Brascamp–Lieb Inequalities on Compact Homogeneous Spaces

Abstract We provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. As an application we obtain sharp integral inequalities on the real unit sphere involving functions with some degree of symmetry.


Introduction
Many well-known multilinear inequalities commonly used in analysis, such as multilinear Hölder's inequality, Loomis-Whitney inequality and the sharp Young convolution inequality, can be seen as instances of a broader family of estimates: the so called Brascamp-Lieb inequalities. These are inequalities of the form where p j ∈ [ , ∞], B j : R n → R n j are linear surjective maps and the functions f j : R n j → R + are measurable, for j = , . . . , m. The constant C in (1.1) is the smallest constant, either nite or in nite, over all measurable inputs f j for which (1.1) holds. This constant depends on the maps B j and the exponents p j and is called the Brascamp-Lieb constant. These inequalities were extensively studied in the last years, starting from the works of Rogers [20] Brascamp, Lieb and Luttinger [7] and Brascamp and Lieb [6], where the authors studied the rank-one case, that is the case where n j = for all j, using rearrangement techniques. In particular they proved that the Brascamp-Lieb constant is the same if one restricts the inputs to Gaussians, a result known as Lieb's Theorem. This result was then extended to the higher rank case by Lieb in [15], then Barthe gave an alternative proof using transportation of mass techniques in [2].
Another approach to the problem was introduced by Carlen, Lieb, Loss who used heat ow methods to prove Lieb's Theorem in the rank one case in [9]. This approach was rediscovered independently and used by Bennett, Carbery, Christ and Tao to prove Lieb's Theorem in the general case in [5]. In particular they were able to prove the following theorem. for all subspaces V ⊆ R n , are satis ed.
The heat ow technique consists in studying the monotonicity properties of a certain function, depending on a nonnegative parameter that can be thought of as time, that is related to the heat evolution of some functions. Comparing this function at di erent times is a way of producing inequalities. For example in [5] the authors study, among other things, the case of the so called geometric Brascamp-Lieb inequality (already studied by Ball in [1] and Barthe in [2]), in which the linear maps B * j are isometries and the condition m j= p − j B * j B j = Id R n (1.4) holds. They show that for nonnegative Schwartz functions f j the quantity is nondecreasing, where u j (t, x) p j is the solution of the heat equation in R n with initial datum f p j j • B j . Inequality (1.1) is then obtained by comparing lim t→ Q(t) with lim t→∞ Q(t).
In this paper we will interpret inequality (1.1) in the following way: we are given a family of functions f j • B j , each one having some degree of symmetry (indeed, these functions are constant on the bers of the maps B j , that are a ne subspaces parallel to ker B j ) and we want to nd exponents p j for which the inequality holds with a nite constant C for all choices of functions. Theorem 1.1 gives a complete answer to this question in the Euclidean setting, relating the exponents p j to the geometry of the maps B j and to the scale invariant structure of R n .
An interesting issue is to extend inequality (1.1) to other settings. This problem was already addressed in the works [9,10] where some inequalities were obtained in the case of real spheres and of the permutation group on d elements S d (see also [3,4] for further results).
In particular in [9, Theorem 1.1] the authors proved that for nonnegative measurable functions f i on the unit sphere S n− of R n depending only on one variable x i (that can be seen as functionsf de ned on the interval [− , ] and pulled-back to the sphere by the projection on the i-th variable π i : S n− → [− , ]), the estimate S n− n j= f j (π j x)dσ ≤ n j= f j • π j L p (S n− ) (1.6) holds, with p ≥ , where dσ is the normalized uniform measure on the sphere. The authors also proved that the inequality is sharp, in the sense that there exist n functions in L p (S n− ) for p < , each depending on a di erent variable, for which the right-hand side of (1.6) is nite and the left-hand side diverges. Inequality (1.6) can be viewed in two ways: • it can be compared with Hölder's inequality, but in (1.6) the sum of the reciprocal of the exponents is bigger than one, a condition that cannot be achieved for general functions just by multilinear Hölder's inequality and continuous embeddings of Lebesgue spaces on the sphere, so that, in this sense, inequality (1.6) can be considered an improvement on Hölder's inequality; • as a Brascamp-Lieb type inequality, plugging in it the estimate f j • π j L p f j L p ([− , ]) .
The proof of inequality (1.6) is based on the heat ow method and relies on the fact that the sphere is a compact homogeneous space of Lie groups. Following ideas of [9] in Section 2 of this paper we nd inequalities similar to (1.6) in the setting of general compact homogeneous spaces of type M = K\G, where G is a connected, unimodular Lie group and K is a closed subgroup of G. We endow M with the unique normalized measure dµ induced by the Haar measure on G. We x a nite set of vector elds I in the Lie algebra of left invariant vector elds g of G satisfying Hörmander's bracket generating condition and we construct the sum of squares sub-Laplacian L, which is a symmetric, negative, essentially self-adjoint, hypoelliptic operator acting on smooth functions de ned on G and on its quotient M. By means of the heat semigroup {e tL } t> , we introduce the nonlinear heat ow v(t, x) = e tL f p /p , where p ≥ and f ∈ C ∞ (M), which is the solution of the nonlinear equation where ∇ is the gradient with respect to the vector elds in I.
Taking m di erent nonnegative functions f i ∈ C ∞ (M) and considering their nonlinear evolutions v i (t, x) one can prove that the function is nondecreasing for p ≥ m. By a comparison between lim t→ ϕ(t) and lim t→∞ ϕ(t), it will then follow that The relevant symmetries in our analysis are those that can be described by means of subsets A of I consisting of vector elds that commute with L. We call a subset A of I maximal if A ∩ I = A, where A is the smallest Lie subalgebra of g containing A. We say that a function f ∈ C ∞ (M) is A-symmetric if Xf = for all vector elds X in A. Functions that are A-symmetric are constant on certain nonintersecting submanifolds that cover the manifold M. The commutation property with L of the vector elds in A ensures that the symmetry is preserved by the heat ow.
The main result of this paper, expressed in Theorem 3.8, says that taking m functions, each A i -symmetric for some maximal subset A i of I, inequality (1.8) holds for p greater than or equal to a criticalp that depends only on the combinatorics of the sets A i and to prove this we use the fact that for p ≥p the function ϕ in (1.7) is nondecreasing. In Theorem 3.10 we obtain an analog of inequality (1.8), but with a di erent exponent p i for each function f i on the right-hand side. In this case we use the fact that the function is nondecreasing if each p i is greater than or equal to a criticalp i that again depends on the combinatorics of the sets A i . As a rst application of this machinery, in Section 4 we study the (abelian) case of the torus T n = R n /Z n . By means of Theorem 3.8 we are able to recover a result by Calderón in [8] and a family of local geometric Brascamp-Lieb inequalities associated to projections to collections of coordinate variables.
In Section 5 we apply the results of Section 2 to the case of the sphere in R n . We recover the result of [9] for functions depending on one variable and extend it to functions depending on k variables, for ≤ k ≤ n − .
A function f of k variables can be understood as a functionf de ned on the k-dimensional unit ball B k of R k and pulled-back to the sphere by the projection π : S n− → B k on the k variables involved.
Let C(n, k) = n k . We prove that if f , . . . , f C(n,k) are nonnegative measurable functions, each depending on a di erent collection of k variables, denoted with xω i , where ω i ⊂ { , . . . , n}, |ω i | = k, the inequality Moreover we prove that this inequality is sharp in the sense of [9]. Since for a function f depending on k variables, , we can interpret (1.10) as a Brascamp-Lieb type inequality.
If we add an additional symmetry to the functions, requiring that each function depends radially on k variables we prove an (again sharp) improvement of inequality (1.10), obtaining a lower critical exponent This case is studied in Section 6. Inequality (1.10) is rst proved for a small range of exponents that is then extended by interpolation. In Section 6 we also study in what range of exponents the inequality can hold, obtaining some optimal result, and provide other examples.

Homogeneous spaces
For an extensive overview of the setting of homogeneous spaces of Lie groups see [13].
Let G be a connected, unimodular Lie group, with bi-invariant Haar measure dµ and let K be a closed subgroup so that the homogeneous space M = K\G is compact and has no boundary. Denote by π : G → K\G the canonical projection. Recall (see [13,  If dµ is the Haar measure on G we have a unique (up to scalars) bi-invariant measure on K\G (cfr. [13, Theorem 1.7, Ch. X]), which we will still denote by dµ, de ned as the push-forward of dµ by means of the projection π. We assume that this measure is normalized, i.e. dµ(M) = . We denote with τg the left translation on G by the element g and by abuse of notation we will also write τg f for f • τg for functions f on G. Recall that a left invariant vector eld is a rst order di erential operator X that commutes with all left translation, i.e. such that for all g ∈ G and f ∈ C ∞ (G). Let g be the Lie algebra of the Lie group G. As usual, we identify X ∈ g with the corresponding left invariant vector eld on G given by the Lie derivative where exp : g → G denotes the exponential mapping and f ∈ C ∞ (G).
There is a one-to-one correspondence between smooth real-valued functions f on the quotient space M = K\G and smooth functionsf on G that are constant on coset spaces, i.e. Left invariant di erential operators act on smooth functions on M via the pushforward of the map π (that we denote with Tπ). We write D instead of Tπ(D) for di erential operators in the universal enveloping algebra U(g) of g acting on C ∞ (M). Note that the integration by parts formula holds on M for X ∈ g and f , g ∈ C ∞ (M). The boundary terms are absent due to the compactness of the quotient.
De nition 2.1. We say that I is a Hörmander system if I = g, where I is the smallest Lie subalgebra containing I.
We now de ne some di erential operators on M = K\G adapted to a Hörmander system I. First of all we de ne a gradient for F ∈ C ∞ (M) l . Finally we de ne a sum of squares operator, which we will sometimes refer to as sub-Laplacian, for f ∈ C ∞ (M). Consider the Hilbert space L (M) with respect to the measure dµ with scalar product ·, · . The operator L is initially de ned in the subspace C ∞ (M), which is dense in L (M) (recall that M is compact). It is immediate to check that the operator −L is symmetric and positive. Since the vector elds in I satisfy Condition 2.1 the operator L is hypoelliptic by Hörmander's theorem. By Nelson's theorem (see [18]) we conclude that the operator L is essentially self-adjoint. Moreover, since M is compact −L has a real discrete nonnegative spectrum Σ ⊂ R + with eigenvalues, counted with multiplicity, with λ k → ∞ as k → ∞. The associated L -normalized eigenfunctions φ i form a complete orthonormal system for L (M). Since the operator L is hypoelliptic, the eigenfunctions are C ∞ (M) and in particular they are bounded. Note that λ has multiplicity 1 and φ = .
The spectral theorem provides a functional calculus for the operator −L, if m ∈ L ∞ (Σ), the operator m(−L) de ned by

. The heat flow
We consider the Cauchy problem for the heat equation on M associated to L with initial datum f , It is known (see [22]) that for all t > the solution at time t of the heat equation with initial datum f ∈ C ∞ (M) is obtained by applying the heat semigroup e tL , which is given by the multiplier e −t(·) : Σ → R + . Explicitly we have Fix p ≥ and consider the evolution for a nonnegative f ∈ C ∞ (M) given by We see that when p > , v(t, x) is a nonlinear evolution since it satis es the nonlinear equation The solutions to Problem (2.4) and its nonlinear version (2.6) enjoy the following properties. Proof. Properties (1) and (2) follow from Hunt's theorem (see [14]) for the group G and pass to the quotient M (see [17,Section 2.5]). Property (3) is obvious. Property (4) follows from the fact that is an eigenvalue for L with constant eigenfunction φ = and that (2.6) converges to (

. A monotonicity result
If the function (2.9) is nondecreasing, the following inequality holds: On the left-hand side we obviously obtain the integral of the product of the initial data. For the right-hand side, by Property (4) of Proposition 2.2, each v i (t, x) converges to f i L p (M) and the result follows since M dµ = .
Remark 2.4. Since the space M is compact, the best constant in inequality (2.10) is 1 and is attained by constant functions, since for a ∈ R + , a L p (M) = a for ≤ p ≤ ∞.

Remark 2.5.
At rst sight the estimate (2.10) looks like Hölder's inequality. Note, however, that in (2.10) there are in general no constraints on the exponent p. In particular m i= p − need not be . In fact, proving the monotonicity of ϕ under certain assumptions that will be made clear later, we will get exponents that do not satisfy Hölder's condition.
We rst nd an explicit formula for the time derivative of ϕ(t). Note that ϕ is di erentiable in time. Since by Property (1) and Proposition 2.6. Under the assumptions above we have Proof. By the Leibniz rule and (2.7) we have We split each integral in the sum into two pieces: For I i (t) we have For II i (t), integrating by parts, we obtain: which, using again the Leibniz rule, gives Finally, taking the sum in i we obtain the result.
Remark 2.7. By manipulating the sums, the time derivative of ϕ can be equivalently written as (2.14) We observe that this expression contains all possible square type terms (X jṽi ) and all possible double products X jṽi X jṽk for j = , . . . , l and i, k = , . . . , m, with i < k. Hence, if there are enough square type terms, i.e. if p is big enough, it is possible to group the terms up in order to get a sum of nonnegative squares of binomials and (possibly) other nonnegative summands.
One could allow each nonnegative f i ∈ C ∞ (M) to evolve with a di erent nonlinear evolution. Indeed, one could choose a di erent p i ≥ for each f i and de ne In this regard, we state a simple modi cation of Proposition 2.6.

Proposition 2.8. In the hypotheses above we have
Proof. The proof is the same as for Proposition 2.6, once noticed that each v i (t, x) solves the equation As a simple consequence of Proposition 2.8 one may obtain multilinear Hölder's inequality for a restricted range of exponents. Indeed, taking p i ≥ m, the time derivative of function (2.9) can be arranged in the form since both summands are nonnegative. Hence by the monotonicity of ϕ the inequality holds.
We conclude the section with the following de nition, which will be useful in what follows.
De nition 2.9. Let f , . . . , fm be nonnegative measurable functions and p i ≥ for i = , . . . , m. We say that the inequality if it does not follow directly from Hölder's inequality and continuous embeddings of Lebesgue spaces.

Inequalities for functions with symmetries
As the proof of multilinear Hölder's inequality suggests, the choice of the exponent p depends in a combinatorial fashion on the number of vector elds and on the number of functions. Multilinear Hölder's inequality from this point of view represents the worst case, in which one considers all vector elds of the family I applied to all functions. In what follows we will investigate the cases where some of the functions are annihilated by a subset of the vector elds in the family I.
which is also an algebra with respect to pointwise multiplication.

Remark 3.2.
Since if Xf = and Yf = , then also [X, Y]f = , we see that if a function is A-symmetric, then it is also A -symmetric. In particular, if A and A are subsets of g such that A = A , we have that Functions that are A-symmetric enjoy invariance properties on the group G, and hence on the manifold M.
Next we de ne a class of sets that will be useful to describe the symmetries we are interested in. Since if X ∈ g commutes with L then e tL X = Xe tL for all t > , we introduce the following notion.
De nition 3.5. Let A ⊂ g and I be a Hörmander system. We say that A is an I-set if L commutes with all the elements in A, i.e. [L, X] = LX − XL = .
We are interested in functions that are A-symmetric, with A some I-set. For these functions we have the following proposition.
Proof. The case p = is immediate. For the case p > it su ces to notice that, if X ∈ A, . Thus the heat ow preserves the symmetry: if A is an I-set and if the initial datum is A-symmetric, then so is its evolution, either linear or nonlinear, under the heat equation.
Given a subalgebra A ⊆ g we can consider the vector space of functions C ∞ A (M) that are annihilated by all vector elds in A. The Lie algebra g has a (nonunique) direct sum decomposition as a vector space where B is a vector subspace of g.
By Proposition 2.6, for the task of proving the inequalities we are interested in, it su ces to take into account only the action of vector elds in the system I. So we can only consider I ∩ A or we may as well consider subalgebras generated by subsets of vectors in I. Di erent subsets of I could generate the same subalgebra and we will not distinguish them. This leads us to the following de nition.

De nition 3.7. Let
In other words a maximal subset A of I contains all possible brackets and linear combinations of its elements that are still in I. For example, if we take the Hörmander system {e , . . . , en , e + e } in the abelian Obviously the length of the multi-index, denoted by |j|, gives the number of sets we are considering in the intersection.
The exponentp has a combinatorial nature and is related to the way the vector elds L i,j are distributed among the sets A i . In the next sections we will give several examples in which this quantity is easily computable. It is interesting to notice that the exponentp found by this method is always an integer.
Proof. Since the functions f i are A i -symmetric, and the sets A i are I-sets, by Proposition . the nonlinear evolutions, de ned in (2.8), are also A i -symmetric. By Proposition 2.6 all possible double products of the form X iṽj X iṽk , with i < k and X i ∈ A c j ∩ A c k , will appear in the time derivative of ϕ. Recall thatp − depends on how many square type elements are needed to complete the squares. In order to have positive derivative, we will needp − to be at least as big as the number of occurrences of the most recurrent vector eld among the A c i .
The following su cient condition to obtain a nontrivial inequality in the sense of De nition 2.9 can be easily proved.

The abelian case
As a rst example, in this section we analyze the inequalities discussed in the previous section when the Lie group is (R n , +). We x an orthonormal basis {e , . . . , en} of R n and consider the corresponding Cartesian coordinates (x , . . . , xn). We take the quotient by the discrete subgroup Z n , where T n is the n-dimensional torus, which can be understood as the cube [ , ] n in R n with identi cations of opposite sides. The Lie algebra of R n is generated by the vector elds X i = ∂x i for i = , . . . , n. Clearly [X i , X j ] = for all i, j = , . . . , n. In this setting a Hörmander system of vector elds must necessarily contain a basis for the Lie algebra, since all commutators are trivial. So let with l ≥ n and let {Y , . . . , Yn} be a basis for the Lie algebra. In this abelian setting all subsets A ⊆ I are I-sets, since every two vectors commute. So we can pick any subset of I and we have the following proposition.
Proof. We know that A ∩ I is maximal. Since g is abelian, vectors in A ∩ I that are not in A are vectors of I that are linearly dependent from the vectors in A. Condition (4.1) ensures that A already contains these vectors.
Let us treat the case l = n, i.e. when I is a basis for g. In this case all subsets of I are maximal, so we have n possible maximal subsets to which we can apply Theorem 3.8. If A is any subset, the vector space sum decomposition g = A ⊕ A c is also a Lie subalgebras decomposition, meaning that A , A c = { } . All subsets have maximal complement and we can directly consider the complements of the annihilating sets. Let where π : T n → T n−s is the linear projection π(x , . . . , xn) = (x s+ , . . . , xn).
We follow the notation of [8], denoting with ω nite subsets of { , . . . , n} and with xω the set of variables We denote with fω a function only depending on xω. Note that for all p ≥ . Let C(n, k) := n k . We have the following proposition. Proof. In the language developed in this section, the sets A c i are given by {X i , . . . , X i k } and they are in correspondence with the collection of variables xω i , where ω i = {i , . . . , i k }. By Theorem 3.8 it su ces to compute the number of occurrences of the most recurrent element among the A c i , or, equivalently, the most recurrent variable x l among the collections xω i . It is easy to see that in this case every variable x l appears exactly n− k− times. Remark 4.3. Proposition 4.2 is a local version of a result due to A. P. Calderón in [8] (see also the work of H. Finner [11] for further results). In the notation above, Calderón proved the inequality which by (4.2) is equivalent to (4.3). The case k = n − is a local version of Loomis-Whitney inequality (see [16]).

All the estimates above (Calderón inequalities, Loomis-Whitney inequality and their local versions) can be proved by a smart iteration of Hölder's inequality.
Another way of proving this kind of inequalities is the heat ow method used in [5]. Recall that a geometric Brascamp-Lieb inequality is an estimate of the type where B i : R n → R n i are surjective linear maps such that B * i is an isometry, i.e. B i B * i = Id R n i , f i : R n i → R + are measurable functions, and the relation is satis ed. In [5] the authors prove that under condition (4.5), inequality (4.4) holds with C = . Restricting the supports of the functions to unit cubes in R n i this local version of the inequality obviously holds  C(n, k). We note that the general condition (4.5) gives rise to exponents that are not covered by Proposition 4.2.

The case of the sphere
In this section we will nd inequalities for functions enjoying some symmetries on spheres. We consider the Euclidean space R n , for n ≥ , with the standard scalar product ·, · and the induced norm |·|. Let {e , . . . , en} be an orthonormal basis and (x , . . . , xn) the associated coordinates. The (n − )-dimensional unit sphere is the set S n− = {x ∈ R n : x + · · · + x n = }, which we endow with the normalized uniform measure dσ. The sphere S n− can be seen as a homogeneous space of the special orthogonal group where SO(n − ) is thought of as a closed subgroup of SO(n) xing one direction. The measure dσ is, up to normalization, the push-forward through the projection map on the quotient S n− of the bi-invariant Haar measure on SO(n).

. Functions depending on k variables
In what follows we will use cartesian coordinates to describe points on the sphere. In particular we will often write f (x , . . . , xn) for functions f : S n− → R, implicitly assuming the condition x + · · · + x n = .
Consider the projection πω : S n− → R k that maps (x , . . . , xn) → (x i , . . . , x i k ). The image of the map πω is the closed unit ball B k in R k .
De nition 5.1. We say that a function f : S n− → R depends on k variables, for ≤ k ≤ n − if there exists a functionf : B k → R such that f =f • πω for some subset ω of { , . . . , n}, with |ω| = k.
By abuse of notation we will often write f (xω) for a function on the sphere depending on k variables, meaningf (xω).
Functions on the sphere depending on k variables enjoy special symmetry properties. Indeed, they are constant on (n − k − )-dimensional subspheres of the original sphere. The ber of a point y ∈ B k is a sphere S n−k− of radius − y − · · · − y k in S n− . Note that for xed ω, π − ω (y) ≠ π − ω (y ) if y ≠ y , so that the subspheres π − ω (y) indexed by y ∈ B k do not intersect each other and cover the whole S n− . For convenience of the reader, in the following proposition we recall a well known integration formula for functions depending on k variables (see for example [12,21]). Let ω = {i , . . . , i k } be a subset of { , . . . , n} with |ω| = k, for ≤ k ≤ n − . Let f : S n− → R be a function depending on the k variables xω. The following integration formula holds:

Proposition 5.2.
The constant c n,k depends only on the dimension n and on the number of variables k.

Remark 5.3.
Let ω be as above, with |ω| = k, for ≤ k ≤ n − , and let f : S n− → R be a function depending on k variables. Since − x i − · · · − x i k ≤ , we have the trivial inequality In this way, we obtain a family of continuous immersions

. The Lie algebra of the special orthogonal group
A basis for so(n), the Lie algebra of SO(n), is given by the vector elds Let δ i,j be the Kronecker delta. The bracket of two basis elements L i,j and L k,l is given by It follows that the commutator of two elements of the basis {L i,j } i<j , if not trivial, is again an element of the basis. This basis will be our Hörmander system I. The corresponding sub-Laplacian is given by The operator L commutes with all the vector elds L i,j , since it is the quadratic Casimir operator, which is an element of the center of the universal enveloping algebra U(so(n)). Note also that L is the Laplace-Beltrami operator on SO(n) with the Riemannian metric induced by the Killing form (see [13]).

. Structure of maximal subsets
We now discuss the structure of maximal subsets of I = {L i,j } i<j . In order to visualize the subsets of I we associate to the vector eld L i,j the pair (i, j). Given a subset A ⊆ {(i, j)} i<j , we can relate to it an undirected simple graph G A = (V , E) where the set of vertices V is given by { , . . . , n} and the edges E are given by the (unordered) pairs (i, j) ∈ A, so that we identify A with E.
The following proposition holds.

Proposition 5.4. Let A be a subset of I. A is a maximal subset if and only if the associated graph G A = (V , E) splits in complete connected components.
Proof. Note that if (a, b) ∈ E and (b, c) ∈ E, then (a, c) ∈ E by the maximality assumption on A and (5.3).
Since connected components are path connected, each connected component of a graph associated to a maximal subset is complete. The converse is straightforward, again by (5.3).
We also have the following result. Proof. Let V = {i , . . . , i k }, with i < · · · < i k , so that | V| = k. Since G is complete, E contains all the edges in E with vertices in V. It is easy to see that, by property (5.3), the map E → so(k) that maps L i j ,i l → L j,l , for i j < i l , is a Lie algebra isomorphism. Moreover the set E is a basis for E . Let us introduce some notation. Let α = (α , . . . , α n ) ∈ { , } n be a multi-index and denote by |α| = α + · · · + α n its length. The scalar product α · β = α β + · · · + α n β n indicates the number of 's in common between α and β, so that two multi-indices are orthogonal if they do not have 's in common.
We will denote with soα the Lie algebra isomorphic to so(|α|) generated by the set {L k,l : α k = α l = }. We can deduce the following theorem describing the structure of subalgebras generated by maximal subsets associated to basis systems of so(n). Theorem 5.6. Let A be a maximal subset of I = {L i,j } i<j . Then there exist multi-indices α , . . . , α N pairwise orthogonal, with |α | ≥ |α | ≥ · · · ≥ |α N | and |α | + · · · + |α N | ≤ n, such that where on the right-hand side we have a direct sum of Lie algebras, i.e. each subalgebra commutes with the others.
Proof. By Proposition 5.4 and Proposition 5.5 the graph associated to A splits in N, say, complete connected components Gα i = (Vα i , Eα i ), where i = , . . . , N, each component describing a graph associated to a basis system of a Lie algebra of type so(k) for some k. Without loss of generality we can assume that |Vα | ≥ · · · ≥ |Vα N | so that |α | ≥ · · · ≥ |α N |. The multi-indices are pairwise orthogonal since the graphs Gα i are disconnected so that Vα i ∩ Vα j = ∅ for i ≠ j. It is clear that |α | + · · · + |α N | = |V| ≤ n. Finally, since the multi-indices are pairwise orthogonal, in view of formula (5.3), xing k ≠ l, ≤ k, l ≤ N, we have that [L i ,j , L i ,j ] = for all i , j such that α i k = α j k = and i , j such that α i l = α j l = , and by linearity the same holds for all brackets between elements of soα l and soα k . Thus the sum in (5.5) is direct.
We now study the properties of functions annihilated by maximal subsets of vectors in I. First we consider the case of a singleton, i.e. A = {L i,j }. for all (x , . . . , xn) ∈ S n− , where byx i we mean that the variable x i is not appearing.
Proof. Clearly, (5.6) implies for all x ∈ S n− , where D denotes the partial derivative with respect to the rst variable off . Conversely, suppose that f satis es L i,j f (x) = for all x ∈ S n− . Since L i,j is the in nitesimal generator of rotations in the x i x j -plane, it xes circles of the type x i + x j = r ; f , being annihilated by L i,j , is constant on these circles, thus it depends on x i and x j through x i + x j .
Proof. The assertion is proved arguing as in Lemma 5.7, once noted that the subalgebra A generates the rotations in the k-plane related to the coordinates x i , . . . , x i k .
By abuse of notation we will just write f in place off .

Remark 5.9.
If a function f ∈ C ∞ (S n− ) is A-symmetric with respect to a maximal subset A of I such that A soα for some multi-index α with |α| = k, the function f is a function of n − k variables in the sense of De nition (5.1). Without loss of generality, assume α i = for i = , . . . , k and zero otherwise. By Lemma 5.8 we have x k+ , . . . , xn), so that f is a function of the n − k variables x k+ , . . . , xn.
A generic maximal subset A of I splits by Theorem 5.6 into N disjoint subsets, labeled by a family of multi-indices α i . Each of these subsets generates a subalgebra of so(n) isomorphic to so(|α i |). In Theorem 5.6 we ordered these subsets by cardinality. We will interpret the splitting in the following way: the subalgebra related to the multi-index α tells us on how many variables the functions annihilated by A depend, as explained in Remark 5.9; the subalgebras related to the multi-indices α i , for ≤ i ≤ N give instead information concerning radiality in the variables. To be more precise, functions in C ∞ (S n− ) that are A-symmetric depend on the n −|α | variables x i for which α i = , and depend radially on the collections of |α i | variables associated to each multi-index α i (that are disjoint by the orthogonality of the multi-indices).  = ( , , , , , , ). A function f ∈ C ∞ (S n− ) that is A-symmetric will depend on the n−|α | = − = variables x , x , x , x and will be radial in the collections of variables x , x associated to α and x , x associated to α . So it will be written as Remark 5.11. We stick to the convention of ordering the subsets by cardinality. We remark that all orderings are equivalent. Indeed, in Example 5.10 one could have considered instead the splitting soα ( ) ⊕ soα ( ) ⊕ soα ( ).
In this point of view, a function f ∈ C ∞ (S n− ) that is A-symmetric is a function of the n − |α | = − = variables x , x , x , x , x , radial in the collections of variables x , x , x associated to α and x , x associated to α . So it can be written as we can reduce the dependence to x + x and x + x , thus obtaining the same numerology as in Example 5.10.
In the rest of the section we will study some interesting instances of multilinear inequalities of the type (2.10) related to the system I = {L i,j } i<j described above. We will obtain nontrivial inequalities in the sense of De nition 2.9. As we saw, functions involved in the inequalities have symmetry properties determined by the maximal system A that annihilates them. We will also be able to show for some of the inequalities that the exponentsp found by means of Theorem 3.8 are sharp in a sense that we make precise with the following de nition.
De nition 5.12. We will say that the exponentp is sharp if the inequality

. Inequalities for functions depending on k variables
The rst inequality we discuss was discovered by Carlen, Lieb and Loss in [9] and it is an inequality for n functions on the sphere S n− each depending on a single di erent variable. The inequality is the following. We will now generalize this result of [9] to functions depending on ≤ k ≤ n − variables and thus obtain Theorem 5.13 as a corollary. The proof for this general case is based on Theorem 3.8, which is in the spirit of the original proof of [9]. We will also give a proof of the sharpness by producing an explicit counterexample.
The case of functions depending on n− variables is the easiest one and we treat it separately. In this case we have n n− = n possible (n − )-tuples of variables, which correspond to empty maximal systems A i , for which A i = { }. Indeed, functions depending on n − variables are almost generic functions, as explained at the beginning of the section, and there is no hope to obtain something better than Hölder's inequality, i.e. p = n. This is con rmed by Theorem 3.8, since each element in each A c i = I occurs in all A c k , for k = , . . . , n. Let us now consider the case of functions depending on ≤ k ≤ n − variables. We have n k := C(n, k) possible choices ok k-tuples out of the set { , . . . , n}. We will label them as ω , . . . , ω C(n,k) following the notation introduced in Section 4. To each collection of variables ω i = {i , . . . , i k } corresponds a maximal subset A i which contains the vector elds L j,l for which j, l ≠ is for all s = , . . . , k.
The subalgebra generated by each maximal subset A i is isomorphic to so(n − k) and the splitting of A i given by Theorem 5.6 has just one direct summand soα i , with α i a multi-index such that α j i = if j ∈ ω i , for j = , . . . , n. By Remark 5.9 a function f ∈ C ∞ (S n− ) that is A i -symmetric is a function of the variable xω i in R k . As we saw we can think of a function depending on xω i as a function de ned on the k-dimensional unit ball B k ⊂ R k , pulled back to the sphere S n− via the projection πω i : S n− → B k , mapping a point x ∈ S n− to xω i . We will write f (xω i ) for f (πω i (x)), with x ∈ S n− . We have the following theorem.
Theorem 5.14. Let f , . . . , f C(n,k) be nonnegative measurable functions, f i : B k → R + . The inequality Moreover inequality (5.9) is sharp in the sense of De nition 5.12.
Remark 5.15. For k = we recover the result of [9]. Note that inequality (5.9) is nontrivial in the sense of De nition 2.9 for n ≥ , sincep < C(n, k).
Proof. By Theorem 3.8, the exponentp is given by the number of occurrences of the most recurrent element among the sets A c i , for i = , . . . , C(n, k). As we said, the elements of A c i are vector elds of type L j,l with either j or l or both j, l in ω i . So an element L j,l will occur in all A c i apart from those for which j, l ∉ ω i . The number of sets ω i made of k elements taken from { , . . . , n} that do not contain two xed elements j, l is n− k . This means that each vector eld will occur in exactlỹ proving the rst half of the theorem. To show thatp is sharp we construct a counterexample. We consider functions f i : B k → R + , where B k is the unit ball in R k , of the form where γ, δ are positive constants to be determined. We remark that for k = this set of functions reduces to the counterexample for Theorem 5.13 contained in [9]. The right-hand side of inequality (5.9) must be nite. We rst compute the L p norm of these functions. Without loss of generality we focus on the case ω = { , , . . . , k} and work with f (x , . . . , x k ). Let p ≥ . We have For the rst term I we have where we used the integration formula (5.1), the fact that ( − x − · · · − x k ) (n−k− )/ ≤ in B k , since k ≤ n − , and also that B k ⊂ [− , ] k . So I is nite if γp < k. For each of the terms I i we have (5.1). So I i is nite whenever γδp < (n − ). We conclude that the right-hand side of (5.9) is nite if To estimate the left-hand side of (5.9) we pass to polar coordinates in the hyperplane R n− with coordinates x , . . . , x n− ; on the sphere S n− the variable |xn| will then be ( − ρ ) / . There are n− k functions not involving the xn variable, and n− k− involving it. For the functions not depending on xn we select the rst summand of (5.10), for those depending on xn we select the summand ( − x n ) −γδ/ .
So for the left-hand side we have:

Further results . Inequalities for radial functions on k variables
In this section we improve on Theorem 5.14 by adding an additional symmetry. We consider functions of k variables, i.e. functions that are de ned on a k-dimensional unit ball and pulled-back to the sphere by means of a projection, that are radial with respect to the variables in the k-dimensional ball, for ≤ k ≤ n − . Given a subset ω i = {i , . . . , i k } of { , . . . , n}, we will use the notation r(xω i ) to denote the radius (x i + · · · + x i k ) / . A functions depending radially on the variables xω is a function f : [ , ] → R pulled back to the sphere via the composition r • πω i . We will write f (r(xω i )) for f ((r(πω i (x)))), with x ∈ S n− . We have n k := C(n, k) possible choices of k-tuples out of the set { , . . . , n}, as in the generic case of functions depending on k variables. We will label the tuples by ω , . . . , ω C(n,k) , as in the previous section. To each collection of variables ω i = {i , . . . , i k } corresponds a maximal subset A i which contains all the vector elds L h,l for which h, l ∉ ω i , but also the vector elds L h,l for which both h, l ∈ ω i , by the radiality assumption.
The subalgebra generated by each maximal subset A i is isomorphic to the direct sum so(n − k)⊕so(k) and has the form A i = soα i ⊕ so β i , where α i is a multi-index such that α j i = if j ∈ ω i and β i = ( , , . . . , ) − α i . Note that by the convention in Theorem 5.6 the splitting should be ordered by the cardinality of multiindices. We can reduce to the cases where k ≤ n . Indeed, consider a function f that depends radially on the k variables {x i , . . . , x i k } and let {x i k+ , . . . , x in } be the remaining n − k variables. It is straightforward that for some function g. There is a correspondence between functions that depend radially on k variables and functions that depend radially on n−k variables. Indeed, the number of possible choices of k-tuples and (n−k)tuples is the same, since n k = n n−k , for k ≤ n . Moreover the splittings of the corresponding associated maximal subsets is the same up to change in the order of the direct summands.
We will stick to the convention that the rst direct summand is related to the longest multi-index, so it su ces to look at the case k ≤ n . The case of n even and k = n/ is a bit di erent and will be treated separately.
We have the following theorem.  [9] is again recovered, since functions that depend radially on one variable are just even functions of one variable. Indeed, for k = we havep = . Note that the exponentp obtained for this type of functions is smaller than that obtained for generic functions of k variables. This in particular implies that inequality (6.1) is nontrivial in the sense of De nition 2.9.
Proof. By Theorem 3.8, the exponentp is given by the number of occurrences of the most recurrent element among the sets A c i , for i = , . . . , C(n, k). The elements of A c i are vector elds of type L h,l with exactly one among h and l in ω i . So an element L h,l will occur in all A c i associated to subsets ω i containing either h but not l, which are n− k− , or l but not h, which are again n− k− . Altogether, each vector eld L h,l will occur n− k− times among the A c i , yielding the exponentp. To prove thatp is sharp we construct an explicit counterexample. Consider functions f i : [ , ] → R + , of the form where γ is a positive constants to be determined. We rst compute the norms on the right-hand side. Without loss of generality we assume that ω = { , . . . , k} and work with f (x + · · · + x k ). Let p ≥ . We have where we used the integration formula (5.1) and then passed to polar coordinates. This integral is nite if γp < k. We control the left-hand side of (6.1) by the trivial bounds (x + · · · + x k ) −γ ≥ (x + · · · + x k + x k+ + · · · + x n− ) −γ , for terms not involving xn, and ( −x −· · ·− x n ) −γ ≥ ( −x n ) −γ , for terms involving xn. We make this distinction to pass to polar coordinates in the hyperplane R n− with coordinates x , . . . , x n− ; on the sphere S n− , |xn| will then just be ( − ρ ) / .
There are n− k terms not involving xn, and n− k− involving it. For the functions not depending on xn we select the rst summand of (6.2), for those depending on xn we select the second one. For the left-hand side we have: Comparing the condition γp < k and (6.3) we see that the right-hand side is nite and the left-hand side divergent if thus proving the optimality of the exponentp.
In the case of n even and functions depending radially on k = n/ variables, the splitting associated to a maximal subset is of type so(n/ ) ⊕ so(n/ ) so that there are two possible orderings. This corresponds to the fact that, given a subset ω i = {i i , . . . , i n/ } of { , . . . , n}, the set {i (n/ )+ , . . . , in} being its complement, a function radial in the variables of ω i is also radial in the variables of its complement, but in this case both sets have cardinality n/ . So one needs to consider a family of (di erent) k-tuples ω i , for i = , . . . , C(n, k)/ , with ω i ∩ ω j ≠ ∅ for all i, j. Di erent choices of families of subsets ω i give equivalent types of functions. We have the following theorem. the point ( / , / , / ) given by Theorem 5.13. Moreover in this case the assumptions of Theorem 6.5 are always ful lled, since given any triple (p − , p − , p − ) ≠ ( / , / , / ) such that p − + p − + p − = / , by pigeonholing there must always be one p i > / . This implies that the point ( / , / , / ) is the only point in the hyperplane p − + p − + p − = / where inequality (6.10) holds. From this we also deduce that inequality (6.10) cannot hold for points in Q such that p − + p − + p − > / . Indeed, by interpolation with points in R this would yield points in the hyperplane p − + p − + p − = / for which the inequality holds, providing a contradiction. This goes in the direction of our conjecture, that R is the optimal region of validity for (6.10). The only points left are those outside of R for which < p − + p − + p − < / . In this range we have the following proposition which leads to a partial improvement towards the sharpness. Proposition 6.7. Suppose that < p − + p − + p − < / and that the condition pa + p b > − pc (6.11) holds for at least one choice of a, b, c in { , , } with a, b, c pairwise distinct. Then inequality (6.10) is false.
Proof. We make the usual construction. Assume for instance that a = , b = , c = . We let for i = , , . As usual the integrability condition for the right-hand side of (6.10) is γ i p i < . For the left-hand side, taking the rst summand for f and f and the second one for f , we get that which diverges for γ + γ + γ = , that is for From the condition γ i p i < we get that we need to have − p < γ + γ < p + p .
Clearly γ + γ can be in this range only when (6.11) holds.
To sum up, we do not know what happens in the range < p − + p − + p − < / , outside of R, where none of the conditions (6.11) is satis ed for any exponent p i . An example of a point in this region is ( / , / , ). n − k = − = . The second subalgebra refers to radiality in two of the variables involved. The ambiguity in the order of the subalgebras is not a problem, since the two possibilities are equivalent in the following sense. If A = {L , , L , } we are considering a function f either of type f (x + x , x ) or a function of type f (x + x , x ) which are equivalent, since x + x = − x − x − x . There are = possible choices for L i,j , and having xed i and j we have = choices for L k,l . By the aforementioned equivalence we have possible maximal subsets.
It is easy to see that in this case the critical exponent given by Theorem 3.8 isp = . The exponent is sharp and this can be easily checked considering the functions f jl i (x i , x j + x l ) = |x i | − / (x j + x l ) − / + ( − x i ) − / . (6.12)