Notes of Gilles Pisier’s lecture nr 5

 

1. Local {k}-structure

 

Let {R(B)=B^{**}/B}, {R_{n+1}(B)=R(R_{n}(B))}.

Theorem 1 (Davis, Johnson, Lindenstrauss) Fix {N\geq 1}. If {B} does not contain {\ell_{1}^{N}} uniformly, then {R_{N-1}(B)=0}.

 

See [DJL], [DL].

This contains James’ theorem. Indeed, if {N=2}, we get uniformly non square implies reflexive. If {B} has non trivial type, then some {R_N (B)} vanishes. Faharat gave examples where {R_N (B)\not=0} but {R_{N+1}(B)=0}, [F].

 

2. UMD

 

Let {(f_n)} be a scalar valued martingale. Let {f^* =\sup|f_n|}. Define the square function {S=S(f)=(\sum_{n=0}^{\infty}|df_n|^2)^{1/2}}

Theorem 2 (Burkholder, Gundy, Davis) Let {1\leq p<\infty}. Then {|f^{*}|_p \sim |S|_p}.

 

Proof: Doob’s inequality. \Box

Corollary 3 Forn all fixed {\epsilon_n =\pm 1}, let {g=\sum_{n=0}^{\infty}\epsilon_n df_n}. Then {g} converges in {L_p} (unconditional convergence), and

\displaystyle  \begin{array}{rcl}  (\textrm{BG}_p) \quad|g|_p \leq C_p \,|f|_p , \end{array}

where {C_p} depends only on {p} ({C_p =p-1} if {p\geq 2}). The opposite inequality holds automatically.

 

Proof: {S(g)=S(f)}. \Box

Definition 4 Let {1<p<\infty}. Say {B} has UMD{_p} if inequality (BG{_p}) holds in {L_p (B)}.

UMD for unconditionality of martingale differences.

Example 1 {L_p} has UMD{_p}.

Proof: Fubini.

Proposition 5 UMD{_p} does not depend on {p\in(1,+\infty)}.

Use stopping times, extrapolation methods, martingale transform: related to singular integrals in harmonic analysis.

Corollary 6 Any {L_p}-space is UMD. Also {L^p (L^r)} 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 {B} is UMD, then

  • B has type {p} iff {B} is isomorphic to a {p}-US space.
  • B has cotype {q} iff {B} is isomorphic to a {q}-UC space.

 

Proof: {p}-US is equivalent to {\exists C} such that for every martingale,

\displaystyle  \begin{array}{rcl}  \sim|d_n|_{L_p (B)}\leq C\,(\sum |d_n|^p \end{array}

Apply this to {f_n =\sum \epsilon_k x_k} and get type {p}.

Conversely, assume that {B} has type {p}. Let {(f_n)} be a martingale. Pointwise,

\displaystyle  \begin{array}{rcl}  (\mathop{\mathbb E}_{\epsilon'}|\sum \epsilon'_{n}df_n (\omega)|p)^{1/p}\leq T_p (B)(\sum |df_n|^p)^{1/p}. \end{array}

Integrate

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}_{\omega}(\mathop{\mathbb E}_{\epsilon'}|\sum \epsilon'_{n}df_n (\omega)|p)^{1/p} &\leq& T_p (B)\mathop{\mathbb E}_{\omega}(\sum |df_n|^p)^{1/p}, \end{array}

UMD{_p} allows, for every fixed sequence {\epsilon'}, to remove the signs, and obtain the martingale characterization of smoothness. \Box

Corollary 8 UMD implies superreflexive.

 

Proof: {L_1} and {\ell_1} fail to have UMD. UMD is a super property. So {\ell_1} 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. \Box

Although small, class UMD is interesting. Lots of analysis works there. Bourgain: UMD equivalent to boundedness of the {B}-valued Hilbert transform.

There is a generalization of the Burkholder, Gundy, Davis inequality. With Kahane’s inequality, one shows that UMD implies

\displaystyle  \begin{array}{rcl}  |f|_{L_p (B)}&\sim& |S(f)|_{L_p}\\ |f^*|_{L_1}&\sim&|S(f)|_{L_1}. \end{array}

 

3. Ribe’s program

 

Recall

Theorem 9 (Ribe) {X}, {Y} Banach spaces. If {X} is uniformly homemorphic to {Y}, then {X} is coarsely finitely representable in {Y} 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

 

3.1. Superreflexivity

 

Let {\tau_n} be the dyadic tree of depth {n}. Index nodes by signs {\epsilon_1,\ldots\epsilon_n}. The following theorem is an answer to a question of M. Gromov.

Theorem 10 (Bourgain) {B} is superreflexive iff {B} does not contain uniformly biLipschitz embedded {\tau_n}‘s.

 

Proof: Easy direction : Assume that {B} is not superreflexive. Then {\exists \theta>0} and a sequence {x_i} such that for all sequences {\alpha_i \in {\mathbb R}},

\displaystyle  \begin{array}{rcl}  \theta \sup_j (|\sum_{i\leq j}\alpha_i|+|\sum_{i>j}\alpha_i|)\leq |\sum\alpha_i x_i|\leq \sum|\alpha_i|. \end{array}

Using {\alpha_i =\pm 1} gives that for all {A}, {A'} contained in disjoint intervals of {[n]},

\displaystyle  \begin{array}{rcl}  \theta(|A|+|A'|)\leq |\sum_{A}x_i -\sum_{A'}x_i|\leq |A|+|A'|. \end{array}

Let {\phi:\tau_n \rightarrow [m]} index lexicographically the leaves of {\tau_n} by integers in {[m]}, {m=2^{n+1} -1}. Then the two subtrees descending from a node go to disjoint intervals. For {t\in\tau_n}, set

\displaystyle  \begin{array}{rcl}  F(t)=\sum_{w\leq t}x_{\phi(w)}. \end{array}

Then {F(s)-F(t)=\sum_{\phi(\beta)}x_i -\sum_{\phi(\beta')}x_i } where {\beta\cup\beta'} is the shortest path from {s} to {t} in the tree, cut at its highest node. Then

\displaystyle  \begin{array}{rcl}  |F(s)-F(t)|\sim|\phi(\beta)|+|\phi(\beta')|=d_{\tau_n}(s,t). \end{array}

Conversely, we show that if a {q}-UC space contains an embedded {\tau_n}, then the distorsion is {\Omega((\log n)^q)}.

Lemma 11 Assume that {B} is {q}-UC.

  1. For all {x_1 ,\ldots,x_m \in B}, \displaystyle  \begin{array}{rcl}  \inf_{0\leq j,\,j+2k<m}\frac{1}{2k}|x_i +x_{j+2k}-2x_{j+k}|\leq {\mathbb C}\,(\log m)^{-1/q}\sup_{0\leq j\leq m}|x_{j+1}-x_j|. \end{array}
  2. For all {F:\tau_n \rightarrow B}, \displaystyle  \begin{array}{rcl}  \inf_{2k \leq N+k\leq n}\frac{1}{k}\mathop{\mathbb E}_{\epsilon,\epsilon',\epsilon''}|F(\tau(\epsilon_1 \cdots\epsilon_N\epsilon'_{N+1}\cdots\epsilon'_{N+k}))-F(\tau(\epsilon_1 ,...,\epsilon_N\epsilon''_{N+1}\ldots\epsilon''_{N+k}))| \leq C\,|F|_{Lip}. \end{array}

 

End of proof of Bourgain’s theorem from the Lemma: We can assume that {|F^{-1}|_{Lip}\leq 1}. With probability {\geq 1/2}, {\epsilon'_{N+1}\not=\epsilon''_{N+1}}, so the distance between the two points in the second statement of the Lemma is equal to {2k}. So the average is {\geq 1}, and this yields a lower bound on distorsion. \Box

Proof: of Lemma. Let us do it for {m=2^m}. Consider dyadic martingale {f_n (\omega)=x_{j+1}-x_j} where {\omega\in I_j}, {j}-th interval of the dyadic decomposition of {[0,1]}. Then all {f_k (\omega)} are differences, and {\frac{1}{2k}(x_i +x_{j+2k}-2x_{j+k})=f_k (\omega)-f_{k-1}(\omega)}.

\displaystyle  \begin{array}{rcl}  \inf_{1\leq k\leq n}\inf_{\omega}|df_{k}|&\leq&\frac{1}{n^{1/q}}(\sum\mathop{\mathbb E}|df_k|^q)^{1/q}\\ &\leq& C\,\frac{1}{n^{1/q}}(\mathop{\mathbb E}|f_n|^q)^{1/q}\\ &\leq& C\,\frac{1}{n^{1/q}}\sup_{0\leq j\leq m}|x_{j+1}-x_j|. \end{array}

For the second statement, set {x_j=F(\tau(\epsilon_1 \cdots\epsilon_{j})\in L_q (B)} and apply 1. to {L_q (B)}. \Box

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 {B} 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 {4}-cycle fixed at opposite vertices.

No direct connection between trees and diamond graphs is known.

 

3.3. Type

 

Definition 14 (Enflo) Let {T} be a metric space. Say {T} has Enflo type {p} if {\exists C>0} such that for all maps {\epsilon:\{\pm 1\}^n \rightarrow T},

\displaystyle  \begin{array}{rcl}  \sum |\textrm{diagonals}|^p \leq C\,\sum|\textrm{edges}|^{p}. \end{array}

Here, diagonals amount to changing all signs, sides to changing one sign.

Theorem 15 (Bourgain, Milman, Wolfson, Pisier) Let {B} be a Banach space.

  • {p}-US implies Enflo type {p}.
  • type {p} implies Enflo type {p_1} for all {p_1 <p}.

 

Proof: The first statement is easy. The second can be found in my Springer Lecture Notes 1206. \Box

Markov type, introduced by Keith Ball. Naor, Peres, Schramm, Sheffield show that it implies type {p}, and is implied by {p}-US. But it is not clea r wether it is more on the US side or on the type {p} side.

 

References

 

[DJL] Davis, W. J.; Johnson, W. B.; Lindenstrauss, J.; The {l^{n}_{1}} problem and degrees of non-reflexivity. Studia Math. 55 (1976), no. 2, 123–139.

[DL] Davis, W. J.; Lindenstrauss, J.; The {l^{n}_{1}} problem and degrees of non-reflexivity. II. Studia Math. 58 (1976), no. 2, 179–196.

[F] Farahat, J. E.; On the problem of {k} structure. Israel J. Math. 28 (1977), no. 1–2, 141–150.

 

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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