Notes of Bernard Maurey’s lecture nr 1

 

Type and cotype of Banach spaces

 

1. Plan of course

 

  1. Gaussian random vectors (Kahane, Landau-Shepp-Fernique). Type and cotype.
  2. Factorization theorems (Grothendieck).
  3. Grothendieck’s inequality, Krivine’s theorem on {\ell_p^n}‘s in Banach spaces.
  4. {K}-convexity (Pisier).
  5. Dvoretzky theorem in cotype {2} spaces.

 

2. Gaussian random vectors

 

2.1. Elementary properties

 

Recall that the characteristic function of a standard Gaussian is

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(e^{it\cdot X})=e^{-\frac{|t|^2}{2}}. \end{array}

It follows that any linear combination of independant Gaussian vectors is a Gaussian, its {L_p} norm only depends on {p} and on the variance. So the infinite dimensional Gaussian vector gives rise to a linear isometric embedding of Hilbert space into {L^p}. Also, orthogonal Gaussian vectors are independent.

 

2.2. Bernoulli variables

 

Khintchin’s theorem is a Bernoulli version of the isometric embedding of Hilbert space.

Theorem 1 (Khintchin) Let {0<p<\infty}. There are constants {A_p} and {B_p} such that, if {\epsilon_j} are independant Bernoulli variables, then for all {a\in\ell_2 ({\mathbb N})},

\displaystyle  \begin{array}{rcl}  A_p \sqrt{\sum_{j}a_j^2}\leq (\mathop{\mathbb E}(|\sum_{j}a_j\epsilon_j|^p))^{1/p}\leq B_p \sqrt{\sum_{j}a_j^2}. \end{array}

 

Proof: In case {p=2n} is an even integer. Expand

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(|\sum_{j}a_j\epsilon_j|^p)&=&\sum_{j_1 ,\ldots,j_n}a_{j_1}\cdots a_{j_n}\mathop{\mathbb E}(\epsilon_{j_1}\cdots \epsilon_{j_n})\\ &=&\sum_{k_1 +\cdots+k_p=2n}a_{j_1}^{2k_1}\cdots a_{j_n}^{2k_p} \end{array}

Expanding {(\sum_{j}a_j^2)^n} leads to a similar expression, up to binomial coefficients. This leads to an upper bound, with {B_{2n}\leq\sqrt{2n}}.

In general, use Laplace transform and inequality {\mathop{\mathbb E}(e^{s\epsilon})=\cosh(s)\leq e^{s^2 /2}}. If {\sum_{j}a_j^2 =1}, then, for {S=\sum a_j \epsilon_j},

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(e^{sS})\leq e^{s^2 /2}. \end{array}

It follows that

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(S\geq t)\leq e^{s^2 /2 -st},\quad \mathop{\mathbb P}(|S|\geq t)\leq 2\,e^{-t^2 /2}, \end{array}

and

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(|S|^p)\leq \int_{0}^{\infty}2p t^{p-1}e^{-t^2 /2}\,dt. \end{array}

The lower bound follows from the upper bound and Hölder’s inequality. For instance,

\displaystyle  \begin{array}{rcl}  \|a\|_2 =\|S\|_2 \leq \|S\|_1^{1/3}\|S\|_4^{2/3}\leq B_4^{2/3}\|a\|_2^{2/3}\|S\|_1^{1/3} \end{array}

\Box

The sharp constants are known, Haagerup.

 

2.3. Kahane’s inequalities

 

Theorem 2 (Kahane) Let {E} be a normed space. There exist constants {a_p}, {b_p} such that for every set of vectors {x_j \in E},

\displaystyle  \begin{array}{rcl}  a_p \|\sum\epsilon_j x_j\|_2 \leq \|\sum\epsilon_j x_j\|_p \leq b_p \|\sum\epsilon_j x_j\|_2 . \end{array}

 

Proof: We use the following fact. Let {C} be a closed convex subset of {C}, let {V} be a {V}-valued random vector with a symmetric distribution. Then if {x\notin C},

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(x+V \notin C)\geq \frac{1}{2}. \end{array}

This implies that if {U} and {V} are independent and symmetric,

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(|U|>t)\leq 2\mathop{\mathbb P}(|U+V|>t). \end{array}

Introduce the martingale {M_k=\sum_{j=1}^{k}\epsilon_j x_j} and prove a maximal inequality (a bit finer than Doob’s inequality)

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(\sup|M_k|>t)\leq 2\mathop{\mathbb P}(|M_n|>t). \end{array}

For this, introduce the stopping time {T=\inf\{k\,;\,|M_k|>t\}}. Note that {M_T} and {Y=M_n -M_T} are independent. Both have symmetric distributions, so {U+V} inequality applies.

Assume that {\mathop{\mathbb E}(|\sum\epsilon_j x_j|^2)=1}. The {U+V} principle implies that if {\mathop{\mathbb P}(|M_n|>4)\leq \frac{1}{16}}, then all {x_j} have norm {\leq 2}. Let {T=\inf\{k\,;\,|M_k|>4\}} and {Y=M_n -M_T}. If {|M_n|>8+t}, then {|M_T|>4} and {|Y|>t}, so

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(|M_n|>8+t)&\leq&\mathop{\mathbb P}(|M_T|>4)\mathop{\mathbb P}(|Y|>t)\\ &\leq&2\mathop{\mathbb P}(|M_T|>4)\mathop{\mathbb P}(|Y|>t)\\ &\leq&4\mathop{\mathbb P}(|M_T|>4)\mathop{\mathbb P}(|Y|>t)\\ &\leq&\frac{1}{4}\mathop{\mathbb P}(|M_n|>t). \end{array}

This shows that the tail of the distribution of {|M_n|} decays exponentially, and gives an estimate on {\mathop{\mathbb E}(|M_n|^p)}. \Box

 

2.4. The Landau-Shepp-Fernique Theorem

 

It is a Gaussian analogue of Kahane’s inequality.

Theorem 3 Let {\tau>0}, {\alpha<1}. There exists a constant {c(\tau,\alpha)} with the following effect. Let {G} be a Gaussian vector in a finite dimensional Banach space {E}. Assume that {\mathop{\mathbb P}(|G|>\tau)<\alpha}. Then

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(e^{c|G|^2})\leq \end{array}

 

Proof: (Inspired from Fernique’s proof). Let {X} and {Y} be independent copies of {G}. Let

\displaystyle  \begin{array}{rcl}  X_{\theta}=\cos\theta X+\sin\theta Y. \end{array}

Note that {X_{\theta}} and {\frac{d}{d\theta}X_{\theta}} are independent. If {f} is a differentiable function on {E},

\displaystyle  \begin{array}{rcl}  f(Y)-f(X)=\int_{0}^{\pi/2}\nabla f(X_{\theta})\cdot Y_{\theta}\,d\theta. \end{array}

By convexity of {x\mapsto e^{sx}}, Jensen’s inequality gives

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(\exp(s\frac{2}{\pi}(f(Y)-f(X))))\leq \frac{2}{\pi}\int_{0}^{\pi/2}\mathop{\mathbb E}(e^{s\nabla f(X_{\theta})\cdot Y_{\theta}})\,d\theta. \end{array}

Lemma 4 Let {X}, {Y} be independent standard Gaussian vectors. If {f} is {1}-Lipschitz on Euclidean space, then

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(e^{\frac{2}{\pi}s(f(Y)-f(X))})\leq e^{s^2 /2}. \end{array}

 

Jensen again gives

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(e^{\frac{2}{\pi}s(f(Y)-\mathop{\mathbb E}(f(X)))})\leq e^{s^2 /2}. \end{array}

This implies

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(|f(Y)-\mathop{\mathbb E}(f(Y))|>t)\leq e^{-(\frac{2}{\pi}t)^2 /2}. \end{array}

Now {f(x)=|x|_E}. So {|\nabla f|_{E^*}\leq 1}. For all {x\in E},

\displaystyle  \begin{array}{rcl}  |\nabla f(x)\cdot Y_{\theta}|\leq|Y_{\theta}|_E . \end{array}

So our assumption gives

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb P}(|\nabla f(x)\cdot Y_{\theta}|>\tau)<\alpha, \end{array}

and this yields a bound on variance

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(|\nabla f(x)\cdot Y_{\theta}|^2)\leq\kappa(\tau,\alpha). \end{array}

Therefore {\mathop{\mathbb E}(\exp(s\frac{2}{\pi}(f(Y)-f(X))))\leq e^{s^2 \kappa^2 /2}}. By independence,

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(\exp(s\frac{2}{\pi}(|Y|)))\leq \end{array}

\Box

 

2.5. Type and cotype

 

Definition 5 Let {X} be a normed space. Say that {X} has type {p} if there exists a constant {T_p} such that for all sets of vectors {x_i \in X},

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(|\sum_{i}\epsilon_i x_i|)\leq T_p (\sum_{i}|x_i|^p)^{1/p}. \end{array}

Say that {X} has cotype {q} if there exists a constant {C_q} such that for all sets of vectors {x_i \in X}, the reverse inequality holds, i.e.

\displaystyle  \begin{array}{rcl}  (\sum_{i}|x_i|^p)^{1/p}\leq C_q \mathop{\mathbb E}(|\sum_{i}\epsilon_i x_i|). \end{array}

 

If all {x_i} are colinear, this boils down to a {1}-dimensional inequalityn which holds iff {1\leq p\leq 2}. So type always belongs to {[1,2]} and similarly cotype always belongs to {[2,\infty)}.

Note that type {p} implies type {r} for {1\leq r\leq p}. Every Banach space has type {1} and cotype {\infty}, by the triangle inequality.

In the {T_p} inequality, thanks to Kahane, one can replace the {L_1}-norm on the left-hand side by an {L_r} norm.

Using Khintchin,

Example 1 If {1\leq r\leq 2}, {L^r} has type {p} and cotype {2}. If {2\leq r<\infty}, {L^r} has type {2} and cotype {r}.

 

These values are sharp. Indeed, {\ell_p} embeds in {L_p (0,1)}. If {(e_n)} is the standard basis of {\ell_p}, then

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(|\sum_{i}\epsilon_i e_i|_{\ell_p})=n^{1/p}, \end{array}

and this shows that type has to be {\leq p}.

 

2.6. Properties of type and cotype

 

  1. Type and cotype pass to subspaces.
  2. Type passes to quotients.
  3. If {X} has type {p}, then {X^*} has cotype {p'=\frac{p}{p-1}}.

Tomorrow, I will discuss ultraproducts, a tool to construct large Banach spaces reflecting properties of much smaller spaces.

 

References

 

 

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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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