1. Local -structure
Let , .
Theorem 1 (Davis, Johnson, Lindenstrauss) Fix . If does not contain uniformly, then .
This contains James’ theorem. Indeed, if , we get uniformly non square implies reflexive. If has non trivial type, then some vanishes. Faharat gave examples where but , [F].
Let be a scalar valued martingale. Let . Define the square function
Theorem 2 (Burkholder, Gundy, Davis) Let . Then .
Proof: Doob’s inequality.
Corollary 3 Forn all fixed , let . Then converges in (unconditional convergence), and
where depends only on ( if ). The opposite inequality holds automatically.
Definition 4 Let . Say has UMD if inequality (BG) holds in .
UMD for unconditionality of martingale differences.
Example 1 has UMD.
Proposition 5 UMD does not depend on .
Use stopping times, extrapolation methods, martingale transform: related to singular integrals in harmonic analysis.
Corollary 6 Any -space is UMD. Also is UMD.
This is nearly all known examples of UMD spaces (up to certain non commutative analogues). The UMD class is pretty small.
Proposition 7 If is UMD, then
- B has type iff is isomorphic to a -US space.
- B has cotype iff is isomorphic to a -UC space.
Proof: -US is equivalent to such that for every martingale,
Apply this to and get type .
Conversely, assume that has type . Let be a martingale. Pointwise,
UMD allows, for every fixed sequence , to remove the signs, and obtain the martingale characterization of smoothness.
Corollary 8 UMD implies superreflexive.
Proof: and fail to have UMD. UMD is a super property. So is not finitely representable in UMD spaces. So UMD spaces have a non trivial type, so they are isomorphic to US smooth spaces, so they are superreflexive.
Although small, class UMD is interesting. Lots of analysis works there. Bourgain: UMD equivalent to boundedness of the -valued Hilbert transform.
There is a generalization of the Burkholder, Gundy, Davis inequality. With Kahane’s inequality, one shows that UMD implies
3. Ribe’s program
Theorem 9 (Ribe) , Banach spaces. If is uniformly homemorphic to , then is coarsely finitely representable in and conversely.
Consider a super property P which is stable by isomorphism. Then P is preserved under uniform homeomorphism.
The (Ribe) program (in fact, popularized by Lindenstrauss) consists in making P visibly metric. This has been successful for the following properties.
- type: Enflo 1970’s, Bourgain-Milman-Wolfson 1974, Mendel-Naor 2000’s
- cotype: Mendel-Naor 2000’s
- superreflexibility: Bourgain 1986, Matousek 1999
Let be the dyadic tree of depth . Index nodes by signs . The following theorem is an answer to a question of M. Gromov.
Theorem 10 (Bourgain) is superreflexive iff does not contain uniformly biLipschitz embedded ‘s.
Proof: Easy direction : Assume that is not superreflexive. Then and a sequence such that for all sequences ,
Using gives that for all , contained in disjoint intervals of ,
Let index lexicographically the leaves of by integers in , . Then the two subtrees descending from a node go to disjoint intervals. For , set
Then where is the shortest path from to in the tree, cut at its highest node. Then
Conversely, we show that if a -UC space contains an embedded , then the distorsion is .
Lemma 11 Assume that is -UC.
- For all ,
- For all ,
End of proof of Bourgain’s theorem from the Lemma: We can assume that . With probability , , so the distance between the two points in the second statement of the Lemma is equal to . So the average is , and this yields a lower bound on distorsion.
Proof: of Lemma. Let us do it for . Consider dyadic martingale where , -th interval of the dyadic decomposition of . Then all are differences, and .
For the second statement, set and apply 1. to .
It turns out that Bourgain’s criterion works for infinite trees. This relies on Bourgain’s theorem, and some more functional analysis.
Theorem 12 (Baudier) If is not superreflexive, then it contains a biLipschitz embedded infinite dyadic tree.
3.2. Diamond graphs
Theorem 13 (Johnson, Schechtman) BiLipschitz embeddings of diamond graphs characterize non superreflexivity.
Diamond graph is obtained by iterating the following procedure: replace an edge by a -cycle fixed at opposite vertices.
No direct connection between trees and diamond graphs is known.
Definition 14 (Enflo) Let be a metric space. Say has Enflo type if such that for all maps ,
Here, diagonals amount to changing all signs, sides to changing one sign.
Theorem 15 (Bourgain, Milman, Wolfson, Pisier) Let be a Banach space.
- -US implies Enflo type .
- type implies Enflo type for all .
Proof: The first statement is easy. The second can be found in my Springer Lecture Notes 1206.
Markov type, introduced by Keith Ball. Naor, Peres, Schramm, Sheffield show that it implies type , and is implied by -US. But it is not clea r wether it is more on the US side or on the type side.