UNIVERSAL INEQUALITIES FOR THE EIGENVALUES OF A POWER OF THE LAPLACE OPERATOR SA ÏD ILIAS AND OLA MAKHOUL

Keywords: Mathematics Subject Classification. 35P15;58C40 eigenvalues, Laplacian, polyharmonic operator, biharmonic operator, clamped plate, Payne-Polya-Weinberger inequality, Hile-Protter inequality, Yang inequality, universal inequalities, commutators, Kohn Laplacian, Heisenberg group

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving different and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.

Introduction

Let Ω be a bounded domain of an n-dimensional Euclidean space R n and consider the following eigenvalue problem for the polyharmonic operator :

   (-∆) l u = λu in Ω,

u = ∂u ∂ν = • • • = ∂ l-1 u ∂ν l-1 = 0 on ∂Ω, (1.1)
where ∆ is the Laplace operator and ν is the outward unit normal.

It is known that this eigenvalue problem has a discrete spectrum,

0 < λ 1 < λ 2 ≤ • • • ≤ λ k ≤ . . . → +∞
In this paper we will be interested in "Universal"(i.e. not depending on the domain) inequalities for the eigenvalues of such a polyharmonic problem and especially we will show how to derive them from a general abstract algebraic formula in the spirit of the work of Harrell, Stubbe, Ashbaugh and Hermi. Let us begin by giving a short and non-exhaustive presentation of the known results in this field.

The first result concerns the Dirichlet Laplacian (i.e. when l = 1). In this case, Polya, Payne and Weinberger (henceforth PPW) proved in 1955 the following bound (see [START_REF] Payne | On the ratio of consecutive eigenvalues[END_REF] for dimension 2 and [START_REF] Thompson | On the ratio of consecutive eigenvalues in N -dimensions[END_REF] for all dimensions), for k = 1, 2, . . .

λ k+1 -λ k ≤ 4 nk k i=1 λ i , (1.2)
This result was improved in 1980 by Hile and Protter [START_REF] Hile | Inequalities for eigenvalues of the Laplacian[END_REF] (henceforth HP) who showed that, for k = 1, 2, . . .

nk 4 ≤ k i=1 λ i λ k+1 -λ i . (1.3)
In 1991, H.C.Yang (see [START_REF] Yang | An estimate of the difference between consecutive eigenvalues[END_REF] and more recently [START_REF] Cheng | Estimates on eigenvalues of laplacian[END_REF]) proved

k i=1 (λ k+1 -λ i ) 2 ≤ 4 n k i=1 λ i (λ k+1 -λ i ), (1.4)
which is, until now, the best improvement of the PPW inequality. From inequality (1.4), we can infer a weaker form

λ k+1 ≤ (1 + 4 n ) 1 k k i=1 λ i . (1.5)
We shall refer to inequality (1.4) as Yang's first inequality (or simply Yang inequality) and to inequality (1.5) as Yang's second inequality.

The comparison of all these inequalities (see [START_REF] Ashbaugh ; Hile-Protter | The universal eigenvalue bounds of Payne-Pólya-Weinberger[END_REF]) can be summarized in Yang 1 =⇒ Yang 2 =⇒ HP =⇒ PPW When l = 2, the eigenvalue problem (1.1) for the bilaplacian is the clamped plate problem. In the same paper as before [START_REF] Payne | On the ratio of consecutive eigenvalues[END_REF], Polya, Payne and Weinberger proved the following analog of the formula (1.2)

λ k+1 -λ k ≤ 8(n + 2) n 2 k k i=1 λ i . (1.6)
And as was noticed by Ashbaugh (see [START_REF] Ashbaugh | Isoperimetric and universal inequalities for eigenvalues[END_REF] inequality (3.56)), there is a better inequality which was implicit in the PPW work,

λ k+1 -λ k ≤ 8(n + 2) n 2 k 2 k i=1 λ 1 2 i 2 .
In 1984, Hile and Yeh [START_REF] Hile | Inequalities for eigenvalues of the biharmonic operator[END_REF] extended the approach used for the Laplacian in [START_REF] Hile | Inequalities for eigenvalues of the Laplacian[END_REF] and proved the sharpest bound

n 2 k 3 2 8(n + 2) ≤ k i=1 λ 1 2 i λ k+1 -λ i k i=1 λ i 1 2
.

(1.7)

Then in 1990, Hook [START_REF] Hook | Domain-independent upper bounds for eigenvalues of elliptic operators[END_REF], Chen and Qian [START_REF] Chen | Estimates for discrete spectrum of the laplacian operator with any order[END_REF] proved independently the following stronger inequality which was again implicit in the work of Hile and Yeh (see also [START_REF] Ashbaugh | Isoperimetric and universal inequalities for eigenvalues[END_REF], [START_REF] Chen | On the difference of consecutive eigenvalues of uniformly elliptic operators of higher orders[END_REF], [START_REF] Chen | On the upper bound of eigenvalues for elliptic equations with higher orders[END_REF] and [START_REF] Chen | Hile-Yeh estimates of eigenvalues of polyharmonic operators[END_REF])

n 2 k 2 8(n + 2) ≤ k i=1 λ 1 2 i λ k+1 -λ i k i=1 λ 1 2 i . (1.8)
Using Chebyshev inequality, Ashbaugh (see [START_REF] Ashbaugh | Isoperimetric and universal inequalities for eigenvalues[END_REF] inequality (3.60)) deduces from the preceding inequality (1.8), the following HP version which is weaker and more esthetically appealing,

n 2 k 8(n + 2) ≤ k i=1 λ i λ k+1 -λ i . (1.9)
Recently, Cheng and Yang [START_REF] Cheng | Inequalities for eigenvalues of a clamped plate problem[END_REF] established the following Yang version

k i=1 (λ k+1 -λ i ) ≤ 8(n + 2) n 2 1 2 k i=1 λ i (λ k+1 -λ i ) 1 2 .
(1.10)

For any l, the PPW inequality is given by

λ k+1 -λ k ≤ 4l(2l + n -2) n 2 k 2 k i=1 λ 1 l i k i=1 λ l-1 l i .
Its HP improvement was proved independently by Hook [START_REF] Hook | Domain-independent upper bounds for eigenvalues of elliptic operators[END_REF] and Chen and Qian [START_REF] Chen | Estimates for discrete spectrum of the laplacian operator with any order[END_REF], it reads

n 2 k 2 4l(2l + n -2) ≤ k i=1 λ 1 l i λ k+1 -λ i k i=1 λ l-1 l i . (1.11)
As in the case l = 2 (inequality (1.9)), this reduces to the weaker form

n 2 k 4l(2l + n -2) ≤ k i=1 λ i λ k+1 -λ i . (1.12)
In 2007, Wu and Cao [START_REF] Wu | Estimates for eigenvalues of Laplacian operator with any order[END_REF] generalized the inequality (1.10) of Cheng and Yang to the polyharmonic problem and obtained

k i=1 (λ k+1 -λ i ) ≤ (1.13) 1 n (4l(n + 2l -2)) 1 2 k i=1 (λ k+1 -λ i ) 1 2 λ l-1 l i 1 2 k i=1 (λ k+1 -λ i ) 1 2 λ 1 l i 1 2
.

This inequality is sharper than inequality (1.11)(see [START_REF] Wu | Estimates for eigenvalues of Laplacian operator with any order[END_REF]). Very recently, Cheng, Ichikawa and Mametsuka [START_REF] Cheng | Inequalities for eigenvalues of laplacian with any order[END_REF] derived the following Yang type inequality for the polyharmonic operator (i.e. such that for l = 1, we have the Yang inequality (1.4))

k i=1 (λ k+1 -λ i ) 2 ≤ 4l(2l + n -2) n 2 k i=1 (λ k+1 -λ i )λ i . (1.14)
All the classical proofs of these inequalities are based on tricky and careful choices of trial functions. For a more comprehensive and general approach, it is important to see if all these inequalities can be deduced using purely algebraic arguments involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting on a Hilbert space. In the case of the Laplacian (i.e. l = 1), this was done by Harrell [START_REF] Harrell | General bounds for the eigenvalues of Schrödinger operators[END_REF][START_REF] Evans | Some geometric bounds on eigenvalue gaps[END_REF], Harrell and Michel [START_REF] Harrell | Erratum to: "Commutator bounds for eigenvalues, with applications to spectral geometry[END_REF][START_REF] Harrell | Commutator bounds for eigenvalues of some differential operators[END_REF], Harrell and Stubbe [START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF], and Ashbaugh and Hermi [START_REF] Ashbaugh | A unified approach to universal inequalities for eigenvalues of elliptic operators[END_REF]. For the polyharmonic problem (i.e. general l), Hook [START_REF] Hook | Domain-independent upper bounds for eigenvalues of elliptic operators[END_REF] generalized the argument of Hile and Protter [START_REF] Hile | Inequalities for eigenvalues of the Laplacian[END_REF] in an abstract setting. Later, this abstract formulation of Hook was simplified and improved by Ashbaugh and Hermi [START_REF] Ashbaugh | Universal inequalities for higher-order elliptic operators[END_REF]. In fact, they obtained the following inequality relating eigenvalues of a self-adjoint operator A, to two families of symmetric operators B ′ p s, skew-symmetric operators T ′ p s, p = 1, . . . , n and their commutators (for a precise statement with detailed assumptions, see Theorem 2.2 of [START_REF] Ashbaugh | Universal inequalities for higher-order elliptic operators[END_REF]), .15) But this abstract inequality, as was observed by Ashbaugh and Hermi in the end of the third paragraph of their article [START_REF] Ashbaugh | Universal inequalities for higher-order elliptic operators[END_REF], could not recover more than the HP version of the universal inequalities (i.e. inequalities (1.11) and (1.12)).

1 4 k i=1 n p=1 [B p , T p ]u i , u i 2 k i=1 n p=1 [A, B p ]u i , B p u i ≤ k i=1 n p=1 T p u i , T p u i λ k+1 -λ i . ( 1
The main goal of the present paper is to prove the following abstract inequality (with the same assumptions as those for the Ashbaugh-Hermi inequality (1.15))which generalizes (1.15) and fills this gap

k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i 2 (1.16) ≤ 4 k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i k i=1 n p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) T p u i 2 ,
where f and g are two functions satisfying some functional conditions (see Definition (2.1)). The family of such couples of functions is large and particular choices for f and g give many of the known universal inequalities. For instance, in the case of the polyharmonic problem, if we take f (x) = g(x) = (λ k+1x)

k i=1 n p=1 f (λ i ) [A, B p ]u i , B p u i 2 ≤ k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i k i=1 n p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) [A, B p ] u i 2 .
(1.17)

Using this last inequality, with particular choices of f and g as before, one can recover many of the known universal inequalities for eigenvalues of Laplace or Schrödinger operators.

In the last section of this paper, we show how one can use the inequality (1.16) to derive new universal bounds, of Yang type, for eigenvalues of the Kohn Laplacian on the Heisenberg group, with any order. These bounds are stronger than the earlier bounds obtained by Niu and Zhang in [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF].

The abstract formulation

Before stating the main result of this section, we introduce a special family of couples of functions which will play an important role in our formulation.

Definition 2.1. Let λ > 0. A couple (f, g) of functions defined on ]0, λ[ belongs to ℑ λ provided that 1. f and g are positive, 2. f and g satisfy the following condition, for any x, y ∈]0, λ[ such that x = y,

f (x) -f (y) x -y 2 + f (x) 2 g(x)(λ -x) + f (y) 2 g(y)(λ -y) g(x) -g(y)
xy ≤ 0.

(2.1)

A direct consequence of our definition is that g must be nonincreasing. If we multiply f and g of ℑ λ by positive constants the resulting functions are also in ℑ λ . In the case where f and g are differentiable, one can easily deduce from (2.1) the following necessary condition:

ln f (x) ′ 2 ≤ -2 λ -x ln g(x) ′ .
This last condition helps us to find many couples (f, g) satisfying the conditions 1) and 2) above. Among them, we mention 1, (λx) α / α ≥ 0 ,

(λ -x), (λ -x) β / β ≥ 1 2 , (λ -x) δ , (λ -x) δ / 0 < δ ≤ 2 .
and (λx) α , (λx) β / α < 0, 1 ≤ β, and α 2 ≤ β .

Let H be a complex Hilbert space with scalar product ., . and corresponding norm . . For any two operators A and B, we denote by [A, B] their commutator, defined by [A, B] = AB -BA.

Theorem 2.1. Let A : D ⊂ H -→ H be a self-adjoint operator defined on a dense domain D, which is semibounded below and has a discrete spectrum λ 1 ≤ λ 2 ≤ λ 3 .... Let {T p : D -→ H} n p=1 be a collection of skew-symmetric operators, and {B p : T p (D) -→ H} n p=1 be a collection of symmetric operators, leaving D invariant. We denote by {u i } ∞ i=1 a basis of orthonormal eigenvectors of A, u i corresponding to λ i . Let k ≥ 1 and assume that λ k+1 > λ k . Then, for any

(f, g) in ℑ λ k+1 k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i 2 (2.2) ≤ 4 k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i k i=1 n p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) T p u i 2 .
Proof of Theorem 2.1. For each i, we consider the vectors φ p i , given by

φ p i = B p u i - k j=1 a p ij u j
where a p ij := B p u i , u j , p = 1, ..., n. We have

φ p i , u j = 0, (2.3)
for all j = 1, ..., k. Taking φ p i as a trial vector in the Rayleigh-Ritz ratio, we obtain

λ k+1 ≤ Aφ p i , φ p i φ p i , φ p i . (2.4)
Since B p is symmetric, for all p = 1, ..., n, we have a p ij = a p ji . Moreover, using the orthogonality conditions (2.3), we obtain

φ p i 2 = φ p i , B p u i - k j=1 a p ij u j = φ p i , B p u i = B p u i - k j=1 a p ij u j , B p u i = B p u i 2 - k j=1 a p ij B p u i , u j = B p u i 2 - k j=1 a p ij 2 (2.5)
and

Aφ p i , φ p i = AB p u i - k j=1 λ j a p ij u j , φ p i = AB p u i , φ p i = AB p u i , B p u i - k j=1 a p ij AB p u i , u j = [A, B p ]u i , B p u i + B p Au i , B p u i - k j=1 λ j a p ij 2 = [A, B p ]u i , B p u i + λ i B p u i 2 - k j=1 λ j a p ij 2 . (2.6)
Hence, inequality (2.4) reduces to

λ k+1 φ p i 2 ≤ [A, B p ]u i , B p u i + λ i B p u i 2 - k j=1 λ j a p ij 2 .
(2.7)

On the other hand, we observe that, for p = 1,

• • • , n, -2 T p u i , φ p i = -2 T p u i , B p u i + 2 T p u i , k j=1 a p ij u j = 2 u i , T p B p u i + 2 k j=1 a p ij T p u i , u j = 2 u i , T p B p u i + 2 k j=1 a p ij c p ij , (2.8)
where

c p ij = T p u i , u j . Note that, since T p is skew-symmetric, we have c p ij = -c p ji for 1 ≤ p ≤ n.
Therefore, using (2.3) and taking the real part of both sides of (2.8), we obtain, for any constant

α i > 0, 2Re T p B p u i , u i + 2 k j=1 Re a p ij c p ij = -2Re φ p i , T p u i = 2Re φ p i , -T p u i + k j=1 c p ij u j ≤ α i φ p i 2 + 1 α i -T p u i + k j=1 c p ij u j 2 = α i φ p i 2 + 1 α i T p u i 2 - k j=1 c p ij 2 .
(2.9)

Multiplying (2.9) by f (λ i ) and taking

α i = α(λ k+1 -λ i )g(λ i ) f (λ i )
, where α is a positive constant and i ≤ k, we infer from (2.7)

2f (λ i ) Re T p B p u i , u i + k j=1 Re a p ij c p ij ≤ α i f (λ i ) φ p i 2 + 1 α i f (λ i ) T p u i 2 - k j=1 c p ij 2 = α(λ k+1 -λ i )g(λ i ) φ p i 2 + 1 α f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 - k j=1 c p ij 2 ≤ αg(λ i ) [A, B p ]u i , B p u i + αg(λ i )λ i B p u i 2 -αg(λ i ) k j=1 λ j a p ij 2 -αg(λ i )λ i φ p i 2 + 1 α f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 - k j=1 c p ij 2 .
(2.10)

Summing over i = 1, • • • , k and using (2.5), we get

2 k i=1 f (λ i )Re T p B p u i , u i + 2 k i,j=1 f (λ i )Re a p ij c p ij ≤ α k i=1 g(λ i ) [A, B p ]u i , B p u i + α k i=1 λ i g(λ i ) B p u i 2 -α k i,j=1 λ j g(λ i )|a p ij | 2 -α k i=1 λ i g(λ i ) φ p i 2 + 1 α k i=1 f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 - k j=1 |c p ij | 2 = α k i=1 g(λ i ) [A, B p ]u i , B p u i + α k i,j=1 (λ i -λ j )g(λ i )|a p ij | 2 + 1 α k i=1 f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 - k j=1 |c p ij | 2 . (2.11)
Since a p ij = a p ji and c p ij = -c p ji , we have

2Re k i,j=1 f (λ i )a p ij c p ij = Re k i,j=1 (f (λ i ) -f (λ j ))a p ij c p ij (2.12) Using that |c p ij | 2 = |c p ji | 2 , we find - 1 α k i,j=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) |c p ij | 2 = -1 2α k i,j=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) + f (λ j ) 2 g(λ j )(λ k+1 -λ j ) |c p ij | 2 .
(2.13) Moreover,

α k i,j=1 g(λ i )(λ i -λ j )|a p ij | 2 = α 2 k i,j=1 g(λ i ) -g(λ j ) (λ i -λ j )|a p ij | 2 .
(2.14)

Thus we infer from (2.11),(2.12),(2.13) and (2.14)

2 k i=1 f (λ i )Re T p B p u i , u i + k i,j=1 f (λ i ) -f (λ j ) Re a p ij c p ij ≤ α k i=1 g(λ i ) [A, B p ]u i , B p u i + 1 α k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) T p u i 2 + α 2 k i,j=1 g(λ i ) -g(λ j ) (λ i -λ j )|a p ij | 2 - 1 2α k i,j=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) + f (λ j ) 2 g(λ j )(λ k+1 -λ j ) |c p ij | 2 . (2.15) But k i,j=1 f (λ j ) -f (λ i ) Re a p ij c ij ≤ α 2 k i,j=1 g(λ j ) -g(λ i ) (λ i -λ j )|a p ij | 2 + 1 2α k i,j=1 f (λ j ) -f (λ i ) 2 (λ i -λ j ) 2 λ i -λ j g(λ j ) -g(λ i ) |c p ij | 2 .
(2.16)

From the condition (2.1) satisfied by f and g, we infer

k i=1 f (λ j )-f (λ i ) Re a p ij c p ij ≤ α 2 k i,j=1 g(λ j ) -g(λ i ) (λ i -λ j )|a p ij | 2 + 1 2α k i,j=1 (f (λ i )) 2 g(λ i )(λ k+1 -λ i ) + (f (λ j )) 2 g(λ j )(λ k+1 -λ j ) |c p ij | 2 .
(2.17)

Hence, taking sum on p, from 1 to n, in (2.15), we find

2 k i=1 n p=1 f (λ i )Re T p B p u i , u i ≤ α k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i + 1 α k i=1 n p=1 f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 .
(2.18)

Since B p is symmetric and T p is skew-symmetric, we have for all p ≤ n,

2Re T p B p u i , u i = T p B p u i , u i + T p B p u i , u i = T p B p u i , u i -u i , B p T p u i = [T p , B p ]u i , u i and inequality (2.18) becomes k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i ≤ α k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i + 1 α k i=1 n p=1 f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 , (2.19)
or equivalently

α 2 k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i -α k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i + k i=1 n p=1 f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 ≥ 0.
(2.20)

To prove inequality (2.2), it suffices to show that

k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i ≥ 0. (2.21)
In fact, if this is the case, the discriminant of the quadratic polynomial (2.20) must be nonpositive, i.e.

k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i 2 -4 k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i k i=1 n p=1 f (λ i ) 2 (λ k+1 -λ i )g(λ i ) T p u i 2 ≤ 0. (2.22)
which yields the theorem. We note that if we replace T p by -T p , inequality (2.20) holds. Thus we can deduce that it holds for all real α and not only α > 0 proving that the coefficient of the quadratic term, i.e.

k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i , is nonnegative. If it is equal to zero, then k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i
is also equal to 0 and the theorem trivially holds.

Remark 2.1.

• In the definition of ℑ λ , the functions f and g can be defined only on a discrete set of eigenvalues.

• One can formulate Theorem 2.1 as in [START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF] for z ∈]λ k , λ k+1 ] (it suffices to replace, in the hypothesis and in the inequality, λ k+1 by z).

• The result of Theorem 2.1 can also be stated, as in [START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF] or [START_REF] Harrell | Universal bounds and semiclassical estimates for eigenvalues of abstract schrödinger operators[END_REF], in the general situation where the spectrum of A is not purely discrete and its point spectrum is nonempty.

• Taking f = g = 1 in (2.
2), we obtain inequality (1.15)

λ i . If for k ≥ 1 we have λ k+1 > λ k , then for any (f, g) ∈ ℑ k , k i=1 n p=1 f (λ i ) [A, B p ]u i , B p u i 2 ≤ k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i k i=1 n p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) [A, B p ] u i 2 .
(2.23)

Remark 2.2.

• As for Theorem 2.1, Corollary 2.1 can be stated in the general case where the spectrum of A is not totally discrete.

• For f (x) = g(x) = (λ k+1x) 2 , inequality (2.23) becomes the abstract inequality which gives the Yang type inequalities for Laplacians and Schrödinger operators (see [START_REF] Ashbaugh | A unified approach to universal inequalities for eigenvalues of elliptic operators[END_REF], [START_REF] Soufi | Universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds[END_REF], [START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF] and [START_REF] Levitin | Commutators, spectral trace identities, and universal estimates for eigenvalues[END_REF]). • For f (x) = g(x) = (λ k+1 -x) α , with α ≤ 2, we recover a Harrell and Stubbe inequality ( [START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF], [START_REF] Ashbaugh | On harrell-stubbe type inequalities for the discrete spectrum of a self-adjoint operator[END_REF])).

• We can easily deduce from the inequality (2.23) new universal inequalities in many different geometric situations (Dirichlet

Laplacian on domains of Submanifolds of Euclidean (or symmetric) spaces as in [START_REF] Soufi | Universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds[END_REF], Hodge de Rham Laplacian or the square of a Dirac operator,and more generally a Laplacian acting on sections of a Riemannian vector bundle on a submanifold of a Euclidean (or symmetric) space).

Application to the polyharmonic operators

In this section, using Theorem 2.1, we will show how to derive universal inequalities for the eigenvalues of a polyharmonic problem. For a power of the Laplacian and with a particular choice of f and g, one can derive inequality (1.13) and inequality (1.14).

In fact, throughout this section we assume that A = Q l , such that Q is a symmetric self-adjoint operator given by

Q = - n p=1 T 2 p ,
where T p are skew-symmetric operators for p = 1, ..., n, with [Q, T p ] = 0 and [T m , B p ] = δ mp .

First we need to calculate the following expressions

n p=1 [A, B p ]u i , B p u i (3.1)
and

n p=1 T p u i , T p u i = Qu i , u i . (3.2)
For this purpose the following two results of Hook (see proposition 3 in [START_REF] Hook | Domain-independent upper bounds for eigenvalues of elliptic operators[END_REF] and Theorem 1 in [START_REF] Hook | Inequalities for eigenvalues of self-adjoint operators[END_REF]) will be useful.

The first one is Lemma 3.1. Under the circumstances stated above, we have

[A, B p ] = [Q l , B p ] = -2lQ l-1 T p and n p=1 [B p , [A, B p ]] = 2l(2l + n -2)Q l-1 .
And the second one is the following Theorem 3.1. Let V be a real or complex inner product space with inner product ., . . Let D be a linear submanifold of V and let Q : D -→ V be a linear operator in V . Suppose l is a positive integer and u is a fixed vector such that for all 0 ≤ r ≤ q ≤ l,

| Q q u, u | = | Q q-r u, Q r u |.
Then, for all integers 0 ≤ r ≤ q ≤ l, when q is even, we have

| Q r u, u | ≤ | Q q u, u | r/q u, u 1-r/q . (3.3)
This inequality is satisfied for q odd and 0 ≤ r ≤ q ≤ l, if in addition to the above, there is a family of operators {T p } n p=1 such that

| Q q u, u | = | n p=1 T p Q q-r u, T p Q r-1 u |
holds for all 0 ≤ r ≤ q ≤ l.

Applying Lemma 3.1, we obtain

n p=1 [A, B p ]u i , B p u i = 1 2 n p=1 [B p , [A, B p ]]u i , u i = l(2l + n -2) Q l-1 u i , u i .
Therefore, if l is odd, then we have

n p=1 [A, B p ]u i , B p u i = l(2l + n -2) Q l-1 2 u i 2 and if l is even, then n p=1 [A, B p ]u i , B p u i = l(2l + n -2) n p=1 T p Q l-2 2 u i 2
The conditions of Theorem 3.1 are satisfied by our operator Q. So inequality (3.3) is valid for all 0 ≤ r ≤ q ≤ l without parity condition on q. Applying this inequality (3.3) with r = l -1 and q = l, we obtain

n p=1 [A, B p ]u i , B p u i = l(2l + n -2) Q l-1 u i , u i ≤ l(2l + n -2) Q l u i , u i l-1 l u i , u i 1-l-1 l = l(2l + n -2)λ l-1 l i (3.4)
and with r = 1 and q = l, we obtain

n p=1 T p u i 2 = Qu i , u i ≤ Q l u i , u i 1 l u i , u i 1-1 l ≤ λ 1 l i . (3.5) Since [T p , B p ] = 1, one gets [T p , B p ]u i , u i = 1. (3.6)
Then using inequalities (3.4), (3.5) and (3.6) together with inequality (2.2), we obtain

n 2 k i=1 f (λ i ) 2 ≤ 4l(2l + n -2) k i=1 g(λ i )λ l-1 l i k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) λ 1 l i or equivalently k i=1 f (λ i ) ≤ 2 n l(2l + n -2) k i=1 g(λ i )λ l-1 l i 1 2 k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) λ 1 l i 1 2
.

(3.7)

Now the operators A = (-∆) l , Q = -∆, B p = x p , p = 1, . . . , n, where x 1 , . . . , x n are Euclidean coordinates, and T p = ∂ ∂xp fit the setup of this section. Thus, taking f (x) = g(x) 2 = (λ k+1x), we can obtain inequality (1.13) of Wu and Cao. Remark 3.1. For the special case l = 2 (i.e the clamped plate problem) and the same values of f and g as above, we obtain inequality (1.10) of Cheng and Yang. We observe that this inequality can also be obtained easily by a simple calculation from our inequality (2.2). In fact, taking A = ∆ 2 , B p = x p , p = 1, . . . , n and T p = ∂ ∂xp , we first observe that [T p , B p ] = 1. Hence, we have

n p=1 [T p , B p ]u i , u i = n, (3.8
)

moreover [A, B p ]u i = [∆ 2 , x p ]u i = 4 ∂ ∂x p ∆u i . Then [B p , [A, B p ]]u i = 4[x p , ∂ ∂x p ∆]u i = -4 ∆ + 2 ∂ ∂x p 2 u i . It follows that n p=1 [A, B p ]u i , B p u i = 1 2 n p=1 [B p , [A, B p ]]u i , u i = 2(n + 2) -∆u i , u i ≤ 2(n + 2) ∆u i 2 u i 2 1 2 (3.9) = 2(n + 2)λ 1 2 i . (3.10) Now n p=1 T p u i , T p u i = -∆u i , u i ≤ λ 1 2 i , (3.11)
where we used the Cauchy-Schwarz inequality to derive (3.9) and (3.11). Substituting On the other hand, if we take f (x) = g(x) = (λ k+1x) 2 , in (3.7), we get the following inequality obtained in [START_REF] Cheng | Inequalities for eigenvalues of laplacian with any order[END_REF] (see inequality (2.27) therein)

k i=1 (λ k+1 -λ i ) 2 2 ≤ 4l(2l + n -2) n 2 k i=1 (λ k+1 -λ i ) 2 λ l-1 l i k i=1 (λ k+1 -λ i )λ 1 l i . (3.12)
Using the following variant of Chebyshev inequality (see Lemma 1 in [START_REF] Cheng | Inequalities for eigenvalues of laplacian with any order[END_REF]), one can deduce a generalized Yang inequality

Lemma 3.2. Let A i , B i and C i , i = 1, . . . , k, verify A 1 ≥ A 2 ≥ . . . ≥ A k ≥ 0, 0 ≤ B 1 ≤ B 2 ≤ . . . ≤ B k and 0 ≤ C 1 ≤ C 2 ≤ . . . ≤ C k , respectively. Then, we have k i=1 A 2 i B i k i=1 A i C i ≤ k i=1 A 2 i k i=1 A i B i C i .
In fact if we apply this Lemma to the right side of inequality (3.12),

with

A i = λ k+1 -λ i , B i = λ l-1 l i and C i = λ 1 l i , we obtain, k i=1 (λ k+1 -λ i ) 2 ≤ 4l(n + 2l -2) n 2 k i=1 (λ k+1 -λ i )λ i . (3.13)
which, in the case where A = (-∆) l , Q = -∆, B p = x p , p = 1, . . . , n and T p = ∂ ∂xp , gives us inequality (1.14) of Cheng, Ichikawa and Mametsuka (see inequality (1.11) in [START_REF] Cheng | Inequalities for eigenvalues of laplacian with any order[END_REF]). Finally, we note that considering other choices of values for the couple (f, g) lead to many new inequalities.

Applications to the Kohn Laplacian on the Heisenberg group

In this section, we consider the 2n + 1-dimensional Heisenberg group H n , which is the space R 2n+1 equipped with the non-commutative group law

(x, y, t)(x ′ , y ′ , t ′ ) = x + x ′ , y + y ′ , t + t ′ + 1 2 ( x ′ , y R n -x, y ′ R n ),
where x, x ′ , y, y ′ ∈ R n , t and t ′ ∈ R. We denote by H n its Lie algebra, it has a basis formed by the following vector fields T = ∂ ∂t ,

X p = ∂ ∂xp + yp 2 ∂ ∂t and Y p = ∂ ∂yp -xp 2 ∂
∂t . We note that the only non-trivial commutators are [Y p , X q ] = T δ pq . Let ∆ H n denote the real Kohn-Laplacian in the Heisenberg group H n . It is given by

∆ H n = n p=1 X 2 p + Y 2 p = ∆ R 2n xy + 1 4 (|x| 2 + |y| 2 ) ∂ 2 ∂t 2 + ∂ ∂t n p=1 y p ∂ ∂x p -x p ∂ ∂y p .
We are concerned here with the following eigenvalue problem:

   (-∆ H n ) l u = λu in Ω, u = ∂u ∂ν = . . . = ∂ l-1 u ∂ν l-1 = 0 on ∂Ω, (4.1)
where Ω is a bounded domain in H n , with smooth boundary ∂Ω, ν is the unit outward normal to ∂Ω and l ≥ 1 is any positive integer. We denote by We orthonormalize the eigenfunctions u i so that; ∀ i, j ≥ 1,

L = -∆ H n and ∇ H n = (X 1 , . . . , X n , Y 1 , . . . , Y n ). We let 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ k ≤ . . . → +∞ denote the eigenvalues of problem (4.1) with corresponding eigenfunc- tions u 1 , u 2 , . . . , u k , . . . in S l,2 0 (Ω). Here S l,2 (Ω) is the Hilbert space of the functions u in L 2 (Ω) such that X p u, Y p u, X 2 p u, Y 2 p u, . . . , X l p (u), Y l p (u) ∈ L 2 (Ω),
u i , u j L 2 = Ω u i u j dxdydt = δ ij .
In all this paragraph, our results can be stated in a general form using functions f and g ∈ ℑ λ k+1 as in the first part of this paper, but we limit ourselves to the case f (x) = g(x) = (λ k+1x) 2 . This gives us new bounds of the Yang type for eigenvalues of problem (4.1) which improve earlier ones obtained by Niu and Zhang [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF]. We also note that we must treat the three following cases independently: the case when l = 1, the case when l = 2 and the case when l ≥ 3. This is essentially due to the difference of the calculations in these three cases.

4.1. The case when l = 1. In this subsection, we are concerned with the case where l = 1. The result we obtain is a result proved earlier by the first author, El Soufi and Harrell in [START_REF] Soufi | Universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds[END_REF] and for which we give here a different proof, more easily adapted to the other cases l = 2 and l ≥ 3.

(λ k+1 -λ i ) 2 ≤ 2 n k i=1 (λ k+1 -λ i )λ i . (4.2)
Proof. We will prove this theorem by applying inequality (2.2) with

A = L = -∆ H n , B 1 = x 1 , . . . , B n = x n , B n+1 = y 1 , . . . , B 2n = y n , T 1 = X 1 , . . . , T n = X n , T n+1 = Y 1 , . . . , T 2n = Y n and f (x) = g(x) = (λ k+1 -x) 2 , namely, n p=1 k i=1 (λ k+1 -λ i ) 2 [X p , x p ]u i , u i L 2 + [Y p , y p ]u i , u i L 2 2 ≤ 4 n p=1 k i=1 (λ k+1 -λ i ) 2 [L, x p ]u i , x p u i L 2 + [L, y p ]u i , y p u i L 2 × n p=1 k i=1 (λ k+1 -λ i ) X p u i 2 L 2 + Y p u i 2 L 2 . (4.3)
By a straightforward calculation, we obtain [L,

x p ]u i = -2X p u i and [L, y p ]u i = -2Y p u i . Hence [L, x p ]u i , x p u i L 2 = -2 Ω X p u i .x p u i = 2 Ω u i .X p (x p u i ) = 2 Ω u 2 i + 2 Ω x p u i .X p u i and [L, y p ]u i , y p u i L 2 = -2 Ω Y p u i .y p u i = 2 Ω u i .Y p (y p u i ) = 2 Ω u 2 i + 2 Ω y p u i .Y p u i , then [L, x p ]u i , x p u i L 2 = [L, y p ]u i , y p u i L 2 = 1. (4.4)
On the other hand, we have

[X p , x p ]u i = [Y p , y p ]u i = u i (4.5)
and

n p=1 X p u i 2 L 2 + n p=1 Y p u i 2 L 2 = Ω |∇ H n u i | 2 = λ i . (4.6)
Thus incorporating (4.4), (4.5) and (4.6) in (4.3), we obtain (4.2).

Remark 4.1. Inequality (4.2) improves the following inequality proved by Niu and Zhang in [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF] (see Remark 5.1 in [START_REF] Soufi | Universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds[END_REF])

λ k+1 -λ k ≤ 2 nk k i=1 λ i .
4.2. The case when l = 2. In this subsection, we will derive the following Theorem 4.2. We have, for each k = 1, 2, . . .,

k i=1 (λ k+1 -λ i ) 2 ≤ 2 √ n + 1 n k i=1 (λ k+1 -λ i )λ 1 2 i 1 2 k i=1 (λ k+1 -λ i ) 2 λ 1 2 i 1 2
.

(4.7)

Proof. The key observation here is to apply Theorem 2.1 with

A = L 2 = (-∆ H n ) 2
, and as before

B 1 = x 1 , B 2 = x 2 , • • • , B n = x n , B n+1 = y 1 , • • • , B 2n = y n , T 1 = X 1 , • • • , T n = X n , T n+1 = Y 1 , • • • , T 2n = Y n and f (x) = g(x) = (λ k+1 -x) 2 . Thus we have n p=1 k i=1 (λ k+1 -λ i ) 2 [X p , x p ]u i , u i L 2 + [Y p , y p ]u i , u i L 2 2 ≤ 4 n p=1 k i=1 (λ k+1 -λ i ) 2 [L 2 , x p ]u i , x p u i L 2 + [L 2 , y p ]u i , y p u i L 2 × n p=1 k i=1 (λ k+1 -λ i ) X p u i 2 L 2 + Y p u i 2 L 2 (4.8) but n p=1 X p u i 2 L 2 + n p=1 Y p u i 2 L 2 = Ω |∇ H n u i | 2 = Ω Lu i .u i ≤ Ω u 2 i 1 2 Ω Lu i 2 1 2 = λ 1 2 i , (4.9)
thus

n p=1 k i=1 (λ k+1 -λ i ) X p u i 2 L 2 + Y p u i 2 L 2 = k i=1 (λ k+1 -λ i )λ 1 2
i . (4.10) Using (4.5), we get

[X p , x p ]u i , u i L 2 = [Y p , y p ]u i , u i L 2 = 1. (4.11)
Thus,

n p=1 k i=1 (λ k+1 -λ i ) 2 ( [X p , x p ]u i , u i L 2 + [Y p , y p ]u i , u i L 2 ) 2 (4.12) = 4n 2 k i=1 (λ k+1 -λ i ) 2 2 .
On the other hand

[L 2 , x p ]u i = L 2 (x p u i ) -x p L 2 u i = L(x p Lu i -2X p u i ) -x p L 2 u i = -2X p Lu i -2L(X p u i ) (4.13)
and the same identity holds with y p and Y p . We infer, using identities (

[L 2 , x p ]u i , x p u i L 2 = -2 Ω X p Lu i .x p u i -2 Ω L(X p u i ).x p u i = -2 Ω X p Lu i .x p u i -2 Ω X p u i .x p Lu i + 4 Ω X p u i .X p u i = 2 Ω Lu i .X p (x p u i ) -2 Ω x p X p u i .Lu i -4 Ω X 2 p u i .u i =2 Ω Lu i .u i -4 Ω X 2 p u i .u i . 4.5) and (4.13)
Similarly, we have

[L 2 , y p ]u i , y p u i L 2 = 2 Ω Lu i .u i -4 Ω Y 2 p u i .u i . (4.15) Since - n p=1 Ω X 2 p u i .u i - n p=1 Ω Y 2 p u i .u i = n p=1 X p u i 2 L 2 + n p=1 Y p u i 2 L 2 = Ω Lu i .u i , we have n p=1 k i=1 (λ k+1 -λ i ) 2 [L 2 , x p ]u i , x p u i L 2 + [L 2 , y p ]u i , y p u i L 2 = 4(n + 1) k i=1 (λ k+1 -λ i ) 2 Ω Lu i .u i ≤ 4(n + 1) k i=1 (λ k+1 -λ i ) 2 Ω u 2 i 1 2 Ω Lu i 2 1 2 = 4(n + 1) k i=1 (λ k+1 -λ i ) 2 λ 1 2 i . ( 4
(λ k+1 -λ i ) 2 ≤ 4(n + 1) n 2 k i=1 (λ k+1 -λ i )λ i . (4.17)
Proof. Inequality (4.7) is equivalent to

k i=1 (λ k+1 -λ i ) 2 2 ≤ 4(n + 1) n 2 k i=1 (λ k+1 -λ i )λ 1 2 i k i=1 (λ k+1 -λ i ) 2 λ 1 2
i .

Now applying Lemma 3.2 with

A i = λ k+1 -λ i and B i = C i = λ 1 2
i , we obtain inequality (4.17). Remark 4.2. Inequality (4.7) is sharper than the following one found by Niu and Zhang [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF]

λ k+1 -λ k ≤ 4(n + 1) n 2 k 2 k i=1 λ 1 2 i 2 .
Proof. We infer from inequality (4.7) and the Chebyshev inequality

k i=1 (λ k+1 -λ i ) 2 2 ≤ 4(n + 1) n 2 k 2 k i=1 (λ k+1 -λ i ) k i=1 (λ k+1 -λ i ) 2 k i=1 λ 1 2 2 , or equivalently k i=1 (λ k+1 -λ i ) 2 ≤ 4(n + 1) n 2 k 2 k i=1 (λ k+1 -λ i ) k i=1 λ 1 2 2 . Thus k i=1 (λ k+1 -λ i ) (λ k+1 -λ i ) - 4(n + 1) n 2 k 2 k i=1 λ 1 2 i 2 ≤ 0. (4.18)
Hence, since λ i ≤ λ k , for all i ≤ k, we can easily deduce the inequality of Niu and Zhang from (4.18).

4.3.

The case when l ≥ 3. We are now concerned with the problem (4.1) for any l ≥ 3. The result depends on the parity of l. In fact, we prove the following Theorem 4.3. For any odd l ≥ 3, we have

k i=1 (λ k+1 -λ i ) 2 ≤ 1 n k i=1 (λ k+1 -λ i )λ 1 l i 1 2 × k i=1 (λ k+1 -λ i ) 2 2l(n + l -1) λ l-1 l i + c 1 (n, l) λ i + λ l-2 l i 1 2 (4.19)
and for any even l ≥ 4, we have

k i=1 (λ k+1 -λ i ) 2 ≤ 1 n k i=1 (λ k+1 -λ i )λ 1 l i 1 2 × k i=1 (λ k+1 -λ i ) 2 2ln + 4(l -1) λ l-1 l i + c 2 (n, l)λ l-1 l i 1 2 , (4.20)
where c 1 (n, l) and c 2 (n, l) are two constants depending on n and l.

Proof. If we apply inequality (2.2) with

A = L l = (-∆ H n ) l , B 1 = x 1 , . . . , B n = x n , B n+1 = y 1 , . . . , B 2n = y n , T 1 = X 1 , . . . , T n = X n , T n+1 = Y 1 , . . . , T 2n = Y n and f (x) = g(x) = (λ k+1 -x) 2 , then we obtain k i=1 n p=1 (λ k+1 -λ i ) 2 [X p , x p ]u i , u i L 2 + [Y p , y p ]u i , u i L 2 2 ≤ 4 k i=1 n p=1 (λ k+1 -λ i ) 2 [L l , x p ]u i , x p u i L 2 + [L l , y p ]u i , y p u i L 2 × k i=1 n p=1 (λ k+1 -λ i ) X p u i 2 L 2 + Y p u i 2 L 2 . (4.21)
And as before, we have

[X p , x p ]u i , u i L 2 = [Y p , y p ]u i , u i L 2 = 1. (4.22)
On the other hand, to calculate n p=1

X p u i 2 L 2 + Y p u i 2
L 2 , we need the following result obtained by Niu and Zhang (see Lemma 2.3 in [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF]) inspired by that of Chen and Qian [START_REF] Chen | Estimates for discrete spectrum of the laplacian operator with any order[END_REF] for the Laplacian: Lemma 4.1. For any d ≥ 1, we have

Ω |∇ d H n u i | 2 1 d ≤ Ω |∇ d+1 H n u i | 2 1 d+1 (4.23) where ∇ d = L d 2 if d is even, ∇ H n L d-1 2 if d is odd.
And as a consequence (see Corollary 2.1 in [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF]), we can easily obtain, for any d ≥ 1

Ω |∇ H n u i | 2 ≤ Ω |∇ d H n u i | 2 1 d . ( 4.24)
Therefore we have

n p=1 X p u i 2 L 2 + Y p u i 2 L 2 = Ω Lu i .u i ≤ Ω L l u i .u i 1 l = λ 1 l i . (4.25) Now we have to calculate n p=1 [L l , x p ]u i , x p u i L 2 + [L l , y p ]u i , y p u i L 2 .
For this purpose, we use the following lemma also obtained by Niu and Zhang in [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF] Lemma 4.2. For any positive integer d, 1 ≤ d ≤ l, we have

L d (x p u i ) = x p L d u i -2 d q=1 L d-q X p L q-1 u i , i = 1, . . . , k, p = 1, . . . ,
n. This is also true for y p and Y p . We infer, using Lemma 4.2,

[L l , x p ]u i = L l (x p u i ) -x p L l u i = -2 l q=1 L l-q X p L q-1 u i . Therefore [L l , x p ]u i , x p u i L 2 = -2 l q=1 Ω L l-q X p L q-1 u i .x p u i = -2 l q=1 Ω X p L q-1 u i .L l-q (x p u i ).
The same identities hold with y p and Y p . Hence we obtain

n p=1 [L l , x p ]u i , x p u i L 2 + [L l , y p ]u i , y p u i L 2 = -2 n p=1 l q=1 Ω X p L q-1 u i .L l-q (x p u i )+ Ω Y p L q-1 u i .L l-q (y p u i ) . ( 4
[L l , x p ]u i , x p u i L 2 + [L l , y p ]u i , y p u i L 2 = -2 n p=1 l q=1 Ω x p L l-q u i .X p L q-1 u i + 4 n p=1 l-1 q=1 Ω X p L l-q-1 u i .X p L q-1 u i + 4 n p=1 l-2 q=1 l-q-1 r=1 Ω L l-q-r X p L r-1 u i .X p L q-1 u i -2 n p=1 l q=1 Ω y p L l-q u i .Y p L q-1 u i + 4 n p=1 l-1 q=1 Ω Y p L l-q-1 u i .Y p L q-1 u i + 4 n p=1 l-2 q=1 l-q-1 r=1 Ω L l-q-r Y p L r-1 u i .Y p L q-1 u i . (4.27)
As in the proof of Theorem 5.1 in [START_REF] Niu | Payne-Polya-Weinberger type inequalities for eigenvalues of nonelliptic operators[END_REF] (see the calculation of the terms I 2 and I ′ 2 ), we can easily obtain, for any odd l ≥ 3,

n p=1 [L l , x p ]u i , x p u i L 2 + [L l , y p ]u i , y p u i L 2 ≤ 2l(n + l -1) λ l-1 l i +c 1 (n, l) λ i + λ l-2 l i (4.28)
and for any even l ≥ 4,

n p=1 [L l , x p ]u i , x p u i L 2 + [L l , y p ]u i , y p u i L 2 ≤ 2ln + 4(l -1) +c 2 (n, l) λ l-1 l i (4.29)
where c 1 (n, 3) = 4, (4.19) is not homogeneous in the eigenvalues λ i (i.e. it is not invariant under the change L → aL, λ i → a l λ i for a > 0). Therefore, using inequality (4.19) for a l L l , we obtain that for any a > 0, which is homogeneous on the eigenvalues λ i .

c 1 (n, l) = 2 l-2 q=1 l-q-1 r=1 l-q-r s=1 s odd 2 s nC s l-q-r (2n -1) s+1 2 + l-q-r s=2 s even 2 s C s l-q-r (2n -1) s 2 for any odd l ≥ 5 and c 2 (n, l) = 4 l-2 q=1 l-q-1 r=1 l-q-r s=1 s odd 2 s nC s l-q-r (2n -1) s+1 2 + l-q-r s=0 s even 2 s C s l-q-r (2n -
k i=1 (λ k+1 -λ i ) 2 ≤ 1 n k i=1 (λ k+1 -λ i )λ
As for the case when l = 2, we can deduce inequalities of Yang-type for l ≥ 3. Similarly, we prove that inequality (4.20) is sharper than (4.36).

S. Ilias, O. Makhoul: Université Franc ¸ois rabelais de Tours, Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France E-mail address: ilias@univ-tours.fr, ola.makhoul@lmpt.univ-tours.fr

g(x) 2 =

 2 (3.8),(3.10) and(3.11) into (2.2) and taking f (x) = (λ k+1x), we obtain inequality(1.10).

2 Su| 2 +

 22 and S l,2 0 denotes the closure of C ∞ 0 (Ω) with respect to the Sobolev norm u |u| 2 dxdydt.

Theorem 4 . 1 .

 41 For any k ≥ 1 k i=1

for any odd l ≥ 3 .

 3 λ i ) 2 2l(n + l -1)λ l-1 l i + c 1 (n, l) aλ i +Optimising with respect to a, we find the following improvement of the inequality (4.[START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF])k i=1 (λ k+1λ i ) 2 ≤ 1 n 2l(n + l -1) + c 1 (n, l)

Corollary 4 . 2 . 4 .

 424 We have, for any odd l ≥ 3, λ i ) 2l(n + l -1)λ i + c 1 (n, l) λ λ i ) 2 ≤ 2ln + 4(l -1) + c 2 (n, l) n 2 k i=1 (λ k+1λ i )λ i . (4.33)where c 1 (n, l) and c 2 (n, l) are explicit constants depending only on n and l.Proof. Applying Lemma 3.2 withA i = λ k+1λ i , B i = 2l(n + l -λ i ) 2 2l(n + l -λ i ) 2 k i=1 (λ k+1λ i ) 2l(n + l -1)λ i + c 1 (n, l) λ 32) can be deduced from (4.19) and (4.34), for any odd l ≥ 3. We proceed in the same way to obtain (4.33), i.e. applying Lemma3.2 but withA i = λ k+1λ i , B i = 2ln + 4(l -1) + c 2 (n, l) λ Inequalities (4.[START_REF] Harrell | On trace identities and universal eigenvalue estimates for some partial differential operators[END_REF]) and(4.20) are sharper than the following inequalities, proved byNiu and Zhang ([25]),λ k+1 -λ k ≤ if l ≥ 3 is odd and λ k+1λ k ≤ if l ≥ 4 is even, c 1 (n, l) and c 2 (n, l) are as in the proof of Theorem 4.3.Proof. By the Chebyshev inequality, we infer from (4.19), for any odd l -λ i ) λ k+1λ iimplies (4.35), since λ i ≤ λ k for i ≤ k.

  If the operators T p are chosen such thatT p = [A, B p ], then [T p , B p ] = [[A, B p ] , B p ]. Applying Theorem 2.1 in this context and using the obvious identity [A, B p ] u i , B p u i = -1 2 [[A, B p ] , B p ] u i , u i ,we obtain Corollary 2.1. Let A : D ⊂ H -→ H be a self-adjoint operator defined on a dense domain D, which is semibounded below and has a discrete spectrum λ 1 ≤ λ 2 ≤ λ 3 .... Let {B p : A(D) -→ H} n p=1 be a collection of symmetric operators, leaving D invariant. We denote by {u i } ∞ i=1 a basis of orthonormal eigenvectors of A, u i corresponding to

	of Ash-
	baugh and Hermi.

  Corollary 4.1. We have, for each k ≥ 1,

	.16)
	Incorporating (4.10), (4.12) and (4.16) in (4.8), we get the result.
	We can easily obtain from inequality (4.7) of Theorem 4.2 an in-
	equality of Yang-type.
	k
	i=1

  [START_REF] Ashbaugh | Isoperimetric and universal inequalities for eigenvalues[END_REF]

	s	.
	2
	Incorporating (4.22), (4.25) and (4.28) in (4.21), we obtain (4.19).
	Similarly, to obtain (4.20), we incorporate (4.22), (4.25) and (4.29)
	in (4.21).
	Remark 4.3. The inequality

Acknowledgments. We thank the referee for the suggestions which allowed us to improve the first version of the paper. This work was partially supported by the ANR (Agence Nationale de la Recherche) through FOG project(ANR-07-BLAN-0251-01).


"UNIVERSAL" INEQUALITIES FOR THE EIGENVALUES OF THE BIHARMONIC OPERATOR SA ÏD ILIAS AND OLA MAKHOUL

Keywords: Mathematics Subject Classification. 35P15, 58J50, 58C40, 58A10 eigenvalues, biharmonic operator, Universal inequalities, submanifolds, eigenmap. 1

In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherical eigenmaps (this includes the compact homogeneous Riemannian spaces) and finally on domains of the hyperbolic space.

Introduction

Let (M, g) be a Riemannian manifold of dimension n and let ∆ be the Laplacian operator on M. In this paper, we will be concerned with the following eigenvalue problem for Dirichlet biharmonic operator, called the clamped plate problem:

   ∆ 2 u = λu in Ω u = ∂u ∂ν = 0 on ∂Ω, (1.1)
where Ω is a bounded domain in M, ∆ 2 the biharmonic operator in M and ν is the outward unit normal. It is well known that the eigenvalues of this problem form a countable family 0 < λ 1 ≤ λ 2 ≤ . . . → +∞.

For the case when M = R n , in 1956, Payne, Polya and Weinberger [START_REF] Payne | On the ratio of consecutive eigenvalues[END_REF] (henceforth PPW) established the following inequality, for each k ≥ 1,

λ k+1 -λ k ≤ 8(n + 2) n 2 k k i=1 λ i .
Implicit in the PPW work, as noticed by Ashbaugh in [START_REF] Ashbaugh | Isoperimetric and universal inequalities for eigenvalues[END_REF], is the better inequality

λ k+1 -λ k ≤ 8(n + 2) n 2 k 2 k i=1 λ 1 2 i 2 . (1.2)
Later, in 1984, Hile and Yeh [START_REF] Hile | Inequalities for eigenvalues of the biharmonic operator[END_REF] extended ideas from earlier work on the Laplacian by Hile and Protter [START_REF] Hile | Inequalities for eigenvalues of the Laplacian[END_REF] and proved the better bound

n 2 k 3 2 8(n + 2) ≤ k i=1 λ 1 2 i λ k+1 -λ i k i=1 λ i 1 2 .
Implicit in their work is the stronger inequality

n 2 k 2 8(n + 2) ≤ k i=1 λ 1 2 i λ k+1 -λ i k i=1 λ 1 2 i ,
which was proved independently by Hook [START_REF] Hook | Domain-independent upper bounds for eigenvalues of elliptic operators[END_REF] and Chen and Qian [START_REF] Chen | Estimates for discrete spectrum of the laplacian operator with any order[END_REF] in 1990 (see also [START_REF] Chen | Hile-Yeh estimates of eigenvalues of polyharmonic operators[END_REF], [START_REF] Chen | On the difference of consecutive eigenvalues of uniformly elliptic operators of higher orders[END_REF] and [START_REF] Chen | On the upper bound of eigenvalues for elliptic equations with higher orders[END_REF]).

In 2003, Cheng and Yang [START_REF] Cheng | Inequalities for eigenvalues of a clamped plate problem[END_REF] obtained the following bound

k i=1 (λ k+1 -λ i ) ≤ 8(n + 2) n 2 1 2 k i=1 λ i (λ k+1 -λ i ) 1 2 . (1.3)
Very recently, Cheng, Ichikawa and Mametsuka [START_REF] Cheng | Inequalities for eigenvalues of laplacian with any order[END_REF] obtained an inequality for eigenvalues of Laplacian with any order l on a bounded domain in R n . In particular, they showed, for l = 2,

k i=1 (λ k+1 -λ i ) 2 ≤ 8(n + 2) n 2 k i=1 (λ k+1 -λ i )λ i . (1.4)
For the case when M = S n , Wang and Xia [START_REF] Wang | Universal bounds for eigenvalues of the biharmonic operator on riemannian manifolds[END_REF] showed

k i=1 (λ k+1 -λ i ) 2 ≤ 1 n k i=1 (λ k+1 -λ i ) 2 n 2 + (2n + 4)λ 1 2 i 1 2 × k i=1 (λ k+1 -λ i ) n 2 + 4λ 1 2 i 1 2
, (1.5) from which they deduced, using a variant of Chebyshev inequality,

k i=1 (λ k+1 -λ i ) 2 ≤ 1 n 2 k i=1 (λ k+1 -λ i ) 2(n+2)λ 1 2 i +n 2 4λ 1 2 i +n 2 . (1.6)
This last inequality was also obtained, by a different method, by Cheng, Ichikawa and Mametsuka (see [START_REF] Cheng | Estimates for eigenvalues of the poly-laplacian with any order in a unit sphere[END_REF]).

On the other hand, Wang and Xia [START_REF] Wang | Universal bounds for eigenvalues of the biharmonic operator on riemannian manifolds[END_REF] also considered the problem (1.1) on domains of an n-dimensional complete minimal submanifold M of R m and proved in this case

k i=1 (λ k+1 -λ i ) 2 ≤ 8(n + 2) n 2 1 2 k i=1 (λ k+1 -λ i ) 2 λ 1 2 i 1 2 k i=1 (λ k+1 -λ i )λ 1 2 i 1 2
, (1.7) from which they deduced the following generalization of inequality (1.4) to minimal Euclidean submanifolds

k i=1 (λ k+1 -λ i ) 2 ≤ 8(n + 2) n 2 k i=1 (λ k+1 -λ i )λ i . (1.8)
Recently, Cheng, Ichikawa and Mametsuka [START_REF] Cheng | Estimates for eigenvalues of a clamped plate problem on riemannian manifolds[END_REF] extended this last inequality to any complete Riemannian submanifold M in R m and showed

k i=1 (λ k+1 -λ i ) 2 ≤ 1 n 2 k i=1 (λ k+1 -λ i ) n 2 δ + 2(n + 2)λ 1 2 i n 2 δ + 4λ 1 2 i ,
(1.9) with δ = Sup Ω |H| 2 , where H is the mean curvature of M. The goal of the first section of this article is to study the relation between eigenvalues of the biharmonic operator and the local geometry of Euclidean submanifolds M of arbitrary codimensions. The approach is based on an algebraic formula (see theorem 2.1) we proved in [START_REF] Ilias | Universal inequalities for the eigenvalues of a power of the laplace operator[END_REF]. This approach is useful for the unification and for the generalization of all the results in the literature. In fact, using this general algebraic inequality, we obtain (see theorem 2.2) the following inequality

k i=1 f (λ i ) ≤ 1 n k i=1 g(λ i ) 2(n + 2)λ 1 2 i + n 2 δ 1 2 × k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) 4λ 1 2 i + n 2 δ 1 2
, (1.10)

where f and g are two functions satisfying some functional conditions (see Definition 2.1), δ = Sup Ω |H| 2 and H is the mean curvature of M. We note that the family of such couples of functions is large. And particular choices for f and g lead to the known results. For instance, if we take f (x) = g(x) = (λ k+1x) 2 , then (1.10) becomes

k i=1 (λ k+1 -λ i ) 2 ≤ 1 n k i=1 (λ k+1 -λ i ) 2 2(n + 2)λ 1 2 i + n 2 δ 1 2 × k i=1 (λ k+1 -λ i ) 4λ 1 2 i + n 2 δ 1 2
, (1.11) which gives easily (see remark 2.1) inequality (1.9) of Cheng, Ichikawa and Mametsuka.

In the second section, we consider the case of manifolds admitting spherical eigenmaps and obtain similar results. As a consequence, we obtain universal inequalities for the clamped plate problem on domains of any compact homogeneous Riemannian manifold.

In the last section, we show how one can easily obtain, from the algebraic techniques used in the previous sections, universal inequalities for eigenvalues of (1.1) on domains of the hyperbolic space H n . We also observe that all our results hold if we add a potential to ∆ 2 (i.e ∆ 2 + q where q is a smooth potential). For instance, in this case instead of inequality (1.10), we obtain

k i=1 f (λ i ) ≤ 1 n k i=1 g(λ i ) 2(n + 2)λ i 1 2 + n 2 δ 1 2 × k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) 4λ i 1 2 + n 2 δ 1 2
, (1.12)

where

λ i = λ i -inf Ω q.
Finally, note that the case of the clamped problem with weight :

   ∆ 2 u = λ ρ u in Ω u = ∂u ∂ν = 0 on ∂Ω, (1.13)
can be easily treated with minor changes.

Euclidean Submanifolds

Before stating the main result of this section, we introduce a family of couples of functions and a theorem obtained earlier in [START_REF] Ilias | Universal inequalities for the eigenvalues of a power of the laplace operator[END_REF], which will play an essential role in the proofs of all our results. Definition 2.1. Let λ > 0. A couple (f, g) of functions defined on ]0, λ[ belongs to ℑ λ provided that 1. f and g are positive, 2. f and g satisfy the following condition, for any

x, y ∈]0, λ[ such that x = y, f (x) -f (y) x -y 2 + f (x) 2 g(x)(λ -x) + f (y) 2 g(y)(λ -y) g(x) -g(y)
xy ≤ 0.

(2.1)

A direct consequence of our definition is that g must be nonincreasing. If we multiply f and g of ℑ λ by positive constants the resulting functions are also in ℑ λ . In the case where f and g are differentiable, one can easily deduce from (2.1) the following necessary condition:

ln f (x) ′ 2 ≤ -2 λ -x ln g(x) ′ .
This last condition helps us to find many couples (f, g) satisfying the conditions 1) and 2) above. Among them, we mention 1, (λx) α / α ≥ 0 ,

(λ -x), (λ -x) β / β ≥ 1 2 , (λ -x) δ , (λ -x) δ / 0 < δ ≤ 2 .
Let H be a complex Hilbert space with scalar product ., . and corresponding norm . . For any two operators A and B, we denote by [A, B] their commutator, defined by [A, B] = AB -BA.

Theorem 2.1. Let A : D ⊂ H -→ H be a self-adjoint operator defined on a dense domain D, which is semibounded below and has a discrete spectrum λ 1 ≤ λ 2 ≤ λ 3 .... Let {T p : D -→ H} n p=1 be a collection of skew-symmetric operators, and {B p : T p (D) -→ H} n p=1 be a collection of symmetric operators, leaving D invariant. We denote by {u i } ∞ i=1 a basis of orthonormal eigenvectors of A, u i corresponding to λ i . Let k ≥ 1 and assume that λ k+1 > λ k . Then, for any

(f, g) in ℑ λ k+1 k i=1 n p=1 f (λ i ) [T p , B p ]u i , u i 2 (2.2) ≤ 4 k i=1 n p=1 g(λ i ) [A, B p ]u i , B p u i k i=1 n p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) T p u i 2 .
Our first result is the following application of this inequality to the eiganvalues of the clamped plate problem (1.1) on a domain of a Euclidean submanifold :

Theorem 2.2. Let X : M -→ R m be an isometric immersion of an n-dimensional Riemannian manifold M in R m .
Let Ω be a bounded domain of M and consider the clamped plate problem (1.1) on it. For any k ≥ 1 such that λ k+1 > λ k and for any (f, g) in ℑ λ k+1 , we have

k i=1 f (λ i ) ≤ 2 n k i=1 g(λ i ) 2(n + 2)λ 1 2 i + n 2 δ 1 2 × k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) λ 1 2 i + n 2 4 δ 1 2 , (2.3)
where δ = sup Ω |H| 2 and H is the mean curvature vector field of the immersion X (i.e which is given by 1 n trace h, where h is the second fundamental form of X).

Proof. To prove this theorem, we apply inequality (2.2) of Theorem 2.1 with A = ∆ 2 , B p = X p and T p = [∆, X p ], p = 1, . . . , m, where X 1 , . . . , X m are the components of the immersion X. This gives

k i=1 m p=1 f (λ i ) [[∆, X p ], X p ]u i , u i L 2 2 (2.4) ≤ 4 k i=1 m p=1 g(λ i ) [∆ 2 , X p ]u i , X p u i L 2 k i=1 m p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) [∆, X p ]u i
where u i are the L 2 -normalized eigenfunctions. First we have, for any p = 1, . . . , m,

[∆ 2 , X p ]u i =∆ 2 X p u i + 2∇∆X p .∇u i + 2∆(∇X p .∇u i ) + 2∆X p ∆u i + 2∇X p .∇∆u i . Thus [∆ 2 , X p ]u i , X p u i L 2 = Ω u 2 i X p ∆ 2 X p + 2 Ω X p u i ∇∆X p .∇u i + 2 Ω X p u i ∆(∇X p .∇u i ) + 2 Ω X p u i ∆X p ∆u i + 2 Ω X p u i ∇X p .∇∆u i (2.5) = Ω ∆X p ∆ X p u 2 i -2 Ω div X p u i ∇u i ∆X p + 2 Ω ∆ X p u i ∇X p .∇u i + 2 Ω X p ∆X p u i ∆u i -2 Ω div X p u i ∇X p ∆u i .
(2.6)

A straightforward calculation gives [∆ 2 , X p ]u i , X p u i L 2 = 4 Ω u i ∆X p ∇X p .∇u i + Ω ∆X p 2 u 2 i + 4 Ω ∇X p .∇u i 2 -2 Ω |∇X p | 2 u i ∆u i . (2.7)
Since X is an isometric immersion, we have

nH = (∆X 1 , . . . ,∆X m ), m p=1 u i ∆X p ∇X p .∇u i = 0 and m p=1 ∇X p .∇u i 2 = |∇u i | 2 . (2.8)
Incorporating these identities in (2.7) and summing on p, from 1 to m, we obtain

m p=1 [∆ 2 , X p ]u i , X p u i L 2 = 4 Ω |∇u i | 2 -2n Ω u i ∆u i + n 2 Ω |H| 2 u 2 i = 2(n + 2) Ω u i (-∆u i ) + n 2 Ω |H| 2 u 2 i ≤ 2(n + 2) Ω (-∆u i ) 2 1 2 Ω u 2 i 1 2 + n 2 Ω |H| 2 u 2 i (2.9) =2(n + 2)λ 1 2 i + n 2 Ω |H| 2 u 2 i ≤2(n + 2)λ 1 2 i + n 2 δ, (2.10)
where we used the Cauchy-Schwarz inequality to obtain (2.9) and where δ = sup Ω |H| 2 . On the other hand, we have

[∆, X p ]u i = 2∇X p .∇u i + u i ∆X p , then m p=1 [∆, X p ]u i 2 L 2 = m p=1 Ω 2∇X p .∇u i + u i ∆X p 2 =4 m p=1 Ω ∇X p .∇u i 2 + 4 m p=1 Ω u i ∆X p ∇X p .∇u i + m p=1 Ω (∆X p ) 2 u 2 i .
Using identities (2.8), we obtain

m p=1 [∆, X p ]u i 2 L 2 =4 Ω |∇u i | 2 + n 2 Ω |H| 2 u 2 i =4 Ω (-∆u i ).u i + n 2 Ω |H| 2 u 2 i ≤4 Ω (-∆u i ) 2 1 2 Ω u 2 i 1 2 + n 2 δ =4λ 1 2 i + n 2 δ. (2.11) A direct calculation gives [[∆, X p ], X p ]u i , u i L 2 = Ω ∆(X 2 p u i )-2X p ∆(X p u i )+X 2 p ∆u i u i = 2 Ω |∇X p | 2 u 2 i . Therefore m p=1 [[∆, X p ], X p ]u i , u i L 2 = 2 m p=1 Ω |∇X p | 2 u 2 i = 2n. (2.12)
To conclude, we simply use the estimates (2.10), (2.11) and (2.12) together with inequality (2.4).

Remark 2.1.

• As indicated in the end of the introduction, Theorem 2.2 holds for a general operator ∆ 2 +q, where q is a smooth potential. Indeed, this is an immediate consequence of the fact that [∆ 2 + q, X p ] = [∆ 2 , X p ] and all the proof of Theorem 2.2 works in this situation. The only modification is in the estimation of the term Ω |∇u i | 2 . In fact, in this case, we have

Ω |∇u i | 2 ≤ Ω (-∆u i ) 2 1 2 Ω u 2 i 1 2 = λ i - Ω qu 2 i 1 2 ≤ λ i 1 2
,

where λ i = λ i -inf Ω q.
Taking into account this modification in inequalities (2.9) and (2.11), we obtain inequality (1.12).

• If f (x) = g(x) = (λ k+1 -x) 2 ,

then inequality (2.3) extends

inequality (1.7) of Wang and Xia to any Riemannian submanifolds of R m . We also observe that, by using a Chebyshev inequality (for instance the one of Lemma 1 in [START_REF] Cheng | Inequalities for eigenvalues of laplacian with any order[END_REF]), inequality (1.9) of Cheng, Ishikawa and Mametsuka can be easily deduced from inequality (2.3).

• If f (x) = g(x) 2 = (λ k+1 -x), then inequality (2.

3) generalizes inequality (1.3) of Cheng and Yang to the case of Euclidean submanifolds.

Using the standard emdeddings of the rank one compact symmetric spaces in a Euclidean space (see for instance Lemma 3.1 in [START_REF] Soufi | Universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds[END_REF] for the values of |H| 2 of these embeddings), we can extend easily the previous theorem to domains or submanifolds of these symmetric spaces and obtain Theorem 2.3. Let M be the sphere S m , the real projective space RP m , the complex projective space CP m or the quaternionic projective space QP m endowed with their respective metrics. Let (M, g) be a compact Riemannian manifold of dimension n and let X : M -→ M be an isometric immersion of mean curvature H. Consider the clamped plate problem on a bounded domain Ω of M. For any k ≥ 1 such that λ k+1 > λ k and for any (f, g) ∈ ℑ λ k+1 , we have

k i=1 f (λ i ) ≤ 2 n k i=1 g(λ i ) 2(n + 2)λ 1 2 i + n 2 δ ′ 1 2 × k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) λ 1 2 i + n 2 4 δ ′ 1 2 , (2.13
)

where δ ′ = sup(|H| 2 + 1) if M = S m , δ ′ = sup(|H| 2 + d(n)), where d(n) =      2(n+1) n if M = RP m , 2(n+2) n if M = CP m , 2(n+4) n if M = QP m .
Remark 2.2.

• We observe (as in Remark 3.2 of [START_REF] Soufi | Universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds[END_REF]) that in some special geometrical situations, the constant d(n) in the inequality of Theorem 2.3 can be replaced by a sharper one. For instance, when 2 , and M is a sphere, (2.13) generalizes to submanifolds inequality (1.5) established by Wang and Xia for spherical domains. • As for Theorem 2.2, the result of Theorem 2.3 holds for a more general operator ∆ 2 + q, with the same modification (i.e λi

M = CP m and -M is odd-dimensional, then one can replace d(n) by d ′ (n) = 2 n (n + 2 -1 n ), -X(M) is totally real, then d(n) can be replaced by d ′ (n) = 2(n+1) n . • When f (x) = g(x) = (λ k+1 -x)
1 2 instead of λ 1 2
i ).

Manifolds admitting spherical eigenmaps

In this section, as before, we let (M, g) be a Riemannian manifold and Ω be a bounded domain of M. A map X : (M, g) → S m-1 is called an eigenmap if its components X 1 , X 2 , . . . , X m are all eigenfunctions associated to the same eigenvalue λ of the Laplacian of (M, g). This is equivalent to say that the map X is a harmonic map from (M, g) into S m-1 with constant energy λ i.e m p=1 |∇X p | 2 = λ . The most important examples of such manifolds M are the compact homogeneous Riemannian manifolds. In fact, they admit eigenmaps for all the positive eigenvalues of their Laplacian (see [START_REF] Li | Eigenvalue estimates on homogeneous manifolds[END_REF]). Theorem 3.1. Let λ be an eigenvalue of the Laplacian of (M, g) and suppose that (M, g) admits an eigenmap X associated to this eigenvalue λ. Let Ω be a bounded domain of M and consider the clamped plate problem (1.1) on it. For any k ≥ 1 such that λ k+1 > λ k and for any (f, g) ∈ ℑ λ k+1 , we have

k i=1 f (λ i ) ≤ k i=1 g(λ i ) λ + 6λ 1 2 i 1 2 k i=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) λ + 4λ 1 2 i 1 2 . (3.1)
Proof. As in the proof of Theorem 2.2, we apply Theorem 2.1 with

A = ∆ 2 , B p = X p and T p = [∆, X p ], p = 1, . . . , m, to obtain k i=1 m p=1 f (λ i ) [[∆, X p ], X p ]u i , u i L 2 2 ≤ 4 k i=1 m p=1 g(λ i ) [∆ 2 , X p ]u i , X p u i L 2 k i=1 m p=1 f (λ i ) 2 g(λ i )(λ k+1 -λ i ) [∆, X p ]u i 2 L 2 , (3.2)
where {u i } ∞ i=1 is a complete L 2 -orthonormal basis of eigenfunctions of ∆ 2 associated to {λ i } ∞ i=1 . As before, we have

m p=1 [[∆, X p ], X p ]u i , u i L 2 = 2 m p=1 Ω |∇X p | 2 u 2 i . Since m p=1 |∇X p | 2 = λ, m p=1 X 2 p = 1 and ∆X p = -λX p , (3.3)
we have

m p=1 [[∆, X p ], X p ]u i , u i L 2 = 2λ (3.4) and m p=1 [∆, X p ]u i 2 L 2 = m p=1 Ω [∆, X p ]u i 2 (3.5) = 4 Ω m p=1 (∇X p .∇u i ) 2 + Ω m p=1 (∆X p ) 2 u 2 i + 4 Ω m p=1 u i ∆X p ∇X p .∇u i ≤ 4 Ω m p=1 |∇X p | 2 |∇u i | 2 + λ 2 Ω m p=1 X 2 p u 2 i -2λ Ω u i ∇ m p=1 X 2 p .∇u i (3.6) = 4λ Ω (-∆u i )u i + λ 2 ≤ 4λ Ω (-∆u i ) 2 1 2 Ω u 2 i 1 2 + λ 2 (3.7) = 4λλ 1 2 i + λ 2 , (3.8)
where we used the Cauchy-Schwarz inequality to obtain (3.6) and (3.7).

Similarly we infer, from identities (2.7) and (3.3),

m p=1 [∆ 2 , X p ]u i , X p u i L 2 =λ 2 Ω u 2 i -λ Ω ∇ m p=1 X 2 p .∇u 2 i + 4 m p=1 Ω ∇X p .∇u i 2 + 2λ Ω (-∆u i )u i ≤ λ 2 + 4 Ω m p=1 |∇X p | 2 |∇u i | 2 + 2λ Ω -∆u 2 1 2 Ω u 2 i 1 2
≤λ 2 + 4λλ

1 2 i + 2λλ 1 2 i =λ 2 + 6λλ 1 2 i , (3.9)
Incorporating (3.4), (3.8) and (3.9) in inequality (3.2), we get the statement of the theorem.

An immediate consequence of Theorem 3.1 is the following Corollary 3.1. Let (M, g) be a compact homogeneous Riemannian manifold without boundary and let λ 1 be the first non zero eigenvalue of its Laplacian . Then inequality of Theorem 3.1 holds with λ = λ 1 .

Remark 3.1. As before, one can get a similar result for the operator ∆ 2 + q.

domains in the hyperbolic space

We turn next to the case of a domain Ω of a hyperbolic space. It is easy to establish a universal inequality for eigenvalues of the clamped plate problem (1.1)on Ω in the vein of the preceding ones. Unfortunately, until now we have not succeeded in obtaining a simple generalization for the case of domains of hyperbolic submanifolds. In what follows, we take the half-space model for H n i.e

H n = {x = (x 1 , x 2 , . . . , x n ) ∈ R n ; x n > 0}
with the standard metric

ds 2 = dx 2 1 + dx 2 2 + . . . + dx 2 n x 2 n .
We note that in terms of the coordinates (x i ) n i=1 , the Laplacian of H n is given by

∆ = x 2 n n j=1 ∂ 2 ∂x j ∂x j + (2 -n)x n ∂ ∂x n . Theorem 4.1. For any k ≥ 1 such that λ k+1 > λ k , the eigenvalues λ ′ i s of the clamped problem (1.1) on the bounded domain Ω of H n must satisfy, for any (f, g) ∈ ℑ λ k+1 , k i=1 f (λ i ) ≤ k i=1 g(λ i ) 6λ 1 2 i -(n -1) 2 1 2 × k i=1 (f (λ i )) 2 g(λ i )(λ k+1 -λ i ) 4λ 1 2 i -(n -1) 2 1 2 , (4.1)
Proof. Theorem 2.1 remains valid for A = ∆ 2 , B p = F = ln x n and T p = [∆, F ], for all p = 1, . . . , n. Thus, denoting by u i the eigenfunction corresponding to λ i , we have

k i=1 f (λ i ) [[∆, F ], F ]u i , u i L 2 2 ≤ 4 k i=1 g(λ i ) [∆ 2 , F ]u i , F u i L 2 k i=1 (f (λ i )) 2 g(λ i )(λ k+1 -λ i ) [∆, F ]u i 2 L 2 . (4.2)
Let us start by the calculation of

[[∆, F ], F ]u i , u i L 2 = Ω [∆, F ](F u i ) -F [∆, F ]u i u i = Ω ∆(F 2 u i ) -2F ∆(F u i ) + F 2 ∆u i u i .
We note that ∆F = 1n and |∇F |

  F ], F ]u i , u i L 2 = 2 , F ]u i =∆ 2 (F u i ) -F ∆ 2 u i =∆(∆F u i + 2∇F.∇u i + F ∆u i ) -F ∆ 2 u i =2(1n)∆u i + 2∆(∇F.∇u i ) + 2∇F.∇∆u i ,

	On the other hand, using again identities (4.3), we obtain
	[∆, F ]u i	2 L 2 =	(∆F u i + 2∇F.∇u i ) 2
		Ω
		=	(∆F ) 2 u 2 i + 4	(∇F.∇u i ) 2 + 4	∆F u i ∇F.∇u i
		Ω			Ω		Ω	(4.5)
		=(1 -n) 2 + 4
							(4.6)
	But
							u 2 i ∆F,
							Ω
	hence			Ω	u i ∇F.∇u i =	n -1 2	.	(4.7)
							1	1
							u 2 i	2	(-∆u i ) 2	2	(4.8)
						Ω	Ω
		=4λ	1 2 i -(n -1) 2		(4.9)
	Now,
	[∆ 2 (4.10)
	thus
					[∆ 2 , F ]u i , F u i L 2
	= 2(1 -n)
							2 = 1.	(4.3)
	Thus a direct calculation gives
		[[∆, Ω	|∇F | 2 u 2 i = 2.	(4.4)

Ω (∇F.∇u i ) 2 + 4(1n) Ω u i ∇F.∇u i . Ω u i ∇F.∇u i = -Ω u i ∇F.∇u i -

Then we infer, from (4.3), (4.6) and (4.7),

[∆, F ]u i 2 L 2 ≤ -(n -1) 2 + 4 Ω |∇F | 2 |∇u i | 2 = -(n -1) 2 + 4 Ω |∇u i | 2 = -(n -1) 2 + 4 Ω u i (-∆u i ) ≤ -(n -1) 2 + 4 Ω F u i ∆u i + 2 Ω F u i ∆(∇F.∇u i ) + 2 Ω F u i ∇F.∇∆u i = 2(1n) Ω F u i ∆u i + 2 Ω ∆(F u i )∇F.∇u i -2 Ω div(F u i ∇F )∆u i = 2 Ω ∆F u i ∇F.∇u i + 4 Ω (∇F.∇u i ) 2 -2 Ω |∇F | 2 u i ∆u i .

We infer, from (4.3) and (4.7),

Inequality (4.2) along with (4.4), (4.9) and (4.11) gives the statement of the theorem.

Remark 4.1.

• It will be interesting to look for an extension of Theorem 4.1 to domains of hyperbolic submanifolds.

• Note that our method works for any bounded domain Ω of a Riemannian manifold admitting a function such that |∇h| is constant and |∆h| ≤ C, where C is a constant. • As before, we observe that we have the same statement as in Theorem 4.1 for the operator ∆ 2 + q (it suffices to replace λ