Chérif Amrouche

Yves Raudin

REFLECTION PRINCIPLES AND KERNELS IN R n + FOR THE BIHARMONIC AND STOKES OPERATORS. SOLUTIONS IN A LARGE CLASS OF WEIGHTED SOBOLEV SPACES

Keywords: AMS Subject Classifications: 35J50, 35J55, 35Q30, 76D07, 76N10. 201

In this paper, we study the Stokes system in the half-space R n + , with n 2. We consider data and give solutions which live in weighted Sobolev spaces, for a whole scale of weights. We start to study the kernels of the biharmonic and Stokes operators. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument.

Introduction

The purpose of this paper is the resolution of the Stokes system

(S + )    -∆u + ∇π = f in R n + , div u = h in R n + , u = g on Γ ≡ R n-1 .
Weighted Sobolev spaces provide a functional framework quite suitable to express the regularity and the behavior at infinity of data and solutions. This paper is the continuation of a previous work in which we only dealt with the basic weights (see [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF]). Here, we are interested in a large class of weights. This leads us to deal with the kernel of the operator associated to this problem and symmetrically with the compatibility condition for the data. So, an important part of this work is devoted to the study of the reflection principles for the biharmonic and Stokes operators. We give weak formulations of these principles with the aim of getting the kernels in some distribution spaces (see Section 2). The main results of [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF] will be naturally included in this paper, but we will not discuss again these particular cases.

We will also base our work on the previously established results on the harmonic and biharmonic operators (see [START_REF] Amrouche | Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition[END_REF], [START_REF] Amrouche | The Neumann problem in the half-space[END_REF], [START_REF] Amrouche | Nonhomogeneous biharmonic problem in the half-space, L p theory and generalized solutions[END_REF], [START_REF] Amrouche | Singular boundary conditions and regularity for the biharmonic problem in the half-space[END_REF]). Among the first works on the Stokes problem in the half-space, we can cite Cattabriga. In [START_REF] Cattabriga | Su un problema al contorno relativo al sistema di equazioni di Stokes[END_REF], he appeals to potential theory to explicitly get the velocity and pressure fields. For the homogeneous problem (f = 0 and h = 0), for instance, he shows that, if g ∈ L p (Γ) and the semi-norm |g| W 1-1/p, p 0 (Γ) < ∞, then ∇u ∈ L p (R n + ) and π ∈ L p (R n + ). We can find similar results in Farwig-Sohr (see [START_REF] Farwig | On the Stokes and Navier-Stokes System for Domains with Noncompact Boudary in L q -spaces[END_REF]) and Galdi (see [START_REF] Galdi | An Introduction to the Mathematical Theory of the Navier-Stokes Equations[END_REF]), who also have chosen the setting of homogeneous Sobolev spaces. On the other hand, Maz'ya-Plamenevskiȋ-Stupyalis (see [START_REF] Maz'ya | The three-dimentional problem of steady-state motion of a fluid with a free surface[END_REF]) work within the suitable setting of weighted Sobolev spaces and consider different sorts of boundary conditions. However, their results are limited to the dimension 3, to the weight zero and to the Hilbertian framework, in which they give generalized and strong solutions. This is also the case with Boulmezaoud (see [START_REF] Boulmezaoud | On the Stokes system and the biharmonic equation in the halfspace: an approach via weighted Sobolev spaces[END_REF]), who only gives strong solutions; however, he suggests an interesting characterization of the kernel that we will get here in another way. Otherwise, always in dimension 3, by Fourier analysis techniques, we can find in Tanaka the case of very regular data, corresponding to velocities which belong to W m+3, 2 2 (R 3 + ), with m 0 (see [START_REF] Tanaka | On the boundary value problem for the stationary Stokes system in the half-space[END_REF]). For any integer n 2, writing a typical point ∈ R n as x = (x ′ , x n ), we denote by R n + the upper half-space of R n and Γ its boundary. We shall use the two basic weights ̺ = (1 + |x| 2 ) 1/2 and lg ̺ = ln(2 + |x| 2 ), where |x| is the Euclidean norm of x. For any integer q, P q stands for the space of polynomials of degree smaller than or equal to q; P ∆ q (respectively P ∆ 2 q ) is the subspace of harmonic (respectively biharmonic) polynomials of P q ; A ∆ q (respectively N ∆ q ) is the subspace of polynomials of P ∆ q , odd (respectively even) with respect to x n , or equivalently, which satisfy the condition ϕ(x ′ , 0) = 0 (respectively ∂ n ϕ(x ′ , 0) = 0), with the convention that these spaces are reduced to {0} if q < 0. For any real number s, we denote by [s] the integer part of s. Given a Banach space B, with dual space B ′ and a closed subspace X of B, we denote by B ′ ⊥ X the subspace of B ′ orthogonal to X. For any k ∈ Z, we shall denote by {1, . . . , k} the set of the first k positive integers, with the convention that this set is empty if k is nonpositive. In the whole text, bold characters are used for the vector and matrix fields.

Let Ω be an open set of R n . For any m ∈ N, p ∈ (1, ∞), (α, β) ∈ R 2 , we define the following space:

W m, p α, β (Ω) = u ∈ D ′ (Ω) : 0 |λ| k, ̺ α-m+|λ| (lg ̺) β-1 ∂ λ u ∈ L p (Ω); k + 1 |λ| m, ̺ α-m+|λ| (lg ̺) β ∂ λ u ∈ L p (Ω) , (1.1)
where k = mn/pα if n/p + α ∈ {1, . . . , m}, and k = -1 otherwise.

In the case β = 0, we simply denote the space by W m, p α (Ω). Note that W m, p α, β (Ω) is a reflexive Banach space equipped with the graph-norm. Now, we define the space

• W m, p α, β (R n + ) = D(R n + ) • W m, p α, β (R n + )
, and its dual is denoted by W -m, p ′ -α, -β (R n + ). In order to define the traces of functions of W m, p α (R n + ) (here we don't consider the case β = 0), for any σ ∈ (0, 1), we introduce the space

W σ, p α (R n ) = u ∈ D ′ (R n ) : w α-σ u ∈ L p (R n ), R n ×R n |̺ α (x) u(x) -̺ α (y) u(y)| p |x -y| n+σp dx dy < ∞ , where w = ̺ if n/p + α = σ and w = ̺ (lg ̺) 1/(σ-α) if n/p + α = σ. For any s ∈ R + , we set W s, p α (R n ) = u ∈ D ′ (R n ) : 0 |λ| k, ̺ α-s+|λ| (lg ̺) -1 ∂ λ u ∈ L p (R n ); k + 1 |λ| [s] -1, ̺ α-s+|λ| ∂ λ u ∈ L p (R n ); ∂ [s] u ∈ W σ, p α (R n ) , where k = s -n/p -α if n/p + α ∈ {σ, . . . , σ + [s]}, with σ = s -[
s] and k = -1 otherwise. In the same way, we also define, for any real number β, the space

W s, p α, β (R n ) = v ∈ D ′ (R n ) : (lg ̺) β v ∈ W s, p α (R n ) .
Let us recall, for any integer m 1 and any real number α, the following trace lemma.

Lemma 1.1. For any integer m 1 and real number α, we have the linear continuous mapping

γ = (γ 0 , γ 1 , . . . , γ m-1 ) : W m, p α (R n + ) -→ m-1 j=0 W m-j-1/p, p α (R n-1 ).

Moreover, γ is surjective and

Kerγ = • W m, p α (R n + ). On the Stokes problem in R n (S) : -∆u + ∇π = f and div u = h in R n ,
let us recall the fundamental results on which we are based in the sequel. First, for any k ∈ Z, we introduce the space

S k = (λ, µ) ∈ P k × P ∆ k-1 : div λ = 0, -∆λ + ∇µ = 0 .
Theorem 1.2 (Alliot-Amrouche [START_REF] Alliot | The Stokes problem in R n : an approach in weighted Sobolev spaces[END_REF]). Let ℓ ∈ Z and assume that n/p ′ / ∈ {1, . . . , ℓ} and n/p / ∈ {1, . . . , -ℓ}. For any

(f , h) ∈ W -1, p ℓ (R n )×W 0, p ℓ (R n ) ⊥ S [1+ℓ-n/p ′ ] , problem (S) admits a solution (u, π) ∈ W 1, p ℓ (R n )×W 0, p ℓ (R n ), unique up to an element of S [1-ℓ-n/p] , with the estimate inf (λ, µ)∈S [1-ℓ-n/p] u + λ W 1, p ℓ (R n ) + π + µ W 0, p ℓ (R n ) C f W -1, p ℓ (R n ) + h W 0, p ℓ (R n ) .
Theorem 1.3 (Alliot-Amrouche [START_REF] Alliot | The Stokes problem in R n : an approach in weighted Sobolev spaces[END_REF]). Let ℓ ∈ Z and m 1 be two integers and assume that n/p ′ / ∈ {1, . . . , ℓ + 1} and n/p / ∈ {1, . . . , -ℓ -m}. For any

(f , h) ∈ W m-1, p m+ℓ (R n ) × W m, p m+ℓ (R n ) ⊥ S [1+ℓ-n/p ′ ] , problem (S) admits a solution (u, π) ∈ W m+1, p m+ℓ (R n ) × W m, p m+ℓ (R n ), unique up to an element of S [1-ℓ-n/p] , with the estimate inf (λ, µ)∈S [1-ℓ-n/p] u + λ W m+1, p m+ℓ (R n ) + π + µ W m, p m+ℓ (R n ) C f W m-1, p m+ℓ (R n ) + h W m, p m+ℓ (R n ) .

Reflection principles and kernels in R n

+

The aim of this section is to characterize the kernel of the Stokes operator with Dirichlet boundary conditions in the half-space. In this geometry, the natural way is to use a reflection principle similar to the well-known Schwarz reflection principle for harmonic functions. Since, in the Stokes system, the velocity field is biharmonic and the pressure is harmonic, it is reasonable to start with the reflection principle for the biharmonic functions. Let us notice that R. Farwig gives these continuation formulae in [START_REF] Farwig | A Note on the Reflection Principle for the Biharmonic Equation and the Stokes system[END_REF], referring to elliptic regularity theory (see Agmon, Douglis and Nirenberg, [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations, I[END_REF]). Let us especially quote R. J. Duffin, who first established in [START_REF] Duffin | Continuation of biharmonic functions by reflection[END_REF] the continuation formula of biharmonic functions in the three-dimensional case and then analogous formulae for the Stokes flow equations. Next, A. Huber extended in [START_REF] Huber | On the Reflection Principle for Polyharmonic Functions[END_REF] this principle to polyharmonic functions. From the classical point of view, the only serious difficulty is the argument at the boundary.

Starting with the biharmonic operator, we will give a weak formulation of the continuation formula, which will allow us to characterize the kernel of this operator, even for very weak solutions.

At first, let us introduce a useful notation. For any function ϕ defined on an open set Ω of R n , we will denote by ϕ * , the composite function

ϕ * = ϕ • r defined on Ω * = r(Ω), of ϕ with the C ∞ -diffeomorphism r : Ω -→ Ω * , x = (x ′ , x n ) -→ x * = (x ′ , -x n ).
Thus, if ϕ ∈ D(R n + ), then ϕ * ∈ D(R n -) and conversely. In the same way, if u ∈ D ′ (Ω), we will denote by u * the distribution in D ′ (Ω * ), defined for any

ϕ ∈ D(Ω * ) by u * , ϕ D ′ (Ω * )×D(Ω * ) = u, ϕ * D ′ (Ω)×D(Ω) . Thus, if u ∈ D ′ (R n + ), then u * ∈ D ′ (R n
-) and conversely. Now, for the convenience of the reader, let us recall the essential tooli.e., the Green formula -in the study of singular boundary conditions for the biharmonic problem (see [START_REF] Amrouche | Singular boundary conditions and regularity for the biharmonic problem in the half-space[END_REF]). For any ℓ ∈ Z, we introduce the space

Y p ℓ, 1 (R n + ) = v ∈ W 0, p ℓ-2 (R n + ) : ∆ 2 v ∈ W 0, p ℓ+2, 1 (R n + )
, which is a reflexive Banach space equipped with its natural norm

v Y p ℓ, 1 (R n + ) = v W 0, p ℓ-2 (R n + ) + ∆ 2 v W 0, p ℓ+2, 1 (R n + ) .
Then we proved in [START_REF] Amrouche | Singular boundary conditions and regularity for the biharmonic problem in the half-space[END_REF]

then the space D(R n + ) is dense in Y p ℓ, 1 (R n + ).
Thanks to this density lemma, we proved in [START_REF] Amrouche | Singular boundary conditions and regularity for the biharmonic problem in the half-space[END_REF], Lemma 4.2, the following result of traces with the Green formula. Lemma 2.2. Let ℓ ∈ Z. Under hypothesis (2.1), the mapping (γ 0 , γ 1 ) :

D(R n + ) -→ D(R n-1 )
2 can be extended to a linear continuous mapping

(γ 0 , γ 1 ) : Y p ℓ, 1 (R n + ) -→ W -1/p, p ℓ-2 (Γ) × W -1-1/p, p ℓ-2 (Γ),
and we have the following Green formula:

∀v ∈ Y p ℓ, 1 (R n + ), ∀ϕ ∈ W 4, p ′ -ℓ+2 (R n + ) such that ϕ = ∂ n ϕ = 0 on Γ, (2.2)
∆ 2 v, ϕ W 0, p ℓ+2, 1 (R n + )×W 0, p ′ -ℓ-2, -1 (R n + ) -v, ∆ 2 ϕ W 0, p ℓ-2 (R n + )×W 0, p ′ -ℓ+2 (R n + ) = v, ∂ N ∆ϕ W -1/p, p ℓ-2 (Γ)×W 1/p, p ′ -ℓ+2 (Γ) -∂ n v, ∆ϕ W -1-1/p, p ℓ-2 (Γ)×W 1+1/p, p ′ -ℓ+2 (Γ)
. Now, we can establish the following result.

Lemma 2.3. Let ℓ ∈ Z with hypothesis (2.1) and u ∈ W 0, p ℓ-2 (R n + ) satisfying ∆ 2 u = 0 in R n + , u = ∂ n u = 0 on Γ;
then there exists a unique biharmonic extension ũ

∈ D ′ (R n ) of u, which is given for all ϕ ∈ D(R n ) by ũ, ϕ D ′ (R n )×D(R n ) = R n + u ϕ -5 ϕ * -6 x n ∂ n ϕ * -x 2 n ∆ϕ * dx. (2.3) Moreover, we have ũ ∈ W -2, p ℓ-4 (R n ) with the estimate ũ W -2, p ℓ-4 (R n + ) C u W 0, p ℓ-2 (R n + ) . (2.4)
Proof.

(1) Let us notice an important point to start with. According to Weyl's lemma, since u is a biharmonic distribution in R n + , we know that u ∈ C ∞ (R n + ) -see e.g. Dautray-Lions [START_REF] Dautray | Analyse mathématique et calcul numérique[END_REF], page 327, Proposition 1. Next, let us remark that the integral in (2.3) is well defined. Indeed, u ∈ W 0, p ℓ-2 (R n + ) and ϕ -thus ϕ * also -has compact support. Now, let us show that ũ belongs to D ′ (R n ) and more precisely the end of our statement. From (2.3) we get the following estimate:

| ũ, ϕ | C u W 0, p ℓ-2 (R n + ) ϕ W 0, p ′ -ℓ+2 (R n ) + ϕ W 1, p ′ -ℓ+3 (R n ) + ϕ W 2, p ′ -ℓ+4 (R n ) . Since n p ′ / ∈ {ℓ -3, ℓ -2}, we have W 2, p ′ -ℓ+4 (R n ) ֒→ W 1, p ′ -ℓ+3 (R n ) ֒→ W 0, p ′ -ℓ+2 (R n ) and then, for any ϕ ∈ W 2, p ′ -ℓ+4 (R n ), | ũ, ϕ | C u W 0, p ℓ-2 (R n + ) ϕ W 2, p ′ -ℓ+4 (R n )
. So, we can deduce that ũ ∈ W -2, p ℓ-4 (R n ) and the estimate (2.4). (2) For the uniqueness, let us consider two biharmonic extensions ũ1 and ũ2 of u which belong to D ′ (R n ) and set U = ũ2 -ũ1 . Then we have ∆ 2 U = 0 in R n and we can deduce that U is analytic in R n . Since U = 0 in R n + , the analytic continuation principle implies that in fact U = 0 in R n .

(3) Evidently, ũ is an extension of u. Indeed, let ϕ ∈ D(R n + ) and φ the zero extension of

ϕ to R n , then ũ, φ D ′ (R n )×D(R n ) = R n + u ϕ dx; that is, ũ| R n + = u. On the other hand, let ϕ ∈ D(R n -) and φ the zero extension of ϕ to R n , then we get ũ, φ D ′ (R n )×D(R n ) = R n + u -5 ϕ * -6 x n ∂ n ϕ * -x 2 n ∆ϕ * dx.
Moreover, we can express ũ, φ by means of an integral in R n -:

I 1 = R n + u ϕ * dx = R n - u * ϕ dx, I 2 = R n + u x n ∂ n ϕ * dx = R n - x n u * ∂ n ϕ dx = - R n - ∂ n (x n u * ) ϕ dx = - R n - (u * + x n ∂ n u * ) ϕ dx, I 3 = R n + u x 2 n ∆ϕ * dx = R n - x 2 n u * ∆ϕ dx = R n - ∆(x 2 n u * ) ϕ dx = R n - (2 u * + 4 x n ∂ n u * + x 2 n ∆u * ) ϕ dx. Hence, ũ, φ D ′ (R n )×D(R n ) = R n - -u * + 2 x n ∂ n u * -x 2 n ∆u * ϕ dx; that is, ũ| R n - = -u * -2 x n (∂ n u) * -x 2 n (∆u) * . So, ũ| R n - ∈ C ∞ (R n -)
and we find the classical formulation obtained by R. J. Duffin (see [START_REF] Duffin | Continuation of biharmonic functions by reflection[END_REF]) in the three dimensional case: for any

x ∈ R n -, ũ(x) = -u -2 x n ∂ n u -x 2 n ∆u (x * ). (2.5)
(4) It remains to show that this extension is actually biharmonic in R n . From the definition (2.3), we obtain the following expression: for all ϕ ∈

D(R n ), ∆ 2 ũ, ϕ D ′ (R n )×D(R n ) = ũ, ∆ 2 ϕ D ′ (R n )×D(R n ) = R n + u ∆ 2 (ϕ -5 ϕ * ) -6 x n ∂ n ∆ 2 ϕ * -x 2 n ∆ 3 ϕ * dx,
that we can rewrite as follows:

∆ 2 ũ, ϕ = R n + u ∆ 2 Φ dx, where Φ = ϕ -ϕ * -x 2 n ∆ϕ * + 2 x n ∂ n ϕ * . Besides, we have Φ = ϕ -ϕ * = 0 on Γ, ∂ n Φ = ∂ n ϕ + ∂ n ϕ * = ∂ n ϕ -(∂ n ϕ) * = 0 on Γ. Then, according to Lemma 2.2, we get ∆ 2 ũ, ϕ = 0 for all ϕ ∈ D(R n ); that is, ∆ 2 ũ = 0 in R n .
As a main consequence of Lemma 2.3, we are going to characterize the kernel K of the biharmonic operator (∆ 2 , γ 0 , γ 1 ) in W 0, p ℓ-2 (R n + ) as a polynomial space. For any q ∈ Z, let us introduce B q as a subspace of P ∆ 2 q :

B q = u ∈ P ∆ 2 q : u = ∂ n u = 0 on Γ . Corollary 2.4. Let ℓ ∈ Z with hypothesis (2.1), then K = B [2-ℓ-n/p] . Proof. Given u ∈ K, thanks to Lemma 2.3, we know that ũ ∈ W -2, p ℓ-4 (R n ) ⊂ S ′ (R n ) and ∆ 2 ũ = 0 in R n .
We can deduce that ũ, and consequently u, is a polynomial. In addition,

u ∈ W 0, p ℓ-2 (R n + ) implies that u ∈ P [2-ℓ-n/p] (see [3]). Remark 2.5. Coming back to Lemma 2.3, since ũ ∈ P [2-ℓ-n/p] , we get in fact ũ ∈ W m+2, p m+ℓ (R n ) for any integer m -4. Indeed, under hypothesis (2.1), we have the imbedding chain W m+2, p m+ℓ (R n ) ֒→ • • • ֒→ W -2, p ℓ-4 (R n ) and besides P [2-ℓ-n/p] ⊂ W m+2, p m+ℓ (R n
). Better, we can see that this kernel does not really depend on the regularity according to the Sobolev imbeddings. More precisely, if we denote by K m the kernel of (∆ 2 , γ 0 , γ 1 ) in W m+2, p m+ℓ (R n + ), identical arguments lead us to the following result.

Corollary 2.6. Let ℓ ∈ Z and m -2 be two integers and assume that n p ′ / ∈ {1, . . . , ℓ + min{m, 2}} and n p / ∈ {1, . . . , -ℓ -m};

(2.6)

then K m = B [2-ℓ-n/p] .
Finally, we showed in [START_REF] Amrouche | Nonhomogeneous biharmonic problem in the half-space, L p theory and generalized solutions[END_REF] that we can link this kernel to those of the Dirichlet and Neumann problems for the Laplacian. With this intention, we defined the two operators Π D and Π N by

∀r ∈ A ∆ k , Π D r = 1 2 xn 0 t r(x ′ , t) dt, ∀s ∈ N ∆ k , Π N s = 1 2 x n xn 0 s(x ′ , t) dt,
satisfying the following properties:

∀r ∈ A ∆ k , ∆Π D r = r in R n + , Π D r = ∂ n Π D r = 0 on Γ, ∀s ∈ N ∆ k , ∆Π N s = s in R n + , Π N s = ∂ n Π N s = 0 on Γ. (2.7)
So we got a second characterization of this kernel:

B [2-ℓ-n/p] = Π D A ∆ [-ℓ-n/p] ⊕ Π N N ∆ [-ℓ-n/p] . (2.8)
Now, we can use these results in the study of the Stokes operator. But to begin with we must establish a result equivalent to Lemma 2.2. Let us denote by T : (u, π) -→ (-∆u + ∇π,div u) the Stokes operator. For any ℓ ∈ Z, we introduce the space

T p ℓ, 1 (R n + ) = (u, π) ∈ W 0, p ℓ-1 (R n + ) × W -1, p ℓ-1 (R n + ) : T (u, π) ∈ W 0, p ℓ+1, 1 (R n + ) × W 0, p ℓ, 1 (R n + )
, which is a reflexive Banach space equipped with the graph-norm. Then we have the following density result. (2.9)

then the space D(R n + ) × D(R n + ) is dense in T p ℓ, 1 (R n + ).
Proof. For every continuous linear form Λ ∈ (T p ℓ,1 (R n + )) ′ , there exists a unique

(f , ϕ, g, ψ) ∈ W 0,p ′ -ℓ+1 (R n + )× • W 1,p ′ -ℓ+1 (R n + ) × W 0,p ′ -ℓ-1,-1 (R n + ) × W 0,p ′ -ℓ,-1 (R n + ), such that, for all (u, π) ∈ T p ℓ, 1 (R n + ), Λ, (u, π) = (f , ϕ), (u, π) + (g, ψ), T (u, π) .
(2.10)

Thanks to the Hahn-Banach theorem, it suffices to show that any Λ which vanishes on

D(R n + ) × D(R n + ) is actually zero on T p ℓ, 1 (R n + ). Let us suppose that Λ = 0 on D(R n + ) × D(R n + ), thus on D(R n + ) × D(R n + ). Then we can deduce from (2.10) that (f , ϕ) + T (g, ψ) = 0 in R n + , hence, T (g, ψ) ∈ W 0, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + )
. Let f , φ, g, ψ be respectively the zero extensions of f , ϕ, g, ψ to R n . By (2.10), it is clear that we have

( f , φ) + T (g, ψ) = 0 in R n , and thus T (g, ψ) ∈ W 0, p ′ -ℓ+1 (R n ) × W 1, p ′ -ℓ+1 (R n ).
According to the results in the whole space (see Theorem 1.3), we can deduce that (g, ψ)

∈ W 2, p ′ -ℓ+1 (R n ) × W 1, p ′ -ℓ+1 (R n ).
Since g and ψ are the zero extensions, it follows that (g, ψ)

∈ • W 2, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ). Then, by density of D(R n + ) × D(R n + ) in • W 2, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ), we can construct a sequence (g k , ψ k ) k∈N ⊂ D(R n + ) × D(R n + ) such that (g k , ψ k ) → (g, ψ) in • W 2, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ). Thus, for any (u, π) ∈ T p ℓ, 1 (R n + ), we have Λ, (u, π) = -T (g, ψ), (u, π) + (g, ψ), T (u, π) = lim k→∞ {-T (g k , ψ k ), (u, π) + (g k , ψ k ), T (u, π) } = 0;
i.e., Λ is identically zero.

Thanks to this density lemma, we can prove the following result.

Lemma 2.8. Let ℓ ∈ Z. Under hypothesis (2.9), we can define the linear continuous mapping (the trace of the velocity field)

τ 0 : T p ℓ, 1 (R n + ) -→ W -1/p, p ℓ-1 (Γ), (u, π) -→ u| Γ = (γ 0 u 1 , . . . , γ 0 u n ).
Moreover, we have the following Green formula:

∀(u, π) ∈ T p ℓ, 1 (R n + ), ∀(ϕ, ψ) ∈ W 2, p ′ -ℓ+1 (R n + ) × W 1, p ′ -ℓ+1 (R n + ) such that ϕ = 0 and div ϕ = 0 on Γ, T (u, π), (ϕ, ψ) W 0, p ℓ+1, 1 (R n + )×W 0, p ℓ, 1 (R n + ), W 0, p ′ -ℓ-1, -1 (R n + )×W 0, p ′ -ℓ, -1 (R n + ) = (u, π), T (ϕ, ψ) W 0, p ℓ-1 (R n + )×W -1, p ℓ-1 (R n + ), W 0, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) -u, (∂ n ϕ ′ , -ψ) W -1/p, p ℓ-1 (Γ)×W 1/p, p ′ -ℓ+1 (Γ)
.

(2.11)

Proof. Let us make three remarks to start. Firstly, the left-hand term in (2.11) is nothing but the integral

R n + T (u, π) • (ϕ, ψ) dx.
Secondly, the reason for the logarithmic factor in the definition of

T p ℓ, 1 (R n + ) is that the imbed- dings W 2, p ′ -ℓ+1 (R n + ) ֒→ W 0, p ′ -ℓ-1, -1 (R n + ) and W 1, p ′ -ℓ+1 (R n + ) ֒→ W 0, p ′ -ℓ, -1 (R n + ) hold without supplementary critical values with respect to (2.9) -whereas the imbedding W 2, p ′ -ℓ+1 (R n + ) ֒→ W 0, p ′ -ℓ-1 (R n + ) fails if n/p ′ ∈ {ℓ, ℓ+1}. Thirdly, for any ϕ ∈ W 2, p ′ -ℓ+1 (R n + )
, the boundary conditions ϕ = 0 and div ϕ = 0 on Γ are equivalent to ϕ = 0 and ∂ n ϕ n = 0 on Γ.

So we can write the following Green formula:

∀(u, π) ∈ D(R n + ) × D(R n + ), ∀(ϕ, ψ) ∈ W 2, p ′ -ℓ+1 (R n + ) × W 1, p ′ -ℓ+1 (R n + ) (2.
12) such that ϕ = 0 and div ϕ = 0 on Γ,

R n + T (u, π) • (ϕ, ψ) dx = R n + (u, π) • T (ϕ, ψ) dx - Γ u • (∂ n ϕ ′ , -ψ) dx ′ .
We can deduce the following estimate:

u, (∂ n ϕ ′ , -ψ) W -1/p, p ℓ-1 (Γ)×W 1/p, p ′ -ℓ+1 (Γ) (u, π) T p ℓ, 1 (R n + ) (ϕ, ψ) W 2, p ′ -ℓ+1 (R n + )×W 1, p ′ -ℓ+1 (R n + ) .
By Lemma 1.1, for any g ∈ W

1-1/p ′ , p ′ -ℓ+1
(Γ), there exists a lifting function

(ϕ, ψ) ∈ W 2, p ′ -ℓ+1 (R n + ) × W 1, p ′ -ℓ+1 (R n + ) such that ϕ = 0, ∂ n ϕ ′ = g ′ , ∂ n ϕ n = 0 and -ψ = g n on Γ, satisfying (ϕ, ψ) W 2, p ′ -ℓ+1 (R n + )×W 1, p ′ -ℓ+1 (R n + ) C g W 1-1/p ′ , p ′ -ℓ+1 (Γ)
,

where C is a constant not depending on (ϕ, ψ) and g. Then we can deduce that

u W -1/p, p ℓ-1 (Γ) C (u, π) T p ℓ, 1 (R n + )
. Thus, the linear mapping τ 0 : (u, π) -→ u| Γ defined on

D(R n + ) × D(R n + ) is continuous for the norm of T p ℓ, 1 (R n + ). Since D(R n + ) × D(R n + ) is dense in T p ℓ, 1 (R n + )
, τ 0 can be extended by continuity to a mapping still called

τ 0 ∈ L T p ℓ, 1 (R n + ); W -1/p, p ℓ-1 (Γ)
. Moreover, we also can deduce the formula (2.11) from (2.12) by density of

D(R n + ) × D(R n + ) in T p ℓ, 1 (R n + ).
We now can give the continuation result for the Stokes operator.

Lemma 2.9. Let ℓ ∈ Z with hypothesis (2.9) and (u, π)

∈ W 0, p ℓ-1 (R n + ) × W -1, p ℓ-1 (R n + ) satisfying -∆u + ∇π = 0 and div u = 0 in R n + , u = 0 on Γ;
then there exists a unique extension (ũ, π)

∈ D ′ (R n ) × D ′ (R n ) of (u, π) satisfying -∆ũ + ∇π = 0 and div ũ = 0 in R n , (2.13
)

which is given for all (ϕ, ψ) ∈ D(R n ) × D(R n ) by ũ, ϕ = R n + [u • (ϕ -ϕ * ) -2 u n ϕ * n + 2 u n x n (div ϕ) * ] dx + π, 2 x n ϕ * n -x 2 n (div ϕ) * W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) , (2.14)
and

π, ψ = π, ψ -ψ * -2 x n ∂ n ψ * W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) + 4 R n + u n ∂ n ψ * dx.
(2.15)

Moreover, we have (ũ, π) ∈ W -2, p ℓ-3 (R n ) × W -2, p ℓ-2 (R n ) with the estimate (ũ, π) W -2, p ℓ-3 (R n )×W -2, p ℓ-2 (R n ) C (u, π) W 0, p ℓ-1 (R n + )×W -1, p ℓ-1 (R n + ) .
(2.16) Remark 2.10. Knowing that u n satisfies the biharmonic problem, see (2.20), naturally we must find (2.3) from (2.14). Indeed, if we take ϕ ′ = 0 in (2.14), we get the following: for all

ϕ n ∈ D(R n ), ũn , ϕ n = R n + [u n (ϕ n -ϕ * n ) -2 u n ϕ * n -2 u n x n ∂ n ϕ * n ] dx + π, 2 x n ϕ * n + x 2 n ∂ n ϕ * n W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + )
.

Since ∆u n = ∂ n π in R n + , we can write π, 2 x n ϕ * n + x 2 n ∂ n ϕ * n W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) = π, ∂ n (x 2 n ϕ * n ) W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) = -∂ n π, x 2 n ϕ * n W -2, p ℓ-1 (R n + )× • W 2, p ′ -ℓ+1 (R n + ) = -∆u n , x 2 n ϕ * n W -2, p ℓ-1 (R n + )× • W 2, p ′ -ℓ+1 (R n + ) = -u n , ∆(x 2 n ϕ * n ) W 0, p ℓ-1 (R n + )×W 0, p ′ -ℓ+1 (R n + ) = -u n , 2 ϕ * n + 4 x n ∂ n ϕ * n -x 2 n ∆ϕ * n W 0, p ℓ-1 (R n + )×W 0, p ′ -ℓ+1 (R n + ) . Hence, ũn , ϕ n = R n + u n ϕ n -5 ϕ * n -6 x n ∂ n ϕ * n -x 2 n ∆ϕ * n dx,
which is exactly the formula (2.3) for u n .

Proof of Lemma 2.9. (1) As for the biharmonic operator, according to (2.14) and (2.15), we can readily check that (ũ, π) ∈ W -2,p ℓ-3 (R n )×W -2,p ℓ-2 (R n ) with the estimate (2.16). Besides, the argument for the uniqueness of the extension also holds for the Stokes operator and it is clear that (2.14) and (2.15) define an extension of (u, π) to R n . Indeed, we have both for all

ϕ ∈ D(R n + ), ũ, φ = R n + u • ϕ dx and for all ψ ∈ D(R n + ), π, ψ = π, ψ W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + )
, where φ and ψ are respectively the zero extensions of ϕ and ψ to R n .

(2) Now, we also can give the functional writing of this extension in R n -. For all ϕ ∈ D(R n -), we have

ũ, φ = R n + -u ′ • ϕ ′ * -3 u n ϕ * n + 2 u n x n (div ϕ) * dx + π, 2 x n ϕ * n -x 2 n (div ϕ) * D ′ (R n + )×D(R n + ) .

Breaking down this expression, we get

R n + -u ′ • ϕ ′ * -3 u n ϕ * n dx = R n - -u ′ * • ϕ ′ -3 u * n ϕ n dx, R n + 2 u n x n (div ϕ) * dx = R n - -2 u * n x n div ϕ dx = R n - 2 ∇(u * n x n ) • ϕ dx = R n - 2 (u * n ϕ n + x n ∇u * n • ϕ) dx, π, 2 x n ϕ * n D ′ (R n + )×D(R n + ) = -2 π * , x n ϕ n D ′ (R n -)×D(R n -) = -2 x n π * , ϕ n D ′ (R n -)×D(R n -) , π, -x 2 n (div ϕ) * D ′ (R n + )×D(R n + ) = -x 2 n π, (div ϕ) * D ′ (R n + )×D(R n + ) = -x 2 n π * , div ϕ D ′ (R n -)×D(R n -) = ∇(x 2 n π * ), ϕ D ′ (R n -)×D(R n -) = 2 x n π * , ϕ n D ′ (R n -)×D(R n -) + x 2 n ∇π * , ϕ D ′ (R n -)×D(R n -) . Hence, ũ, φ = R n - -u ′ * • ϕ ′ -u * n ϕ n + 2 x n ∇u * n • ϕ dx + x 2 n ∇π * , ϕ D ′ (R n -)×D(R n -) .
Let us notice that here we always can replace the duality brackets by integrals. Indeed, ϕ has a compact support in R n -, in addition to u * and π * belonging to

C ∞ (R n -), thus to L 1 loc (R n -). So, we get ũ′ , φ′ = R n - -u ′ * + 2 x n ∇ ′ u * n + x 2 n ∇ ′ π * • ϕ ′ dx; i.e., ∀x ∈ R n -, ũ′ (x) = -u ′ + 2 x n ∇ ′ u n + x 2 n ∇ ′ π (x * ) and ũn , φn = R n - -u * n + 2 x n ∂ n u * n + x 2 n ∂ n π * ϕ n dx; i.e., ∀x ∈ R n -, ũn (x) = -u n -2 x n ∂ n u n -x 2 n ∂ n π (x * ). Likewise, for all ψ ∈ D(R n -), we have π, ψ = π, -ψ * -2 x n ∂ n ψ * D ′ (R n + )×D(R n + ) + 4 R n + u n ∂ n ψ * dx.
Separately, we get

π, -ψ * D ′ (R n + )×D(R n + ) = -π * , ψ D ′ (R n -)×D(R n -) , π, -2 x n ∂ n ψ * = -2 x n π, ∂ n ψ * D ′ (R n + )×D(R n + ) = -2 x n π * , ∂ n ψ D ′ (R n -)×D(R n -) = 2 ∂ n (x n π * ), ψ D ′ (R n -)×D(R n -) = 2 π * + x n ∂ n π * , ψ D ′ (R n -)×D(R n -) , R n + u n ∂ n ψ * dx = - R n - u * n ∂ n ψ dx = R n - ∂ n u * n ψ dx.
Hence, π, ψ

D ′ (R n )×D(R n ) = π * + 2 x n ∂ n π * + 4 ∂ n u * n , ψ D ′ (R n -)×D(R n -) ; i.e., ∀x ∈ R n -, π(x) = (π -2 x n ∂ n π -4 ∂ n u n ) (x * ). So, we find the classical continuation formulae: for all x ∈ R n -,    ũ′ (x) = -u ′ + 2 x n ∇ ′ u n + x 2 n ∇ ′ π (x * ), ũn (x) = -u n -2 x n ∂ n u n -x 2 n ∂ n π (x * ), π(x) = (π -2 x n ∂ n π -4 ∂ n u n ) (x * ).
(3) Finally, it remains to show that this extension satisfies (2.13) -that is the Stokes system in the whole space. For all (ϕ, ψ)

∈ D(R n ) × D(R n ), we have R n T (ũ, π) • (ϕ, ψ) dx = (ũ, π), T (ϕ, ψ) D ′ (R n )×D ′ (R n ), D(R n )×D(R n ) = ũ, -∆ϕ + ∇ψ D ′ (R n )×D(R n ) -π, div ϕ D ′ (R n )×D(R n ) .
Then, according to (2.14) and (2.15), we get

R n T (ũ, π) • (ϕ, ψ) dx = - R n + [u • ∆(ϕ -ϕ * ) -2 u n ∆ϕ * n + 2 u n x n (div ∆ϕ) * ] dx -π, 2 x n ∆ϕ * n -x 2 n (div ∆ϕ) * W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) + R n + [u • ∇(ψ -ψ * ) + 4 u n ∂ n ψ * + 2 u n x n ∆ψ * ] dx + π, -2 x n ∂ n ψ * -x 2 n ∆ψ * W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) -π, div(ϕ -ϕ * ) + 2 ∂ n ϕ * n -2 x n ∂ n (div ϕ) * W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) -4 R n + u n ∂ n (div ϕ) * dx.
With the intention of showing that R n T (ũ, π) • (ϕ, ψ) dx = 0, we are going to rewrite this expression as follows:

R n + u • (-∆Φ + ∇Ψ) dx + π, -div Φ W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) = (u, π), T (Φ, Ψ) W 0, p ℓ-1 (R n + )×W -1, p ℓ-1 (R n + ), W 0, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + )
,

where (Φ, Ψ) ∈ W 2, p ′ -ℓ+1 (R n + ) × W 1, p ′ -ℓ+1 (R n + )
, with Φ = 0 and ∂ n Φ n = 0 on Γ. Then, the zero of (u, π), T (Φ, Ψ) will be a straightforward consequence of the Green formula (2.11).

Let us construct the functions Φ and Ψ. We will start with the terms π, • . Noticing that (div ∆ϕ) * = ∆(div ϕ) * , we find π, -2

x n ∆ϕ * n + x 2 n ∆(div ϕ) * -2 x n ∂ n ψ * -x 2 n ∆ψ * -div(ϕ -ϕ * ) -2 ∂ n ϕ * n + 2 x n ∂ n (div ϕ) * Then, R n T (ũ, π) • (ϕ, ψ) dx = 0, for all (ϕ, ψ) ∈ D(R n ) × D(R n ); that is, T (ũ, π) = (0, 0).
Now, we can characterize the Stokes kernel. For ℓ ∈ Z, let us denote by K S the kernel of the Stokes operator (T, τ 0 ) in W 0, p ℓ-1 (R n + )×W -1, p ℓ-1 (R n + ) and for any k ∈ Z, introduce the polynomial space (2.17)

S + k = (λ, µ) ∈ P ∆ 2 k × P ∆ k-1 : -∆λ + ∇µ = 0 and div λ = 0 in R n + , λ = 0 on Γ . Let (u, π) ∈ K S .
then K m S = S + [1-ℓ-n/p] .
We can be more specific about polynomials which build up this kernel. The idea of this characterization is due to T.Z. Boulmezaoud (see [START_REF] Boulmezaoud | On the Stokes system and the biharmonic equation in the halfspace: an approach via weighted Sobolev spaces[END_REF]). We give it with a completely different proof, based on the kernels of the Dirichlet and Neumann problems for the Laplacian and the one of the biharmonic problem with Dirichlet boundary conditions in the half-space. Lemma 2.13. Let ℓ ∈ Z. Then (u, π) ∈ S + [1-ℓ-n/p] if and only if there exists

ϕ ∈ A ∆ [1-ℓ-n/p] such that u = ϕ -∇ Π D div ′ ϕ ′ + Π N ∂ n ϕ n , (2.18) π = -div ϕ. (2.19)
Proof. Given (u, π) ∈ S + [1-ℓ-n/p] , then we also have div u = 0 on Γ and thus ∂ n u n = 0 on Γ. Moreover ∆π = 0 in R n + and thus ∆ 2 u n = 0 in R n + . So we get the biharmonic problem

∆ 2 u n = 0 in R n + and u n = ∂ n u n = 0 on Γ.
(2.20)

Hence, u n ∈ B [1-ℓ-n/p] and there exists (r, s)

∈ A ∆ [-1-ℓ-n/p] × N ∆ [-1-ℓ-n/p] such that u n = Π D r + Π N s.
We can deduce from (2.7) that ∂ n π = ∆u n = r + s in R n + and thus π satisfies ∆π = 0 in R n + and ∂ n π = s on Γ.

Then, there exists ψ ∈ N ∆ [-ℓ-n/p] (see [START_REF] Amrouche | Nonhomogeneous biharmonic problem in the half-space, L p theory and generalized solutions[END_REF]) such that

π = ψ + Ks in R n + , (2.21)
where

Ks(x ′ , x n ) = xn 0 s(x ′ , t) dt.

So, we have ∆u

n = r + s = ∂ n π = ∂ n ψ + s in R n + , thus r = ∂ n ψ. Hence, u n = Π D ∂ n ψ + Π N s in R n + . (2.22)
From (2.21), we get, for every i ∈ {1, . . . , n -1},

∆u i = ∂ i π = ∂ i ψ + ∂ i Ks ∈ N ∆ [-1-ℓ-n/p] ⊕ A ∆ [-1-ℓ-n/p] = ∆Π N ∂ i ψ + ∆Π D ∂ i Ks.
Then,

w i = u i -Π N ∂ i ψ -Π D ∂ i Ks satisfies ∆w i = 0 in R n +
and w i = 0 on Γ.

Hence, we have the existence of ϕ i ∈ A ∆ [1-ℓ-n/p] (see [START_REF] Amrouche | Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition[END_REF]), such that w i = ϕ i ; i.e.,

u i = Π N ∂ i ψ + Π D ∂ i Ks + ϕ i .
Thereby, writing ϕ ′ = (ϕ 1 , . . . , ϕ n-1 ), we get

div ′ u ′ = Π N ∆ ′ ψ + Π D ∆ ′ Ks + div ′ ϕ ′ = -Π N ∂ 2 n ψ -Π D ∂ 2 n Ks + div ′ ϕ ′ = -1 2 x n ∂ n ψ -1 2 (x n ∂ n Ks -Ks) + div ′ ϕ ′ = -1 2 x n ∂ n ψ -1 2 (x n s -Ks) + div ′ ϕ ′ .
In addition, by (2.22), we have

∂ n u n = ∂ n Π D ∂ n ψ + ∂ n Π N s = 1 2 x n ∂ n ψ + 1 2 x n s + xn 0 s(x ′ , t) dt = 1 2 x n ∂ n ψ + 1 2 (x n s + Ks) .
Since div u = 0, we can deduce that div ′ ϕ ′ = -Ks and thus (2.21) can be rewritten as

π = ψ -div ′ ϕ ′ . Now, if we set ϕ n = - xn 0 ψ(x ′ , t) dt, then we have ψ = -∂ n ϕ n and ϕ n ∈ A ∆ [1-ℓ-n/p] . So, we obtain π = -div ϕ; i.e., (2.19), with ϕ = (ϕ ′ , ϕ n ) ∈ A ∆ [1-ℓ-n/p] .
Coming back to the velocity field, we get, for every i ∈ {1, . . . , n -1},

u i = ϕ i -∂ i Π N ∂ n ϕ n -∂ i Π D div ′ ϕ ′ .
(2.23)

Likewise, for the normal component, (2.22) yields

u n = -Π D ∂ 2 n ϕ n + Π N ∂ n Ks = 1 2 (ϕ n -x n ∂ n ϕ n ) + 1 2 x n Ks = ϕ n -1 2 x n ∂ n ϕ n -1 2 ϕ n -1 2 x n div ′ ϕ ′ = ϕ n -∂ n Π N ∂ n ϕ n -∂ n Π D div ′ ϕ ′ .
So, combining this with (2.23), we get u = ϕ -∇ Π N ∂ n ϕ n + Π D div ′ ϕ ′ ; i.e., the statement (2.18).

Conversely, we can verify that such a pair (u, π) belongs to S + [1-ℓ-n/p] .

Generalized solutions to the Stokes system

In this section, we will establish the central result on the generalized solutions to the Stokes system in the half-space, with Theorem 3.3. We will be interested in the existence of a solution (u, π)

∈ W 1, p ℓ (R n + ) × W 0, p ℓ (R n + ) to (S + ), for data f ∈ W -1, p ℓ (R n + ), h ∈ W 0, p ℓ (R n + ) and g ∈ W 1-1/p, p ℓ
(Γ). To avoid troubles with the compatibility conditions, we will start with the study of the negative weights. For this, as for the weight ℓ = 0 in [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF], we will adapt a method used by Farwig-Sohr in [START_REF] Farwig | On the Stokes and Navier-Stokes System for Domains with Noncompact Boudary in L q -spaces[END_REF]. Then, we get back the positive weights by a duality argument, and the compatibility condition naturally comes from the kernel of the dual case.

First, we will establish the result for the homogeneous problem in the case of negative weights. Lemma 3.1. Let ℓ be a negative integer and assume that n/p / ∈ {1, . . . , -ℓ}. For any g ∈ W 1-1/p, p ℓ (Γ), the homogeneous Stokes problem

-∆u + ∇π = 0 in R n + , (3.1)
div u = 0 in R n + , (3.2)
u = g on Γ (3.3) has a solution (u, π) ∈ W 1, p ℓ (R n + ) × W 0, p ℓ (R n + ), unique up to an element of S + [1-ℓ-n/p] , with the estimate inf (λ, µ)∈S + [1-ℓ-n/p] u + λ W 1, p ℓ (R n + ) + π + µ W 0, p ℓ (R n + ) C g W 1-1/p, p ℓ (Γ) .
Proof. The operator associated to this problem is clearly continuous, moreover its kernel is known. The last point concerns its surjectivity, then the final estimate will be a straightforward consequence of the Banach theorem. So, we only must prove the existence of a solution (u, π).

(1) Firstly, we will show that system (3.1)-(3.3) can be reduced to a set of three problems on the fundamental operators ∆ 2 and ∆.

Applying the operator div to the first equation (3.1), we obtain

∆π = 0 in R n + . (3.4)
Now, applying the operator ∆ to the same equation (3.1), we deduce

∆ 2 u = 0 in R n + . (3.5)
From the boundary condition (3.3), we take out

u n = g n on Γ, (3.6)
and moreover div ′ u ′ = div ′ g ′ on Γ, where div

′ u ′ = n-1 i=1 ∂ i u i . Since div u = 0 in R n
+ , we also have div u = 0 on Γ, then we can write (2) Next, we will solve these three problems.

∂ n u n + div ′ u ′ = 0 on Γ, hence ∂ n u n = -div ′ g ′ on Γ. (3.7) Combining (3.5), (3.6) and (3.7)
Step 1: Problem (P ). Since g ∈ W 1-1/p, p ℓ (Γ), we have g n ∈ W

1-1/p, p ℓ (Γ) and div ′ g ′ ∈ W -1/p, p ℓ (Γ), so (P ) is a homogeneous biharmonic problem with singular boundary conditions. Since ℓ < 0, we know that problem (P ) has a solution u n ∈ W 1, p ℓ (R n + ), unique up to an element of B [1-ℓ-n/p] (see [START_REF] Amrouche | Singular boundary conditions and regularity for the biharmonic problem in the half-space[END_REF], Theorem 4.5).

Step 2: Problem (Q). Since ∆ 2 u n = 0 in R n + , according to an appropriate trace result (see [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF], Lemma 3.7), we can deduce that ∆u n | Γ ∈ r ∈ A ∆ [1-ℓ-n/p] , we still have u ⋄ 1 = u 1r a solution to the first component of the equations (3.1) and (3.3). Consequently, the pair (u ⋄ , π) now completely satisfies the problem (3.1)-(3.3). Remark 3.2. If g is sufficiently smooth; i.e., g ∈ D(Γ), using a potentialtheoretic method, it has been shown (see [START_REF] Cattabriga | Su un problema al contorno relativo al sistema di equazioni di Stokes[END_REF], [START_REF] Chang | The steady Navier-Stokes problem for low Reynolds number viscous jets into a half-space[END_REF]) that there exists a unique solution of (3.1)-(3.3) with a finite Dirichlet integral. In that case, we can see that this solution is naturally coming in the functional setting of Lemma 3.1. Now, we can give the following general result.

For any f ∈ W -1, p ℓ (R n + ), h ∈ W 0, p ℓ (R n + ) and g ∈ W 1-1/p, p ℓ (Γ), satisfying the compatibility condition ∀ϕ ∈ A ∆ [1+ℓ-n/p ′ ] , f -∇h, ϕ W -1, p ℓ (R n + )× • W 1, p ′ -ℓ (R n + ) + div f , Π D div ′ ϕ ′ + Π N ∂ n ϕ n W -2, p ℓ (R n + )× • W 2, p ′ -ℓ (R n + ) + g, ∂ n ϕ W 1-1/p, p ℓ (Γ)×W -1/p ′ , p ′ -ℓ (Γ) = 0, (3.9)
problem (S + ) admits a solution (u, π) ∈ W 1, p ℓ (R n + ) × W 0, p ℓ (R n + ), unique up to an element of S + [1-ℓ-n/p] , and there exists a constant C such that inf

(λ, µ)∈S + [1-ℓ-n/p] u + λ W 1, p ℓ (R n + ) + π + µ W 0, p ℓ (R n + ) C f W -1, p ℓ (R n + ) + h W 0, p ℓ (R n + ) + g W 1-1/p, p ℓ (Γ) .
Proof.

(1) First, we still assume that ℓ < 0. We write f = div F, where F = (F i ) 1 i n ∈ W 0, p ℓ (R n + ), with the estimate

F W 0, p ℓ (R n + ) C f W -1, p ℓ (R n + ) . Let us respectively denote by F = ( F i ) 1 i n ∈ W 0, p ℓ (R n ) and h ∈ W 0, p ℓ (R n
) the zero extensions of F and h to R n . By Theorem 1.2, we know that there exists a solution (ũ, π) ∈ W 1, p ℓ (R n ) × W 0, p ℓ (R n ) to the problem ( S) :

-∆ũ + ∇π = div F and div ũ = h in R n .

Consequently, we can reduce the system (S + ) to the homogeneous problem (S ♯ ) :

-∆v + ∇ϑ = 0 and div v = 0 in R n + , v = g ♯ on Γ,
Moreover, we have the Green formula ∆u g , λ

W -1, p ℓ (R n + )× • W 1, p ′ -ℓ (R n + ) = R n + u g • ∆λ dx + g, ∂ n λ Γ , = R n + u g • ∆λ dx + g ′ , ∂ n λ ′ Γ ,
because ∂ n λ n = 0 on Γ, according to the definition of the kernel. Next, we have another Green formula

div u g , µ W 0, p ℓ (R n + )×W 0, p ′ -ℓ (R n + ) = - R n + u g • ∇µ dx -g n , µ Γ .
Finally, since -∆λ + ∇µ = 0, we have

R n + u g • ∆λ dx - R n + u g • ∇µ dx = 0,
we then get a first formulation for this compatibility condition:

∀(λ, µ) ∈ S + [1+ℓ-n/p ′ ] , f , λ -h, µ + g ′ , ∂ n λ ′ Γ -g n , µ Γ = 0. Now, according to the characterization (2.18)-(2.19), we can replace each pair (λ, µ) ∈ S + [1+ℓ-n/p ′ ] by ϕ -∇(Π D div ′ ϕ ′ + Π N ∂ n ϕ n ), -div ϕ , where ϕ belongs to A ∆ [1+ℓ-n/p ′ ] . Then we have f , λ W -1, p ℓ (R n + )× • W 1, p ′ -ℓ (R n + ) = f , ϕ -f , ∇(Π D div ′ ϕ ′ + Π N ∂ n ϕ n ) , = f , ϕ + div f , Π D div ′ ϕ ′ + Π N ∂ n ϕ n , because (Π D div ′ ϕ ′ + Π N ∂ n ϕ n )| Γ = 0. Likewise, h, µ W 0, p ℓ (R n + )×W 0, p ′ -ℓ (R n + ) = h, -div ϕ W 0, p ℓ (R n + )×W 0, p ′ -ℓ (R n + ) = ∇h, ϕ W -1, p ℓ (R n + )× • W 1, p ′ -ℓ (R n + )
.

Moreover, we can remark that, on the one hand , µ = -∂ n ϕ n on Γ, and on the other hand, according to (2.7), we have

∂ n λ ′ = ∂ n ϕ ′ on Γ, hence the equivalent formulation ∀ϕ ∈ A ∆ [1+ℓ-n/p ′ ] , f -∇h, ϕ + div f , Π D div ′ ϕ ′ + Π N ∂ n ϕ n + g, ∂ n ϕ Γ = 0,
i.e., the compatibility condition (3.9).

W m-1, p m+ℓ (R n + ), and then problem (R) has a solution u ′ ∈ W m+1, p m+ℓ (R n + ), unique up to an element of A ∆ [1-ℓ-n/p] (see [START_REF] Amrouche | Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition[END_REF], Corollary 3.4). Likewise, point (3) is unchanged with respect to the proof of Lemma 3.1.

Proof of Theorem 4.1. (1) Assume that ℓ -2. The proof is quite similar to the one of Theorem 3.3. Here again, the only question is the surjectivity of the Stokes operator for such data. For that, we must simply replace Theorem 1.2 by Theorem 1.3 and Lemma 3.1 by Lemma 4.2 in the proof of the existence of a solution for negative weights in Theorem 3.3.

(2) Assume that ℓ > 0. We simply extend the regularity argument used in [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF] for the cases ℓ = 0 and ℓ = -1. Now, hypothesis (4.1) is reduced to

n/p ′ / ∈ {1, . . . , ℓ + 1}. (4.3) Since n/p ′ = ℓ + 1, we have the imbedding W m-1, p m+ℓ (R n + ) ֒→ W -1, p ℓ (R n + ); moreover, W m, p m+ℓ (R n + ) ֒→ W 0, p ℓ (R n + ) and W m+1-1/p, p m+ℓ (Γ) ֒→ W 1-1/p, p ℓ
(Γ) hold. So, thanks to Theorem 3.3, we know that problem (S + ) admits a unique solution (u, π) ∈ W 1, p ℓ (R n + ) × W 0, p ℓ (R n + ). We will show by induction that

(f , h, g) ∈ W m-1, p m+ℓ (R n + ) × W m, p m+ℓ (R n + ) × W m+1-1/p, p m+ℓ (Γ) =⇒ (u, π) ∈ W m+1, p m+ℓ (R n + ) × W m, p m+ℓ (R n + ). (4.4)
For m = 0, (4.4) is true. Now, assume that (4.4) is true for 0, 1, . . . , m and suppose that (f , h, g)

∈ W m, p m+ℓ+1 (R n + ) × W m+1, p m+ℓ+1 (R n + ) × W m+2-1/p, p m+ℓ+1 (Γ). Let us prove that (u, π) ∈ W m+2, p m+ℓ+1 (R n + ) × W m+1, p m+ℓ+1 (R n + ). Since we also have the imbeddings W m, p m+ℓ+1 (R n + ) ֒→ W m-1, p m+ℓ (R n + ), W m+1, p m+ℓ+1 (R n + ) ֒→ W m, p m+ℓ (R n + ) and W m+2-1/p, p m+ℓ+1 (Γ) ֒→ W m+1-1/p, p m+ℓ
(Γ), according to the induction hypothesis, we can deduce that the solution (u, π) ∈ W m+1, p m+ℓ (R n + ) × W m, p m+ℓ (R n + ). Now, for any i ∈ {1, . . . , n -1}, we have

-∆(̺ ∂ i u) + ∇(̺∂ i π) = ̺ ∂ i f + 2 ̺ x.∇∂ i u + n -1 ̺ + 1 ̺ 3 ∂ i u + 1 ̺ x∂ i π. Thus, -∆(̺ ∂ i u) + ∇(̺ ∂ i π) ∈ W m-1, p m+ℓ (R n + ). Moreover, div(̺ ∂ i u) = 1 ̺ x ∂ i u + ̺ ∂ i h. Thus, div(̺ ∂ i u) ∈ W m, p m+ℓ (R n + ). We also have γ 0 (̺ ∂ i u) = ̺ ′ ∂ i γ 0 u = ̺ ′ ∂ i g ∈ W m+1-1/p, p m+ℓ (
Γ). So, by the induction hypothesis, we can deduce that ∀i ∈ {1, . . . , n -1}, (∂ i u,

∂ i π) ∈ W m+1, p m+ℓ+1 (R n + ) × W m, p m+ℓ+1 (R n + ). It remains to prove that (∂ n u, ∂ n π) ∈ W m+1, p m+ℓ+1 (R n + ) × W m, p m+ℓ+1 (R n + ).
For that, let us observe that, for any i ∈ {1, . . . , n -1}, we have

∂ i ∂ n u = ∂ n ∂ i u ∈ W m, p m+ℓ+1 (R n + ), ∂ 2 n u i = -∆ ′ u i + ∂ i π -f i ∈ W m, p m+ℓ+1 (R n + ), ∂ 2 n u n = ∂ n h -∂ n div ′ u ′ ∈ W m, p m+ℓ+1 (R n + ), ∂ n π = f n + ∆u n ∈ W m, p m+ℓ+1 (R n + ). Hence, ∇(∂ n u) ∈ W m, p m+ℓ+1 (R n + ) and, knowing that ∂ n u ∈ W m, p m+ℓ (R n + ), we can deduce that ∂ n u ∈ W m+1, p m+ℓ+1 (R n + )
, according to definition (1.1). Consequently, we have ∇u ∈ W m+1, p m+ℓ+1 (R n + ). Likewise, ∇π ∈ W m, p m+ℓ+1 (R n + ) and, finally, we can conclude that (u, π) ∈ W m+2, p m+ℓ+1 (R n + ) × W m+1, p m+ℓ+1 (R n + ).

Very weak solutions

The aim of this section is to come back to the homogeneous Stokes system (3.1)-(3.3), but with singular data on the boundary. In [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF], we solved it in the cases g ∈ W -1/p, p -1 (Γ) and g ∈ W -1/p, p 0 (Γ). Here, we will extend these results to the other weights, introducing the question of the kernel and, by duality, the compatibility condition. Thanks to Lemma 2.8, the proof will be more direct.

f W 0, p ′ -ℓ+1 (R n + ) + h W 1, p ′ -ℓ+1 (R n + ) .
In other words, Λ is continuous on W 0, p ′ -ℓ+1 (R n + )×

• W 1, p ′ -ℓ+1 (R n + ) ⊥ S + [1-ℓ-n/p] , and according to the Riesz representation theorem, we can deduce that there exists a unique (u, π) ∈ W 0, p ℓ-1 (R n + ) × W -1, p ℓ-1 (R n + )/S + [1-ℓ-n/p] , which is the dual space of W 0, p ′ -ℓ+1 (R n + )×

• W 1, p ′ -ℓ+1 (R n + ) ⊥ S + [1-ℓ-n/p] , such that ∀(f , h) ∈ W 0, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ), Λ(f , h) = u, f W 0, p ℓ-1 (R n + )×W 0, p ′ -ℓ+1 (R n + ) + π, -h W -1, p ℓ-1 (R n + )× • W 1, p ′ -ℓ+1 (R n + )
;

i.e., the pair (u, π) satisfies (5.2) and the kernel of the associated operator is S + [1-ℓ-n/p] .

Lemma 2 . 7 .

 27 Let ℓ ∈ Z such that n/p ′ / ∈ {1, . . . , ℓ -1} and n/p / ∈ {1, . . . , -ℓ + 1};

  , we obtain the biharmonic problem (P ) : ∆ 2 u n = 0 in R n + , u n = g n and ∂ N u n = -div ′ g ′ on Γ. Then, combining (3.4) with the trace on Γ of the n th component in the equations (3.1), we obtain the Neumann problem (Q) : ∆π = 0 in R n + and ∂ n π = ∆u n on Γ. Lastly, if we consider the n -1 first components of the equations (3.1) and (3.3), we can write the Dirichlet problem (R) : ∆u ′ = ∇ ′ π in R n + and u ′ = g ′ on Γ.

Theorem 3 . 3 .

 33 Let ℓ ∈ Z and assume that n/p ′ / ∈ {1, . . . , ℓ} and n/p / ∈ {1, . . . , -ℓ}. (3.8)

Theorem 5 . 1 . 1 ( 3 ) 1 ( 1 ( 1 ( 1 ( 1 ( 1 (

 5113111111 Let ℓ ∈ Z with hypothesis (2.9). For any g ∈ W -1/p, p ℓ-1 (Γ), satisfying the compatibility condition∀ϕ ∈ A ∆ [1+ℓ-n/p ′ ] , g, ∂ n ϕ W -1/p, p ℓadmits a solution (u, π) ∈ W 0, p ℓ-1 (R n + ) × W -1, p ℓ-1 (R n + ), unique up to an element of S + [1-ℓ-n/p] ,and there exists a constant C such thatinf (λ, µ)∈S + [1-ℓ-n/p] u + λ W 0, p ℓ-1 (R n + ) + π + µ W -1, p ℓ-1 (R n + )To start with, let us observe that Lemma 2.8 gives a meaning to these boundary conditions. Besides, thanks to the Green formula (2.11), we get the equivalence between problem (3.1)-(3.3) and the variational formulation:Find (u, π) ∈ T p ℓ, 1 (R n + ) satisfying ∀(v, ϑ) ∈ W 2, p ′ -ℓ+1 (R n + ) × W 1, p ′ -ℓ+1 (R n + ), (5.2) such that v = 0 and div v = 0 on Γ, (u, π), T (v, ϑ) W 0, p ℓ-1 (R n + )×W -1, p ℓ-1 (R n + ), W 0, p ′ -ℓ+1 (R n + (∂ n v ′ , -ϑ) W -1/p, pℓus solve problem (5.2). By Theorem 4.1, we know that under hypothesis (2.9), for all (f , h)∈ W 0, p ′ -ℓ+1 (R n + )× • W 1, p ′ -ℓ+1 (R n + ) ⊥ S + [1-ℓ-n/p] , there exists a unique solution (v, ϑ) ∈ W 2, p ′ -ℓ+1 (R n + )×W 1, p ′ -ℓ+1 (R n + )/S + [1+ℓ-n/p ′ ] to -∆v + ∇ϑ = f and div v = h in R n + , v = 0 on Γ, with the estimate (v, ϑ) W 2, p ′ -ℓ+1 (R n + )×W 1, p ′ -ℓ+1 (R n + )/S + [1+ℓ-n/p ′ ] C f W 0, p ′ -ℓ+1 (R n + ) + h W 1, p ′ -ℓ+1 (R n + ) . Consider the linear form Λ : (f , h) -→ g, (∂ n v ′ , -ϑ) W -1/p, p ℓℓ-n/p]. By (5.1), we have for any ϕ ∈ A ∆ [1+ℓ-n/p ′ ] , or, equivalently, for any (λ, µ)∈ S + [1+ℓ-n/p ′ ] , |Λ(f , h)| = g, (∂ n v ′ , -ϑ) + ∂ n ϕ W -1/p, p ℓ-Γ)×W 1/p, p ′ -ℓ+1 (Γ) = g, (∂ n [v ′ + λ ′ ], -[ϑ + µ]) W -1/p, p ℓ-Γ) (v, ϑ) + (λ, µ) W 2, p ′ -ℓ+1 (R n + )×W 1, p ′ -ℓ+1 (R n + ) . Thus, |Λ(f , h)| C g W -1/p, p ℓ-Γ) (v, ϑ) W 2, p ′ -ℓ+1 (R n + )×W 1, p ′ -ℓ+1 (R n + )/S + [1+ℓ-n/p ′ ]

  , Lemma 4.1, the following result.

	Lemma 2.1. Let ℓ ∈ Z such that
	n p ′ / ∈ {1, . . . , ℓ -2} and	n p	/ ∈ {1, . . . , -ℓ + 2};	(2.1)

  By Lemma 2.9, we can see that π and ũ are respectively harmonic and biharmonic tempered distributions in R n , thus polynomials. Hence, we have the following result.Corollary 2.11. Let ℓ ∈ Z with hypothesis (2.9); then K S = S + [1-ℓ-n/p] . Again, this kernel does not depend on the regularity. That is, if we denote by K m

S the kernel of the Stokes operator (T,

τ 0 ) in W m+1, p m+ℓ (R n + ) × W m, p m+ℓ (R n + ),

we have the following result. Corollary 2.12. Let ℓ ∈ Z and m -1 be two integers and assume that n p ′ / ∈ {1, . . . , ℓ + min{m, 1}} and n p / ∈ {1, . . . , -ℓ -m};

= π,div ϕϕ * + 2 x n ∇ϕ * n + x 2 n ∇(ψ * -(div ϕ) * ) , hence we put Φ = ϕϕ * + 2 x n ∇ϕ * n + x 2 n ∇(ψ * -(div ϕ) * ). Next, we can group together the remaining terms in R n + (A+B) dx, where

. We can rewrite B by means of e n = (0, . . . , 0, 1) as follows:

Using the identity e n ∆ξ = ∇(x n ∆ξ) -∆(x n ∇ξ) + 2 ∇∂ n ξ, we get

under hypothesis (2.9). In addition,

(Γ). As ℓ < 0, we know that problem (Q) has a solution π ∈ W 0, p ℓ (R n + ), unique up to an element of N ∆ [-ℓ-n/p] (see [START_REF] Amrouche | The Neumann problem in the half-space[END_REF], Theorem 3.4).

Step 3: Problem (R). Thanks to the previous result, we can deduce that [START_REF] Amrouche | Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition[END_REF], Theorem 3.2).

(3) In order, we have found u n , π and u ′ , non-unique, which satisfy (3.3) and partially satisfy (3.1), more precisely such that

It remains to show we can choose them satisfying (3.2) and the n th component of (3.1); i.e., -∆u n + ∂ n π = 0 in R n + . Consider such a pair (u, π) satisfying problems (P ), (Q) and (R). Thanks to the first equations of (P ) and (Q), we obtain

Thus, with the boundary condition of (Q), we can deduce that the distribu- [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF], Theorem 3.5). Moreover, we can write µ = ∆Π D µ, with andu † n is still a solution to problem (P ).

Note that π is unchanged with u † n , because ∆q = µ = 0 on Γ. Thus, if we set u † = (u ′ , u † n ), the pair (u † , π) completely satisfies (3.1). Next, as ∆π = 0 in R n + , we also have ∆ div u † = 0 in R n + . Moreover, from the boundary condition in (R), we obtain div ′ u ′ = div ′ g ′ on Γ. Then, with the boundary condition in (P ), we can write [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF], Theorem 3.8). If we take for instance r(x) =

x 1 0 ν(t, x 2 , . . . , x n ) dt, we have ν = ∂ 1 r and, thus, ν = div r, with r = (r, 0, . . . , 0). Setting u ⋄ = u †r, we get div u ⋄ = 0 in R n + and, as where we have set

Next, thanks to Lemma 3.1, we know that (S ♯ ) admits a solution (v, ϑ)

(2) We now assume that ℓ > 0. We will reason by duality from the case ℓ < 0. So, we have established that, under hypothesis (3.8), the Stokes operator

is an isomorphism for any integer ℓ < 0 and real number p > 1. Thus, replacing p by p ′ and -ℓ by ℓ, we deduce that its adjoint operator

is an isomorphism for any integer ℓ > 0 and real number p > 1, always under hypothesis (3.8). Moreover, by a density argument, we can readily show that

So, we have proved that, for any ℓ > 0, problem (S + ) with g = 0 admits a unique solution provided (f , h) ⊥ S + [1+ℓ-n/p ′ ] . Now, it remains to show that the general problem (S + ) can be reduced to the particular case with g = 0, by means of a lifting function; and then that the orthogonality condition on the lifted problem is equivalent to the compatibility condition (3.9).

First, by Lemma 1.1, there exists a lifting function u g ∈ W 1, p ℓ (R n + ) of g, i.e., u g = g on Γ, such that

) is equivalent to the following, with homogeneous boundary conditions:

So, provided (f +∆u g , -h+div u g ) ⊥ S + [1+ℓ-n/p ′ ] , we know that (S ⋆ ) admits a unique solution. This condition is written in the following way:

Strong solutions and regularity

In this section, we are interested in the existence of strong solutions, i.e., of solutions (u, π) ∈ W 2, p ℓ+1 (R n + ) × W 1, p ℓ+1 (R n + ); and more generally, in the regularity of solutions to the Stokes system (S + ) according to the data. For any f ∈ W m-

We have already proved this result for ℓ = 0 and ℓ = -1 in our previous work (see [START_REF] Amrouche | Very weak, generalized and strong solutions to the Stokes system in the half-space[END_REF], Corollaries 5.5 and 5.7). We will use similar arguments for the other negative weights, with the aim of minimizing the set of critical values, thanks to the known results on the harmonic and biharmonic operators in the half-space. Then, for the positive weights, we will use a regularity argument to avoid the compatibility conditions which would naturally appear in the auxiliary problems with the previous method.

At first, we adapt Lemma 3.1 and its proof for more regular data.


Esmaeil Najafi Aghdam

Philippe Benabes
email: philippe.benabes@supelec.fr

Javad Abbasszadeh

Completely first order and tone free partitioned data weighted averaging technique used in a multibit delta sigma modulator

Keywords: Data weighted averaging, DWA, DEM, ADC, DAC

In order to reduce mismatch error of a DAC used in a multibit delta-sigma some dynamic element matching (DEM) algorithms have been proposed before, from which Data-Weighted-Averaging (DWA) method is more hardware efficient and widely used.

Unfortunately, DWA technique loses its functionalities for periodic signals which cannot be practically avoided. Many improvements have been suggested to minimize in band tone generated by DWA algorithm, but they cause limited performance compared with an ideal first order DEM. This paper presents a new modified DWA structure, which results in a completely first order mismatch noise shaping while solving in band tone problem.

Simulations are presented for a 3rd-order lowpass delta sigma. However, its modified version can also be used in bandpass applications.

I. INTRODUCTION

Single bit delta-sigma modulators lead to very good linearity characteristics due to the 1 -bit DAC inherent linearity. Today, however, telecommunications applications require large bandwidth thus leading to high oversampling ratios for singlebit topologies. Reducing this ratio can be compensated by using a multibit quantizer so that theoretically, the signal-tonoise-ratio (SNR) increases by 6dB per bit. Furthermore, multibit quantization enhances system stability by making possible the use of higher open-loop filter gain. It is also important to note that using multibit structure is one of the best ways to decrease jitter sensitivity and amplifiers constraints especially in the continuous time delta sigma modulators [START_REF] Schreier | Understanding Delta-sigma data converters[END_REF][START_REF] Story | Digital to analogue converter adapted to select input sources based on a preselected algorithm once per cycle of of a sampling signal[END_REF]. Unfortunately, a DAC with very good linearity characteristics is required in multibit DSM. This leads to accurate component matching that is not generally compatible with CMOS standard technology. Some algorithms were proposed to correct non-linearity errors in the internal DAC such as digital correction and dynamic element matching [START_REF] Baird | Linearity enhancement of multi-bit a/d and d/a converters using data weighted averaging[END_REF][START_REF] Aghdam | A new mixed stable dem algorithm for bandpass multibit delta sigma adc[END_REF][START_REF] Aghdam | A hardware efficient 3-bit second-order dynamic element matching circuit clocked at 300mhz[END_REF][START_REF] Aghdam | Higher order dynamic element matching by shortened tree-structure in delta-sigma modulators[END_REF][START_REF] Welz | Necessary and sufficient conditions for mismatch shaping in a general class of multibit dacs[END_REF]. Randomization barrel shuffler can spread mismatch errors over the whole frequency range [START_REF] Schreier | Understanding Delta-sigma data converters[END_REF]. For high resolution application the circuit errors must be shaped and moved out of the band of interest. The mostly used first order DACs error shaping algorithm is Data Weighted Averaging in the feedback path of a Delta-Sigma modulator as shown in Fig. 1. Such a selection scheme results in a first order mismatch shaping as will be denoted in the next section. However, the DWA error reduction technique suffers more from periodic tones appeared in its band of interest which is usually accrued for some input levels. Many modifications have been developed to solve tonal behavior of the DWA [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF][START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF][START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]. Most of them scarify DWA full performance to avoid affecting desired band from those tones [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]. In other words, recently proposed modifications cause the DWA results error shaping behavior lower than its ideally first order function. In this work, a new modification will be proposed which cause no functionality restriction, so the DWA algorithm can still acts as an ideal first order noise shaping system. The proposed algorithm uses a simple first order digital divider box and then a partitioned DWA structure.

In section II, the concepts of a conventional DWA is reviewed, in section III the tonal behavior of DWA scheme is presented, then, the modification proposed before will be addressed. The proposed new method is described in section IV, followed by simulation results. Some conclusions are finally mentioned in section V.

II. CONVENTIONAL DWA ALGORITHM

As mentioned above, the DWA is a technique to remove DAC cells mismatch errors out of the band of interest [START_REF] Baird | Linearity enhancement of multi-bit a/d and d/a converters using data weighted averaging[END_REF][START_REF] Aghdam | A new mixed stable dem algorithm for bandpass multibit delta sigma adc[END_REF]. In essence, DWA tries to choose different DAC cells for consecutive codes. In this way, less common units participate in consecutive codes and DAC error signal shows fast changes in time domain which is an indication of high-frequency content in frequency domain [START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF].

Although, a conventional DWA block, shown in Fig. 1, doesn't have any filter structure [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF], use a suitable element selection strategy which can cause mismatched nonlinear errors to be filtered like an ideal first order high pass filter. The element selection strategy is depicted in table one for some consecutive input levels, where DAC uses a 3-bit thermometric structure. As indicated on the table, DAC cells (example: D 1 , D 2 ... D 8 ) must be selected circularly based its consecutive input value, v(n). From one period to another, the status of selected cells is simply memorized by a pointer called Ptr(n) described at end column on the table. Mathematically, it was proven that the nonlinear part of DAC mismatch error on its output can be estimated as follow [START_REF] Baird | Linearity enhancement of multi-bit a/d and d/a converters using data weighted averaging[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]:

∑ ∑ - = - = = - = - = - = 1 0 1 0 1 ) ( 1 / ) ) ( ( ) ( : )) ( ( ) 1 ( N n mean Ptr n n mean mean error n v N V V V n V ptr IM Where z Ptr IM z DAC (1)
Nonlinearity errors for a DAC derived by DWA element selection scheme can then be analytically calculated using [START_REF] Schreier | Understanding Delta-sigma data converters[END_REF]. Such an error shaping is similar to main delta-sigma modulators, where an M-cell DAC presents a non linear error term on its output which can be formulated as follow [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]:

12 ) 1 1 ( 3 2 2 3 2 M M OSR DAC error - ⋅ = σ α π (2)
Where σ α 2 is variance of DAC cells mismatch error, for example (0.01) 2 , means one percent mismatch.

Typical implementation circuit for the conventional DWA is shown in Fig. 2 [11].

DAC input value DAC Cells 3-bit thermometric structure) (

III. TONAL BEHAVIOUR OF DWA

The conventional DWA is able to shape DAC mismatch errors as a first order high pass filter, but for some periodic inputs and some input DC-levels.

The tonal behavior of a conventional DWA scheme can also be described by system level simulation as Fig.

3 [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF].

Where, the undesired tones comeback to the band of interest while the input level goes some smaller levels, which has not been already predicted by [START_REF] Schreier | Understanding Delta-sigma data converters[END_REF]. More simulation examples which are monitoring tonal behavior of a conventional DWA algorithm are presented in [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]. The tones frequency can be estimated as below [START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF][START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]

: ,... 2 , 1 , = = k f M GCD k f s tons (3)
Term GCD means the greatest common devisor and M is the number of DAC cells. Practically in order to avoid these tones, the over sampling ratio (OSR) of the delta-sigma modulator must satisfy certain condition as denoted below for the same k, which is not suitable in general case specially for wide-band applications. ,.... 2 , 1 , ) 2

1 ( = + ≥ k M k OSR (4)

IV. PREVENTION OF TONAL BEHAVIOUR OF DWA

Based our researches, there are at least 9 different methods which have been already proposed to reduce in band tones resulting from DWA algorithm application. They are only summarized on table-II; one can refer to the related references. Between proposed tones prevention methods, the partitioned technique (PDWA) is more attractive because of its relatively better performance, simplicity and generality for lowpass and bandpass applications [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF][START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF][START_REF] Fishov | Segmented mismatch-shaping d/a conversion[END_REF][START_REF] Fishov | Segmented mismatch-shaping d/a conversion[END_REF]. Fig. 4 shows PDWA architecture, where DAC input code is divided into two parts using a simple divider and adding rest of division on only one definitive part of the results. By this means, the difference of mean value between two sub DACs (upper M/2 cells and lower one) cannot be shaped by two separated DWA algorithms. So, theoretically, its performance will be decreased even for non periodic inputs. Furthermore, its tonal behavior may occur because of limited perturbation in its global algorithm but with smaller probability compared with conventional DWA implementation. Figure 3. Output spectrum(log-scale versa normalized frequency) for different input levels where DAC mismatch errors made inband tones be appeared [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF] Table 1) Technique to prevent tonal behavior of DWA Tone-prevention technique for DWA Characteristic and Inconvenience 1 dither [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF] The simplest, less performance 2 rotation, RDWA [START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF] First idea, limited performance, need to be optimized 3 RnDWA [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF] Randomization based, limited performance, difficult to be implemented 4 DIA [START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF][START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF] double index, Compatible with SC, no general 5 IDWA [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF] Incremented cells, limited performance, need to further DAC cells 6 bidirectionnelle BI-DWA [START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF] Two pointer, limited performance, need to further DAC cells 7 PDWA [START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF][START_REF] Fishov | Segmented mismatch-shaping d/a conversion[END_REF] Segmentation, limited performance, the best 8 DWApartielle [START_REF] Welz | Necessary and sufficient conditions for mismatch shaping in a general class of multibit dacs[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF] Applied incompletely, Limited performance 9 pseudo -DWA [START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF] Applied incompletely, Limited performance, Equation below, analytically estimate the remained unshaped DAC error which correspond mainly the LSB of input code and difference between the mean value of upper half DAC and lower half one, in case of a simple partitioned DWA.

] 2 2 [ 2 ) ( 2 / 1 2 / 1 ) ( ∑ ∑ + - = Δ M M i M i n v PDWA M M LSB n error α α (5)
This theoretical prediction of extra error terms in PDWA is verified by simulation as shown in Fig. 5. As seen on this spectrum, the DEM performance is decreased by increasing inband noise level, but its tones are moved far from the band of interest.

V. PROPOSED METHOD TO PREVENT TONAL BEHAVIOUR OF DWA ALGORITHM

As mentioned above, in order to implement a multibit delta-sigma modulators using a DWA mismatch noise shaping technique, the partitioned DWA algorithm is the one of the best choice to prevent its tonal behavior [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF][START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF][START_REF] Vadipour | Techniques for preventing tonal behavior of data weighted averaging algorithm in sigma delta modulator[END_REF][START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]. However, its performance is limited mainly by divider function which has no more mismatch shaping action in its previously proposed structure [START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF][START_REF] Fishov | Segmented mismatch-shaping d/a conversion[END_REF][START_REF] Aghdam | Reducing multibit DAC circuits errors by a simplified dynamic element matching algorithm used in delta-sigma converters[END_REF]. In this work, a new filter based divider is proposed which causes divider to shape division rest. In fact, such an algorithm must select a suitable sequence as the difference between two outputs codes in a manner that there is no any power remained in the band of interest while respecting to the number conservation rule. Fig. 6 shows a desired first order filter based divider structure for lowpass applications. It has a switching block (SB) including a first order highpass digital filter in a delta-sigma like feedback architecture which can form its output R(n).

Where, v(n) is the input code, v h (n) and v l (n) are output of divider. R(n) is the filter output indicates that the rest of division must be added in what part of DWA. The case R(n)>0

corresponds to the upper selection and vice versa. r(n) is a 1bit random number used when R(n) has no definitive indication, e.g. in its zero case. The switching sequence signal named S(n), is calculated at each period depending on filter output, DAC input code and randomly generated 1-bit r(n). Its transfer function is in general proposed as below, but may be simplified further with elimination of the cases S(n) =±2.

-Z -1 1+Z -1 S (n) LP v(n) + - +2 +1 0 - 1 -2 . . . . . R(n) Dither v (n) l v (n) h r(n) (a) (b)
V(n)

⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ - ≤ ≤ - - ≤ ≤ + - = + = cases other all n R even and M n v n R even and M n v n R odd is n v n R odd is n v n r n R odd is n v n S 0 0 ) ( ) 2 ( ) ( 2 2 0 ) ( ) 2 ( ) ( 2 2 0 ) ( ) ( 1 0 ) ( ) ( ) ( 0 ) ( ) ( 1 ) ( ≺ ≺ (6)
Tones moved away By this means, two half DAC inputs can be calculated as: [START_REF] Welz | Necessary and sufficient conditions for mismatch shaping in a general class of multibit dacs[END_REF] Since, the switching sequence S(n) has a first order highpass transfer function depending on filter output R(n), the LSB (or two LSBs) is shaped by the same manner, which results in a first order mismatch shaping on [START_REF] Aghdam | A hardware efficient 3-bit second-order dynamic element matching circuit clocked at 300mhz[END_REF]. Total DAC error using the proposed partitioned DWA algorithm can now be written as [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF] for lowpass applications, where α h and α l are the mean value of mismatch error for upper half DAC and lower one respectively. It can be derived following (1) [9,11]. Two former terms are directly given by ( 1) and the third term can be considered as an extra gain error which is linear with the DAC input having less importance [START_REF] Schreier | Understanding Delta-sigma data converters[END_REF]. However, the fourth term is a nonlinear error depending on the difference between average gains of two half DACs and shape of S(n) sequence which is produced by a desired highpass filter. Therefore, the only inconvenience rest in the proposed structure relayed on the forth term. This term can be compared with the whole of the right hand of (1), where it is clearly decreased by the shape of S(n) produced by the desired filter. [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF] As theoretically predicted by [START_REF] Vleugels | A 2.5-v sigma-delta modulator for broadband communication applications[END_REF], the overall resolution of the proposed modification on PDWA is obtained nearly identical to the similar case of an ideal DWA block. The modified PDWA performance in terms of effective number of bit (ENOB) can be improved by 0.22-bit in best case or can be decreased as low as 0.5-bit in the worth case input sequences as described more in [START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF]. Based (1) and ( 8), we can derive the performance of the proposed algorithm as below: [START_REF] Radke | A 12-bit current-mode sigma delta dac based upon rotated data weighted averaging[END_REF] Where ENOB DWA is related to the performance of a fully data weighted averaging algorithm without any tones.

The difference between two sides of (9) comes from correlation between v h (n) and v l (n) . The correlation term is about zero for lower input, i.e. there is no quotient form division operation and it is high for medium or upper input level. In case of low level input, the correlation is low so the proposed algorithm appears even a little more efficient the simple DWA. Practically, the performance of the proposed algorithm is obtained about 0.2-bit lower than that of an ideal DWA without tones as verified by different simulations, there is no saturation on integrator output R(n) [START_REF] Aghdam | Nouveles Techniques D'appariement Dynamique Dans Un CNA Multibit Pour Les Convertisseurs Sigma-Delta[END_REF].

VI. SIMULATION RESULTS

The proposed modification on PDWA algorithm is used in a third order lowpass multibit DSM with a 4-bit quantizer. A zoomed view of its output spectrum is shown in Fig. 7. It can be seen that the proposed technique has a better function suppressing the in band tones and moving them far from band of interest by the desired filter. This figure is simulated supposing a given 1 percent DAC cells error. The amount of improvement depends on stochastic properties of input signal.

VII. CONCLUSION

Some aspects of the conventional DWA algorithm using as a dynamic element matching strategy were studied. Its major disadvantage is tonal behavior for periodic input levels. There were nine modification proposed before but no one operates completely as a first order error shaping predicted by an ideal DWA. We have proposed another modification which leads to definitively better results giving nearly a complete first order mismatch error shaping function with a little extra circuit. The analytical analysis has been derived and verified by system level simulations. Simulations are presented for a third order lowpass delta sigma. However, its modified version can also be used in bandpass applications.

Figure 1 .

 1 Figure 1. Multibit Delta-Sigma modulator using a DAW algorithm

Figure 2 .

 2 Figure 2. a suitable circuit block diagram to implement a 4-bit DWA

Figure 5 .

 5 Figure 5. DAC output spectrum using PDWA with structure shown Fig.4

Figure 6 .Figure 4 .

 64 Figure 6. a)Modified PDWA block diagram, b) proposed lowpass

Figure 7 .

 7 Figure 7. DAC output spectrum(zoomed), improving tones behavior and mismatch shaping function by a modified partitioned DWA proposed in this work

Table 1 .

 1 Lowpass DWA selection scheme


Sébastien Aubert
email: sebastien.aubert@stnwireless.com

Fabienne Nouvel
email: fabienne.nouvel@insa-rennes.fr

Amor Nafkha
email: amor.nafkha@supelec.fr

Complexity gain of QR Decomposition based Sphere Decoder in LTE receivers

Keywords: MIMO detection, Sphere Decoder, QR Decomposition, Householder, LTE

It has been widely shown that the Sphere Decoding can be used to find the Maximum Likelihood (ML) solution with an expected complexity that is roughly cubic in the dimensions of the problem. However, the computational complexity becomes prohibitive if the Signal-to-Noise Ratio is too low and/or if the dimension of the problem is too large. That is why another technique denoted as Fixed-complexity Sphere Decoder (FSD) is an interesting approach. This algorithm needs a preprocessing step, and in this paper the QR-Decomposition-based preprocessing technique, which is not inconsequential, will be studied. Two different techniques are exposed, including the classical Gram Schmidt orthonormalization process. Their computational complexities and their impacts on the FSD computational complexity are studied. In the LTE context, the overall computational complexities of the two detection techniques are quantified and are shown to be dependent on the constellation size.

I. INTRODUCTION AND PROBLEM STATEMENT

One of the main challenges in receiver design for wireless communication systems lies in the non-orthogonality of the transmission channel which implies a performance degradation of the system due to Inter-Layer Interferences (ILI). So, to ensure high reliability of data transmission, a special attention has to be paid to the receiver block. The system model, made of n T and n R transmit and receive antennas respectively, reads

y = Hx + n, (1)
where x ∈ C n T and y ∈ C n R denote the system input and output respectively, the matrix H ∈ C n R ×n T representing the equivalent channel matrix, and n ∈ C n R the additive white gaussian noise. Typically, the noise components are independent and identically distributed as zero-mean Gaussian random variables with a variance σ 2 . The entries of x are assumed to be uniformly distributed over a finite set of symbols ξ n T . The channel matrix H is assumed to be perfectly known at the receiver end. The optimum Maximum Likelihood (ML) detection [START_REF] Verdú | Multiuser Detection[END_REF] consists in finding the signal vector x ∈ ξ n T that minimizes the Euclidean distance with respect to the received signal vector y. x is the closest lattice point [START_REF] Agrell | Closest point search in lattices[END_REF] in the Euclidean space x = argmin

x∈ξ n T y -Hx 2 2 . (2)
An exhaustive search over all feasible elements ξ n T is necessary to solve ML problems [START_REF] Agrell | Closest point search in lattices[END_REF]. However, the computational complexity is exponential in the number of possible constellation points, making this technique unsuitable for practical purposes when aiming at high modulation orders and numbers of transmit antennas. Nevertheless, for low dimension problems with low modulation order schemes such as BPSK and QPSK, exhaustive search methods have been shown to be feasible [START_REF] Burg | A 50 mbps 4x4 maximum likelihood decoder for multiple-input multiple-output systems with qpsk modulation[END_REF]. For higher problem dimension, the ML detection problem (2) can be resolved with a reasonable complexity using an efficient search method like the universal lattice decoding algorithm [START_REF] Mow | Maximum likelihood sequence estimation from the lattice viewpoint[END_REF], also called Sphere Decoder (SD). The SD search algorithm is based on the Finke-Pohst enumeration to explore all the lattice points inside a hypersphere centered at the received symbol vector [START_REF] Fincke | Improved methods for calculating vectors of short length in a lattice, including a complexity analysis[END_REF]. The interest in lattice decoding has steadily grown in the last few years. One of the most efficient known SD algorithms has been proposed by Viterbo and Boutros (VB) in [START_REF] Viterbo | A universal lattice code decoder for fading channels[END_REF]. However, one important drawback in the VB algorithm and other similar SD algorithms is the choice of the initial value of the search radius. If this radius is chosen too small, there may be no solution for the algorithm (no point inside the hypersphere) and if the radius is chosen too large, the number of explored points may be very high and the algorithm will be inefficient, acting like the ML algorithm. Although there are some methods for deciding on the initial value of the radius, there is not yet any automatic method that works well in all different applications. In all proposed methods, it is still possible that no valid point is found inside the hypersphere. In those cases, the search procedure must be repeated with a larger radius. This reiteration considerably increases the computation time. In some cases, like signal detection in fading channels, the choice of this radius is even more difficult [START_REF] Viterbo | A universal lattice code decoder for fading channels[END_REF]: because of the rapid changes in the channel condition, the initial value of this radius must be determined more frequently. The average complexity of the SD algorithm is shown to be polynomial (roughly cubic) in the number of transmit antennas over a certain range of Signal-to-Noise Ratio (SNR) and number of transmit antennas, while the worst case complexity is still exponential [START_REF] Jalden | On the complexity of sphere decoding in digital communications[END_REF]. From a hardware implementation point of view, the SD algorithm exhibits two major weaknesses. First, its complexity coefficients can become large when the problem dimension is high, i.e. at the spectral efficiency demanded by future devices. Second, the variance of its computation time can also be large, leading to undesirable highly variable decoding delays.

For the sake of lower computational complexity and simplicity for the hardware implementation, the use of the Fixedcomplexity SD (FSD) is interesting. This detector has been previously proposed for the detection in uncoded MIMO systems using Quadrature Amplitude Modulation (QAM) constellations [START_REF] Barbero | Performance analysis of a fixedcomplexity sphere decoder in high-dimensional mimo systems[END_REF]. It overcomes the two main drawbacks of the SD from an implementation point of view, i.e. its variable complexity depending on the noise level, and the sequential nature of its tree search phase. The next Section briefly reviews the state of the art in the existing FSD algorithm. Subsequently, in Section III, the contribution is exposed with the description of two efficient preprocessing methods through their computational complexity and stability [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF]. To demonstrate the efficiency of our approach to reduce the computational complexity of the FSD, several simulation results and comparative analysis are provided in Section IV. The paper concludes with a summary in Section V.

II. FIXED-COMPLEXITY SPHERE DECODER

The FSD was introduced in [START_REF] Barbero | Performance analysis of a fixedcomplexity sphere decoder in high-dimensional mimo systems[END_REF], [START_REF] Barbero | Fpga design considerations in the implementation of a fixed-throughput sphere decoder for mimo systems[END_REF] to overcome some of the problems faced by the original SD in relation to its hardware implementation. Specifically, it addressed the variable complexity of the SD and the sequentiality of certain parts of the SD algorithm which made a pipelined implementation difficult. The FSD can in essence be described as a partial search through a small subset of the constellation using a specific detection ordering. In [START_REF] Barbero | Performance analysis of a fixedcomplexity sphere decoder in high-dimensional mimo systems[END_REF], it was shown that the FSD achieves maximal diversity in reception when applied to the two by two independent and identically distributed Rayleigh fading multiple antenna channel. The FSD performs a search over a fixed number of lattice points Hx, generated by a small subset S included in ξ n T , around the received vector y. The transmitted vector x which belongs to S associated to the smallest Euclidean distance is then selected as the estimate (2). The processing and preprocessing parts will be presented in the following Subsections.

A. Preprocessing of FSD

The channel matrix H is first decomposed using a QR Decomposition (QRD) method. Hence, H = QR, where Q is a n R × n T unitary matrix, i.e. Q H Q = I, and R is a n T × n T upper triangular matrix. Then, the received symbol vector y is multiplied with Q H prior to the symbol detection step, and (1) reads

Q H y = Q H QRx + Q H n = Rx + Q H n. (3)
The matrix R structure being upper triangular, the transmitted symbol vector x could be estimated from the vector Q H y denoted as ỹ and the matrix R by employing the Gauss elimination algorithm [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] or the proposed Householder algorithm (see Section III). Denoting Q H n as ñ, (3) reads

ỹ = Rx + ñ. (4)
Note that Q being an unitary matrix, the covariance of the noise term remains unaffected. Assuming the QRD is realized and due to the shape of R, (1) can be written in an iterative monotonically increasing form

x = argmin x∈ξ n T ỹ -Rx 2 2 = argmin x∈ξ n T n T i=1 δ i ≤ C ( 5
)
where C is the sphere constraint and δ i denotes the Partial Euclidean Distance (PED) corresponding to the symbol transmitted on the i-th layer and such as

δ i = ỹi - n T j=i r ij x j 2 2 (6)
for all 1 ≤ i ≤ n T .

B. Processing of FSD

Thanks to the PED introduction, the optimal symbols vector search can be depicted as a tree search [START_REF] Hassibi | On the sphere-decoding algorithm i. expected complexity[END_REF], with the first node corresponding to n T -th symbol at the top level. Figure 1 illustrates the tree construction for two transmit antennas. The FSD algorithm idea is to apply a full ML search at the top layer and a linear equalizer on the rest of the layers. Concerning the tree exploration, whether the sphere constraint is respected or not, the following explored symbol vector is tested by stepping down the tree from the root node, according to the depth-first search principle. The Reference FSD (RFSD) algorithm is detailed in [START_REF] Eckert | Investigations on different approaches for equalization in mimo ofdm systems in terms of performance and complexity[END_REF]. The basic principle induces some optimizations such as Schnorr-Euchner (SE) enumeration which was introduced in [START_REF] Guo | A vlsi architecture of the schnorr-euchner decoder for mimo systems[END_REF], [START_REF] Agrell | Closest point search in lattices[END_REF]. It was shown in [START_REF] Chan | A new reduced-complexity sphere decoder for multiple antenna systems[END_REF] that enumerating the symbols at the top-level of the tree in an ascending order according to their distance to the Babai point [START_REF] Agrell | Closest point search in lattices[END_REF] will expedite the tree search, as depicted in Figure 1. Thus, the processing root Node not processed Node processed with δ 1 < C Node processed with δ 2 ≥ C complexity is decreased by reducing the number of visited nodes. In addition, the problematic setting of a proper initial radius becomes useless: once the Babai point is obtained, the radius is shrunk to the total computed Euclidean distance.

The processing complexity can also be decreased through the preprocessing step, which is presented in the following Subsections. The preprocessing algorithm stability [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] point of view is also an important aspect since the loss of orthogonality in the matrix Q (due to the round-off error) induces ILI and reduces the FSD performance.

In this paper, Gram-Schmidt (GS) orthogonalization and Householder (HH) transformation are described as well as their computational complexities and their impact on the processing computational complexity.

III. ENHANCED PREPROCESSING STEP A. Gram-Schmidt

The GS algorithm [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] is an orthogonalization procedure to get an orthonormal basis Q = q 1 , q 2 from any invertible matrix H = h 1 , h 2 , as described in Figure 2 which geometrically illustrates the first two steps of the orthogonalization procedure. The (h 1 , h 2 ) basis is transformed into the orthonormal (q 1 , q 2 ) basis thanks to the orthogonal projection of h 2 onto the subspace generated by q 1 (which is the same as the subspace generated by h 1 ). q 2 is then defined as h 2 -proj h1 (h 2 ), with proj u (v) the projection of v on u such that proj u (v) = u H v u H u u. The obtained set of orthogonal vectors finally has to be normalized, so that a set of orthonormal vectors is obtained. With this technique, the matrix Q is determined first, then the matrix R is deduced using Q and H. It can be shown that R = r 11 r 12 0 r 22 , with r 11 ∈ R + and r 12 , r 22 ∈ C.

h 1 h 2 q 2 proj h 1 (h2)
Another technique, which results in real positive coefficients rather than complex ones, may be used.

B. Householder

The HH algorithm [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] is a transformation procedure to get an upper triangular matrix R, as described in Figure 3 which geometrically illustrates the first two steps of the reflection procedure. A Householder matrix, which is also known as a Householder reflector, is a matrix of the form

h 1 h 2 span(h1) ⊥ Ph 2
P v = I -2 vv H v H v , with v ∈ C 2 *
. The application of P v to a vector x yields P 3 illustrates the reflection of h 2 with respect to the hyperplane span(h 1 ) ⊥ . Householder matrices are a mathematical tool for introducing zeros into vectors, such as any matrix can be made upper triangular with an iterative processing, without need of computing explicitly the Householder matrix. With this technique, the matrix R is determined first, then the matrix Q is deduced using R and H. It can be shown that R = r 11 r 12 0 r 22 , with r 11 , r 22 ∈ R + and r 12 ∈ C.

v x = x -2 v H x v H v v = x -2 • proj v (x). Figure
The detector, associated with the HH-based preprocessing, which takes into account the occurrence of a real positive coefficient in the matrix R is denoted as the Enhanced FSD (EFSD). Even if the HH algorithm implies a fixed increase complexity of the QRD, the EFSD complexity is largely decreased compared to the RFSD. In Section IV, the overall complexity changes will be quantified.

C. Householder and Gram Schmidt numerical stability

It has been shown in [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] that the GS-based QRD computational complexity is lower than the HH-based QRD's. In particular, the GS method produces Q explicitly unlike the HH method which holds the matrix Q in factored form. Nevertheless, it is shown in Figure 4 that the GS-based QRD produces a loss of orthogonality of the matrix Q ( Q H Q -I F = 0, where • F denotes the Frobenius norm), especially in ill-conditioned channel matrix case, commonly observed in LTE systems because of the presence of correlation between antennas. The ratio

Cond(H) = |λ max | |λ min | (7)
is called condition number, with λ max and λ min respectively denoting maximal and minimal eigenvalues of H. It measures the degree of singularity of H. In particular, when Cond(H) is large, the matrix H is ill-conditioned and when Cond(H) is very small (close to 1) the matrix H becomes more orthogonal.

In Figure 4, the ratio of the loss of orthogonality with the GS technique by the loss of orthogonality with the HH technique is depicted versus the condition number, in a complex 5 × 5 matrix case. Thus, the HH tehnique is more stable than the stable version GS technique [START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF], even in the case of a QRD of a quasi-orthogonal matrix.

In the sequel, the two preprocessing results are developed and compared, in particular in terms of computational complexities.

D. Householder and Gram Schmidt complexities

It can be shown that GS orthonormalization of Cond(H)

H = h 11 h 12 h 21 h 22 gives H = QR with Q = h11 h1 2 h * 21 h1 2 h21 h1 2 -h * 11 h1 2 , (8)
Q H GS Q GS -I F Q H HH Q HH -I
R = h 1 2 h * 11 h 12 + h * 21 h 22 0 h 12 h 21 -h 11 h 22 . (9)
The HH reflection results are given below

Q =    h11 h1 2 h 21 h 2 2 h21 h1 2 h 22 h 2 2    , (10)
R = h 1 2 k h 1 2 0 h 2 2 , (11) with k
= h H 1 h2 h H 1 h1 and h 2 = h 2 -kh 1 .
A complexity change due to HH-based QRD and to EFSD adaptation can be quantified, in order to come to a conclusion for the overall complexity evolution in LTE context. In Table I, an approximation (real additions are ignored) of the computational complexity in terms of real multiplications, denoted as MUL, of the GS-based and the HH-based QRD are presented in a complex two by two MIMO case. Thanks to the occurrence of a real positive coefficient in the GS HH 94 141 matrix R diagonal, during the preprocessing step (instead of a complex coefficient), the processing complexity is impacted. More generally, the FSD computational complexity is variable and depends on the number of explored branches of the tree, called in this paper Paths Explorations (PE). A PE denotes the path from the root to a leaf of the tree (see Figure 1). In Table II, RFSD and EFSD computational complexities in terms of MUL are given for different PE numbers.

IV. SIMULATION RESULTS

In this Section, simulation results in terms of complexity and performance are shown. step and the variable computational complexity decrease in the detection step imply a switching point between the GS-RFSD and the HH-EFSD techniques. When less than 5 PE are performed, the computational complexity of the GS-RFSD is the lowest, whereas when more than 5 PE are performed, the computational complexity of the HH-EFSD is the lowest. In order to highlight this threshold, the ratio of overall computational complexities of GS-RFSD by overall computational complexities of HH-EFSD is plotted in Figure 6 and shows that the overall computational complexity gain grows with the number of PE. The plot confirms that the proposed technique overall computational complexity is lower than the reference technique as soon as 5 PE or more are needed. Therefore the proposed detector is more efficient in term of computational complexity for the 10 PE calibration point, which is considered to be the medium PE number in LTE context [START_REF]Technical Specification Group RAN[END_REF].

B. Fixed Complexity Sphere Detector performance overview in LTE context

Considering the proposed detector, this solution performance is compared to the ML and the reference detector performances. The three techniques are plotted in Figure 7 without channel coding. The GS-RFSD and the HH-EFSD performances in terms of Bit Error Rate (BER) are superposed in order to check the equivalence of the two methods. Simulation conditions of ML, GS-RFSD and HH-EFSD are given in Table IV. Figure 7 shows how curves are lay- ered. Therefore, the obtained performances are exactly the same with GS-RFSD and with HH-EFSD for the announced complexity. Moreover, it is possible to check the HH-EFSD performance is very similar to ML performance. Indeed, BER performances are exactly the same in high SNR case, and a tiny offset is observed in low SNR case which recalls the FSD is suboptimal.

V. CONCLUSION

In this paper, the HH-EFSD technique has been presented. The presented technique performance in terms of BER has been shown to be very similar to ML technique and its computational complexity has been shown to be decreased thanks to the HH-based QRD as soon as 5 or more PE are needed. This enhancement during the preprocessing step implies an effective gain of complexity in LTE context in which it has been shown that the mean PE is higher than this announced threshold.

Fig. 1 .

 1 Fig. 1. Tree representation of FSD.

Fig. 2 .

 2 Fig. 2. GS orthogonalization principle.

Fig. 3 .

 3 Fig. 3. HH transformation principle.

Fig. 4 .

 4 Fig. 4. Loss of orthogonality in the computed Q versus Condition Number.

Fig. 5 .

 5 Fig. 5. Complexity gain between GS-RFSD and HH-EFSD versions.

Fig. 6 .

 6 Fig. 6. ML, GS-RFSD and HH-EFSD techniques comparison for QPSK, 16QAM and 64QAM modulations.

TABLE I QRD

 I COMPLEXITIES.

ACKNOWLEDGMENT

The authors would like to thank Marc Soler and Andrea Ancora for their helpful discussions. This work was supported by ST-NXP Wireless.