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<p<\infty}, {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<u<1}. 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<p<q\leq 2} 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}

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