We introduce the notion of the scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p

Mendel, Manor

Naor, Assaf

English

Caltech Authors - Main

Bull. London Math. Soc. 39 (2007) 493–498 C2007 London Mathematical Society
doi:10.1112/blms/bdm016
SCALED ENFLO TYPE IS EQUIVALENT TO RADEMACHER TYPE
MANOR MENDEL and ASSAF NAOR
Abstract
We introduce the notion of the 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
A Banach space X is said to have Rademacher type p > 0 (see [7]) if there exists a constant
T < ∞ such that for every n ∈ N and x1, . . . , xn ∈ X,
Eε
∥∥∥∥
n∑
j=1
εjxj
∥∥∥∥
p
X
 T p
n∑
j=1
‖xj‖pX , (1.1)
where here, and in what follows, Eε denotes the expectation with respect to uniformly chosen
ε = (ε1, . . . , εn) ∈ {−1, 1}n. The inﬁmum over all constants T for which (1.1) holds is denoted
by Tp(X).
An important theorem of Ribe [12] states that Banach spaces which are uniformly
homeomorphic must have the same isomorphic local properties. In other words, any property
which remains valid under linear isomorphisms, and whose deﬁnition involves statements about
ﬁnitely many vectors, is preserved under bijections which are uniformly continuous in both
directions. This fact suggests that local properties of Banach spaces have a purely metric
reformulation. In the past three decades this point of view fuelled research on the metric
structure of Banach spaces, and led to the development of analogs of the local theory of
Banach spaces in the context of general metric spaces. We refer to the discussion in [2, 8, 9]
and the references therein for more information on this topic. In particular, as explained in the
above references, motivated by the search for concrete versions of Ribe’s theorem for various
fundamental local properties of Banach spaces, several researchers proposed non-linear notions
of type, which make sense in the setting of arbitrary metric spaces (see [5, 3, 1]). Speciﬁcally,
following Enﬂo [5], we say that a metric space (M, dM) has Enﬂo type p if there exists a
constant K such that for every n ∈ N and every mapping f : {−1, 1}n → M,
EεdM(f(ε), f(−ε))p
 Kp
n∑
j=1
EεdM (f(ε1, . . . , εj−1, εj , εj+1, . . . , εn), f(ε1, . . . , εj−1,−εj , εj+1, . . . , εn))p . (1.2)
For Banach spaces, (1.1) follows from (1.2) if we consider the function ε → ∑nj=1 εjxj . The
question of 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 type 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 another notion
of non-linear type). In [10] it is shown that for UMD Banach spaces (see [4] for a discussion
of this notion) Rademacher type p is equivalent to Enﬂo type p.
Received 10 June 2005; revised 10 May 2006; published online 4 May 2007.
2000 Mathematics Subject Classiﬁcation 46B20, 51F99.
494 MANOR MENDEL AND ASSAF NAOR
Motivated by our recent work on metric cotype [8], we introduce below the notion of the
scaled Enﬂo type of a metric space (which is, in a sense, ‘opposite’ to the notion of metric
cotype deﬁned in [8]), and we 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. Let (M, dM) be a metric space, and let 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 : Znm → M,
Eε
∫
Znm
dM
(
f
(
x +
m
2
ε
)
, f(x)
)p
dμ(x)  τpmp
n∑
j=1
∫
Znm
dM (f(x + ej), f(x))
p
dμ(x), (1.3)
where μ is the uniform probability measure on Znm, and {ej}nj=1 is the standard basis of Rn.
The inﬁmum over all constants τ for which (1.3) holds is denoted by τp(M).
Theorem 1.1. Let X be a Banach space and let p ∈ [1, 2]. Then X has Rademacher type
p if and only if X has scaled Enﬂo type p. More precisely,
1
2π
Tp(X)  τp(X)  5Tp(X).
The heart of Theorem 1.1 is the upper bound on τp(X) in terms of the Rademacher type p
constant Tp(X) (the lower bound in Theorem 1.1 is easy to prove — see Lemma 2.1 below). If
f in (1.3) were a ‘linear mapping’, then (1.3) would trivially imply (1.1). Therefore in the proof
of Theorem 1.1 we apply (1.1) to a ‘smoothed’ version of f obtained by convolving it with the
normalized indicator of a large ball in Znm. However, such an averaging necessarily results in a
‘loss of information’, which makes it diﬃcult to relate the bound on the averaged f obtained
from (1.1) to the original function f . This problem is overcome by increasing the value of m,
and is precisely why we need to scale by a factor of m/2 in (1.3). The details of this argument,
which relies in part on some technical estimates from [8], are presented in Section 2.
Remark 1. If a metric space has Enﬂo type p, then it also has scaled Enﬂo type p. This
follows from a straightforward modiﬁcation of [8, Lemma 2.4]. We do not know whether 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, positively resolving
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 p and Enﬂo type p coincide for UMD Banach
spaces.
Remark 2. The idea of scaling by 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 (that is, 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.
2. Proof of Theorem 1.1
We start by showing that scaled Enﬂo type p implies Rademacher type p.
Lemma 2.1. Let X be a Banach space, and let p ∈ [1, 2]. Then Tp(X)  2π · τp(X).
SCALED ENFLO TYPE 495
Proof. Let X be a Banach space, and assume that τp(X) < ∞ for some p ∈ [1, 2]. Fix
τ > τp(X), v1, . . . , vn ∈ X, and let m be an even integer. Deﬁne f : Znm → X by
f(x1, . . . , xn) =
n∑
j=1
e2πixj/mvj .
Then
n∑
j=1
∫
Znm
‖f (x + ej) − f(x)‖pX dμ(x) =
∣∣e2πi/m − 1∣∣p · n∑
j=1
‖vj‖pX

(2π
m
)p
·
n∑
j=1
‖vj‖pX , (2.1)
and
Eε
∫
Znm
∥∥∥∥f (x + m2 ε
)
− f(x)
∥∥∥∥
p
X
dμ(x) = 2p
∫
Znm
∥∥∥∥
n∑
j=1
e2πixj/mvj
∥∥∥∥
p
X
dμ(x). (2.2)
We recall the contraction principle (see [6]), which states that for every a1, . . . , an ∈ R,
Eε
∥∥∥∥
n∑
j=1
εjajvj
∥∥∥∥
p
X

(
max
1jn
|aj |
)p
· Eε
∥∥∥∥
n∑
j=1
εjvj
∥∥∥∥
p
X
.
Thus ∫
Znm
∥∥∥∥
n∑
j=1
e2πixj/mvj
∥∥∥∥
p
X
dμ(x) =
∫
Znm
Eε
∥∥∥∥
n∑
j=1
e(2πi/m)(xj+m(1−εj)/4)vj
∥∥∥∥
p
X
dμ(x)
=
∫
Znm
Eε
∥∥∥∥
n∑
j=1
εje
2πixj/mvj
∥∥∥∥
p
X
dμ(x)
 1
2p
Eε
∥∥∥∥
n∑
j=1
εjvj
∥∥∥∥
p
X
. (2.3)
Combining (2.1), (2.2) and (2.3) 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 : Znm → X and ε ∈ {−1, 1}n. Deﬁne A(k)f : Znm → X by
A(k)f(x) = 1
kn
∑
z∈(−k,k)n∩(2Z)n
f(x + z).
Lemma 2.2. For p  1 and every f : Znm → X, we have∫
Znm
∥∥A(k)f(x) − f(x)∥∥p
X
dμ(x)  (k − 1)pnp−1
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x).
Proof. For every t ∈ R let s(t) be the sign of t (with the convention that s(0) = 0). For
every z ∈ Znm, we have
‖f(x + z) − f(x)‖pX
 ‖z‖p−11 ·
n∑
j=1
|zj |∑
=1
∥∥∥∥f
(
x +
j−1∑
t=1
ztet +  · s(zj) · ej
)
− f
(
x +
j−1∑
t=1
ztet + ( − 1) · s(zj) · ej
)∥∥∥∥
p
X
.
496 MANOR MENDEL AND ASSAF NAOR
Observe that since k is odd, |(−k, k)n ∩ (2Z)n| = kn. Thus
∫
Znm
∥∥∥A(k)f(x) − f(x)∥∥∥p
X
dμ(x)
 1
kn
∑
z∈(−k,k)n∩(2Z)n
∫
Znm
‖f(x + z) − f(x)‖pX dμ(x)
 1
kn
∑
z∈(−k,k)n∩(2Z)n
∫
Znm
‖z‖p−11
n∑
j=1
|zj |∑
=1
∥∥∥∥f
(
x +
j−1∑
t=1
ztet + s(zj)ej
)
− f
(
x +
j−1∑
t=1
ztet + ( − 1)s(zj)ej
)∥∥∥∥
p
X
dμ(x)
 1
kn
∑
z∈(−k,k)n∩(2Z)n
n∑
j=1
‖z‖p−11 |zj |
∫
Znm
‖f(y + s(zj)ej) − f(y)‖pX dμ(x)
 (k − 1)pnp−1
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x).
This completes the proof of Lemma 2.2.
Proof of Theorem 1.1. Fix an odd integer k ∈ N, with k < m/2. As in [8], given j ∈
{1, . . . , n} we deﬁne S(j, k) ⊆ Znm by
S(j, k) := {x ∈ [−k, k]n ⊆ Znm : yj ≡ 0 (mod 2) and ∀ = j, y ≡ 1 (mod 2)} .
For f : Znm → X, deﬁne
E(k)j f(x) =
(
f ∗ 1S(j,k)
μ(S(j, k))
)
(x)
=
1
μ(S(j, k))
∫
S(j,k)
f(x + y) dμ(y). (2.4)
In [8] (see equation (39) there) it is shown that for every x ∈ Znm and ε ∈ {−1, 1}n,
( k
k + 1
)n−1(A(k)f(x + ε) − A(k)f(x − ε))
=
n∑
j=1
εj
[E(k)j f(x + ej) − E(k)j f(x − ej)]+ U(x, ε) + V (x, ε),
where, by [8, inequalities (41) and (42)], for every ε ∈ {−1, 1}n,
max
{ ∫
Znm
‖U(x, ε)‖pX dμ(x),
∫
Znm
‖V (x, ε)‖pX dμ(x)
}
 8
pn2p−1
kp
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX .
SCALED ENFLO TYPE 497
Thus, for every T > Tp(X), we have( k
k + 1
)p(n−1)
Eε
∫
Znm
∥∥∥A(k)f(x + ε) − A(k)f(x − ε)∥∥∥p
X
dμ(x)
 3p−1Eε
∫
Znm
∥∥∥∥
n∑
j=1
εj
[
E(k)j f(x + ej) − E(k)j f(x − ej)
]∥∥∥∥
p
X
dμ(x)
+
24pn2p−1
kp
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x)
 3p−1T p
n∑
j=1
∫
Znm
∥∥∥E(k)j f(x + ej) − E(k)j f(x − ej)∥∥∥p
X
dμ(x)
+
24pn2p−1
kp
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x)
 3p−1T p
n∑
j=1
∫
Znm
‖f(x + ej) − f(x − ej)‖pX dμ(x)
+
24pn2p−1
kp
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x) (2.5)

(
6p
3
T p +
24pn2p−1
kp
) n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x), (2.6)
where in (2.5) we used the fact that E(k)j is an averaging operator, and hence has norm 1.
On the other hand
Eε
∫
Znm
∥∥∥f (x + m
2
ε
)
− f(x)
∥∥∥p
X
dμ(x)
 3p−1Eε
∫
Znm
∥∥∥A(k)f (x + m
2
ε
)
− A(k)f(x)
∥∥∥p
X
dμ(x)
+ 3p−1Eε
∫
Znm
∥∥∥f (x + m
2
ε
)
− A(k)f
(
x +
m
2
ε
)∥∥∥p
X
dμ(x)
+ 3p−1Eε
∫
Znm
∥∥∥A(k)f(x) − f(x)∥∥∥p
X
dμ(x)
 3p−1
[(m
4
)p−1
Eε
∫
Znm
m/4∑
t=1
∥∥∥A(k)f (x + 2tε) − A(k)f(x + (2t − 2)ε)∥∥∥p
X
dμ(x)
+ 2Eε
∫
Znm
∥∥∥A(k)f(x) − f(x)∥∥∥p
X
dμ(x)
]
 3p−1
[(m
4
)p
Eε
∫
Znm
m/4∑
t=1
∥∥∥A(k)f (y + ε) − A(k)f(x − ε)∥∥∥p
X
dμ(x)
+ 2kpnp−1
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x)
]
(2.7)

[
3p−1
(m
4
)p(
1 +
1
k
)p(n−1)(6p
3
T p +
24pn2p−1
kp
)
+
2(3kn)p
3n
]
·
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x) (2.8)
498 SCALED ENFLO TYPE
 5pmpT p
n∑
j=1
∫
Znm
‖f(x + ej) − f(x)‖pX dμ(x), (2.9)
where in (2.7) we used Lemma 2.2, in (2.8) we used (2.6), and (2.9) is true provided that
4n2−1/p  k  3m/2n1−1/p, which is a valid choice of k if m  3n3−2/p.
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Manor Mendel
California Institute of Technology
Pasadena, CA
USA
Assaf Naor
Microsoft Research
One Microsoft Way
Redmond WA 98052
USA
anaor@microsoft.com
Current Address:
The Open University of Israel
108 Ravutzki St.
Ra’anana 43107
Israel
mendelma@gmail.com
Current Address:
New York University
251 Mercer Street
New York, NY 10012
USA
naor@cims.nyu.edu


Scaled Enflo type is equivalent to Rademacher type

Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces’,

Markov chains, Riesz transforms and Lipschitz maps’,

Martingales and singular integrals

Metric cotype’,

On inﬁnite-dimensional topological groups’,

On type of metric spaces’,

On uniformly homeomorphic normed spaces’,

Probabilistic methods in the geometry of Banach spaces’, Probability and analysis,

Scaled Enflo type is equivalent to Rademacher type

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