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<p}, 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<p<2}. 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

 

 

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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