Hao-Guang Li
email: lihaoguang@mail.scuec.edu.cn

And Chao-Jiang Xu

Chao-Jiang Xu
email: chao-jiang.xu@univ-rouen.fr

CAUCHY PROBLEM FOR THE SPATIALLY HOMOGENEOUS LANDAU EQUATION WITH SHUBIN CLASS INITIAL DATUM AND GELFAND-SHILOV SMOOTHING EFFECT

Keywords: Mathematics Subject Classification. 35H20, 35E15, 35B65, 76P05, 82C40 Spatially homogeneous Landau equation, spectral decomposition, ultra-analytic smoothing effect, Shubin space, Gelfand-Shilov space

In this work, we study the nonlinear spatially homogeneous Landau equation with Maxwellian molecules, by using the spectral analysis, we show that the non linear Landau operators is almost linear, and we prove the existence of weak solution for the Cauchy problem with the initial datum belonging to Shubin space of negative index which conatins the probability measures. Based on this spectral decomposition, we prove also that the Cauchy problem enjoys S 1 2 1 2 -Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy problem with Shubin class initial datum is ultra-analytics and exponential decay for any positive time.

Introduction

In this work, we study the spatially homogeneous Landau equation

∂ t f = Q L (f, f ), f | t=0 = f 0 , (1.1)
where f = f (t, v) ≥ 0 is the density distribution function depending on the variables v ∈ R 3 and the time t ≥ 0. The Landau bilinear collision operator is given by

Q L (g, f )(v) = ▽ v • R 3 a(v -v * ) g(v * )(▽ v f )(v) -(▽ v g)(v * )f (v) dv * ,
where a(v) = (a i,j (v)) 1≤ i,j≤3 stands for the non-negative symmetric matrix

a(v) = (|v| 2 I -v ⊗ v)|v| γ ∈ M 3 (R), -3 < γ < +∞.
In this work, we only consider the Cauchy problem (1.1) with the Maxwellian molecules, that means γ = 0. For the non-negative initial datum f 0 , we suppose

R 3 f 0 (v)dv = 1, R 3 v j f 0 (v)dv = 0, j = 1, 2, 3, R 3 |v| 2 f 0 (v)dv = 3. (1.2)
We shall study the linearization of the Landau equation (1.1) near the absolute Maxwellian distribution µ(v) = (2π) -3 2 e -|v| 2 2 . Considering the fluctuation of density distribution function

f (t, v) = µ(v) + √ µ(v)g(t, v),
since Q L (µ, µ) = 0, the Cauchy problem (1.1) is reduced to the Cauchy problem

∂ t g + L(g) = L(g, g), t > 0, v ∈ R 3 , g| t=0 = g 0 , (1.3)
with g 0 (v) = µ -1 2 f 0 (v) -√ µ, where

L(g) = -µ -1 2 Q L ( √ µg, µ) + Q L (µ, √ µg) , L(g, g) = µ -1 2 Q L ( √ µg, √ µg).
The linear operator L is non-negative (see [START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF]) with the null space

N = span √ µ, v 1 √ µ, v 2 √ µ, v 3 √ µ, |v| 2 √ µ .
Then the assumption (1.2) on the initial datum f 0 reduces to

               R 3 √ µ(v)g 0 (v)dv = 0, R 3 v j √ µ(v)g 0 (v)dv = 0, j = 1, 2, 3, R 3 |v| 2 √ µ(v)g 0 (v)dv = 0.
This shows that g 0 ∈ N ⊥ . We recall the spectral decomposition of the linear Landau operator (see Apendix 6 and [START_REF] Bobylev | The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules[END_REF], [START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF]).

L(ϕ n,l,m ) = λ n,l ϕ n,l,m , n, l ∈ N, -l ≤ m ≤ l (1.4)
where {ϕ n,l,m } n,l∈N,|m|≤ l is an orthonormal basis of L 2 (R 3 ) composed by eigenvectors of the harmonic oscillator H = -△ v + |v| 2 4 and the Laplace-Beltrami operator on the unit sphere S 2 , H(ϕ n,l,m ) = (2n + l + 3 2 ) ϕ n,l,m , -∆ S 2 (ϕ n,l,m ) = l(l + 1)ϕ n,l,m .

The eigenvalues of (1.4) satisfies : λ 0,0 = λ 0,1 = λ 1,0 = 0, λ 0,2 = 12 and for 2n + l > 2, λ n,l = 2(2n + l) + l(l + 1). (1.5) Using this spectral decomposition, the definition of the operators e cH s and H α are then classical. We introduce the following function spaces: Gelfand-Shilov spaces, for 0 < s ≤ 1,

S 1 2s 1 2s (R 3 ) = u ∈ S ′ (R 3 ); ∃ c > 0, e cH s u ∈ L 2 (R 3 ) ;
and the Shubin spaces, for β ∈ R, (see [START_REF] Shubin | Pseudodifferential Operators and Spectral theory[END_REF], Ch. IV, 25.3),

Q β (R 3 ) = u ∈ S ′ (R 3 ); u Q β (R 3 ) = H β 2 u L 2 (R 3 ) < +∞ .
We have

Q β (R 3 ) ⊂ H β (R 3 ), ∀β ≥ 0, H β (R 3 ) ⊂ Q β (R 3 ), ∀β < 0,
where H β (R 3 ) is the usual Sobolev spaces. In particular, for β < -3 2 the Shubin space Q β (R 3 ) contains the probability mesures (see [START_REF] Cho | Probability measures with finite moments and the homogeneous Boltzmann equation[END_REF][START_REF] Morimoto | A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules[END_REF][START_REF] Morimoto | Smoothing effect of the homogeneous Boltzmann equation with measure initial datum[END_REF][START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas[END_REF] and [START_REF] Li | The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect[END_REF]). See Appendix 6 for more properties of Gelfand-Shilov spaces and the Shubin spaces.

It is showed in [START_REF] Morimoto | A remark on the ultra-analytic smoothing properties of the spatially homogeneous Boltzmann equation[END_REF] that, for g 0 ∈ L 2 (R 3 ) with f 0 = µ + √ µg 0 ≥ 0, the solution of the Cauchy problem (1.3) obtained in [START_REF] Villani | On the spatially homogeneous Landau equation for Maxwellian molecules[END_REF] belongs to S (R 3 ) for any t > 0. In this work, we consider the initial datum which belongs to the Shubin spaces of negative index. The main theorem of this paper is in the following.

Theorem 1.1. Let α ≤ 0, there exists c 0 > 0 such that for any initial datum

g 0 ∈ Q α (R 3 ) ∩ N ⊥ with S 2 g 0 L 2 (R 3 ) ≤ c 0 , (1.6)
the Cauchy problem (1.3) admits a global weak solution

g ∈ L +∞ ([0, +∞[; Q α (R 3 )).
Moreover, we have the Gelfand-Shilov smoothing effect of Cauchy problem, and there exists c 1 > 0 such that for any t > 0,

e c1tH H α 2 g(t) L 2 (R 3 ) ≤ g 0 Q α (R 3 ) .
Remark 1.2. . 1) The orthogonal projectors {S N , N ∈ N} is defined, for g ∈ S ′ (R 3 ),

S N g = N k=0 2n+l=k |m|≤ l
g, ϕ n,l,m ϕ n,l,m ∈ S(R 3 ).

(1.7)

2) The constant c 0 in (1.6) is not asked to be very small, see the Remark 4.2. On the other hand, the condition (1.6) is a restriction for the initial datum on S 2 g 0 , but not a smallness hypothesis for the initial datum g 0 .

3) For the Landau equation (also Boltzmann equation), a physics condition on the initial datum is f 0 = µ + √ µg 0 ≥ 0 which implies the non-negativity of solution

f = µ + √ µg.
On the other hand, from the partial differential equations point of view, for the Cauchy problem (1.1) (also (1.3)), we don't need to impose this nonnegative condition. So that, in the Theorem 1.1, we do not ask for the initial datum f 0 = µ + √ µg 0 to be non-negative. 4) Combining this Theorem with the results of [START_REF] Villani | On the spatially homogeneous Landau equation for Maxwellian molecules[END_REF] and [START_REF] Morimoto | A remark on the ultra-analytic smoothing properties of the spatially homogeneous Boltzmann equation[END_REF] (see also [START_REF] Morimoto | Ultra-analytic effect of Cauchy problem for a class of kinetic equations[END_REF][START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF]), we get a complete result for the Cauchy problem

(1.3) with initial datum g 0 ∈ Q β (R 3 ) ∩ N ⊥ , β ∈ R:
The existence of global (weak) solution and S

f 0 = δ 0 - 3 2 - |v| 2 2 µ
be the initial datum of the Cauchy problem (1.1), then

f 0 = µ + √ µg 0 with g 0 = 1 √ µ δ 0 - 5 2 - |v| 2 2 √ µ ∈ Q α (R 3 ) ∩ N ⊥ , α < - 3 2 , (1.8)
and S 2 g 0 L 2 (R 3 ) = 0. Then Theorem 1.1 imply that the Cauchy problem (1.1) admits a global solution

f = µ + √ µg ∈ L +∞ ([0, +∞[; Q α (R 3 )) ∩ C 0 (]0, +∞[; S 1 2 1 2 (R 3 )).
This paper is arranged as follows : In the Section 2, we introduce the spectral analysis of the Landau operators and prove that the nonlinear Landau operator is almost diagonal. By using this decomposition, we can present explicitly the formal solutions to the Cauchy problem (1.3) by transforming it into an infinite system of ordinary differential equations. In the Section 3, we establish an upper bounded estimates for the nonlinear operators. We prove the main theorem 1.1 in the Section 4, and collect the main technical computations in the Section 5. In the Section 6, we give the proof of the Example 1.1 and the characterization of the Gelfand-Shilov spaces and the Shubin spaces.

Spectral analysis and formal solutions

In this section, we study the algebra property of the nonlinear Landau operators on the orthonormal basis {ϕ n,l,m } of L 2 (R 3 ),

L(ϕ ñ, l, m, ϕ n,l,m ). Recall, for n, l ∈ N, m ∈ Z, |m| ≤ l, ϕ n,l,m (v) = n! √ 2 Γ(n + l + 3/2) 1/2 |v| √ 2 l e -|v| 2 4 L (l+1/2) n |v| 2 2 Y m l v |v| ,
where Γ( • ) is the standard Gamma function, and -L

n is the Laguerre polynomial of order α and degree n,

L (α) n (x) = n r=0 (-1) n-r Γ(α + n + 1) r!(n -r)!Γ(α + n -r + 1)
x n-r ; -Y m l (σ) is the orthonormal basis of spherical harmonics

Y m l (σ) = N l,m P |m| l (cos θ)e imφ , |m| ≤ l,
where σ = (cos θ, sin θ cos φ, sin θ sin φ) and N l,m is the normalisation factor. It is obviously that, the conjugate of Y m l (σ) satisfies

Y m l (σ) = Y -m l (σ).
-

P |m| l
is the Legendre functions of the first kind of order l and degree |m|

P |m| l (x) = (1 -x 2 ) |m| 2 d |m| dx 1 2 l l! d l dx l (x 2 -1) l .
Then, {ϕ n,l,m } ⊂ S(R 3 ) the Schwartz function space, and

ϕ 0,0,0 (v) = √ µ, ϕ 0,1,0 (v) = v 1 √ µ, ϕ 0,1,1 (v) = v 2 + iv 3 √ 2 √ µ, ϕ 0,1,-1 (v) = v 2 -iv 3 √ 2 √ µ, ϕ 1,0,0 (v) = 2 3 3 2 - |v| 2 2 √ µ ,
and N = span {ϕ 0,0,0 , ϕ 0,1,0 , ϕ 0,1,1 , ϕ 0,1,-1 , ϕ 1,0,0 } .

We have also the explicit form of the eigenfunctions {ϕ 0,2,m2 , |m 2 | ≤ 2} :

ϕ 0,2,0 (v) = 1 3 3 2 v 2 1 -1 2 |v| 2 √ µ, ϕ 0,2,1 (v) = v1v2+iv1v3 √ 2 √ µ, ϕ 0,2,-1 (v) = v1v2-iv1v3 √ 2 √ µ, ϕ 0,2,2 (v) = v 2 2 -v 2 3 2 √ 2 + i v2v3 √ 2 √ µ, ϕ 0,2,-2 (v) = v 2 2 -v 2 3 2 √ 2 -i v2v3 √ 2 √ µ.
(2.1)

We have the following algebraic identities :

Proposition 2.1. For n, l ∈ N, |m| ≤ l, we have (i) L(ϕ 0,0,0 , ϕ n,l,m ) = -(2(2n + l) + l(l + 1)) ϕ n,l,m ; (ii) L(ϕ 0,1,m1 , ϕ n,l,m ) = A - n,l,m,m1 ϕ n+1,l-1,m1+m + A + n,l,m,m1 ϕ n,l+1,m1+m , ∀ |m 1 | ≤ 1; (iii) L(ϕ 1,0,0 , ϕ n,l,m ) = 4 3(n + 1)(2n + 2l + 3) 3 ϕ n+1,l,m ; (iv) L(ϕ 0,2,m2 , ϕ n,l,m ) = A 1 n,l,m,m2 ϕ n+2,l-2,m+m2 + A 2 n,l,m,m2 ϕ n+1,l,m+m2 + A 3 n,l,m,m2 ϕ n,l+2,m+m2 , ∀ |m 2 | ≤ 2; (v) L(ϕ ñ, l, m, ϕ n,l,m ) = 0, ∀ 2ñ + l > 2, | m| ≤ l.
where the coefficients will be precisely defined in Section 5.

The proof of this Proposition and the estimates of A 1 n,l,m,m2 , A 2 n,l,m,m2 and A 3 n,l,m,m2 are the main technic parts of this paper, we will give it in the Section 5. Now we come back to the Cauchy problem (1.3), we search a solution of the form

g(t) = +∞ n=0 +∞ l=0 l m=-l g n,l,m (t)ϕ n,l,m , g n,l,m (t) = g(t), ϕ n,l,m (2.2)
with initial data

g| t=0 = g 0 = +∞ n=0 +∞ l=0 l m=-l g 0 n,l,m ϕ n,l,m , g 0 n,l,m = g 0 , ϕ n,l,m .
The hypothesis

g 0 ∈ Q α (R 3 ) ∩ N ⊥ is equivalent to, g 0 0,0,0 = g 0 0,1,1 = g 0 0,1,0 = g 0 0,1,-1 = g 0 1,0,0 = 0, and
g 0 2 Q α = +∞ n=0 +∞ l=0 l m=-l (2n + l + 3 2 ) α |g 0 n,l,m | 2 < ∞.
See Appendix 6 for the norm of Shubin space. It follows from Proposition 2.1 that, we have the almost diagonalization of non linear Landau operators, meaning that for the function f, g define by the series

(2.2), for n, l ∈ N, m ∈ Z, |m| ≤ l L(f, g), ϕ n,l,m L 2 = -(2(2n + l) + l(l + 1)) f 0,0,0 (t)g n,l,m (t) + |m * |≤ l+1,|m1|≤1 m1+m * =m A - n-1,l+1,m * ,m1 f 0,1,m1 (t)g n-1,l+1,m * (t) + |m * |≤ l-1,|m1|≤1 m1+m * =m A + n,l-1,m * ,m1 f 0,1,m1 (t)g n,l-1,m * (t) + 4 3n(2n + 2l + 1) 3 f 1,0,0 (t)g n-1,l,m (t) + |m * |≤ l+2,|m2|≤2 m * +m2=m A 1 n-2,l+2,m * ,m2 f 0,2,m2 (t)g n-2,l+2,m * (t) + |m * |≤ l,|m2|≤2 m * +m2=m A 2 n-1,l,m * ,m2 f 0,2,m2 (t)g n-1,l,m * (t) + |m * |≤ l-2,|m2|≤2 m * +m2=m A 3 n,l-2,m * ,m2 f 0,2,m2 (t)g n,l-2,m * (t), (2.3)
with the conventions g n,l,m ≡ 0, if n < 0 or l < 0 , and L(g), ϕ n,l,m

L 2 = λ n,l g n,l,m (t), n, l ∈ N, m ∈ Z, |m| ≤ l.
We remark from (2.3) that,

∀ f, g ∈ N ⊥ ⇒ L(f, g) ∈ N ⊥ . (2.4)
So that, formally, if g is a solution of the Cauchy problem (1.3), we find that the family of functions {g n,l,m (t); n, l ∈ N, |m| ≤ l}, satisfy the following infinite system of the differential equations, n, l ∈ N, |m| ≤ l,

∂ t g n,l,m (t) + λ n,l g n,l,m (t) = L(g, g), ϕ n,l,m L 2 , t > 0; g n,l,m | t=0 = g 0 , ϕ n,l,m = g 0 n,l,m (2.5)
where L(g, g), ϕ n,l,m L 2 was precisely defined in (2.3). We have firstly,

Proposition 2.2. Let g 0 ∈ Q α (R 3 )∩N ⊥
, assume that g is a solution of the Cauchy problem (1.3) of the form (2.2), then we have

g 0,0,0 (t) = g 0,1,0 (t) = g 0,1,1 (t) = g 0,1,-1 (t) = g 1,0,0 (t) = 0, ∀ t ≥ 0, (2.6)
and g 0,2,m (t) = e -12t g 0 0,2,m , t ≥ 0, |m| ≤ 2.

(2.7)

Proof.

(1) Substituting n = 0, l = 0, m = 0 into the above infinite ODE system (2.5), one has ∂ t g 0,0,0 (t) + λ 0,0 g 0,0,0 (t) = 0.

We remind that λ 0,0 = 0, then g 0,0,0 (t) = g 0 0,0,0 = 0. (2) Now we set n = 0, l = 1, and |m| ≤ 1, the ODE system (2.5) turn out to be ∂ t g 0,1,m (t) + λ 0,1 g 0,1,m (t) = -4g 0,0,0 (t)g 0,1,m (t) + A + 0,0,0,m g 0,1,m (t)g 0,0,0 (t) By using the known results λ 0,1 = 0, g 0,0,0 (t) = 0, one can verify that g 0,1,m (t) = g 0 0,1,m = 0, ∀|m| ≤ 1. (3) Take now n = 1, l = 0, m = 0 in (2.5), we have

∂ t g 1,0,0 (t) + λ 1,0 g 1,0,0 (t) = -4g 0,0,0 (t)g 1,0,0 (t) + |m * |≤ 1,|m1|≤1 m1+m * =0 A - 0,1,m * ,m1 g 0,1,m1 (t)g 0,1,m * (t)
+ 4g 1,0,0 (t)g 0,0,0 (t) + A 2 0,0,0,0 g 0,2,0 (t)g 0,0,0 (t). Then λ 1,0 = 0, g 0,0,0 (t) = 0, g 0,1,m ′ (t) = 0, ∀|m ′ | ≤ 1, imply g 1,0,0 (t) = g 0 1,0,0 = 0. (4) Furthermore, for n = 0, l = 2 and |m| ≤ 2 in (2.5), we have that g 0,0,0 (t) = 0, g 0,1,m ′ (t) = 0, ∀|m ′ | ≤ 1, imply ∂ t g 0,2,m (t) + λ 0,2 g 0,2,m (t) = 0.

Recalled that λ 0,2 = 12 in (1.5), we obtain, g 0,2,m (t) = e -12t g 0 0,2,m , ∀|m| ≤ 2. This ends the proof of Proposition 2.2. Substituting (2.6) and (2.7) into the infinite system of the differential equations (2.5), we have, for all 2n

+ l > 2, |m| ≤ l,                                      ∂ t g n,l,m (t) + λ n,l g n,l,m (t) = + |m * |≤ l+2,|m2|≤2 m * +m2=m A 1 n-2,l+2,m * ,m2 e -12t g 0 0,2,m2 g n-2,l+2,m * (t) + |m * |≤ l,|m2|≤2 m * +m2=m A 2 n-1,l,m * ,m2 e -12t g 0 0,2,m2 g n-1,l,m * (t) + |m * |≤ l-2,|m2|≤2 m * +m2=m A 3 n,l-2,m * ,m2 e -12t g 0 0,2,m2 g n,l-2,m * (t), g n,l,m | t=0 = g 0 n,l,m , (2.8)
with the convention

A 1 n-2,l+2,m * ,m2 = 0, if n -2 < 0; A 2 n-1,l,m * ,m2 = 0, if n -1 < 0. (2.9)
We can solve this infinite differential equation by induction.

In fact, for n = 0, l ≥ 3, |m| ≤ l, the following system

             ∂ t g 0,l,m (t) + λ 0,l g 0,l,m (t) = |m * |≤ l-2,|m2|≤2 m * +m2=m A 3 0,l-2,m * ,m2 e -12t g 0 0,2,m2 g 0,l-2,m * (t), g 0,l,m (0) = g 0 0,l,m .
can be solved by induction on l start from l = 3 since g 0,1,m * (t) ≡ 0 for all |m * | ≤ 1.

For the general case of l, the index of the right hand side are l -2, which have been already known by induction.

Then we solve the differential equations (2.8) for all n ≥ 1, l ≥ 0 and |m| ≤ l. We also prove by induction on n and for fixed n induction on l. Since for the first two terms on the right hand side of (2.8), the first index are less than n -1, and for the last terms on the right hand side, the second index are less than l -2, which have been already known by induction. So that in each steps of the induction, the right hand side of (2.8) is already known by induction hypothesis. Then the differential equations are linear differential equations, and can be solved explicitly with any initial detum g 0 n,l,m . We get then the formal solution of Cauchy problem (1.3) by solve the differential system (2.8), and we have :

Theorem 2.3. Let {g 0
n,l,m ; n, l ∈ N, |m| ≤ l} be a complex sequence with g 0 0,0,0 = g 0 0,1,1 = g 0 0,1,0 = g 0 0,1,-1 = g 0 1,0,0 = 0. Then the system (2.8) admits a sequence of solutions {g n,l,m (t); 2n+l > 2, |m| ≤ l}.

For all N ≥ 2, we note that

g N (t) = N k=2 2n+l=k n+l≥2 |m|≤ l g n,l,m (t)ϕ n,l,m (2.10)
with g 0,2,m (t) = e -12t g 0 0,2,m , |m| ≤ 2, t > 0. Then g N satisfies the following Cauchy problem        ∂ t g N + L(g N ) = S N L(g N , g N ), g N | t=0 = 2≤2n+l≤ N n+l≥2 |m|≤ l g 0 n,l,m ϕ n,l,m . (2.11)
The proof of the existence of weak solution of Theorem 1.1 is reduced to prove the convergence of the sequences {g N ; N ∈ N} in the function space Q α (R 3 ). Namely,

g N → g(t) = +∞ k=2 2n+l=k n+l≥2 |m|≤ l g n,l,m (t)ϕ n,l,m ∈ Q α (R 3 ), as N → +∞.
The Gelfand-Shilov regularity is reduced to prove: there exists a constant c 1 > 0, such that

∀t > 0, e c1tH H α 2 g(t) 2 L 2 (R 3 ) = e c1t(2n+l+ 3 2 ) (2n + l + 3 2 ) α |g n,l,m (t)| 2 < ∞.
This will be the main jobs of the Section 3 and Section 4.

The trilinear estimates for non linear operator

To prove the convergence of the formal solution obtained in Theorem 2.3, we need to estimate the following trilinear terms

(L(f, g), h) L 2 (R 3 ) , f, g, h ∈ S (R 3 ) ∩ N ⊥ .
We need firstly the following estimates for the coefficients A 1 , A 2 and A 3 (see their definition (5.10) in Section 5) of the Proposition 2.1. Proposition 3.1. For the coefficients of the Proposition 2.1 defined in (5.10), we have the following estimates:

1) For n, l ∈ N, n ≥ 2, max |m * |≤l |m|≤ l+2,|m2|≤2 m+m2=m * A 1 n-2,l+2,m,m2 2 ≤ 16n(n -1) 3 . (3.1) 2) For n, l ∈ N, n ≥ 1, A 2 n-1,0,0,0 = 0; max |m * |≤l |m|≤ l,|m2|≤2 m+m2=m * A 2 n-1,l,m,m2 2 ≤ 4n(2n + 2l + 1) 3 , ∀ l ≥ 1. (3.2) 3) For n, l ∈ N, l ≥ 2, max |m * |≤l |m|≤ l-2,|m2|≤2 m+m2=m * A 3 n,l-2,m,m2 2 ≤ (2n + 2l + 1)(2n + 2l -1) 2 . (3.3)
We will give the proof of this Proposition in the Section 5.

We now present the trilinear estimation for the nonlinear Landau operator L,

for g ∈ S ′ (R 3 ) ∩ N ⊥ , N > 2, we note SN g = 2≤2n+l≤ N n+l≥2 |m|≤ l g n,l,m ϕ n,l,m , g n,l,m = g, ϕ n,l,m . (3.4)
Then we have the following trilinear estimates:

Proposition 3.2. Let f, g, h ∈ Q α (R 3 ) ∩ N ⊥ with α ≤ 0, then for any N ≥ 2, |(L( SN f, SN g), H α SN h) L 2 | ≤ 4 √ 3 3 + √ 2 S2 f L 2 H α+1 2 SN-2 g L 2 H α+1 2 SN h L 2 ,
and also for any c > 0, t ≥ 0,

|(L( SN f, SN g), e 2ctH H α SN h) L 2 | ≤ 4 √ 3 3 + √ 2 e 2ct S2 f L 2 e ctH H α+1 2 SN-2 g L 2 e ctH H α+1 2 SN h L 2 .
The proof of this Proposition is similar to Lemma 3.5 in [START_REF] Lerner | Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff[END_REF], Proposition 3.2 in [START_REF] Li | The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect[END_REF] and Section 3 in [START_REF] Glangetas | Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation[END_REF].

Proof. Let f, g, h ∈ Q α (R 3 ) ∩ N ⊥ with α ≤ 0.
For N ≥ 2, by using the orthogonal property of {ϕ n,l,m ; n, l ∈ N, |m| ≤ l}, we can deduce from Proposition 2.1 and (2.9) that

(L( SN f, SN g), H α SN h) = 2≤2n+l≤ N n≥2 |m|≤ l+2,|m2|≤2 |m+m2|≤l A 1 n-2,l+2,m,m2 (2n + l + 3 2 ) α f 0,2,m2 g n-2,l+2,m h n,l,m+m2 + 2≤2n+l≤ N n≥1,n+l≥2 |m|≤ l,|m2|≤2 |m+m2|≤ l A 2 n-1,l,m,m2 (2n + l + 3 2 ) α f 0,2,m2 g n-1,l,m h n,l,m+m2 + 2≤2n+l≤ N l≥2 |m|≤ l-2,|m2|≤2 |m+m2|≤ l A 3 n,l-2,m,m2 (2n + l + 3 2 ) α f 0,2,m2 g n,l-2,m h n,l,m+m2 ≤ B 1 + B 2 + B 3 .
For the first term B 1 , we have

B 1 ≤ 2≤2n+l≤ N n≥2 (2n + l + 3 2 ) α × |m2|≤2 |m|≤ l+2 |m+m2|≤ l |f 0,2,m2 | A 1 n-2,l+2,m,m2 g n-2,l+2,m h n,l,m+m2
, by using the Cauchy-Schwarz inequality

|m2|≤2 |f 0,2,m2 | |m|≤ l+2 |m+m2|≤ l A 1 n-2,l+2,m,m2 g n-2,l+2,m h n,l,m+m2 ≤ S2 f L 2 |m2|≤2 |m|≤ l+2 |m+m2|≤ l A 1 n-2,l+2,m,m2 g n-2,l+2,m h n,l,m+m2 2 1 2 ≤ S2 f L 2 |m|≤ l+2 |g n-2,l+2,m | 2 1 2 × |m2|≤2 |m|≤ l+2 |m+m2|≤ l A 1 n-2,l+2,m,m2 h n,l,m+m2 2 1 2
.

By changing the order of summation

|m2|≤2 |m|≤ l+2 |m+m2|≤ l A 1 n-2,l+2,m,m2 h n,l,m+m2 2 = |m * |≤l h n,l,m * 2 |m|≤ l+2,|m2|≤2 m+m2=m * A 1 n-2,l+2,m,m2 | 2 ≤ 16n(n -1) 3 |m * |≤l h n,l,m * 2
Substituting back to the estimation of B 1 , one can verify that

B 1 ≤ S2 f L 2 2≤2n+l≤ N n≥2 (2n + l + 3 2 ) α 16n(n -1) 3 × |m|≤ l+2 |g n-2,l+2,m | 2 1 2 |m * |≤ l |h n,l,m * | 2 1 2 ≤ 2 √ 3 3 S2 f L 2 2≤2n+l≤ N n≥2 (2n + l - 1 2 ) α+1 |m|≤ l+2 |g n-2,l+2,m | 2 1 2 × 2≤2n+l≤ N n≥2 (2n + l + 3 2 ) α+1 |m * |≤ l |h n,l,m * | 2 1 2 ≤ 2 √ 3 3 S2 f L 2 H α+1 2 SN-2 g L 2 H α+1 2 SN h L 2 ,
where we use the estimation (2n + l + 3 2 ) α (2n -2) ≤ (2n + l -1 2 ) α+1 when α ≤ 0. Now we turn back to estimate B 2 , B 3 . By using the Cauchy-Schwarz inequality

|m2|≤2 |f 0,2,m2 | |m|≤ l |m+m2|≤ l A 2 n-1,l,m,m2 g n-1,l,m h n,l,m+m2 ≤ S2 f L 2 |m|≤ l |g n-1,l,m | 2 1 2 |m|≤ l,|m2|≤2 |m+m2|≤ l A 2 n-1,l,m,m2 2 h n,l,m+m2 2 1 2 ≤ S2 f L 2 |m|≤ l |g n-1,l,m | 2 1 2 × |m * |≤ l h n,l,m * 2 |m|≤ l,|m2|≤2 m+m2=m * A 2 n-1,l,m,m2 2 1 2
,

and |m2|≤2 |f 0,2,m2 | |m|≤ l-2 |m+m2|≤ l A 3 n,l-2,m,m2 g n,l-2,m h n,l,m+m2 ≤ S2 f L 2 |m|≤ l-2 |g n,l-2,m | 2 1 2 |m|≤ l-2,|m2|≤2 |m+m2|≤ l A 3 n,l-2,m,m2 h n,l,m+m2 2 1 2 ≤ S2 f L 2 |m|≤ l-2 |g n,l-2,m | 2 1 2 × |m * |≤ l h n,l,m * 2 |m|≤ l-2,|m2|≤2 m+m2=m * A 3 n,l-2,m,m2 2 1 2
.

Substituting the estimations (3.2) and (3.3) in B 2 , B 3 , it follows that

B 2 ≤ S2 f L 2 2≤2n+l≤ N n≥1,l≥1 (2n + l + 3 2 ) α 4n(2n + 2l + 1) 3 × |m|≤ l |g n-1,l,m | 2 1 2 |m * |≤ l |h n,l,m * | 2 1 2 ≤ 2 √ 3 3 S2 f L 2 H α+1 2 SN-2 g L 2 H α+1 2 SN h L 2 ,
here for n ≥ 1, we use

(2n + l + 3 2 ) α (n + l + 1 2 ) ≤ (2n + l + 3 2 ) α+1 ; 2n(2n + l + 3 2 ) α ≤ (2n + l - 1 2 ) α+1 , for l ≥ 1, α ≤ 0.
And

B 3 ≤ S2 f L 2 2≤2n+l≤ N l≥2 (2n + l + 3 2 ) α (2n + 2l + 1)(2n + 2l -1) 2 ×   |m|≤ l-2 |g n,l-2,m | 2   1 2   |m * |≤ l |h n,l,m * | 2   1 2 ≤ √ 2 S2 f L 2 H α+1 2 SN-2 g L 2 H α+1 2 SN h L 2 ,
here for l ≥ 2 and n ∈ N, we use

(2n + l + 3 2 ) α (n + l + 1 2 ) ≤ (2n + l + 3 2 ) α+1 ; (2n + l + 3 2 ) α (n + l - 1 2 ) ≤ (2n + l - 1 2 ) α+1 , for α ≤ 0.
Therefore,

|(L( SN f, SN g), H α SN h) L 2 | ≤ 4 √ 3 3 + √ 2 S2 f L 2 H α+1 2 SN-2 g L 2 H α+1 2 SN h L 2 .
This is the first result of Proposition 3.2.

For the second inequality of the Proposition 3.2, we just to use,

e ct(2n+l+ 3 2 ) = e ct(2(n-2)+(l+2)+ 3 2 ) e 2ct = e ct(2(n-1)+l+ 3 2 ) e 2ct = e ct(2n+(l-2)+ 3 2 ) e 2ct .
This ends the proof of Proposition 3.2.

The convergence of the formal solution

In this section, we study the convergence of the solutions {g N ; N ∈ N} defined by (2.10) in Theorem 2.3 where the initial data is the sequence

{g 0 n,l,m = g 0 , ϕ n,l,m ; n, l ∈ N, |m| ≤ l} with g 0 ∈ Q α ∩ N ⊥ . Note that g N ∈ S (R 3 ) ∩ N ⊥ ,
from the definition of (1.7) and (3.4), we have

S N g N (t) = SN g N (t) = g N (t) ∈ S (R 3 ) ∩ N ⊥ . (4.1)
In particular, for N = 2, we have

g 2 (t) L 2 (R 3 ) = S2 g 2 (t) L 2 (R 3 ) ≤ e -12t S2 g 0 L 2 (R 3 ) . (4.2)
Moreover, we recall the result (2.4) that

S N L(g N (t), g N (t)) = SN L(g N (t), g N (t)).
Therefore, we can rewrite the Cauchy problem (2.11) as follows:

       ∂ t g N + L(g N ) = SN L(g N , g N ), g N | t=0 = 2≤2n+l≤ N n+l≥2 |m|≤ l g 0 , ϕ n,l,m ϕ n,l,m . (4.3)
Now for N > 2, c > 0, taking the inner product of e 2ctH H α g N (t) in L 2 (R 3 ) on both sides of (4.3), we have

∂ t g N (t), e 2ctH H α g N (t) L 2 (R 3 ) + Lg N (t), e 2ctH H α g N (t) L 2 (R 3 ) = SN L(g N , g N ), e 2ctH H α g N (t) L 2 (R 3 )
, where g N is defined in (2.10). Since

λ 0,2 = 12 > 16 11 2 + 3 2 , λ n,l = 2(2n + l) + l(l + 1) ≥ 16 11 2n + l + 3 2 , ∀ 2n + l > 2.
The orthogonality of the basis {ϕ n,l,m } {n,l∈N, m∈Z,|m|≤ l} imply that

Lg N (t), e 2ctH H α g N (t) L 2 (R 3 ) ≥ 16 11 e ctH H α+1 2 g N (t) 2 L 2 (R 3 ) .
On the other hand

2 ∂ t g N (t), e 2ctH H α g N (t) L 2 (R 3 ) + 2c g N (t), H e 2ctH H α g N (t) L 2 (R 3 ) = d dt g N (t), e 2ctH H α g N (t) L 2 (R 3 ) = d dt e ctH H α 2 g N (t) 2 L 2 (R 3 ) .
Therefore, we have 1 2

d dt e ctH H α 2 g N (t) 2 L 2 (R 3 ) + 16 11 -c e ctH H α+1 2 g N (t) 2 L 2 (R 3 ) = L( SN g N , SN g N ), e 2ctH H α SN g N (t) L 2 .
It follows from Proposition 3.2 and the inequality (4.2) that, for any N > 2, t > 0, 1 2

d dt e ctH g N (t) 2 Q α (R 3 ) + 16 11 -c e ctH g N (t) 2 Q α+1 (R 3 ) ≤ 4 √ 3 3 + √ 2 e -(12-c)t S2 g 0 L 2 (R 3 ) e ctH g N -2 Q α+1 (R 3 ) e ctH g N Q α+1 (R 3 ) ≤ 4 √ 3 3 + √ 2 e -(12-c)t S2 g 0 L 2 (R 3 ) e ctH g N 2 Q α+1 (R 3 ) (4.4)
where we used the definition of the shubin spaces Q α+1 (R 3 ) that

e ctH g N 2 Q α+1 (R 3 ) = 2≤2n+l≤ N n+l≥2 |m|≤ l e 2ct(2n+l+ 3 2 ) (2n + l + 3 2 ) α+1 |g n,l,m | 2 .
Proposition 4.1. There exists c 0 > 0, c 1 > 0 such that for all g 0 ∈ Q α (R 3 ) ∩ N ⊥ with α ≤ 0, and

S2 g 0 L 2 (R 3 ) =   |m2|≤2 | g 0 , ϕ 0,2,m2 | 2   1 2 ≤ c 0 , if {g n,l,m (t); n, l ∈ N, m ∈ Z, |m| ≤ l} is the solution of (2.8) with inital datum {g 0 n,l,m = g 0 , ϕ n,l,m ; n, l ∈ N, |m| ≤ l}, then, for any N ≥ 2, t > 0, e c1tH g N (t) 2 Q α (R 3 ) + c 1 t 0 e c1τ H g N (τ ) 2 Q α+1 (R 3 ) dτ ≤ g 0 2 Q α (R 3 ) . (4.5)
We have also, for any t ≥ 0 and any N ≥ 2, SN L(g

N (t), g N (t)) Q α-2 (R 3 ) ≤ 2 g 0 Q α (R 3 ) . (4.6) Remark 4.2.
It is enough to take 0 < c 1 << 1 very small such that

0 < c 0 = 16 11 -3 2 c 1 4 √ 3 3 + √ 2 < 16 11 4 √ 3 3 + √ 2 ≈ 0.39.
Proof. For N = 2, it follows from Proposition 2.2 and α ≤ 0 that

e c1tH H α 2 g 2 (t) 2 L 2 (R 3 ) + c 1 t 0 e c1τ H H α+1 2 g 2 (τ ) 2 L 2 (R 3 ) dτ = 48 -21c 1 48 -14c 1 e 7c1t-24t |m|≤2 7 2 α |g 0 0,2,m | 2 ≤ |m|≤2 7 2 α |g 0 0,2,m | 2 ≤ g 0 2 Q α (R 3 ) .
Then for N > 2, we can deduce from (4.4) with c = c 1 and the hypothesis S2

g 0 L 2 (R 3 ) ≤ c 0 that d dt e c1tH H α 2 g N (t) 2 L 2 (R 3 ) + c 1 e c1tH g N (t) 2 Q α+1 (R 3 ) ≤ 0.
This ends the proof of (4.5) by integration over [0, t]. Now we prove the estimate (4.6). For h ∈ Q -α+2 (R 3 ) with α ≤ 0, by using the first inequality in the Proposition 3.2 with α replacing by α-1 where α-1 ≤ -1 < 0, and the notation of (4.1)

| SN L(g N , g N ), h | =|(L( SN g N , SN g N ), H α-1 SN H 1-α h) L 2 (R 3 ) | ≤ 4 √ 3 3 + √ 2 S2 g N L 2 (R 3 ) H α 2 SN-2 g N L 2 (R 3 ) SN h Q -α+2 (R 3 ) ≤ 4 √ 3 3 + √ 2 g 2 L 2 (R 3 ) H α 2 g N -2 L 2 (R 3 ) h Q -α+2 (R 3 ) .
Then by the definition of the norm and (4.5), we have

SN L(g N (t), g N (t)) Q α-2 (R 3 ) = sup h Q 2-α (R 3 ) =1 | S N L(g N , g N ), h | ≤ 4 √ 3 3 + √ 2 S2 g 0 L 2 H α 2 g N -2 L 2 (R 3 ) ≤ 16 11 H α 2 g 0 L 2 (R 3 )
which ends the proof of the Proposition 4.1.

In particulary, we get the following surprise results

Corollary 4.3. For any f, g ∈ Q α (R 3 ) ∩ N ⊥ with α ≤ 0, we have L(f, g) ∈ Q α-2 (R 3 ) ∩ N ⊥ and L(f, g) Q α-2 (R 3 ) ≤ 4 √ 3 3 + √ 2 S2 f L 2 (R 3 ) g Q α (R 3 ) .
Convergence in Shubin space. We prove now the convergence of the sequence

g N (t) → g(t) = +∞ k=2 2n+l=k n+l≥2 |m|≤ l g n,l,m (t)ϕ n,l,m
where for all N ≥ 2, g N was defined in (2.10) with the coefficients {g n,l,m (t)} defined in (2.8), the inital datum is {g 0 n,l,m = g 0 , ϕ n,l,m ; n, l ∈ N, |m| ≤ l} with g 0 ∈ Q α (R 3 ) ∩ N ⊥ . By Proposition 4.1 and the orthogonality of the basis {ϕ n,l,m },

2≤2n+l≤ N n+l≥2 |m|≤ l e 2c1t(2n+l+ 3 2 ) (2n + l + 3 2 ) α |g n,l,m (t)| 2 ≤ g 0 2 Q α (R 3 ) .
It follows that for all t ≥ 0,

g N (t) 2 Q α (R 3 ) = H α 2 g N (t) 2 L 2 (R 3 ) = 2≤2n+l≤ N n+l≥2 |m|≤ l (2n + l + 3 2 ) α |g n,l,m (t)| 2 ≤ g 0 2 Q α (R 3 ) .
By using the monotone convergence theorem, we have

g N → g(t) ∈ Q α (R 3 ).
Moreover, for any T > 0, lim

N →∞ g N -g L ∞ ([0,T ];Q α (R 3 )) = 0 .
On the other hand, using (4.6) and Corollary 4.3 , we have also

SN L(g N , g N ) → L(g, g), in Q α-2 (R 3 ).
We recall the definition of weak solution of (1.3):

Definition 4.4. Let g 0 ∈ S ′ (R 3 ), g(t, v
) is called a weak solution of the Cauchy problem (1.3) if it satisfies the following conditions:

g ∈ C 0 ([0, +∞[; S ′ (R 3 )), g(0, v) = g 0 (v), L(g) ∈ L 2 ([0, T [; S ′ (R 3 )), L(g, g) ∈ L 2 ([0, T [; S ′ (R 3 )), ∀T > 0, g(t), φ(t) -g 0 , φ(0) + t 0 Lg(τ ), φ(τ ) dτ = t 0 g(τ ), ∂ τ φ(τ ) dτ + t 0 L(g(τ ), g(τ )), φ(τ ) dτ, ∀t ≥ 0, For any φ(t) ∈ C 1 [0, +∞[; S (R 3 ) .
We prove now the main Theorem 1.1.

Existence of weak solution.

Let {g n,l,m , n, l ∈ N, n + l ≥ 2, |m| ≤ l} be the solution of the infinite system (2.8) in Theorem 2.3 with the initial datum give in the Proposition 4.1, then for any N ≥ 2, g N satisfy the equation (4.3).

We have, firstly, from the Proposition 4.1, there exists positive constant C > 0, for any N ≥ 2 and any T > 0,

g N L ∞ ([0,T ];Q α (R 3 )) ≤ g 0 Q α (R 3 ) , L(g N ) L 2 ([0,T ];Q α-3 (R 3 )) ≤ C g 0 Q α (R 3 ) , SN L(g N , g N ) L 2 ([0,T ];Q α-2 (R 3 )) ≤ C g 0 Q α (R 3 ) .
So that the equation (4.3) implies that the sequence

{ d dt SN g(t)} is uniformly bounded in Q α-3 (R 3 ) with respect to N ∈ N and t ∈ [0, T ]. The Arzelà-Ascoli Theorem implies that g N → g ∈ C 0 ([0, +∞[; Q α-2 (R 3 )) ⊂ C 0 ([0, +∞[; S ′ (R 3 )),
and g(0) = g 0 .

Secondly, for any φ(t) ∈ C 1 R + , S (R 3 ) , the Cauchy problem (4.3) can be rewrite as follows

g N (t), φ(t) -g N (0), φ(0) - t 0 g N (τ ), ∂ τ φ(τ ) dτ = - t 0 Lg N (τ ), φ(τ ) dτ + t 0 SN L(g N (τ ), g N (τ )), φ(τ ) dτ
Let N → +∞, we conclude that,

g(t), φ(t) -g 0 , φ(0) - t 0 g(τ ), ∂ τ φ(τ ) dτ = - t 0 L(g(τ )), φ(τ ) dτ + t 0 L(g(τ ), g(τ )), φ(τ ) dτ, which shows g ∈ L ∞ ([0, +∞[; Q α (R 3 )) is a global weak solution of Cauchy problem (1.3).
Regularity of the solution. For N ≥ 2, we deduce from the formulas (4.5) and the orthogonality of the basis (ϕ n,l,m ) that

e c1tH H α 2 g N (t) L 2 (R 3 ) ≤ H α 2 g 0 L 2 (R 3 ) , ∀ N ≥ 2, t ≥ 0,
by using the monotone convergence theorem, we conclude that, such that

e c1tH H α 2 g(t) L 2 (R 3 ) ≤ g 0 Q α (R 3 ) , ∀ t ≥ 0.
The proof of Theorem 1.1 is completed.

The techincal computations

The proof of the main technic part was presented in this section. More precisely, we prepare to prove Proposition 2.1 in Section 2 and Proposition 3.1 in Section 3.

To this ends, we need to state some Lemmas and new notations. Recall firstly

v k v j √ µ ∈ span {ϕ 0,2,0 , ϕ 0,2,±1 , ϕ 0,2,±2 , ϕ 1,0,0 , ϕ 0,0,0 } . (5.1)
This relation is important in the expansion of the nonlinear operators. Setting Ψ n,l,m (v) = √ µ(v)ϕ n,l,m (v).

Recalled Lemma 7.2 of [START_REF] Glangetas | Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation[END_REF] that the Fourier transformation is ,

Ψ n,l,m (ξ) = B n,l |ξ| 2n+l e -|ξ| 2 2 Y m l ( ξ |ξ| ), (5.2)
where

B n,l = (-i) l (2π) 3 4 1 √ 2n!Γ(n + l + 3 2 )2 2n+l 1 2
.

(5.3)

For the Laplace-Beltrami operator on the unit sphere S 2 , see also Section 4.2 in [6], we have

1≤ k,j≤3 k = j v 2 j ∂ 2 v k -v k v j ∂ vj ∂ v k -2 3 k=1 v k ∂ v k = ∆ S 2 . (5.4) And for n, l ∈ N, m ∈ Z, |m| ≤ l, 3 k=1 ∂ 2 v k + v k ∂ v k Ψ n,l,m = -(2n + l + 3)Ψ n,l,m . (5.5)
Recall that the family Y m l (σ) l≥0,|m|≤ l is the orthonormal basis of L 2 (S 2 , dσ)

(see [START_REF] Slater | Quantum theory of atomic structure[END_REF] of Chap.1 in [START_REF] Jones | Spherical harmonics and tensors for classical field theory[END_REF]). We have the following addition lemma,

Lemma 5.1. For any ω ∈ S 2 , l, l ∈ N, |m| ≤ l, | m| ≤ l, Y m l (ω)Y m l (ω) = 0≤ p≤min(l, l) S 2 Y m l (ω)Y m l (ω)Y -m- m l+ l-2p (ω)dω Y m+ m l+ l-2p (ω)
where we always define Y -m-

m l+ l-2p (ω) ≡ 0, if |m + m| > l + l -2p.
Proof. For the proof of Lemma 5.1, we refer to (86) in Chap. 3 of [START_REF] Jones | Spherical harmonics and tensors for classical field theory[END_REF] or Gaunt's formula from of Chap.13 in [START_REF] Slater | Quantum theory of atomic structure[END_REF].

In particular, for l = 1 or l = 2 in Lemma 5.1, we have

Corollary 5.2. For all ω ∈ S 2 , l ∈ N, |m| ≤ l, |m 1 | ≤ 1, |m 2 | ≤ 2, Y m1 1 (ω)Y m l (ω) = 0≤ p≤min(1,l) Cm1,m l,l+1-2p Y m1+m l+1-2p (ω); Y m2 2 (ω)Y m l (ω) = 0≤ p≤min(2,l) C m2,m l,l+2-2p Y m2+m l+2-2p (ω) where Cm1,m l,l+1-2p = S 2 Y m1 1 (ω)Y m l (ω)Y -m1-m l+1-2p (ω)dω, C m2,m l,l+2-2p = S 2 Y m2 2 (ω)Y m l (ω)Y -m2-m l+2-2p (ω)dω.
More explicitly, for any l ∈ N, |m| ≤ l

Y m1 1 (ω)Y m l (ω) = Cm1,m l,l+1 Y m1+m l+1 (ω) + Cm1,m l,l-1 Y m1+m l-1 (ω), and
Y m2 2 (ω)Y m l (ω) = C m2,m l,l+2 Y m2+m l+2 (ω) + C m2,m l,l Y m2+m l (ω) + C m2,m l,l-2 Y m2+m l-2 (ω)
where, for convenience, we note

Y m1+m -1 (ω) = 0, Y m2+m -2 (ω) = 0. Lemma 5.3. Let Ψ n,l,m = √ µϕ n,l,m , then for v ∈ R 3 , we have 1) For |m 1 | ≤ 1, R 3 v * (v • v * )Ψ 0,1,m1 (v * )dv * = 4π 3 |v|Y m1 1 (σ). (5.6) 2) For |m 2 | ≤ 2, R 3 v * (v • v * ) 2 Ψ 0,2,m2 (v * )dv * = 16π 15 |v| 2 Y m2 2 (σ).
(5.7)

3) and

R 3 v * (v • v * ) 2 Ψ 1,0,0 (v * )dv * = - √ 6 3 |v| 2 .
(5.8)

Proof. Set σ * = v * |v * | , σ = v |v| ∈ S 2 , then R 3 v * (v • v * )Ψ 0,1,m1 (v * )dv * = R 3 v * |v||v * |(σ • σ * )Ψ 0,1,m1 (v * )dv * R 3 v * (v • v * ) 2 Ψ 0,2,m2 (v * )dv * = R 3 v * |v| 2 |v * | 2 (σ • σ * ) 2 Ψ 0,2,m2 (v * )dv * R 3 v * (v • v * ) 2 Ψ 1,0,0 (v * )dv * = R 3 v * |v| 2 |v * | 2 (σ • σ * ) 2 Ψ 1,0,0 (v * )dv *
By using the formulas ( 53 ) in Sec.1, Chap. III of [START_REF] Sansone | Orthogonal Functions[END_REF] that

P 1 (x) = x, P 2 (x) = 3 2 x 2 - 1 2 .
We apply the addition theorem of spherical harmonics (7 -34) in Chapter 7 of [START_REF] Slater | Quantum theory of atomic structure[END_REF] (see also (VII) in Sec.19, Chapter III of [START_REF] Sansone | Orthogonal Functions[END_REF]) that,

P k (σ * • σ) = 4π 2k + 1 |m k |≤ k Y -m k k (σ * )Y m k k (σ).
Then

σ * • σ = P 1 (σ * • σ) = 4π 3 | m1|≤1 Y -m1 1 (σ * )Y m1 1 (σ); (σ * • σ) 2 = 2 3 P 2 (σ * • σ) + 1 3 = 8π 15 | m2|≤2 Y -m2 2 (σ * )Y m2 2 (σ) + 1 3 .
Substituting this expansion into the integral and using the orthogonal of the eigenfunctions ϕ n,l,m in L 2 (R 3 ), one has

R 3 v * |v||v * |(σ • σ * )Ψ 0,1,m1 (v * )dv * = 4π 3 |v| Y m1 1 (σ) R 3 v * |v| 2 |v * | 2 (σ • σ * ) 2 Ψ 0,2,m2 (v * )dv * = 16π 15 |v| 2 Y m2 2 (σ) R 3 v * (v • v * ) 2 Ψ 1,0,0 (v * )dv * = 1 3 R 3 v * |v * | 2 Ψ 1,0,0 (v * )dv * |v| 2 = - √ 6 3 |v| 2 ,
where we used the explicit formula of ϕ 0,1,m1 , ϕ 1,0,0 and ϕ 0,2,m2 in (2.1) in section 2. This ends the proof of Lemma 5.3.

We prove now the 5 parts of the Proposition of 2.1 by the following 5 Lemmas Lemma 5.4. For n, l ∈ N, |m| ≤ l, we have

L(ϕ 0,0,0 , ϕ n,l,m ) = -(2(2n + l) + l(l + 1)) ϕ n,l,m .
Proof. Since ϕ 0,0,0 = √ µ, then for all n, l ∈ N, |m| ≤ l, one has

L(ϕ 0,0,0 , ϕ n,l,m ) = 1 µ(v) Q(µ, Ψ n,l,m ) = 1 µ(v) 1≤ k,j≤3 ∂ v k R 3 a k,j (v -v * ) µ(v * )∂ vj Ψ n,l,m (v) -∂ v * j µ(v * )Ψ n,l,m (v) dv * where a k,k (v -v * ) = 1≤ j≤3 j = k (v j -v * j ) 2 ; a k,j (v -v * ) = -(v k -v * k )(v j -v * j ) when k = j. It follows that L(ϕ 0,0,0 , ϕ n,l,m ) = L 1 (ϕ 0,0,0 , ϕ n,l,m ) -L 2 (ϕ 0,0,0 , ϕ n,l,m )
where

L 1 (ϕ 0,0,0 , ϕ n,l,m ) = 1 µ(v) 1≤ k,j≤3 k = j ∂ v k R 3 v 2 j -2v j v * j + (v * j ) 2 µ(v * )dv * ∂ v k Ψ n,l,m (v) - 1 µ(v) 1≤ k,j≤3 k = j ∂ v k R 3 v 2 j -2v j v * j + (v * j ) 2 ∂ v * k µ(v * )dv * Ψ n,l,m (v) ; L 2 (ϕ 0,0,0 , ϕ n,l,m ) = 1 µ(v) 1≤ k,j≤3 k = j ∂ v k R 3 v k v j -v * k v j -v k v * j + v * k v * j µ(v * )dv * ∂ vj Ψ n,l,m (v) - 1 µ(v) 1≤ k,j≤3 k = j ∂ v k R 3 v k v j -v * k v j -v k v * j + v * k v * j ∂ v * j µ(v * )dv * Ψ n,l,m (v) .
Then by using (5.2), we have,

R 3 ∂ v * k µ(v * )dv * = iξ k µ(ξ)| ξ=0 = iξ k Ψ 0,0,0 (ξ) ξ=0 = 0, k = 1, 2, 3; R 3 v * j ∂ v * k µ(v * )dv * = -ξ k ∂ ξ * j Ψ 0,0,0 (ξ) ξ=0 = 0, k = j; R 3 (v * j ) 2 ∂ v * k µ(v * )dv * = -iξ k ∂ 2 ξ * j Ψ 0,0,0 (ξ) ξ=0 = 0, k = j; R 3 v * j ∂ v * j µ(v * )dv * = -Ψ 0,0,0 (ξ) -ξ j ∂ ξ * j Ψ 0,0,0 (ξ) ξ=0 = -1; R 3 v * k v * j ∂ v * j µ(v * )dv * = - R 3 v * k µ(v * )dv * = 0.
It is obviously that

R 3 µ(v * )dv * = (ϕ 0,0,0 , ϕ 0,0,0 ) L 2 (R 3 ) = 1, R 3 v * k µ(v * )dv * = 0, R 3 v * k v * j µ(v * )dv * = 0, when k = j.
Then we can deduce that

L(ϕ 0,0,0 , ϕ n,l,m ) = L 1 (ϕ 0,0,0 , ϕ n,l,m ) -L 2 (ϕ 0,0,0 , ϕ n,l,m ) = 1 µ(v)     1≤ k,j≤3 k = j v 2 j ∂ 2 v k -v k v j ∂ vj ∂ v k + 6     Ψ n,l,m (v) + 1 µ(v) 1≤ k,j≤3 k = j R 3 (v * j ) 2 µ(v * )dv * ∂ 2 v k Ψ n,l,m (v) = 1 µ(v)     1≤ k,j≤3 k = j v 2 j ∂ 2 v k -v k v j ∂ vj ∂ v k + 6     Ψ n,l,m (v) + 1 µ(v) 2 3 R 3 |v * | 2 µ(v * )dv * 3 k=1 ∂ 2 v k Ψ n,l,m (v) = 1 µ(v)     1≤ k,j≤3 k = j v 2 j ∂ 2 v k -v k v j ∂ vj ∂ v k + 6 + 2∆ v     Ψ n,l,m (v) By using (5.4) and (5.5) that, 
   1≤ k,j≤3 k = j v 2 j ∂ 2 v k -v k v j ∂ vj ∂ v k + 6 + 2∆ v     Ψ n,l,m (v) = 2 3 k=1 v k ∂ v k + 6 + 2∆ v Ψ n,l,m (v) + ∆ S 2 Ψ n,l,m (v) = -2(2n + l)Ψ n,l,m (v) -l(l + 1)Ψ n,l,m (v)
where we used the Laplace-Beltrami operator property

∆ S 2 Ψ n,l,m = -l(l + 1)Ψ n,l,m .
Therefore, we conclude with the notation Ψ n,l,m = √ µϕ n,l,m that

L(ϕ 0,0,0 , ϕ n,l,m ) = -(2(2n + l) + l(l + 1)) ϕ n,l,m .
We end the proof of Lemma 5.4 .

Lemma 5.5. For n, l ∈ N, |m| ≤ l, we have

L(ϕ 0,1,m1 , ϕ n,l,m ) =A - n,l,m,m1 ϕ n+1,l-1,m1+m + A + n,l,m,m1 ϕ n,l+1,m1+m , ∀ |m 1 | ≤ 1 with A - n,l,m,m1 = 4 π 3 (l -1) 2(n + 1) Cm1,m l,l-1 ; A + n,l,m,m1 = 4 π 3 (l + 2) √ 2n + 2l + 3 Cm1,m l,l+1
where Cm1,m l,l-1 , Cm1,m l,l+1 were defined in the Corollary 5.2.

Proof. For |m 1 | ≤ 1, for all n, l ∈ N, |m| ≤ l, one has

L(ϕ 0,1,m1 , ϕ n,l,m ) = 1 µ(v) Q(Ψ 0,1,m1 , Ψ n,l,m ) = 1 µ(v) 1≤ k,j≤3 ∂ v k R 3 a k,j (v -v * ) × Ψ 0,1,m1 (v * )∂ vj Ψ n,l,m (v) -∂ v * j Ψ 0,1,m1 (v * )Ψ n,l,m (v) dv * . =L 1 (ϕ 0,1,m1 , ϕ n,l,m ) -L 2 (ϕ 0,1,m1 , ϕ n,l,m ). Recall, R 3 ∂ v * k Ψ 0,1,m1 (v * )dv * = iξ k Ψ 0,1,m1 (ξ) ξ=0 = 0, k = 1, 2, 3; R 3 v * j ∂ v * k Ψ 0,1,m1 (v * )dv * = -ξ k ∂ ξ * j Ψ 0,1,m1 (ξ) ξ=0 = 0, k = j; R 3 (v * j ) 2 ∂ v * k Ψ 0,1,m1 (v * )dv * = -iξ k ∂ 2 ξ * j Ψ 0,1,m1 (ξ) ξ=0 = 0, k = j; R 3 v * j ∂ v * j Ψ 0,1,m1 (v * )dv * = -Ψ 0,1,m1 (ξ) -ξ j ∂ ξ * j Ψ 0,1,m1 (ξ) ξ=0 = 0;
R 3 v * k v * j ∂ v * j Ψ 0,1,m1 (v * )dv * = - R 3 v * k Ψ 0,1,m1 (v * )dv * .
By using the relation (5.1), one can verify that

R 3 Ψ 0,1,m1 (v * )dv * = (ϕ 0,1,m1 , ϕ 0,0,0 ) L 2 (R 3 ) = 0, R 3 v * k v * j Ψ 0,1,m1 (v * )dv * = 0, ∀ 1 ≤ k, j ≤ 3.
We can conclude that

L(ϕ 0,1,m1 , ϕ n,l,m ) = L 1 (ϕ 0,1,m1 , ϕ n,l,m ) -L 2 (ϕ 0,1,m1 , ϕ n,l,m ) = 1 µ(v) 1≤ k,j≤3 k = j -2 R 3 v * j Ψ 0,1,m1 (v * )dv * v j ∂ 2 v k Ψ n,l,m (v) + R 3 v * k Ψ 0,1,m1 (v * )dv * v j ∂ v k ∂ vj Ψ n,l,m (v) + R 3 v * j Ψ 0,1,m1 (v * )dv * v k ∂ v k ∂ vj Ψ n,l,m (v) .
By Fourier transformation, we have

F [ √ µ L(ϕ 0,1,m1 , ϕ n,l,m )](ξ) = 2i 3 j=1 R 3 v * j Ψ 0,1,m1 (v * )dv * ∂ ξj |ξ| 2 Ψ n,l,m (ξ) -2i R 3 (v * • ξ)Ψ 0,1,m1 (v * )dv * |ξ|∂ |ξ| + 4 Ψ n,l,m (ξ) = H 1 (ξ) -H 2 (ξ) where v * • ξ = 3 k=1 v * k ξ k .
We simplify the calculation into the following equality

L(ϕ 0,1,m1 , ϕ n,l,m ) = 1 √ µ [H 1 (v) -H 2 (v)] .
(5.9)

By using the explicit formula of the Fourier transform of Ψ n,l,m that,

Ψ n,l,m = B n,l |ξ| 2n+l e -|ξ| 2 2 Y m l ( ξ |ξ| )
with B n,l defined in (5.3), we have

|ξ| 2 Ψ n,l,m (ξ) = B n,l |ξ| 2(n+1)+l e -|ξ| 2 2 Y m l ( ξ |ξ| ) = B n,l B n+1,l Ψ n+1,l,m .
It follows from the inverse Fourier transformation of H 1 (ξ) and the equality (5.6) in Lemma 5.3,

H 1 (v) = 2 B n,l B n+1,l 3 j=1 R 3 v * j Ψ 0,1,m1 (v * )dv * v j Ψ n+1,l,m (v) = 2 B n,l B n+1,l R 3 (v * • v)Ψ 0,1,m1 (v * )dv * Ψ n+1,l,m (v) = 2 4π 3 B n,l B n+1,l |v| Y m1 1 v |v| Ψ n+1,l,m (v) = 2 4π 3 (n + 1)!B n,l (-i) l 2 (n+1) |v| l+1 e -|v| 2 2 L (l+1/2) n+1 |v| 2 2 × Cm1,m l,l+1 Y m1+m l+1 (σ) + Cm1,m l,l-1 Y m1+m l-1 (σ)
Using the formulas (10 ′ ), [START_REF] Morimoto | Ultra-analytic effect of Cauchy problem for a class of kinetic equations[END_REF], [START_REF] Morimoto | Smoothing effect of the homogeneous Boltzmann equation with measure initial datum[END_REF] of Sec.1, Chap.IV in [START_REF] Sansone | Orthogonal Functions[END_REF] for x = |v| 2 2 , we have L (l+ 1 2 )

n+1 (x) = L (l+ 3 2 ) n+1 (x) -L (l+ 3 2 ) n (x), n = 0, 1, • • • and xL (l+ 1 2 ) n+1 (x) = (n + l + 3 2 )L (l-1 2 ) n+1 (x) -(n + 2)L (l-1 2 )
n+2 (x). Direct calculation shows that

H 1 (v) = 4 π 3 Cm1,m l,l+1
2(n + 1)(2n + 2l + 3)(2n + 2l + 5)Ψ n+1,l+1,m1+m (v)

-4 π 3 Cm1,m l,l+1 2(n + 1) √ 2n + 2l + 3Ψ n,l+1,m1+m (v) + 4 π 3 Cm1,m l,l-1 (2n + 2l + 3) 2(n + 1)Ψ n+1,l-1,m1+m (v) -4 π 3 Cm1,m l,l-1 4(n + 1)(n + 2)(2n + 2l + 3)Ψ n+2,l-1,m1+m (v)
4(n + 1)(n + 2)(2n + 2l + 3) F (Ψ n+2,l-1,m1+m ).
Then by the inverse Fourier transform of H 2 (ξ) and substituting the equalities of H 1 , H 2 into (5.9), we conclude that

L(ϕ 0,1,m1 , ϕ n,l,m ) = 1 √ µ [H 1 (v) -H 2 (v)]
= A - n,l,m,m1 ϕ n+1,l-1,m1+m + A + n,l,m,m1 ϕ n,l+1,m1+m .

Lemma 5.6. For n, l ∈ N, |m| ≤ l, we have L(ϕ 1,0,0 , ϕ n,l,m ) = 4 3(n + 1)(2n + 2l + 3) 3 ϕ n+1,l,m .

Proof. Firstly L(ϕ 1,0,0 , ϕ n,l,m ) = L 1 (ϕ 1,0,0 , ϕ n,l,m ) -L 2 (ϕ 1,0,0 , ϕ n,l,m ).

Using

R 3 ∂ v * k Ψ 1,0,0 (v * )dv * = iξ k Ψ 1,0,0 (ξ) ξ=0 = 0, k = 1, 2, 3; R 3 v * j ∂ v * k Ψ 1,0,0 (v * )dv * = -ξ k ∂ ξ * j Ψ 1,0,0 (ξ) ξ=0 = 0, k = j; R 3 (v * j ) 2 ∂ v * k Ψ 1,0,0 (v * )dv * = -iξ k ∂ 2 ξ * j Ψ 1,0,0 (ξ) ξ=0 = 0, k = j; R 3 v * j ∂ v * j Ψ 1,0,0 (v * )dv * = -Ψ 1,0,0 (0) -ξ j ∂ ξ * j Ψ 1,0,0 (ξ) ξ=0 = 0; R 3 v * k v * j ∂ v * j Ψ 1,0,0 (v * )dv * = - R 3 v * k Ψ 1,0,0 (v * )dv * ,
and

R 3 Ψ 1,0,0 (v * )dv * = (ϕ 1,0,0 , ϕ 0,0,0 ) L 2 (R 3 ) = 0, R 3 v * k Ψ 1,0,0 (v * )dv * = 0, ∀ 1 ≤ k ≤ 3.
We can conclude that

L(ϕ 1,0,0 , ϕ n,l,m ) = L 1 (ϕ 1,0,0 , ϕ n,l,m ) -L 2 (ϕ 1,0,0 , ϕ n,l,m ) = 1 µ(v) 1≤ k,j≤3 k = j R 3 |v * j | 2 Ψ 1,0,0 (v * )dv * ∂ 2 v k Ψ n,l,m (v) + 1 µ(v) 1≤ k,j≤3 k = j R 3 v * k v * j Ψ 1,0,0 (v * )dv * ∂ v k ∂ vj Ψ n,l,m (v).
Again by the Fourier transformation of √ µ L(ϕ 1,0,0 , ϕ n,l,m ), we have

F [ √ µ L(ϕ 1,0,0 , ϕ n,l,m )](ξ) = R 3 v * (ξ • v * ) 2 Ψ 1,0,0 (v * )dv * Ψ n,l,m (ξ) -|v * | 2 √ µ, ϕ 1,0,0 L 2 (R 3 ) |ξ| 2 Ψ n,l,m (ξ).
We deduce from the equality (5.8) in Lemma 5.3 that

R 3 v * (ξ • v * ) 2 Ψ 1,0,0 (v * )dv * = - √ 6 3 |ξ| 2 .
From the definition of eigenfunctions ϕ 1,0,0 , ϕ 0,0,0 , one can calculate that

|v| 2 √ µ = 3ϕ 0,0,0 - √ 6ϕ 1,0,0 Then -|v * | 2 √ µ, ϕ 1,0,0 L 2 (R 3 ) = √ 6.
This implies that

F [ √ µ L(ϕ 1,0,0 , ϕ n,l,m )](ξ) = 2 √ 6 3 |ξ| 2 Ψ n,l,m (ξ) = 4 3(n + 1)(2n + 2l + 3) 3 Ψ n+1,l,m (ξ).
We end the proof of the Lemma by inverse Fourier transformation.

Lemma 5.7. For n, l ∈ N, |m| ≤ l, we have

L(ϕ 0,2,m2 , ϕ n,l,m ) = A 1 n,l,m,m2 ϕ n+2,l-2,m+m2 + A 2 n,l,m,m2 ϕ n+1,l,m+m2 + A 3 n,l,m,m2 ϕ n,l+2,m+m2 , ∀ |m 2 | ≤ 2 with A 1 n,l,m,m2 = -4 π 15 4(n + 2)(n + 1) S 2 Y m2 2 (ω)Y m l (ω)Y -m2-m l-2 (ω)dω; A 2 n,l,m,m2 = 4 π 15 2(n + 1)(2n + 2l + 3) × S 2 Y m2 2 (ω)Y m l (ω)Y -m2-m l (ω)dω; A 3 n,l,m,m2 = -4 π 15 (2n + 2l + 5)(2n + 2l + 3) × S 2 Y m2 2 (ω)Y m l (ω)Y -m2-m l+2
(ω)dω.

(5.10)

Proof. For |m 2 | ≤ 2, for all n, l ∈ N, |m| ≤ l, one has L(ϕ 0,2,m2 , ϕ n,l,m ) = L 1 (ϕ 0,2,m2 , ϕ n,l,m ) -L 2 (ϕ 0,2,m2 , ϕ n,l,m ). Using now R 3 ∂ v * k Ψ 0,2,m2 (v * )dv * = iξ k Ψ 0,2,m2 (ξ) ξ=0 = 0, k = 1, 2, 3; R 3 v * j ∂ v * k Ψ 0,2,m2 (v * )dv * = -ξ k ∂ ξ * j Ψ 0,2,m2 (ξ) ξ=0 = 0, k = j; R 3 (v * j ) 2 ∂ v * k Ψ 0,2,m2 (v * )dv * = -iξ k ∂ 2 ξ * j Ψ 0,2,m2 (ξ) ξ=0 = 0, k = j; R 3 v * j ∂ v * j Ψ 0,2,m2 (v * )dv * = -Ψ 0,2,m2 (0) -ξ j ∂ ξ * j Ψ 0,2,m2 (ξ) ξ=0 = 0; R 3 v * k v * j ∂ v * j Ψ 0,2,m2 (v * )dv * = - R 3 v * k Ψ 0,2,m2 (v * )dv * , and
R 3 Ψ 0,2,m2 (v * )dv * = (ϕ 0,2,m2 , ϕ 0,0,0 ) L 2 (R 3 ) = 0, R 3 v * k Ψ 0,2,m2 (v * )dv * = 0, ∀ 1 ≤ k ≤ 3.
We can conclude that

L(ϕ 0,2,m2 , ϕ n,l,m ) = L 1 (ϕ 0,2,m2 , ϕ n,l,m ) -L 2 (ϕ 0,2,m2 , ϕ n,l,m ) = 1 µ(v) 1≤ k,j≤3 k = j R 3 |v * j | 2 Ψ 0,2,m2 (v * )dv * ∂ 2 v k Ψ n,l,m (v) + 1 µ(v) 1≤ k,j≤3 k = j R 3 v * k v * j Ψ 0,2,m2 (v * )dv * ∂ v k ∂ vj Ψ n,l,m (v).
By the Fourier transformation, we have

F [ √ µ L(ϕ 0,2,m2 , ϕ n,l,m )](ξ) = -|v * | 2 √ µ, ϕ 0,2,m2 L 2 (R 3 ) |ξ| 2 Ψ n,l,m (ξ) + R 3 v * (ξ • v * ) 2 Ψ 0,2,m2 (v * )dv * Ψ n,l,m (ξ).
By using

|v * | 2 √ µ, ϕ 0,2,m2 L 2 (R 3 ) = 3ϕ 0,0,0 - √ 6ϕ 1,0,0 , ϕ 0,2,m2 L 2 (R 3 ) = 0.
and by using the equality (5.7) with v = ξ in Lemma 5.3 that

R 3 v * (ξ • v * ) 2 Ψ 0,2,m2 (v * )dv * = 16π 15 |ξ| 2 Y m2 2 ( ξ |ξ| ),
we obtain

F [ √ µ L(ϕ 0,2,m2 , ϕ n,l,m )](ξ) = 4 π 15 |ξ| 2 Y m2 2 ξ |ξ| Ψ n,l,m (ξ).
We apply Corollary 5.

2 with l = 2, ω = ξ |ξ| that Y m2 2 (ω)Y m l (ω) = C m2,m l,l-2 Y m2+m l-2 (ω) + C m2,m l,l Y m2+m l (ω) + C m2,m l,l+2 Y m2+m l+2 (ω). Recalled that Ψ n,l,m (ξ) = B n,l |ξ| 2n+l e -|ξ| 2 2 Y m l ( ξ |ξ| ),
we have

F [ √ µ L(ϕ 0,2,m2 , ϕ n,l,m )](ξ) = 4 π 15 B n,l |ξ| 2n+l+2 e -|ξ| 2 2 × C m2,m l,l-2 Y m2+m l-2 (ω) + C m2,m l,l Y m2+m l (ω) + C m2,m l,l+2 Y m2+m l+2 (ω) = 4 π 15 B n,l B n+2,l-2 C m2,m l,l-2 Ψ n+2,l-2,m (ξ) + 4 π 15 B n,l B n+1,l C m2,m l,l-2 Ψ n+1,l,m (ξ) + 4 π 15 B n,l B n,l+2 C m2,m l,l-2 Ψ n,l+2,m (ξ).
Direct calculation and the inverse Fourier transform implies that

L(ϕ 0,2,m2 , ϕ n,l,m ) = A 1 n,l,m,m2 ϕ n+2,l-2,m+m2 + A 2 n,l,m,m2 ϕ n+1,l,m+m2 + A 3 n,l,m,m2 ϕ n,l+2,m+m2
. This ends the proof of Lemma. Lemma 5.8. For n, l ∈ N, |m| ≤ l, we have

L(ϕ ñ, l, m, ϕ n,l,m ) = 0, ∀ 2ñ + l > 2, | m| ≤ l.
Proof. For any n, l ∈ N, m ∈ Z, and 2ñ + l > 2, | m| ≤ l, we have again

L(ϕ ñ, l, m, ϕ n,l,m ) = L 1 (ϕ ñ, l, m, ϕ n,l,m ) -L 2 (ϕ ñ, l, m, ϕ n,l,m ). Using the facts R 3 ∂ v * k Ψ ñ, l, m(v * )dv * = iξ k Ψ ñ, l, m(ξ) ξ=0 = 0, k = 1, 2, 3; R 3 v * j ∂ v * k Ψ ñ, l, m(v * )dv * = -ξ k ∂ ξ * j Ψ ñ, l, m(ξ) ξ=0 = 0, k = j; R 3 (v * j ) 2 ∂ v * k Ψ ñ, l, m(v * )dv * = -iξ k ∂ 2 ξ * j Ψ ñ, l, m(ξ) ξ=0 = 0, k = j; R 3 v * j ∂ v * j Ψ ñ, l, m(v * )dv * = -Ψ ñ, l, m(0) -ξ j ∂ ξ * j Ψ ñ, l, m(ξ) ξ=0 = 0; R 3 v * k v * j ∂ v * j Ψ ñ, l, m(v * )dv * = - R 3 v * k Ψ ñ, l, m(v * )dv * .
By the relation (5.1), one can verify that, for any 2ñ + l > 2, | m| ≤ l,

R 3
Ψ ñ, l, m(v * )dv * = (ϕ ñ, l, m, ϕ 0,0,0 ) L 2 (R 3 ) = 0,

R 3 v * k Ψ ñ, l, m(v * )dv * = 0, ∀ 1 ≤ k ≤ 3 R 3 v * k v * j Ψ ñ, l, m(v * )dv * = 0, ∀ 1 ≤ k, j ≤ 3.
We can conclude that, for all 2ñ + l > 2, | m| ≤ l, L(ϕ ñ, l, m, ϕ n,l,m ) ≡ 0.

For the proof of the Proposition 3.1, we recall the elementary result about the Legendre polynomial in the following. P l-1 (x), P 2 (x)P l (x) = 3(l + 2)(l + 1) 2(2l + 3)(2l + 1) P l+2 (x) + (l + 1)l (2l + 3)(2l -1) P l (x) + 3l(l -1) 2(2l + 1)(2l -1) P l-2 (x) for l ≥ 2.

For more general case, we can refer to the Example 11 in Chap.XV in [START_REF] Whittaker | A course of modern analysis[END_REF] or (1.4) in Appendix 1 in [START_REF] Jones | Spherical harmonics and tensors for classical field theory[END_REF].

Proof of the Proposition 3.1

Recalled from Lemma 5.7 that We recalled the addition theorem (7-34) of Chapter 7 in [START_REF] Slater | Quantum theory of atomic structure[END_REF], (VIII) of Sec.19, Chap. III in [START_REF] Sansone | Orthogonal Functions[END_REF] or Theorem 1 of Sec.4, Chap. 1 in [START_REF] Müller | Analysis of spherical symmetries in Euclidean spaces[END_REF] that, for σ, κ ∈ S ).

For the harmonic oscillator H = -△ + |v| 2 4 of 3-dimension and s > 0, we have

H k 2 H α = (λ α ) k 2 H α , λ α = 3 j=1 (α j + 1 2 ), k ∈ N, α ∈ N 3 .
Shubin spaces. We refer the reader to the works [START_REF] Gramchev | Classes of degenerate elliptic operators in Gelfand-Shilov spaces[END_REF][START_REF] Shubin | Pseudodifferential Operators and Spectral theory[END_REF] for the Shubin spaces.

Let τ ∈ R, The Shubin spaces Q τ (R 3 ) can be also characterized through the decomposition into the Hermite basis :

f ∈ Q τ (R 3 ) ⇔ f ∈ S ′ (R 3 ), H τ 2 f L 2 < +∞; ⇔ f ∈ S ′ (R 3 ), (|α| + 3 2 ) τ /2 (f, H α ) L 2 α∈N 3 l 2 < +∞ ,
and for τ > 0,

Q τ (R 3 ) H τ (R 3 )
where H τ (R 3 ) is the usuel Sobolev space. In fact, Hf ∈ L 2 (R 3 ) ⇒ △f, |v| 2 f ∈ L 2 (R 3 ).

So that for the negative index, we have,

H -τ (R 3 ) Q -τ (R 3 ).
See more details in the Appendix in [START_REF] Morimoto | A remark on the ultra-analytic smoothing properties of the spatially homogeneous Boltzmann equation[END_REF].

  Shilov smoothing effect of Cauchy problem. 5) It is well known that the single Dirac mass on the origin is a stationary solution of the Cauchy problem (1.1). The following example is somehow surprise. Example 1.1. Let

Lemma 5 . 9 .P 2 (P 1

 5921 Let l ∈ N be nonnegative integer, P l (x) is the Legendre polynomial, we have                             P 2 (x)P 0 (x) = P 2 (x), (x)P l (x) = l + 1 2l + 1 P l+1 (x) + l 2l + 1

2 Y

 2 -2,m,m2 = -4 π 15 (2n + 2l + 1)(2n + 2l -1) S m2 2 Y m l-2 Y -m2-m l dω.

2 , 2 3 . 2 n 2 2 κP 2 ǫ0|α| 1 2ν 4 ) k 2 f

 2232222142 , ∀ k ∈ N.Therefore, for any m * ∈ Z and |m * | ≤ l, (κ• σ)P l+2 (κ • σ)Y m * l (κ)Y -m * l (σ)dκ dσ.By using Lemma 5.9 and the orthogonal of {Y m l } l∈N,|m|≤ l on S 2 , we have, for n ≥ 2, This is the estimation (3.1). Similar to the proof of (3.1), one can deduce also from Lemma 5.9 and the orthogonal of {Y m l } l∈N,|m|≤ l on S 2 thatA (κ • σ)P 0 (κ • σ)Y 0 0 (κ)Y 0 0 (σ)dκ dσ (κ • σ)Y 0 0 (κ)Y 0 0 (σ)dκ dσ = 0, ∀ n ≥ 1,and for any m * ∈ Z and |m * | ≤ l + 1)(2l -1)≤ (2n + 2l + 1)(2n + 2l -1) 2 ∀ n ∈ N, l ≥ 2. ⇔ f ∈ L 2 (R 3 ), ∃ ǫ 0 > 0, e (f, H α ) L 2 α∈N 3 l 2 < +∞; ⇔ ∃ C > 0, A > 0, (-△ + |v| 2 L 2 (R 3 ) ≤ AC k (k!) ν , k ∈ N where H α (v) = H α1 (v 1 )H α2 (v 2 )H α3 (v 3 ), α ∈ N 3 ,and for x ∈ R,

  .By using the Corollary 5.2 and the explicit formula of Ψ n,l,m , one can verify that

	H 2 (ξ) = 2i	4π 3	(2n + l + 4)B n,l |ξ| 2n+l+1 e -|ξ| 2 2
	× Cm1,m l,l+1 Y m1+m l+1	+ Cm1,m l,l-1 Y m1+m l-1
	-2i	4π 3	2(n + 1)(2n + 2l + 3)B n+1,l |ξ| 2n+l+3 e -|ξ| 2 2
	× Cm1,m l,l+1 Y m1+m l+1	+ Cm1,m l,l-1 Y m1+m l-1
	= -4	π 3		Cm1,m l,l+1 (2n + l + 4) √ 2n + 2l + 3 F (Ψ n,l+1,m1+m )
	+ 4	π 3		Cm1,m l,l-1 (2n + l + 4) 2(n + 1) F (Ψ n+1,l-1,m1+m )
	+ 4	π 3		Cm1,m l,l+1	2(n + 1)(2n + 2l + 3) √	2n + 2l + 5 F (Ψ n+1,l+1,m1+m )
	-4	π 3		Cm1,m l,l-1
	Now we calculate H 2 . We can deduce from the equality (5.6) in Lemma 5.3 with
	v = ξ that
	H 2 (ξ) = 2i	4π 3 × [(2n + l + 4) Ψ n,l,m -2(n + 1)(2n + 2l + 3) Ψ n+1,l,m ] |ξ| Y m1 1 ( ξ ) |ξ|

Acknowledgements. The first author is supported by the Natural Science Foundation of China under Grant No.11701578. The research of the second author was supported partially by "The Fundamental Research Funds for Central Universities of China".

The estimations (3.2) and (3.3) follow. We end the proof of Proposition 3.1.

Appendix

The proof of the example and the characterization of the Gelfand-Shilov spaces and the Shubin spaces are presented in this section. For the self-content of paper, we will present some proof here.

The proof of the Example 1.1. Now we prove that the function g 0 defined in (1.8) is belongs to

One can calculate directly that g 0 , ϕ 0,0,0 = δ 0 , 1ϕ 0,0,0 , ϕ 0,0,0 = 0;

Since g 0 is radial, we can verify that

This shows that g 0 ∈ N ⊥ and S 2 g L 2 (R 3 ) = 0. Now we prove that g 0 ∈ Q α (R 3 ) for α < -3 2 . Since g 0 ∈ N ⊥ and radial, we can write g 0 in the form

where we can calculate in details that,

By using the Stirling equivalent

Therefore, for any α < -3 2 ,

This implies that g 0 ∈ Q α (R 3 ), we end the proof of the Example.

Gelfand-Shilov spaces. The symmetric Gelfand-Shilov space S ν ν (R 3 ) can be characterized through the decomposition into the Hermite basis {H α } α∈N 3 and the harmonic oscillator H = -△ + |v| 2 4 . For more details, see Theorem 2.1 in [START_REF] Gramchev | Classes of degenerate elliptic operators in Gelfand-Shilov spaces[END_REF] f ∈ S ν ν (R 3 ) ⇔ f ∈ C ∞ (R 3 ), ∃ τ > 0, e τ H 1 2ν f L 2 < +∞;


Jérôme Darmont
email: jerome.darmont@univ-lyon2.fr

Data-Centric Benchmarking

The aim of this chapter is to present an overview of the major past and present state-of-the-art data-centric benchmarks. Our review includes the TPC standard benchmarks, but also alternative or more specialized benchmarks. We survey benchmarks from three families: transaction benchmarks aimed at On-Line Transaction Processing (OLTP), decision-support benchmarks aimed at On-Line Analysis Processing (OLAP) and big data benchmarks. Eventually, we discuss the issues, tradeoffs and future trends in data-centric benchmarking.

INTRODUCTION

In data management, both system designers and users casually resort to performance evaluation. On one hand, designers need to test architectural features and hypotheses regarding the actual (vs. theoretical) behavior of a system, especially in terms of response and scalability. Performance tuning also necessitates accurate performance evaluation. On the other hand, users are also keen on comparing the efficiency of different technologies before selecting a software solution. Thence, performance measurement tools are of premium importance in the data management domain.

Performance evaluation by experimentation on a real system is generally referred to as benchmarking. It consists in performing a series of tests on a given system to estimate its performance in a given setting. Typically, a data-centric benchmark is constituted of two main elements: a data model (conceptual schema and extension) and a workload model (set of read and write operations) to apply on this dataset, with respect to a predefined protocol. Both models may be parameterized. Most benchmarks also include a set of simple or composite performance metrics such as response time, throughput, number of input/output operations, disk or memory usage, etc.

The Transaction Processing Performance Council (TPC), a non-profit organization founded in 1988, plays a preponderant role in data-centric benchmarking. Its mission is to issue standard benchmarks, to verify their correct application by the industry, and to publish performance test results. TPC members include all the major industrial actors from the database field.

BACKGROUND Transaction Processing Benchmarks

The first TPC benchmark for relational, transactional databases, TPC-C (TPC, 2010), has been in use since 1992. TPC-C features a complex business database (a classical customer-order-product-supplier model with nine types of tables bearing various structures and sizes) and a workload of diversely complex transactions that are executed concurrently. The performance metric in TPC-C is transaction throughput. As all TPC benchmarks, TPC-C's only parameter is a scale factor SF that determines data size. TPC-C was complemented in 2007 by TPC-E (TPC, 2015a), which simulates a brokerage firm with the aim of being representative of more modern OLTP systems. In its principles and features, TPC-E is otherwise very similar to TPC-C.

There are few alternatives to TPC-C and TPC-E for relational applications. Yet, some benchmarks fit niches where there is no standard benchmark. For instance, OO7 [START_REF] Dewitt | The OO7 benchmark[END_REF] and OCB [START_REF] <jrn>darmont | Benchmarking OODBs with a Generic Tool[END_REF] are object-oriented database benchmarks modeling engineering applications, e.g., computer-aided design or software engineering. However, their complexity makes both these benchmarks hard to understand and implement. Moreover, with objects in databases being more commonly managed in object-relational systems nowadays, object-relational benchmarks such as BUCKY [START_REF] Dewitt | The BUCKY Object-Relational Benchmark[END_REF] and BORD (Lee et al., 2000) now seem more relevant. Such benchmarks focus on queries implying object identifiers, inheritance, joins, class and object references, mul-tivalued attributes, query unnesting, object methods, and abstract data types. However, typical object navigation is considered already addressed by object-oriented benchmarks and is not taken into account. Moreover, object-relational database benchmarks have not evolved since the early 2000's, whereas object-relational database systems have.

Similarly, XML benchmarks aim at comparing the various XML storage and querying solutions developed since the late nineties. From the early so-called XML application benchmarks that implement a mixed XML database that is either data-oriented (structured data) or document-oriented (in general, random texts built from a dictionary), XBench [START_REF] Özsu | XBench Benchmark and Performance Testing of XML DBMSs[END_REF] stands out. XBench is indeed the only benchmark proposing a true mixed dataset (i.e., data and document-oriented) and helping evaluate all the functionalities offered by XQuery. FlexBench (Vranec & Mlýnková, 2009) also tests a large set of data characteristics and proposes query templates that allow modeling multiple types of applications. Finally, Schmidt et al. (2009) and [START_REF] <conf>zhang | Renda-RX: A Benchmark for Evaluating XML-Relational Database System[END_REF] propose benchmarks that are specifically tailored for testing logical XML model-based systems, namely native XML and XMLrelational database management systems, respectively.

Decision-Support Benchmarks

TPC-H (TPC, 2014a) has long been the only standard decision-support benchmark. It exploits a classical product-order-supplier database schema, as well as a workload that is constituted of twenty-two SQL-92, parameterized, decision-support queries and two refreshing functions that insert tuples into and delete tuples from the database. Query parameters are randomly instantiated following a uniform law. Three primary metrics describe performance in terms of power, throughput, and a combination of power and throughput.

However, TPC-H's database schema is not a star-like multidimensional schema that is typical in data warehouses. Furthermore, its workload does not include any true OLAP query. TPC-DS (TPC, 2015b) now fills in this gap. Its schema represents the decision-support functions of a retailer under the form of a constellation schema with several fact tables and shared dimensions. TPC-DS' workload is constituted of four classes of queries: reporting queries, ad-hoc decision-support queries, interactive OLAP queries, and extraction queries. SQL-99 query templates help randomly generate a set of about five hundred queries, following non-uniform distributions. TPC-DS features one primary throughput metric that takes both query execution and data warehouse maintenance into account.

Given the primordial importance of data integration in many data-centric (including data warehousing) scenarios, TPC-H was recently complemented by TPC-DI (TPC, 2014b). TPC-DI focuses on Extract, Load and Transform (ETL) processes. Data are first generated in a staging area as if they were extracted from a virtual retail brokerage firm's operational databases. Then, data are transformed through, e.g., type conversions, attribute splits or merges, and error checks. Finally, data are loaded into a warehouse constituted of five fact tables and eight dimension tables. There are two load phases: an initial, so-called historical load, and then incremental updates. Transformations are different in these two phases. TPC-DI's main metric is a combination of throughputs from the historical load and two incremental updates.

There are, again, few decision-support benchmarks out of the TPC, but with TPC-DS having had an eight-year long development, alternative data warehouse benchmarks were proposed. Published by the OLAP council, a now inactive organization founded by OLAP vendors, APB-1 (OLAP Council, 1998) was the first of them and actually predates TPC-DS. APB-1 has been intensively used in the late nineties. However, APB-1 is very simple and rapidly proved limited to evaluate the specificities of various activities and functions. Thus, more elaborate alternatives were proposed, such as DWEB [START_REF] <jrn>darmont | Benchmarking Data Warehouses[END_REF], which can be parameterized to generate various ad-hoc synthetic data warehouses and workloads that include typical OLAP queries, and SSB [START_REF] <conf>o'neil | The Star Schema Benchmark and Augmented Fact Table Indexing[END_REF], which is based on TPC-H's database remodeled as a star schema and features a query workload that provides both functional and selectivity coverage.

It is also worth noting that TPC-DS is a complex benchmark. Thence, simpler benchmarks are still in use, especially for testing OLAP scenarios in the cloud. For instance, TPC-H was used to benchmark Hadoop and Pig [START_REF] Moussa | TPC-H Benchmark Analytics Scenarios and Performances on Hadoop Data Clouds[END_REF] and SSB for testing the efficiency of view materialization in the cloud (Perriot et al., 2014). Niche benchmarks also rely a lot on TPC-H. XWeB [START_REF] <conf>mahboubi | XWeB: The XML Warehouse Benchmark[END_REF] proposes a unified reference model for XML warehouses and its associate XQuery decision-support workload. RTDW-bench [START_REF] <conf>jedrzejczak | RTDW-bench: Benchmark for Testing Refreshing Performance of Real-Time Data Warehouse[END_REF] is designed for testing the ability of a real-time data warehouse to handle a transaction stream without delay, given an arrival rate. Eventually, [START_REF] <conf>bär | Towards benchmarking stream data warehouses[END_REF] propose a benchmark for stream data warehouses that measures the freshness of materialized views.

Finally, a couple of benchmarks are even more specific (and unrelated from TPC-H), e.g., Spadawan (Lopes Siqueira et al., 2010), which allows performance evaluation of specific, complex operations in spatial data warehouses, and BenchDW [START_REF] <conf>triplet | BenchDW: a generic framework for biological data warehouse benchmarking[END_REF], which targets biological data warehouses and particularly focuses on performance metrics, with twenty-two different metrics such as documentation quality, accuracy and response time.

Big Data Benchmarks

In the timely trend of big data analytics, benchmarking needs are as high as ever to compare alternative systems, including the many NoSQL database systems. In this context of data variety, volume and velocity, adaptability and scalability are premium features that must be evaluated. The TPC promotes two benchmarks for big data. TPC-VMS (TPC, 2013) is actually a benchmarking environment for virtualized databases that allows running TPC-C, TPC-E, TPC-H or TPC-DS on three virtual machines (VMs). Its metric is the minimum value of the selected benchmark's metric on all VMs. TPCx-HS (TPC, 2015c) focuses on Hadoop and MapReduce-based applications. It models a simple application (with no data model directly available) and features, in addition to classical metrics, availability and energy metrics.

Before the TPC could issue its big data benchmarks, and still in parallel, there are many industrial and academic initiatives. MalStone (Open Cloud Consortium, 2009) is a benchmark for assessing data intensive parallel processing. It features MalGen, a synthetic data generator that produces large datasets generated probabilistically following specified distributions. In the same line, HiBench (Huang et al., 2010) is a set of Hadoop programs, ranging from data sorting to clustering, aimed at measuring metrics such as response time, HDFS bandwidth consumption and data access patterns. SWIM (Chen et al., 2013) also measures the performance of Hadoop/MapReduce systems. SWIM contains suites of workloads of thousands of jobs, with complex data, arrival, and computation patterns, and therefore provides workload-specific optimizations. Finally, HcBench (Saletore et al., 2013) models real Hadoop usages in a datacenter. HcBench features various job types and data sizes.

By contrast, YCSB [START_REF] <conf>cooper | Benchmarking cloud serving systems with YCSB[END_REF] is a framework that focuses on data, and more specifically on performance evaluation of key-value stores. YCSB defines several metrics and workloads to measure system behavior in different situations, or the same system when using different configurations. OLTP-Bench (Curino et al., 2012) is the first true benchmarking framework designed for cloud transactional database systems as a service. OLTP-Bench actually features a set of existing microbenchmarks (i.e., designed to test one very specific aspect of performance, e.g., ResourceStresser), popular benchmarks (e.g., TPC-C) and real-world applications (e.g., Wikipedia).

Regarding big data analytics, PRIMEBALL [START_REF] <conf>ferrarons | PRIMEBALL: a Parallel Processing Framework Benchmark for Big Data Applications in the Cloud[END_REF] aims at providing a reallife context to cloud data warehouse benchmarking. Its authors provide the specifications of a fictitious news site hosted in the cloud that is to be managed by the framework under analysis, together with several objective use cases and measures for evaluating system performance. The Big Data Benchmark (Amplab, 2014) goes one step further by actually implementing existing analytical workload models by [START_REF] <conf>pavlo | A comparison of approaches to large-scale data analysis[END_REF]. Its only metric is the response time of such relational queries as scans, aggregations and joins. It can be used for both MapReduce-based systems (such as Shark and Hive) and classical parallel database systems. BigBench (Rabl et al., 2015) is a so-called a specification-based benchmark that is independent from technology. BigBench relies a lot on TPC-DS, borrowing its data model and part of its workload model. The remainder of the workload is adapted from big data use cases issued by the McKinsey Global Institute. Finally, yet other benchmarks, i.e., CloudSuite [START_REF] <conf>yasin | Deep-dive analysis of the data analytics workload in CloudSuite[END_REF] and DCBench (Jia et al., 2013), feature machine learning and data mining-oriented workload models that mostly run on Hadoop and exploit the Mahout library.

Eventually, BigDataBench [START_REF] Wang | Benchmarking Replication and Consistency Strategies in Cloud Serving Databases: HBase and Cassandra[END_REF] aims at providing the widest possible scope of big data models and workloads. It includes nineteen benchmarks representing a large variety of data models, workload models and application scenarios from search engines, social networks, ecommerce, multimedia analytics and bioinformatics. Workload models cover OLTP, "cloud OLTP" and OLAP. As BigBench, BigDataBench also allows alternative implementations, e.g., using MapReduce or Spark. [START_REF]The Benchmark Handbook for Database and Transaction Processing Systems[END_REF] defines four primary criteria to specify a "good" benchmark.

ISSUES AND TRADEOFFS IN DATA PROCESSING BENCHMARKS

1. Relevance: The benchmark must deal with aspects of performance that appeal to the largest number of potential users.

2. Portability: The benchmark must be reusable to test the performances of different DBMSs.

3. Simplicity: The benchmark must be feasible and must not require too many resources.

4. Scalability: The benchmark must adapt to small or large computer architectures.

In their majority, existing benchmarks aim at comparing the performances of different systems in given experimental conditions. This helps vendors position their products relatively to their competitors', and users achieve strategic and costly software choices based on objective information. These benchmarks invariably present fixed data and workload models. Gray's scalability factor is achieved through a reduced number of parameters that mainly allows varying database size in predetermined proportions. All TPC benchmarks notably feature a single scale factor parameter. This solution is simple (still according to Gray's criteria), but the relevance of such benchmarks is inevitably reduced to the test cases that are explicitly modeled. For instance, the typical customer-order-product-supplier data model from TPC benchmarks is unsuitable to many application domains. This leads benchmark users to design more or less elaborate variants of standard tools, when they feel these are not generic enough to fulfill particular needs. Such users are generally not confronted to software choices, but to architectural choices or performance optimization tradeoffs within a given system or family of systems. In this context, it is essential to multiply experiments and test cases, and a monolithic benchmark is of reduced relevance.

To enhance the relevance of benchmarks aimed at system designers, three solutions are possible. The first one is to design an ad-hoc benchmark for a particular application, e.g., RTW-bench, Spadawan and BenchDW, for real-time, spatial and biological data warehouses, respectively. However, the benchmark's application span is necessarily quite narrow. One alternative is to resort to benchmark generators, also called tunable or generic benchmarks, such as OCB, DWEB or FlexBench, which help generate various data or workload models, and thus allow experiments to run in various conditions. The other alternative, which is preferred in recent benchmarks such as YSCB, OLTP-Bench and BigDataBench, is to offer a unifying framework that includes a comprehensive set of stateof-the-art benchmarks. However, the two latter approaches are mechanically detrimental to simplicity, which is a primordial criterion. It is thus necessary to devise benchmark suites that to not sacrifice simplicity too much.

FUTURE RESEARCH DIRECTIONS

The previous section showed that classical transaction and decision-oriented benchmarks are well established. However, big data benchmarking, which predominantly uses cloud technologies, faces a new paradigm and must measure new features. Thus, in addition to [START_REF]The Benchmark Handbook for Database and Transaction Processing Systems[END_REF] criteria for building a good benchmark, [START_REF] <conf>folkerts | Benchmarking in the Cloud: What It Should, Can, and Cannot Be[END_REF] propose that the quality criteria that are commonly accepted by the benchmarking community must be revisited.

Although the cloud inherits from a long legacy of distributed systems, important issues are unique to the cloud. For instance, the concept of elasticity applied to data management may translate in the ability to bring in new data sources dynamically to meet emerging needs [START_REF]Research challenges for cloud intelligence[END_REF]. Thus, cloud benchmark data models should be dynamic. Moreover, the three or four Vs (if we include veracity) of big data are unequally addressed in current benchmarks, which mostly focus on scaling (volume) and, to a lesser extent, on variety with multi-benchmark suites such as BigDataBench. Among recent benchmarks, only HcBench features inter-job arrival rates that can simulate data streams. Yet, it is a quite low-level benchmark. Thus, RTDW-Bench and Bär and Golab's (2012) stream warehouse benchmark could be welcome additions in multi-benchmark suites.

With respect to veracity, specific security issues appeared in the new framework of the cloud, e.g., cloud provider or subcontractor espionage, cost-effective defense of availability or uncontrolled mashups (Chow et al., 2009). Such features are important to assess, for security is one of the top concern of cloud users and would-be users. Data consistency is also a concern, e.g., PRIMEBALL's metrics do not only target transaction performance and storage costs, but also data consistency. [START_REF] <conf>bermbach | Towards Comprehensive Measurement of Consistency Guarantees for Cloud-Hosted Data Storage Services[END_REF] further advocate for a standard comprehensive benchmark for quantifying the consistency guarantees of eventually consistent storage systems. Moreover, for web application-based benchmarks, data quality assessment is also of premium importance. To the best of our knowledge, this intricate task has not been included yet in any big data benchmark.

Finally, the economic model of the cloud is fundamentally new. Instead of a costly initial investment, pay-as-you-go models allow users to pay a small amount per use, e.g., of a dataset, in return for a one-time advantage [START_REF]Research challenges for cloud intelligence[END_REF]. Thus, cost is also a key criterion when benchmarking cloud/big data solutions. TPC benchmarks typically feature a cost metric, but it is presumably too high-level for fine-grained cost analyses.

CONCLUSION

Benchmarking is a small field, but it is nonetheless essential to data-centric research and industry. It serves both engineering and research purposes, when designing systems or validating solutions; as well as marketing purposes, when monitoring competition and comparing commercial products.

We subdivide benchmarks in three classes. First, standard, general-purpose benchmarks such as the TPC's do an excellent job in evaluating the global performance of systems. They are well suited to software selection by users and marketing battles by vendors, who try to demonstrate the superiority of their product at one moment in time. However, their relevance drops for some particular applications that exploit data or workloads models that are radically different from those they implement. Adhoc benchmarks are a solution. They either are adaptations of general-purpose benchmarks, or specifically designed benchmarks. Designing myriads of narrow-band benchmarks is not time-efficient, though, and trust in yet another new benchmark might prove limited in the community. Hence, the last alternative is to use generic or multi-benchmarks that feature a common framework for generating various experimental possibilities. The drawback of this approach is that benchmark complexity must be mastered. In conclusion, before starting a benchmarking experiment, users' needs must be carefully assessed so that the right benchmark or benchmark class is selected, and test results are meaningful.

It is nonetheless clear that the TPC plays a primordial role in the data benchmarking community, not only by issuing standards, but also by structuring and leading the community, e.g., by organizing the annual Technology Conference on Performance Evaluation and Benchmarking (TPCTC). This event does not only promote the TPC's activity, but also greatly encourages industrial and academic advances in the field of performance evaluation and benchmarking, whether they are related to the TPC or TPC benchmarks or not.

KEY TERMS AND DEFINITIONS

Benchmark: A standard program that runs on different systems to provide an accurate measure of their performance.

Cloud Benchmarking: Use of cloud services in the respective (distributed) systems under test [START_REF] <conf>folkerts | Benchmarking in the Cloud: What It Should, Can, and Cannot Be[END_REF].

Database Benchmark: A benchmark specifically aimed at evaluating the performance of Database Management Systems (DBMSs) or DBMS components.

Data Model: In a data-centric benchmark, a database schema and a protocol for instantiating this schema, i.e., generating synthetic data or reusing real-life data.

Performance Metrics: Simple or composite metrics aimed at expressing the performance of a system. Synthetic Benchmark: A benchmark in which the workload model is artificially generated, as opposed to a real-life workload.

Workload Model: In a data-centric benchmark, a set of predefined read and write operations or operation templates to apply on the benchmark's database, following a predefined protocol.


Somayeh Sobati Moghadam

Jérôme Darmont
email: jerome.darmont@univ-lyon2.fr

Gérald Gavin
email: gerald.gavin@univ-lyon1.fr

S4: A New Secure Scheme for Enforcing Privacy in Cloud Data Warehouses

Outsourcing data into the cloud becomes popular thanks to the pay-as-you-go paradigm. However, such practice raises privacy concerns. The conventional way to achieve data privacy is to encrypt sensitive data before outsourcing. When data are encrypted, a tradeoff must be achieved between security and efficient query processing. Existing solutions that adopt multiple encryption schemes induce a heavy overhead in terms of data storage and query performance, and are not suited for cloud data warehouses. In this paper, we propose an efficient additive encryption scheme (S4) based on Shamir's secret sharing for securing data warehouses in the cloud. S4 addresses the shortcomings of existing approaches by reducing overhead while still enforcing good data privacy. Experimental results show the efficiency of S4 in terms of computation and storage overhead with respect to existing solutions.

Introduction

Data warehouses (DWs) provide a consolidated view of organizations and businesses' data, optimized for reporting and analysis. greatly enhance decision making. DWs consolidate historical data from different sources and allow on-line analytical processing (OLAP). Nowadays, data outsourcing scenarios tremendously grow with the advent of cloud computing that offers both cost savings and service benefits. One of the most notable cloud outsourcing services is Database-as-a-Service, where individuals and organizations outsource data storage and management to a Cloud Service Provider (CSP) [START_REF] Xiong | Preserving data privacy in outsourcing data aggregation services[END_REF]. Naturally, such services allow outsourcing a DW and running OLAP queries [START_REF] Amanatidis | Provably-secure schemes for basic query support in outsourced databases[END_REF]. Yet, data outsourcing brings out privacy concerns since sensitive data are stored, maintained and processed by an external third party that may not be fully trusted.

A typical solution to preserve data privacy is encrypting data locally before sending them to an external server. Secure database management systems (SDBMSs) such as CryptDB [START_REF] Popa | CryptDB: protecting confidentiality with encrypted query processing[END_REF] implement cryptographic schemes. Paillier's partially homomorphic encryption scheme [START_REF] Paillier | Public-key cryptosystems based on composite degree residuosity classes[END_REF] is notably used in CryptDB to provide high security. However, it induces a high storage and computation overhead. Hence, in this paper, we propose a new Secure Secret Splitting Scheme (S4) that aims at replacing Paillier's scheme in systems such as CryptDB. S4 is based on the idea of secret sharing [START_REF] Shamir | How to share a secret[END_REF] and is efficient both in terms of storage and computing, without sacrificing privacy too much.

In the remainder of this paper, Section 2 discusses related works about SDBMSs, homomorphic encryption and secret sharing. Section 3 details and discusses S4. Section 4 provides an experimental validation of S4 against Paillier's scheme. Finally, section 5 concludes the paper and hints as future research.

Related Works

Secure Database Management Systems

CryptDB brings together powerful cryptographic tools to handle query processing on encrypted data without decryption [START_REF] Popa | CryptDB: protecting confidentiality with encrypted query processing[END_REF]. Encryption in CryptDB is like onion layers that store multiple ciphertexts, i.e., encrypted data, within each other. Each onion layer enables certain kind of query processing and a given security level provided by one encryption scheme. For instance, order-preserving encryption (OPE) enables range queries and additive homomorphic encryption enables addition over encrypted data. Yet, CryptDB is not perfectly secure since schemes such as OPE reveal some statistical information about plaintext [START_REF] Naveed | Inference attacks on property-preserving encrypted databases[END_REF].

MONOMI builds upon CryptDB to allow the execution of analytical workloads over encrypted data outsourced to the cloud [START_REF] Tu | Processing analytical queries over encrypted data[END_REF]. MONOMI aims at improving CryptDB's query processing capability and efficiency based on split client/server execution. A designer also optimizes physical data layout.

Eventually, using a local trusted hardware at the CSP's, such as TrustedDB [START_REF] Bajaj | TrustedDB: a trusted hardware based database with privacy and data confidentiality[END_REF] and CipherBase [START_REF] Arasu | Transaction processing on confidential data using cipherbase[END_REF], is an alternative approach to query encrypted data. However, trusted hardware is limited in computation ability and memory capacity, and also very expensive.

Homomorphic Encryption

Fully homomorphic encryption (FHE) allows performing arbitrary arithmetic operations over encrypted data without decryption [START_REF] Gentry | A fully homomorphic encryption scheme[END_REF]. FHE provides semantic security, i.e., it is computationally impossible to distinguish two ciphertexts encrypted from the same plaintext. However, FHE requires so much computing power that it cannot be used in practice.

Partially homomorphic encryption (PHE) is more efficient than FHE. Paillier's [START_REF] Paillier | Public-key cryptosystems based on composite degree residuosity classes[END_REF] the most efficient additive FHE. With Paillier's scheme, multiplying the encryption of two values results in an encryption of the sum of the values, i.e., Enc k (x) × Enc k (y) = Enc k (x + y), where the multiplication is performed modulo some public-key k [13]. Paillier's scheme is, however, still computationally intensive and induces as large ciphertext sizes as 2048 bits. Additionally, modular multiplications become computationally expensive on a large number of records, such as in the fact table of a DW [START_REF] Tu | Processing analytical queries over encrypted data[END_REF][START_REF] Sion | Towards Secure Data Outsourcing[END_REF].

Secret Sharing

Secret sharing divides a secret piece of data into so-called shares that are stored at n participants'. A subset of k ≤ n participants is required to reconstruct the secret. In Shamir's, the first secret sharing scheme [START_REF] Shamir | How to share a secret[END_REF], to share a secret v j , a random polynomial P vj (x) of degree k -1 is first built. The owner of the secret chooses a prime p > v j and k -1 random numbers a 1 , a 2 , ..., a k-1 from F p ; and sets a 0 = v j (Equation 1). P vj (x) passes through the point (0, v j ).

P vj (x) = a k-1 x k-1 + ... + a 1 x + a 0 mod p (1)
To build n points over P vj (x), a set of n distinct elements in F p , X = {x 1 , x 2 , . . . , x n }, is chosen such that x i = 0 ∀i = 1, ..., n. For each participant i, the corresponding share is v i,j = P vj (x i ). For each secret v j , there are n points (x i , v i,j ) through which the polynomial P vj (x) passes [START_REF] Hadavi | Security and searchability in secret sharing-based data outsourcing[END_REF]. Any k shares form k points (x i , v i,j ) i = 1, . . . k, from which polynomial P vj (x) can be reconstructed using Lagrange interpolation [START_REF] Dautrich | Security limitations of using secret sharing for data outsourcing[END_REF] (Equation 2).

P vj (x) = k i=1 v i,j i (x) mod p (2) i (x) = 1≤j≤k,j =i (x -x j )(x i -x j ) -1 mod p
where (x ix j ) -1 is the multiplicative inverse of (x ix j ) modulo p [START_REF] Dautrich | Security limitations of using secret sharing for data outsourcing[END_REF]. Eventually, the secret is the constant term of the polynomial:

v j = P vj (0) = k i=1 v i,j i (0) mod p. (3)

S4

S4's driving idea is based on secret sharing, but instead of sharing secrets to n participants' or CSPs', they are stored at one single CSP's. Thus, we avoid the high storage overhead of secret sharing. In S4, each secret v j is divided into

n = k splits v 1,j , ..., v k,j . k -1 splits, v 1,j , ..., v k-1,j
, are stored at the CSP's and v k,j is stored in a trusted machine, e.g., at the user's (Figure 1). In order to reduce storage overhead at the user's, v k,j is set to be the same for all secrets.

Splitting and Reconstruction Processes

First, x k and v k are randomly set up from F p , where p is a big prime number, i.e., greater than the greatest possible query answer. For any secret v j , a random polynomial P vj (x) is built that passes through (0, v j ) and (x k , v k ). To this end, ), (0, v j ) and (x k , v k ), polynomial P vj (x) is built using Equation 2. Storing the k -2 random points is unnecessary because they are not needed for secret reconstruction.

To divide v j into k -1 splits (since (x k , v k ) is already fixed), a set of k -1 distinct elements X = {x 1 , x 2 , . . . , x k-1 } is chosen from F p such that x i = 0 and x i = x k ∀i = 1, ..., k -1. Then, splits are v i,j = P vj (x i ). K=(X, (x k , v k )) is considered as a private key for S4 and must be kept hidden from the CSP. To reconstruct secret v j , its k-1 splits must be retrieved from the CSP. Given points (x i , v i,j ), i = 1, ..., k -1 and (x k , v k ), which is stored at the user's, polynomial P vj (x) can be reconstructed using Equation 2. Its constant term is v j .

Summation Queries

Let a relational table T consist of one attribute A (additional attributes, if any, can be processed similarly). Suppose T has m records. We denote by v j the j th value of A. For attribute A in T , k -1 attributes A i , i = 1, ..., k -1 are created in table T at the CSP's, where each attribute A i stores the i th splits. Without loss of generality, we assume integer data type for A. Other data types can be transformed into integers before splitting. S4 allows summation queries to be computed directly at the CSP's. Consider a query that sums q values of A.

SUM= 1≤j≤q v j , v j ∈ dom(A) ∀j = 1, ..., q.
The CSP computes the sum of the splits stored in A i as

SUM i ∀i = 1, . . . k -1 such that SUM i = 1≤j≤q,1≤i≤k-1 v i,j mod p.
Then, SUM 1 , SUM 2 ,..., SUM k-1 are shipped back to the user and polynomial P SUM (x) is built using Equation 2 using k points

(x i , SUM i ) i=1,...,k-1 , (x k , q j=1 v k = q × v k ).
P SUM (x)'s constant term is SUM. S4 does not alter the number of records. Hence, COUNT queries can be processed normally, thus also allowing AVG queries.

Security Analysis

Paillier's PHE is semantically secure, but it is too expensive in terms of ciphertext storage space and query response time. S4 proposes a classical trade-off with a lower level of security, but better storage and response time efficiency [START_REF] Shamir | How to share a secret[END_REF]. Let us consider a scenario where the CSP is said honest but curious, which is a widely used adversary model for cloud data outsourcing [START_REF] Sanamrad | Query log attack on encrypted databases[END_REF]. Such a CSP faithfully complies to any service-level agreement and, in our particular case, stores data, runs queries and provides results without alteration, malicious or otherwise. Yet, the CSP may access data and infer information from queries and results.

Privacy in S4 relies on the fact that a secret value is only retrievable by the user via private key K. As in secret sharing, it is indeed guaranteed that at least k splits and X are necessary to reconstruct a secret, while the CSP has access to only k -1 splits. Both X and the k th split, i.e., K, are stored at the user's. However, the CSP still has access to linear combinations of splits (Equation 2), which provide some information. Still, the higher k is, the more difficult it is to interpret linear combinations of splits. Thus, k is the prime security parameter in S4. Experiments in Section 4 provide hints for choosing k.

Moreover, if some secrets are known by the CSP, e.g., through public communication of a company to its shareholders, solving Equation 3 becomes possible. For example, if the CSP knows secrets v 1 , ..., v k-1 . Also knowing the corresponding splits v 1,j , ..., v k-1,j ∀j ∈ [1, k -1], the CSP can recover the Lagrange basis polynomials i (0) ∀i ∈ [1, k] and solve Equation 3 to recover all secrets. However, the CSP must know at least k -1 secrets to do so. Moreover, we also propose leads to address this problem in Section 5.

Experimental Evaluation

Experimental Setup

We implement S4 in C using compiler gcc 4.8.2. S4's source code is freely available on-line 1 . Experiments related to Paillier's PHE exploit the libpaillier standard C library [START_REF] Bethencourt | Paillier library[END_REF]. All mathematical computations use the GNU Multiple Precision Arithmetic Library (GMP) [START_REF]Free Software Foundation: GNU Multiple Precision Arithmetic library[END_REF]. Eventually, we conduct our experiments on an Intel Core i7 3.10 GHz PC with 16 GB of RAM running Linux Ubuntu 15.05.

1 http://eric.univ-lyon2.fr/download/libS4.zip

We compare S4 and Paillier's PHE using simple synthetic datasets, i.e., 32bit unsigned integers generated uniformly at random from the integer range [10 3 , 10 4 [. We scale up the number of records m such that m ∈ (10 3 , 10 4 , 10 5 , 10 6 ), forming four distinct datasets.

In S4, we vary k from 8 to 64, higher values of k inducing too long execution times. Prime p must be greater than the greatest query answer, e.g., p > m j=1 v j . In Paillier's PHE, we use a key size of 1024 bits, which induces ciphertexts of 2048 bits. Such key size is the absolute minimum to achieve security [START_REF] Jost | Encryption Performance Improvements of the Paillier Cryptosystem[END_REF][START_REF] Yildizli | Distributed privacy preserving clustering via homomorphic secret sharing and its application to (vertically) partitioned spatio-temporal data[END_REF].

Encryption and Decryption Time

Figure 2 plots the time secret splitting in S4 and secret encryption in Paillier's scheme with respect to m. It shows that encryption time in S4 is lower than Paillier's when k ≤ 16, and then becomes higher when k ≥ 16. Secret splitting consists in building a random polynomial by randomly choosing k -2 points. Hence, splitting time increases with k. Figure 2 actually illustrates the tradeoff between S4's security and encryption efficiency with respect to Paillier's PHE.

Figure 3 plots the time of secret reconstruction in S4 and secret decryption in Paillier's scheme with respect to m. With the selected values of k, decryption is faster with S4 than with Paillier's PHE. This is mainly because Paillier's scheme needs m expensive modular multiplications of large, 2048-bit numbers for decryption, while secret reconstruction in S4 works by polynomial interpolation over k points and evaluating the polynomial in one single point.

Conclusion

In this paper, we introduce S4, a new cryptographic scheme that supports summation queries in cloud-based OLAP. We experimentally show that S4 is much more efficient than Paillier's PHE in terms of query response time and space overhead. Thus, replacing Paillier's scheme with S4 in secure DBMSs such as CryptDB and MONOMI can improve analytical query processing in cloud DWs. Moreover, we also plan a variant of S4 for computing multiplications. However, we achieve performance gains through a slight degradation of security, especially when an adversary has knowledge of secret values. Although it is definitely acceptable in some cloud DW and OLAP scenarios, e.g., public aggregate data might not actually yield secrets, i.e., fine-grained data, we will devote future research to strengthen S4 against such threats. More precisely, we plan to introduce noise, as in many cryptographic problems such as approximate-GCD [START_REF] Liu | Practical Fully Homomorphic Encryption without Noise Reduction[END_REF] or LWE [START_REF] Regev | On lattices, learning with errors, random linear codes, and cryptography[END_REF]. For instance, instead of sharing v j , we could share 10 r × v j + noise. By doing so, security is intuitively enhanced while the whole process remains correct, provided r is sufficiently large and noise sufficiently small.

Fig. 1 :

 1 Fig. 1: S4 secret splitting

Figure 4 Figure 5 Fig. 5 :

 455 Fig. 2: Splitting/encryption time