Scaled Enflo type is equivalent to Rademacher type

Manor Mendel California Institute of Technology

Assaf Naor Microsoft Research

Abstract We introduce the notion of scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo

type p is equivalent to Rademacher type p.

1 Introduction

Recall that a Banach space X is said to have Rademacher type p > 0 (see [7]) if there exists a constant T < ∞ such that for every x_{1}, …, x_{n} ∈ X,

ε∥∑_{j=1}^{n} ε_{j}x_{j} ∥_{X}^{p} ≤ T^{p} ∑_{j=1}^{n} ∥x_{j}∥_{X}^{p},  (1)

where here, and in what follows, _{ε} denotes the expectation with respect to uniformly chosen ε = (ε_{1}, …, ε_{n}) ∈ {−1, 1}^{n}. The infimum over all constants T for which (1) holds is denoted T_{p}(X).

Motivated by the search for concrete versions of Ribe’s theorem [12] for various fundamental local properties of Banach spaces (see the discussion in [2, 9, 8]), several researchers proposed non-linear notions of type, which make sense in the setting arbitrary metric spaces (see [5, 3, 1]). In particular, following Enflo [5] we say that a metric space (M, d_{M}) has Enflo type p if there exists a constant K such that for every n ∈ and every f : {−1, 1}^{n} → M,

ε d_{M}(f (ε), f (−ε))^{p} ≤ T^{p} ∑_{j=1}^{n} _{ε} d_{M} (f (ε_{1}, …, ε_{j−1} , ε_{j}, ε_{j+1} , …, ε_{n}), f (ε_{1}, …, ε_{j−1} , −ε_{j}, ε_{j+1} , …, ε_{n}))^{p} .  (2)

For Banach spaces (1) follows from (2) by considering the function ε ↦ ∑ _{j=1}^{n} ε_{j}x_{j}. The question whether in the category of Banach spaces Rademacher type p implies Enflo type p was posed by Enflo in [5], and in full generality remains open. In [11] Pisier showed that if a Banach space has Rademacher p then it has Enflo type p′ for every p′ < p (see also the work of Bourgain, Milman and Wolfson [3] for a similar result which holds for a another notion of non-linear type). In [10] it was shown that for UMD Banach spaces (see [4]) Rademacher type p is equivalent to Enflo type p.

Motivated by our recent work on metric cotype [8], we introduce below the notion of scaled Enflo type of a metric space (which is, in a sense, “opposite” to the notion of metric cotype defined in [8]), and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p. This settles the long standing problem of finding a purely metric formulation of the notion of type (though Enflo’s problem described above remains open). Modulo some of the results of [8], the proof of our main theorem is very simple.

Definition 1.1 (Scaled Enflo type). Let (M, d_{M}) be a metric space and p > 0. We say that M has scaled Enflo type p with constant τ if for every integer n there exists an even integer m such that for every f : _{m}^{n} → M,

ε ∫_{_{m}^{n}} d_{M} (f (x + \frac{m}{2}ε) , f (x))^{p} dμ(x) ≤ τ^{p}m^{p} ∑_{j=1}^{n} ∫_{_{m}^{n}} d_{M} (f (x + e_{j}), f (x))^{p} dμ(x),  (3)

where μ is the uniform probability measure on _{m}^{n}, and {e_{j}}_{j=1}^{n} is the standard basis of ^{n}. The infimum over all constants τ for which (3) holds is denoted τ_{p}(M).

Theorem 1.2. Let X be a Banach space and p ∈ [1, 2]. Then X has Rademacher type p if and only if X has scaled Enflo type p. More precisely,

\frac{1}{2π}T_{p}(X) ≤ τ_{p}(X) ≤ 5T_{p}(X).

2 Proof of Theorem 1.2

We start by showing that scaled Enflo type p implies Rademacher type p. Lemma 2.1. Let X be a Banach space and p ∈ [1, 2]. Then T_{p}(X) ≤ 2πτ_{p}(X). Proof. Let X be a Banach space and assume that τ_{p}(X) < ∞ for some p ∈ [1, 2]. Fix τ > τ_{p}(X), v_{1}, …, v_{n} ∈ X, and let m be an even integer. Define f : _{m}^{n} → X by f (x_{1}, …, x_{n}) = ∑ _{j=1}^{n} e^{\frac{2πix_{j}}{m}} v_{j}. Then

∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥ _{X}^{p}dμ(x) = |e^{\frac{2πi}{m}} − 1|^{p} · ∑_{j=1}^{n} ∥v_{j}∥_{X}^{p} ≤ (\frac{2π}{m})^{p} · ∑_{j=1}^{n} ∥v_{j}∥_{X}^{p},  (4)

and

ε ∫_{_{m}^{n}} ∥f (x + \frac{m}{2}ε) − f (x)∥_{X}^{p}dμ(x) = 2^{p} ∫_{_{m}^{n}} ∥∑_{j=1}^{n} e^{\frac{2πix_{j}}{m}} v_{j} ∥_{X}^{p}dμ(x).  (5)

We recall the contraction principle (see [6]), which states that for every a_{1}, …, a_{n} ∈ ,

ε∥∑_{j=1}^{n} ε_{j}a_{j}v_{j} ∥_{X}^{p} ≤ (max_{1≤j≤n} |a_{j}|)^{p} · _{ε}∥∑_{j=1}^{n} ε_{j}v_{j} ∥_{X}^{p}.

Thus,

∫_{_{m}^{n}} ∥∑_{j=1}^{n} e^{\frac{2πix_{j}}{m}} v_{j} ∥_{X}^{p}dμ(x) = ∫_{_{m}^{n}} _{ε}∥∑_{j=1}^{n} e^{^{\frac{2πi}{m}}(x_{j}+ \frac{m(1−ε_{j})}{4})}v_{j} ∥_{X}^{p}dμ(x) = ∫_{_{m}^{n}} _{ε}∥∑_{j=1}^{n} ε_{j}e^{\frac{2πix_{j}}{m}} v_{j} ∥_{X}^{p}dμ(x) ≥ \frac{1}{2^{p}} _{ε}∥∑_{j=1}^{n} ε_{j}v_{j} ∥_{X}^{p}. (6)

Combining (4), (5) and (6) yields the required result.

Let X be a Banach space with type p, m an integer divisible by 4, and k an odd integer. Fix f : _{m}^{n} → X and ε ∈ {−1, 1}^{n}. Define A^{(k)}f : _{m}^{n} → X by

A^{(k)}f (x) = \frac{1}{k^{n}} ∑_{z∈(−k,k)^{n}∩(2 )^{n}} f (x + z).

Lemma 2.2. For p ≥ 1 and every f : _{m}^{n} → X

∫_{_{m}^{n}} ∥A^{(k)}f (x) − f (x)∥ _{X}^{p}dμ(x) ≤ (k − 1)^{p}n^{p−1} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x).

Proof. For every t ∈ let s(t) be the sign of t (with convention that s(0) = 0). For every z ∈ _{m}^{n},

∥f (x + z) − f (x)∥_{X}^{p} ≤ ∥z∥_{1}^{p−1} · ∑_{j=1}^{n} ∑_{ℓ=1}^{|z_{j}|} ∥f (x + ∑_{t=1}^{j−1} z_{t}e_{t} + ℓ · s(z_{j}) · e_{j}) − f (x + ∑_{t=1}^{j−1} z_{t}e_{t} + (ℓ − 1) · s(z_{j}) · e_{j})∥_{X}^{p}.

Observe that since k is odd, |(−k, k)^{n} ∩ (2 )^{n}| = k^{n}. Thus

∫_{_{m}^{n}} ∥A^{(k)}f (x) − f (x)∥ _{X}^{p}dμ(x) ≤ \frac{1}{k^{n}} ∑_{z∈(−k,k)^{n}∩(2 )^{n}} ∫_{_{m}^{n}} ∥f (x + z) − f (x)∥_{X}^{p}dμ(x)

≤ \frac{1}{k^{n}} ∑_{z∈(−k,k)^{n}∩(2 )^{n}} ∫_{_{m}^{n}} ∥z∥_{1}^{p−1} ∑_{j=1}^{n} ∑_{ℓ=1}^{|z_{j}|} ∥f (x + ∑_{t=1}^{j−1} z_{t}e_{t} + ℓs(z_{j})e_{j}) − f (x + ∑_{t=1}^{j−1} z_{t}e_{t} + (ℓ − 1)s(z_{j})e_{j})∥_{X}^{p}dμ(x)
≤ \frac{1}{k^{n}} ∑_{z∈(−k,k)^{n}∩(2 )^{n}} ∑_{j=1}^{n} ∥z∥_{1}^{p−1}|z_{j}| ∫_{_{m}^{n}} ∥f (y + s(z_{j})e_{j}) − f (y)∥_{X}^{p}dμ(x)
≤ (k − 1)^{p}n^{p−1} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x).

Proof of theorem 1.2. Fix an odd integer k ∈ , with k < \frac{m}{2}. As in [8], given j ∈ \{1, …, n\} we define S(j, k) ⊆ _{m}^{n} by

S(j, k) {y ∈ [−k, k]^{n} ⊆ _{m}^{n} : y_{j} ≡ 0 mod 2 and ∀ ℓ j, y_{ℓ} ≡ 1 mod 2} .

For f : _{m}^{n} → X we define

E_{j}^{(k)}f (x) = (f ∗ \frac{1_{S(j,k)}}{μ(S(j, k))} ) (x) = \frac{1}{μ(S(j, k))} ∫_{S(j,k)} f (x + y)dμ(y).  (7)

In [8] (see equation (39) there) it is shown that for every x ∈ _{m}^{n} and ε ∈ {−1, 1}^{n},

(\frac{k}{k + 1})^{n−1} (A^{(k)}f (x + ε) − A^{(k)}f (x − ε)) = ∑_{j=1}^{n} ε_{j} [E_{j}^{(k)}f (x + e_{j}) − E_{j}^{(k)}f (x − e_{j})] + U(x, ε) + V(x, ε),

where, by inequalities (41) and (42) in [8], for every ε ∈ {−1, 1}^{n},

max\{∫_{_{m}^{n}} ∥U(x, ε)∥_{X}^{p}dμ(x), ∫_{_{m}^{n}} ∥V(x, ε)∥_{X}^{p}dμ(x)\} ≤ \frac{8^{p}n^{2p−1}}{k^{p}} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥ _{X}^{p}.

Thus, for every T > T_{p}(X),

(\frac{k}{k + 1})^{p(n−1)} _{ε} ∫_{_{m}^{n}} ∥A^{(k)}f (x + ε) − A^{(k)}f (x − ε)∥ _{X}^{p}dμ(x)

≤ 3^{p−1} _{ε} ∫_{_{m}^{n}} ∥∑_{j=1}^{n} ε_{j} [E_{j}^{(k)}f (x + e_{j}) − E_{j}^{(k)}f (x − e_{j})] ∥_{X}^{p}dμ(x) + \frac{24^{p}n^{2p−1}}{k^{p}} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x)
≤ 3^{p−1}T^{p} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥E_{j}^{(k)}f (x + e_{j}) − E_{j}^{(k)}f (x − e_{j})∥ _{X}^{p}dμ(x) + \frac{24^{p}n^{2p−1}}{k^{p}} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x)
≤ 3^{p−1}T^{p} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x − e_{j})∥ _{X}^{p}dμ(x) + \frac{24^{p}n^{2p−1}}{k^{p}} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x)  (8)
≤ (\frac{6^{p}}{3}T^{p} + \frac{24^{p}n^{2p−1}}{k^{p}} ) ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x),  (9)

where in (8) we used the fact that E_{j}^{(k)} is an averaging operator, and hence has norm 1.

On the other hand

ε ∫_{_{m}^{n}} ∥f (x + \frac{m}{2}ε) − f (x)∥ _{X}^{p}dμ(x) ≤ 3^{p−1} _{ε} ∫_{_{m}^{n}} ∥A^{(k)}f (x + \frac{m}{2}ε) − A^{(k)}f (x)∥ _{X}^{p}dμ(x) +

3^{p−1} _{ε} ∫_{_{m}^{n}} ∥f (x + \frac{m}{2}ε) − A^{(k)}f (x + \frac{m}{2}ε)∥ _{X}^{p}dμ(x) + 3^{p−1} _{ε} ∫_{_{m}^{n}} ∥A^{(k)}f (x) − f (x)∥ _{X}^{p}dμ(x)

≤ 3^{p−1} [(\frac{m}{4})^{p−1} _{ε} ∫_{_{m}^{n}} ∑_{t=1}^{m/4} ∥A^{(k)}f (x + 2tε)− A^{(k)}f (x + (2t − 2)ε)∥ _{X}^{p}dμ(x) + 2 _{ε} ∫_{_{m}^{n}} ∥A^{(k)}f (x) − f (x)∥ _{X}^{p}dμ(x)]

≤ 3^{p−1} [(\frac{m}{4})^{p} _{ε} ∫_{_{m}^{n}} ∥A^{(k)}f (x + ε) − A^{(k)}f (x − ε)∥ _{X}^{p}dμ(x) + 2k^{p}n^{p−1} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x)]  (10)

≤ [3^{p−1} (\frac{m}{4})^{p} (1 + \frac{1}{k})^{p(n−1)} (\frac{6^{p}}{3}T^{p} + \frac{24^{p}n^{2p−1}}{k^{p}} ) + \frac{2(3kn)^{p}}{3n}] ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x)  (11)

≤ 5^{p}m^{p}T^{p} ∑_{j=1}^{n} ∫_{_{m}^{n}} ∥f (x + e_{j}) − f (x)∥_{X}^{p}dμ(x),  (12)

where in (10) we used Lemma 2.2, in (11) we used (8), and (12) is true if 4n^{2−1/p} ≤ k ≤ \frac{3m}{2n^{1−1/p}}, which is a valid choice of k if m ≥ 3n^{3−2/p}.

Remark 2.3. If a metric space has Enflo type p then it also has scaled Enflo type p. This follows from a straightforward modification of Lemma 2.4 in [8]. We do not know if scaled Enflo type p implies Enflo type p. In the category of Banach spaces, a positive answer to this question would show that Enflo type p is equivalent to Rademacher type p, resolving positively Enflo’s problem [5]. We do know that for Banach spaces, scaled Enflo type p implies Enflo type p′for all p′ < p, and that scaled Enflo type and Enflo type coincide for UMD Banach spaces.

Remark 2.4. The idea of scaling by \frac{m}{2} in the definition of scaled Enflo type originates from the definition of metric cotype introduced in [8], which involves a similar scaling procedure. In the case of non-linear type it is possible that this scaling is not necessary, i.e. that Enflo type is equivalent to Rademacher type. However, as shown in [8], in the context of metric cotype the scaling is necessary- we refer to [8] for more details.

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