aa r X i v : . [ m a t h . F A ] F e b A SUBMETRIC CHARACTERIZATION OF ROLEWICZ’SPROPERTY ( β ) SHENG ZHANG
Abstract.
The main result is a submetric characterization of the class ofBanach spaces admitting an equivalent norm with Rolewicz’s property ( β ). Asapplications we prove that up to renorming, property ( β ) is stable under coarseLipschitz embeddings and coarse quotients. Introduction
Metric characterization of Banach space properties is one of the most intrigu-ing problems in nonlinear geometry of Banach spaces. Being a part of the Ribeprogram, it aim to seek pure metric interpretation of important concepts in Ba-nach space theory. The first step in this direction is Bourgain’s characterizationof superreflexivity in terms of Lipschitz embeddability of binary trees [5]. Metriccharacterizations of other properties such as RNP, reflexivity, Rademacher’s typeand cotype have been discovered thereafter. We refer to the survey paper [13]and references therein for a list of Banach space properties that are known tohave such characterization.The main concern of this paper is Rolewicz’s property ( β ), which is an as-ymptotic property of Banach spaces introduced by Rolewicz [14]. We recall theequivalent definition from [10] that a Banach space X is said to have property( β ) if for every t ∈ (0 , a ], where a ∈ [1 ,
2] depends only on the space X , thereexists δ = δ ( t ) ∈ (0 ,
1) such that whenever x, x n ∈ B X with inf i = j k x i − x j k ≥ t ,there exists k ∈ N such that k x − x k k ≤ − δ. A modulus for the property ( β ) is defined by¯ β X ( t ) = 1 − sup (cid:26) inf n ≥ (cid:26) k x − x n k (cid:27) : x, x n ∈ B X , inf i = j k x i − x j k ≥ t (cid:27) . Then X has property ( β ) if and only if ¯ β X ( t ) > t > β ) can be characterized in terms of Lipschitz embeddability of thecountably branching tree. In this paper, our goal is to provide a different metric The author was supported by National Natural Science Foundation of China, grant numbers11801469 and 12071389. characterization of property ( β ). The characterization we obtained, in the lan-guage of Ostrovskii [12], is called a submetric characterization; it combines ideasfrom [4, 7] and a self-improvement argument first appeared in [9].We gather some necessary definitions and preliminary results in section 2. Sec-tion 3 is devoted to the submetric characterization of property ( β ). The lastsection contains two applications regarding the stability of property ( β ) undercoarse Lipschitz embeddings and nonlinear quotients. We use standard nota-tions: B X ( x, r ) denotes the closed ball in a metric space X with center x andradius r . If X is a Banach space, B X := B X (0 ,
1) is the closed unit ball of X .All Banach spaces are assumed to be real.2. Preliminaries
Uniform and coarse continuity.
Let f : X → Y be a map between metric spaces X and Y . The modulus ofcontinuity of f is defined by ω f ( t ) := sup { d Y ( f ( x ) , f ( y )) : d X ( x, y ) ≤ t } . The map f is said to be uniformly continuous if lim t → ω f ( t ) = 0; f is said tobe coarsely continuous if ω f ( t ) < ∞ for all t >
0. The Lipschitz constant of f isdefined by Lip( f ) := sup (cid:26) d Y ( f ( x ) , f ( y )) d X ( x, y ) : x, y ∈ X, x = y (cid:27) , and f is called a Lipschitz map if Lip( f ) < ∞ . We say f is a Lipschitz embeddingif there exists constants C, L > x, y ∈ X ,1 C d X ( x, y ) ≤ d Y ( f ( x ) , f ( y )) ≤ Ld X ( x, y ) . If X is unbounded, we defineLip d ( f ) := sup (cid:26) d Y ( f ( x ) , f ( y )) d X ( x, y ) : x, y ∈ X, d X ( x, y ) ≥ d (cid:27) , and say that f is Lipschitz for large distances if Lip d ( f ) < ∞ for all d >
0. Ifthere exists d > d ( f ) < ∞ , then we say f is coarse Lipschitz. f is said to be a coarse Lipschitz embedding if there exists constants d ≥ A, B > x, y ∈ X with d X ( x, y ) ≥ d ,1 A d X ( x, y ) ≤ d Y ( f ( x ) , f ( y )) ≤ Bd X ( x, y ) . A metric space X is said to be metrically convex if for every x , x ∈ X and 0 <λ <
1, there exists x λ ∈ X such that d ( x , x λ ) = λd ( x , x ) and d ( x , x λ ) = (1 − λ ) d ( x , x ). Uniformly continuous maps and coarsely continuous maps definedon metrically convex metric spaces must be Lipschitz for large distances. SUBMETRIC CHARACTERIZATION OF ROLEWICZ’S PROPERTY ( β ) 3 The countably branching tree.
The countably branching tree of infinite height, denoted by T , is an unweightedrooted tree each of whose vertices has countably infinite edges incident to it. Westill use T to denote its vertex set, then T has a one-to-one correspondence withall finite subsets of N , where ∅ is the root of T , and every other vertices can berepresented as ( n < n < ... < n k ) for some k ∈ N . The ancestor-to-descendantrelations between vertices is defined as follows: the root ∅ is the ancestor of allthe other vertices; a vertex I = ( m < m < ... < m l ) is said to be an ancestor ofa vertex J = ( n < n < ... < n k ) (or J is a descendant of I ), written as I < J ,if l < k and m j = n j for all 1 ≤ j ≤ l . If I < J and k = l + 1 then we say J isan immediate descendant of I . The height of a vertex J , denoted by | J | , is thenumber of edges jointing J and the root, which is also the number of elementsin J viewed as a finite subset of N ; here we set |∅| = 0 by convention. Thenotation max J and min J denote the largest and the smallest natural number in J respectively. T will always be equipped with the shortest path metric, i.e., thedistance between two vertices I, J ∈ T is defined by d T ( I, J ) := | I | + | J | − | gca( I, J ) | , where gca( I, J ) is the common ancestor of I and J with the greatest height. T has an important property that it is self-contained, meaning that T containsproper subsets that are rescaled isometric copies of itself. In the lemma below wechoose a sequence of such subsets in a way that vertices of the same height are“separated”. Lemma 2.1.
There exists a sequence {T n } ∞ n =0 of subsets of T that satisfies thefollowing:(1) T n ⊇ T n +1 for every n ∈ N ∪ { } .(2) For every n ∈ N ∪{ } , T n is n -isometric to T , i.e., there exists a surjectivemap i n : T → T n such that d T ( i n ( I ) , i n ( J )) = 2 n d T ( I, J ) for all I, J ∈ T .Therefore, T n can be viewed as a weighted countably branching tree eachof whose edges has weight n .(3) For every n ∈ N ∪{ } and every vertex J ∈ T n , the immediate descendantsof J in T n , denoted by { J ∪ J i } ∞ i =1 where | J i | = 2 n and max J < min J i forall i , satisfies max J i < min J i +1 for all i .Proof. Set T = T . Suppose T n has been defined that satisfies (1)–(3), we define T n +1 as follows: first choose the root ∅ . Suppose J ∈ T n has been chosen. Let { J ∪ J i } ∞ i =1 be the immediate descendants of J in T n , where | J i | = 2 n , max J < min J i and max J i < min J i +1 for all i . Choose an arbitrary immediate descendantof J ∪ J in T n , denoted by J ∪ ¯ J , where | ¯ J | = 2 n +1 and J < ¯ J . Set k = 1and suppose J ∪ ¯ J k j has been chosen, since max J i < min J i +1 for all i , thereexists k j +1 ∈ N such that min J k j +1 > max ¯ J k j . Choose an arbitrary immediatedescendant of J ∪ J k j +1 in T n , denoted by J ∪ ¯ J k j +1 , where | ¯ J k j +1 | = 2 n +1 and J k j +1 < ¯ J k j +1 . Then { J ∪ ¯ J k j } ∞ j =1 is a sequence of descendants of J with height | J | + 2 n +1 and satisfies max ¯ J k j < min ¯ J k j +1 for all j . Now the vertex set of T n +1 are those vertices chosen in this induction process. It is easy and left to the readerto check that {T n } ∞ n =0 is a sequence that satisfies (1)–(3). (cid:3) SHENG ZHANG
We call such T n ’s pruned isometric subsets of T , where the subscript n means T n is 2 n -isometric to T in the sense of (2), and T = T . In Lemma 2.1 thesequence {T n } ∞ n =0 are built from T , but in the sequel we will need to build such asequence starting from a fix pruned isometric subset T k , i.e., there is a sequenceof subsets of T k , denoted by {T k + n } ∞ n =0 , which satisfies the following:(1’) T k + n ⊇ T k + n +1 for every n ∈ N ∪ { } .(2’) For every n ∈ N ∪ { } , T k + n is 2 k + n -isometric to T .(3’) For every n ∈ N ∪ { } and every vertex J ∈ T k + n , the immediate descendantsof J in T k + n , denoted by { J ∪ J i } ∞ i =1 where | J i | = 2 k + n and max J < min J i for all i , satisfies max J i < min J i +1 for all i . T also plays a significant role in the metric characterization of Banach spaceproperties. As mentioned in the introduction, it was proved in [3] that within theclass of reflexive Banach spaces, Rolewicz’s property ( β ) can be characterized byLipschitz embeddability of T . Theorem 2.2 ([3]) . A reflexive Banach space X DOES NOT have an equivalentnorm with property ( β ) if and only T admits a Lipschitz embedding into X . A submetric characterization of property ( β ) The following is a submetric characterization of the class of Banach spacesadmitting an equivalent norm with Rolewicz’s property ( β ): Theorem 3.1.
Let X be a Banach space. The following assertions are equivalent:(i) X DOES NOT have an equivalent norm with property ( β ).(ii) There exist a pruned isometric subset T k of T , k ∈ N ∪ { } , a Lipschitzmap φ : T k → X and a constant γ > , such that for every J ∈ T k , andevery pair of descendants ¯ J = J ∪ J , ¯ J = J ∪ J of J in T k that satisfies | J | = | J | and max J < min J , k φ ( ¯ J ) − φ ( ¯ J ) k ≥ γd T ( ¯ J , ¯ J ) . Proof. (i) ⇒ (ii): If X is reflexive but does not have an equivalent norm with prop-erty ( β ), then it follows from Theorem 2.2 that T admits a Lipschitz embeddinginto X . Note that the Lipschitz embedding satisfies (ii) for k = 0, and we canassume that it has Lipschitz constant 1 so that γ ∈ (0 , X is non-reflexive, fix θ ∈ (0 , { x j } ∞ j =1 ⊆ B X and { x ∗ j } ∞ j =1 ⊆ B X ∗ such that x ∗ m ( x n ) = θ if m ≤ n and x ∗ m ( x n ) = 0 if m > n . Define φ : T → X by φ ( J ) = P j ∈ J x j , J ∈ T . Next we show that φ has Lipschitz constant at most 1and satisfies (ii) for k = 0 and γ = θ/ J, ¯ J ∈ T and J < ¯ J , let l := min( ¯ J \ J ), then we have k φ ( J ) − φ ( ¯ J ) k = (cid:13)(cid:13)(cid:13) X j ∈ ¯ J \ J x j (cid:13)(cid:13)(cid:13) ≤ | ¯ J | − | J | = d T ( J, ¯ J ) , and k φ ( J ) − φ ( ¯ J ) k = (cid:13)(cid:13)(cid:13) X j ∈ ¯ J \ J x j (cid:13)(cid:13)(cid:13) ≥ x ∗ l (cid:16) X j ∈ ¯ J \ J x j (cid:17) = ( | ¯ J | − | J | ) θ = θd T ( J, ¯ J ) . SUBMETRIC CHARACTERIZATION OF ROLEWICZ’S PROPERTY ( β ) 5 Let J ∈ T k , ¯ J = J ∪ J and ¯ J = J ∪ J are descendants of J in T k that satisfies | J | = | J | and max J < min J , let k := min J , then k φ ( ¯ J ) − φ ( ¯ J ) k = (cid:13)(cid:13)(cid:13) X j ∈ J x j − X j ∈ J x j (cid:13)(cid:13)(cid:13) ≥ x ∗ k (cid:16) X j ∈ J x j − X j ∈ J x j (cid:17) = | J | θ = θd T ( ¯ J , ¯ J )2 . (ii) ⇒ (i): Suppose that X has property ( β ), and there exist a pruned isometricsubset T k of T , k ∈ N ∪ { } , a Lipschitz map φ : T k → X and a constant γ > φ ) = 1.Next we show that there is a sequence of subsets of T k , denoted by {T k + n } ∞ n =0 ,that satisfies (1’)-(3’) in Section 2 and Lip( φ | T k + n ) ≤ τ n for every n ∈ N ∪ { } ,where τ := 1 − ¯ β X (2 γ ).The sequence {T k + n } ∞ n =0 is defined in a similar way as that in Lemma 2.1.Suppose T k + n has been defined so that it has the above properties, we define T k + n +1 as follows: first choose the root ∅ . Suppose J ∈ T k + n has been chosen. Let { J ∪ J i } ∞ i =1 be the immediate descendants of J in T k + n , where | J i | = 2 k + n , max J < min J i and max J i < min J i +1 for all i . Consider the immediate descendants of J ∪ J in T k + n , denoted by { J ∪ J ∪ J ,i } ∞ i =1 , where | J ,i | = 2 k + n , max J < min J ,i and max J ,i < min J ,i +1 for all i . Then we have k φ ( J ) − φ ( J ∪ J ) k ≤ τ n k + n , k φ ( J ∪ J ∪ J ,i ) − φ ( J ∪ J ) k ≤ τ n k + n , i ∈ N k φ ( J ∪ J ∪ J ,i ) − φ ( J ∪ J ∪ J ,j ) k ≥ γd T ( J ∪ J ∪ J ,i , J ∪ J ∪ J ,j )= 2 k + n +1 γ, i = j, Thus there exists k ∈ N such that k φ ( J ) − φ ( J ∪ J ∪ J ,k ) k ≤ τ n k + n · (cid:16) −
12 ¯ β X (cid:16) γτ n (cid:17)(cid:17) ≤ τ n +1 k + n +1 , where the second inequality follows from the fact that ¯ β X ( · ) is non-decreasing;denote J ∪ J ∪ J ,k := J ∪ ¯ J k . Suppose J ∪ ¯ J k j has been chosen, since max J i < min J i +1 for all i , there exists k j +1 ∈ N such that min J k j +1 > max ¯ J k j . Again wecan choose an immediate descendant of J ∪ J k j +1 in T k + n , denoted by J ∪ ¯ J k j +1 with | ¯ J k j +1 | = 2 k + n +1 and J k j +1 < ¯ J k j +1 , that satisfies k φ ( J ) − φ ( J ∪ ¯ J k j +1 ) k ≤ τ n +1 k + n +1 . Then the sequence { J ∪ ¯ J k j } ∞ j =1 we have chosen consists of descendants of J withheight | J | + 2 k + n +1 , and for every j ∈ N one has max ¯ J k j < min ¯ J k j +1 and k φ ( J ) − φ ( J ∪ ¯ J k j ) k ≤ τ n +1 k + n +1 . The vertex set of T k + n +1 are those vertices chosen in this induction process. Nowit is easy to see that the sequence {T k + n } ∞ n =0 has the desired properties, and itfollows that γ ≤ τ n for all n ∈ N ∪ { } . Note that τ ∈ (0 ,
1) since X has property( β ), we get a contradiction by letting n → ∞ . (cid:3) SHENG ZHANG
Remark . The original submetric characterization we have is the following:(iii) There exist a pruned isometric subset T k of T , k ∈ N ∪ { } , and a map φ : T k → X that satisfies the following:(a) φ is an ancestor-to-descendant Lipschitz embedding, i.e., there exists C, L >
J < ¯ J in T k ,1 C d T ( J, ¯ J ) ≤ k φ ( J ) − φ ( ¯ J ) k ≤ Ld T ( J, ¯ J ) . (b) There exists γ > {T k + n } ∞ n =0 of subsets of T k that satisfy (1’)-(3’) in section 2, and every n ∈ N ∪ { } , I, J ∈ T k + n with I = J and | I | = | J | , k φ ( I ) − φ ( J ) k ≥ k + n γ. Shortly after we obtained the above submetric characterization (iii), we hada discussion with Florent Baudier. He made the observation that an improvedversion ((ii) in Theorem 3.1) can actually be derived from the work of Dilworth,Kutzarova, and Randrianarivony in [7]. He also realized that the characterizationcan be proved using the notion of beta convexity, which is a metric analogue ofproperty ( β ) introduced in his on-going work with Chris Gartland [2].4. Applications
Stability of property ( β ) under coarse Lipschitz embeddings. Our first application of the submetric characterization is the following stabilityresult under coarse Lipschitz embeddings.
Theorem 4.1.
Let X be a Banach space that has an equivalent norm with prop-erty ( β ). If a Banach space Y admits a coarse Lipschitz embedding into X , then Y also has an equivalent norm with property ( β ).Proof. Assume that Y does not have an equivalent norm with property ( β ), thenit follows from the proof of (i) ⇒ (ii) in Theorem 3.1 that there exist γ ∈ (0 , φ : T → Y that satisfy (a) and (b) as follows:(a) For all J < ¯ J in T , γd T ( J, ¯ J ) ≤ k φ ( J ) − φ ( ¯ J ) k ≤ d T ( J, ¯ J ) . (b) For every J ∈ T , and every pair of descendants ¯ J = J ∪ J , ¯ J = J ∪ J of J that satisfies | J | = | J | and max J < min J , k φ ( ¯ J ) − φ ( ¯ J ) k ≥ γd T ( ¯ J , ¯ J ) . Denote φ ( J ) := v J for J ∈ T .Let f : Y → X be a coarse Lipschitz embedding, so there exists d ≥ A, B > x, y ∈ Y with k x − y k ≥ d ,1 A k x − y k ≤ k f ( x ) − f ( y ) k ≤ B k x − y k . Choose k ∈ N such that 2 k γ > d . Let T k be a pruned isometric subset of T . Weclaim that the map g := f ◦ φ | T k : T k → X , g ( J ) := u J , J ∈ T k satisfies SUBMETRIC CHARACTERIZATION OF ROLEWICZ’S PROPERTY ( β ) 7 (1) For all J < ¯ J in T k , k g ( J ) − g ( ¯ J ) k ≤ Bd T ( J, ¯ J ) . (2) For every J ∈ T k , and every pair of descendants ¯ J = J ∪ J , ¯ J = J ∪ J of J in T k that satisfies | J | = | J | and max J < min J , k g ( ¯ J ) − g ( ¯ J ) k ≥ γd T ( ¯ J , ¯ J ) A .
Proof of the claim.
Note that for
J < ¯ J in T k , we have k v J − v ¯ J k ≥ γd T ( J, ¯ J ) ≥ k γ > d, and thus k u J − u ¯ J k ≤ B k v J − v ¯ J k ≤ Bd T ( J, ¯ J ) , which implies that Lip( g ) ≤ B .On the other hand, let J ∈ T k , and ¯ J = J ∪ J and ¯ J = J ∪ J are descendantsof J in T k that satisfies | J | = | J | and max J < min J , we have k v ¯ J − v ¯ J k ≥ γd T ( ¯ J , ¯ J ) ≥ k +1 γ > d, which implies that k u ¯ J − u ¯ J k ≥ A k v ¯ J − v ¯ J k ≥ γd T ( ¯ J , ¯ J ) A , and this finishes the proof of the claim.Now it follows from Theorem 3.1 (ii) ⇒ (i) that X does not have an equivalentnorm with property ( β ); this is a contradiction. (cid:3) Stability of property ( β ) under nonlinear quotients. The notion of property ( β ) has been found very useful in the study of nonlinearquotients of Banach spaces. Lima and Randrianarivony [11] first brought inproperty ( β ) to study unifrom quotients of ℓ p spaces. They proved that ℓ q is not auniform quotient of ℓ p for 1 < p < q , and the main technique used is call the “forkargument”. The “fork” here describes a configuration of points just as the x and { x n } ∞ n =1 in the definition of property ( β ), and the “fork argument” is a processof nonlinear lifting of points sitting approximately in such a position. Later,Baudier and Zhang [4] proved the same result by estimating the ℓ p -distortionof the countably branching trees. Their proof shows that the nonexistence ofnonlinear quoitent maps from ℓ p to ℓ q is actually due to the discrepancy betweentheir modulus of property ( β ). Then it becomes a natural question whether theproperty ( β ) is preserved (up to renorming) under nonlinear quotients. Dilworth,Kutzarova and Randrianarivony [7] gave an affirmative answer to the question inthe uniform category. They applied the “fork argument” to uniform quotientmaps onto a diamond-type graph called parasol graph, and proved that if aseparable Banach space Y is a uniform quotient of a Banach space admittingan equivalent norm with property ( β ), then Y also has an equivalent norm withproperty ( β ). Recently it was shown in [6] that the separable condition can beremoved. SHENG ZHANG
The goal of this subsection is to provide a stability theorem of property ( β )under nonlinear quotients in the coarse category. Our approach, which works aswell in the context of uniform quotient, does not require the use of the parasolgraph or the “fork argument”. Instead, we will apply the submetric characteriza-tion of property ( β ) and use the same technique as that in the proof of Theorem4.1.We recall from [1] that a map f : X → Y between metric spaces X and Y iscalled co-uniformly continuous if for every d >
0, there exists δ = δ ( d ) > x ∈ X , B Y ( f ( x ) , δ ) ⊆ f ( B X ( x, d )) . If the δ satisfies δ ≥ d/C for some constant C > d , then the map f is said to be co-Lipschitz. A uniform (resp. Lipschitz) quotient map is a co-uniformly continuous (resp. co-Lipschitz) map that is also uniformly continuous(resp. Lipschitz), and we say that Y is a uniform (resp. Lipschitz) quotient of X if there exists a uniform (resp. Lipschitz) quotient map from X to Y .The notion of coarse quotient was introduced by Zhang in [15]. A map f : X → Y between metric spaces X and Y is called co-coarsely continuous if thereis a constant K ≥ d > K , there exists δ = δ ( d ) > x ∈ X , B Y ( f ( x ) , d ) ⊆ f ( B X ( x, δ )) K . Here for a set A in a metric space X , A K := S a ∈ A B X ( a, K ). A coarse quotientmap is a co-coarsely continuous map that is also coarsely continuous, and we saythat Y is a coarse quotient of X if there exists a coarse quotient map from X to Y .Co-Lipschitz maps are surjective. A co-uniformly continuous map f : X → Y is surjective if the target space Y is connected. For a co-coarsely continuous map f : X → Y , we only have Y = f ( X ) K for some K ≥
0. Therefore, if the targetspace of a coarse quotient map f is unbounded, one must have Lip d ( f ) > d > Lemma 4.2.
Let f : X → Y be a map from a metric space X to a metrciallyconvex metric space Y .(1) If f is co-uniformly continuous, then for every d > , there exists C = C ( d ) > such that for all x ∈ X and r ≥ d , B Y ( f ( x ) , r/C ) ⊆ f ( B X ( x, r )) . (2) If f is co-coarsely continuous, then there exists a constant K ≥ thatsatisfies the following: for every d > K , there exists C = C ( d ) > suchthat for all x ∈ X and r ≥ d , B Y ( f ( x ) , r ) ⊆ f ( B X ( x, Cr )) K . Corollary 4.3.
Let X and Y be two metric spaces and f : X → Y be a mapthat is co-uniformly continuous or co-coarsely continuous. Assume that Y is SUBMETRIC CHARACTERIZATION OF ROLEWICZ’S PROPERTY ( β ) 9 metrically convex, then there exist K > and C > such that for all x ∈ X and r > , B Y ( f ( x ) , r ) ⊆ f ( B X ( x, Cr )) K . Proof. If f is co-uniformly continuous, we fix d >
0, then it follows from Lemma4.2 (1) that there exists C = C ( d ) > x ∈ X and r ≥ d , B Y ( f ( x ) , r/C ) ⊆ f ( B X ( x, r )) ⊆ f ( B X ( x, r )) d/C . Thus for all x ∈ X and r > B Y ( f ( x ) , r/C ) ⊆ f ( B X ( x, r )) d/C , and the result follows.Similarly, if f is co-coarsely continuous, note that the constant C in Lemma4.2 (2) satisfies B Y ( f ( x ) , r ) ⊆ f ( B X ( x, Cr )) d for all x ∈ X and r >
0, again theresult follows. (cid:3)
Theorem 4.4.
Let X be a Banach space that has an equivalent norm with prop-erty ( β ). If a Banach space Y is a uniform or coarse quotient of a subset of X , where the quotient map is Lipschitz for large distances, then Y also has anequivalent norm with property ( β ).Proof. Assume that Y does not have an equivalent norm with property ( β ), thenit follows from the proof of (i) ⇒ (ii) in Theorem 3.1 that there exist γ ∈ (0 , φ : T → Y that satisfy (a) and (b) as follows:(a) For all J < ¯ J in T , γd T ( J, ¯ J ) ≤ k φ ( J ) − φ ( ¯ J ) k ≤ d T ( J, ¯ J ) . (b) For every J ∈ T , and every pair of descendants ¯ J = J ∪ J , ¯ J = J ∪ J of J that satisfies | J | = | J | and max J < min J , k φ ( ¯ J ) − φ ( ¯ J ) k ≥ γd T ( ¯ J , ¯ J ) . Denote φ ( J ) := v J for J ∈ T .Let S be a subset of X and f : S → Y be a uniform or coarse quotient mapthat is Lipschitz for large distances. By Corollary 4.3, there exist K >
C > x ∈ S and r > B Y ( f ( x ) , r ) ⊆ f ( B S ( x, Cr )) K . (4.1)Let d > d ( f ) ∈ (0 , ∞ ) and ω f ( d ) < ∞ . Choose k ∈ N such that2 k γ > ω f ( d ) + 2 K . Let T k be a pruned isometric subset of T . We define a map g : T k → X as follows: first choose any u ∅ ∈ S such that k v ∅ − f ( u ∅ ) k ≤ K .Suppose u J ∈ S has been defined for J ∈ T k such that k v J − f ( u J ) k ≤ K . Let ¯ J be an immediate descendant of J in T k . It follows from (4.1) that v ¯ J ∈ B Y (cid:0) f ( u J ) , k v ¯ J − v J k + K (cid:1) ⊆ f (cid:0) B S (cid:0) u J , C (cid:0) k v ¯ J − v J k + K (cid:1)(cid:1)(cid:1) K , so there exists u ¯ J ∈ S such that k u ¯ J − u J k ≤ C (cid:0) k v ¯ J − v J k + K (cid:1) and k v ¯ J − f ( u ¯ J ) k ≤ K . This induction process defines a u J ∈ S for every J ∈ T k ; thus g : T k → X , g ( J ) := u J , J ∈ T k is a well-defined map. We claim that the map g satisfies (i) For all J < ¯ J in T k , k g ( J ) − g ( ¯ J ) k ≤ Cd T ( J, ¯ J ) . (ii) For every J ∈ T k , and every pair of descendants ¯ J = J ∪ J , ¯ J = J ∪ J of J in T k that satisfies | J | = | J | and max J < min J , k g ( ¯ J ) − g ( ¯ J ) k ≥ γd T ( ¯ J , ¯ J )2Lip d ( f ) . Proof of the claim.
If ¯ J is an immediate descendant of J in T k , we have k u J − u ¯ J k ≤ C (cid:0) k v J − v ¯ J k + K (cid:1) ≤ C (2 k + K ) < k +1 C = 2 Cd T ( J, ¯ J ) , which implies that g is Lipschitz with Lipschitz constant at most 2 C .On the other hand, let J ∈ T k , and ¯ J = J ∪ J and ¯ J = J ∪ J are descendantsof J in T k that satisfies | J | = | J | and max J < min J , we must have k u ¯ J − u ¯ J k ≥ d , otherwise2 k +1 γ ≤ γd T ( ¯ J , ¯ J ) ≤ k v ¯ J − v ¯ J k ≤ k f ( u ¯ J ) − f ( u ¯ J ) k + 2 K ≤ ω f ( d ) + 2 K, which contradicts the choice of k . Therefore, k u ¯ J − u ¯ J k ≥ d ( f ) k f ( u ¯ J ) − f ( u ¯ J ) k≥ d ( f ) (cid:0) k v ¯ J − v ¯ J k − K (cid:1) ≥ d ( f ) ( γd T ( ¯ J , ¯ J ) − K ) ≥ γd T ( ¯ J , ¯ J )2Lip d ( f ) , and this finishes the proof of the claim.Now it follows from Theorem 3.1 (ii) ⇒ (i) that X does not have an equivalentnorm with property ( β ); this is a contradiction. (cid:3) Remark . In Theorem 4.4, if Y is a coarse quotient of a subset of X , then itis enough to assume that the coarse quotient map is coarse Lipschitz rather thanLipschitz for large distances. Corollary 4.6.
Let X be a Banach space admitting an equivalent norm withproperty ( β ). If a Banach space Y is (i) a Lipschitz quotient of a subset of X ; or(ii) a uniform quotient of X ; or (iii) a coarse quotient of X , then Y also has anequivalent norm with property ( β ). Acknowledgement.
The author would like thank Florent Baudier and ChrisGartland for very helpful discussion which improved Theorem 3.1. See Remark3.2.
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