Correlation and Brascamp-Lieb inequalities for Markov semigroups

TL;DR

This paper develops a unifying Markov semigroup framework for Brascamp-Lieb inequalities across different settings and explains the combinatorial reasons behind unexpected exponents in these inequalities.

Contribution

It introduces a general framework based on Markov generators and elucidates the combinatorial origins of unusual exponents in Brascamp-Lieb inequalities.

Findings

Unified Markov generator framework for inequalities

Explanation of unexpected exponents in inequalities

Extension beyond previous Euclidean and spherical settings

Abstract

This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres and symmetric groups). Our aim is twofold. Firstly, we provide a general, unifying, framework based on Markov generators, in order to cover a variety of examples of interest going beyond previous investigations. Secondly, we put forward the combinatorial reasons for which unexpected exponents occur in these inequalities.

Equations

$${\int }_H\prod _{i=1}^m{f}_i(B_{i}x{)}^{c_i}dx\le C\prod _{i=1}^m{\left({\int }_{H_i}{f}_i(y)dy\right)}^{c_i}$$

$${\int }_{{\mathbb{R}}^n}\prod _{i=1}^m{f}_i(P_{E_i}x{)}^{c_i}dx\le \prod _{i=1}^m{\left({\int }_{E_i}{f}_i\right)}^{c_i}.$$

$${\int }_{{\mathbb{S}}^{n-1}}\prod _{i=1}^n{f}_i(x_i)d\sigma (x)\le \prod _{i=1}^m{\left({\int }_{{\mathbb{S}}^{n-1}}{f}_i(x_i{)}^{2}d\sigma (x)\right)}^{\frac{1}{2}},$$

$$L(g\circ T_i)=\tilde{g}\circ T_i$$

$$L(g\circ T_i)=(L_{i}g)\circ T_i.$$

$${P_{t}\left(\prod_{i=1}^{m}f_{i}^{c_{i}}\circ T_{i}\right)\leq\prod_{i=1}^{m}\big{(}P_{t}(f_{i}\circ T_{i})\big{)}^{c_{i}}}$$

$$e^{-H}L(e^H)\le \sum _{i=1}^m{c}_i{e}^{-F_i}L(e^{F_i}).$$

$$P_{t}\bigg{(}\prod_{i=1}^{m}f_{i}^{c_{i}}\circ T_{i}\bigg{)}\leq\prod_{i=1}^{m}\big{(}P_{t}(f_{i}\circ T_{i})\big{)}^{c_{i}}.$$

$$\alpha(s)=P_{s}\bigg{(}\exp\bigg{(}\sum_{i=1}^{m}c_{i}\log P_{t-s}(f_{i}\circ T_{i})\bigg{)}\bigg{)},\quad 0\leq s\leq t.$$

$$\alpha^{\prime}(s)=P_{s}\bigg{(}L(e^{H})-e^{H}\sum_{i=1}^{m}c_{i}\,e^{-F_{i}}L(e^{F_{i}})\bigg{)}.$$

$$\int\prod_{i=1}^{m}f_{i}^{c_{i}}\circ T_{i}\,d\mu\leq\prod_{i=1}^{m}\bigg{(}\int f_{i}\circ T_{i}\,d\mu\bigg{)}^{c_{i}}.$$

$$\beta^{\prime}(t)=\int e^{H}\left(\sum_{i=1}^{m}c_{i}\,e^{-F_{i}}L(e^{F_{i}})\right)d\mu=-\int\bigg{(}L(e^{H})-e^{H}\sum_{i=1}^{m}c_{i}\,e^{-F_{i}}L(e^{F_{i}})\bigg{)}d\mu.$$

$$\sum_{i=1}^{m}c_{i}\,{\cal E}\big{(}e^{H-F_{i}},e^{F_{i}}\big{)}\leq 0.$$

$$\underset{t\to \infty }{\lim }{t}^{n/2}{P}_{t}f=\kappa \int fd\mu .$$

$$\int\prod_{i=1}^{m}f_{i}^{c_{i}}\circ T_{i}\,d\mu\leq\kappa^{-1}\prod_{i=1}^{m}\bigg{(}\kappa_{i}\int f_{i}\,d\mu_{i}\bigg{)}^{c_{i}}.$$

$$\Gamma (f,g)=\frac{1}{2}(L(fg)-fLg-gLf).$$

$$e^{-H}L(e^H)-\sum _{i=1}^m{c}_i{e}^{-F_i}L(e^{F_i})=\Gamma (H)-\sum _{i=1}^m{c}_i\Gamma (F_i).$$

$$\Gamma\bigg{(}\sum_{i=1}^{m}c_{i}\,f_{i}\circ T_{i}\bigg{)}=\sum_{i,j=1}^{m}c_{i}c_{j}\,\Gamma(f_{i}\circ T_{i},f_{j}\circ T_{j})\leq\sum_{i=1}^{m}c_{i}\,\Gamma(f_{i}\circ T_{i}).$$

$$\Big{|}\sum_{i=1}^{m}c_{i}\nabla(f_{i}\circ T_{i})(x)\Big{|}^{2}\leq\sum_{i=1}^{m}c_{i}\big{|}\nabla(f_{i}\circ T_{i})(x)\big{|}^{2}.$$

$$\mathcal{E}_{i}(x):=\big{\{}\nabla(f_{i}\circ T_{i})(x);f_{i}:E_{i}\to\mathbb{R}\big{\}}\subset T_{x}E.$$

$$\forall v_{i}\in\mathcal{E}_{i},\quad\Big{|}\sum_{i=1}^{m}c_{i}\,v_{i}\Big{|}^{2}\leq\sum_{i=1}^{m}c_{i}\,|v_{i}|^{2}\quad\Longleftrightarrow\quad\forall v\in\mathcal{E},\quad\displaystyle\sum_{i=1}^{m}c_{i}\,\big{|}P_{\mathcal{E}_{i}}v\big{|}^{2}\leq|v|^{2}$$

$$\sum _{i=1}^m{c}_i{P}_{{\mathcal{E}}_i(x)}\le {\mathrm{Id}}_{T_{x}E}.$$

$$\sum _{i=1}^m{c}_i{P}_{E_i}={\mathrm{Id}}_{{\mathbb{R}}^n}$$

$${\int }_{{\mathbb{R}}^n}\prod _{i=1}^m{f}_i(P_{E_i}x{)}^{c_i}{e}^{-|x{|}^2/2}\frac{dx}{(2\pi {)}^{n/2}}$$

$${\int }_{{\mathbb{R}}^n}\prod _{i=1}^m{g}_i(P_{E_i}x{)}^{c_i}dx\le \prod _{i=1}^m{\left({\int }_{E_i}{g}_i(y)dy\right)}^{c_i}.$$

$$\sum _{i=1}^m{c}_i{P}_{E_i}\le \text{mbox}{Id}_{{\mathbb{R}}^n}.$$

$${\int }_{{\mathbb{S}}^{n-1}}\prod _{i=1}^m{f}_i^{c_i/2}d\sigma \le \prod _{i=1}^m{\left({\int }_{{\mathbb{S}}^{n-1}}{f}_{i}d\sigma \right)}^{c_i/2}.$$

$$\Gamma (f)=|\nabla f{|}^2,$$

$$f(xg)=f(x),\forall g\in G_i,\forall x\in G.$$

$${\mathcal{E}}_i(x)=\{\nabla f(x);\text{ f:G→R is Gi-right-invariant}\}.$$

Full text

1. 1 Introduction

2. 2 The abstract argument: commuting maps and BL-condition

3. 3 Examples of diffusion semigroups

4. 3.1 Riemannian manifolds

5. 3.2 Riemannian Lie groups

6. 3.2.1 Calculations in $SO(n)$

7. 3.2.2 Passing to quotients

8. 3.3 Dirichlet distributions and their relatives

9. 4 Discrete models

10. 4.1 Abstract criterion

11. 4.2 Examples

12. 4.2.1 Homomorphisms of finitely generated groups

13. 4.2.2 Coordinates of the symmetric group

14. 4.2.3 Slices of the discrete cube and multivariate hypergeometric distributions

15. 4.2.4 Product spaces and Finner’s theorem

16. 5 Sums of squares

17. 6 Entropy of marginals

1 Introduction

A celebrated inequality of Brascamp and Lieb [8, 18] asserts that given linear surjective maps between Euclidean spaces $B_{i}:H\to H_{i}$, $i=1,\ldots,m,$ and given positive coefficients $(c_{i})_{i=1}^{m},$ the best constant $C$ such that for all non-negative measurable functions $f_{i}:H_{i}\to\mathbb{R}$ it holds

[TABLE]

can be computed by requiring the inequality on centered Gaussian functions only (i.e. of the form $f_{i}=e^{-Q_{i}}$ where $Q_{i}$ is a positive definite quadratic form). This far-reaching extension of Hölder’s inequality found applications in harmonic analysis but also in convex geometry. Indeed, a particular case called the geometric Brascamp-Lieb inequality, put forward by Ball [2] when $\mathrm{dim}(H_{i})=1$, leads to many precise volume estimates. The general geometric version corresponds to the case when for all $i=1,\ldots,m$, $B_{i}B_{i}^{*}=\mathrm{Id}_{H_{i}}$ and $\sum_{i}c_{i}B_{i}^{*}B_{i}=\mathrm{Id}_{H}$, where $B_{i}^{*}$ is the adjoint of $B_{i}$. Under these hypotheses the optimal constant in the Brascamp-Lieb inequality is $C=1$. More concretely: let $E_{1},\ldots,E_{m}$ be vector subspaces of $\mathbb{R}^{n}$ with its canonical Euclidean structure. Denoting by $P_{E_{i}}$ the orthogonal projection onto $E_{i}$, if $\sum_{i}c_{i}P_{E_{i}}=\mathrm{Id}_{\mathbb{R}^{n}}$ then for all measurable functions $f_{i}:\mathbb{R}_{i}\to\mathbb{R}^{+}$ it holds

[TABLE]

There exist by now many different proofs of the Brascamp-Lieb theorem: symmetrization when $\mathrm{dim}(H_{i})=1$ [8], study of Gaussian kernels [18], optimal transport [3]. Heat flow derivation were presented in the recent works [11] for $\mathrm{dim}(H_{i})=1$ and [7] in general: the geometric Brascamp-Lieb inequality is established by interpolating between the left and right hand side of the inequality, thanks to the Heat semigroup. The case when optimal Gaussian functions exist follows from the geometric case by a clever change of variables and turns out to be generic (the non-trivial remaining cases are in a sense “boundary” cases and can be decomposed into simpler ones). So the geometric case is also essential from a theoretical viewpoint. The Heat flow proofs required a more precise study of the structure of the problem, since the finiteness of the constant and the existence of Gaussian maximizers have to be treated beforehand. They lead to a complete treatment of the equality cases [11, 7, 20]. They were also flexible enough to adapt to other ambient spaces, as observed by Carlen, Lieb and Loss [11] who discovered the following Young type inequality on the Euclidean sphere $\mathbb{S}^{n-1}$: for all measurable functions $f_{i}:[-1,1]\to\mathbb{R}^{+}$, it holds

[TABLE]

where $\sigma$ is the uniform probability measure on $\mathbb{S}^{n-1}$. This inequality can be understood as a correlation inequality: the coordinates of a uniform random vector on the sphere are not independent, so there is no Fubini equality. Instead, inequality (1) holds and is a lot better than Hölder’s inequality, which would involve $L^{n}$-norms of the functions. In a sense, the exponent $2$, which turns out to be optimal, shows that the coordinate functions are not too far from being independent. The above inequality was extended to a spherical version of the geometric Brascamp-Lieb inequality in [5]. Carlen, Lieb and Loss also proved a similar inequality for the set of permutations of a finite set and coordinate functions [12].

In this paper we provide a general framework based on Markov generators which allows us to unify the existing results, derive extensions and clarify the conditions which are required to prove correlation inequalities. Decompositions of the identity as (10) play an important role. In the case of functions depending of blocks of coordinates, we put forward a general set of conditions, which is similar to the hypotheses of Finner’s theorem for product probability spaces [15], but applies to particular non-product spaces. See e.g. Propositions 10, 20 and Section 4.2.4.

The structure of the exposition is as follows. The abstract framework is described in Section 2 where a general condition is stated. The next sections provide concrete illustrations of Proposition 1. Section 3 deals with the case where our Markov generator is a diffusion, as it is the case in some classical geometric and probabilistic situations. In particular, we shall put forward the algebraic content of our condition in the case of Riemannian Lie groups (with emphasis on the orthogonal group $SO(n)$) and their quotients. We study discrete models and their combinatorics in Section 4, and the case where the generator is a sum of squares in Section 5. The final section is devoted to related – dual, more precisely – entropy inequalities for the marginals of a probability distribution. We state there an abstract superadditive inequality for the associated Fisher information, that leads to a somewhat different route to Brascamp-Lieb inequalities.

2 The abstract argument: commuting maps and BL-condition

The basic input is a measurable space $E$ and a Markov semigroup $(P_{t})_{t\geq 0}$ acting on functions on $E$, with generator $L$. We do not discuss here the various questions related to the underlying domain of $L$ and its associated carré du champ operator (see below) as well as the classes of functions under consideration. When a given inequality on functions is stated, it is always understood relatively to the suitable domains of $(P_{t})_{t\geq 0}$, $L$ or $\Gamma$. These are clear in all the continuous or discrete illustrations in this work. We refer to [1] for an introduction and further details in this respect and to [13] for the discrete setting.

The general framework of our study is the following. We introduce $m\geq 1$ measurable spaces $E_{i}$ and maps $T_{i}:E\to E_{i}$, $i=1,\ldots,m$. We assume that, for each $i=1,\ldots,m$, that the map $T_{i}$ commutes to $P_{t}$ or $L$ in the sense that for every $g:E_{i}\to\mathbb{R}$, $L(g\circ T_{i})$ factors through $T_{i}$:

[TABLE]

for some $\tilde{g}:E_{i}\to\mathbb{R}$. In other words, $L$ (or $P_{t}$) leaves invariant the algebra of functions on $E$ of the form $g\circ T_{i}$. This means that $P_{t}$ or $L$ may be projected on $E_{i}$ and there exists a Markov generator $L_{i}$ on $E_{i}$ such that

[TABLE]

We denote below by ${(P^{i}_{t})}_{t\geq 0}$ the semigroup with generator $L_{i}$. If follows that $P_{t}(g\circ T_{i})=(P^{i}_{t}g)\circ T_{i}$.

We aim at understanding how the “geometry” or the “combinatorics” of the $T_{i}$’s and of appropriate choice of constants $c_{i}>0$ ensure that

[TABLE]

for all $f_{i}:E_{i}\to\mathbb{R}^{+}$, $i=1,\ldots,m$. Since $(P_{t}(F^{1/c}))^{c}\leq(P_{t}(F^{1/d}))^{d}$ for $c\geq d>0$, we would like to pick the largest possible constants $c_{i}$’s. Also, for obvious reasons (pick all the $f_{i}$ but one to be identically $1$), the $c_{i}$’s will belong to $(0,1]$ and the inequalities we consider can be rewritten in terms of $L^{p_{i}}$-norms for $p_{i}=\frac{1}{c_{i}}$.

This problem is of course reminiscent of the Brascamp-Lieb convolution inequalities described in the introduction, and it can as well be interpreted as a correlation problem. This correlation problem has many ramifications, as we shall see.

We will, in this general framework, be dealing with inequalities which are valid for the measures $P_{t}(.)(x)$, uniformly on the point $x$. The following main equivalence is implicit in [11], [4].

Definition-Proposition 1**.**

Let $c_{i}$ be non-negative reals and $T_{i}:E\to E_{i}$ maps commuting with $L$, for $i=1,\ldots,m$. We say $\{c_{i},T_{i}\}$ satisfy the BL-condition if: For all functions $F_{i}:E\to\mathbb{R}$, $i=1,\ldots,m,$ of the form $F_{i}=g_{i}\circ T_{i}$, setting $H=\sum_{i=1}^{m}c_{i}F_{i}$, it holds

[TABLE]

Then, the following are equivalent:

• •

For all non-negative functions $f_{i}:E_{i}\to\mathbb{R}$, $i=1,\ldots,m$, and every $t\geq 0$,

[TABLE]

• •

The $\{c_{i},T_{i}\}$ satisfy the BL-condition

Proof.

Let $f_{i}:E_{i}\to\mathbb{R}$, $i=1,\ldots,m$, be bounded positive functions. Let $t\geq 0$ and consider

[TABLE]

Set $F_{i}=\log P_{t-s}(f_{i}\circ T_{i})$, $i=1,\ldots,m$, and $H=\sum_{i=1}^{m}c_{i}F_{i}$. Direct calculations give

[TABLE]

Next, by the commutation property (2), $F_{i}=\log P_{t-s}(f_{i}\circ T_{i})$ is a function of $T_{i}$ so that, under (3), $\alpha^{\prime}(s)\leq 0$ and thus $\alpha(0)\geq\alpha(t)$. Hence (4) follows from (3). The converse implication is obtained by differentiating (4) at $t=0$. ∎

Remark 2*.*

Given maps $T_{i}:E\to E_{i}$, $i=1,\ldots,m$, one may not always be able to check the BL-condition  (3). It might be necessary to consider further bijective maps $R:E\to E$, $R_{i}:E_{i}\to E_{i}$ and to deal with $\tilde{T}_{i}=R_{i}\circ T_{i}\circ R:E\to E_{i}$, $i=1,\ldots,m,$ (still assumed to commute with $P_{t}$) instead of $T_{i}$. This is exemplified by the paper [7] where the Gaussian-extremizable cases of the Euclidean Brascamp-Lieb inequality are reduced to the geometric Brascamp-Lieb inequality. Actually this change of variables is also implicit in [11] where the functions $f_{i}$ are evolving according to different semigroups.

It is usually of more interest to state Brascamp-Lieb type inequalities with respect to the invariant measure $\mu$ of the semigroup $(P_{t})_{t\geq 0}$. When $(P_{t})_{t\geq 0}$ is ergodic with invariant probability measure $\mu$, we may let $t\to\infty$ in the local inequality (4) and get inequalities of the type

[TABLE]

Actually this can be viewed directly by studying $\beta(t)=\int\prod_{i}P_{t}(f_{i}\circ T_{i})^{c_{i}}d\mu$. Indeed with the notation of the above proof

[TABLE]

Hence integrating from [math] to $\infty$, the BL-condition (3) yields (5). Note that the condition $\beta^{\prime}(t)\geq 0$ may be rewritten in terms of the Dirichlet form ${\cal E}(f,g):=\int f(-Lg)d\mu$ as

[TABLE]

Remark 3*.*

If $(P_{t})_{t\geq 0}$ has an infinite invariant measure $\mu$, more hypotheses are needed to get a meaningful limit to the local bounds as $t\to\infty$. Assume that $(P_{t})_{t\geq 0}$ is of dimension $n$, and size $\kappa>0$, in the sense that for every $\mu$-integrable function $f:E\to\mathbb{R}$, at any point,

[TABLE]

If the semigroups ${(P_{t}^{i})}_{t\geq 0}$ have invariant measures $\mu_{i}$, dimensions $n_{i}$ and sizes $\kappa_{i}$, $i=1,\ldots,m,$ and if in addition $\sum_{i=1}^{m}c_{i}n_{i}=n,$ we may use $P_{t}(f_{i}\circ T_{i})=P_{t}^{i}(f_{i})\circ T_{i}$ and let $t\to\infty$ in (4) to get

[TABLE]

3 Examples of diffusion semigroups

This section is devoted to several examples of illustration of the preceding abstract scheme in case the generator $L$ satisfies a chain rule formula. Recall that the carré du champ of the generator $L$ is defined on some suitable algebra of functions by

[TABLE]

For simplicity one writes $\Gamma(f)$ for $\Gamma(f,f)$. If $L$ is a diffusion generator (i.e. a linear differential operator of order 2 without constant term), then the chain rule yields L(e^{f})=e^{f}\big{(}Lf+\Gamma(f)\big{)}. So for $H=\sum_{i=1}^{m}c_{i}F_{i}$,

[TABLE]

Hence, we have:

Fact 4** (BL-condition in the diffusion case).**

If $L$ is a diffusion operator, then the BL-condition (3) is equivalent to saying that for every functions $f_{i}:E_{i}\to\mathbb{R}$, $i=1,\ldots,m$,

[TABLE]

Depending on the structure, this condition may be expressed more intrinsically in terms of the operators $T_{i}$. We investigate several instances below.

3.1 Riemannian manifolds

Let us assume that $E$ is a Riemannian manifold and that $\Gamma(f)=|\nabla f|^{2}$. This is in particular the case if $P_{t}$ is the Heat equation on $E$ associated to the Riemannian Laplacian $\Delta$. We also assume that the maps $T_{i}$ are differentiable. Then condition (7) amounts to the fact that for every $x\in E$, and for all smooth functions $f_{i}$,

[TABLE]

For each $x\in E$, we introduce the subspace of $T_{x}E$, the tangent space at $x$,

[TABLE]

This is the orthogonal of the kernel of $DT_{i}(x)$, so it is orthogonal to the tangent directions of the level set $\{y\in E;\;T_{i}(y)=T_{i}(x)\}$. We denote by $P_{\mathcal{E}_{i}(x)}$ the orthogonal projection on $\mathcal{E}_{i}(x)$ in the Euclidean space $T_{x}E$. We can reformulate (8) using the following well known equivalence, which relies on the fact that a linear map and its adjoint have the same norm: For $\mathcal{E}$ a Euclidean space, $\mathcal{E}_{i}$, $i=1,\ldots,m,$ Euclidean subspaces of $\mathcal{E}$ and $c_{1},\ldots,c_{m}>0$ we have:

[TABLE]

writing $P_{\mathcal{E}_{i}}$ for the orthogonal projection onto $\mathcal{E}_{i}$. More concisely, denoting the identity map by $\mathrm{Id}_{\mathcal{E}}$, the latter condition rewrites as an inequality between symmetric maps: $\sum_{i=1}^{m}c_{i}P_{\mathcal{E}_{i}}\leq\mathrm{Id}_{\mathcal{E}}.$

Therefore, we see that BL-condition amounts here to a “moving decomposition of the identity” inequality in all tangent spaces.

Fact 5** (BL-condition in the Riemannian case).**

In the setting described above, the BL-condition (3) is equivalent to saying that for all $x\in E$,

[TABLE]

Next, we present instances of such decompositions in the case of model spaces.

Geometric Brascamp-Lieb inequality in Euclidean space. In $\mathbb{R}^{n}$, let, for $i=1,\ldots,m$, $E_{i}$, be vector subspaces of dimension $n_{i}\geq 1$ and let $c_{i}\geq 0$, such that

[TABLE]

We take of course $T_{i}:\mathbb{R}^{n}\to E_{i}$ such that $T_{i}(x)=P_{E_{i}}x$, $x\in\mathbb{R}^{n}$, $i=1,\ldots,m$.

If $B$ is a linear map, $\nabla(f\circ B)(x)={{}^{t}}\!B\nabla f(Bx)$ and \Delta(f\circ B)(x)=\mbox{Tr}\big{(}{{}^{t}}B\mbox{Hess}f(Bx)B\big{)}. It is then clear that the generator $L=\Delta-x\cdot\nabla$ of the Ornstein-Uhlenbeck semigroup commutes with the $T_{i}^{\prime}s$. Also for all $x\in\mathbb{R}^{n}$, the spaces $\mathcal{E}_{i}(x)$ are simply $E_{i}$. Hence  (10) is guaranteed by the decomposition of the identity induced by the $E_{i}$’s. Thus, we get a Brascamp-Lieb inequality for the standard Gaussian measure, which is ergodic for the Ornstein-Uhlenbeck semigroup:

[TABLE]

Note that the decomposition of identity rewrites as $\sum_{i=1}^{m}c_{i}|P_{E_{i}}x|^{2}=|x|^{2}$, hence setting $g_{i}(y)=f_{i}(y)\exp(-|y|^{2}/2)$ and using the condition $n=\sum c_{i}n_{i}$ (take traces in the decomposition of the identity), we obtain the Euclidean inequality

[TABLE]

Alternatively we could have used the Heat semigroup (with generator $\Delta$) to get a local inequality and pass to the limit using the dimension of this semigroup, as explained in the Remark 3.

Geometric Brascamp-Lieb inequality on the sphere. The first inequality of this type was established by Carlen, Lieb and Loss [11] for coordinate functions on the sphere. It involves an unexpected exponent 2. A natural extension in the spirit of the latter Euclidean inequality was given in [5]. It reads as follows: If $x\in\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$ (the standard $(n-1)$-sphere), set as before $T_{i}(x)=P_{E_{i}}(x)$, $i=1,\ldots,m$, where $E_{i}\subset\mathbb{R}^{n}$ are subspaces for which we have

[TABLE]

Then, whenever $f_{i}$ are non-negative measurable functions on the sphere, such that $f_{i}$ depends only on $E_{i}$ (that is $f_{i}(x)=g_{i}(P_{E_{i}}(x)$), for the uniform probability measure $\sigma$ on $\mathbb{S}^{n-1}$ we have,

[TABLE]

It is easy to see that the Laplacian on $\mathbb{S}^{n-1}$ commutes to the operators $T_{i}$. The strategy in [5] is to derive decompositions of the identity in all tangent hyperplanes to the sphere, thus fulfilling Condition (10). Another approach based on analysis on the orthogonal group will be given next.

Hyperbolic space. It is natural to ask for an hyperbolic analogue of the previous statement. Let us explain, in two dimensions, why the method does not give any interesting correlation inequality. The natural functionals $T_{i}$ to consider are the Busemann functions (which basically are the coordinate in the direction of a point at infinity), they commute with the Laplace operator. In the disk model, choose $b_{1},\ldots,b_{m}$ on the unit circle and let $T_{i}$ be the corresponding Busemann functions. At a point $x$ in the disk the directions $\mathcal{E}_{i}(x)$ are simply the lines spanned by the gradients of the $T_{i}^{\prime}s$ (the tangent to the geodesic passing through $x$ and going to $b_{i}$). When $x$ tends to a point at infinity $b$ which is not one of the $b_{i}^{\prime}$s, it is clear that the lines $\mathcal{E}_{i}(x)$ become asymptotically parallel to the line $\mathbb{R}b$. Hence if a decomposition of the identity exists in all tangent planes we get that $\sum c_{i}\leq 1$. But in this case the decomposition (10) is trivial since $P_{\mathcal{E}_{i}(x)}\leq\mathrm{Id}$, and the inequality that we get is nothing else than Hölder’s inequality.

3.2 Riemannian Lie groups

In the case of Lie groups (and their quotients), the geometric structure required to have Brascamp-Lieb type inequalities is very clear and elegant.

The algebraic structure of the problem appears clearly when functions depending only on some variables are seen as functions invariant under the (right) action of subgroups of isometries. For instance, a function $f(x)$ on $\mathbb{R}^{n}$ is a function of $x_{1}$ if and only $f$ is invariant under all translation leaving $e_{1}=(1,0,\ldots,0)$ invariant. Note also that a function $f(x)$ on the sphere $S^{n-1}\subset\mathbb{R}^{n}$ is a function of $x_{1}$ if and only if $f$ is invariant under all rotations leaving $e_{1}$ invariant. In this section, we shall extensively use this point of view in the case of compact Riemannian Lie group.

Let $G$ be a connected compact Riemannian Lie group with unit element denoted by $e$. Let $\mathcal{G}=T_{e}M$ be associated Lie algebra ; by assumption, $\mathcal{G}$ is a Euclidean space. Let $\mu$ be the normalized bi-invariant Haar measure on $G$. Here we will work with the Laplace-Beltrami operator $\Delta$ as Markov generator, for which we indeed have that

[TABLE]

as required in the previous section.

Let $G_{i}$ be a connected Lie subgroup of $G$, with Lie algebra $\mathcal{G}_{i}\subset\mathcal{G}$. A function $f:G\to\mathbb{R}$ is said to be $G_{i}$-right-invariant if

[TABLE]

Equivalently $f$ is of the form $g\circ T_{i}$ where $T_{i}:G\to G/G_{i}$ is the canonical projection onto the right-quotient, defined by $T_{i}(x)=xG_{i}$. In other words, using notation (9), we are interested in the case where, for $x\in G$,

[TABLE]

If $f$ is $G_{i}$-right-invariant, then for all $v\in\mathcal{G}_{i}$ and all $t\in\mathbb{R}$,

[TABLE]

If $f$ is differentiable, we get that

[TABLE]

where $L_{x}:G\to G$ is the left-multiplication by $x$. Since $L_{x}$ is an isometry of $G$, its differential at $e$, $d(L_{x})_{e}$, is an isometry between the Euclidean spaces $T_{e}G=\mathcal{G}$ and $T_{x}G$. In particular, we will exploit the invariance property in the following form

[TABLE]

Roughly speaking, a $G_{i}$-right-invariant function $f$ “depends” only on $\mathcal{G}_{i}^{\perp}$ in the sense that the gradient $\nabla f(x)$ is in the direction $\mathcal{G}_{i}^{\perp}$ transported on $T_{x}M$:

[TABLE]

setting $\mathcal{E}_{i}:=\mathcal{G}_{i}^{\bot}$. With this formalism, the condition to have a Brascamp-Lieb inequality boils down to the existence of a decomposition of the identity in the Lie algebra:

Theorem 6**.**

Let $G$ be a connected compact Riemannian Lie group. Let $(G_{i})_{i=1}^{m}$ be connected Lie subgroups and let $\mathcal{E}_{i}:=\mathcal{G}_{i}^{\bot}$be the orthogonal complements in the Lie algebra $\mathcal{G}$ of $G$ of their Lie algebras $(\mathcal{G}_{i})_{i=1}^{m}$. Assume that for given $d_{1},\ldots,d_{m}>0$ the following inequality holds between symmetric linear maps of $\mathcal{G}$:

[TABLE]

Then the BL-condition (3) is satisfied. In particular, if for $i=1,\ldots,m$, $f_{i}:G\to\mathbb{R}^{+}$ is $G_{i}$-right-invariant, it holds

[TABLE]

Proof.

We consider the Heat kernel on $G$. The Laplace Beltrami operator commutes with right multiplication by the elements of the group so that the commutation relation is verified, in particular $P_{t}f_{i}$ is again $G_{i}$-invariant. Next let us check condition (3) in the form (8) put forward in the beginning of the Riemannian case. If for $i\leq n$, $h_{i}$ is a differentiable $G_{i}$-invariant function then, then, rewriting (11) as

[TABLE]

we get $P_{\mathcal{E}_{i}}d(L_{x^{-1}})_{e}\,\nabla h_{i}(x)=d(L_{x^{-1}})_{e}\,\nabla h_{i}(x)$. Using the fact that $d(L_{x^{-1}})_{e}$ is an isometry between $T_{x}M$ and $\mathcal{G}$ and the decomposition of the identity in $\mathcal{G}$, we see that

[TABLE]

The result follows. Equivalently, we could have said that the isometry $dL_{x}$ pushes forward the decomposition (12) from $\mathcal{G}=T_{e}G$ to the decomposition (10) on $T_{x}G$. ∎

3.2.1 Calculations in $SO(n)$

We consider subgroups related to the natural action of $SO(n)$ on $\mathbb{R}^{n}$ and study the relationship between decompositions of the identity of $\mathbb{R}^{n}$ and the ones induced on $\mathcal{A}_{n}=so(n)$, the set of antisymmetric $n\times n$ matrices which is the Lie algebra of $SO(n)$. The Euclidean structure on $\mathcal{A}_{n}$ is given by the Hilbert-Schmidt norm and the corresponding scalar product $\langle A,B\rangle=\mathrm{Tr}({{}^{t}}AB)=-\mathrm{Tr}(AB)$.

We will consider as before functions on $SO(n)$ which are right-invariant with respect to subgroups. There exists two natural subgroups associated to a subspace $E\subset\mathbb{R}^{n}$: $\mathrm{Fix}(E)$ and $\mathrm{Stab}(E)$.

Lemma 7**.**

Let $E$ be a vector subspace of $\mathbb{R}^{n}$. Consider the group

[TABLE]

and let $\mathcal{H}$ be its Lie algebra. We have $\mathcal{H}=\{A\in\mathcal{A}_{n};\ A_{|E}=0\}$ and if $P_{\mathcal{E}}:\mathcal{A}_{n}\to\mathcal{A}_{n}$ denotes the orthogonal projection onto $\mathcal{E}:=\mathcal{H}^{\bot}$, we have that

[TABLE]

Moreover a function $f:G\to\mathbb{R}$ is $H$-right-invariant means that $f(U)$ is actually a function of $U_{|E}$.

Proof.

The equality $\mathcal{H}=\{A\in\mathcal{A}_{n};\;A_{|E}=0\}$ is obvious. Let us check that the orthogonal projection of $A\in\mathcal{A}_{n}$ onto $\mathcal{H}$ is $P_{E^{\bot}}AP_{E^{\bot}}$. Indeed the latter is clearly antisymmetric and vanishes on vectors of $E$, so it belongs to $\mathcal{H}$. It remains to check the orthogonality condition: if $B\in\mathcal{H}$,

[TABLE]

Since $B$ vanishes on $E$, $B=B(P_{E}+P_{E^{\bot}})=BP_{E^{\bot}}$ and taking adjoints $P_{E^{\bot}}B=B$. It is then clear that $\mathrm{Tr}(BP_{E^{\bot}}AP_{E^{\bot}})=\mathrm{Tr}(BA)$. The orthogonality follows.

Since $\mathcal{E}=\mathcal{H}^{\bot}$ and denoting for shortness $P$ instead of $P_{E}$, and $I$ instead of $\mathrm{Id}_{\mathbb{R}^{n}}$, we have

[TABLE]

Eventually, since $P_{\mathcal{E}}$ is a self-adjoint involution

[TABLE]

The statement on $H$-right-invariant functions is easy. Such a function can be viewed as a function on $SO(n)/H\approx SO(n)/SO(E^{\bot})$ which can be identified to the Stieffel manifold of orthogonal frames of size $\mathrm{dim}(E)$ in $\mathbb{R}^{n}$. More explicitly, $U_{1}H=U_{2}H$ is equivalent to $U_{2}^{-1}U_{1}\in H$, that is for all $x\in E$, $U_{1}(x)=U_{2}(x)$. Hence the restriction of $U$ to $E$ characterizes the class of $U$ in the quotient. ∎

Lemma 8**.**

Let $E$ be a vector subspace of $\mathbb{R}^{n}$. Consider the group

[TABLE]

and let $\mathcal{H}$ be its Lie algebra. If $P_{\mathcal{E}}:\mathcal{A}_{n}\to\mathcal{A}_{n}$ denotes the orthogonal projection onto $\mathcal{H}^{\bot}$, it holds

[TABLE]

Moreover a function $f:G\to\mathbb{R}$ is $H$-right-invariant means that $f(U)$ is actually a function of $U(E)$.

Proof.

The argument is very similar to the one of the previous lemma. First note that

[TABLE]

For a $H$ right-invariant function $f$, $f(U)$ depends only on $UH$. Since $U_{1}H=U_{2}H$ is equivalent to $U_{1}(E)=U_{2}(E)$, the quantity $f(U)$ depends on $U(E)$. In other words $f$ factors through the Grassmann manifold of spaces of dimension $\mathrm{dim}(E)$ in $\mathbb{R}^{n}$.

One easily checks that $\mathcal{H}=\{A\in\mathcal{A}_{n};\;A(E)\subset E\mbox{ and }A(E^{\bot})\subset E^{\bot}\}$. The orthogonal projection for $A\in\mathcal{A}_{n}$ onto $\mathcal{H}$ is $P_{E}AP_{E}+P_{E^{\bot}}AP_{E^{\bot}}$. Indeed this is clearly an antisymmetric map for which $E$ and $E^{\bot}$ are stable. Moreover for $B\in\mathcal{H}$, it is clear that $B=P_{E}BP_{E}+P_{E^{\bot}}BP_{E^{\bot}}$. Hence

[TABLE]

Eventually, since $\mathcal{E}=\mathcal{H}^{\bot}$, $P_{\mathcal{E}}(A)=A-P_{E}AP_{E}+P_{E^{\bot}}AP_{E^{\bot}}$. So calculating as in the previous lemma, we have $P_{\mathcal{E}}(A)=PA+AP-2PAP$ and

[TABLE]

∎

The connection between decompositions of identity of $\mathbb{R}^{n}$ and of $\mathcal{A}_{n}$ is explained next.

Proposition 9**.**

For $i=1,\ldots,m$, let $c_{i}>0$, $E_{i}$ be a vector subspace of $\mathbb{R}^{n}$ and let $G_{i}$ be either $\mathrm{Fix}(E_{i})$ or $\mathrm{Stab}(E_{i})$. Denote by $\mathcal{E}_{i}=\mathcal{G}_{i}^{\perp}$ the orthogonal of $\mathcal{G}_{i}$ (the Lie algebra of $G_{i}$) in $\mathcal{A}_{n}$. We have

[TABLE]

As a consequence, if $\sum_{i=1}^{m}c_{i}P_{E_{i}}\leq\mathrm{Id}_{\mathbb{R}^{n}}$ then inequality (13) holds on $G=SO(n)$ (equipped with its uniform probability measure $\mu$) whenever each $f_{i}(U)$ is a function of $U(E_{i})$ or of $U_{|E_{i}}$, $i=1,\ldots,m$.

Proof.

By Lemma 7 and Lemma 8, for any $A\in\mathcal{A}_{n}$, $\|P_{\mathcal{E}_{i}}(A)\|^{2}\leq 2\|P_{E_{i}}A\|^{2}.$ Hence

[TABLE]

∎

Note that we have not used the full strength of Lemmata 7 and 8, since we have discarded the terms $\|P_{E_{i}}AP_{E_{i}}\|^{2}$. However, in the case where the $E_{i}$’s are one dimensional subspaces of $\mathbb{R}^{n}$, these terms vanishes, since in this particular case we have

[TABLE]

So, if $E_{i}=\mathbb{R}u_{i}$ where the $u_{i}$’s are norm $1$ vectors satisfying the decomposition of the identity

[TABLE]

where $u_{i}\otimes u_{i}=P_{E_{i}}$, then we have, with the notation of the Proposition,

[TABLE]

We do not loose in the passage to the Lie algebra. A particular case of interest is when $m=n$, $c_{1}=\ldots=c_{n}=1$ and $(u_{1},\ldots,u_{n})$ is an orthonormal basis of $\mathbb{R}^{n}$.

For higher dimensional $E_{i}$’s, it is possible, in some specific situations, to recombine the terms $\|P_{E_{i}}AP_{E_{i}}\|^{2}$ to recover a multiple of $\|A\|^{2}$ and to improve the exponents in the correlation inequality. This is easily seen for coordinate subspaces, i.e. spaces spanned by vectors of the canonical basis $(e_{1},\ldots,e_{n})$ of $\mathbb{R}^{n}$ (or of any given orthonormal basis, of course). The following proposition puts forward a typical set of conditions in order that BL-condition (3) is fulfilled. It will appear later in similar forms.

Proposition 10**.**

Let $\mathcal{I}$ be a collection of subsets of $\{1,\ldots,n\}$. Assume that it is written as a disjoint union $\mathcal{I}=\mathcal{I}_{1}\cup\mathcal{I}_{2}$. For each nonempty subset $I\in\mathcal{I}$, let $c_{I}\geq 0$, $E_{I}:=\mathrm{span}(e_{i};\;i\in I)$ and $f_{I}:SO(n)\to\mathbb{R}^{+}$ such that

• •

if $I\in\mathcal{I}_{1}$ then for all $U$, $f_{I}(U)$ only depends on $U_{|E_{I}}$,

• •

if $I\in\mathcal{I}_{2}$ then for all $U$, $f_{I}(U)$ only depends on $U(E_{I})$.

If for all $1\leq i,j\leq n$ with $i\neq j$ it holds:

[TABLE]

then BL-condition(3) is satisfied and in particular

[TABLE]

Proof.

Simply note that for $A=(a_{i,j})_{1\leq i,j\leq n}\in\mathcal{A}_{n}$, $\|P_{E_{I}}AP_{E_{I}}\|^{2}=\sum_{i,j\in I}a_{i,j}^{2}$ and

[TABLE]

Let us set $\lambda_{I}:=1$ if $I\in\mathcal{I}_{1}$, $\lambda_{I}:=2$ if $I\in\mathcal{I}_{2}$. Using Lemmata 7 and 8, and the antisymmetry of $A\in\mathcal{A}_{n}$, we have

[TABLE]

The latter is upper bounded by $\|A\|^{2}$ as soon as for all $i\neq j$,

[TABLE]

which is exactly our hypothesis on the coefficients $(c_{I})_{I\in\mathcal{I}}$. Hence $\sum_{I}c_{I}P_{\mathcal{E}_{I}}\leq\mathrm{Id}_{\mathcal{A}_{n}}$ and Theorem 6 yields the claim. ∎

Let us restate the previous result in the case we are looking to inequalities involving identical $c_{i}$’s.

Proposition 11**.**

Let $\mathcal{I}$ be a family of subsets of $\{1,\ldots,n\}$, and consider

[TABLE]

then for all non-negative functions $g_{I},h_{I}$ defined on suitable spaces,

[TABLE]

Let us put forward two particular cases of application of the previous result:

• •

Blocks of coordinates: if $\mathcal{I}$ is a non-trivial partition of $\{1,\ldots,n\}$ then each pair $\{i,j\}$ meets at most two sets in the family and we get $p=q=2$.

• •

Loomis-Whitney inequality: if $\mathcal{I}$ is the family of all subsets of $\{1,\ldots,n\}$ of size $k$, then any pair meets ${n\choose k}-{n-2\choose k}$ sets. Hence we have

[TABLE]

However the number of sets of cardinality $k$ which intersect a given pair in exactly one point is ${n\choose k}-{n-2\choose k}-{n-2\choose k-2}=2{n-2\choose k-1}$. So we get a smaller exponent

[TABLE]

It is worth noting that a direct application of Proposition 9 would have given worst estimates (when $k\geq 2$), in both cases. Indeed, if we denote by $P^{I}$ the projection onto a subspace spanned by $\{e_{i},\ i\in I\}$ for $I\subset\{1,\ldots n\}$, we have

[TABLE]

and therefore we would get exponent $p$ and $q$ equal to $2\frac{k}{n}{n\choose k}=2{n-1\choose k-1}$.

Remark 12*.*

On can take advantage of the terms $\|P_{E}AP_{A}\|^{2}$ in more general situations. They have to be rather symmetric though. Letting $2\leq k\leq n-1$, one instance is given by the family of all the spaces spanned by any $k$ vertices of a regular simplex in $\mathbb{R}^{n}$ with center of mass at the origin.

3.2.2 Passing to quotients

So far, we have taken advantage of right-invariances of the functions $f_{i}$. Plainly, similar results hold if all the functions are left-invariant instead. It would be very interesting to get better inequalities when the functions $f_{i}$ enjoy left and right invariances together (this would encompass functions on $SO(n)$ depending on matrices $U$ only through submatrices). Unfortunately, our approach does not give interesting general results in this direction (nothing better than what one gets by applying first Hölder’s inequality in order to get two integrals; each of these integrals is then upper-bounded by using only one-sided invariance). In the specific case when the functions have different right-invariances and a common left invariance, our results can be stated instead on the left-quotient. This is a way to get inequalities for homogeneous spaces corresponding to a compact Riemmanian Lie group.

Let us illustrate this remark for the sphere: if $E_{i}$ is a subspace of $\mathbb{R}^{n}$ and $f_{i}:\mathbb{S}^{n-1}\to\mathbb{R}^{+}$ is of the form $f_{i}(x)=g_{i}(P_{E_{i}}x)$, we may introduce $F_{i}:SO(n)\to\mathbb{R}^{+}$ defined by $F_{i}(U)=g_{i}(P_{E_{i}}{{}^{t}}Ue_{1})=g_{i}({{}^{t}}(UP_{E_{i}})e_{1}).$ Then $F_{i}$ is $\mathrm{Fix}(E_{i})$-right-invariant and also $\mathrm{Fix}(\mathbb{R}e_{1})$-left-invariant. Applying our results on $SO(n)$ and using the fact that the law of ${{}^{t}}Ue_{1}$ under the Haar probability measure on $SO(n)$ is the uniform distribution on the sphere recovers the main result of [5] which extends inequality (1): if $\sum_{i}c_{i}P_{E_{i}}\leq\mathrm{Id}_{\mathbb{R}^{n}}$ then

[TABLE]

Moreover, if $f:\mathbb{S}^{n-1}\to\mathbb{R}^{+}$ is of the form f(x)=g\big{(}|P_{E}x|\big{)}, then the function $F:SO(n)\to\mathbb{R}^{+}$ defined by F(U)=g\big{(}|P_{E}{{}^{t}}Ue_{1}|\big{)} is $\mathrm{Stab}(E_{i})$-right-invariant and $\mathrm{Fix}(\mathbb{R}e_{1})$-left-invariant. This allows us to transfer all of our $SO(n)$ results to the sphere.

Actually, a more general route is to note that BL-condition, in the form (12), passes to quotient.

Lemma 13**.**

Let $E$ be a Riemannian homogeneous space and $G$ a compact Riemannian Lie group of isometries acting transitively on $E$. Assume we are in the situation of Theorem 6. A function $f:E\to\mathbb{R}$ is said $G_{i}$-invariant if $f(g\cdot x)=f(x)$ for every $x\in E$ and $g\in G_{i}$. We can consider the associated $\tilde{T}_{i}:E\to E/G_{i}$ or more simply, with the notation (9),

[TABLE]

If condition (12) holds on $G$, then the BL-condition holds in $E$ in the equivalent form (10).

Proof.

Fix $x\in E$ and let $G_{x}=\{g\in G\,;\ g\cdot x=x\}$. Then, if we decompose the algebra $\mathcal{G}=T_{\mathrm{Id}}G$ (equipped with its Euclidean structure) as an orthonormal sum $\mathcal{G}=\mathcal{G}_{x}\oplus\mathcal{G}_{x}^{\perp}$ where $\mathcal{G}_{x}$ is the Lie algebra associated to $G_{x}$, we have that $\mathcal{G}_{x}^{\perp}$ is isometric to $T_{x}M$ by the isometry map

[TABLE]

We see that $\pi(\mathcal{G}_{i})\subset\mathcal{E}_{i}(x)^{\perp}$ and therefore $\mathcal{E}_{i}(x)\subset\pi(\mathcal{E}_{i})$. Note that $G_{x}\subset G_{i}$ and $\mathcal{E}_{i}\subset\mathcal{G}_{x}^{\perp}$. Since $P_{\pi(\mathcal{E}_{i})}=\pi P_{\mathcal{E}_{i}}~\pi^{-1}$, we get from (12) that

[TABLE]

∎

It is sometimes necessary to work directly on quotients, in particular for quotients of finite measure with a cover of infinite measure. We briefly discuss the example of the flat torus $(\mathbb{R}/\mathbb{Z})^{n}$. We consider for $i=1,\ldots,m$, rational vectors $u_{i}\in\mathbb{Q}^{n}$. For each $i$ let $\ell_{i}$ be the largest common divisor of the numbers $\langle u_{i},e_{1}\rangle,\ldots,\langle u_{i},e_{n}\rangle$. In order to define the map $x\mapsto\langle x,u_{i}\rangle$ on the torus, one has to identify $\langle u_{i},e_{k}\rangle$ to [math] for all $k$. This amounts to quotient $\mathbb{R}$ by $\sum_{k=1}^{m}\langle u_{i},e_{k}\rangle\mathbb{Z}=\ell_{i}\mathbb{Z}$. Let $T_{i}:(\mathbb{R}/\mathbb{Z})^{n}\to\mathbb{R}/\ell_{i}\mathbb{Z}$ be the map defined by $T_{i}(x)=\langle x,u_{i}\rangle\mod\ell_{i}$. One easily checks that the Laplacian commutes with $T_{i}$ (same calculation as in $\mathbb{R}^{n}$). Since for every $x$, $\nabla(f_{i}\circ T_{i})(x)$ is a multiple of $u_{i}$, if $\sum_{i=1}^{m}c_{i}u_{i}\otimes u_{i}\leq\mathrm{Id}_{\mathbb{R}^{n}}$ it follows that

[TABLE]

3.3 Dirichlet distributions and their relatives

For $x\in\mathbb{R}^{n}$, we set $S(x)=x_{1}+\cdots+x_{n}$. Let $\alpha\in(0,+\infty)^{n}$, then by definition the Dirichlet law $D_{n-1}(\alpha)$ is the distribution of

[TABLE]

where $X_{1},\ldots,X_{n}$ are independent random variables such that for each $i$, $X_{i}$ is $Gamma(\alpha_{i})$ distributed. More precisely it is supported on $T_{n-1}=\{y\in\mathbb{R}_{+}^{n-1};\;y_{1}+\cdots+y_{n-1}\leq 1\}$ and

[TABLE]

In order to get more symmetric results, we prefer to work with another representation: we consider the law $\widetilde{D}_{n-1}(\alpha)$ of

[TABLE]

It is supported on the regular simplex $\Delta_{n-1}=\{y\in\mathbb{R}_{+}^{n};\;y_{1}+\cdots+y_{n}=1\}$ and its density with respect to Lebesgue measure on $\Delta_{n-1}$ is proportional to $y\mapsto\prod_{i\leq n}y_{i}^{\alpha_{i}-1}$. Recall that some Dirichlet distributions are closely related to uniform spherical measures. Indeed if $G_{i}$ are independent variables with distribution $\exp(-t^{2})dt/\sqrt{\pi}$, then the uniform measure on $\mathbb{S}^{N}$ coincides with the law of

[TABLE]

Note that $G_{i}^{2}$ has distribution $Gamma(1/2)$. Write $N=k_{1}+\cdots+k_{n}$. It is then clear that the image of the uniform probability on $\mathbb{S}^{N-1}$ by the map

[TABLE]

is $\widetilde{D}_{n-1}(k_{1}/2,\ldots,k_{n}/2).$ This allows us to transfer some of our spherical results, but only to Dirichlet laws with half integer coefficients. In order to deal with general coefficients the following direct study is needed.

The measure $D_{n-1}(\alpha)$ is known (see [14, 19]) to be reversible and ergodic for the following Fleming-Viot operator

[TABLE]

In the symmetric representation associated to $\widetilde{D}_{n-1}(\alpha)$, it is natural to consider the operator $\widetilde{L}_{\alpha}$ defined for smooth functions $f:\mathbb{R}^{n}\to\mathbb{R}^{+}$ and for $x\in\Delta_{n-1}$ by

[TABLE]

It is not hard to check that $\widetilde{L}_{\alpha}f$ only depends on the restriction of $f$ to $\Delta_{n-1}$ (in the intrinsic formulation $\partial_{i}g$ is to be understood as $Dg\cdot P_{H}e_{i}=Dg\cdot(e_{i}-\mathbbm{1}/n)$, where $\mathbbm{1}=(1,\ldots,1)\in\mathbb{R}^{n}$ and $H=\mathbbm{1}^{\bot}$). However it is convenient to be able to apply $\widetilde{L}_{\alpha}f$ to functions $f$ defined on the whole space. For example if we write $f(y)=g(y_{1},\ldots,y_{n-1})$, $y\in\Delta_{n-1}$ then it is clear that $\widetilde{L}_{\alpha}f(y)=L_{\alpha}g(y_{1},\ldots,y_{n-1})$; hence the properties of $L_{\alpha}$ will pass to $\widetilde{L}_{\alpha}f$ (in particular $\widetilde{D}_{n-1}(\alpha)$ is reversible and ergodic for the semigroup generated by $\widetilde{L}_{\alpha}f$).

The carré du champ of $\widetilde{L}_{\alpha}$ can be expressed in the following convenient form, for $x\in\Delta_{n-1}$:

[TABLE]

where we have noted that $\Gamma(f)$ is actually a variance with respect to the probability measure $\sum x_{i}\delta_{i}$. The last formula comes from the representation $\mathrm{Var}(X)=\frac{1}{2}E((X-X^{\prime})^{2})$ where $X^{\prime}$ is an independent copy of $X$. We are ready to establish

Proposition 14**.**

Let $\mathcal{I}$ be a collection of subsets of $\{1,\ldots,n\}$. Assume that it is written as a disjoint union $\mathcal{I}=\mathcal{I}_{1}\cup\mathcal{I}_{2}$. For each nonempty subset $I\in\mathcal{I}$, let $c_{I}\geq 0$, and $f_{I}:\Delta_{n-1}\to\mathbb{R}^{+}$ such that

• •

if $I\in\mathcal{I}_{1}$ then for all $x$, $f_{I}(x)$ only depends on $(x_{k})_{k\in I}$,

• •

if $I\in\mathcal{I}_{2}$ then for all $x$, $f_{I}(x)$ only depends on $\sum_{k\in I}x_{k}$.

If for all $1\leq i,j\leq n$ with $i\neq j$ it holds:

[TABLE]

then the BL-condition (3) is satisfied and if $X$ is $\widetilde{D}_{n-1}(\alpha)$ distributed

[TABLE]

Proof.

First, we check the commutation relations. Since the coordinates play symmetric roles, we may assume that $I=\{1,\ldots,k\}$. Also we may extend our functions to $\mathbb{R}_{+}^{n}$. If for all $x$, $g(x)=f(x_{1}+\cdots+x_{n})$ it is obvious that

[TABLE]

It is then clear that $\widetilde{L}_{\alpha}g(x)$ is a function of $x_{1}+\cdots+x_{k}$. Similarly, if $g(x)=h(x_{1},\ldots,x_{k})$ then $\widetilde{L}_{\alpha}g(x)$ is a function of $(x_{i})_{i\leq k}$.

Next, we have to check the analogue of Condition (7), namely

[TABLE]

In view of the above expression of $\Gamma$, this amounts to show that for all $x\in\Delta_{n-1}$,

[TABLE]

Hence it is sufficient to show that for all $i\neq j$, it holds

[TABLE]

If $f_{I}(x)$ only depends on $(x_{k})_{k\in I}$ then $\partial_{i}f_{I}-\partial_{j}f_{I}=0$ if $\{i,j\}\cap I=\emptyset$. Moreover if $f_{I}(x)=g(\sum_{k\in I}x_{k})$ then $\partial_{i}f_{I}-\partial_{j}f_{I}=0$ also if $\{i,j\}\subset I$. Hence the summations on $I$ actually only involve the sets $I\in\mathcal{I}_{1}$ such that $\{i,j\}\cap I\neq\emptyset$ and the sets $I\in\mathcal{I}_{2}$ such that $\mathrm{card}(\{i,j\}\cap I)=1$. By hypothesis the sum of the corresponding coefficients $c_{I}$ is at most one, so that the required inequality is a mere consequence of the convexity of the square function. Hence Condition (7) holds true and we get the local inequality. By ergodicity the inequality passes to the measure $\widetilde{D}_{n-1}(\alpha)$. ∎

Let $p>0$. Let $B_{p}^{n}=\{x\in\mathbb{R}^{n};\;\sum_{i}|x_{i}|^{p}\leq 1\}$ be the unit ball for the $\ell_{p}$ norm on $\mathbb{R}^{n}$. On the corresponding unit sphere $\partial B_{p}^{n}=\{x\in\mathbb{R}^{n};\;\sum_{i}|x_{i}|^{p}=1\}$, one often considers the cone measure $\mu_{p}^{n}$ defined by $\mu_{p}^{n}(A)=\mathrm{Vol}_{n}([0,1].A)/\mathrm{Vol}_{n}(B_{p}^{n})$, $A\subset\partial B_{p}^{n}$. Here $[0,1]\cdot A$ is the intersection of $B_{p}^{n}$ with the cone of apex at the origin spanned by $A$.

Corollary 15**.**

Let $X$ be a random vector on $\mathbb{R}^{n}$. Assume that it is either uniformly distributed on $B_{p}^{n}$ or distributed according to the cone measure on $\partial B_{p}^{n}$. Then for all even functions $f_{i}:[-1,1]\to\mathbb{R}^{+}$

[TABLE]

Proof.

This is deduced from a particular case of the previous result on Dirichlet distributions, which ensures that for $Y$ distributed according to $\widetilde{D}_{n-1}(\alpha)$, and $g_{i}:[0,1]\to\mathbb{R}^{+},$ a similar inequality holds: E\prod g_{i}(Y_{i})\leq\prod\big{(}Eg_{i}^{2}(Y_{i})\big{)}^{1/2}. Indeed the uniform measure on $B_{p}^{n}$ and the cone measure on $\partial B_{p}^{n}$ can be viewed as symmetrized versions of the images of Dirichlet laws by maps of the form $T(x_{1},\ldots,x_{n})=(T_{1}(x_{1}),\ldots,T_{n}(x_{n}))$. Hence if we choose $g_{i}=f_{i}\circ T_{i}$ in the latter inequality, we get the claim. Let us make this strategy explicit in the case of the cone measure. Let $\varepsilon_{i}$, $G_{i}$, $i=1,\ldots,n,$ be independent random variables. Assume that $\varepsilon_{1}$ is uniform on $\{-1,1\}$ and $G_{i}$ distributed according to $e^{-t^{p}}dt/\Gamma(1+1/p)$. Then it is known that the vector

[TABLE]

is distributed according to the cone measure. Hence $|X_{i}|^{p}=G_{i}^{p}/(G_{1}^{p}+\cdots+G_{n}^{p})$ where $G_{i}^{p}$ is $Gamma(1/p)$-distributed. So applying the Brascamp-Lieb inequality for $f_{i}(x)=g_{i}(x_{i}^{1/p})$ yields the claim.

A similar approach is possible for the uniform distribution on $B_{p}^{n}$ thanks to the representation provided in [6]. ∎

Remark 16*.*

The cone measure on $\partial B_{2}^{n}$ is simply the uniform measure on $\mathbb{S}^{n-1}$, for which a similar inequality holds for general functions $f_{i}$ (i.e. it is not necessary to assume that they are even). Hence one way ask whether the symmetry assumption in the previous corollary is really needed. In order to remove it one would need a result for symmetrized Dirichlet laws, namely for measures on $\partial B_{1}^{n}$ with density with respect to Lebesgue measure proportional to $\prod_{i}|x_{i}|^{\alpha_{i}-1}$. At first sight, there does not seem to be any problem to extend our approach. However the ergodicity of these measures is a delicate issue. Indeed the fact that the density vanishes inside the domain may, in terms of the corresponding random process, create potential barriers that may not be crossed or potential wells into which the process may get stuck. On the technical level, the domain of the operator may be too small to contain enough non-symmetric functions.

Remark 17*.*

Proposition 14 and many results of this work, involve two kinds of functions which depend only on some coordinates $(x_{k})_{k\in I}$ (some depend on all these coordinates and some depend on them only through their sum). It is possible to consider more general dependences. We have not tried to reach the highest generality in this respect. Let us briefly mention a quite general extension of Proposition 14: we could consider functions $f_{I}$ where $I=(I_{1},\ldots,I_{K})$ is a collection of disjoint subsets of $\{1,\ldots,n\}$, such that $f_{I}(x)$ only depends on

[TABLE]

One can check that the map $T_{I}$ commutes with the Fleming-Viot operator (this uses the disjointness of $I_{1},\ldots,I_{K}$). If one considers now a collection of functions $(f_{I})_{I\in\mathcal{I}}$ and corresponding coefficients $(f_{I})_{I\in\mathcal{I}}$, then a Brascamp-Lieb inequality holds provided for all $i\neq j$ in $\{1,\ldots,n\}$, $\sum_{I\in A_{i,j}}c_{I}\leq 1$, where

[TABLE]

The proof follows the same arguments as the one of Proposition 14. We omit the details. Note that several results of this paper can be extended in an analogous way.

4 Discrete models

In this section, we deal with discrete models, and in particular we have to use the BL-condition in its brute form (3) since we are no longer working with diffusion generators. We nevertheless provide a simple criterion which can be worked out for a number of discrete models of interest.

4.1 Abstract criterion

Throughout this paragraph, $E$ will thus be a finite or countable state space. Let $K$ be a Markov kernel on $E$, that is, $K:E\times E\to[0,\infty)$ is such that for every $x\in E$, $\sum_{y\in E}K(x,y)=1$. If $f:E\to\mathbb{R}$ is bounded, set $Kf(x)=\sum_{y\in E}K(x,y)f(y)$, $x\in E$. As before, for given maps $T_{i}:E\to E_{i}$, $i=1,\ldots,m$, we say they commute with $K$ if for any function $f:E\to\mathbb{R}$, $K(f\circ T_{i})$ is a function of $T_{i}$. Again, this amounts to the existence of a Markov kernel $K_{i}$ on $E_{i}$ such that $K(f\circ T_{i})=K_{i}(f)\circ T_{i}$. This definition is of course equivalent to abstract one of §2 in terms of the associated Markov generator

[TABLE]

The next proposition provides a simple equivalent criterion for the BL-condition (3) in this context.

Proposition 18** (BL-condition in the discrete case).**

For distinct $x,y\in E$ such that $K(x,y)>0$, set

[TABLE]

Let $c_{i}\geq 0$, $i=1,\ldots,m$. Then the BL-condition (3) holds if and only if

[TABLE]

Therefore, under this condition, for every non-negative functions $f_{i}:E_{i}\to\mathbb{R}$, $i=1,\ldots,m$, and every $t\geq 0$,

[TABLE]

In particular, if $K$ has an ergodic invariant probability measure $\mu$ and if for all $x,y\in E$ distinct with $K(x,y)>0$, it holds ${\rm card}\,\{i=1,\ldots,m;\;T_{i}(x)\not=T_{i}(y)\}\leq p$, then choosing $c_{i}=\frac{1}{p}$, $i=1,\ldots,m$, we have that

[TABLE]

Proof.

At fixed $x\in E$, condition (3) may be written as

[TABLE]

The sums over $i$ on both sides only run over $i\in I_{x,y}$ so that the preceding inequality is equivalent to saying that

[TABLE]

where $\varphi(u)=e^{u}-1$. Since $\varphi(0)=0$, we can restrict the previous sum over $y\in E\setminus\{x\}$, and of course we can ask that $K(x,y)\neq 0$. Now, for fixed $x,y\in E$ with $x\neq y$ and $K(x,y)\neq 0$, we argue that the Condition (15) on the $c_{i}^{\prime}s$ combines with the convexity of $\varphi$ to give (pointwise) the desired inequality.

Conversely, if (16) holds for all choices of $f_{i}$, $i=1,\ldots,m,$ we choose $f_{i}(z)=\theta\mathbbm{1}_{z\neq T_{i}(x)}$ where $\theta\in\mathbb{R}^{+}$. Letting $\theta\to+\infty$ and comparing the orders of the terms in (16) shows that for each $y\neq x$ with $K(x,y)\neq>0$, we must have $\sum_{i\in I_{x,y}}c_{i}\leq 1$. ∎

Remark 19* (Extension to non-finite settings).*

The careful reader has probably noticed that the finiteness (or countability) of $E$ is not central in the argument. All the argument works as soon as we can express $L+I=:K$ in terms of a Markov kernel. Indeed, this allows us to reduce the problem to a pointwise inequality.

We next illustrate instances of the preceding result.

4.2 Examples

4.2.1 Homomorphisms of finitely generated groups

Let for example $G$, $G_{i}$, $i=1,\ldots,m$, be finite or countable groups and $T_{i}:G\to G_{i}$ be homomorphisms. Let $K$ be a Markov kernel on $G$. It is clear that each $T_{i}$ commutes with $K$.

Assume furthermore that $K$ is left-invariant in the sense that $K(gx,gy)=K(x,y)$ for all $x,y,g\in G$. We may let for example $G$ be finitely generated with generating set $S$, and $K(x,y)={\rm Card}\,(S)^{-1}{\rm 1}_{S}(y^{-1}x)$, $x,y\in G$. Then, condition (15) of Proposition 18 amounts to

[TABLE]

for every $z\in S$ where $I_{z}=\{i=1,\ldots,m;z\notin{\rm Ker}\,(T_{i})\}$.

4.2.2 Coordinates of the symmetric group

Let $E$ be the symmetric group $\mathcal{S}_{n}$ over $n$ elements $\{1,\ldots,n\}$, $n\geq 2$. This set is the discrete analogue of $SO(n)$. Unlike the continuous setting, there are several possible choices for the kernel $K$. However in view of the latter proposition, where each couple $(x,y)$ with $K(x,y)>0$ leads to a linear constraint on the exponents $c_{i}$, it is natural to take a small (or even minimal) generating set $S$ and to consider:

[TABLE]

We choose for $S$ the set of all transpositions. The following calculation will show that it is the best choice, since it minimizes the size of the support $\mathrm{supp}(\tau)=\{j;\;\tau(j)\neq j\}$.

The normalized counting measure $\mu$ is invariant for $K$. Actually $S$ being stable by inverse it is also reversible:

[TABLE]

Let $I$ be a subset of $\{1,\ldots,n\}$. We consider the map $T_{I}$ defined by

[TABLE]

Then $T_{I}$ commutes with $K$; indeed

[TABLE]

and $T_{I}(\tau x)=(\tau\circ x)_{|I}=\tau\circ x_{|I}$ depends only on $T_{I}(x)$. The result of Proposition 18 involves the condition $T_{I}(x)\neq T_{I}(y)$ for $K(x,y)>0$. Let us formulate it in a more concrete manner:

[TABLE]

Note that since the proposition involves this condition for all $x\in\mathcal{S}_{n}$, the set $x^{-1}(\mathrm{supp}(\tau))$ can be any set with the size of the support of $\tau$. Choosing transpositions then clearly appears as the most economical choice.

For $I\subset\{1,\ldots,n\}$ we may also consider the map $R_{I}$ defined by

[TABLE]

Then $R_{I}$ also commutes with $K$ and for any $x$ and any transposition $\tau$, $R_{I}(x)\neq R_{I}(\tau x)$ happens if and only if $\tau$ moves one point in $x(I)$ outside $x(I)$. Hence

[TABLE]

Combining these observations with Proposition 18 yields a discrete analogue to Proposition 10:

Proposition 20**.**

Let $\mathcal{I}$ be a collection of subsets of $\{1,\ldots,n\}$. Assume that it is written as a disjoint union $\mathcal{I}=\mathcal{I}_{1}\cup\mathcal{I}_{2}$. For each nonempty subset $I\in\mathcal{I}$, let $c_{I}\geq 0$ and $f_{I}:\mathcal{S}_{n}\to\mathbb{R}^{+}$ such that

• •

if $I\in\mathcal{I}_{1}$ then for all $x$, $f_{I}(x)$ only depends on $x_{|I}$,

• •

if $I\in\mathcal{I}_{2}$ then for all $x$, $f_{I}(x)$ only depends on $x(I)$.

If for all $1\leq i,j\leq n$ with $i\neq j$ it holds:

[TABLE]

then the BL-condition(3) is satisfied and

[TABLE]

The examples given after Proposition 10 transfer to $\mathcal{S}_{n}$. For a family $\mathcal{I}$ of subsets of $\{1,\ldots,n\}$, introduce the exponents:

[TABLE]

Then, for functions $g_{I}$ and $h_{I}$ defined on suitable sets, we have

[TABLE]

A particular case of interest (where these two cases coincide) is when \mathcal{I}=\big{\{}\{1\},\ldots,\{n\}\big{\}}. Then, $p=q=2$ and we recover the inequality on permanents given in [12].

4.2.3 Slices of the discrete cube and multivariate hypergeometric distributions

For $n\geq k\geq 0$, let

[TABLE]

equipped with uniform measure. These sets are discrete analogues of the sphere $S^{n-1}$. Two elements $x,y$ in $\Omega_{n,k}$ are neighbors if and only if they differ on exactly two coordinates, a relation written as $x\sim y$. Let $K$ be the nearest neighbor random walk on $\Omega_{n,k}$ (known as the Bernoulli-Laplace model) defined by

[TABLE]

It is easy to check that $Kf(x)$ only depends on the $i$’th coordinate $x_{i}$ of $x$ if this is the case for $f$. Indeed, the number of neighbors $y$ of $x$ such that $y_{i}=x_{i}$ is equal to $(k-x_{i})(n-1-k+x_{i})$, whereas when $y_{i}=1-x_{i}$, this number is equal to the number of coordinates $x_{j}$, $j\not=i$, such that $x_{j}=1-x_{i}$. For the coordinate maps $T_{i}(x)=x_{i}$, $1\leq i\leq n$, we are thus in the preceding setting of commuting operators so that Proposition 18 applies with $p=2$.

Alternatively one can use the following observation, which was pointed out to us by P. Caputo. The uniform probability measure on $\Omega_{n,k}$ is the image of the uniform probability measure on the permutation group $\mathcal{S}_{n}$ by the map $x\in\mathcal{S}_{n}\mapsto({\bf 1}_{x(i)\leq k})_{1\leq i\leq n}$. Consequently the correlation inequalities derived on $\mathcal{S}_{n}$ for functions depending on blocks of coordinates pass to $\Omega_{n,k}$ to yield the same result. Such a reasoning may be extended in order to encompass more general distributions. Consider integer numbers $K\leq M$ and $m=(m_{i})_{1\leq i\leq n}$ such that $\sum_{i}m_{i}=M$. The multivariate hypergeometric distribution $\mathcal{H}(m,K)$ is defined on $\mathbb{N}^{n}$ by

[TABLE]

if $k_{1}+\cdots+k_{n}=K$ and for all $i$, $k_{i}\leq m_{i}$ and $H(m,K)(\{(k_{1},\ldots,k_{n})\})=0$ otherwise. Given an urn containing $M$ balls of $n$ different colors, and more precisely $m_{i}$ of the $i^{\mathrm{th}}$ color, if one draws $K$ balls (uniformly) at random then the $n$-tuple $(X_{1},\ldots,X_{n})$