## Notes of Bernard Maurey’s lecture nr 3

1. Factorization

Theorem 1 Let ${X}$ have type 2 and ${Y}$ cotype ${2}$. Let ${X_0}$ be a subspace of ${X}$. Every ${u:X_0 \rightarrow Y}$ factors through a Hilbert space.

Corollary 2 One recovers Kwapien’s theorem: if ${X}$ has type ${2}$ and cotype ${2}$, then ${X}$ is isomorphic to a Hilbert space.

Corollary 3 Let ${X_0}$ be a Hilbert space isometrically embedded in ${X}$. Assume ${X}$ has type ${2}$. Then ${X_0}$ is complemented in ${X}$.

The proof of the factorization theorem relies on the following Lemma, proved yesterday.

Lemma 4 There exists a measure on the unit sphere ${S}$ of ${E^*}$ such that for all ${x\in E}$,

$\displaystyle \begin{array}{rcl} \frac{1}{C_{\phi}}\phi(x)\leq(\int_{S}(t\cdot x)^2 \,d\mu(t))^{1/2}\leq T_{2,E}|x|. \end{array}$

Proof: Let ${\mathcal{E}}$ be the set of finite dimensional subspaces of ${X}$. For ${E\in\mathcal{E}}$, let ${\phi_{E}:E\rightarrow{\mathbb R}_+}$ be equal to ${|u|}$ on ${E_0 =E\cap X_0}$ and to ${0}$ elsewhere. Then ${\phi_E}$ has cotype ${2}$, so there is a corresponding measure ${\mu_E}$ on the unit sphere ${S_E}$ of ${E^*}$. Define

$\displaystyle \begin{array}{rcl} u_{1,E}:E\rightarrow C(S_E)\rightarrow H_E =L_2 (S_E ,\mu_E),\quad x\mapsto (t\mapsto t(x)). \end{array}$

For ${y\in H_E}$, there is a unique ${x\in E_0}$ such that ${u_{1,E}(x)=y}$. Set ${v(y)=u(x)}$. The ultraproduct ${\mathcal{H}=\prod_{E\in\mathcal{E}}H_E /\mathcal{U}}$ is again a Hilbert space. The collection of maps ${u_{1,E}}$ defines a map ${u_1 : X\rightarrow \mathcal{H}}$. Let ${H}$ be the closure of the image ${u_1 (X_0)}$. $\Box$

1.1.

Rosenthal 1972, Nikishin 1970, 1972, Maurey, revisited by Pisier in 1985. Rosenthal and Maurey look for factorizations of maps ${X\rightarrow L_1 (\mu)}$ through multiplication operators ${L_p (\mu)\rightarrow L_1 (\mu)}$. Nikishin considers maps ${X\rightarrow L_0 (\Omega,\mathop{\mathbb P})}$, removes a set of small measure so that map factors to weak ${L_{\infty}}$, ${L_{p,\infty}(\Omega\setminus B)}$.

Definition 5 Weak ${L_{p}}$, denoted by ${L_{p,\infty}(\Omega,\mu)}$, is the set of functions ${f:\Omega\rightarrow{\mathbb R}}$ such that

$\displaystyle \begin{array}{rcl} \mu\{|f|>t\}\leq\frac{C^p}{t^p}. \end{array}$

The best ${C}$ is ${\|f\|_{p,\infty}}$.

If ${r, a function in ${L_{p,\infty}}$ belongs to ${L^r}$,

$\displaystyle \begin{array}{rcl} \|f\|_r \leq \|f\|_{p,\infty}\frac{1}{1-\frac{r}{p}}\mu(A)^{1-\frac{r}{p}}. \end{array}$

Therefore, for ${p>1}$, an alternative norm on ${L_{p,\infty}}$ is

$\displaystyle \begin{array}{rcl} \sup_{A}\frac{1}{\mu(A)^{p'}}\int_{A}|f|\,d\mu. \end{array}$

More generally, the Lorentz space ${L^{p,q}}$ is the set of functions whose rearrangement ${f^*}$ satisfies

$\displaystyle \begin{array}{rcl} s^{1/p}f^* (s)\in L^{p'}({\mathbb R}_+). \end{array}$

Theorem 6 (Pisier) For ${\mathcal{X}\subset L_{1}^{+}(\Omega,\mu)}$, the following are equivalent:

1. There exists a probability density ${f}$ such that for ${d\nu=f\,d\mu}$, the set ${\{x/f \,;\,x\in \mathcal{X}\}}$ is bounded in ${L^{p,\infty} (\nu)}$.
2. ${\exists C>0}$ such that for every ${x_1 ,\ldots,x_n \in\mathcal{X}}$ and ${a_1 ,\ldots,a_n\geq0}$, $\displaystyle \begin{array}{rcl} \int\max(a_i x_i)\,d\mu \leq C(\sum a_i^{p})^{1/p}. \end{array}$

Proof: For every density ${f}$, let ${A_j}$ be the set where ${\max(a_i x_i)}$ is achieved by ${a_j x_j}$. Estimate

$\displaystyle \begin{array}{rcl} \int_{A_i}\max(a_i x_i)\leq K a_i \nu(A_i)^{1/p'} \end{array}$

sum up and use Hölder inequality and the fact that ${\nu}$ is a probability measure.

Conversely, assume for simplicity that ${p=2}$. We shall maximize the integral functional on the set ${F_0 \subset L_1}$,

$\displaystyle \begin{array}{rcl} F=\{\max(a_i x_i)\,;\,\sum a_i^2 \leq 1,\,x_i \in\mathcal{X}\}. \end{array}$

Assume that ${g\mapsto\int g\,d\mu}$ achieves its maximum at some ${f_0}$ and (rescale if necessary) that the maximum equals ${1}$. Let ${u}$, ${v>0}$, ${x\in\mathcal{X}}$,

$\displaystyle \begin{array}{rcl} \int\max(ux,vf_0)\,d\mu\leq 1, \end{array}$

….

$\displaystyle \begin{array}{rcl} \nu\{\frac{x}{f_0}>2t\}\leq\frac{1}{t^2}, \end{array}$

this show that the set ${\{x/f \,;\,x\in \mathcal{X}\}}$ is bounded in ${L_{2,1}}$.

What if the maximum is not achieved ? Use Ekeland’s ${\epsilon}$-minimization. If ${X}$ is a complete metric space, and ${f:X\rightarrow{\mathbb R}}$ bounded below. Then, ${\forall\epsilon>0}$, there exists an ${x_{\epsilon}}$ such that for all ${y\in X}$,

$\displaystyle \begin{array}{rcl} f(y)\geq f(x_{\epsilon})-\epsilon d(y,x_{\epsilon}). \end{array}$

Proof is by transfinite induction.

Pick an ${\epsilon}$-maximum ${f_0}$. Then for all ${u}$, ${v}$, ${x\in\mathcal{X}}$,

$\displaystyle \begin{array}{rcl} \int\max(ux,vf_0)\,d\mu\leq 1+\epsilon\|f_0 -\max (ux,vf_0\| \end{array}$

and proof continues as before… $\Box$

1.2. Sublinear maps

Say an operator ${X\rightarrow L^1}$ is sublinear if ${|u(\lambda x)|=|\lambda||u(x)|}$ and ${|u(x+y)|\leq|u(x)|+|v(y)|}$.

Example 1 Given ${v:X\rightarrow L_1 (Z)}$, ${x\mapsto |v(x)|_Z}$ has this property.

The previous theorem implies

Proposition 7 Let ${u:X\rightarrow L^1}$ be sublinear and satisfy ${|u(x)|\leq|u||x|}$. The following are equivalent:

1. ${u}$ factors through ${X\rightarrow L_{p,\infty}(\nu)\rightarrow L_1 (\mu)}$ where the second arrow is multiplication by ${f}$, ${M_f}$.
2. ${\exists C>0}$ such that for every ${x_1 ,\ldots,x_n \in\mathcal{X}}$, $\displaystyle \begin{array}{rcl} \int\max|u(x_i)|\,d\mu \leq C(\sum |x_i|^{p})^{1/p}. \end{array}$

Theorem 8 (Pisier 1985) Let ${1. Let ${u:X\rightarrow L_1}$ be sublinear and bounded. If ${X}$ has type ${p}$, then ${u}$ factors through ${X\rightarrow L_{p,\infty}(\nu)\rightarrow L_1 (\mu)}$.

Proof: One expresses ${\int\max|u(x_i)| \,d\mu}$ in terms of max. $\Box$

1.3. Back to linear maps

In the linear case, one case dualize. Maps ${C(K)\rightarrow Y}$ dualize to ${Y^* \rightarrow \mathcal{M}(K)}$. If ${Y^*}$ is separable, the image is contained in some ${L_1 (\mu)}$.

Proposition 9 Let ${v:C(K)\rightarrow Y}$ be bounded and linear. The following are equivalent:

1. ${v}$ factors through ${v:C(K)\rightarrow L_{q,1}(K,\nu)\rightarrow Y}$ for some probability measure ${\nu}$.
2. For all ${\phi_1 ,\ldots,\phi_n}$, ${\sum|v(\phi)|^q)^{1/q} \leq K_2 \|\sum|\phi_i|\|_{\infty}}$.

Theorem 10 If ${Y}$ has cotype ${q}$, and ${v=C(K)\rightarrow Y}$ is bounded and linear, then ${v}$ factors through ${v:C(K)\rightarrow L_{q,1}(K,\nu)\rightarrow Y}$ for some probability measure ${\nu}$.

1.4. Applications

This is what motivated Rosenthal in 1972.

Corollary 11 A reflexive subspace of ${L_{1}[0,1]}$ cannot contain ${\ell_{1}^{n}}$ for all ${n}$.

If ${\ell_{1}^{n}}$ ${(1+\epsilon)}$-embeds in ${L_1}$, then the basis functions ${f_i}$ have nearly disjoint supports (there are disjoint subsets ${A_i}$ such that ${\int_{A_i}f_i >1-\epsilon}$. According to factorization theorem, up to a change of measure, ${Y\subset L_r}$ for some ${r>1}$.

Let ${\mathbb{T}}$ be the circle. Let ${\Lambda\subset {\mathbb Z}}$ be a family of characters. This defines a set ${L_{1}^{\Lambda}\subset L_1 (\mathbb{T})}$ which is translation invariant. Since the change of measure should be translation invariant, there is no change of measure. This ${\exists \epsilon>0}$ such that ${L_{1}^{\Lambda}\subset L_{1+\epsilon} (\mathbb{T})}$ (Ebenstein-Bachelis).

1.5. Comparison of Gaussian and Bernoulli averages

Theorem 12 Let ${g_i}$ (resp. ${\epsilon_i}$) be independant Gaussian (resp. Bernoulli) variables. The following are equivalent:

1. ${\exists C>0}$ such that for all ${x_1 ,\ldots,x_n}$, ${\mathop{\mathbb E}|\sum g_i x_i|\leq C\mathop{\mathbb E}|\sum\epsilon_i x_i|}$.
2. ${X}$ has non trivial cotype.

Note that the reverse inequality always holds:

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}|\sum g_i x_i|=\mathop{\mathbb E}_{\epsilon}\mathop{\mathbb E}_{g}|\sum \epsilon_i g_i x_i|\geq\mathop{\mathbb E}_{\epsilon}|\sum\epsilon_i(\mathop{\mathbb E}|g_i|)x_i| =\sqrt{\frac{2}{\pi}}\mathop{\mathbb E}|\sum\epsilon_i x_i|. \end{array}$

Proof: If ${Y}$ has trivial cotype, then it ${1+\epsilon}$-contains ${\ell_{\infty}}$ for which comparison fails. Conversely, factorize the map ${v:C([n])=\ell_{\infty}^{n}\rightarrow L_2 (X)}$, ${(t_1 ,\ldots,t_n)\rightarrow \sum t_i \epsilon_i x_i}$. $\Box$

On monday, ${K}$-convexity.

References