Notes of Bernard Maurey’s lecture nr 4

Scribe: Yanqi Qiu.

1. ${K}$-convexity

1.1. Definition

Let ${G_n =\{\pm 1\}^n}$, viewed as a group and a measure space. Let ${\epsilon_j}$ denote the ${j}$-th coordinate.

Definition 1 Define the Rademacher subspace ${Rad(B)\subset L_2 (B)}$ as the linear span of functions of the form ${\epsilon_j x}$, ${j\in[n]}$, ${x\in B}$.

Note that ${B}$ has cotype ${q}$ iff the natural map ${Rad(B)\rightarrow\ell_{q}(B)}$ is continuous. Or dually, that the map ${\ell_p (B^*) \rightarrow L_2 (G_n ,B^*)/Rad(B)^{\bot}}$ is continuous. This gives an elegant proof of the fact that ${L_1}$ has cotype ${2}$.

Definition 2 Say that ${B}$ is ${K}$-convex if there exists ${C>0}$ such that for all ${n}$, the orthogonal projection ${L_2 (G_n ,B)\rightarrow Rad(B)}$ has norm ${\leq C}$.

Thanks to Khintchin’s inequality, one could replace ${L_2}$ with ${L_p}$ in this definition.

Example 1 If ${1, ${L_p}$ is ${K}$-convex.

Theorem 3 (Pisier) If ${B}$ has type ${p>1}$, then ${B}$ is ${K}$-convex.

1.2. Proof of Pisier’s theorem

One uses Fourier analysis on ${G_n}$. Every function ${f:G_n \rightarrow B}$ can be written uniquely

$\displaystyle \begin{array}{rcl} f=\sum_{A\subset[n]}\hat{f}_{A}w_A , \end{array}$

where ${(w_A)_{A\subset [n]}}$ is the Fourier-Walsh basis, ${w_A =\prod_{i\in A}\epsilon_i}$. The orthogonal projection of ${L_2 (B)}$ onto ${Rad(B)}$ is

$\displaystyle \begin{array}{rcl} f\mapsto\sum_{A\subset[n],\,|A|=1}\hat{f}_{A}w_A . \end{array}$

It can be recovered, by a contour integration, from the holomorphic function

$\displaystyle \begin{array}{rcl} s\mapsto\sum_{A\subset[n]}s^{|A|}\hat{f}_{A}w_A . \end{array}$

Thus one needs estimates on this function or on its semi group version

$\displaystyle \begin{array}{rcl} T_t f=\sum_{A\subset[n]}e^{-t|A|}\hat{f}_{A}w_A . \end{array}$

Note that ${T_t}$ is a convex combination of contractive commuting projections.

Lemma 4 Let ${M\in\mathcal{L}(B)}$. Assume that ${M}$ is a convex combination of contractive commuting projections. Let ${a_n (B)}$ be the best constant in the inequality

$\displaystyle \begin{array}{rcl} |\sum_{j=1}^{n}\epsilon_j x_j |_{L_2 (B)}\leq\sqrt{n}a_n (B)\,(\sum_{j=1}^{n}|x_j|^2)^{1/2}. \end{array}$

Then for all ${n}$,

$\displaystyle \begin{array}{rcl} \|M^n -M^{n+1}\|_{\mathcal{L}(B)}\leq 2 a_n (B). \end{array}$

Note that ${a_n (L_2 (B))=a_n (B)}$. Applying the Lemma to ${T_t}$ leads to the following estimate on the generator ${G}$ of the semi-group (${Gf=\sum|A|\hat{f}_{A}w_A}$),

$\displaystyle \begin{array}{rcl} \|(G-\lambda)^{-1}\|\leq 8n. \end{array}$

The contour integral

$\displaystyle \begin{array}{rcl} Rad=\frac{1}{2i\pi}\int_{\gamma}(z-G)^{-1}\,dz \end{array}$

gives the estimate

$\displaystyle \begin{array}{rcl} \|Rad\|\leq K\,8n e^{\pi^2 n}. \end{array}$

2. Hypercontractivity

Let us take the opportunity to discuss a related inequality. It is based again on analysis on the discrete cube.

Let ${0. Then ${\phi_u =1+u\epsilon_1}$ is a nonnegative function on ${G_1}$. Let ${M_u}$ denote convolution with ${\phi_u}$. Then ${T_t =M_{e^{-t}}}$.

Theorem 5 (Bonami, Beckner) If ${1 and ${u=\sqrt{\frac{p-1}{q-1}}}$, then

$\displaystyle \begin{array}{rcl} \|M_u\|_{L^p (G_1)\rightarrow L^q (G_1)}\leq 1. \end{array}$

This boils down to the inequality

$\displaystyle \begin{array}{rcl} (\frac{|1+ut|^q +|1-ut|^q}{2})^{1/q}\leq (\frac{|1+t|^p +|1-t|^p}{2})^{1/p}, \end{array}$

which is proven by brute force, see Johnson and Lindenstrauss’ article in the Handbook in Geometry of Banach Spaces.

This property tensorizes: Applying BB inequality ${n}$ times (combining it with Hölder-Minkowski’s inequality) gives a BB inequality on ${G_n}$. Since ${\phi_u \geq 0}$,

$\displaystyle \begin{array}{rcl} |f*\phi_{u}|_{B}\leq |f|*\phi_u \end{array}$

for every Banach space ${B}$, so BB inequality is valid ${L^p (G_1 ,B)\rightarrow L^q (G_1 ,B)}$.

2.1. Proof of Kahane’s inequality

Here is a simple proof (due to Christer Borrell) of Kahane’s inequality from BB. Let ${f=\sum \epsilon_j x_j}$. Then ${M_u f=uf}$, so BB implies (in particular)

$\displaystyle \begin{array}{rcl} u|f|_{L_q (B)}\leq|f|_{L_2 (B)}, \end{array}$

with ${u=1/\sqrt{q-1}}$. The constant ${b_q}$ in Kahane’s inequality is ${O(\sqrt{q})}$. This implies a sharpening of a result of Landau-Shepp-Fernique,

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(|\sum \epsilon_j x_j|^2)\leq 1 \Rightarrow \mathop{\mathbb E}(e^{C|\sum \epsilon_j x_j|^2})\leq e. \end{array}$