## Notes of Gilles Pisier’s lecture nr 4

I have announced that superreflexivity is equivalent to being isomorphic to ${q}$-UC for some ${q<\infty}$. It implies that superreflexivity is equivalent to being isomorphic to a space which is uniformly non square. Indeed, James’ theorem states that non reflexive implies uniformly non square. The latter is a super property, the former is isomorphism invariant, so isomorphic to a space which is uniformly non square implies superreflexive. Conversely, uniform convexity implies non square.

1. End of equivalence of superreflexivity and being isomorphic to UC/US

There remains to prove that superreflexive implies

(*) ${\exists q>0}$, ${\exists C>0}$ such that for all dyadic martingales convergent in ${L_q (B)}$,

$\displaystyle \begin{array}{rcl} (\sum|df_n |^{q}_{L_q (B)})^{1/q} \leq C\,|\sum df_n |_{L_q (B)}. \end{array}$

This relies on ideas due to James and independently to Gurarii-Gurarii. The keyword is basic sequence. A sequence ${x_i}$ in a Banach space is ${\lambda}$-basic if for all ${n\leq N}$, for all sequences of coeffcients ${\alpha_i}$,

$\displaystyle \begin{array}{rcl} \sup_n |\sum_{i=1}^{n}\alpha_i x_i |\leq\lambda|\sum_{i=1}^{N}\alpha_i x_i | \end{array}$

For instance, a martingale is a ${\lambda=1}$-basic sequence.

1.1. Step 1

Theorem 1 (Gurarii-Gurarii, James) The following are equivalent:

1. ${B}$ is superreflexive.
2. ${\forall \lambda>1}$, ${\exists C}$, ${\exists q<\infty}$ such that for all ${n}$, all ${\lambda}$-basic sequences in ${B}$ (and any quotient of ${B}$) such that $\displaystyle \begin{array}{rcl} (\sum|x_i|^q)^{1/q}\leq C\,\sum|x_i|. \end{array}$
3. ${\forall \lambda>1}$, ${\exists C}$, ${\exists p>1}$ such that for all ${n}$, all ${\lambda}$-basic sequences in ${B}$ (and any quotient of ${B}$) such that $\displaystyle \begin{array}{rcl} \sum|x_i|\leq C\,(\sum|x_i|^p)^{1/p}. \end{array}$

Proof: 1${\Rightarrow}$2. Blocking argument. ${\lambda}$-basic is stable under blocking, i.e. grouping consecutive terms of the sequence in blocks. Thus the best constant in the wanted inequality (for ${\lambda}$-basic sequences) is submultiplicative. Either this constant decays polynomially (and we are done) or it stays ${\geq 1}$. In that case, we have a sequence with ${|y_i|\geq 1}$ and ${|\sum y_i |\leq 1+\epsilon}$. One can choose a biorthogonal sequence ${\xi_j \in B^*}$ (i.e. ${\xi_j (y_i)=\delta_{ij}}$) such that ${|\xi_j|\leq 2\lambda}$. Set ${x_i=\sum_{j\leq i} y_j}$. Then ${\xi_j (x_i)=0}$ or ${1}$ depending wether ${i or not, and ${|x_i| \leq \lambda(1+\epsilon)}$, this contradicts superreflexivity.

Observe that if ${S\subset B}$ is a closed subspace, ${\theta_n (B/S)\leq\theta_n (B)}$. So the result holds also in quotients of ${B}$.

2${\Rightarrow}$3. By duality.

3${\Rightarrow}$1. If ${B}$ is non reflexive, there exists sequences ${x_i}$, ${\xi_j}$ such that ${\xi_j (x_i)=0}$ or ${\theta}$ depending wether ${i or not. We can assume that ${\xi_j}$ is ${\lambda}$-basic. Thus

$\displaystyle \begin{array}{rcl} n\theta=(\sum_{i=1}^{n}\xi_j)(x_n)\leq|\sum_{i=1}^{n}\xi_j| \end{array}$

cannot be bounded by ${(\sum|x_i|^p)^{1/p}}$, which is ${O(n^{1/p})}$. $\Box$

1.2. Step 2

Prove that ${B}$ superreflexive implies ${L_r (B)}$ superreflexive.

We already know hat superreflexive implies ${J}$-convex. This easily implies that the average over admissible

$\displaystyle \begin{array}{rcl} (average_{admis}|\sum\epsilon_i x_i|^r)^{1/r}\leq (n-\delta)(average|x_i|^r)^{1/r}. \end{array}$

(since the constant is nearly ${n}$, fixed), which passes through integration.

1.3. Step 3

Apply Step 1 to ${L_2 (B)}$.

1.4. Step 4

See the notes.

2. Relationship between UC and type, cotype

Recall that ${q}$-UC implies cotype ${q}$. Converse fails (${\ell_1}$ and ${L_1}$ have cotype ${2}$). We shall see that the converse holds in the family of UMD spaces.

2.1. Interpolation

Theorem 2 (Pisier, Xu) Let ${1. There exists a Banach space ${W_p}$ such that ${\mathcal{V}_1 \subset W_p \subset \ell_{\infty}}$, ${\mathrm{dim}(W_{p}^{**}/W_p)=1}$, ${W_{p}^{p}}$ isomorphic to ${W_{p'}}$, and ${W_p}$ has the same type and cotype as ${L_p}$, except if ${p=2}$.

We use (Lions-Peetre) real interpolation, ${W_p =(\mathcal{V}_1 ,\ell_{\infty})_{\theta}}$, with ${\theta =1-\frac{1}{p}}$.

Definition 3 Let ${B_0}$, ${B_1}$ be Banach spaces with continuous inclusions in a third Banach space ${X}$. Define

$\displaystyle \begin{array}{rcl} K_t (x)=\inf_{x=x_0 +x_1}\{|x|_{B_0}+t|x|_{B_1}\} \end{array}$

On ${B_0 +B_1}$, put the norm

$\displaystyle \begin{array}{rcl} |x|_{\theta,q}=(\int_{0}^{\infty}(t^{-\theta}K_t (x))^q \frac{dt}{t})^{1/q}. \end{array}$

and complete to get ${(B_0 ,B_1)_{\theta,q}}$.

Example 1 ${(L_1 ,L_{\infty})_{\theta,q}=L_{p_{\theta},q}}$ is a Lorentz space.

If ${q=p_{\theta}}$, ${L_{p_{\theta},q}=L_q}$ coincides with the ordinary Lebesgue space.

If ${q=\infty}$, ${L_{p,\infty}}$ is the weak ${L_p}$ space.

Indeed, ${K_t (x)=\int_{0}^{t}x^* (s)\,ds}$, where ${x^*}$ is the increasing rearrangement of ${|x|}$, and by definition,

$\displaystyle \begin{array}{rcl} |x|_{L_{p,q}}=(\int_{0}^{\infty}(t^{-\theta}(\int_{0}^{t}x^* (s)\,ds))^q \frac{dt}{t})^{1/q}. \end{array}$

If ${q=p}$,

$\displaystyle \begin{array}{rcl} |x|_{L_{q,q}}=(\int_{0}^{\infty}(\int_{0}^{t}x^* (s)\,ds)^q \,dt)^{1/q}\sim |x|_{L_q} \end{array}$

by Hardy inequality applied to ${x^*}$.

Example 2 Similarly, ${(\ell_1 ,\ell_{\infty})_{\theta,q}=\ell_{p_{\theta},q}}$ is a Lorentz sequence space.

If ${q=p_{\theta}}$, ${\ell_{p_{\theta},q}=\ell_q}$ coincides with the ordinary sequence space.

If ${q=\infty}$, ${\ell_{p,\infty}}$ is the weak ${\ell_p}$ space.

Proposition 4 More generally, ${(L_{p_{0}},L_{p_{1}})_{\theta,q}=L_{p,q}}$ if ${\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}}$. Also, for all ${B}$, ${(L_{p_{0}}(B),L_{p_{1}}(B))_{\theta,q}=L_{p,q}(B)}$

${L_{r}((B_0 ,B_1)_{\theta,q})\subset(L_{r}(B_0),L_{r}(B_1))_{\theta,q}}$ if ${q\geq r}$,

${(L_{r}(B_0),L_{r}(B_1))_{\theta,q}\subset L_{r}((B_0 ,B_1)_{\theta,q})}$ if ${q\leq r}$.

This follows from the Hölder-Minkowski inequality ${L_q (L_r)\subset L_r (L_q)}$ if ${q\leq r}$.

2.2. Comparison spaces ${\mathcal{V}_p}$

Here is a ${2}$-parameter family

Definition 5 For a sequence ${x=(x_n)}$ of real numbers, let

$\displaystyle \begin{array}{rcl} |x|_{\mathcal{V}_p}\sup_{0=n(0)

Replacing the ${\ell_p}$ norm in the definition by ${\ell_{p,q}}$, we get ${\mathcal{V}_{p,q}}$.

${\mathcal{V}_2}$ was discovered by James. This was the first example of a Banach space such that ${\mathrm{dim}(B^{**}/B)=1}$. ${\mathcal{V}_p}$ contains ${\ell_{\infty}^{n}}$‘s uniformly, so ${\mathcal{V}_p}$ has trivial type and cotype. However, the dual ${\mathcal{V}_p^*}$ has cotype ${2}$ or ${p'}$ depending wether ${p'\leq 2}$ or ${\geq 2}$.

Lemma 6

$\displaystyle \begin{array}{rcl} W_p \subset V_p \subset W_{p,\infty}=\mathcal{V}_{p,\infty}. \end{array}$

More generally, ${W_{p,q}\subset\mathcal{V}_{p,q}}$.

Proof: The trick for ${W_{p,\infty}=\mathcal{V}_{p,\infty}}$ is a property of ${K_t}$ for integer ${t}$ (this suffices for sequence spaces). For ${t=N}$,

$\displaystyle \begin{array}{rcl} K_N (x,\mathcal{V}_1 ,\ell_{\infty})\sim \end{array}$

The main point of interpolation theory is the continuity of operators. If ${T:B_0 \rightarrow C_0}$ and ${T:B_1 \rightarrow C_1}$ has norm ${\leq 1}$, then so does ${T:(B_0 ,B_1)_{\theta,q}\rightarrow (C_0 ,C_1)_{\theta,q}}$.

Apply this to ${T: (x_n) \mapsto (y_k)}$, where ${y_k =x_{n(k)}-x_{n(k-1)}}$. This gives

$\displaystyle \begin{array}{rcl} |x_{n(k)}-x_{n(k-1)}|_{\ell_{p,q}}\leq 2|x|_{W_{p,q}}, \end{array}$

i.e. ${W_{p,q}\subset \mathcal{V}_{p,q}}$. $\Box$

2.3. Reiteration

This is the abstract version of Marcinkiewicz’ theorem in harmonic analysis.

Proposition 7 Let ${X_0 =(B_0 ,B_1)_{\theta_0,q_0}}$, ${X_1 =(B_0 ,B_1)_{\theta_1,q_1}}$. Then

$\displaystyle \begin{array}{rcl} (X_0 ,X_1)_{\alpha,q}=(B_0 ,B_1)_{\theta,q}, \end{array}$

where ${\theta=(1-\alpha)\theta_0 +\alpha \theta_1}$ does not depend on the ${q_i}$‘s.

Corollary 8 Let ${1. Then ${W_p =(\mathcal{V}_r ,\ell_{\infty})_{\alpha,p}}$ where ${\frac{1-\alpha}{r}=\frac{1}{p}}$.

Proof: Apply reiteration lemma to inclusions ${W_p \subset V_p \subset W_{p,\infty}}$. Get

$\displaystyle \begin{array}{rcl} (W_{p,p},\ell_{\infty})_{\alpha,p} \subset (V_p ,\ell_{\infty})_{\alpha,p}\subset (W_{p,\infty},\ell_{\infty})_{\alpha,p}, \end{array}$

i.e.

$\displaystyle \begin{array}{rcl} W_p \subset (V_p ,\ell_{\infty})_{\alpha,p}\subset W_p . \end{array}$

$\Box$

Lemma 9 Let ${1. The family ${W_p}$ is ${r}$-convex in the following sense: ${L_r (W_p)\subset W_p (L_r)}$. It is ${s}$-concave in the following sense: ${W_p (L_s)\subset L_s (W_p)}$.

Proof: By definition, ${W_p (B)=(\mathcal{V}_r (B),\ell_{\infty}(B))_{\alpha,p}}$. Since sup of integral is less than integral of sup, ${L_r (\mathcal{V}_r)\subset \mathcal{V}_r (L_r)}$, ${L_r (\ell_{\infty})\subset \ell_{\infty} (L_r)}$. Interpolate and apply Hölder-Minkowski.

Concavity is by duality, since ${W_{p}(B)^*\sim W_{p'}(B^*)}$. $\Box$

2.4. Proof that ${W_p}$ has type ${r}$ for all ${2\geq r

Let ${\Omega=\{\pm 1\}^{{\mathbb N}}}$. Consider thet map ${T:\ell_r \rightarrow L_r (\Omega)}$, ${(\alpha_n)\mapsto \sum\alpha_n \epsilon_n}$. For ${x\in W_p}$,

Khintchin implies that ${T}$ is bounded. Up to constant,

$\displaystyle \begin{array}{rcl} |x|_{\ell_r (W_p)}\geq|x|_{W_p (\ell_r)}\geq|(T\otimes id)(x)|_{W_p (L_r)}\geq |(T\otimes id)(x)|_{L_s (W_p)}. \end{array}$

Taking expectation gives the type inequality.