Notes on Alexandros Eskenazis Rennes lectures january 26th 2023

When does ${L_p}$ embed into ${L_q}$ ?

The question goes back to Banach. This will be an excuse to review some modern tools. Banach space theory has evolved into metric geometry. The leitmotiv will be: how does one prove non-embeddability?

1. Linear embeddings

1.1. Banach’s question

Notation. ${L_p=L_p(0,1)=}$ measurable functions on ${(0,1)}$ . ${\ell_p =}$ ${p}$ -summable sequences.

${\ell_p^n={\mathbb R}^n}$ in its ${\ell_p}$ norm.

Definition 1 A linear operator ${T:X\rightarrow Y}$ between normed spaces is a ${D}$ -isomorphic embedding, where ${D\ge 1}$ , if there exists ${s>0}$ such that ${\forall x\in X}$ ,

$\displaystyle s\|x\|_X \le \|Tx\|_Y\le DS\|x\|_X.$

Example 1 ${\ell_p}$ embeds in ${L_p}$ isometrically.

Remark. ${\ell_p}$ is not isomorphic to ${L_p}$ when ${p\not=2}$ . This is a nontrivial fact, probably going back to Banach.

Question (Banach). For which ${p\not=q\in[1,\infty)}$ does ${L_p}$ embed isomorphically into ${L_q}$ ?

Theorem 2 (Banach 1932, Paley 1936) ${L_p}$ does not embed into ${L_q}$ unless ${p=2}$ or ${1\le q<p<2}$ .

Theorem 3 (Kadec 1958) ${L_p}$ embeds isometrically into ${L_q}$ if ${p=2}$ or ${1\le q<p<2}$ .

Theorem 2 is much harder than Theorem 3 (although older).

1.2. Proof of Kadec’ theorem

Let ${g_1,g_2,\ldots}$ be a sequence of iid standard gaussian random variables on a ${\sigma}$ -finite probability space.

Gaussians have the following property: if standard gaussian r.v. ${g}$ and ${g'}$ are independent, the random variable ${\lambda g+\mu g'}$ has the same distribution as ${\sqrt{\lambda^2+\mu^2}g}$ . Indeed, both have the same characteristic function.

Define the embedding ${T:\ell_2\rightarrow L_q}$ as follows,

$\displaystyle \forall a=(a_n)_n \in\ell_2,\quad Ta=\sum a_i g_i.$

Then ${Ta}$ has the same distribution as ${\|a\|_2 g}$ . Therefore

$\displaystyle \|Ta\|_p=\|a\|_2\|g\|_p .$

Question. Why doesn’t this work for other values of ${p}$ ? In other words, do there exist iid symmetric random variables ${X,X'}$ such that ${\lambda g+\mu g'}$ has the same distribution as ${(\lambda^p+\mu^p)^{1/p}g}$ ?

The answer arises in Paul Lévy’s book.

Theorem 4 (Levy 1951) Such random variables exist iff ${0<p\le 2}$ . They are called standard ${p}$ -stable random variables, and for ${0<p<2}$ , they satisfy

$\displaystyle \lim_{t\rightarrow\infty}t^p\mathop{\mathbb P}\{|x|\ge t\}=\sigma_p\in(0,\infty).$

Therefore,

$\displaystyle \mathop{\mathbb E}(|X|^q) =\mathop{\mathbb E}(\int_0^\infty qt^{q-1}\mathbf{1}_{\{t\le|x|\}}dt) =q\int_0^\infty t^{q-1}\mathop{\mathbb P}\{|x|\ge t\}\,dt$

is finite iff ${q<p}$ .

This shows that ${\ell_p}$ embeds isometrically into ${L_q}$ for ${1\le q<p<2}$ .

Remark 1 To prove that ${L_p}$ embeds into ${L_q}$ in this range, one needs two more nontrivial results:

${L_p}$ is finitely representable into ${\ell_p}$ .

If ${X}$ is finitely representable in ${\ell_q}$ and is separable, then ${X}$ embeds into ${L_q}$ .

This ends our excursion in the construction of embeddings. From now on, we switch to nonembeddability results.

1.3. Linear distorsion

Definition 5 The linear distorsion of ${X}$ into ${Y}$ is the smallest ${D\ge 1}$ such that there exists a ${D}$ -isomorphic embedding of ${X}$ into ${Y}$ . It is denoted by

$\displaystyle c_Y^{lin}(X),$

and if ${Y=L_q}$ , by

$\displaystyle c_q^{lin}(X).$

Thus we have stated and partly proved that

$\displaystyle c_q(L_p)=\begin{cases} 1 & \text{ if }p=2 \text{ or }1\le q<p<2, \\ \infty & \text{otherwise}. \end{cases}$

Goal. Understand the asymptotics of ${c_q^{lin}(\ell_p^n)}$ when ${p,q}$ are in the second range.

Theorem 6 Let ${1\le p\not=q<\infty}$ . Then

$\displaystyle c_q^{lin}(\ell_p^n)=_{p,q}~1 \text{ if }1\le q<p\le 2,\quad \text{ (Kadec)}$

$\displaystyle n^{1/p - 1/q} \text{ if }1\le p<q\le 2, \quad \text{ (type range)}$

$\displaystyle n^{1/p - 1/2} \text{ if }1\le p\le 2\le q,$

$\displaystyle n^{1/q - 1/p} \text{ if }1\le p<q\le 2,\quad \text{ (cotype range)}$

$\displaystyle n^{1/2 - 1/p} \text{ if }1\le p<q\le 2,$

$\displaystyle n^{\frac{(q-p)(p-2)}{p^2(q-2)}} \text{ if }1\le p<q\le 2,\quad (X_p \text{ range)}.$

Easy. The upper bounds are easy, the embedding is identity or identity to ${\ell_2^n}$ followed with Kadec’s isometric embedding.

Our goal is to prove lower bounds, by designing invariants.

2. Smoothness and convexity in ${L_p}$ spaces

2.1. Smoothness and convexity

Definition 7 (Ball – Carlen – Lieb 1993) Fix ${1\le p\le 2}$ . Say a Banach space ${X}$ is ${p}$ -uniformly smooth with constant ${s}$ if

$\displaystyle \forall x,y\in X,\quad \frac{\|x\|^p+\|y\|^p}{2}\le \|\frac{x+y}{2}\|^p+s^p\|\frac{x-y}{2}\|.$

Fix ${2\le q\le \infty}$ . Say a Banach space ${X}$ is ${q}$ -uniformly convex with constant ${K}$ if

$\displaystyle \forall x,y\in X,\quad \|\frac{x+y}{2}\|^p+\frac{1}{K^{p}}\|\frac{x-y}{2}\|\le \frac{\|x\|^q+\|y\|^q}{2}.$

The best constants are denoted by ${s_p(X)}$ and ${K_q(X)}$ .

Exercise. (Lindenstrauss’ duality formula à la Ball-Carlen-Lieb). For any normed space ${X}$ and ${\frac{1}{p}+\frac{1}{p'}=1}$ ,

$\displaystyle s_p(X)=K_{p'}(X^*)\quad\text{ and }\quad s_p(X^*)=K_{p'}(X).$

2.2. The case of ${L_p}$ spaces

Theorem 8 (Clarkson’s inequality) For ${1\le p\le 2}$ , ${s_p(L_p)=1}$ .

For ${2\le q <\infty}$ , ${K_q(L_q)=1}$ .

Proof. Since the inequality involves only ${\|.\|_q^q}$ , is suffices to prove that for all ${a,b\in{\mathbb R}}$ ,

$\displaystyle |\frac{a+b}{2}|^q+|\frac{a-b}{2}|^q\le\frac{|a|^q+|b|^q}{2}.$

By monotonicity of ${\ell^q}$ norms and the parallelogram identity,

$\displaystyle (|\frac{a+b}{2}|^q+|\frac{a-b}{2}|^q)^{1/q}\le(|\frac{a+b}{2}|^2+|\frac{a-b}{2}|^2)^{1/2}=(\frac{a^2+b^2}{2})^{1/2}\le(\frac{|a|^q+|b|^q}{2})^{1/q}.$

Theorem 9 For ${1<p\le 2}$ , ${K_2(L_p)\le\frac{1}{\sqrt{p-1}}}$ .

For ${2\le q<\infty}$ , ${s_2(L_q)\le\sqrt{q-1}}$ .

2.3. Two lemmata

The proof of Theorem 10 requires two lemmata, the first one is Bonami’s hypercontractivity inequality on the ${1}$ -cube.

Lemma 10 (Bonami’s two-point inequality) Let ${1<p\le 2}$ . For ${a,b\in{\mathbb R}}$ ,

$\displaystyle (a^2+(p-1)b^2)^{1/2}\le(\frac{|a+b|^p+|a-b|^p}{2})^{1/p}.$

Proof. One can assume that ${a=1}$ and ${b=x\le 1}$ . Using Taylor’s expansion, one checks that

$\displaystyle (1+(p-1)x^2)^{p/2}\le \frac{(1+x)^p+(1-x)^p}{2}.$

The second lemma illustrates a general principle: an inequality for ${L_p}$ with constant one can be nothing but a relaxation of the parallelogram identity in Euclidean space.

Lemma 11 (Hanner’s inequality) Let ${1\le p\le 2}$ . For ${f,g\in L_p}$ ,

$\displaystyle |\|f\|_p-\|g\|_p|^p +(\|f\|_p+\|g\|_p)^p\le \|f+g\|_p^p+\|f-g\|_p^p.$

One first checks that for ${r\in[0,1]}$ , the numbers

$\displaystyle \alpha(r)=(1+r)^{p-1}+(1-r)^{p-1}\quad \text{and}\quad \beta(r)=\frac{(1+r)^{p-1}-(1-r)^{p-1}}{r^{p-1}}$

satisfy

$\displaystyle \forall A,B\in{\mathbb R},\quad \max_{r\in[0,1]}\{\alpha(r)|A|^p+\beta(r)B^p\}\le |A+B|^p+|A-B|^p.$

(again, one can assume that ${A=1}$ and ${|B|\le 1}$ ; then it amounts to the monotonicity of a function).

Hanner’s inequality follows: assuming that ${\|f\|_p\ge \|g\|_p}$ , set ${A=|f(x)|}$ , ${B=|g(x)|}$ , ${r=\frac{\|g\|_p}{\|f\|_p}}$ to get

$\displaystyle |f(x)+g(x)|^p+|f(x)-g(x)|^p\ge \left( (1+\frac{\|g\|_p}{\|f\|_p})^{p-1}+(1-\frac{\|g\|_p}{\|f\|_p})^{p-1} \right)|f(x)|^p$

$\displaystyle +\frac{(1+\frac{\|g\|_p}{\|f\|_p})^{p-1}-(1-\frac{\|g\|_p}{\|f\|_p})^{p-1}}{(\frac{\|g\|_p}{\|f\|_p})^{p-1}}|g(x)|^p.$

Integrating with respect to ${x}$ yields

$\displaystyle \|f+g\|_p^p+\|f-g\|_p^p \ge\left( |\|f\|_p+\|g\|_p|^{p-1} +|\|f\|_p-\|g\|_p|^{p-1} \right)\|f\|_p$

$\displaystyle + \left((\|f\|_p+\|g\|_p|)^{p-1} -|\|f\|_p-\|g\|_p|^{p-1} \right)\|g\|_p$

$\displaystyle =(\|f\|_p+\|g\|_p)^{p}+|\|f\|_p-\|g\|_p|^{p}.$

2.4. Proof of Theorem 10

Let ${f=\frac{x+y}{2}}$ , ${g=\frac{x-y}{2}}$ . We show that

$\displaystyle \|f\|_p^2+(p-1)\|g\|_p^2 \le \frac{\|f+g\|^2_p+\|f-g\|^2_p}{2}.$

Indeed, first apply Bonami’s inequality.

Conjecture. Does Hanner’s inequality hold in Schatten class, i.e. for matrices where the ${\ell_p}$ norm of singular values is used?

Known for ${p\ge 4}$ and ${p\le \frac{4}{3}}$ (Ball-Carlen-Lieb). Heinavaara 2022 proves this for ${p=3}$ .

Conjecture. For ${1<p\le 2}$ ,

$\displaystyle \mathop{\mathbb E}|\sum\epsilon_i\|f_i\|_p|^p\le \mathop{\mathbb E}\|\sum\epsilon_i f_i\|_p^p$

Theorem 12 (D. Schechtman 1995) Yes for ${p\ge 3}$ .

3. Martingales in Banach spaces

Definition 13 Let ${X}$ be a Banach space. A (Paley-Walsh) martingale with values in ${X}$ is a sequence of functions ${M_k:\{-1,1\}^k\rightarrow X}$ such that for all ${\epsilon\in\{-1,1\}^k}$ ,

$\displaystyle M_k(\epsilon_1,\ldots,\epsilon_k)=\frac{M_{k+1}(\epsilon_1,\ldots,\epsilon_k,1)+M_{k+1}(\epsilon_1,\ldots,\epsilon_k,0)}{2}.$

The basic example is

$\displaystyle M_k(\epsilon)=\sum\epsilon_ix_i,$

for given vectors ${x_1, x_2,\ldots\in X}$ .

Remark. If ${X={\mathbb R}}$ , the set ${\{M_k-M_{k-1}\}}$ is orthogonal, so

$\displaystyle \mathop{\mathbb E}|M_n-M_0|^2=\sum\mathop{\mathbb E}|M_k-M_{k-1}|^2.$

Definition 14 (Pisier 1975) A Banach space ${X}$ has martingale type ${p}$ , ${1\le p\le 2}$ , il for every ${X}$ -valued martingale ${\{M_k\}}$ ,

$\displaystyle \mathop{\mathbb E}\|M_n-M_0\|_X^p \le T_p(X)^p\sum\mathop{\mathbb E}\|M_k-M_{k-1}|_X^p.$

${X}$ has martingale cotype ${q}$ , ${2\le q\le \infty}$ , il for every ${X}$ -valued martingale ${\{M_k\}}$ ,

$\displaystyle \mathop{\mathbb E}\|M_n-M_0\|_X^q \ge\frac{1}{c_q(X)^q}\sum\mathop{\mathbb E}\|M_k-M_{k-1}|_X^q.$

These properties can be thought of as relaxations of the identity that holds in Hilbert space.

3.1. Smoothness/convexity versus type/cotype

These properties follow from the convexity properties introduced earlier.

Proposition 15 (Pisier) ${p}$ -smoothness (resp. ${q}$ -convexity) implies martingale type ${p}$ (resp. cotype ${q}$ ) with the same constant.

Pisier’s renorming theorem states that the converse is true, up to changing for an equivalent norm.

The advantage of the type/cotype formulation is that the error is multiplicative, hence these properties are isomorphism invariant.

3.2. Proof of Pisier’s “smoothness implies type”

Let ${\{M_k\}}$ be an ${X}$ -valued martingale with ${M_0=0}$ . Then

$\displaystyle \mathop{\mathbb E}\|M_n(\epsilon)\|^p = \mathop{\mathbb E}_{\epsilon_1,\ldots,\epsilon_{n-1}}(\frac{\|M_n(\epsilon_1,\ldots,\epsilon_{n-1},1)\|^p+\|M_n(\epsilon_1,\ldots,\epsilon_{n-1},0)\|^p}{2})$

$\displaystyle \le \mathop{\mathbb E}_{\epsilon_1,\ldots,\epsilon_{n-1}}\left( \|\frac{M_n(\epsilon_1,\ldots,\epsilon_{n-1},1)+M_n(\epsilon_1,\ldots,\epsilon_{n-1},0)}{2} \|^p \right.$

$\displaystyle \left.+S_p(X)^p\|\frac{M_n(\epsilon_1,\ldots,\epsilon_{n-1},1)-M_n(\epsilon_1,\ldots,\epsilon_{n-1},0)}{2}\|^p \right)$

$\displaystyle =\mathop{\mathbb E}(\|M_{n-1}\|^p+s_p(X)^p\|M_{n}-M_{n-1}\|^p)$

$\displaystyle \le \mathop{\mathbb E}(\|M_{n-2}\|^p+s_p(X)^p(\|M_{n}-M_{n-1}\|^p+\|M_{n-1}-M_{n-2}\|^p)$

$\displaystyle \le ...$

$\displaystyle \le s_p(X)^p \sum\mathop{\mathbb E}\|M_k-M_{k-1}\|^p,$

thus ${X}$ has martingale type ${p}$ with constant ${s_p(X)}$ .

Corollary 16 Let ${x_1,\ldots,x_n\in L_q}$ .

If ${q\le 2}$ , then

$\displaystyle \mathop{\mathbb E}\|\sum\epsilon_i x_i\|_q^q \le \sum\|_i\|_q^q, \quad (\text{Rademacher type }q)$

$\displaystyle \mathop{\mathbb E}\|\sum\epsilon_i x_i\|_q^2 \ge(q-1) \sum\|_i\|_2^q,\quad (\text{Rademacher cotype }2).$

If ${q\ge 2}$ , then

$\displaystyle \mathop{\mathbb E}\|\sum\epsilon_i x_i\|_q^2 \le (q-1)\sum\|_i\|_q^2, \quad (\text{Rademacher type }2)$

$\displaystyle \mathop{\mathbb E}\|\sum\epsilon_i x_i\|_q^q \ge \sum\|_i\|_q^q,\quad (\text{Rademacher cotype }q).$

Beware that ${L_1}$ has Rademacher cotype ${2}$ with constant ${\sqrt{2}}$ but no nontrivial martingale cotype.

3.3. Proof of Theorem 7, type and cotype range

Let ${T:\ell_p^n\rightarrow L_q}$ be an embedding of distorsion ${D}$ . Apply type ${q}$ of ${L_q}$ to ${x_i=Te_i}$ . Then

$\displaystyle \sum\|Te_i\|_q^q\le nD^q.$

On the other hand,

$\displaystyle \mathop{\mathbb E}\|T(\sum\epsilon_i e_i)\|_q^q \ge \mathop{\mathbb E}\|\sim\epsilon_ie_i\|_p^p=n^{q/p}.$

This yields ${D\ge n^{1/p - 1/q}}$ .

The proofs of the three other lower bounds on ${c_q^{lin}(\ell_p^n)}$ are similar. The results are sharp.

Note that the argument used linearity very strongly.

4. Nonlinear embeddings

Given metric spaces ${M}$ and ${N}$ , one can speak of the biLipschitz distorsion of ${M}$ into ${N}$ , denoted by ${c_N(M)}$ , as the least ${D}$ such that there exists ${s>0}$ and ${f:M\rightarrow N}$ satisfying

$\displaystyle \forall x,x'\in M,\quad s\,d(x,x')\le d(f(x),f(x'))\le Ds\,d(x,x').$

Now we discretize spaces. Let ${[m]_p^n=(\{1,2,\ldots,n\},\|.\|_p)}$ , viewed as an approximation of ${\ell_p^n}$ . What can one say about ${c_q([m]_p^n)}$ ?

Theorem 17 Let ${m,n\ge 2}$ . Then ${c_q([m]_p^n)}$ is of the order of

$\displaystyle 1 \text{ if }1\le q<p\le 2,\quad \text{ (Kadec)}$

$\displaystyle n^{1/p - 1/q} \text{ if }1\le p<q\le 2, \quad \text{ (metric type range)}$

$\displaystyle n^{1/p - 1/2} \text{ if }1\le p\le 2\le q,$

$\displaystyle ?? \text{ if }2\le q\le p,$

$\displaystyle n^{1/q - 1/p} \text{ if }1\le p<q\le 2,\quad \text{ (metric cotype range)}$

$\displaystyle n^{1/2 - 1/p} \text{ if }1\le p<q\le 2,$

$\displaystyle n^{\frac{(q-p)(p-2)}{p^2(q-2)}} \text{ if }1\le 2\le p<q,\quad (\text{metric }X_p \text{ range)}.$

So we see that a phase transition occurs when ${q\le 2\le p}$ .

In certain cases, the upper bounds are smart, but I will not focus on them.

In the unknown range ${2\le q\le p}$ , the best we know now is

$\displaystyle \min\{n^{1/q - 1/p},m^{1- q/p}\}\le_{p,q}c_q([m]_p^n)\le_{p,q}\min\{n^{1/q - 1/p},m^{1- 2/p}\}.$

The left-hand side would be sharp if the following was true: for every ${r>2}$ and every ${0<\theta<1}$ , the metric space ${(L_r,\|x-y\|_r^{\theta})}$ has a biLipschitz embedding into ${L_r}$ .

Theorem 18 (Bretagnolle – Dacunha-Castelle – Krivine 1965) This is true for ${r\le 2}$ .

In my thesis, we have the following result:

Theorem 19 (Eskenazis – Naor 2016) For ${r>2}$ , ${0<\theta<1}$ , ${0<\alpha<\theta}$ , the space

$\displaystyle (L_r,\frac{\|x-y\|_r^{\theta}}{1+\log^\alpha(1+\|x-y\|_r)})$

does not embed in ${L_r}$ .

5. Metric type

5.1. Enflo type

Definition 20 (Enflo 1969, modern terminology) A metric space ${M}$ has Enflo type ${p}$ with constant ${T>0}$ if for all ${n\in{\mathbb N}}$ , for all ${f:\{-1,1\}^n\rightarrow M}$ ,

$\displaystyle \mathop{\mathbb E} d(f(\epsilon),f(-\epsilon))^p\le T^p\sum_i \mathop{\mathbb E} d(f(\epsilon),f(\epsilon_1,\ldots,\epsilon_{i-1},-\epsilon_i,\epsilon_{i+1},\ldots,\epsilon_n))^p.$

Remark. If ${X}$ is a normed space and ${f(\epsilon)=\sum\epsilon_i x_i}$ , then the lefthand side is

$\displaystyle 2^p\mathop{\mathbb E}\|\sum\epsilon_i x_i\|^p$

and the righthand side is

$\displaystyle 2^p\sum\|x_i\|^p,$

so this is really a metric analogue of Rademacher type: for linear spaces,

$\displaystyle \text{Enflo type }p \Rightarrow \text{Rademacher type }p.$

Theorem 21 (Khot – Naor 2006) For normed linear spaces,

$\displaystyle \text{Martingale type }p \Rightarrow \text{Enflo type }p.$

Proof. Let ${f:\{-1,1\}^n\rightarrow X}$ which has martingale type ${p}$ . Define a martingale

$\displaystyle M_k(\epsilon)=\mathop{\mathbb E}_\epsilon f(\epsilon_1,\ldots,\epsilon_k,\delta_{k+1},\ldots,\delta_n).$

From martingale type, we know that ${\mathop{\mathbb E}\|f-\mathop{\mathbb E} f\|p\le T^p\sum_k\mathop{\mathbb E}\|M_k-M_{k+1}\|^p}$ . But

$\displaystyle M_k-M_{k+1}=\frac{1}{2}\mathop{\mathbb E}_\delta(f(\epsilon,\delta)-f(\epsilon',\delta)),$

where ${\epsilon'}$ has one sign changed. So

$\displaystyle \mathop{\mathbb E}\|M_k-M_{k+1}\|^p\le \frac{1}{2^p}\mathop{\mathbb E}_{\epsilon}\|f(\epsilon)-f(\epsilon')\|^p$

where ${\epsilon}$ has length ${n}$ and ${\epsilon'}$ one sign flipped at position ${k}$ . So summing over ${k}$ yields the expected righthand side. Now add and subtract the expectation,

$\displaystyle \mathop{\mathbb E}\|f(\epsilon)-f(-\epsilon)\|^p\le 2^{p-1}(\mathop{\mathbb E}\|f(\epsilon)-\mathop{\mathbb E} f\|^p +\mathop{\mathbb E}\|\mathop{\mathbb E} f-f(-\epsilon)\|^p)$

$\displaystyle \le 2^p\mathop{\mathbb E}\|f(\epsilon)-\mathop{\mathbb E} f\|^p\le 2^p\sum_k\mathop{\mathbb E}\|M_k-M_{k+1}\|^p$

$\displaystyle \le \sum_k\mathop{\mathbb E}_{\epsilon}\|f(\epsilon,\epsilon')\|^p,$

which completes the proof.

The converse is a recent breakthrough, still in the linear case:

Theorem 22 (Ivanisvili – von Handel – Volberg 2020) For linear spaces, Rademacher type implies Enflo type.

5.2. Proof of distorsion lower bounds for ${p\le 2}$

The upper bound is easy (use global embedding of ${L_p}$ ).

Here is another easy lower bound:

$\displaystyle c_q([m]_p^n)\ge c_q([2]_p^n)=c_q(\{-1,1\}^n,\|.\|_p).$

Assume that ${1\le p\le q\le 2}$ (the other case is similar). Let ${f:\{-1,1\}^n\rightarrow L_q}$ have distorsion ${D}$ with rescaling ${s}$ . Since ${L_q}$ has Enflo type ${q}$ with constant ${1}$ ,

$\displaystyle \mathop{\mathbb E}\|f(\epsilon)-f(-\epsilon)\|_q^q \le \sum\|f(\epsilon)-f(\epsilon')\|_q^q,$

(one sign change in ${\epsilon'}$ ), so the righthand side is bounded above by ${(sD)^q n 2^q}$ , and the lefthand side is bounded below by ${s^q\mathop{\mathbb E}\|\epsilon-(-\epsilon)\|_p^q=2^q s^q n^{q/p}}$ , so ${D\ge n^{1/p -1/q}}$ .

6. Metric cotype

Formally, Rademacher cotype is the reverse of Rademacher type, but a naive delinearization does not work. The cube does not suffice, one needs an extra scaling factor ${m}$ .

Definition 23 (Mendel – Naor 2008) A metric space ${M}$ has metric cotype ${q>0}$ with constant ${c>0}$ if for any ${n}$ , there exists ${m=m(n,M)}$ such that any function ${f:{\mathbb Z}_{2m}^n\rightarrow M}$ satisfies

$\displaystyle \sum_{t=1}^n \mathop{\mathbb E}_{x\in{\mathbb Z}_{2m}^n} d(f(x+me_i),f(x))^q \le C^q m^q \mathop{\mathbb E}_x\mathop{\mathbb E}_\epsilon d(f(x+\epsilon),f(x)).$

Remark. As soon as ${M}$ has at least 2 points, ${m(n,M)\ge n^{1/q}}$ .

Theorem 24 (Mendel – Naor 2008, Giladi – Mendel – Naor 2011) For normed spaces, Rademacher cotype is equivalent to metric cotype, and one can always take ${m\le n^{1+ 1/q}}$ .

The major open question is wether one can take ${m=n^{1/q}}$ . Indeed, this would have many geometric applications.

Theorem 25 (Eskenazis – Mendel – Naor 2019) For normed spaces, martingale cotype implies metric cotype, with ${m\le n^{1/q}}$ .

6.1. Proof of the remaining distorsion lower bounds in Theorem 18

I will cheat and identify ${{\mathbb Z}_{2m}^{n}}$ with ${[m]_p^n}$ . This is not a serious matter, since

$\displaystyle [m]_p^n \subset {\mathbb Z}_{m}^{n}\subset [m+1]_p^{2n}.$

We must show that

$\displaystyle c_q([m]_p^n)\ge_{p,q} m^{1-\frac{q}{p}} \quad \text{ if }m\le n^{1/q},$

$\displaystyle c_q([m]_p^n)\ge_{p,q} n^{\frac{1}{q}-\frac{1}{p}} \quad \text{ if }m\ge n^{1/q}.$

Since decreasing ${m}$ decreases the ${L^p}$ -distorsion, one can assume that ${m=n^{1/q}}$ . Let ${f:\rightarrow L_q}$ have distorsion ${D}$ . Then the lefthand side of metric cotype assumption is ${\ge s^qm^qn}$ , whereas the right hand side is ${\le m^q(sD)^q n^{q/p}}$ , so ${D\ge n^{\frac{1}{q}-\frac{1}{p}}}$ .

We see that we really need ${m=n^{1/q}}$ .

6.2. Proof of Theorem 26

The strategy is inspired from hypercontractivity: smoothing the space using an averaging operator puts functions closer to linear functions, i.e. leads us closer to the linear setting.

Given ${h:\{-1,1\}^n \rightarrow X}$ , we define a martingale

$\displaystyle E_i h(\epsilon)=\mathop{\mathbb E}_{\delta}h(\epsilon,\delta),$

where ${\epsilon}$ has length ${i}$ . We view ${E_i}$ as a smoothing operator.

If ${f:{\mathbb Z}_{4m}^n\rightarrow X}$ and ${x\in {\mathbb Z}_{4m}^n}$ , let ${f_x(\epsilon)=f(x+\epsilon)}$ . Then

$\displaystyle \sum_i \mathop{\mathbb E}\|f(x+me_i)-f(x)\|^p = \sum_i \mathop{\mathbb E}_{x,\epsilon}\|f_{x+2me_i}(\epsilon)-f(\epsilon)\|^p$

$\displaystyle =\sum_i \mathop{\mathbb E}_{x,\epsilon}(\|f_{x+2me_i}(\epsilon)-E_i f_{x+2me_i}(\epsilon)\|+\|E_i f_{x+2me_i}(\epsilon)-E_i f_x(\epsilon)\|+\|E_i f_{x}(\epsilon)-f_x(\epsilon)\|)^p$

$\displaystyle \le (1)+(2),$

where (1) is an approximation term,

$\displaystyle (1)=\sum_i \|E_i f_{x}(\epsilon)-f_x(\epsilon)\|^p,$

and (2) is a smoothing term,

$\displaystyle (2)=\sum_i \|E_i f_{x+2me_i}(\epsilon)-E_if_x(\epsilon)\|^p$

We first estimate (1) with Hölder,

$\displaystyle (1)\le \sum_i (\mathop{\mathbb E}_{x,\epsilon}\|E_if_x(\epsilon)-f(\epsilon)\|^p +\|f_x(\epsilon)-f(x)\|^p).$

Jensen’s inequality gives

$\displaystyle \mathop{\mathbb E}_\epsilon\|E_i h(\epsilon)\|^p\le \mathop{\mathbb E}\|h(\epsilon)\|^p.$

In other words, ${E_i}$ contracts ${L_p}$ , so

$\displaystyle (1)\le \sum_i \mathop{\mathbb E}_{x,\epsilon}\|f_x(\epsilon)-f(x)\|^p=n\mathop{\mathbb E}_{x,\epsilon}\|f_x(\epsilon)-f(x)\|^p,$

which is ok since ${n=m^p}$ , this what we want to see on the righthand side.

Next we bound (2). We can replace ${2m}$ with ${-2m}$ mod ${4m}$ . For fixed ${i}$ , write the summand in (2) as a telescopic sum,

$\displaystyle \mathop{\mathbb E}_{x,\epsilon}\|E_i f_{x-2me_i}(\epsilon)-E_if_x(\epsilon)\|^p \le \mathop{\mathbb E}_{x,\epsilon}\|\sum_\ell(E_i f_{x-2\ell e_i}(\epsilon)-E_if_{x-2(\ell-1)e_i}(\epsilon)\|^p$

$\displaystyle \le m^{p-1}\sum_\ell\mathop{\mathbb E}_{x,\epsilon}\|E_i f_{x+2\ell e_i}(\epsilon)-E_i f_{x-2(\ell-1)e_i}(\epsilon)\|^p$

$\displaystyle =m^p\mathop{\mathbb E}_{x,\epsilon}\|E_i f_{x+2 e_i}(\epsilon)-E_i f_{x}(\epsilon)\|^p$

But

$\displaystyle \|E_i f_{x+2e_i}(\epsilon)-E_i f_{x}(\epsilon)\|=2\|E_i f_x(\epsilon)-E_{i-1}f_x(\epsilon)\|^p.$

By martingale cotype,

$\displaystyle (2)\le m^p \mathop{\mathbb E}_{x,\epsilon}\|E_n f_x (\epsilon)-E_0 f_x(\epsilon)\|^p$

$\displaystyle =m^p\mathop{\mathbb E}_{x,\epsilon}\|f(x+\epsilon)-\mathop{\mathbb E}_\delta f(x+\delta)\|^p\le m^p\mathop{\mathbb E}_{x,\epsilon}\|f(x+\epsilon)-f(x)\|^p.$

Adding ${(1)+(2)}$ gives the expected bound ${m^p\mathop{\mathbb E}_{x,\epsilon}\|f(x+\epsilon)-f(x)\|^p}$ .

6.3. Final comment

What ${m=n^{1/q}}$ does for you is that it provides invariance under coarse embeddings, and not merely biLipschitz embeddings. So the story is far from being finished.

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Notes on Alexandros Eskenazis Rennes lectures january 26th 2023

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