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PROPERTIES IN BANACH SPACES
A Fovelle
To cite this version:
A Fovelle. HAMMING GRAPHS AND CONCENTRATION PROPERTIES IN BANACH SPACES.
2021. �hal-03267848�
BANACH SPACES
A. FOVELLE
Abstract. In this note, we study some concentration properties for Lipschitz maps de- fined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of quasi-reflexive spaces satisfying upper`ptree estimates to the setting of`p-sums of such spaces. We also give a sufficient condition for a space to be asymptotic-c0 in terms of a concentration property, as well as relevant counterexamples.
1. Introduction
In 2008, in order to show that Lpp0,1q is not uniformly homeomorphic to `p ``2 for p P p1,8qzt2u, Kalton and Randrianarivony [26] introduced a new technique based on a certain class of graphs and asymptotic smoothness ideas. To be more specific, they introduced a concentration property for Lipschitz maps defined on Hamming graphs into a reflexive asymptotically uniformly smooth (AUS) Banach space X (we refer the reader to section 2 for the definitions), that prevents the coarse embedding of certain other spaces into X. Their result was used by Kalton himself to deduce some information about the spreading models of a space that coarse Lipschitz embeds into a reflexive AUS space (see [24]), and was later extended to the quasi-reflexive case by Lancien et Raja [30], who introduced a weaker concentration property. Soon after, Causey [11] proved that this same weaker concentration property also applies to quasi-reflexive spaces with so-called upper `p tree estimates.
The purpose of this paper is to start a general study of these concentration properties, together with new ones. In particular, we will address the question of their stability under sums of Banach spaces.
Our main result will be the following (see the definitions of concentration properties HIC in section 2).
Theorem 1.1. Let pP p1,8q, λ¡0, pXnqnPN a sequence of Banach spaces with property λ-HICp,d.
Let E be a reflexive Banach space with a normalized 1-unconditionalp-convex basispenqnPN
with convexity constant 1.
Then X p°
nPNXnqE has property pλ 2 εq-HICp,d for every ε¡0.
As a consequence, we get that an`q-sum of a quasi-reflexive Banach space satisfying `p upper tree estimates, 1 q, p 8, cannot equi-Lipschitz contain the Hamming graphs.
This is a generalization of the result mentionned above by Causey (see [11]), who proved it for quasi-reflexive Banach spaces satisfying `p upper tree estimates. This is the first result of this type for non quasi-reflexive Banach spaces.
Key words and phrases. Banach spaces, Hamming graphs, Asymptotic structure, nonlinear embeddings, concentration properties.
The author was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (con- tract ANR-15-IDEX-03).
1
In order to show this result, we introduce the notions and the terminology we will use later in the second section of this paper while section 3 is dedicated to the proof itself.
Recently, Baudier, Lancien, Motakis and Schlrumprecht [4] proved that any quasi-reflexive asymptotic-c0 Banach spaceX (see section 4 for the definition of asymptotic-c0) has prop- erty HIC8. Even though we don’t know if a Banach space with this property is quasi- reflexive, we prove in the fourth and last section of this paper that property HIC8 implies asymptotic-c0. In particular, the space TpTq, where T is the original Banach space con- structed by Tsirelson in [39], cannot have this concentration property.
We also give an exemple of a separable dual asymptotic-c0 space that does not Lipschitz contain `1 and without any of the concentration properties introduced in this paper. This example is based on a generalization of the construction of Lindenstrauss spaces, that we owe to Schlumprecht. This construction is detailed in Appendix A.
2. Definitions and notation
2.1. Basic definitions. All Banach spaces in these notes are assumed to be real and infinite dimensional unless otherwise stated. We denote the closed unit ball of a Banach space X by BX, and its unit sphere by SX. Given a Banach space X with norm }.}X, we simply write }.}as long as it is clear from the context on which space it is defined. We recall that a Banach space is said to bequasi-reflexive if the image of its canonical embedding into its bidual is of finite codimension in its bidual.
We say that a basic sequencepeiq of a Banach spaceE is c-unconditional, for some c¥1, if, for any paiq P c00 (the vector space of all real sequences with finite support) and any pσiq t1u, we have :
¸8 i1
aiei
¤c
¸8 i1
σiaiei
.
Let pXnqnPN be a sequence of Banach spaces. Let E penqnPN be a 1-unconditional basic sequence in a Banach space E with norm }.}E. We define the sum p°
nPNXnqE to be the space of sequences pxnqnPN, wherexn PXn for all n PN, such that °
}xn}Xnen converges inE, and we set
}pxnqnPN}
¸
nPN
}xn}Xnen E
8. One can check that p°
nPNXnqE, endowed with the norm }.} defined above, is a Banach space. We can, in a similar way, define finite sums °n
j1Xj
E for allnPN, and, in case n2, we will write X1À
E
X2. If it is implicit what is the basis E of the Banach space E that we are working with, we write p°
nPNXnqE orX1
À
E
X2. Also, if the Xn’s are all the same, say XnX, for all nPN, we writeEpXq.
Let us finish this section with the following definition.
LetpP p1,8qandE a Banach space with a 1-unconditional basis penqnPN. We say that the basispenqnPN is p-convex with convexity constantC if :
¸
jPN
p|x1j|p |xkj|pq1pej
p
¤Cp
¸k n1
}xn}p
for all x1 °8
j1
x1jej, ,xk °8
j1
xkjej PE.
2.2. Hamming graphs. Before introducing concentration properties, we need to define special metric graphs that we shall callHamming graphs. LetM be an infinite subset ofN. We denote rMsω the set of infinite subsets ofM. ForMP rNsω and kPN, we note
rMsk tn pn1, , nkq PMk;n1 nku, rMs¤k
¤k j1
rMsjY t∅u, and
rMs ω ¤8
k1
rMskY t∅u. Then we equip rMsk with theHamming distance:
dHpn, mq |tj;nj mju|
for all n pn1, , nkq, m pm1, , mkq P rMsk.
Let us mention that this distance can be extended torMs ω by letting
dHpn, mq |tiP t1, ,minpl, jqu;nimiu| maxpl, jq minpl, jq for all n pn1, , nlq, m pm1, , mjq P rMs ω (with possibly l0 orj 0).
We also need to introduceIkpMq, the set of strictly interlaced pairs in rMsk: IkpMq tpn, mq rMsk;n1 m1 nk mku and, for eachjP t1, , ku, let us denote
HjpMq tpn, mq rMsk;@ij, nimi and nj mju.
Note that, for pn, mq PIkpMq,dHpn, mq k andnXm∅.
Let us mention that, in this paper, we will only be interested in the Hamming distance but originally, when Hamming graphs were used in [26], it could be equally replaced (unless for their last Theorem 6.1) by the symmetric distance, defined by
d∆pn, mq 1 2|n4m|
for all n, mP rNs ω, wheren4m denotes the symmetric difference betweennand m.
2.3. Asymptotic properties. We now define the uniform asymptotic properties of norms that will be considered in this paper. Let pX,}.}q be a Banach space. Following Milman (see [33]), we introduce the following modulus : for allt¥0, let
ρXptq sup
xPSX
infY sup
yPSY
p}x ty} 1q
whereY runs through all closed linear subspaces ofX of finite co-dimension.
We say that }.} is asymptotically uniformly smooth (in short AUS) if limtÑ0 ρXptq
t 0. If pP p1,8q,}.} is said to be p-AUS if there is a constant C¡0 such that, for all tP r0,8q, ρXptq ¤Ctp. IfX has an equivalent norm for whichX is AUS (resp. p-AUS),X is said to be AUSable (resp. p-AUSable). Every asymptotically uniformly smooth Banach space is p-AUSable for somepP p1,8q, this was first proved for separable Banach spaces by Knaust, Odell and Schlumprecht (see [27]) and later generalized by Raja for any Banach space (see [36], Theorem 1.2).
LetX be a Banach space,B X and MP rNsω. A familypxnqnPrMs¤k inB is a tree in B ofheight k. This treepxnqnPrMs¤k is said to beweakly null if the sequencepxn,nqn¡maxpnq
is weakly null for everynP rMsk1Y t∅u (with maxp∅q 0).
Let 1 p ¤ 8,C ¡0. We say thatX satisfies upper `p tree estimates with constant C if for anykPNand any weakly null tree pxnqnPrNs¤k inBX, there exists nP rNsk such that
@a pa1, , akq PRk,
¸k j1
ajxn1,,nj
¤C}a}`kp.
We say that X satisfies upper `p tree estimates if X satisfies upper `p tree estimates with constantC for someC ¡0.
We say thatXhas the tree-p-Banach-Saks property with constant Cif for anykPNand any weakly null treepxnqnPrNs¤k inBX, there existsnP rNsk such that
¸k j1
xn1,,nj
¤Ck1{p with the convention 1{p0 ifp 8.
We say thatX has thetree-p-Banach-Saks property (tree-p-BS) ifX has the tree-p-Banach- Saks property with constantC for someC ¡0.
Let pP p1,8q. We denote Tp the class of allp-AUSable Banach spaces, Ap the class of all Banach spaces satisfying upper`p tree estimates, Np the class of all Banach spaces with tree-p-BS property and Pp
1 r pTr. It is known that (see [19] and [7]) Tp Ap NpPp
and Causey proved in [11] that these inclusions are strict (even among reflexive spaces).
It is also known (see [7]) that N8 A8.
2.4. Metric embeddings. Let us recall some definitions on metric embeddings.
LetpX, dXqand pY, dYqtwo metric spaces, f a map from X toY. We define the compression modulus of f by
@t¥0, ρfptq inftdYpfpxq, fpyqq;dXpx, yq ¥tu; and theexpansion modulus off by
@t¥0, ωfptq suptdYpfpxq, fpyqq;dXpx, yq ¤tu.
We adopt the convention infp∅q 8. Note that, for everyx, yPX, we have ρfpdXpx, yqq ¤dYpfpxq, fpyqq ¤ωfpdXpx, yqq.
We say that f is a Lipschitz embedding if there existA, B in p0,8q such thatρfptq ¥ At andωfptq ¤Btfor allt¥0. If there exists a such embeddingf, we notepX, dXq ãÑ
L pY, dYq. If the metric spaces are unbounded, the mapf is said to be acoarse embeddingif limtÑ8ρfptq 8 and ωfptq 8for all t¡0.
If one is given a family of metric spacespXiqiPI, one says thatpXiqiPI equi-Lipschitz embeds intoY if there existA, Binp0,8qand, for alliPI, mapsfi :XiÑY such thatρfiptq ¥At and ωfiptq ¤ Bt for all t¥0. One also says that the family pXiqiPI equi-coarsely embeds intoY if there exist non-decreasing functions ρ, ω:r0,8q Ñ r0,8qand for all iPI, maps fi :Xi ÑY such that ρ¤ρfi,ωfi ¤ω, limtÑ8ρptq 8and ωptq 8for allt¡0.
Besides, we say that f is a coarse Lipschitz embedding if there exist A, B, C, D in p0,8q
such thatρfptq ¥AtC and ωfptq ¤Bt Dfor all t¥0. IfX and Y are Banach spaces, this is equivalent to the existence of numbers θ¥0 and 0 c1 c2 so that :
c1}xy}X ¤ }fpxq fpyq}Y ¤c2}xy}X
for all x, yPX satisfying }xy}X ¥θ.
Finally, a way to refine the scale of coarse embeddings is to talk about compression expo- nents, introduced by Guentner and Kaminker in [20]. Let X andY to Banach spaces. The compression exponent ofX inY, denotedαYpXq, is the supremum of allαP r0,1qfor which there exists a coarse embeddingf :XÑY and A, C inp0,8qso thatρfptq ¥AtαC for all t¡0.
2.5. Definitions of concentration properties. In this subsection, we introduce all the concentration properties mentioned in this paper. Before doing so, let us recall a version of Ramsey’s Theorem we will use several times.
Theorem 2.1 (Ramsey’s Theorem [37]). Let kPN andA rNsk. There exists MP rNsω such that either rMskAor rMskXA∅.
The following properties are studied in [26], [30] and [4]. We use the convention 1{8 0.
Definition 2.2. LetpX, dq be a metric space,λ¡0, pP p1,8s.
We say thatX has propertyλ-HFCp (Hamming Full Concentration) if, for all kPN, for every Lipschitz functionf :prNsk, dHq ÑX, one can find MP rNsω such that
@n, mP rMsk, dpfpnq, fpmqq ¤λk1pLippfq.
We say that X has propertyHFCp ifX has propertyλ-HFCp for someλ¡0.
We say thatXhas propertyλ-HICp(Hamming Interlaced Concentration) if, for allkPN, for every Lipschitz functionf :prNsk, dHq ÑX, one can find pn, mq PIkpNq satisfying
dpfpnq, fpmqq ¤λk1pLippfq.
We say that X has propertyHICp ifX has propertyλ-HICp, for someλ¡0.
Remark 2.3. 1qLet us notice that, by Ramsey theory (IkpNq can be identified withrNs2k), a metric spacepX, dq has propertyλ-HICp if and only if, for all k PN, for every Lipschitz functionf :prNsk, dHq ÑX, one can find MP rNsω that satisfies
@pn, mq PIkpMq, dpfpnq, fpmqq ¤λk1pLippfq.
2qBaudier, Lancien, Motakis and Schlrumprecht showed that property HFC8 is equivalent for a Banach space to being reflexive and asymptotic-c0 (see [4] for the proof of this result and section 4 for the definition of asymptotic-c0).
We now introduce a property that seems weaker than the previous one but is enough to prevent the equi-Lipschitz embedding (or equi-coarse embedding for the case p 8) of Hamming graphs. We will show later that this property actually coincides with property HICp,pP p1,8s.
Definition 2.4. Let pX, dq be a metric space, λ ¡ 0 p P p1,8s. We say that X has property λ-HCp if, for all k PN, for every Lipschitz function f : prNsk, dHq Ñ X, one can find n, mP rNsk satisfying nXm∅and
dpfpnq, fpmqq ¤λk1pLippfq.
We say that X has propertyHCp ifX has propertyλ-HCp, for someλ¡0.
It is easy to check that all these concentration properties are stable under coarse Lips- chitz embeddings and that properties HFC8, HC8 and HIC8 are even stable under coarse embeddings, when the embedded space is a Banach space.
Let us now introduce the last concentration properties we will study here, more precise than HCp and HICp,pP p1,8q, where directionnal Lipschitz constants take part, hence the
”d” in subscript in the acronyms below.
Definition 2.5. LetpX, dq be a metric space,λ¡0, pP p1,8q.
We say that X has property λ-HFCp,d (resp. λ-HICp,d) if, for every k P N and every Lipschitz function f :prNsk, dHq ÑX, there existsMP rNsω such that
dpfpnq, fpmqq ¤λ k
¸
j1
αpj 1p
for all n, mP rMsk (resp. pn, mq PIkpMq), where, for eachj P t1, , ku αj sup
pn,mqPHjpNqdpfpnq, fpmqq.
We say that X has property HFCp,d (resp. HICp,d) if X has property λ-HFCp,d (resp. λ- HICp,d), for some λ¡0.
Similary, we say that X has property λ-HCp,d if, for every k P N and every Lipschitz functionf :prNsk, dHq ÑX, one can find n, mP rNsk satisfying nXm∅and
dpfpnq, fpmqq ¤λ k
¸
j1
αpj 1p
where the αj,jP t1, , ku, are defined as above.
We say that X has propertyHCp,d ifX has propertyλ-HCp,d, for someλ¡0.
It is important to note that Theorem 6.1 [26] and Theorem 5.2 [11] can be rephrased as follows: for p P p1,8q, a reflexive (resp. quasi-reflexive) Banach space satisfying upper
`p tree estimates has property HFCp,d (resp. HCp,d) and a reflexive (resp. quasi-reflexive) Banach space with tree-p-Banach-Saks property has property HFCp (resp. HCp). Even though Kalton and Randrianarivony [26] proved their theorem for reflexive p-AUS Banach spaces, their proof implicitely contains this result. Let us also note that a Banach space with property HFCp is necessarily reflexive (see [2]). In 2017, Lancien and Raja [30] proved that all quasi-reflexivep-AUS Banach spaces have property HCp,d. It was later extended as mentionned by Causey [11].
The stability of these last properties under coarse Lipschitz embeddings when the em- bedded space is a Banach space is a bit less clear so we include a proof for completeness.
Proposition 2.6. Let pP p1,8q, X a Banach space and pY, dYq a metric space.
If Y has property HFCp,d (resp. HICp,d, resp. HCp,d) andX coarse Lipschitz intoY, then X has property HFCp,d (resp. HICp,d, resp. HCp,d).
Proof. We only prove the stability of HFCp,d, the proof for the other two properties is similar.
Let us assume that Y has property λ-HFCp,d for a λ ¡ 0 and that there exist a map ϕ:X ÑY and A, B, C, D¡0 such that ρϕptq ¥AtB and ωϕptq ¤Ct D for all t¥0.
LetkPN,f :prNsk, dHq ÑX a Lipschitz function with Lippfq ¡0.
Without loss of generality, we can assume that, for all jP t1, , ku, we have αj sup
pn,mqPHjpMq}fpnq fpmq} ¡0.
Indeed, dH is a graph metric so maxjPt1,,kuαj Lippfq ¡ 0 and if αj 0 for some jP t1, , ku, then the expression of f does not depend on this jth coordinate.
Therefore
α min
1¤j¤kαj P p0,Lippfqs.
Let us note that ωϕptq ¤ pC Dqt for all t ¥ 1 so, for all j P t1, , ku and for all pn, mq PHjpNq, we have
dY
ϕ 1
αfpnq
, ϕ 1
αfpmq
¤ωϕ
αj
α ¤ C D
α αj. Now, by assumption on Y, we can find MP rNsω so that
dY
ϕ 1
αfpnq
, ϕ 1
αfpmq
¤ λpC Dq α
k
¸
j1
αpj 1p
for all n, mP rMsk. Thus
}fpnq fpmq} ¤ λpC Dq A
k
¸
j1
αpj 1p
αB
A ¤ λpC Dq B A
k
¸
j1
αpj 1p
for all n, mP rMsk. Consequently, X has property HFCp,d. As promised, the next proposition shows that properties HCp and HICp, pP p1,8s, are equivalent. This explains why we will only talk about property HC8 in the last section.
Before proving this result, let us introduce some vocabulary. Let M P rNsω. For n, m P rMsk satisfying nXm ∅, we denote φ the unique increasing bijection from nYm onto t1, ,2ku. If
I tA t1, ,2ku;|A| ku, we say that pn, mqis in position API ifφpnq A.
Thus, we note that pn, mq rMsk, with nXm ∅, can be in 2kk11
possible different positions if we ask n1 to be the first element (and we can do it without loss of generality).
We denote these positionsPikpMq,iP!
1, , 2kk11)
. Let us remark that each one of these positions can be identified with rMs2k, which will allow us to use Ramsey theory.
Proposition 2.7. For every p P p1,8s, properties HCp and HICp are equivalent. More precisely, a metric space with property λ-HICp, for some λ¡0, has property λ-HCp and a metric space with property λ-HCp has property 2λ-HICp.
Proof. For every p P p1,8s, λ ¡0, the implication λ-HICp ùñ λ-HCp is clear so let us show the other implication.
We will do it withp 8, the other cases can be treated similarly.
Let pX, dq be a metric space with property λ-HC8 for some λ ¡ 0. Let k P N, f : prNsk, dHq ÑX a Lipschitz function.
For eachMP rNsω, there existiP!
1, , 2kk11)
andpn, mq PPikpMqsuch thatdpfpnq, fpmqq ¤ λLippfq.
Let us show that there exist iP!
1, , 2kk11)
and MP rNsω such that dpfpnq, fpmqq ¤ λLippfq for all pn, mq PPikpMq.
By Ramsey theory, if A1 tpn, mq P P1kpNq;dpfpnq, fpmqq ¤ λLippfqu P1kpNq, there existsM1P rNsω such thatP1kpM1q A1 orP1kpM1q XA1 ∅.
IfP1kpM1qXA1 ∅, we apply the same result withA2 tpn, mq PP2kpM1q;dpfpnq, fpmqq ¤ λLippfqu P2kpM1q and we getM2P rM1sω such thatP2kpM2q A2 orP2kpM2q XA2∅.
...
As X has property λ-HC8, we cannot repeat this operation for all 2kk11
positions so there exist i P !
1, , 2kk11)
and M P rNsω such that dpfpnq, fpmqq ¤ λLippfq for all pn, mq PPikpMq.
Let us show that there existspn, mq PIkpNq such that dpfpnq, fpmqq ¤2λLippfq.
For that, we denoteM tq1 q2 qj u.
Now, we just have to observe that we can choose pn, pq P PikpMq such that n1 p1 and n, p tq1, q2k 1, , q2kp2k1q 1u. This leaves us enough space to get an element mP rMsk so that pn, mq PIkpMq and pm, pq PPikpMq.
The result follows from the triangle inequality.
Remark 2.8. With a similar proof, we can prove that properties HCp,d and HICp,d are equivalent.
3. Stability under sums
3.1. Statements. In order to prove the stability of property HCp,d, p P p1,8q, under `p sums, the idea is to adapt Braga’s proof of [7] Proposition 7.2 with property HICp,d instead of property p-Banach-Saks.
To do so, we need the following proposition. We chose to state it with property HICp,d, which we recall is equivalent to property HCp,d, but the same result can be shown for property HFCp,d with a similar proof.
Proposition 3.1. Let p P p1, 8q, λ ¡ 0, E be a Banach space with a normalized 1- unconditional p-convex basis penqnPN with convexity constant 1.
For every n P N and every finite sequence pXjqnj1 of Banach spaces having property λ- HICp,d, the space °n
j1Xj
E has property pλ εq-HICp,d for each ε¡0.
Proof. It is enough to prove this result forXX1À
E
X2.
LetkPN,MP rNsω,ε¡0,h pf, gq:prMsk, dHq ÑX a Lipschitz function.
For eachjP t1, , ku, let us denote γj sup
pn,mqPHjpMq}hpnq hpmq}. There exists ε1 ¡0 such that
λp
¸k j1
pγj 2ε1qp ¤ pλ εqp
¸k j1
γjp.
Letα1 inf
M1PrMsω sup
pn,mqPH1pM1q}fpnq fpmq}.
There exists M1 P rMsω so that }fpnq fpmq} ¤α1 ε1 for everypn, mq PH1pM1q. Letβ1 inf
M11PrM1sω sup
pn,mqPH1pM11q}gpnq gpmq}.
There exists M11 P rM1sω so that }gpnq gpmq} ¤β1 ε1 for everypn, mq PH1pM11q. ...
Letαk inf
MkPrM1k1sω sup
pn,mqPHkpMkq}fpnq fpmq}.
There exists MkP rM1k1sω so that}fpnq fpmq} ¤αk ε1 for everypn, mq PHkpMkq.
Letβk inf
M1kPrMksω sup
pn,mqPHkpM1kq}gpnq gpmq}.
There exists M1kP rMksω so that }gpnq gpmq} ¤βk ε1 for everypn, mq PHkpM1kq.
Let us begin by showing that}αje1 βje2} ¤γj for all jP t1, , ku.
For that, assume that there existsjP t1, , ku such that}αje1 βje2} ¡γj. Then, there existsη¡0 so that}pαjηqe1 pβjηqe2} ¡γj.
If there existspn, mq PHjpM1kqsuch that}fpnqfpmq} ¥αjηand}gpnqgpmq} ¥βjη, then}hpnq hpmq} ¡γj, which is impossible.
So}fpnq fpmq} ¤αj η or}gpnq gpmq} ¤βj η for all pn, mq PHjpM1kq.
Now we note that HjpM1kq can be identified with rM1ksk 1 so, by Ramsey theory, we get M1 P rM1ksωsuch that}fpnqfpmq} ¤αjηfor allpn, mq PHjpM1qor}gpnqgpmq} ¤βjη for all pn, mq PHjpM1q. This contradicts the definition ofαj orβj.
Thus}αje1 βje2} ¤γj for allj P t1, , ku. By assumption, there existsM1 P rMk1sω so that
}fpnq fpmq} ¤λ k
¸
j1
pαj ε1qp 1p
and }gpnq gpmq} ¤λ k
¸
j1
pβj ε1qp p1
for allpn, mq PIkpM1q. Let us denotexn pαn ε1qe1 pβn ε1qe2 for eachnP t1, , ku. Usingp-convexity, we get :
}hpnq hpmq}p ¤λp
k
¸
j1
pαj ε1qp 1
p
e1
k
¸
j1
pβj ε1qp 1
p
e2
p
¤λp
¸k n1
}xn}p λp
¸k j1
}pαj ε1qe1 pβj ε1qe2}p
¤λp
¸k j1
pγj 2ε1qp for all pn, mq PIkpM1q.
Therefore
}hpnq hpmq}p¤ pλ εq k
¸
j1
γjp 1
p
for all pn, mq PIkpM1q,i.eX haspλ εq-HICp,d.
From this property about finite sums, we can deduce our main result. In order to do so, let us remark that a Banach spaceE that has ap-convex basis with constant 1 satisfies the following: ifxPE and pxnqnPN is a weakly null sequence inE, then
lim sup}x xn}p ¤ }x}p lim sup}xn}p.
Therefore, we deduce from the proof of Theorem 4.2 [26] that ifE is in addition reflexive, then for every k P N, every M P rNsω, every ε ¡ 0 and every Lipschitz function f : prMsk, dHq ÑE, there exist M1 P rMsω and uPE so that
}fpnq u} ¤ k
¸
j1
αpj 1
p
ε for all nP rM1sk, whereαj sup
pn,mqPHjpMq}fpnq fpmq} for allj P t1, , ku. We now prove Theorem 1.1.
Proof of Theorem 1.1. Letε¡0,MP rNsω,kPN,f :prMsk, dHq ÑXa Lipschitz function.
There exists ε1 ¡0 such that pλ 2 ε1q
k
¸
j1
αpj 1p
4ε1 ¤ pλ 2 εq k
¸
j1
αpj 1p
where
αj sup
pn,mqPHjpMq}fpnq fpmq}
for all jP t1, , ku. The well-defined map
φ:
$&
%
X Ñ E
pxnqnPN ÞÑ °8
n1
}xn}en
satisfies Lippφq ¤1 and }φpxq} }x} for allxPX, thus sup
pn,mqPHjpMq}φfpnq φfpmq} ¤αj
for everyj P t1, , ku.
From the previous remark, we getuPE and M1 P rMsω such that }φfpnq u} ¤
k
¸
j1
αpj 1p
ε1 for all nP rM1sk. LetN PNsuch that°8kN 1}uk}ek ¤ε1.
For each n P N, let us denote Pn the projection from X onto Xn and Πn the projection from X onto p°n
k1XkqE. We have
¸8 nN 1
}Pnfpnq}en
¤
¸8 nN 1
}Pnfpnq}en
¸8 nN 1
}un}en
ε1
¤
¸8 nN 1
p}Pnfpnq} }un}qen
ε1
¤ }φfpnq u} ε1
¤ k
¸
j1
αpj 1p
2ε1 for all nP rM1sk.
Moreover, according to the previous result, we get an infinite subsetM2P rM1sω such that }ΠN fpnq ΠN fpmq} ¤ pλ ε1q
k
¸
j1
αpj p1
for all pn, mq PIkpM2q. We deduce
}fpnq fpmq} ¤ }ΠNpfpnq fpmqq} }pIΠNq fpnq} }pIΠNq fpmq}
¤ pλ ε1q k
¸
j1
αpj
1p k
¸
j1
αpj 1p
2ε1 k
¸
j1
αpj 1p
2ε1
¤ pλ 2 εq k
¸
j1
αpj 1
p
for all pn, mq PIkpM2q. The result follows.
Remark 3.2. With this result and Proposition 2.7, we immediately deduce the following: if eachXn,nPN, has propertyλ-HCp, w, thenp°
nPNXnqE has propertyp2λ 2 εq-HCp, w for everyε¡0.
Once again, we chose to state this theorem with property HICp,d, but the result stays true for property HFCp,d, with a similar proof.
Remark 3.3. Of course, the condition that all spaces have property HFCp,d with the same constant is essential because
prNs ω, dHq ãÑ
L Xω 8
¸
n1
`n1p`2q
`2
even though `n1p`2q has property HFC2,w (it is reflexive and 2-AUS) for everynPN. To see that, let us note that, for every kPN,prNs¤k, dHq isometrically embeds into`k1p`2q.
Then, the barycentric gluing technique by Baudier (see [1]) gives us a Lipschitz embedding from prNs ω, dHq intoXω.
In [6], Braga asked the following (Problem 3.7): if a Banach space X has the Banach- Saks property, i.eevery bounded sequence inX admits a subsequence whose Ces`aro means converge in norm, does it follow that prNs ω, d∆q does not Lipschitz embed into X ? The answer to this question is negative. Indeed, let ppnqnPN p1,8qbe a decreasing sequence such that lim
nÑ 8pn1 and denote X °8
n1`pn
`2. With a similar argument, or simply appealing to Ribe’s Theorem [38] (that implies that`1 coarse Lipschitz embeds intoX), we see that
prNs ω, dHq ãÑ
L X and prNs ω, d∆q ãÑ
L X
even though the space X has the Banach-Saks property (see [35]).
Before we write a direct consequence of this theorem, let us briefly recall the definition of the James sequence spaces.
LetpP p1,8q. The James space Jp is the real Banach space of all sequencesx pxpnqqnPN
of real numbers with finite p-variation and verifying limnÑ8xpnq 0. The space Jp is endowed with the following norm
}x}Jp sup
$&
% k1
¸
i1
|xppi 1q xppiq|p 1
p
; 1¤p1 p2 pk
,.
-.
The space JJ2, constructed by James in [22], is the historical example of a quasi-reflexive Banach space which is isomorphic to its bidual. In fact, Jp can be seen as the space of all sequences of real numbers with finite p-variation, which is Jp`Re, where edenotes the constant sequence equal to 1.
Besides of being quasi-reflexive, the space Jp has the property of beingp-AUSable (see [34], Proposition 2.3) and its dual Jp is q-AUSable, where q denotes the conjugate exponent of p (see [29] and references therein).
We can now state the following corollary.
Corollary 3.4. Let p, q P p1,8q, p1 the conjugate exponent of p, s minpp, qq and t minpp1, qq.
If X a quasi-reflexive Banach space satisfying upper `p tree estimates, then the space `qpXq has property HCs,w.
In particular, `qpJpq has property HCs,w and`qpJpq has property HCt,w.
Let us mention that we stated this corollary for`q-sums but we could have done it with any reflexiveq-convexification of a Banach space with a 1-unconditional basis (such as Tq, theq-convexification of Tsirelson space, orSq, theq-convexification of Schlumprecht space, see [6] and references therein).
Withp2, we get that the spaces`2pJqand `2pJq have property HC2 and thus cannot contain equi-Lipschitz the family of Hamming graphs. In fact, property HCp provides more information than an obstruction to the equi-Lipschitz embedding of Hamming graphs, it also gives us an estimation of some compression exponents, given by the result below. Before stating it, we need the following definition.
Definition 3.5. Let q P p1,8q and X be a Banach space. We say that X has the q- co-Banach-Saks property if for every semi-normalized weakly null sequence pxnqnPN in X, there exists a subsequence px1nqnPN of pxnqnPN and c ¡ 0 such that, for all k P N and all k¤n1 nk, we have
}x1n1 x1nk} ¥ck1{q.
Theorem 3.6. Let1 q pinp1,8q. Assume Xis an infinite dimensional Banach space with the q-co-Banach-Saks property and Y is a Banach space with property HCp. Then X does not coarse Lipschitz embed into Y. More precisely, the compression exponent αYpXq of X into Y satisfies the following:
piq if X contains an isomorphic copy of`1, then αYpXq ¤ 1p; piiq otherwise,αYpXq ¤ qp.
In particular, if X isq-AUC (see [30] for the definition), then αYpXq ¤ qp.
We refer the reader to Theorem 3.5 and Corollary 3.6 of [30] for a proof of this result.
Let us note that Proposition 3.2 of [30] also stays true by replacing ”quasi reflexive AUS”
by ”having property HCp for somepP p1,8q”.
We also would like to mention the following: we could define symmetric concentration propertiesSFCp, SIPp and SCp, corresponding respectively to properties HFCp, HICp and HCp by asking the functionf to be Lipschitz for the symmetric distance in the definitions of these properties (instead of being Lipschitz for the Hamming distance). Then, it is known that a reflexive (resp. quasi-reflexive) p-AUS Banach space, for p P p1,8q, would have property SFCp (resp. SCp). However, the analogue of the directionnal Lipschitz constants is not clear for the symmetric distance.
We conclude this subsection by mentioning the following result.
Proposition 3.7. There exists a space X with property HFCp that does not have property HCp,d, for a fixed p P p1,8q, and there exists a space Y with property HFCs,d for every sP p1, pq that does not have property HCp.
Proof. To see it, if we denoteqP p1,8qthe conjugate exponent ofp, we letXTiq, where Ti is the Tirilman space, andY Sq (see [11] and references therein for more information about these spaces). Both spaces are reflexive, X PNp and Y PPp (see [11]) hence X has property HFCp andY has property HFCs,d for everysP p1, pq. Now, fora pajqkj1 PB`k
p, we let f :
"
rNsk Ñ X n ÞÑ °k
j1ajenj , with penq the coordinate functionals associated with the canonical basispenqof Ti.
Let us assumeXhas property λ-HCp,d for someλ¡0. Then, there existsn, mP rNsk such thatnXm∅ and
¸k j1
ajej
}fpnq} ¤ }fpnq fpmq} ¤2λ
because of properties ofX. This is impossible, for the same reasons used by Causey [11] to proveXRAp. Using the same function intoY and properties ofY (see [8]), the arguments used by Causey [11] to proveY RNp give us the result forY. 3.2. Related questions. The following questions about Theorem 1.1 come up naturally.
Problem 1. Can we replace property HFCp,d (resp. HICp,d) by property HFCp (resp.
HICp) in Theorem 1.1?
Problem 2. Can the conclusion of Theorem 1.1 be improved so thatX p°
nPNXnqE has property pλ εq-HICp,d for everyε¡0?
Problem 3. Letp P p1,8q,X a p-AUS Banach space so thatX is complemented in X and thatX{X is reflexive andp-AUSable. Does X have property HCp?
A positive answer to the second question would provide us, for eachp P p1,8q, with an example of a reflexive Banach space, not AUSable, with property HFCp,d. Indeed, following Braga’s proof of Theorem 7.1 [7], the space Xp,`1,T would be such an example (see [7] and references therein for more information about this space).
Moreover, let us recall that Kalton proved the existence of a Banach spaceX that is not p-AUSable but that is uniformly homeomorphic to ap-AUS Banach space (see [23]). Thus, the space X has property HFCp,d, p P p1,8q, even though it is notp-AUSable. However, the following problem remains open.
Problem 4. Is there a Banach space that has property HFCp (or HFCp,d/HCp/HCp,d) without being AUSable? If a Banach spaceX coarse Lipschitz embeds into a Banach space Y that is reflexive and AUS, does it follow thatX is AUSable?
We will finish this section by saying a few words about a natural class of spaces to study here: the Lindenstrauss spaces (see [31]). For any Banach space X, we will note ZX the Lindenstrauss space associated toX. In [7], Braga showed that neitherZc0,Z`1 orZX
ω can have any of the concentration properties we introduced, even though they are 2-AUSable (see [13]). The key point of the proof for the spaces Zc0 and ZX
ω is that they satisfy the assumptions of the following property, that can be deduced from [7].
Proposition 3.8. Let X be a Banach space such that X is separable.
Assume that there exist A, C ¥ 1, pzk,j,n qkPN,jPt1,,ku,nPN CBX such that for every kPN, the map
Fk:
$&
%
rNsk Ñ X n ÞÑ °k
j1
zk,j,n
j
satisfies
1
AdHpn, mq ¤ }Fkpnq Fkpmq} ¤AdHpn, mq for all n, mP rNsk.
Then, the space X does not have any of the concentration properties introduced before.
We can therefore ask ourselves the following question.
Problem 5. Can we find an infinite-dimensional Banach space X and a p P p1,8q such thatZX orZX have property HCp?
Finally, by Aharoni’s Theorem, we know that the Hamming graphs equi-Lipschitz embed into Zc0 {Zc0. Does it mean that these graphs can be Lipschitz embedded into Zc0 ? Into Zc0?
4. Asymptotic-c0 spaces
Before stating the last result of this paper, we recall the definition of an asymptotic-c0
space. The following definition is due to Maurey, Milman and Tomczak-Jaegermann [32].
Definition 4.1. LetX be a Banach space. We denote by cofpXq the set of all its closed finite codimensional subspaces.
ForC ¥1, we say thatX is C-asymptotically c0 if, for any kPN, we have
DX1 PcofpXq @x1 PSX1 DX2PcofpXq @x2PSX2 DXkPcofpXq @xk PSXk,
@pa1, , akq PRk,
¸k i1
aixi
¤C max
1¤i¤k|ai|
We say that X is asymptotically c0 (or asymptotic-c0) if it isC-asymptotically c0 for some C¥1.
LetX be a Banach space. A familypxpjiq;i, jPNq Xis called an infinite array. For an infinite arraypxpjiq;i, jPNq, we call the sequencepxpjiqqjPN thei-th row of the array. We call an arrayweakly null if all rows are weakly null. A subarray of pxpjiq;i, j PNq is an infinite
array of the formpxpjiq
s;i, sPNq, wherepjsq Nis a subsequence. Thus, for a subarray, we are taking the same subsequence in each row.
The following notion, introduced by Halbeisen and Odell ([21]), can be seen as a gener- alization of spreading models.
Definition 4.2. A basic sequencepeiq is called an asymptotic model of a Banach spaceX if there exist an infinite arraypxpjiq;i, jPNq SX and a null-sequencepεnq p0,1q, so that, for all nPN, all paiqni1 r1,1s and alln¤k1 k2 kn
¸n i1
aixpkiq
i
¸n i1
aiei εn.
The following proposition concerning this notion was prooved in [21].
Proposition 4.3 ([21], Proposition 4.1 and Remark 4.7.5). Assume that pxpjiq;i, j PNq SX is an infinite array, all of whose rows are normalized and weakly null. Then there is a subarray of pxpjiq;i, jPNq which has a 1-suppression unconditional asymptotic model peiq.
We call a basic sequence peiq c-suppression unconditional, for some c ¥ 1, if, for any paiq c00 and any AN, we have :
¸
iPA
aiei
¤c
¸8 i1
aiei
.
Note that a c-unconditional basic sequence isc-suppression unconditional.
As for the proof of the fact that every Banach space with property HFC8 is asymptotic- c0 (see [4]), the key ingredient will be the following theorem of Freeman, Odell, Sari and Zheng.
Theorem 4.4 ([18], Theorem 4.6). If a separable Banach space X does not contain any isomorphic copy of `1 and all the asymptotic models generated by normalized weakly null arrays are equivalent to the c0 unit vector basis, then X is asymptotically c0.
We now have all the tools to prove our result.
Theorem 4.5. If a Banach space has property HC8, then it is asymptotic-c0.
Proof. Let X be a Banach space with property HC8. Then X has property λ-HIC8, for some λ ¡0, by Proposition 2.7. Let us note that we can assume that X is separable by Proposition 11 of [15], and that X cannot contain an isomorphic copy of `1 since`1 does not have this property.
Assume by contradiction that X is not asymptotic-c0. By the Theorem 4.4, there exists a 1-suppression unconditional sequencepeiq that is not equivalent to the unit vector basis of c0, and hence λk °k
i1p1qiei Õ 8, ifkÕ 8, and that is generated as an asymptotic model of a normalized weakly null array pxpjiq;i, jPNq inX. Let kPN such that λ2k
4 ¡λ, and δ λ2k
2 . After passing to appropriate subsequences of the array, we may assume that, for any 2k¤j1 j2k and anya1, , a2k P r1,1s, we have
¸2k i1
aixpjiq
i
¸2k i1
aiei δ.
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