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<j} 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<j} 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<p<\infty}. 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)<n(1)<\cdots} (\sum_{k=0}^{\infty}|x_{n(k)}-x_{n(k-1)}|^p)^{1/p}. \end{array}

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<r<p<\infty}. 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<r<p<s<\infty}. 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<p}

 

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.

 

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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