the following slight refinement of a result of Bourgain [Bou86].
holds true.To see this, one can "draw" the binary tree and label the vertices from top to bottom along an arbitrary path.After reaching a leaf, one should return to the nearest ancestor with an unlabeled child and continue labeling along an arbitrary downwards path starting at this child.This process should continue until the whole tree has been labeled.Since E is nonsuperreflexive, by a classical theorem of Pták [Pis16, Theorem 11.10] (which is often attributed to James), there exists vectors {x k } 2 d+1 −1 k=1 such that for every scalars a 1 , . . ., a 2 d+1 −1 , where On the other hand, by the property (119) of σ, we can assume without loss of generality that max σ(s 1 ), . . ., σ(s j+1 ) < min σ(t 1 ), . . ., σ(t k+1 ) .
Proof of Theorem 12.It follows from definition (29) that if a metric space M has Talagrand type (p, ψ) with constant τ ∈ (0, ∞) and another metric space N is such that every finite subset of N embeds bi-Lipscitzly in M with distortion at most K ∈ [1, ∞), then N has Talagrand type (p, ψ) with constant τK.Let (M, g) be a Riemannian manifold of pinched negative curvature equipped with its Riemannian distance d M .Then, by Theorem 36, there exists N ∈ N and D ∈ (0, ∞) such that (M, d M ) embeds with distortion at most D in a product of N binary R-trees of infinite depth.In particular, every finite subset X of M embeds with distortion at most D in a product of N binary R-trees of depth d, for some d depending on the cardinality of X.Therefore, by Proposition 37 (see also the discussion following Theorem 2.1 in [Ost14]), X embeds with distortion at most K = K(N, D) ∈ (0, ∞) in every nonsuperreflexive Banach space.In particular, X embeds with distortion at most K in the classical exotic Banach space (J, • J ) of James [Jam78], which has Rademacher type 2 yet is not superreflexive.By Theorem 6, there exists a universal constant C ∈ (0, ∞) such that for every ε ∈ (0, 1), (J, • J ) has Talagrand type (2, ψ 2,1−ε ) with constant C/ √ ε and thus the same holds for the Riemannian manifold (M, d M ).
Khot and Naor provide lower bounds for the L 1 -distortion of quotients of C n by linear codes and by the action of transitive subgroups of the symmetric group S n .As the proofs of [KN06] rely on delicate properties of both these structured quotients and L p spaces, it seems improbable that they can be easily modified to give nonembeddability results into spaces with given Talagrand type.

Concluding remarks and open problems
In this final section, we shall present a few remarks regarding the preceeding results and indicate some potentially interesting directions of future research.9.1.Talagrand type and linear type.In order to highlight the relation of our results with Talagrand's original inequality (6), we decided to state Theorem 1, 2, 6 and 7 only for spaces of Rademacher or martingale type 2. In the terminology of Definition 10, one has the following more general results for spaces of Rademacher or martingale type s.Here and throughout, we will denote by ψ s,δ : [0, ∞) → [0, ∞) a Young function with ψ s,δ (t) = t s log −δ (e + t) for large enough t > 0.
Question 1.Does every Banach space of Rademacher type s also have Talagrand type (s, ψ s,s/2 )? 9.2.Talagrand type of L 1 (µ).It is worth emphasizing that the proofs of both Theorems 40 and 41 crucially rely on the fact that s > 1 due to the use of Bonami's hypercontractive inequalities [Bon70].In the following theorem, we establish the Talagrand type of L 1 .It is worth emphasizing the somewhat surprising fact that Theorem 42 below shows that a stronger property than the trivial "Enflo type 1" inequality holds true in L 1 .
Proof.Since Talagrand type is a local invariant, it clearly suffices to consider the case that µ is the counting measure on N and thus L 1 (µ) is isometric to 1 .We will employ a classical result of Schoenberg [Sch38], according to which there exists a function s : R → 2 such that s(0) = 0 and Fix n ∈ N and a function f : C n → 1 .Consider the composition g : C n → 2 ( 2 ) given by g = s • f .Then, we have where the last inequality follows from Theorem 7. Combining this with the pointwise identity 1/2 1 and the fact that for every h : {−1, h L 1 (log L) −1 (σn) , we deduce that This concludes the proof of the theorem.
The argument used in the proof of Theorem 42 to derive the Talagrand type of 1 from the Talagrand type of 2 is very specifically tailored to L 1 (µ) spaces.It remains an interesting open problem to investigate the Talagrand type of noncommutative L 1 -spaces.
Therefore, the following question seems natural.
In the case of Gauss space, it has been shown by Pisier (see [Pis88]) that dimension-free Riesz transform inequalities hold true provided that the target space E has the UMD property.In particular, this means that in the case of UMD spaces, Theorem 30 can be improved as follows.In Theorems 11 and 12, we showed that Gromov hyperbolic groups and complete Riemannian manifolds of pinched negative curvature have Talagrand type (2, ψ 1−ε ) for every ε ∈ (0, 1).On the other hand, a classical inductive argument essentially going back to Enflo [Enf69] shows that all Alexandrov spaces of nonpositive curvature have Enflo type 2, which is closely related to Talagrand type (2, ψ 2,0 ).We believe that the following question deserves further investigation.
Question 4. Does there exist some δ ∈ (0, 1] such that every Alexandrov space of nonpositive curvature has Talagrand type (2, ψ 2,δ )?More ambitiously, does every Alexandrov space of nonpositive curvature have Talagrand type (2, ψ 2,1 )? 9.5.CAT(0) spaces as test spaces for superreflexivity.In Proposition 37, we showed that all binary R-trees of finite depth embed with uniformly bounded distortion into any nonsuperreflexive Banach space.It was communicated to us by Florent Baudier that using this proposition and the barycentric gluing technique (see [Bau07] and the survey [Bau20]), one can in fact prove that the binary R-tree of infinite depth admits a bi-Lipschitz embedding into any nonsuperreflexive Banach space.Then, an inductive argument (see, e.g., [Ost14, Remark 2.2]) shows that any finite product of binary R-trees also embeds bi-Lipschitzly into any nonsuperreflexive Banach space.Therefore, one deduces from Theorem 36 that every finite-dimensional complete simply connected Riemannian manifold of pinched negative curvature embeds bi-Lipschitzly into any nonsuperreflexive Banach space.Conversely, since all binary trees embed in the hyperbolic plane H 2 , if a Banach space E bi-Lipschitzly contains H 2 , then E cannot be superreflexive by Bourgain's theorem [Bou86].In conclusion, we deduce the following characterization.
Theorem 45.A Banach space (E, • E ) is nonsuperreflexive if and only if for every n ∈ N, every n-dimensional complete, simply connected Riemannian manifold (M, g) of pinched negative curvature equipped with the Riemannian distance d M admits a bi-Lipschitz embedding in E.
In recent years, there have been plenty of such characterizations in the literature, although one can argue that this is not a particularly novel one due to its close relation to Bourgain's characterization in terms of trees.We believe the following stronger question deserves further investigation.
Question 5. Which Alexandrov spaces of nonpositive curvature admit a bi-Lipschitz embedding into every nonsuperreflexive Banach space?
There are plenty of CAT(0) spaces which do not embed into finite products of binary R-trees and in order to prove that they embed into all nonsuperreflexive Banach spaces, one may need to employ interesting structural properties of such spaces.On the other hand, there exist CAT(0) spaces which do not embed into L 1 , which is of course nonsuperreflexive.Indeed, if every CAT(0) space admitted a bi-Lipschitz embedding into L 1 , then every classical expander (which is also an expander with respect to L 1 by Matoušek's extrapolation lemma for Poincaré inequalities, see [Mat97]), would be an expander with respect to all CAT(0) spaces and this is known to be false by important work of Mendel and Naor [MN15].9.6.General hypercontractive semigroups.In [CEL12], Cordero-Erausquin and Ledoux established versions of Talagrand's (scalar-valued) inequality (6) in the setting of hypercontractive Markov semigroups satisfying some minimal assumptions.At first glance, the arguments which we use in the present paper to obtain vector-valued extensions of (6) seem to rely more heavily in specific properties of the Hamming cube, such as identity (54) from [IvHV20] or the Eldan-Gross process [EG19].Nevertheless, we strongly believe that there are versions of our results for other hypercontractive Markov semigroups satisfying some fairly general assumptions.

5. 1 ..
A stronger theorem in Gauss space.For a smooth function f : R n → E, where (E, • E ) is a Banach space, and p ∈ [1, ∞) we will use the shorthand notation ∇f Lp(γn;E) In [Pis86, Corollary 2.4], Pisier presented an argument of Maurey showing that for every Banach space

=
A⊆{1,...,n} f (A)w A (ε), where the Walsh function w A : C n → {−1, 1} is given by w A (ε) = i∈A ε i for ε ∈ C n .In this basis, the action of the hypercube Laplacian on f can be written as ∆f = A⊆{1,...,n} |A| f (A)w A .
R be a Young function with ψ p,δ (x) = t p log −δ (e + x) for x large enough.
asserts that a Banach space (E, • E ) has nontrivial Rademacher type if and only if for every p ∈ (1, ∞), there exist M p = M p (E) ∈ (0, ∞) such that for every n ∈ N and k ∈ {1, ..., n}, every f :C n → E satisfies Rad k f Lp(σn;E) M k p f Lp(σn;E) .6.1.Proof of Theorem 5.Although Theorem 5 is a formal consequence of Theorem 8 and Lemma 17, we present a short self-contained proof.

)
Let B d be the binary R-tree of depth d.For a point a ∈ B d suppose that a belongs in the edge {v, w} of B d and that v is closer to the root than w.Consider the embedding ψ :B d → E given by + d B d (v, a) • x σ(w) .Let a, b ∈ B d and suppose that c is their least common ancestor.Then, there are downwards paths {s 1 , . . ., s j+1 }, {t 1 , . . ., t k+1 } in B d such that a ∈ [s j , s j+1 ), b ∈ [t k , t k+1 ) and s 1 , t 1 are the two distinct children of c.In this notation, the embedding ψ satisfies