## 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. 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
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 .

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.