R Alexandre

Y Morimoto
email: morimoto@math.h.kyoto-u.ac.jp

S Ukai
email: ukai@kurims.kyoto-u.ac.jp

AND C.-J Xu

T Yang

BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF IN THE WHOLE SPACE: III, QUALITATIVE PROPERTIES OF SOLUTIONS

Keywords: Mathematics Subject Classification. 35A02, 35B40, 35B65, 35H10, 35S05, 35Q20, 82B40 Boltzmann equation, non-cutoff cross sections, hypoellipticity, uniqueness, non-negativity, convergence rate

. Let us just recall herein that the Boltzmann bilinear collision operator is given by

where f ′ * = f (t, x, v ′ * ), f ′ = f (t, x, v ′ ), f * = f (t, x, v * ), f = f (t, x, v), and for σ ∈ S 2 , the pre-and post-collisional velocities are linked by the relations

v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ .
The non-negative cross section B(z, σ) depends only on |z| and the scalar product z |z| • σ. As in the previous parts, we assume that it takes the form

B(|v -v * |, cos θ) = Φ(|v -v * |)b(cos θ), cos θ = v -v * |v -v * | • σ , 0 ≤ θ ≤ π 2 ,
where

(1.2) Φ(|z|) = Φ γ (|z|) = |z| γ , b(cos θ) ≈ θ -2-2s when θ → 0+,
for some γ > -3 and 0 < s < 1.

In the present work, we are concerned with qualitative properties of classical solutions to the Boltzmann equation, under the previous assumptions. By qualitative properties, we mean specifically regularization properties, positivity, uniqueness of solutions and asymptotic trend to global equilibrium.

Let us recall that in a close to equilibrium framework, the existence of such classical solutions was proven in our series of papers [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] and using a different method, by Gressmann and Strain [START_REF] Gressman | Global classical solutions of the Boltzmann with long-range interactions[END_REF][START_REF] Gressman | Global strong solutions of the Boltzmann equation without angular cut-off[END_REF][START_REF] Gressman | Global classical solutions of the Boltzmann equation with long-range interactionsss and soft potentials[END_REF]. We refer also to [START_REF] Alexandre | Bounded solutions of the Boltzmann equation in the whole space[END_REF] for bounded local solutions.

The first qualitative property which will be addressed here is concerned with regularization properties of classical solutions, that is, the immediate smoothing effect on the solution. For the homogeneous Boltzmann equation, after the works of Desvillettes [START_REF] Desvillettes | About the regularization properties of the non cut-off Kac equation[END_REF][START_REF] Desvillettes | Regularization for the non-cutoff 2D radially symmetric Boltzmann qquation with a velocity dependant cross section[END_REF][START_REF] Desvillettes | Regularization properties of the 2-dimensional non radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules[END_REF], this issue has now a long history [START_REF] Alexandre | Littlewood-Paley decomposition and regularity issues in Boltzmann homogeneous equations.I. Non cutoff and Maxwell cases[END_REF][START_REF] Alexandre | Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules[END_REF][START_REF] Chen | The Gevrey hypoellipticity for a class of kinetic equations[END_REF][START_REF] Desvillettes | Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff[END_REF][START_REF] Huo | Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff[END_REF][START_REF] Morimoto | Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff[END_REF][START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF][START_REF] Ukai | Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff[END_REF]. All these works deal with smoothed type kinetic part for the cross sections, which therefore rules out the more physical assumption above, that is, including the singular behavior for relative velocity near 0. We refer the reader to our forthcoming work [START_REF] Alexandre | Smoothing effect for spatially homogeneous Boltzmann equation[END_REF] for this issue.

Regularization effect for the inhomogeneous Boltzmann equation was studied in our previous works [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF][START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF], but for Maxwellian type molecules or smoothed kinetic parts for the cross section. Nevertheless, we have introduced many technical tools, some of which are helpful for tackling the singular assumption above. In particular, by improving the pseudo-differential calculus and functional estimates from [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF][START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF], we shall be able to prove our regularity result.

We shall use the following standard weighted Sobolev space defined, for k, ℓ ∈ R, as

H k ℓ = H k ℓ (R 3 v ) = { f ∈ S ′ (R 3 v ); W ℓ f ∈ H k (R 3 v )} and for any open set Ω ⊂ R 3 x H k ℓ (Ω × R 3 v ) = { f ∈ D ′ (Ω × R 3 v ); W ℓ f ∈ H k (Ω × R 3 v )} where W ℓ (v) = v ℓ = (1 + |v| 2 ) ℓ/2
is always the weight for v variables. Herein, (•, •) L 2 = (•, •) L 2 (R 3 v ) denotes the usual scalar product in L 2 = L 2 (R 3 ) for v variables. Recall that L 2 ℓ = H 0 ℓ . Theorem 1.1. Assume (1.2) holds true, with 0 < s < 1, γ > max{-3, -3/2 -2s}, 0 < T ≤ +∞. Let Ω be an open domain of R 3

x . Let f ∈ L ∞ ([0, T ]; H 5 ℓ (Ω × R 3 )), for any ℓ ∈ N, be a solution of Cauchy problem (1.1). Moreover, assume that f satisfies the following local coercivity estimate : for any compact K ⊂ Ω and 0 < T 1 < T 2 < T , there exist two constants η 0 > 0, C 0 > 0 such that

(1.3) -(Q( f, h), h) L 2 (R 7 ) ≥ η 0 h 2 H s γ/2 (R 7 ) -C 0 h 2 L 2 γ/2+s (R 7 )
for any h ∈ C 1 0 (]T 1 , T 2 [; C ∞ 0 (K; H +∞ ℓ (R 3 ))). Then we have f ∈ C ∞ (]0, T [×Ω; S(R 3 )) .

Classical solutions satisfying such a local coercivity estimate do exist [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF], see Corollary 2.15 in next section.

Our next result is related to uniqueness of solutions. We shall consider function spaces with exponential decay in the velocity variable, for m ∈ R Ẽm 0 (R 6 ) = g ∈ D ′ (R 6 x,v ); ∃ ρ > 0 s.t. e ρ<v> 2 g ∈ L ∞ (R 3

x ; H m (R 3 v )) ,

and for T > 0 Ẽm ([0, T ] × R 6 x,v ) = f ∈ C 0 ([0, T ]; D ′ (R 6 x,v )); ∃ ρ > 0 s.t. e ρ v 2 f ∈ L ∞ ([0, T ] × R 3 x ; H m (R 3 v )) .
Theorem 1.2. Assume that 0 < s < 1 and max{-3, -3/2 -2s} < γ < 2 -2s. Let f 0 ≥ 0 and f 0 ∈ Ẽ0 0 (R 6 ). Let 0 < T < +∞ and suppose that f ∈ Ẽ2s ([0, T ] × R 6

x,v ) is a non-negative solution to the Cauchy problem (1.1). Then any solution in the function space Ẽ2s ([0, T ] × R 6

x,v ) coincides with f . Remark.

1) Note that the solutions considered above are not necessarily classical ones. Moreover, Theorem 1.2 does not require the coercivity. On the other hand, if we suppose the coercivity, then we can get the uniqueness in the function space Ẽs ([0, T ] × R 6

x,v ), without the non-negativity assumption, see precisely Theorem 4.1 in Section 4. 2) We can also remove the restriction γ + 2s < 2, if we consider the small perturbation around Maxwellian, see precisely Theorem 4.3 in Section 4.

3) Finally, in the soft potential case γ + 2s ≤ 0, we can refine the above uniqueness results which can be applied to the solution of Theorem 1.4 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], see precisely Theorem 4.4 in Section 4.

Our next issue is about the non-negativity of solutions. We shall use the following modified weighted Sobolev spaces: For k ∈ N, ℓ ∈ R

Hk ℓ (R 6 ) = f ∈ S ′ (R 6 x,v ) ; f 2 Hk ℓ (R 6 ) = |α|+|β|≤N Wℓ-|β| ∂ α β f 2 L 2 (R 6 ) < +∞ ,
where Wℓ = (1 + |v| 2 ) |s+γ/2| ℓ/2 . Combining with the existence results of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] and the above Theorem 1.2, one has Theorem 1.3. Let 0 < s < 1, γ > max{-3, -3/2 -2s}, k ≥ 6. There exist ε 0 > 0 and ℓ 0 such that the Cauchy problem (1.1) admits a unique global solution f = µ + µ 1/2 g for initial datum f 0 = µ + µ 1/2 g 0 satisfying 1) g ∈ L ∞ ([0, +∞[; H k ℓ 0 (R 6 ))), if γ + 2s > 0 and g 0 H k ℓ 0 (R 6 ) ≤ ε 0 . 2) g ∈ L ∞ ([0, +∞[; Hk ℓ 0 (R 6 )), if γ + 2s ≤ 0 and g 0 Hk ℓ 0 (R 6 ) ≤ ε 0 . If f 0 = µ + µ 1/2 g 0 ≥ 0, then the above solution f = µ + µ 1/2 g ≥ 0.

Remark. The existence of global solution was proved in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF], while the uniqueness follows from Theorem 1.2, more precisely Theorem 4.3,in Section 4. One of the basic issues in the mathematical theory for Boltzmann equation theory is about the convergence of solutions to equilibrium. This topic has been recently renewed and complemented by proofs of optimal convergence rates in the whole space, see for example [START_REF] Duan | Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications[END_REF][START_REF] Glassey | The Cauchy problem in kinetic theory[END_REF][START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF][START_REF] Yang | Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space[END_REF] and references therein. This is closely related to the study of the hypocoercivity theory that is about the interplay of a conservative operator and a degenerate diffusive operator which gives the convergence to the equilibrium. Note that this kind of interplay also gives the full regularization.

For later use, denote

N = span{µ 1 2 , µ 1
2 v i , µ 1
2 |v| 2 , i = 1, 2, 3}, as the null space of the linearized Boltzmann collision operator, and P the projection operator to N in L 2 (R 3 v ). For the problem considered in this paper, we have the following convergence rate estimates. Theorem 1.4. Let 0 < s < 1 and f = µ + µ 1/2 g be a global solution of the Cauchy problem (1.1) with initial datum f 0 = µ + µ 1/2 g 0 . We have the following two cases: 1) Let γ + 2s > 0, N ≥ 6, ℓ > 3/2 + 2s + γ. There exists

ε 0 > 0 such that if g 0 2 L 1 (R 3 x ;L 2 (R 3 v )) + g 0 2 H N ℓ (R 6 ) ≤ ε 0 and g ∈ L ∞ ([0, +∞[ ; H N ℓ (R 6
)), then we have for all t > 0, g(t) (1 + t) -5/2 .

2) Let max{-3, -3 2 -2s} < γ ≤ -2s, N ≥ 6, ℓ ≥ N + 1. There exists ε 0 > 0 such that if g 0 2 H N ℓ (R 6 ) ≤ ε 0 and g ∈ L ∞ ([0, +∞[ ; H N ℓ (R 6 )), then we have for all t > 0, sup

x∈R 3 g(t) 2 H N-3 (R 3 v ) (1 + t) -1 .
We emphasize that the above convergence rate for the hard potential case is optimal in the sense that it is the same for the linearized problem through either spectrum analysis in [START_REF] Pao | Boltzmann collision operator with inverse power intermolecular potential, I[END_REF], or direct Fourier transform using the compensating function introduced in [START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF]. However, the convergence rate for soft potential is not optimal. In fact, how to obtain an optimal convergence rate even for the cutoff soft potential is still an unsolved problem [START_REF] Strain | Exponential decay for soft potentials near Maxwellian[END_REF][START_REF] Ukai | On the Cauchy problem for the Boltzmann equation with a soft potential[END_REF].

We also would like to mention that the above convergence rate is for the whole space setting. If the problem is instead considered on the torus with small perturbation, then the exponential decay for hard potential can be obtained, and this point is a direct consequence of the energy estimates given in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] by using Poincaré inequality (this is for example the case considered in [START_REF] Gressman | Global strong solutions of the Boltzmann equation without angular cut-off[END_REF]).

Before presenting the plan of the paper we want to give some comments on our proofs. First of all, our proof of regularization property applies to the classical solutions obtained in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF]. Note that from those existence theorems, one can show that if the initial data satisfying g 0 H k l ≤ ǫ k for k ≥ 6 and l ≥ l 0 for some l 0 , the solution is also in H k when ǫ k is small. However, the current existence theory does not yield that g ∈ H k+N , under the condition that g 0 ∈ H k+N for N > 0 if ||g 0 || H k+N is not small. Therefore, we can not just mollify the initial data to study the full regularity by working formally on the smooth solution. Instead, we need analytic tools from peudo-differential theory and harmonic analysis to study the gain of regularity rigorously. In fact, it is a standard technic for the hypoellipticity of linear differential operators [START_REF] Hörmander | The analysis of Linear Partial Differential Operators III[END_REF][START_REF] Morimoto | Hypoelliticity for a class of kinetic equations[END_REF][START_REF] Olejnik | Second order equations with non-negative characteristic form[END_REF]. The same comments apply for the uniqueness and positivity issues for which we give also rigorous proofs.

The paper is organized as follows. In Section 2, we give the functional analysis of the collision operator, including upper bounds, commutators estimates and coercivity. In Section 3, we prove Theorem 1.1 giving the regularization of solutions. Section 4 is devoted to precise versions of uniqueness results related to Theorem 1.2, while Section 5 proves the non-negativity of solutions. Finally the last Section proves Theorem 1.4 about the convergence of solutions to equilibrium.

Notations: Herein, letters f , g, • • • stand for various suitable functions, while C, c, • • • stand for various numerical constants, independent from functions f , g, • • • and which may vary from line to line. Notation A B means that there exists a constant C such that A ≤ CB, and similarly for A B. While A ∼ B means that there exist two generic constants C 1 , C 2 > 0 such that

C 1 A ≤ B ≤ C 2 A.

Functional analysis of the collision operator

In this section, we study the upper bound and commutators estimates for the collision operator Q( •, • ). Since it is only an operator with respect to velocity variable, in this section, our analysis is on R 3 v , forgetting variable x. In what follows, we denote Φγ by Φγ (z) = (1 + |z| 2 ) γ/2 . Q Φγ will denote the collision operator defined with the modified kinetic factor Φγ .

2.1. Upper bound estimate. For 0 < s < 1, γ ∈ R, we proved the following upper bounded estimate (Theorem 2.1 of [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF])

(2.1) |(Q Φγ ( f, g), h)| || f || L 1 ℓ + +(γ+2s) + ||g|| H m+s (ℓ+γ+2s) + h H s-m -ℓ
, for any m, ℓ ∈ R, and the estimate of commutators with weight (Lemma 2.4 of [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF])

(2.2) W ℓ Q Φγ ( f, g) -Q Φγ ( f, W ℓ g), h f L 1 ℓ+(2s-1) + +γ + g H (2s-1+ǫ) + ℓ+(2s-1) + +γ + h L 2 ,
for any ℓ ∈ R.

For the singular type of kinetic factors considered herein |vv * | γ , we need to take into account the singular behavior close to 0. Therefore, we decompose the kinetic factor in two parts. Let 0 ≤ φ(z) ≤ 1 be a smooth radial function with value 1 for z close to 0, and 0 for large values of z. Set

Φ γ (z) = Φ γ (z)φ(z) + Φ γ (z)(1 -φ(z)) = Φ c (z) + Φ c(z).
And then correspondingly we can write

Q( f, g) = Q c ( f, g) + Q c( f, g),
where the kinetic factor in the collision operator is defined according to the decomposition respectively. Since Φ c(z) is smooth, and Φ c(z) ≤ Φγ (z), Q c( f, g) has similar properties as for Q Φγ ( f, g) as regards upper bounds and commutators estimatations, which means that (2.1) and (2.2) hold true for Q c( f, g).

From now on, we concentrate on the study the singular part Q c ( f, g), referring for the smooth part Q c( f, g) to [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF]. Note that in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], the same decomposition was also used, but for the modified operator Γ( f, g). Here, the absence of the gaussian factor slightly adds some more difficulties.

Proposition 2.1. Let 0 < s < 1, γ > max{-3, -2s -3/2} and m ∈ [s -1, s]. Then we have |(Q c ( f, g), h)| f L 2 ||g|| H s+m h H s-m .
Remark 2.2. As will be clearer from the proof below, the following precise estimates are also available: if

γ + 2s > 0, we have |(Q c ( f, g), h)| || f || L 1 ||g|| H s+m h H s-m .
and moreover if γ + 2s > -1, we have

|(Q c ( f, g); h)| f L 3/2 ||g|| H s+m h H s-m .
For the proof of Proposition 2.1, we shall follow some of the arguments form [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF]. First of all, by using the formula from the Appendix of [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF], and as in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], one has

(Q c ( f, g), h) = b ξ |ξ| • σ [ Φc (ξ * -ξ -) -Φc (ξ * )] f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dξdξ * dσ. = |ξ -|≤ 1 2 ξ * • • • dξdξ * dσ + |ξ -|≥ 1 2 ξ * • • • dξdξ * dσ =A 1 ( f, g, h) + A 2 ( f, g, h) .
Then, we write A 2 ( f, g, h) as

A 2 = b ξ |ξ| • σ 1 |ξ -|≥ 1 2 ξ * Φc (ξ * -ξ -) f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dξdξ * dσ. - b ξ |ξ| • σ 1 |ξ -|≥ 1 2 ξ * Φc (ξ * ) f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dξdξ * dσ = A 2,1 ( f, g, h) -A 2,2 ( f, g, h) .
While for A 1 , we use the Taylor expansion of Φc at order 2 to have

A 1 = A 1,1 ( f, g, h) + A 1,2 ( f, g, h)
where G,H) is the remaining term corresponding to the second order term in the Taylor expansion of Φc . The A i, j with i, j = 1, 2 are estimated by the following lemmas.

A 1,1 = b ξ -• (∇ Φc )(ξ * )1 |ξ -|≤ 1 2 ξ * f (ξ * ) ĝ(ξ -ξ * ) h(ξ)dξdξ * dσ, and A 1,2 (F,
Lemma 2.3. We have |A 1,1 | + |A 1,2 | f L 2 || f || H s+m h H s-m .
Proof. Considering firstly A 1,1 , by writing

ξ -= |ξ| 2 ξ |ξ| • σ ξ |ξ| -σ + 1 - ξ |ξ| • σ ξ 2 ,
we see that the integral corresponding to the first term on the right hand side vanishes because of the symmetry on S 2 . Hence, we have

A 1,1 = R 6 K(ξ, ξ * ) f (ξ * ) ĝ(ξ -ξ * ) h(ξ)dξdξ * ,
where

K(ξ, ξ * ) = S 2 b ξ |ξ| • σ 1 - ξ |ξ| • σ ξ 2 • (∇ Φc )(ξ * )1 |ξ -|≤ 1 2 ξ * dσ . Note that |∇ Φc (ξ * )| 1 ξ * 3+γ+1 , from the Appendix of [9]. If √ 2|ξ| ≤ ξ * , then |ξ -| ≤ ξ * /2
and this imply the fact that 0 ≤ θ ≤ π/2, and we have

|K(ξ, ξ * )| π/2 0 θ 1-2s dθ ξ ξ * 3+γ+1 1 ξ * 3+γ ξ ξ * .
On the other hand, if

√ 2|ξ| ≥ ξ * , then |K(ξ, ξ * )| π ξ * /(2|ξ|) 0 θ 1-2s dθ ξ ξ * 3+γ+1 1 ξ * 3+γ ξ ξ * 2s-1 .
Hence we obtain

|K(ξ, ξ * )| 1 ξ * 3+γ        ξ ξ * 1 √ 2|ξ|≤ ξ * + 1 √ 2|ξ|≥ ξ * ≥|ξ|/2 + ξ ξ * 2s 1 ξ * ≤|ξ|/2        . (2.3) Notice that (2.4)            ξ ξ * ∼ ξ -ξ * on supp 1 ξ * ≥ √ 2|ξ| ξ ∼ ξ -ξ * on supp 1 ξ * ≤|ξ|/2 ξ ∼ ξ * ξ -ξ * on supp 1 √ 2|ξ|≥ ξ * ≥|ξ|/2 .
Replacing the factors ξ / ξ * and ( ξ / ξ * ) 2s on the right hand side of (2.3) by

ξ -ξ * ξ * s+m ξ ξ * s-m and ξ s+m ξ -ξ * s-m ξ * 2s
, respectively, we obtain

|K(ξ, ξ * )| ξ s-m ξ -ξ * s+m ξ * 3+γ+2s + 1 ξ-ξ * ξ * ξ * 3+γ+s-m ξ -ξ * s+m ξ s-m ξ -ξ * s+m . (2.5)
Putting g(ξ) = ξ s+m ĝ(ξ), h(ξ) = ξ s-m ĥ(ξ), we have by the Cauchy-Schwarz inequality

|A 1,1 | 2 R 6 | f (ξ * )| ξ * 3+γ+2s | g(ξ -ξ * )| 2 dξdξ * R 6 | f (ξ * )| ξ * 3+γ+2s | h(ξ)| 2 dξdξ * + R 6 | f (ξ * )| 2 ξ * 6+2γ+2s-2m 1 ξ-ξ * ξ * ξ -ξ * 2s+2m dξdξ * R 6 | g(ξ -ξ * )| 2 | h(ξ)| 2 dξdξ * = AB + DE .
Since ξ * -(3+γ+2s) ∈ L 2 , the Cauchy-Schwarz inequality again shows

A R 3 | f (ξ * )| ξ * 3+γ+2s dξ * g H s+m f L 2 g H s+m , B f L 2 h H s-m .
Note that

1 ξ-ξ * ξ * ξ -ξ * 2s+2m dξ          1 ξ * -3+2s+2m if s + m < 3/2 log ξ * if s + m ≥ 3/2 . Since 3 + 2(γ + 2s) > 0 and 6 + 2γ + 2(s -m) > 0, we get D ≤ f 2 L 2 , which concludes the desired bound for A 1,1 . Remark that if γ + 2s > 0 then we obtain |A 1,1 | F L 1 G H s+m H H s-m because F L ∞ ≤ F L 1 . If 0 ≥ γ + 2s > -3/2 then we can just estimate |A 1,1 | F L 2 G H s+m H H s-m . If 0 ≥ γ + 2s > -1 then |A 1,1 | F L 3/2 G H s H H s .
Those follow from the Hölder inequality and F L p ≤ F L q with 1/p + 1/q = 1. Now we consider A 1,2 ( f, g, h), which comes from the second order term of the Taylor expansion. Note that

A 1,2 = b ξ |ξ| • σ 1 0 dτ(∇ 2 Φc )(ξ * -τξ -) • ξ -• ξ -F(ξ * ) Ĝ(ξ -ξ * ) H(ξ)dσdξdξ * .
Again from the Appendix of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], we have

|(∇ 2 Φc )(ξ * -τξ -)| 1 ξ * -τξ -3+γ+2 1 ξ * 3+γ+2 , because |ξ -| ≤ ξ * /2. Similar to A 1,1 , we can obtain |A 1,2 | R 6 K(ξ, ξ * ) f (ξ * ) ĝ(ξ -ξ * ) h(ξ)dξdξ * ,
where K(ξ, ξ * ) has the following upper bound

K(ξ, ξ * ) min(π/2, π ξ * /(2|ξ|)) 0 θ 1-2s dθ ξ 2 ξ * 3+γ+2 (2.6) 1 ξ * 3+γ        ξ ξ * 2 1 √ 2|ξ|≤ ξ * + 1 √ 2|ξ|≥ ξ * ≥|ξ|/2 + ξ ξ * 2s 1 ξ * ≤|ξ|/2        ,
from which we obtain the same inequality as (2.5) for K(ξ, ξ * ). Hence we obtain the desired bound for A 1,2 .

And this completes the proof of the lemma.

Lemma 2.4. We have also

|A 2,1 | + |A 2,2 | f L 2 || f || H s+m h H s-m . Proof. In view of the definition of A 2,2 , the fact that |ξ| sin(θ/2) = |ξ -| ≥ ξ * /2 and θ ∈ [0, π/2] imply √ 2|ξ| ≥ ξ * .
We can then directly compute the spherical integral appearing inside A 2,2 together with Φ as follows:

b ξ |ξ| • σ Φ(ξ * )1 |ξ -|≥ 1 2 ξ * dσ 1 ξ * 3+γ ξ 2s ξ * 2s 1 √ 2|ξ|≥ ξ * (2.7) ξ s-m ξ -ξ * s+m ξ * 3+γ+2s + 1 ξ-ξ * ξ * ξ * 3+γ+s-m ξ -ξ * s+m ξ s-m ξ -ξ * s+m ,
which yields the desired estimate for A 2,2 .

We now turn to

A 2,1 = b 1 |ξ -|≥ 1 2 ξ * Φc (ξ * -ξ -) f (ξ * ) ĝ(ξ -ξ * ) h(ξ)dσdξdξ * . Firstly, note that we can work on the set |ξ * • ξ -| ≥ 1 2 |ξ -| 2 .
In fact, on the complementary of this set, we have

|ξ * • ξ -| ≤ 1 2 |ξ -| 2 so that |ξ * -ξ -| |ξ * |
, and in this case, we can proceed in the same way as for A 2,2 . Therefore, it suffices to estimate

A 2,1,p = b 1 |ξ -|≥ 1 2 ξ * 1 |ξ * • ξ -|≥ 1 2 |ξ -| 2 Φc (ξ * -ξ -) f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dσdξdξ * . By 1 = 1 ξ * ≥|ξ|/2 1 ξ-ξ * ≤ ξ * -ξ -+ 1 ξ * ≥|ξ|/2 1 ξ-ξ * > ξ * -ξ -+ 1 ξ * <|ξ|/2
we decompose

A 2,1,p = A (1) 2,1,p + A (2) 2,1,p + A (3) 2,1,p .
On the sets for above integrals, we have

ξ * -ξ - ξ * , because |ξ -| |ξ * | that follows from |ξ -| 2 ≤ 2|ξ * • ξ -| |ξ -| |ξ * |.
Furthermore, on the sets for A (1) 2,1,p and A (2) 2,1,p we have ξ ∼ ξ * , so that sup b

1 |ξ -|≥ 1 2 ξ * 1 ξ * ≥|ξ|/2 1 |ξ -|≤|ξ|/ √ 2 and ξ * -ξ -
ξ . Hence we have, in view of sm ≥ 0,

|A (1) 2,1,p | 2 | Φc (ξ * -ξ -)| 2 | f (ξ * )| 2 ξ * -ξ -2s-2m 1 ξ-ξ * ≤ ξ * -ξ - ξ -ξ * 2s+2m dξdξ * dσ × | ξ -ξ * s+m ĝ(ξ -ξ * )| 2 | ξ s-m ĥ(ξ)| 2 dσdξdξ * .
If γ + 2s > 0 then by the change of variables ξ *ξ -→ u we have

|A (1) 2,1,p | 2 F 2 L ∞ u -(6+2γ+2s-2m) 1 w ≤ u w 2s+2m dwdu G 2 H s+m H 2 H s-m F 2 L 1 G 2 H s+m H 2 H s-m .
If γ + 2s > -3/2 then with u = ξ *ξ -we have

|A (1) 2,1,p | 2 | f (ξ * )| 2 sup u u -(6+2γ+2s-2m) 1 ξ + -u ≤ u ξ + -u 2s+2m dξ + dξ * g 2 H s+m h 2 H s-m f 2 L 2 g 2 H s+m h 2 H s-m , because dξ ∼ dξ + on the support of 1 |ξ -|≤|ξ|/ √ 2 .
In the case γ +2s > -1, by the Hölder inequality and the change of variables u = ξ *ξ -we have

|A (1) 2,1,p | 2 | f (ξ * )| 3 dξ * 2/3 ×        u -(6+2γ+2s-2m) 1 ξ + -u ≤ u ξ + -u 2s+2m dξ + 3 du        1/3 g 2 H s+m h 2 H s-m f 2 L 3/2 g 2 H s+m h 2 H s-m .
As for A (2) 2,1,p we have by the Cauchy-Schwarz inequality

|A (2) 2,1,p | 2 | Φc (ξ * -ξ -)|| f (ξ * )| ξ * -ξ -2s | ξ -ξ * s+m ĝ(ξ -ξ * )| 2 dσdξdξ * × | Φc (ξ * -ξ -)|| f (ξ * )| ξ * -ξ -2s | ξ s-m ĥ(ξ)| 2 dσdξdξ * .
Since we have

| Φc (ξ * -ξ -)|| f (ξ * )| ξ * -ξ -2s dξ * dσ          f L 1 if γ + 2s > 0 f L 2 if γ + 2s > -3/2 , f L 3/2 if γ + 2s > -1 ,
we have the desired estimates for A (2) 2,1,p . On the set A (3) 2,1,p we have ξ ∼ ξξ * . Hence

|A (3) 2,1,p | 2 b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)|| f (ξ * )| ξ 2s | ξ -ξ * s+m ĝ(ξ -ξ * )| 2 dσdξdξ * × b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)|| f (ξ * )| ξ 2s | ξ s-m ĥ(ξ)| 2 dσdξdξ * .
We use the change of variables in ξ

* , u = ξ * -ξ -. Note that |ξ -| ≥ 1 2 u + ξ -implies |ξ -| ≥ u / √ 10. If γ + 2s > 0 then we have b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)|| f (ξ * )| ξ 2s dσdξ * f L ∞ |ξ| u 2s u -(3+γ) ξ -2s du f L ∞ .
On the other hand, if γ + 2s > -3/2 ( or 0 ≥ γ + 2s > -1 ) then this integral is upper bounded by

b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)| ξ 2s/p ξ * -ξ -2s/q ξ * 2s/q | f (ξ * )| ξ 2s/q dσdξ * ≤ b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)| p ξ 2s ξ * -ξ -2sp/q dσdξ * 1/p b 1 |ξ -|≥ 1 2 ξ * ξ * 2s | f (ξ * )| q ξ 2s dσdξ * 1/q ≤ b 1 |ξ -| u | Φc (u)| p ξ 2s u 2sp/q dσdu 1/p f L q du u p(3+γ+2s) f p ,
where 1/p + 1/q = 1, p = 2 ( or p = 3/2). Hence we also obtain the desired estimates for A (3) 2,1,p . The proof of the lemma is complete Proposition 2.1 is then a direct consequence of Lemmas 2.3 and 2.4, while the statements of Remark 2.2 are mentioned in the proof of the two previous lemmas.

Estimate of commutators with weights.

The following estimation on commutators will now be proved. Because of the weight loss related to the Bolzmann equation, test functions involve these weights, and therefore, this estimation is quite necessary.

Proposition 2.5. Let 0 < s < 1, γ > max{-3, -2s -3/2}. For any ℓ, β, δ ∈ R W ℓ Q c ( f, g) -Q c ( f, W ℓ g), h f L 2 ℓ-1-β-δ g H (2s-1+ǫ) + β ||h|| L 2 δ .
The next two lemmas are a preparation for the complete proof of this Proposition.

Lemma 2.6. If λ < 3/2 then |v-v * |≤1 | f (v * )| |g(v)| 2 |v -v * | λ dvdv * f L 2 g 2 L 2 . (2.8) If 3/2 < λ < 3 then |v-v * |≤1 | f (v * )| |g(v)| 2 |v -v * | λ dvdv * f L 2 g H λ 2 -3 4 .
Proof. Since |v * | -λ 1 |v * |≤1 ∈ L 2 for λ < 3/2, it follows from the Cauchy-Schwarz inequality that if λ < 3/2 then

|v-v * |≤1 | f * | |g| 2 |v -v * | λ dvdv * ≤ |g| 2 |v-v * |≤1 |v -v * | -2λ dv * 1/2 | f * | 2 dv * 1/2 dv f L 2 g 2 L 2 . It follows from the Hardy-Littlewood-Sobolev inequality that if 3/2 < λ < 3 then |v-v * |≤1 | f * | |g| 2 |v -v * | λ dvdv * f L 2 g 2 L p with 1 p = 3 2 - λ 3 < 1 f L 2 g 2 H λ 2 -3 4
because of the Sobolev embedding theorem.

Lemma 2.7. Let 0 < s < 1 and γ > max{-3, -2s -3/2}. Then bΦ γ c | f (v * )||g(v) -g(v ′ )| 2 dvdv * dσ f L 2 g 2 H s .
Proof. Note that

Q Φ γ c (| f |, g), g = - 1 2 bΦ γ c | f * ||g -g ′ | 2 dvdv * dσ + 1 2 bΦ γ c | f * | g ′2 -g 2 dvdv * dσ .
Since Proposition 2.1 with m = 0 is applicable to the left hand side, it suffices to consider the second term of the right hand side. It follows from the cancellation lemma of [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] (more precisely the formula (29) there) that

bΦ γ c | f * | g ′2 -g 2 dvdv * dσ = | f (v * )|S (v -v * )g(v) 2 dvdv * ,
where

S (v -v * ) =Φ γ (v -v * )φ(v -v * ) 2π π/2 0 b(cos θ) sin θ 1 cos 3+γ+1 (θ/2) -1 dθ + Φ γ (v -v * ) 2π π/2 0 b(cos θ) sin θ cos 3+γ+1 (θ/2) φ v -v * cos(θ/2) -φ(v -v * ) dθ .
The integral of the second term on the right hand side can be written as φ(vv * ) whose support is contained in

{0 < |v -v * | 1}. Since s > -γ/2 -3/4
, the estimation for the first term just follows from Lemma 2.6 because the case γ = -3/2 can be treated as γε for any small ε > 0.

Proof of Proposition 2.5. We write

W ℓ Q c ( f, g) -Q c ( f, W ℓ g), h = bΦ c f ′ * g ′ W ℓ (v) -W ℓ (v ′ ) hdvdv * dσ = bΦ c W ℓ (v ′ ) -W ℓ (v) f * g ′ h ′ dvdv * dσ + bΦ c W ℓ (v ′ ) -W ℓ (v) f * g -g ′ h ′ dvdv * dσ =J 1 + J 2 . Set v τ = v + τ(v ′ -v) for τ ∈ [0, 1] and notice that |W ℓ (v ′ ) -W ℓ (v)| 1 0 W ℓ-1 (v τ )dτ|v -v * | sin(θ/2) .
On the support of φ(vv * ) we have for a large C > 0

v * 1 2 + 1 C (|v * | -|v -v * |) ≤ 1 2 + 1 C (|v * | -|v τ -v * |) ≤ v τ ≤ 1 + |v * | + |v τ -v * | ≤ 1 + |v * | + |v -v * | v * , so that v τ ∼ v * ∼ v ∼ v ′ . The Cauchy-Schwarz inequality shows |J 2 | 2 b(cos θ) sin 2s+2ε (θ/2)Φ γ+2s c | v * ℓ-1-β-δ f * | | v ′ δ h ′ | 2 dvdv * dσ × b(cos θ) sin 2-2s-2ε (θ/2)Φ γ+2-2s c | v * ℓ-1-β-δ f * || v β g -( v β g) ′ | 2 dvdv * dσ + b(cos θ) sin 2-2s-2ε (θ/2)Φ γ+2-2s c | v * ℓ-1-β-δ f * | v β -v ′ β 2 |g ′ | 2 dvdv * dσ = J 2,1 × J (1) 2,2 + J (2) 2,2 . Take the change of variables v → v ′ for J 2,1 . Since -(γ + 2s) < 3/2, it follows from (2.8) that J 2,1 |v ′ -v * | 1 | v * ℓ-1-β-δ f * | | v ′ δ h ′ | 2 |v ′ -v * | -γ-2s dv ′ dv * f L 2 ℓ-1-β-δ h 2 L 2 δ Apply Lemma 2.7 with s = (2s -1 + ε) + and γ = γ + 2 -2s to J (1) 2,2 . Then J (1) 2,2 f L 2 ℓ-1-β-δ g 2 H (2s-1+ε) + β because max{(2s -1 + ε) + , -γ 2 + s -1 -3 4 } = (2s -1 + ε) + . Since (2.8) also implies J (2) 2,2 |v ′ -v * | 1 |v ′ -v * | γ+4-2s | v * ℓ-1-β-δ-2 f * || v ′ β g ′ | 2 dv ′ dv * f L 2 ℓ-3-β-δ g 2 L 2 β ,
we obtain the desired bound for J 2 . As for J 1 we use the Taylor expansion

v ′ ℓ -v ℓ = ∇ v ′ ℓ • (v ′ -v) + 1 2 1 0 ∇ 2 v τ ℓ dτ(v ′ -v) 2 .
Then, it follows from the symmetry that the integral corresponding to the first term vanishes, so that we have

J 1 b(cos θ)Φ c sin 2 (θ/2)|v ′ -v * | 2 ∇ 2 v τ ℓ | f * g ′ h ′ |dvdv * dσ |v ′ -v * | 1 |v ′ -v * | γ+2 | v * ℓ-2-β-δ f * | |( v β g) ′ ||( v δ h) ′ |dv ′ dv * f L 2 ℓ-2-β-δ g 2 L 2 β h 2 L 2 δ ,
which completes the proof of the of Proposition 2.5. Now using (2.2) with Q c( f, g) and the Proposition 2.5, we get Proposition 2.8. Let 0 < s < 1, γ > max{-3, -2s -3/2}. For any ℓ ∈ R,

(2.9) W ℓ Q( f, g) -Q( f, W ℓ g), h f L 2 ℓ+3/2+(2s-1) + +γ + +ǫ g H (2s-1+ǫ) + ℓ+(2s-1) + +γ + ||h|| L 2 .
We can now prove the upper bound estimate with weights.

Proposition 2.9. Let 0 < s < 1, γ > max{-3, -2s -3/2}. Then we have, for any ℓ ∈ R and m

∈ [s -1, s], Q( f, g), h f L 1 ℓ + +(γ+2s) + + f L 2 g H max{s+m, (2s-1+ǫ) + } (ℓ+γ+2s) + h H s-m -ℓ . Proof. Using (2.1), for any m, ℓ ∈ R, Q c( f, g), h || f || L 1 ℓ + +(γ+2s) + ||g|| H m+s (ℓ+γ+2s) + h H s-m -ℓ .
On the other hand, for any ℓ ∈ R we have

Q c ( f, g), h ≤ Q c ( f, W ℓ g), W -ℓ h + W ℓ Q c ( f, g) -Q c ( f, W ℓ g), W -ℓ h , then Proposition implies, for m ∈ [s -1, s] Q c ( f, W ℓ g), W -ℓ h f L 2 g H s+m ℓ h H s-m -ℓ
and Proposition 2.5, for any ℓ, β, δ ∈ R

W ℓ Q c ( f, g) -Q c ( f, W ℓ g), W -ℓ h f L 2 ℓ-1-β-δ g H (2s-1+ǫ) + β W -ℓ h L 2 δ .
We choose δ = 0, β = ℓ, since for m ∈ [s -1, s], sm ≥ 0, ending the proof of Proposition.

Coercivity of collision operators.

We study now the coercivity estimate for a small perturbation of µ. For any 0 < s < 1 and γ > -3, we recall the non-isotropic norm associated with the cross-section B(vv * , σ) introduced in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF] (2.10)

|||g||| 2 Φ γ = B(v -v * , σ)µ * g ′ -g 2 + B(v -v * , σ)g 2 * µ ′ - √ µ 2 = J 1 (g) + J 2 (g)
where the integration is over

R 3 v × R 3 v * × S 2 σ .
The following link with weighted Sobolev norm was shown previously

C 1 g 2 H s γ/2 (R 3 v ) + g 2 L 2 s+γ/2 (R 3 v ) ≤ |||g||| 2 Φ γ ≤ C 2 g 2 H s s+γ/2 (R 3 v ) , (2.11)
where C 1 , C 2 > 0 are two generic constants. Recall that in the definition of the non-isotropic norm, we obtain an equivalent norm if we replace µ by any positive power of µ.

The coercivity of the linearized operator -Q(µ, h) is given by the next result Proposition 2.10. There exists C > 0 such that

-Q(µ, h), h L 2 (R 3 v ) ≥ 1 2 |||h||| 2 Φ γ -C h 2 L 2 γ/2+s
.

Proof. Though the statement follows from [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], we give a proof for the convenience of the reader. By definition,

-Q(µ, h), h L 2 (R 3 v ) = - B(v -v * , σ) µ ′ * h ′ -µ * h hdσdv * dv = - B(v -v * , σ)µ * h h ′ -h dσdv * dv = 1 2 B(v -v * , σ)µ * h ′ -h 2 dσdv * dv + 1 2 B(v -v * , σ)µ * h 2 -h ′2 dσdv * dv = 1 2 J 1 (h) + I = 1 2 |||h||| 2 Φ γ - 1 2 J 2 (h) + I . Then we have from [9] |J 2 (h)| ≤ C 1 h 2 L 2 γ/2+s
, and the cancellation Lemma [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] implies

|I| ≤ C h 2 L 2 γ/2+s
, thus proving proposition 2.10.

Let us note than another proof is also possible by using instead the Appendix.

Lemma 2.11. Let 0 < s < 1 and γ > max{-3, -2s -3/2}. If we put D( √ µ f, g) = B √ µ f * g -g ′ 2 dvdv * dσ , then there exists a C > 0 such that Q( √ µ f, g), g L 2 (R 3 ) + 1 2 D( √ µ f, g) ≤ C                f L 2 g 2 L 2 γ/2 if γ > -3/2 f L 2 g 2 H s ′ γ/2 if -3/2 ≥ γ f H 2s ′ g 2 L 2 γ/2+s ′ if -3/2 ≥ γ (2.12)
for any s ′ ∈]0, s[ satisfying γ + 2s ′ > -3/2 and s ′ < 3/4.

Proof. The left hand side of (2.12) equals

1 2 B √ µ * f * g 2 -g ′2 dσdvdv * ,
and by the cancellation lemma of [2]

√ µ * | f * | |v -v * | γ g 2 dvdv * = J( f, g) .
Divide the integral to {|vv * | ≤ 1} and another region, if necessary. Then it follows from Lemma 2.6 that we obtain the first two estimates. The third estimate is a direct consequence of Pitt's inequality,

J( f, g) |v -v * | 2(γ+2s ′ ) µ * dv * 1/2 f 2 * |v -v * | 4s ′ dv * 1/2 |g| 2 dv f H 2s ′ g 2 L 2 γ/2+s ′ ,
where we choose 0 < 2s ′ < 3/2.

Lemma 2.12. Let 0 < s < 1 and γ > max{-3, -3/2 -2s}. Then for any N ∈ N we have

D( √ µ | f | , g) f L 2 -N |||g||| 2 Φ γ . (2.13) Proof. Put F = √ µ| f |. Then (2.13) in the case γ ≥ 0 follows from Lemma 3.2 of [9] with f 2 = F. Suppose that γ < 0. In view of Φ γ = Φ c + Φ c , we write D(F, g) = b Φ c + Φ c F * g -g ′ 2 dvdv * dσ = D c (F, g) + D c (F, g) .
We have

D c (F, g) bΦ c v * |γ| F * v γ/2 g -v ′ γ/2 g ′ 2 dvdv * dσ + bΦ c v * |γ| F * v ′ γ/2 -v γ/2 2 g 2 dvdv * dσ = D (1) c (F, g) + D (2) c (F, g) , because v * ∼ v + τ(v ′ -v) for any τ ∈ [0, 1] on the support of φ(v -v * ), as stated in the proof of Proposition 2.5. Since v ′ γ/2 -v γ/2 1 0 v + τ(v ′ -v) γ/2-1 dτ|v -v * | sin θ/2 v γ/2 |v -v * | sin θ/2
on the support of φ(vv * ), it follows from the Cauchy-Schwarz inequality that

D (2) c (F, g) |v-v * |≤1 |v -v * | γ+2 v * |γ| F * dv * v γ/2 g 2 dv F L 2 |γ| g 2 L 2 γ/2
.

By means of Lemma 2.7 we have

D (1) c (F, g) F L 2 |γ| g 2 H s γ/2
. On the other hand, noting that

Φ c v -v * γ ≤ v * |γ| v γ we have D c (F, g) b v * |γ| F * v γ/2 g -v ′ γ/2 g ′ 2 dvdv * dσ + b v * |γ| F * v ′ γ/2 -v γ/2 2 g 2 dvdv * dσ = D (1) c (F, g) + D (2)
c (F, g) . It follows from Lemma 3.2 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF] with γ = 0 together with Proposition 2.4 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF] that

D (1) c (F, g) v |γ| F L 1 2s ||| v γ/2 g||| 2 Φ 0 F L 1 |γ|+2s |||g||| 2 Φ γ . Since with v τ = v + τ(v ′ -v) we have v ′ γ/2 -v γ/2 1 0 v τ γ/2-1 dτ|v -v * | sin θ/2 v * |γ|/2+1 1 0 v τ -v * γ/2-1 |v -v * | sin θ/2 v * |γ|/2+1 v -v * γ/2-1 |v -v * | sin θ/2 v * |γ|+1 v γ/2 sin θ/2
we see that

D (2) c (F, g) v * 3|γ|+2 F * v γ/2 g 2 dvdv * F L 1 3|γ|+2 g 2 L 2 γ/2 .
Summing up above four estimates, in view of (2.11) we also obtain the desired estimate (2.13) in the case γ < 0.

By means of Lemma 2.11 and Lemma 2.12, in view of (2.11) we get the following upper bounded estimate, which is needed in order to prove the non linear coercivity for small perturbative solution.

Proposition 2.13. Let 0 < s < 1 and γ > max{-3, -3/2 -2s}. Then we have

Q( √ µ g, h), h L 2 (R 3 v ) g L 2 |||h||| 2 Φ γ . Remark 2.
14. If we proceed as in the proof of Proposition 3.1 from [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], we can prove

Q( √ µ g, f ), h L 2 (R 3 v ) |||g||| Φ γ + g W 1,∞ ||| f ||| Φ γ |||h||| Φ γ
for any γ > -3.

From Proposition 2.10, Proposition 2.13 and (2.11), we can deduce the following non linear coercivity for the small perturbation g. Corollary 2.15. Let 0 < s < 1 and γ > max{-3, -3/2 -2s}. There exist η 0 > 0, ǫ 0 > 0 and C > 0 such that if g L 2 (R 3 v ) ≤ ǫ 0 ,then we have

-Q(µ + √ µ g, h), h L 2 (R 3 v ) ≥ η 0 |||h||| 2 Φ γ -C h 2 L 2 γ/2+s η 0 ||h|| 2 H s γ/2 -C h 2 L 2 γ/2+s
.

Estimate of commutators with pseudo-differential operators.

We study now the commutators with pseudo-differential operators: again in the next Sections, these will be used as a rigorous replacement of formal derivatives, and when the operator is a smoothed one, as completely justified test functions.

Proposition 2.16. Let M λ (ξ) = ξ λ for λ ≥ 0. Assume that 0 < s < 1 and γ > max{-3, -3 2 -2s}. Let α, β, ρ ≥ 0 satisfy 3 + γ + α + β + ρ > 3/2 , (2.14) α + β ≥ 2s -1 , (2.15) β ≤ 1 , (2.16) α + β + ρ ≥ s . (2.17) If α + λ < 3/2 then we have M λ (D) Q c ( f, g) -Q c ( f, M λ (D) g), h f H ρ ||M λ (D)g|| H α ||h|| H β . If α + λ ≥ 3/2 then ρ = (λ -β) + satisfies (2.14) and we have M λ (D) Q c ( f, g) -Q c ( f, M λ (D) g), h f H (λ-β) + ||M λ (D)g|| H α ||h|| H β .
Proof. We recall (2.4), that is,

           ξ ξ * ∼ ξ -ξ * on supp 1 ξ * ≥ √ 2|ξ| ξ ∼ ξ -ξ * on supp 1 ξ * ≤|ξ|/2 ξ ∼ ξ * ξ -ξ * on supp 1 √ 2|ξ|≥ ξ * ≥|ξ|/2 .
Since M λ (ξ) is increasing function of |ξ|, we have

M λ (ξ) -M λ (ξ -ξ * ) M λ (ξ -ξ * )1 ξ * ≥|ξ|/2 + ξ * ξ M λ (ξ -ξ * )1 ξ * <|ξ|/2 (2.18) + M λ (ξ -ξ * ) M λ (ξ * ) ξ -ξ * λ 1 ξ-ξ * ≤ ξ * ,
where we have used the mean value theorem to gain the second term of the right hand side. Since we have

(Q c ( f, g), h) = b ξ |ξ| • σ [ Φc (ξ * -ξ -) -Φc (ξ * )] f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dξdξ * dσ, it follows that M λ (D) Q c ( f, g) -Q c ( f, M λ (D) g), h = b ξ |ξ| • σ Φc (ξ * -ξ -) -Φc (ξ * ) M λ (ξ) -M λ (ξ -ξ * ) f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dξdξ * dσ. = |ξ -|≤ 1 2 ξ * • • • dξdξ * dσ + |ξ -|≥ 1 2 ξ * • • • dξdξ * dσ = B 1 ( f, g, h) + B 2 ( f, g, h) .
The estimations for B 1 ( f, g, h) and B 2 ( f, g, h) are almost similar as those for A 1 ( f, g, h) and A 2 ( f, g, h) in the proof of Lemma 2.2, by adding the extra factor

M λ (ξ) -M λ (ξ -ξ * ). Indeed, for B 1, j ( f, g, h) corresponding to A 1, j ( f, g, h), ( j = 1, 2
), note that it follows from (2.3), (2.6) and (2.18) that

|K(ξ, ξ * )| + | K(ξ, ξ * )| M λ (ξ) -M λ (ξ -ξ * ) ξ * ρ ξ -ξ * α ξ β ξ * 3+γ+α+β+ρ M λ (ξ -ξ * )        1 ξ * ≥|ξ|/2 + ξ * ξ 1+α+β-2s 1 ξ * <|ξ|/2        + ξ * ρ ξ -ξ * α ξ β ξ * 3+γ+β+ρ M λ (ξ -ξ * ) M λ (ξ * ) ξ -ξ * α+λ + 1 ξ -ξ * α 1 ξ-ξ * ≤ ξ * ,
where we have estimated the factor ξ / ξ * of K on the supp

1 ξ * ≥ √ 2|ξ| by ( ξ / ξ * ) β because of (2.16). Noting (2.15) we have |K(ξ, ξ * )| + | K(ξ, ξ * )| M λ (ξ) -M λ (ξ -ξ * ) ξ * ρ ξ -ξ * α ξ β ξ * 3+γ+α+β+ρ M λ (ξ -ξ * )        ξ * ξ 1+α+β-2s + M λ (ξ * ) ξ -ξ * α+λ + 1 ξ -ξ * α 1 ξ-ξ * ≤ ξ *        .
The Cauchy-Schwarz inequality shows

|B 1,1 | 2 + |B 1,2 | 2 R 6 |K(ξ, ξ * )| + | K(ξ, ξ * )| M λ (ξ) -M λ (ξ -ξ * ) | f (ξ * )|| ĝ(ξ -ξ * )|| h(ξ)|dξdξ * 2 R 3 | ξ * ρ f (ξ * )| ξ * 3+γ+α+β+ρ dξ * M λ (D)g 2 H α R 3 ξ * ρ | f (ξ * )| ξ * 3+γ+α+β+ρ dξ * h 2 H β + R 6 | ξ * ρ f (ξ * )| 2 ξ * 6+2(γ+β+ρ) M λ (ξ * ) 2 1 ξ-ξ * ξ * ξ -ξ * 2(α+λ) + 1 ξ-ξ * ξ * ξ -ξ * 2α dξdξ * M λ (D)g 2 H α h 2 H β .
Since it follows from (2.14) that ξ * -(3+γ+α+β+ρ) ∈ L 2 , the first term has the upper bound

f 2 H ρ M λ (D)g 2 H α h 2 H β . If α + λ < 3/2 then 1 ξ-ξ * ξ * ξ -ξ * 2(α+λ) dξ 1 ξ * -3+2λ+2α ,
which gives the same upper bound for the second term. If α + λ ≥ 3/2 then the condition γ > -3 implies

R 6 | ξ * ρ f (ξ * )| 2 ξ * 6+2(γ+β+ρ) M λ (ξ * ) 2 1 ξ-ξ * ξ * ξ -ξ * 2(α+λ) + 1 ξ-ξ * ξ * ξ -ξ * 2α dξdξ * ξ * λ-β f (ξ * ) 2 L 2 .
Thus B 1, j ( j = 1, 2) have the desired upper bound. The estimation for B 2,2 ( f, g, h) are almost the same as above, in view of (2.7). As for B 2,1 , it remains only to estimate

B 2,1,p = b 1 ξ * ≤ √ 2|ξ| 1 |ξ -|≥ 1 2 ξ * 1 |ξ * • ξ -|≥ 1 2 |ξ -| 2 Φc (ξ * -ξ -) M λ (ξ) -M λ (ξ -ξ * ) × f (ξ * ) ĝ(ξ -ξ * ) ĥ(ξ)dσdξdξ * .
By

1 = 1 ξ * ≥|ξ|/2 1 ξ-ξ * ≤ ξ * -ξ -+ 1 ξ * ≥|ξ|/2 1 ξ-ξ * > ξ * -ξ -+ 1 ξ * <|ξ|/2
we decompose

B 2,1,p = B (1) 2,1,p + B (2) 2,1,p + B (3) 2,1,p .
On the sets for above integrals, we have

ξ * -ξ - ξ * , because |ξ -| |ξ * | that follows from |ξ -| 2 ≤ 2|ξ * • ξ -| |ξ -| |ξ * |.
Furthermore, on the sets for B (1) 2,1,p and B (2) 2,1,p we have ξ ∼ ξ * , so that sup b

1 |ξ -|≥ 1 2 ξ * 1 ξ * ≥|ξ|/2 1 |ξ -|≤|ξ|/ √
2 and ξ *ξ - ξ . Hence we have

|B (1) 2,1,p | 2 | Φc (ξ * -ξ -)| 2 | ξ * ρ f (ξ * )| 2 ξ * -ξ -2β+2ρ M λ (ξ * ) 2 1 ξ-ξ * ≤ ξ * -ξ - ξ -ξ * 2α+2λ + 1 ξ-ξ * ≤ ξ * -ξ - ξ -ξ * 2α dξdξ * dσ × | ξ -ξ * α M λ (ξ -ξ * ) ĝ(ξ -ξ * )| 2 | ξ β ĥ(ξ)| 2 dσdξdξ * . Note that ξ * ∼ ξ ∼ ξ + ξ + -u + u with u = ξ * -ξ -. Since 3 + 2(γ + α + β + ρ) > 0 then if α + λ < 3/2 we have |B (1) 2,1,p | 2 | ξ * ρ f (ξ * )| 2 ×        sup u u -(6+2γ+2β+2ρ) 1 ξ + -u ≤ u ( ξ + -u 2λ + u 2λ ) ξ + -u 2(α+λ) dξ +        dξ * M λ (D)g 2 H α h 2 H β f 2 H ρ M λ (D)g 2 H α h 2 H β because dξ ∼ dξ + on the support of 1 |ξ -|≤|ξ|/ √ 2 .
The case α + λ ≥ 3/2 can be considered by the same arguments as above. As for B (2) 2,1,p , we first note that ξ

+ = ξ -ξ * + u implies ( M λ (ξ * ) ∼ ) M λ (ξ + ) ξ -ξ * λ + u λ ξ -ξ * λ
on the integral set, and hence we have by the Cauchy-Schwarz inequality

|B (2) 2,1,p | 2 | Φc (ξ * -ξ -)|| ξ * ρ f (ξ * )| ξ * -ξ -α+β+ρ | ξ -ξ * α M λ (ξ -ξ * ) ĝ(ξ -ξ * )| 2 dσdξdξ * × | Φc (ξ * -ξ -)|| ξ ρ f (ξ * )| ξ * -ξ -α+β+ρ | ξ β ĥ(ξ)| 2 dσdξdξ * f 2 H ρ M λ (D)g 2 H α h 2 H β because Φc (u) u -(α+β+ρ) ∈ L 2 .
On the integral set of B (3) 2,1,p we have ξ ∼ ξξ * and

|M λ (ξ) -M λ (ξ -ξ * )| ξ * ξ M λ (ξ -ξ * ) , so that |B (3) 2,1,p | 2 b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)|| ξ * ρ f (ξ * )| ξ α+β+1 ξ * ρ-1 | ξ -ξ * α M λ (ξ -ξ * ) ĝ(ξ -ξ * )| 2 dσdξdξ * × b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)|| ξ * ρ f (ξ * )| ξ α+β+1 ξ * ρ-1 | ξ β ĥ(ξ)| 2 dσdξdξ * .
We use the change of variables in

ξ * , u = ξ * -ξ -. Note that |ξ -| ≥ 1 2 u + ξ -implies |ξ -| ≥ u / √ 10. Since ξ * -ξ -+ ξ *
ξ , in view of (2.15) and (2.17) we have

b 1 |ξ -|≥ 1 2 ξ * | Φc (ξ * -ξ -)|| ξ * ρ f (ξ * )| ξ α+β+1 ξ * ρ-1 dσdξ * = b 1 |ξ -|≥ 1 2 ξ * | ξ * -ξ -s Φc (ξ * -ξ -)| ξ s ξ * -ξ -α+β+ρ ξ * s | ξ * ρ f (ξ * )| ξ s × ξ * -ξ -α+β+ρ-s ξ * 1-s-ρ ξ α+β+1-2s dσdξ * b 1 |ξ -| u u 2s | Φc (u)| 2 ξ 2s u 2(α+β+ρ) dσdu 1/2 b 1 |ξ -|≥ 1 2 ξ * ξ * 2s | ξ * ρ f (ξ * )| 2 ξ 2s dσdξ * 1/2 f H ρ ,
from which we also can obtain the desired bound for B (3) 2,1,p . The proof of the proposition is complete. We give an application of Proposition 2.16. Let S ∈ C ∞ 0 (R) satisfy 0 ≤ S ≤ 1 and

S (τ) = 1, |τ| ≤ 1; S (τ) = 0, |τ| ≥ 2. Set Λ λ N (D v ) = M λ (D v ) S N (D v ) = D v λ S N (D v ).
Corollary 2.17. Assume that 0 < s < 1 and γ > max{-3, Proof. In the proof of Proposition 2.16, instead of (2.18) we use

-3 2 -2s}. If 0 ≤ λ < 3/2 then for any max{2s - 1, s/2} ≤ s ′ < s satisfying γ + 2s ′ > -3/2 we have Λ λ N Q( f, g) -Q( f, Λ λ N g), h f H s ′ 3/2+(
Λ λ N (ξ) -Λ λ N (ξ -ξ * ) M λ (ξ -ξ * )1 ξ * ≥|ξ|/2 + ξ * ξ M λ (ξ -ξ * )1 ξ * <|ξ|/2 + M λ (ξ -ξ * ) M λ (ξ * ) ξ -ξ * λ 1 ξ-ξ * ≤ ξ * , and choose (α, β, ρ) = (0, s ′ , s ′ ) or (0, s ′ , λ -s ′ ) , we can get , Λ λ N Q c ( f, g) -Q c ( f, Λ λ N g), h f H s ′ ||M λ (D v )g|| L 2 ||h|| H s ′ .
On the other hand, using Proposition 2.9 of [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF],

Λ λ N Q c( f, g) -Q c( f, Λ λ N g), h = Λ -s ′ Λ λ N Q c( f, g) -Q c( f, Λ λ N g) , Λ s ′ h ≤ Λ λ-s ′ N Q c( f, g) -Q c( f, Λ λ-s ′ N g), Λ s ′ h + Q c( f, Λ λ-s ′ N g) -Λ -s ′ N Q c( f, Λ λ N g) , Λ s ′ h f L 2 3/2+(2s-1) + +γ + +ǫ ||g|| H λ-s ′ +(2s-1) + (2s-1) + +γ + ||h|| H s ′ ,
which completes the proof of Corollary.

Full regularity of solutions

Let f ∈ L ∞ ([0, T [ ; H 5 ℓ (Ω × R 3 )
), for any ℓ ∈ N be a solution of Cauchy problem (1.1). The regularity of f will now be considered. First of all, note that f ∈ C 1 ([0, T [; H 1 ℓ (Ω × R 3 )) by using the equation. For α ∈ N 6 , we recall the Leibniz formula

∂ α Q(g, f ) = α 1 +α 2 =α C α 1 α 2 Q(∂ α 1 g, ∂ α 2 f ).
Here and below, φ denotes a cutoff function satisfying φ ∈ C ∞ 0 and 0 ≤ φ ≤ 1. Notation φ 1 ⊂⊂ φ 2 stands for two cutoff functions such that φ 2 = 1 on the support of φ 1 .

Take some smooth cutoff functions ϕ, ϕ

2 , ϕ 3 ∈ C ∞ 0 (]T 1 , T 2 [) and ψ, ψ 2 , ψ 3 ∈ C ∞ 0 (K) such that ϕ ⊂⊂ ϕ 2 ⊂⊂ ϕ 3 and ψ ⊂⊂ ψ 2 ⊂⊂ ψ 3 . Set f 1 = ϕ(t)ψ(x) f , f 2 = ϕ 2 (t)ψ 2 (x) f and f 3 = ϕ 3 (t)ψ 3 (x) f , so here we can suppose that 0 < T < ∞. For α ∈ N 6 , |α| ≤ 5, denote F = ∂ α x,v (ϕ(t)ψ(x) f ) ∈ L ∞ (]T 1 , T 2 [; L 2 ℓ (R 6 )
). Then the Leibniz formula yields the following equation :

(3.1) F t + v • ∂ x F -Q( f, F) = G, (t, x, v) ∈ R 7 ,
where

(3.2) G = α 1 +α 2 =α, 1≤|α 1 | C α 1 α 2 Q ∂ α 1 f 2 , ∂ α 2 f 1 + ∂ α ϕ t ψ(x) f + v • ψ x (x)ϕ(t) f + [∂ α , v • ∂ x ](ϕ(t)ψ(x) f ) ≡ (A) + (B) + (C).
Note carefully that a priori F is not regular enough, and therefore at that point, taking it as test function in the equation of (3.1) is not allowed. This is one of the main difficulties alluded to in the Introduction. Therefore, as in [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF], we need to mollify F. This mollification process of course complicates the analysis below, but is necessary if we want to avoid formal proofs. The previous set of tools related to commutators estimations will then be used. For this purpose, let S ∈ C ∞ 0 (R) satisfy 0 ≤ S ≤ 1 and

S (τ) = 1, |τ| ≤ 1; S (τ) = 0, |τ| ≥ 2. Then S N (D x )S N (D v ) = S (2 -2N |D x | 2 )S (2 -2N |D v | 2 ) : H -∞ ℓ (R 6 ) → H +∞ ℓ (R 6
), is a regularization operator such that

S N (D x )S N (D v ) f -f L 2 ℓ (R 6 ) → 0, as N → ∞. Set P N, ℓ = ψ 2 (x)W ℓ S N (D x ) S N (D v ). Then P N, ℓ F ∈ C 1 0 (]T 1 , T 2 [; C ∞ 0 (K; H +∞ (R 3 )
)), and we can take

h = P ⋆ N, ℓ (P N, ℓ F) ∈ C 1 (R; H +∞ (R 6
)) as a test function for equation (3.1).

It follows by integration by parts on

R 7 = R 1 t × R 3 x × R 3 v that [S N (D v ), v] • ∇ x S N (D x )F, W ℓ P N,ℓ F L 2 (R 7 ) -P N,ℓ Q( f 2 , F), P N,ℓ F L 2 (R 7 ) = G, h L 2 (R 7 )
, where we used the fact

(∂ t + v • ∇ x ) P N,ℓ F, P N,ℓ F L 2 (R 7 ) = 0.
We get then

-Q( f, P N, ℓ F), P N, ℓ F L 2 (R 7 ) = -[S N (D v ), v] • ∇ x S N (D x )F, W ℓ P N,ℓ F L 2 (R 7 ) + P N, ℓ Q( f 2 , F) -Q( f 2 , P N, ℓ F), P N, ℓ F L 2 (R 7 ) + G, h L 2 (R 7 )
.

Next, we follow the main steps in our previous works [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF], but need to be careful due to the singular behavior of the relative velocity part of the kernel.

Gain of regularity in v.

In this subsection, we will prove a partial smoothing effect on the weak solution F in the velocity variable v .

Proposition 3.1. Assume that 0 < s < 1, γ > max{-3, -2s -3/2}. Let f ∈ L ∞ ([0, T ]; H 5 ℓ (Ω x × R 3 v ))
, for any ℓ ∈ N be a solution of the equation (1.1) satisfying the coercivity condition (1.3). Then one has

(3.3) Λ s v ϕ(t)ψ(x) f ∈ L 2 (R t ; H 5 ℓ (R 6 )), for any big ℓ > 0 and any cut off function ϕ ∈ C ∞ 0 (]T 1 , T 2 [), ψ ∈ C ∞ 0 (K). Similarly to [6], [S N (D v ), v] • ∇ x S N (D x ) F, W ℓ P N,ℓ F L 2 (R 7 ) ≤ C f 1 2 L 2 ([0,T ];H 5 ℓ (R 6 )) .
Then the coercivity assumption (1.3) implies

η 0 ||Λ s v W γ/2 P N, ℓ F|| 2 L 2 (R 7 ) ≤ C f 1 2 L 2 (R;H 5 ℓ+(γ/2+s) + (R 6 )) + G, h L 2 (R 7 )
(3.4)

+ P N, ℓ Q( f 2 , F) -Q( f 2 , P N, ℓ F), P N, ℓ F L 2 (R 7 )
.

The proof of Proposition 3.1 will be completed by estimating the last two terms in (3.4) through the following three Lemmas.

Lemma 3.2. Let f 1 ∈ L ∞ ([0, T ]; H 5 ℓ (R 6
)), ℓ ≥ 0. Then, we have, for any ε > 0,

G, h L 2 (R 7 ) ≤ C ε f 2 4 L ∞ ([0,T ];H 5 ℓ+2+(γ+2s) + (R 6 )) + ε P N, ℓ F 2 L 2 (R 4 t,x ,H s γ/2 (R 3 v ))
. Proof. By using the decomposition in (3.2), it is obvious that

(B) = ∂ α ϕ t ψ(x) f + v • ψ x (x)ϕ(t) f ∈ L 2 ℓ (R 7 ), and (B) L 2 ℓ (R 7 ) ≤ C f 2 L ∞ (R,H 5 ℓ+1 (R 6 )) . Note that [∂ α , v • ∂ x ] is a differential operator of order |α| so that we have (C) L 2 ℓ (R 7 ) ≤ C f 2 L ∞ (R,H 5 ℓ (R 6 )
) . For the term (A), recall that α 1 + α 2 = α, |α| ≤ 5 and |α 2 | ≤ 4. Here we use the following upper bounded estimate from Proposition 2.9

(3.5) Q( f, g), h L 2 (R 3 v ) ≤ C f L 2 ℓ-γ/2+2+(γ+2s) + (R 3 v ) g H s ℓ-γ/2+(γ+2s) + (R 3 v ) h H s -ℓ+γ/2 (R 3 v )
. Then, by separating the cases |α 1 | ≤ 3 and |α 1 | > 3, we get

Q ∂ α 1 f 2 , ∂ α 2 f 1 , P ⋆ N,ℓ P N,ℓ F L 2 (R 7 ) ≤ C ∂ α 1 f 2 L 2 ℓ-γ/2+2+(γ+2s) + (R 3 v ) ∂ α 2 f 1 H s ℓ-γ/2+(γ+2s) + (R 3 v ) P ⋆ N,ℓ P N,ℓ F H s -ℓ+γ/2 (R 3 v ) dxdt ≤ C f 2 L ∞ (R,H 5 ℓ+2+(γ+2s) + (R 6 )) f 1 L 2 (R,H 4+s ℓ+2+(γ+2s) + (R 6 )) P N,ℓ F L 2 (R 4 t,x ,H s γ/2 (R 3 v )) .
Here we used the fact that W -ℓ P ⋆ N,ℓ is a uniformly (with respect to N, ℓ) bounded operator. This ends the proof of Lemma 3.2 by Cauchy Schwarz inequality.

We turn now to estimating the commutators of the regularization operator with the collision operator which are given in the following two Lemmas.

The next lemma is about the commutator of the collision operator with a mollification w.r.t. x variable.

Lemma 3.3. Let 0 < s < 1, γ > max{-3, -2s -3/2}. For any suitable functions f and h with the following norms well defined, one has 3 ) uniformly with respect to N. Then for any smooth function h, one has

S N (D x )Q( f, g) -Q( f, S N (D x ) g), h L 2 (R 7 ) (3.6) ≤ C2 -N ∇ x f L ∞ (R 4 t,x , L 2 ℓ+2-γ/2+(2s+γ) + (R 3 v )) g L 2 (R 4 t,x , H s ℓ-γ/2+(2s+γ) + (R 3 v )) h L 2 (R 4 t,x , H s -ℓ+γ/2 (R 3 v )) . for a constant C independent of N. Proof. Let us introduce KN (z) = 2 3N Ŝ (2 N z)2 N z. Note that KN ∈ L 1 (R
S N (D x ) Q( f, g) -Q( f, S N (D x )g) , h L 2 (R 7 ) = 1 0 R t R 3 x ×R 3 y KN (x -y) × Q ∇ x f (t, +τ(x -y), • ), 2 -N g(t, y, • ) , h(t, x, • ) L 2 (R 3 v )
dtdxdy dτ.

By applying (3.5), the right hand side of this equality can be estimated from above by

C sup t,x ||∇ x f (t, x, • )|| L 2 ℓ+2-γ/2+(2s+γ) + (R 3 v ) × R t R 3 x | KN | * ||2 -N g(t, • )|| H s ℓ-γ/2+(2s+γ) + (R 3 v ) (x)||h(t, x, •)|| H s -ℓ+γ/2 (R 3 v ) dxdt ≤ C2 -N ∇ x f L ∞ (R 4 t,x ;L 2 ℓ+2-γ/2+(2s+γ) + (R 3 v )) g L 2 (R 4 t,x ; H s ℓ-γ/2+(2s+γ) + (R 3 v )) ||h|| L 2 (R 4 t,x ; H s -ℓ+γ/2 (R 3 v ))
, which completes the proof of the lemma.

We now apply (3.6) with g ∼ S N (D v )g, and use the fact that a regularization operator S N (D v ) w.r.t. v variable has the property that, for any p

2 -N S N (D v )g(t, x, • ) H s p (R 3 v ) ≤ 2 -N S N (D v )g(t, x, • ) H 1 p (R 3 v ) ≤ C g(t, x, • ) L 2 p (R 3 v ) ,
where C is a constant independent on N. It follows that

S N (D x )Q( f, S N (D v )g) -Q( f, S N (D x ) S N (D v )g), h L 2 (R 7 ) (3.7) ≤ C ∇ x f L ∞ (R 4 t,x , L 2 ℓ+2-γ/2+(2s+γ) + (R 3 v )) g L 2 (R 4 t,x , L 2 ℓ-γ/2+(2s+γ) + (R 3 v )) h L 2 (R 4 t,x , H s -ℓ+γ/2 (R 3 v )) .

Completion of proof of Proposition 3.1.

As regards the commutator terms in (3.4), we have

P N,ℓ Q( f, F) -Q( f, P N,ℓ F), P N,ℓ F L 2 (R 7 ) = S N (D v ) Q( f, F) -Q( f, S N (D v ) F), S ⋆ N (D x )ψ 2 (x)W ℓ P N,ℓ F L 2 (R 7 ) + S N (D x ) Q( f, S N (D v ) F) -Q( f, S N (D x )S N (D v ) F), ψ 2 (x)W ℓ P N,ℓ F L 2 (R 7 ) + W ℓ Q( f, S N (D x )S N (D v ) F) -Q( f, P N,ℓ F), ψ 2 (x)P N,ℓ F L 2 (R 7 )
.

= (1) + (2) + (3).
Note that [W ℓ , S N (D v )] is also a uniformly bounded operator from L 2 to L 2 ℓ-1 with respect to the parameter N. Using Corollary 2.17 with λ = 0, we have, for 0

< s ′ < s, γ + 2s ′ > -3/2, |(1)| ≤ C f 2 L ∞ (R 4 t,x , H s ′ 3/2+γ + +(2s-1) + (R 3 v )) F L 2 (γ + +2s-1) + (R 7 ) W ℓ P N, ℓ F L 2 (R 4 t,x , H s ′ (R 3 v )) ≤ ε Λ s v W γ/2 P N,ℓ g 2 L 2 (R 7 ) + C ε f 3 4 H 5 2ℓ (R 7 )
. We can use (3.7) to show that

|(2)| ≤≤ C ∇ x f 2 L ∞ (R 4 t,x , L 2 ℓ+2-γ/2+(2s+γ) + (R 3 v )) F L 2 (R 4 t,x , L 2 ℓ-γ/2+(2s+γ) + (R 3 v )) W ℓ P N, ℓ F L 2 (R 4 t,x , H s -ℓ+γ/2 (R 3 v )) ≤ ε Λ s v W γ/2 P N, ℓ g 2 L 2 (R 7 t,x,v ) + C ε f 3 4 H 5 2ℓ (R 7 )
. Finally, (2.9)) implies that

|(3)| ≤ C f 2 L ∞ (R 4 t,x , L 2 ℓ+2+(2s-1) + +γ + (R 3 v )) S N (D x ) S N (D v ) F L 2 (R 4 t,x , H (2s-1+δ) + ℓ+(2s-1) + +γ + (R 3 v )) P N, ℓ F L 2 (R 7 ) ≤ ε Λ s v W γ/2 P N, ℓ F 2 L 2 (R 7 t,x,v ) + C ε f 3 4 H 5 2ℓ (R 7
) . In summary, we have obtained the following estimate for the second term on the right hand side of (3.4)

P N,ℓ Q( f 2 , F) -Q( f 2 , P N,ℓ F), P N,ℓ g L 2 (R 7 ) ≤ ε Λ s v W γ/2 P N, ℓ F 2 L 2 (R 7 t,x,v ) + C ε f 3 4 H 5 2ℓ (R 7
) . Finally, it holds that from (3.4) and (2.11) that

Λ s v W γ/2 P N,ℓ F 2 L 2 (R 7 ) ≤ C 1 + f 3 4 H 5 2ℓ (R 7 )
, where the constant C is independent of N. Therefore, Proposition 3.1 is proved by taking the limit N → ∞.

Gain of regularity in (t, x).

In [START_REF] Alexandre | Uncertainty principle and kinetic equations[END_REF], by using a generalized uncertainty principle, we proved a hypo-elliptic estimate, as regards a transport equation in the form of (3.8)

f t + v • ∇ x f = g ∈ D ′ (R 2n+1 ), where (t, x, v) ∈ R 1+n+n = R 2n+1 . Lemma 3.4. Assume that g ∈ H -s ′ (R 2n+1 ), for some 0 ≤ s ′ < 1. Let f ∈ L 2 (R 2n+1
) be a weak solution of the transport equation (3.8), such that

Λ s v f ∈ L 2 (R 2n+1 ) for some 0 < s ≤ 1. Then it follows that Λ s(1-s ′ )/(s+1) x f ∈ L 2 -ss ′ s+1 (R 2n+1 ), Λ s(1-s ′ )/(s+1) t f ∈ L 2 -s s+1 (R 2n+1 ),
where

Λ • = (1 + |D • | 2 ) 1/2 .
As mentioned earlier, this hypo-elliptic estimate together with Proposition 3.1 are used to obtain the partial regularity in the variable (t, x). With this partial regularity in (t, x), by applying the Leibniz type estimate on the fractional differentiation on the solution, we will show some improved regularity in all variables, v and (x, t). Then the hypo-elliptic estimate can be used again to get higher regularity in the variable (x, t). This procedure can be continued to obtain at least one order higher differentiation regularity in (t, x) variable.

To proceed, recall (see for example [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF] a Leibniz type formula for fractional derivatives with respect to variable (t, x). Let 0 < λ < 1. Then there exists a positive constant C λ 0 such that for any f ∈ S(R n ), one has (3.9)

|D y | λ f (y) = F -1 |ξ| λ f (ξ) = C λ R n f (y) -f (y + h) |h| n+λ dh.
First of all, we have the following proposition on the gain of regularity in the variable (t, x) through uncertainty principle as in [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF]. Proposition 3.5. Under the hypothesis of Theorem 1.1, one has

(3.10) Λ s 0 t,x f 1 ∈ L 2 ([0, T ]; H 5 ℓ (R 6 )), for any ℓ ∈ N and 0 < s 0 = s(1-s) (s+1) . Therefore, under the hypothesis f ∈ L ∞ ([0, T ]; H 5 ℓ (R 6
)) for all ℓ ∈ N, it follows that for any ℓ ∈ N we have (3.11)

Λ s v (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6 )), Λ s 0 t,x (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6 )
) . This partial regularity in (t, x) variable will now be improved.

Proposition 3.6. Let 0 < λ < 1. Suppose that f ∈ L ∞ ([0, T ]; H 5 ℓ (Ω × R 3 )
) for all ℓ ∈ N is a solution of the equation (1.1), and for any cutoff functions ϕ, ψ, we have

(3.12) Λ s v (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6 )), Λ λ t,x (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6 )). Then, one has Λ s v Λ λ t,x (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6
)), for any ℓ ∈ N and any cutoff functions ϕ, ψ.

Set F N,ℓ = P N,ℓ F = ψ 2 (x)S N (D x ) W ℓ S N (D v )∂ α (ϕ(t)ψ(x) f
), where α ∈ N 6 , |α| ≤ 6 and ℓ ∈ N. Then (3.12) yields

Λ s v F N,ℓ L 2 (R 7 ) ≤ C Λ s v ∂ α (ϕ(t)ψ(x) f ) L 2 ℓ (R 7 ) , and
Λ λ t,x F N,ℓ L 2 (R 7 ) ≤ C Λ λ t,x ∂ α (ϕ(t)ψ(x) f ) L 2 ℓ (R 7 )
, where the constant C is independent on N.

It follows that F N,ℓ satisfies the following equation

(3.13) ∂ t (F N,ℓ ) + v • ∂ x (F N,ℓ ) = Q( f, F N,ℓ ) + G N,ℓ ,
where G N,l is given by

G N,ℓ =W ℓ S N (D v ), v • ∇ x S N (D x )F + P N,ℓ Q f 2 , F -Q f 2 , P N,ℓ F + P N,l G,
with G defined in (3.2). We now choose |D t,x | λ |D t,x | λ F N,ℓ as a test function for equation (3.13). It follows that

Λ s v Λ λ t,x P N,ℓ F L 2 (R 7 ) ≤ C |D t,x | λ Q( f 2 , F N,ℓ ) -Q( f 2 , |D t,x | λ F N,ℓ ), |D t,x | λ F N,l L 2 (R 7 ) + |D t,x | λ G N,ℓ , |D t,x | λ F N,l L 2 (R 7 )
.

Using the formula (3.9), the proof of the Proposition 3.6 is similar to the corresponding result in [START_REF] Alexandre | Regularizing effect and local existence for non-cutoff Boltzmann equation[END_REF], here we omit the cut-off function, it is easy to trait as before.

We can then get the following regularity result on the solution with respect to the (t, x) variable.

Proposition 3.7. Under the hypothesis of Theorem 1.1, one has

(3.14) Λ 1+ε t,x (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6
)), for any ℓ ∈ N and some ε > 0.

Proof. By fixing s 0 = s (1-s) (s+1) , then (3.11) and Proposition 3.6 with λ = s 0 imply

Λ s v Λ s 0 t,x F ∈ L 2 ℓ (R 7 ). It follows that, (Λ s 0 t,x F) t + v • ∂ x (Λ s 0 t,x F) + L 1 (Λ s 0 t,x F) = Λ s 0 t,x Q( f 2 , F) + Λ s 0 t,x G ∈ H -s ℓ (R 7
). By applying Lemma 3.4 with s ′ = s, we can deduce that

Λ s 0 +s 0 t,x (F) ∈ L 2 ℓ (R 7
), for any ℓ ∈ N. If 2s 0 < 1, by using again Proposition 3.6 with λ = 2s 0 and Lemma 3.4 with s ′ = s, we have

Λ s v (ϕ(t)ψ(x) f ), Λ 2s 0 t,x (F) ∈ L 2 ℓ (R 7 ) ⇒ Λ 3s 0 t,x (F) ∈ L 2 ℓ (R 7 ). Choose k 0 ∈ N such that k 0 s 0 < 1, (k 0 + 1)s 0 = 1 + ε > 1.
Finally, (3.14) follows from (3.10) and Proposition 3.6 with λ = k 0 s 0 by induction. And this completes the proof of the proposition 3.7.

Full regularity of solution.

The above preparations will be used for the proof of the full regularity of solution in Theorem 1.1, by using an induction argument.

From Proposition 3.6 and Proposition 3.7, it follows that for any α ∈ N, |α| ≤ 5 and any ℓ ∈ N,

Λ s v ∂ α (ϕ(t)ψ(x) f ), Λ t,x ∂ α (ϕ(t)ψ(x) f ) ∈ L 2 ℓ (R 7
). These will be used to get the high order regularity with respect to the variable v.

Λ λ v ∂ α (ϕ(t)ψ(x) f ), Λ t,x ∂ α (ϕ(t)ψ(x) f ) ∈ L 2 ℓ (R 7
). Then, for all cutoff function and all α ∈ N, |α| ≤ 5, ℓ ∈ N,

(3.16) Λ λ+s v ∂ α (ϕ(t)ψ(x) f ) ∈ L 2 ℓ (R 7 ). Proof. Recall that, for |α| ≤ 5, F = ∂ α (ϕ(t)ψ(x) f ) is the weak solution of the equation : ∂F ∂t + v • ∂ x F -Q( f, F) = G, (t, x, v) ∈ R 7 ,
where G is given in (3.2). Set

P N,ℓ,λ = ψ 2 (x)W ℓ S N (D x ) S N (D v ) Λ λ v , we take now P ⋆ N,ℓ,λ P N,ℓ,λ F = P ⋆ N,ℓ,λ F N,ℓ,λ ∈ C 1 0 ([T 1 , T 2 ]; H +∞ p (R 6
)) as test function. Then, one has

P N,ℓ,λ , v • ∂ x F, P N,ℓ,λ F L 2 (R 7 ) -Q( f, F N,ℓ,λ ), F N,ℓ,λ L 2 (R 7 ) = P N,ℓ,λ Q( f, F) -Q( f, P N,ℓ,λ F), P N,ℓ,λ F L 2 (R 7 ) + P N,ℓ,λ G, P N,ℓ,λ F L 2 (R 7 ) . Since Λ λ v , v • ∂ x = λΛ λ-2 v ∂ v • ∂ x , and Λ λ-2 v ∂ v are bounded operators in L 2 ,
for any 0 < λ < 1, we have, by using the hypothesis (3.15) that

P N,ℓ,λ , v • ∂ x F, P N,ℓ,λ F L 2 (R 7 ) ≤ C Λ λ v F L 2 l (R 7 ) Λ x F L 2 l (R 7 ) .
Using the coercivity (1.3), we get as (3.4),

η 0 ||Λ s v W γ/2 F N, ℓ,λ || 2 L 2 (R 7 ) ≤ C Λ λ v F L 2 l (R 7 ) Λ x F L 2 l (R 7 ) + P N,ℓ,λ G, P N,ℓ,λ F L 2 (R 7 )
(3.17)

+ P N,ℓ,λ Q( f 2 , F) -Q( f 2 , P N,ℓ,λ F), P N,ℓ,λ F L 2 (R 7 )
.

We conclude the proof of Proposition 3.8 by using the following Lemma.

Lemma 3.9. Let f ∈ L ∞ ([0, T ]; H 5 ℓ (Ω × R 3 )), ℓ ≥ ℓ 0 (large). Then, we have, for any ε > 0,

P N,ℓ,λ G, P N,ℓ,λ F L 2 (R 7 ) ≤ ε||Λ s v W γ/2 F N, ℓ,λ || 2 L 2 (R 7 ) + C ε Λ λ v f 3 2 L 2 ([0,T ];H 5 2ℓ (R 6 )) Λ 1 t f 3 2 L 2 ([0,T ];H 5 2ℓ (R 6 )) + Λ 1 x f 3 2 L 2 ([0,T ];H 5 2ℓ (R 6
)) . Proof. By using the decomposition in (3.2), it is obvious that for the linear terms

P N,ℓ,λ ((B) + (C)), P N,ℓ,λ F L 2 (R 7 ) ≤ C Λ λ v f 2 2 L 2 ([0,T ];H 5 ℓ+1 (R 6 )) .
For the term (A), recall that α 1 + α 2 = α, |α| ≤ 5 and |α 2 | < 5. Then, by separating the cases |α 1 | ≤ 3 and

|α 1 | > 3, we get, with Λ λ N (D v ) = Λ λ v S N (D v ), P N,ℓ,λ Q ∂ α 1 f 2 , ∂ α 2 f 1 , P N,ℓ,λ F L 2 (R 7 ) = Λ λ N (D v ) Q ∂ α 1 f 2 , ∂ α 2 f 1 , W ℓ S ⋆ N (D x )ψ 2 F N,ℓ,λ L 2 (R 7 ) ≤ Λ λ N (D v ) Q ∂ α 1 f 2 , ∂ α 2 f 1 -Q ∂ α 1 f 2 , Λ λ N (D v )∂ α 2 f 1 , W ℓ S ⋆ N ψ 2 (D x )F N,ℓ,λ L 2 (R 7 ) + Q ∂ α 1 f 2 , Λ λ N (D v )∂ α 2 f 1 , W ℓ S ⋆ N (D x )ψ 2 F N,ℓ,λ L 2 (R 7 )
.

Using Corollary 2.17, we have

Λ λ N (D v ) Q ∂ α 1 f 2 , ∂ α 2 f 1 -Q ∂ α 1 f 2 , Λ λ N (D v )∂ α 2 f 1 , W ℓ S ⋆ N (D x )ψ 2 F N,ℓ,λ L 2 (R 7 ) ≤ C ∂ α 1 f 2 H s ′ 2+γ + +(2s-1) + (R 3 v ) ∂ α 2 f 1 H λ γ + +(2s-1) + (R 3 v ) W ℓ S ⋆ N (D x )ψ 2 F N,ℓ,λ H s ′ (R 3 v ) dxdt ≤ C        ∂ α 1 f 2 L ∞ (R 4 t,x ;H s ′ 2+γ + +(2s-1) + (R 3 v )) Λ 5 x,v Λ λ v f 1 L 2 γ + +(2s-1) + (R 7 ) Λ s ′ v F N,ℓ,λ L 2 ℓ (R 7 ) , |α 1 | ≤ 3 ∂ α 1 f 2 L 2 (R 4 t,x ;H s ′ 2+γ + +(2s-1) + (R 3 v )) Λ 1+3/2+δ x,v Λ λ v Λ 1/2+δ t f 1 L 2 γ + +(2s-1) + (R 7 ) Λ s ′ v F N,ℓ,λ L 2 ℓ (R 7 ) , |α 1 | > 3 ≤ C Λ 5 x,v Λ t f 2 L 2 2+γ + +(2s-1) + (R 7 ) Λ 5 x,v Λ λ v f 2 L 2 γ + +(2s-1) + (R 7 ) Λ s ′ v F N,ℓ,λ L 2 ℓ (R 7 ) ≤ ǫ Λ s v W γ/2 F N,ℓ,λ 2 L 2 (R 7 ) + C ǫ Λ 5 x,v Λ t f 2 2 L 2 2ℓ (R 7 ) Λ 5 x,v Λ λ v f 2 2 L 2 2ℓ (R 7 ) ,
Proposition 2.9 with m = 0 and Sobolev embedding for x ∈ R 3 and t ∈ R give

Q ∂ α 1 f 2 , Λ λ N (D v )∂ α 2 f 1 , W ℓ S ⋆ N (D x )ψ 2 F N,ℓ,λ L 2 (R 7 ) ≤ C ∂ α 1 f 2 L 2 2+(γ+2s) + +ℓ-γ/2 (R 3 v ) ∂ α 2 f 1 H λ+s ℓ-γ/2+(γ+2s) + (R 3 v ) W ℓ S ⋆ N (D x )ψ 2 F N,ℓ,λ H s -ℓ+γ/2 (R 3 v ) dxdt ≤ ǫ Λ s v W γ/2 F N,ℓ,λ 2 L 2 (R 7 ) + C ǫ Λ 5 x,v Λ t f 2 2 L 2 2ℓ (R 7 ) Λ 5 x,v Λ λ v f 2 2 L 2 2ℓ (R 7 ) .
This ends the proof of Lemma 3.9.

Recall Λ λ N (D v ) = Λ λ v S N (D v ),
we have for the last term of (3.17), (the term involving µ is omitted since is easier than f 2 ),

P N,ℓ,λ Q( f 2 , F) -Q( f 2 , P N,ℓ,λ F), P N,ℓ,λ F L 2 (R 7 ) = Λ λ N (D v )Q( f 2 , F) -Q( f 2 , Λ λ N (D v )F), W ℓ S ⋆ N (D x ) P N,ℓ,λ F L 2 (R 7 ) + S N (D x ) Q( f 2 , Λ λ N (D v )F) -Q( f 2 , S N (D x ) Λ λ N (D v )F), W ℓ P N,ℓ,λ F L 2 (R 7 ) + W ℓ Q( f 2 , S N (D x ) Λ λ N (D v )F) -Q( f 2 , P N,ℓ,λ F), P N,ℓ,λ F L 2 (R 7 ) = (I) + (II) + (III) .
Using again Corollary 2.17, we have by interpolation,

|(I)| ≤ C f 2 H s ′ 2+(2s-1) + +γ + F H λ (2s-1) + +γ + W ℓ S ⋆ N (D x ) P N,ℓ,λ F H s ′ dxdt ≤ C Λ t Λ 2 x Λ s ′ v f 2 L 2 2+(2s-1) + +γ + (R 7 ) Λ λ v F L 2 (2s-1) + +γ + Λ s ′ v W ℓ S ⋆ N (D x ) P N,ℓ,λ F L 2 (R 7 ) ≤ C Λ t Λ 4 x,v f 2 L 2 2+(2s-1) + +γ + (R 7 ) Λ λ v F L 2 (2s-1) + +γ + ǫ Λ s v W γ/2 P N,ℓ,λ F L 2 (R 7 ) + C ǫ Λ λ v F L 2 3ℓ (R 7 ) ≤ ǫ Λ s v W γ/2 P N,ℓ,λ F 2 L 2 (R 7 ) + C ǫ Λ t Λ 4 x,v f 2 2 L 2 2+(2s-1) + +γ + (R 7 ) Λ λ v Λ 5 x,v f 1 2 L 2 2ℓ (R 7
) . Using now (3.6), similarly as for (3.7), we have

|(II)| ≤ C2 -N ∇ x f 2 L ∞ (R 4 t,x , L 2 ℓ+2-γ/2+(2s+γ) + (R 3 v )) M λ (D v )F L 2 (R 4 t,x , H s ℓ-γ/2+(2s+γ) + (R 3 v )) W ℓ P N,ℓ,λ F L 2 (R 4 t,x , H s -ℓ+γ/2 (R 3 v )) ≤ C Λ t Λ 3 x f 2 L 2 ℓ+2-γ/2+(2s+γ) + (R 7 ) Λ λ v F L 2 ℓ-γ/2+(2s+γ) + (R7) Λ s v W γ/2 P N,ℓ,λ F L 2 (R 7 ) .
For the term (III), we use (2.9)

|(III)| = W ℓ Q( f 2 , S N (D x ) Λ λ N (D v )F) -Q( f 2 , P N,ℓ,λ F), P N,ℓ,λ F L 2 (R 7 ) ≤ C f 2 L ∞ (R 4 t,x ;L 2 ℓ+2+(2s-1) + +γ + (R 3 v )) S N (D x ) Λ λ N (D v )F L 2 (R 4 t,x ;H (2s-1+ǫ) + ℓ+(2s-1) + +γ + (R 3 v )) P N,ℓ,λ F L 2 (R 7 ) ≤ ǫ Λ s v W γ/2 P N,ℓ,λ F 2 L 2 (R 7 ) + C ǫ Λ λ v F 2 L 2 2ℓ (R7) + Λ t Λ 2 x f 2 2 L 2 ℓ+2-γ/2+(2s+γ) + (R 7 ) Λ λ v F 2 L 2
ℓ (R7) . Finally, from (3.17), choose ǫ > 0 small enough, we get for big ℓ,

η 0 2 ||Λ s v W γ/2 F N, ℓ,λ || 2 L 2 (R 7 ) Λ λ v Λ 5 x,v f 2 2 L 2 2ℓ (R 7 ) + Λ x Λ 5 x,v f 2 2 L 2 2ℓ (R 7 ) + Λ t Λ 5 x,v f 2 2 L 2 2ℓ (R 7 ) Λ λ v Λ 5 x,v f 2 2 L 2
2ℓ (R 7 ) . Taking the limit N → ∞, we have proved (3.16), and ended the proof of Proposition 3.8.

We can now conclude that the following regularity result with respect to the variable v holds true.

Proposition 3.10. Under the hypothesis of Theorem 1.1, one has

(3.18) Λ 1+ε v (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6
)), for any ℓ ∈ N and some ε > 0.

Again, this result is indeed obtained by noticing that there exists k 0 ∈ N such that

k 0 s < 1, (k 0 + 1)s = 1 + ε > 1.
Then we get (3.18) from (3.3), Proposition 3.8 with λ = k 0 s by induction.

High order regularity by iterations

From Proposition 3.7 and Proposition 3.10, we can now deduce that, for any ℓ ∈ N, and any cutoff functions ϕ(t) and ψ(x),

Λ v (ϕ(t)ψ(x) f ), Λ t,x (ϕ(t)ψ(x) f ) ∈ L 2 ([0, T ]; H 5 ℓ (R 6 )) , which is ϕ(t)ψ(x) f ∈ L 2 ([0, T ]; H 6 ℓ (R 6 )) ∩ H 1 ([0, T ]; H 5 ℓ (R 6
))

The proof of full regularity is then completed by induction for (x, v) variable

ϕ(t)ψ(x) f ∈ L 2 ([0, T ]; H 7 ℓ (R 6 )) ∩ H 1 ([0, T ]; H 6 ℓ (R 6
)) and using the equation to prove the regularity for t variable.

Uniqueness of solutions

In this section, we prove precise versions for uniqueness results which will cover more general cases than those presented in Theorem 1.2.

We need the coercive estimate in a global version: For suitable function f , we say that f satisfies the global coercive estimate, if there exist constants c 0 > and C > 0 independent of t ∈]0, T [ such that

-(Q( f (t), h), h) L 2 (R 6 ) ≥ c 0 h 2 L 2 (R 3 x ;H s γ/2 (R 3 v )) -C h 2 L 2 (R 3 x ;L 2 (γ/2+s) + (R 3 v )) (4.1)
for any h ∈ L 2 (R 3

x ; S(R 3 v )). Using the notations introduced in Section 1, we prove the following precise version of Theorem 1.2, where we do not assume that solution is a perturbation around a normalized Maxwellian. Theorem 4.1. Assume that 0 < s < 1 and max -3, -3/2 -2s < γ < 2 -2s. Let f 0 ∈ Ẽ0 0 (R 6 ), 0 < T < +∞ and suppose that f ∈ Ẽm ([0, T ] × R 6

x,v ), m ≥ s is a weak solution to the Cauchy problem (1.1). If f is non-negative, then solution f is unique in the function space Ẽ2s ([0, T ] × R 6

x,v ). Moreover, if f is non-negative and satisfies the global coercive estimate (4.1), then the solution f is unique in the function space Ẽs ([0, T ] × R 6

x,v ). The same conclusion holds without the non-negativity of f if the term h 2

L 2 (R 3 x ;H s γ/2 (R 3 v )) in the condition (4.1) is replaced by |||h||| 2 Φ γ dx .
Remark 4.2. In the case where γ > -3/2 and f ∈ Ẽs ([0, T ] × R 6

x,v ) is non-negative, it follows that f coincides with any another solution f 2 ∈ Ẽ2s ([0, T ] × R 6

x,v ) without the coercivity condition (4.1). The next result proves the uniqueness of perturbative solutions around a normalized Maxwellian obtained in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] where we do not assume the non-negativity of solutions. Theorem 4.3. Assume that 0 < s < 1, max{-3, -3/2 -2s} < γ. Let ℓ 1 > 3/2 + max{(γ + 2s) + , |γ|/2}. Then there exists an ε 0 > 0 such that if f 1 (t), f 2 (t) ∈ Ẽs ([0, T ] × R 6

x,v ) are two solutions of the Cauchy problem (1.1) with the properties

µ -1/2 f j (t) -µ ∈ L ∞ ([0, T ] × R 3
x ; H s ℓ 1 ) , j = 1, 2 , and the smallness condition for f 1 (4.2)

||µ -1/2 f 1 (t) -µ || L ∞ ([0,T ]×R 3 x ;L 2 (R 3 )) ≤ ε 0 , then f 1 (t) ≡ f 2 (t) for all t ∈ [0, T ].
To study the uniqueness of solutions constructed in Theorem 1.4 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], we define another function space with exponential decay in the velocity variable as follows: For m ∈ R and for T > 0, set

Bm ([0, T ] × R 6 x,v ) = f ∈ C 0 ([0, T ]; D ′ (R 6 x,v )); ∃ ρ > 0 s.t. e ρ v 2 f ∈ L ∞ ([0, T ] × R 3 x ; L 2 (R 3 v )) ∩ L 2 ([0, T ]; L ∞ (R 3 x ; H m (R 3 v ))
) . We get the following refinement of the last part of Theorem 4.1, in the case γ + 2s ≤ 0. Theorem 4.4. Assume that 0 < s < 1 and max -3, -3/2 -2s < γ ≤ -2s. Let 0 < T < +∞ and suppose that f

1 (t) ∈ Bs ([0, T ] × R 6 x,v
) is a solution to the Cauchy problem (1.1) satisfying the global coercivity estimate (4.1) with the term h 2

L 2 (R 3 x ;H s γ/2 (R 3 v )) replaced by |||h||| 2 Φ γ dx . Then f 1 (t) coincides with any another solution f 2 (t) ∈ Bs ([0, T ] × R 6 x,v ).
If the Cauchy problem (1.1) admits two solutions f 1 (t), f 2 (t) ∈ Ẽs ([0, T ]×R 6 x,v ), then there exist ρ 0 , ρ 1 , ρ 2 > 0 such that

e ρ 0 v 2 f 0 ∈ L ∞ (R 3 x ; L 2 (R 3 v )), e ρ 1 v 2 f 1 , e ρ 2 v 2 f 2 ∈ L ∞ ([0, T ] × R 3 x ; H s (R 3 v )) .
Take 0 < ρ < min{ρ 0 , ρ 1 , ρ 2 } and κ > 0 sufficiently small such that ρ 2κ > T . Then we have

g 0 = e ρ v 2 f 0 ∈ L ∞ (R 3 x ; L 2 l (R 3 v )), g 1 = e (ρ-κt) v 2 f 1 , g 2 = e (ρ-κt) v 2 f 2 ∈ L ∞ ([0, T ] × R 3 x ; H s l (R 3 v
)) for any l ∈ N, and g 1 , g 2 are two solutions of the following Cauchy problem (4.3) gt

+ v • ∇ x g + κ(1 + |v| 2 ) g = Γ t ( g, g), g| t=0 = g 0 ,
where

Γ t (g, h) = µ κ (t) -1 Q(µ κ (t)g, µ κ (t)h) , µ κ (t) = µ(t, v) = e -(ρ-κt)(1+|v| 2 )
.

Set g = g 1g 2 . Then we have (4.4)

g t + v • ∇ x g + κ(1 + |v| 2 )g = Γ t (g 1 , g) + Γ t (g, g 2 ), g| t=0 = 0 .
4.1. Estimates for modified collisional operator. We now prepare several lemmas concerning the estimates for Γ t ( f, g), h L 2 , where L 2 = L 2 (R 3 v ). In this subsection, variables t and x are regarded as parameters. For the brevity we often write Γ and µ instead of Γ t and µ κ (t, v), respectively. All constants in estimates are uniform with respect to t ∈ [0, T ] and moreover they hold with µ k (t) replaced by µ 1/2 . Lemma 4.5. Let 0 < s < 1 and γ > max{-3, -2s -3/2}. Then for any β ∈ R we have

Γ t ( f, g) , h L 2 -Q(µ κ f, g), h L 2 (4.5) D(µ κ | f |, v β g) 1/2 f 1/2 L 2 h L 2 s+γ/2-β + f L 2 g L 2 s+γ/2+β h L 2 s+γ/2-β . Proof. We write Γ( f, g) , h L 2 -Q(µ f, g), h L 2 = B µ ′ * -µ * f * gh ′ dσdv * dv = 2 B (µ ′ * ) 1/2 -µ 1/2 * µ 1/2 * f * g ′ h ′ dσdv * dv + B (µ ′ * ) 1/2 -µ 1/2 * 2 f * gh ′ dσdv * dv + 2 B (µ ′ * ) 1/2 -µ 1/2 * h ′ f * µ 1/2 * g -g ′ dσdv * dv = D 1 + D 2 + D 3 .
By the Cauchy-Schwarz inequality we have for any β ∈ R

|D 3 | B (µ ′ * ) 1/2 -µ 1/2 * 2 | f * | v -β h ′ 2 dσdv * dv 1/2 × Bµ * | f * | v ′ 2β g -g ′ 2 dσdv * dv 1/2 = D 3 ( f, v -β h) 1/2 D β (µ f, g) 1/2 .
We have

D β (µ f, g) ≤ 2 D(µ | f |, v β g) + Bµ * | f * | v β -v ′ β 2 g 2 dvdv * dσ D(µ | f |, v β g) + f L 2 g 2 L 2 β+γ/2
, because it follows from the same arguments in the proof of Lemma 2.12 that

(4.6) v β -v ′ β sin θ 2 v β v * 2|β|+1 1 |v-v * |>1 + v β-1 |v -v * |1 |v-v * |≤1 . Note that (µ ′ * ) 1/2 -µ 1/2 κ, * 2 (µ ′ * ) 1/4 -µ 1/4 * ) 2 (µ ′ * ) 1/2 + µ 1/2 * min(|v ′ -v * |θ, 1) min(|v ′ -v ′ * |θ, 1)(µ ′ * ) 1/2 + min(|v ′ -v * |θ, 1) 2 µ 1/2 * .
By this decomposition we estimate

D 3 ( f, v -β h) D (1)
3 + D (2) 3 . It follows from the Cauchy-Schwarz inequality that

D (1) 3 2 B|v ′ -v ′ * | γ+2s min(|v ′ -v ′ * |θ, 1) 2 µ ′ * W -β h ′2 v ′ -(γ+2s) dσdv * dvdx × B|v ′ -v * | -(γ+2s) min(|v ′ -v * |θ, 1) 2 f 2 * W -β h ′2 v ′ γ+2s dσdv * dv h 2 L 2 s+γ/2-β f 2 L 2 h 2 L 2 s+γ/2-β
.

Here, we have taken the change of variables (v, v * ) → (v ′ , v ′ * ) and v → v ′ in the first and second factors, respectively, and moreover, in view of 2(γ + 2s) > -3, we have used the fact that

B|v -v * | γ+2s min(|v -v * | 2 θ 2 , 1)dσ µ * dv * |v -v * | 2(γ+2s) µ * dv * v 2(γ+2s) .
Since the estimation of D (2) 3 is quite similar as D (1) 3 we obtain

(4.7) D 3 ( f, v -β h) f L 2 h 2 L 2 s+γ/2-β
and hence

|D 3 | D(µ| f |, v β g) 1/2 f 1/2 L 2 h 2 L 2 s+γ/2-β + f L 2 g L 2 β+γ/2 h L 2 s+γ/2-β
The Cauchy-Schwarz inequality shows

|D 2 | 2 ≤ D 3 ( f, v -β h) B (µ ′ * ) 1/2 -µ 1/2 * 2 | f * | v ′ β g 2 dσdv * dv ,
so that it is easy to see, in view of (4.6),

|D 2 | f L 2 g L 2 s+γ/2+β h L 2 s+γ/2-β . Take the change of variables (v ′ , v ′ * ) → (v, v * ) and (v, v * ) → (v * , v) for D 1 .
Then we consider

D 1 = 2 B µ 1/2 -µ ′ ) 1/2 µ 1/2 f ) ′ (gh) * dσdv * dv = 2 B ∇µ 1/2 (v ′ ) • (v -v ′ ) µ 1/2 f ′ (gh) * dσdv * dv + 1 0 B ∇ 2 µ 1/2 (v ′ + τ(v -v ′ ))(v -v ′ ) 2 µ 1/2 f ′ (gh) * dσdv * dvdτ = D 1,1 + D 1,2 ,
by using the Taylor formula

µ 1/2 -µ ′1/2 = ∇µ 1/2 (v ′ ) • (v -v ′ ) + 1 2 1 0 ∇ 2 µ 1/2 (v ′ + τ(v -v ′ ))(v -v ′ ) 2 dτ . Note | ∇ 2 µ 1/2 κ (v ′ + τ(v -v ′ ))(v -v ′ ) 2 | |v ′ -v * | 2 (1 -cos θ).
and devide

D 1,1 = 2 {v ′ -v * | 2 (1-cos θ)≤1} + {v ′ -v * | 2 (1-cos θ)>1}
.

Then it follows from the spherical symmetry that the first term of the decomposition D 1,1 vanishes, so that we can estimate by the change of variables

v → v ′ |D 1 | |v ′ -v * | γ µ 1/2 | f | ′ S 2 b(cos θ) min θ 2 |v ′ -v * | 2 , 1 dσ (gh) * dv ′ dv * |v ′ -v * | 2(γ+2s) µ ′ g 2 * v * -(γ+2s)+β dv ′ dv * 1/2 × f ′2 h 2 * v * γ+2s-β dv ′ dv * 1/2 f L 2 g L 2 s+γ/2+β h L 2 s+γ/2-β
Summing up above estimates we obtain the desired estimate.

Since Lemma 2.11 holds with √ µ replaced by µ κ (t, v), the combination of Lemma 2.11 and Lemma 4.5 with

β = 0 implies Lemma 4.6. Let 0 < s < 1, γ > max{-3, -2s -3/2}. If f ≥ 0 then we have Γ t ( f, g), g L 2 ≤ - 1 4 D(µ κ f, g) (4.8) + C min f L 2 g 2 H s ′ γ/2 + g 2 L 2 s+γ/2 , f H 2s ′ g 2 L 2 s+γ/2 for any s ′ ∈]0, s[ satisfying γ + 2s ′ > -3/2 and s ′ < 3/4.
Furthermore, if γ > -3/2 then the second term on the right hand side can be replaced by C f L 2 g 2

L 2 s+γ/2 . Lemma 4.7. Let 0 < s < 1, γ > max{-3, -2s -3/2}. For any ℓ ∈ R and m ∈ [0, s] we have Γ t ( f, g), h L 2 f L 2 g H s+m ℓ+(γ+2s) + h H s-m -ℓ . (4.9) Furthermore Γ t ( f, g), g L 2 f L 2 |||g||| 2 Φ γ . (4.10)
Proof. Since Lemma 2.12 holds with √ µ replaced by µ κ (t, v), in view of (2.11) we have

D(µ κ f, g) f L 2 |||g||| 2 Φ γ f L 2 g 2 H s
(s+γ/2) + . Applying this to the right hand side of (4.5), by Proposition 2.9 we obtain the desired estimate (4.9). The second estimate can be obtained by using Proposition 2.13 instead of Proposition 2.9.

For α > 3/2, set ϕ(v, x) = (1 + |v| 2 + |x| 2 ) α/2 and W ϕ,l = v l ϕ(v, x) = (1 + |v| 2 ) ℓ/2 (1 + |v| 2 + |x| 2 ) α/2 . Lemma 4.8. If ϕ(v, x) = (1 + |v| 2 + |x| 2 ) α/2 for α > 3/2 and W ϕ,l = v l ϕ(v, x) = (1 + |v| 2 ) ℓ/2 (1 + |v| 2 + |x| 2 ) α/2 .
for ℓ ∈ R then we have

|∂ β 1 x ∂ β 2 v W ϕ,ℓ (v)| v -|β 1 |-|β 2 | W ϕ,ℓ (v) , (4.11) W ϕ,ℓ (v ′ ) -W ϕ,ℓ (v) sin θ 2 |v -v ′ * | v ′ * α+|ℓ-1| v W ϕ,ℓ (v) , (4.12) ∇ 2 W ϕ,ℓ (v + τ(v ′ -v)) v * α+|ℓ-2| v 2 W ϕ,ℓ (v) , τ ∈ [0, 1] . (4.13)
Proof. The first inequality follows from the direct calculation. Since

W ϕ,ℓ (v ′ ) -W ϕ,ℓ (v) = 1 0 ∇ v W ϕ,ℓ (v + τ(v ′ -v))dτ • (v ′ -v) and since |v -v ′ | = sin(θ/2)|v -v * |, |v -v * | ∼ |v -v ′ * |
, for the proof of (4.12) it suffices to show

(4.14) ∇W ϕ,ℓ (v τ ) W ϕ,ℓ (v τ ) v τ v ′ * α+(ℓ-1) + v W ϕ,ℓ (v) , v τ = v + τ(v ′ -v) .
For a ∈ R we have

1 + a 2 + |v| 2 1 + a 2 + |v -v ′ * | 2 + |v ′ * | 2 1 + a 2 + |v τ -v ′ * | 2 + |v ′ * | 2 1 + a 2 + |v τ | 2 + |v ′ * | 2 (1 + a 2 + |v τ | 2 ) v ′ * 2 , from which we get ϕ(v τ , x) -1 ϕ(v, x) -1 v ′ *
α by setting a = |x|. Putting a = 0 in the above inequality we have

v v τ v ′ * . Since v τ v v ′ * holds similarly we have v τ ℓ-1 v ℓ-1 v ′ * |ℓ-1|
, which concludes the second inequality of (4.14). (4.13) also follows from the similar observation. Lemma 4.9. Let 0 < s < 1 and γ > max{-3, -2s -3/2}. Then for any ℓ ≥ 0 we have

W ϕ,ℓ Γ t ( f, g) -Γ t ( f, W ϕ,ℓ g), h L 2 D(µ κ | f |, h) 1/2 f 1/2 L 2 W ϕ,ℓ g L 2 γ/2 + f H (2s ′ -1) + W ϕ,ℓ g L 2 s ′ +γ/2 h L 2 s+γ/2 , for any s ′ ∈](2s -1) + , s[ satisfying γ + 2s ′ > -3/2. Proof. Note that W ϕ,ℓ Γ( f, g) -Γ( f, W ϕ,ℓ g), h L 2 = B f * µ * h ′ g W ′ ϕ,l -W ϕ,l dvdv * dσ = B f * µ * 1/2 h ′ -µ * 1/2 h µ ′ * 1/2 g W ′ ϕ,l -W ϕ,l dvdv * dσ + B f * µ * 1/2 h µ ′ * 1/2 -µ * 1/2 g W ′ ϕ,l -W ϕ,l dvdv * dσ + B f * µ * 1/2 h µ * 1/2 g W ′ ϕ,l -W ϕ,l dvdv * dσ = A 1 + A 2 + A 3 .

It follows from the Cauchy-Schwarz inequality that

A 2 1 ≤ B | f * | µ * 1/2 h ′ -µ * 1/2 h 2 dvdv * dσ × B | f * | µ ′ * 1/2 g W ′ ϕ,l -W ϕ,l 2 dvdv * dσ =A 1,1 × A 1,2 .
Writing

µ * 1/2 h ′ -µ * 1/2 h = µ * 1/2 h ′ -h + h ′ µ ′ * 1/2 -µ * 1/2
we obtain easily

A 1,1 ≤ 2 B(µ | f |) * h ′ -h 2 dvdv * dσ + B | f * | µ ′ * 1/2 -µ * 1/2 2 h ′2 dvdv * dσ = 2(D(µ | f |, h) + D 3 ( f, h)) D(µ | f |, h) + f L 2 h 2 L 2 s+γ/2
, where we have used (4.7). By (4.12) and the Cauchy-Schwarz inequality we have

A 1,2 2 b sin 2 (θ/2) |v -v ′ * | γ+2 v 2 µ ′ * 1/2 | f * | W ϕ,l g 2 dvdv * dσ 2 b sin 2 (θ/2) |v -v ′ * | 2(γ+2) v γ+4 µ ′ * W ϕ,l g 2 dσdv ′ * dv × b sin 2 (θ/2) f 2 * v γ W ϕ,l g 2 dσdv * dv W ϕ,l g 2 L 2 γ/2 f 2 L 2 W ϕ,l g 2 L 2 γ/2
, where we have used the change of variables v * → v ′ * . Hence we have

|A 1 | D(µ| f |, h) + f 1/2 L 2 h L 2 s+γ/2 f 1/2 L 2 W ϕ,l g L 2 γ/2
. By using the similar formula as (4.12) with v ′ * replaced by v * we have

µ 1/4 * µ ′ * 1/2 -µ * 1/2 W ′ ϕ,l -W ϕ,l min |v -v * | 2 θ 2 , |v -v * |θ W ϕ,l v -1 ,
so that for any δ > 0

|A 2 | |v -v * | γ |v -v * | 2s + 1 |v-v * | 1 |v -v * | (2s-1+δ) + +1 µ 1/4 * f * W ϕ,l g h v dvdv * f L 2 W ϕ,l g L 2 (2s-1+δ) + /2+γ/2 h L 2 (2s-1+δ) + /2+γ/2 .
In order to estimate A 3 we use the Taylor expansion for W ϕ,l -W ′ ϕ,l of second order. Then we have

A 3 = B f * µ * gh ∇ v W ϕ,l (x, v) • (v ′ -v)dvdv * dσ + 1 2 1 0 dτ B f * µ * gh ∇ 2 v W ϕ,l (x, v + τ(v ′ -v))(v ′ -v) 2 dvdv * dσ =A 3,1 + A 3,2 . Setting k = v-v * |v-v * | and writing v ′ -v = 1 2 |v -v * | σ -(σ • k)k + 1 2 ((σ • k) -1)(v -v * ),
we have

A 3,1 = 1 2 B f * µ * gh ∇ v W ϕ,l (x, v) • (v -v * ) cos θ -1 dvdv * dσ because it follows from the symmetry that S 2 b(σ • k) σ -(σ • k)k dσ = 0. Therefore, for any 0 ≤ s ′ < s satisfying γ + 2s ′ < -3/2 we have, in view of (4.11), |A 3,1 | |v -v * | 2(γ+2s ′ ) µ 2 * dv * 1/2 f 2 * |v -v * | 2(2s ′ -1) + dv * 1/2 |W ϕ,l-1 g||h|dv f H (2s ′ -1) + W ϕ,l g L 2 s ′ +γ/2 h L 2 s ′ +γ/2 , by means of |∇ v (W ϕ,l )(x, v)| W ϕ,l-1 .
The better bound holds for |A 3,2 | since it follows from (4.13) that

| ∇ 2 v W ϕ,l (v + τ(v ′ -v))(v ′ -v) 2 | v * α+|ℓ-2| W ϕ,l-2 θ 2 |v -v * | 2 .
Therefore we have

|A 3 | f H (2s ′ -1) + W ϕ,l g L 2 s ′ +γ/2
h L 2 s+γ/2 . Summing up above estimates we obtain the conclusion.

Proofs of the uniqueness Theorems.

Using the notations introduced in subsection ??, the proof of Theorem 4.1 is reduced to Proposition 4.10. Assume that 0 < s < 1 and max -3, -3/2-2s < γ < 2-2s. Let ℓ 0 > 3/2+max{1, (γ+2s) + } and g 0 ∈ L ∞ (R 3

x ; L 2 ℓ 0 (R 3 v )). Suppose that the Cauchy problem (4.3) admits two solutions

g 1 , g 2 ∈ L ∞ ([0, T ] × R 3 x ; H m ℓ 0 (R 3 v )) . Then g 1 ≡ g 2 in [0, T ], 1) if m = 2s and g 1 ≥ 0. When γ > -3/2 , we can suppose g 1 ∈ L ∞ ([0, T ] × R 3 x ; H s ℓ 0 (R 3 v )).
2) if m = s and the coercivity inequality (4.1) is satisfied for f 1 = µ κ (t)g 1 ≥ 0.

3) if m = s and f = µ κ (t)g 1 satisfies the following strong coercivity estimate

-(Q( f (t), h), h) L 2 (R 6 ) ≥ c 0 |||h||| 2 Φ γ dx -C h 2 L 2 (R 3 x ;L 2 (γ/2+s) + (R 3 v )) . (4.15) Proof. Let S (τ) ∈ C ∞ 0 (R) satisfy 0 ≤ S ≤ 1 and S (τ) = 1, |τ| ≤ 1 ; S (τ) = 0, |τ| ≥ 2. Set S N (D x ) = S (2 -2N |D x | 2
) and multiply W ϕ,l S N (D x ) 2 W ϕ,l g to (4.4), where we choose ℓ, α such that

ℓ 0 -max{1, (γ + 2s) + } > ℓ > α > 3/2.
Integrating and letting N → ∞, we have 1 2

d dt W ϕ,l g(t) 2 L 2 (R 6 ) + κ W ϕ,l+1 g(t) 2 L 2 (R 6 ) = W ϕ,l Γ t (g 1 , g) + W ϕ,l Γ t (g, g 2 ) , W ϕ,l g L 2 (R 6 ) -(v • ∇ x (ϕ -1 )W l g, W ϕ,l g) L 2 (R 6 ) , because (v • ∇ x S N (D x )W ϕ,l g, S N (D x )W ϕ,l g) L 2 (R 6 ) = 0.
The second term on the right hand side is estimated by W ϕ,l g 2 L 2 (R 6 ) because of (4.11). Write the first term on the right hand side as

Γ t (g 1 , W ϕ,l g) , W ϕ,l g L 2 (R 6 ) + W ϕ,l Γ t (g 1 , g) -Γ t (g 1 , W ϕ,l g) , W ϕ,l g L 2 (R 6 ) + Γ t (g, g 2 ) , W ϕ,l W ϕ,l g L 2 (R 6 ) = B 1 + B 2 + B 3 .
If g 1 ≥ 0 then it follows from Lemma 4.6 that

B 1 ≤ - 1 4 D(µ κ g 1 , W ϕ,l g)dx (4.16) + C min g 1 L ∞ t,x (L 2 ) W ϕ,l g 2 L 2 x (H s ′ γ/2 ) + W ϕ,l g 2 L 2 x (L 2 s+γ/2 ) , g 1 L ∞ t,x (H 2s ′ ) W ϕ,l g 2 L 2 x (L 2 s+γ/2
) . We notice that the last term can be replaced by

g 1 L ∞ t,x (L 2 ) W ϕ,l g 2 L 2 x (L 2 s+γ/2 ) if γ > -3/2.
By means of Lemma 4.9 we obtain for any δ > 0

B 2 ≤ δ D(µ κ |g 1 | , W ϕ,l g)dx + C δ g 1 L ∞ t,x (H (2s-1) + ) W ϕ,l g L 2 x (L 2 s ′ +γ/2 ) W ϕ,l g L 2 x (L 2 s+γ/2 ) . (4.17) Lemma 4.7 with ℓ = ℓ -γ/2 implies that for m = s, 0 B 3 x -α g L 2 (R 3 ) g 2 H s+m ℓ-γ/2+(2s+γ) + (R 3 ) x α v ℓ W ϕ,l W ϕ,l g H s-m γ/2 dx (4.18) g 2 L ∞ t,x (H s+m ℓ-γ/2+(2s+γ) + ) W ϕ,α g L 2 x (L 2 s+γ/2 ) W ϕ,ℓ g L 2 x (H s-m γ/2 ) , because x -α ≤ W ϕ,α and x α v -ℓ W ϕ,l is a bounded operator on H s-m . Note that for any δ > 0 W ϕ,ℓ+(s+γ/2) + g(t) 2 L 2 (R 6 ) ≤ δ W ϕ,ℓ+1 g(t) 2 L 2 (R 6 ) + C δ W ϕ,ℓ g(t) 2 L 2 (R 6 ) . If g 1 , g 2 ∈ Ẽ2s ([0, T ] × R 6
) and g 1 ≥ 0 then by summing up (4.16), (4.17) and (4.18) with m = s we have

d dt W ϕ,l g(t) 2 L 2 (R 6 ) ≤ C κ g 1 L ∞ t,x (H 2s ) + g 2 L ∞ t,x (H 2s ℓ-γ/2+(2s+γ) + ) W ϕ,l g(t) 2 L 2 (R 6 ) ,
where g 1 L ∞ t,x (H 2s ) can be replaced by

g 1 L ∞ t,x (L 2 ) if γ > -3/2.
Here it should be noted that the term D(µ κ g 1 , W ϕ,l g) ≥ 0 follows from the non-negativity of g 1 . Therefore, W ϕ,l g(0) L 2 (R 6 ) = 0 implies W ϕ,l g(t) L 2 (R 6 ) = 0 for all t ∈ [0, T ]. And this gives g 1 = g 2 , and concludes the part 1) of Proposition 4.10.

For the part 2) of Proposition 4.10, using h 2

H s ′ γ/2 ≤ δ h 2 H s γ/2 + C δ h 2 L 2 γ/2
and summing up (4.16), (4.17) and (4.18) with m = 0, we have

d dt W ϕ,l g(t) 2 L 2 (R 6 ) ≤ - 1 16 D( f 1 (t), W ϕ,l g(t))dx + κ W ϕ,ℓ+1 g(t) 2 L 2 (R 6 ) + δ g 1 L ∞ t,x (H (2s-1) + ) + g 2 L ∞ t,x (H s ℓ-γ/2+(2s+γ) + ) W ϕ,l g(t) 2 L 2 x (H s γ/2 ) + C κ,δ g 1 L ∞ t,x (H (2s-1) + ) + g 2 L ∞ t,x (H s ℓ-γ/2+(2s+γ) + ) W ϕ,l g(t) 2 L 2 (R 6 ) ,
then the coercivity condition (4.1)(with (γ/2 + s) + < 1) together with (2.12) leads us to

d dt W ϕ,l g(t) 2 L 2 (R 6 ) g 1 L ∞ t,x (H (2s-1) + ) + g 2 L ∞ t,x (H s ℓ-γ/2+(2s+γ) + ) W ϕ,l g(t) 2 L 2 (R 6 ) , (4.19)
where it should be noted that (2.12) holds with µ 1/2 replaced by µ κ (t, v). Thus, the part 2) of Proposition 4.10 is proved.

When g 1 is not necessarily non-negative, by using Lemma 4.5 we obtain

B 1 ≤ Q(µ κ g 1 , W ϕ,l g) , W ϕ,l g L 2 (R 6 ) + δ D(µ κ |g 1 | , W ϕ,l g)dx + C δ W ϕ,l g 2 L 2 x (L 2 s+γ/2 )
instead of (4.16). Since Lemma 2.12 holds with √ µ replaced by µ κ , by means of (4.15) we get

B 1 ≤ -(c 0 -δ) |||W ϕ,l g)||| 2 Φ γ dx + C δ W ϕ,l g 2 L 2 x (L 2 (s+γ/2) + ) . (4.20)
This estimate and (4.18), together with (4.17) applied by Lemma 2.12, imply (4.19). Hence the part 3) of Proposition 4.10 is also proved.

Proof of Theorem 4.3 :

If we set g j (t) = µ -1/2 f j (t) ( j = 1, 2) and g = g 1g 2 , then we have

g t + v • ∇ x g = Γ(g 1 , g) + Γ(g, g 2 ), g| t=0 = 0 ,
where Γ(g, h) = µ -1/2 Q(µ 1/2 g, µ 1/2 h). Take the inner product with W ϕ,ℓ S N (D x ) 2 W ϕ,ℓ g where we choose ℓ, α such that ℓ 1 -(γ + 2s) > ℓ > α > 3/2. Then we obtain 1 2

d dt W ϕ,ℓ g(t) 2 L 2 (R 6 ) = W ϕ,ℓ Γ(g 1 , g) + W ϕ,ℓ Γ(g, g 2 ) , W ϕ,ℓ g L 2 (R 6 ) -(v • ∇ x (ϕ -1 )W ℓ g, W ϕ,ℓ g) L 2 (R 6 ) ,
where the second term on the right hand side is estimated by W ϕ,ℓ g 2 L 2 (R 6 ) because of (4.11). We write the first term on the right hand side as

Γ(g 1 , W ϕ,ℓ g) , W ϕ,ℓ g L 2 (R 6 ) + B 2 + B 3 ,
where B 2 , B3 are defined by the same way as the above B 2 , B 3 with Γ t replaced by Γ and satisfy the similar estimates as (4.17) and (4.18), respectively, that is,

B 2 ≤ δ D(µ 1/2 |g 1 |, W ϕ,ℓ g)dx + C δ g 1 L ∞ t,x (H (2s-1) + ) W ϕ,ℓ g L 2 x (L 2 s ′ +γ/2 ) W ϕ,ℓ g L 2 x (L 2 s+γ/2 ) , B 3 g 2 L ∞ t,x (H s ℓ-γ/2+(2s+γ) + ) W ϕ,α g L 2 x (L 2 γ/2 ) W ϕ,ℓ g L 2 x (H s γ/2 ) g 2 L ∞ t,x (H s ℓ-γ/2+(2s+γ) + δ |||W ϕ,ℓ g||| 2 Φ γ dx + C δ W ϕ,ℓ g L 2 (R 6 ) ,
where the non-isotropic norm ||| • ||| Φ γ is recalled in (2.10) . By means of Lemma 2.12, we have

B 2 ≤ δ g 1 L ∞ t,x (H (2s-1) + ) |||W ϕ,ℓ g||| 2 Φ γ dx + C ′ δ g 1 L ∞ t,x (H (2s-1) + ) W ϕ,ℓ g 2 L 2 (R 6 ) .
On the other hand, it follows from Proposition 2.1 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF] and (4.10) that for suitable C 1 , C 2 > 0 we have

Γ(g 1 , W ϕ,ℓ g) , W ϕ,ℓ g L 2 (R 6 ) = -L 1 W ϕ,ℓ g , W ϕ,ℓ g L 2 (R 6 ) + Γ( g1 , W ϕ,ℓ g) , W ϕ,ℓ g L 2 (R 6 ) (4.21) ≤ -C 1 |||W ϕ,ℓ g||| 2 Φ γ dx + C 2 sup [0,T ]×R 3 x g1 L 2 (R 3 v ) |||W ϕ,ℓ g||| 2 Φ γ dx + ||W ϕ,ℓ g|| 2 L 2 (R 6 ,
where g 1 = √ µ + g1 . Therefore, (2.11) and the smallness condition (4.2) imply

d dt W ϕ,ℓ g(t) 2 L 2 (R 6 ) ≤ -C 1 -C 2 ε 0 -δ g 1 L ∞ t,x (H (2s-1) + ) + g 2 L ∞ t,x (H s ℓ+2s+γ/2 ) |||W ϕ,ℓ g||| 2 Φ γ dx + C δ g 1 L ∞ t,x (H (2s-1) + ) + g 2 L ∞ t,x (H s ℓ+2s+γ/2 ) W ϕ,ℓ g(t) 2 L 2 (R 6 )
. which shows g(t) = 0 for all t ∈ [0, T ] if ε 0 < C 1 /C 2 . Thus we have proved Theorem 4.3.

Proof of Theorem 4.4

Let now f j (t) ∈ Bs ([0, T ] × R 6

x,v ), ( j = 1, 2) and set g j (t) = µ κ (t) -1 f j (t) for a suitable µ κ (t) = e -(ρ-κt)(1+|v| 2 ) . Then we have for any ℓ ∈ N

g j (t) ∈ L ∞ ([0, T ] × R 3 x ; L 2 ℓ (R 3 v )) ∩ L 2 ([0, T ]; L ∞ (R 3 x ; H m ℓ (R 3 v ))).
The proof of Theorem 4.4 is reduced to Proposition 4.11. Assume that 0 < s < 1 and max{-3, -3/2 -2s} < γ ≤ -2s. Let 0 < T < +∞ and ℓ 2 ≥ 3. Suppose that the Cauchy problem (4.3) admits two solutions

g 1 , g 2 ∈ L ∞ ([0, T ] × R 3 x ; L 2 ℓ 2 (R 3 v )) ∩ L 2 ([0, T ]; L ∞ (R 3 x ; H s ℓ 2 (R 3 v ))) . If (4.15) is satisfied for f = µ κ (t)g 1 then g 1 (t) ≡ g 2 (t) for all t ∈ [0, T ] .
Proof. Noting γ + 2s ≤ 0, we estimate more carefully B 2 , B 3 in the proof of Proposition 4.10. It follows from Lemma 4.9 and Lemma 2.12 that

B 2 ≤ δ |||W ϕ,l g||| Φ γ dx + C δ g 1 (t) L ∞ x (H s ) W ϕ,l g(t) 2 L 2 (R 6 ) .
Lemma 4.7 with ℓ = ℓγ/2 and m = 0 yields

B 3 g 2 (t) L ∞ x (H s ℓ+|γ|/2 ) W ϕ,α g(t) L 2 (R 6 ) W ϕ,ℓ g(t) L 2 x (H s γ/2 ) ≤ δ W ϕ,ℓ g(t) 2 L 2 x (H s γ/2 ) + C δ g 2 (t) 2 L ∞ x (H s ℓ+|γ|/2 ) W ϕ,α g(t) 2 L 2 (R 6 ) .
Above estimates for B j ( j = 2, 3) and (4.20) imply that max

t∈[0,T 1 ] W ϕ,l g(t) 2 L 2 (R 6 ) ≤ W ϕ,l g(0) 2 L 2 (R 6 ) + ε(T 1 ) max t∈[0,T 1 ] W ϕ,l g(t) 2 L 2 (R 6 ) ,
where

ε(T 1 ) T 1 + T 1 g 1 L 2 ([0,T 1 ];L ∞ x (H s )) + g 2 2 L 2 ([0,T 1 ];L ∞ x (H s ℓ+|γ|/2 ))
. By assumption ε(T 1 ) → 0 as T 1 → 0 . Therefore there exists a T * > 0 such that g(t) ≡ 0 for t ∈ [0, T * ]. Replacing the initial time 0 by T * if needed, we finally obtain g(t) ≡ 0 for t ∈ [0, T ]. 4.3. Uniqueness of known solutions. Firstly we consider the uniqueness of global solutions given in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF]. Theorem 4.3 is applicable to show the uniqueness of global solutions in Theorem 1.5 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], and also solutions in Theorem 1.1 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] because the global solutions given there are of the form µ + √ µg with

g(t, x, v) ∈ L ∞ ([0, ∞[; H m ℓ (R 6
)) for m ≥ 6 and a suitable ℓ. It should be noted that the uniqueness holds under the smallness condition (4.2) of the perturbation g, without the non-negativity of solution µ + µ 1/2 g.

It follows from Corollary 2.15 that the smallness condition (4.2) implies (4.15) for the global solution given in Theorem 1.4 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], because g there satisfies g L ∞ ([0,∞);H 3 (R 3

x ;L 2 (R 3 v )) < ε 0 and for any 0

< T < +∞ T 0 |α|≤3 |||∂ α x g(t, x)||| 2 Φ γ dx dt < +∞ .
Therefore Theorem 4.4 shows the uniqueness of the solution given in Theorem 1.4 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF] by means of the Sobolev embedding.

In [START_REF] Alexandre | Bounded solutions of the Boltzmann equation in the whole space[END_REF], bounded solutions of the Boltzmann equation in the whole space have been constructed without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, it has been shown that if the initial data is non-negative and belongs to a uniformly local Sobolev space with the Maxwellian decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. Since solutions there are non-negative and belong to E 2s ([0, T ] × R 6

x,v ), Theorem 4.1 yields their uniqueness.

Non-negativity of solutions

The purpose of this section is to show the non-negativity of solutions constructed in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF], where the solution f = µ + √ µg is a perturbation around a normalized Maxwellian distribution µ(v), that means g is solution of following Cauchy problem :

(5.1)

∂ t g + v • ∇ x g + L(g) = Γ(g, g), g| t=0 = g 0 , where L(g) = -Γ( √ µ, g) -Γ(g, √ µ) = L 1 (g) + L 2 (g).
It is the limit of a sequence constructed successively by the following linear Cauchy problem, (5.2)

∂ t f n+1 + v • ∇ x f n+1 = Q( f n , f n+1 ), f n+1 | t=0 = f 0 = µ + µ 1/2 g 0 ≥ 0 ,
if one returns to the original Boltzmann equation. Hence the non-negativity of solution comes from the following induction argument: Let f 0 = f 0 = µ + µ 1/2 g 0 ≥ 0, suppose that (5.3)

f n = µ + µ 1/2 gn ≥ 0 ,
for some n ∈ N. Then (5.3) is true for n + 1.

Proposition 5.1. Assume that max{-3,

-3 2 -2s} < γ < 2 -2s. Let { f n } is a sequence of solutions of Cauchy problem (5.2) with ∃ρ > 0 ; e ρ v 2 f n (t, x, v) ∈ L ∞ ([0, T ] × R 3 x ; H N (R 3 v )) for ∀n = 1, 2, 3, • • • , for some N ≥ 4.
Then for any n ∈ N, f n ≥ 0 on [0, T ] implies f n+1 ≥ 0 on the same interval.

Proof. Taking a κ > 0 such that ρ 2κ > T , we set g n (t, x, v) = µ κ (v) -1 f n (t, x, v) with µ κ (t) = e -(ρ-κt) v 2 then it follows from (5.2) that (5.4)

∂ t g n+1 + v • ∇ x g n+1 + κ v 2 g n+1 = Γ t (g n , g n+1 ) .
We notice that for any ℓ ∈ N

g n ∈ L ∞ ([0, T ] × R 3 x ; H N ℓ (R 3 v ))
so that sup t,x |||g n ||| Φ γ < ∞. If g satisfies |||g||| < ∞ and if g ± = ± max(±g, 0), then we have

|||g + ||| 2 Φ γ + |||g -||| 2 Φ γ ≤ |||g||| 2 Φ γ , because |||g||| 2 Φ γ = Bµ * (g ′ + + g ′ -) -(g + + g -) 2 + B(g * ,+ + g * ,-) 2 µ ′ - √ µ 2 =|||g + ||| 2 Φ γ + |||g -||| 2 Φ γ -2 b(cos θ)Φ(|v -v * |)µ * g ′ + g -+ g + g ′ -
and the third term is non-negative. Therefore

g n -∈ L ∞ t,x (H s ℓ (R 3 )). Take the convex function β(s) = 1 2 (s -) 2 = 1 2 s (s -) with s -= min{s, 0}. Let ϕ(v, x) = (1 + |v| 2 + |x| 2 ) α/2
with α > 3/2, and notice that

β s (g n+1 )ϕ(v, x) -2 = d ds β (g n+1 )ϕ(v, x) -2 = g n+1 -ϕ(v, x) -2 ∈ L ∞ ([0, T ]; L 1 (R 3 x ; L 2 (R 3 v )), because gn+1 ∈ L ∞ t (H N (R 6 )) with N ≥ 4 implies g n+1 ∈ L ∞ t,x (L 2 v ). Multiplying (5.4) by β s (g n+1 )ϕ(v, x) -2 = g n+1 -ϕ(v, x) -2 we have d dt R 6 β(g n+1 )ϕ(v, x) -2 dxdv + κ R 6 v 2 β(g n+1 )ϕ(v, x) -2 dxdv = R 6 Γ t (g n , g n+1 ) β s (g n+1 )ϕ(v, x) -2 dxdv - R 6 v • ∇ x (β(g n+1 )ϕ(v, x) -2 )dxdv - R 6 ϕ(v, x) 2 v • ∇ x ϕ(v, x) -2 β(g n+1 )ϕ(v, x) -2 dxdv,
where the first term on the right hand side is well defined because g n+1 , g n+1 -∈ L ∞ t,x (H s ℓ ). Since the second term vanishes and |v • ∇ x ϕ(v, x) -2 | ≤ Cϕ(v, x) -2 , we obtain

d dt R 6 β(g n+1 )ϕ(v, x) -2 dxdv + κ R 6 v 2 β(g n+1 )ϕ(v, x) -2 dxdv (5.5) ≤ R 6 Γ t (g n , g n+1 )β s (g n+1 )ϕ(v, x) -2 dxdv + C R 6 β(g n+1 )ϕ(v, x) -2 dxdv.
The first term on the right hand side is equal to

R 6 Γ t (g n , g n+1 -)g n+1 -ϕ(v, x) -2 dxdv + B µ κ, * (g n * ) ′ (g n+1 + ) ′ g n+1 -ϕ(v, x) -2 dvdv * dσdx = A 1 + A 2 .
From the induction hypothesis, the second term A 2 is non-positive.

On the other hand, we have

A 1 = (Γ t (g n , g n+1 -), ϕ(v, x) -2 g n+1 -) L 2 (R 3 v ) dx = (Γ t (g n , ϕ(v, x) -1 g n+1 -), ϕ(v, x) -1 g n+1 -) L 2 (R 3 v ) dx + (ϕ(v, x) -1 Γ t (g n , g n+1 -) -Γ t (g n , ϕ(v, x) -1 g n+1 -), ϕ(v, x) -1 g n+1 -) L 2 (R 3 v ) dx. = A 1,1 + A 1,2 .
It follows from Lemma 4.6 that

A 1,1 ≤ - 1 4 D(µ κ g n , ϕ(v, x) -1 g n+1 -)dx + C g n L ∞ ([0,T ]×R 3 x ;H 2s ) ϕ(v, x) -1 g n+1 - 2 L 2 s+γ/2 dx.
By means of Lemma 4.9 we have

|A 1,2 | ≤ δ D(µ κ g n , ϕ(v, x) -1 g n+1 -)dx + g n L ∞ ([0,T ]×R 3 x ;H s ) ϕ(v, x) -1 g n+1 - 2 L 2 s+γ/2 dx + C δ g n L ∞ ([0,T ]×R 3 x ;H s ) ϕ(v, x) -1 g n+1 - 2 L 2 dx.
Therefore

A 1 g n L ∞ ([0,T ]×R 3 x ;H 2s ) ϕ(v, x) -1 g n+1 - 2 L 2 s+γ/2 dx ,
where we have used the fact that D(µ κ g n , ϕ(v, x) -1 g n+1 -) ≥ 0 because of the induction hypothesis µ κ g n = f n ≥ 0. If γ + 2s < 2 then we have

ϕ(v, x) -1 g n+1 - 2 L 2 s+γ/2 dx ≤ δ R 6 v 2 β(g n+1 )ϕ(v, x) -2 dxdv + C δ R 6 β(g n+1 )ϕ(v, x) -2 dxdv.
Therefore, from (5.5) we have 6 . This implies that f n+1 ≥ 0 and then it completes the proof of the proposition.

d dt R 6 β(g n+1 )ϕ(v, x) -2 dxdv 1 + g n L ∞ ([0,T ]×R 3 x ;H 2s ) R 6 β(g n+1 )ϕ(v, x) -2 dxdv . (5.6) Since β(g n+1 )| t=0 = 0 we obtain R 6 β(g n+1 )ϕ(v, x) -2 dxdv = 0 for all t ∈ [0, T ], which implies that g n+1 (t, x, v) ≥ 0 for (t, x, v) ∈ [0, T ] × R
Proposition 5.2. Assume that γ ≥ 2 -2s. Let { f n } with f n = µ + µ 1 2 gn be sequence of solutions of Cauchy problem (5.2) with sup [0,T ]×R 3 x ||| gn ||| Φ γ being sufficiently small uniform in n. If e 1 2 v 2 f n (t, x, v) ∈ L ∞ ([0, T ] × R 3 x ; H N (R 3 v )) for ∀n = 1, 2, 3, • • • , for some N ≥ 4, then for any n ∈ N, f n ≥ 0 on [0, T ] implies f n+1 ≥ 0 on the same interval.
Proof. This case can be treated by the same way as the proof of Theorem 4.3. In fact, if we put g n = µ -1/2 f n , then we have

d dt R 6 β(g n+1 )ϕ(v, x) -2 dxdv ≤ R 6 Γ(g n , g n+1 )β s (g n+1 )ϕ(v, x) -2 dxdv + C R 6 β(g n+1 )ϕ(v, x) -2 dxdv
instead of (5.5). We need to estimate A 1,1 and A 1,2 defined by replacing Γ t by Γ in above A 1,1 and A 1,2 . By the same way as in (4.21) we have for suitable

C 1 , C 2 > 0 A 1,1 ≤ -C 1 |||ϕ(v, x) -1 g n+1 -||| 2 Φ γ dx + C 2 sup [0,T ]×R 3 x ||| gn ||| Φ γ |||ϕ(v, x) -1 g n+1 -||| 2 Φ γ dx + ||ϕ(v, x) -1 g n+1 -|| 2 L 2 dx .
It follows from Lemma 4.9 and Lemma 2.12 that

| A 1,2 | ≤ δ 1 + sup [0,T ]×R 3 x || gn || L 2 |||ϕ(v, x) -1 g n+1 -||| 2 Φ γ dx + C δ g n L ∞ ([0,T ]×R 3 x ;H s ) ϕ(v, x) -1 g n+1 - 2 L 2 dx.
If ||| gn ||| Φ γ is sufficiently small, then both estimates lead us to (5.6). Hence we have f n+1 ≥ 0 and then it completes the proof of the proposition.

Completion of the proof of Theorem 1.3.

We recall now the existence and convergence of the sequence {g n } constructed in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF][START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] for different cases of index :

The hard potential case γ+2s > 0. (Theorem 1.1 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF]) Let g 0 ∈ H k ℓ 0 (R 6 ) for some k ≥ 6, ℓ 0 > 3/2+2s+γ. There exists ε

0 > 0, such that if g 0 H k ℓ 0 (R 6 ) ≤ ε 0 , then the sequence {g n } converges in L ∞ ([0, +∞[ ; H k ℓ 0 (R 6 )) to a global solution g with g L ∞ ([0,+∞[;H k ℓ 0 (R 6 )) ≤ Cε 0 .
The soft potential case γ + 2s ≤ 0. (Theorem 1.5 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF]) Assume γ > max{-3, -3 2 -2s}. Let g 0 ∈ Hk k (R 6 ) for some k ≥ 6. There exits ε 0 > 0 such that if g 0 Hk k (R 6 ) ≤ ε 0 , then the sequence

{g n } converges in L ∞ ([0, +∞[ ; Hk k (R 6
)) to a global solution g. Remark that the approximate sequence

{g n } is convergent in L ∞ ([0, T ]; H k (R 6 )).
So in both cases, the sequence f n = µ + µ 1/2 gn satisfies the conditions of Propositions 5.1 and 5.2 with N = k -2, which implies that the limit f = µ + µ 1/2 g ≥ 0. We have proved Theorem 1.3.

Convergence to the equilibrium state

In this section, the convergence rates of the solutions to the equilibrium will be discussed for both the soft and hard potentials. Precisely, for the hard potential, the optimal convergence rates in the Sobolev space can be obtained by combining the energy estimates proven previously and the L p -L q estimate on the solution operator of the linearized equation. Such L p -L q estimate can be obtained either by spectrum analysis [START_REF] Ukai | Solutions of the Boltzmann equation, Pattern and Waves -Qualitative Analysis of Nonlinear Differential Equations[END_REF] or by using the compensating functions introduced by Kawashima [START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF]. On the other hand, for soft potential, the convergence rate presented here is solely based on the energy estimate and is not optimal.

6.1. Hard potential. In this subsection, we will combine the compensating function and the energy estimate to obtain the optimal convergence rate for the hard potential case γ + 2s > 0, that is, the first part of Theorem 1.4. Note that the decay estimate in the theorem can be generalized to the case when the initial lies in Z q (R 6 ) with 1 < q < 2, where Z q (R 6 ) = L q (R 3

x ; L 2 v (R 3 )). The compensating function is useful in deriving L p -L q estimates for linear dissipative kinetic equations in the form of (6.1) (ii) iS (ω) is self-adjoint on L 2 (R 3 ) for all ω ∈ S 2 . (iii) There exist constants λ > 0 and c 0 > 0 such that for all g ∈ L 2 (R 3 ) and ω ∈ S 2 , (6.2)

g t + v • ∇ x g + Lg = h,
Re(S (ω)(v • ω)g, g) L 2 (R 3 v ) + (Lg, g) L 2 (R 3 v ) ≥ c 0 ( Pg 2 L 2 (R 3
v ) + |||(I -P)g||| 2 ). The construction of S (ω) was given in [START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF], but for completeness and the convenience of the readers, we recall some basic derivation and estimates.

Let W be the subspace spanned by the thirteen moments containing the null space N of L and the images of N under the mappings g(v) → v j g(v) ( j = 1, 2, 3) denoted by:

W = span{e j | j = 1, 2, • • • , 13}.
Here, the orthonormal set of functions e j is given by

e 1 = µ 1 2 , e i+1 = v i µ 1 2 , i = 1, 2, 3, e 5 = 1 √ 6 (|v| 2 -3)µ 1 2
, and

e j+4 = 3 i=1 c ji √ 2 (v 2 i -1)µ 1 2 , j = 2, 3 e 8 = v 1 v 3 µ 1 2 , e 9 = v 2 v 3 µ 1 2 , e 10 = v 3 v 1 µ 1 2 , e i+10 = 1 √ 10 (|v| 2 -5)v i µ 1 2 , i = 1, 2, 3, where the constant vectors c i = (c i1 , c i2 , c i3 ), i = 2, 3 together with c 1 = ( 1 √ 3 , 1 √ 3 , 1 √
3 ) form an orthonormal basis of R 3 .

Let P 0 be the orthogonal projection from L 2 (R 3 v ) onto W, that is,

P 0 g = 13 k=1 (g, e k ) L 2 (R 3 v ) e k . Set W k = f, e k , k = 1, 2, • • • , 13
∂ t W + j V j ∂ x j W + LW = h + R,
where V j ( j = 1, 2, 3) and L are the symmetric matrices defined by

L = {(Le l , e k ) L 2 (R 3 v ) } 13 k,l=1 , V j = {(v j e k , e l ) L 2 (R 3 v ) } 13 k,l=1
,

and h = [(h, e 1 ) L 2 (R 3 v ) , ..., (h, e 13 ) L 2 (R 3 v ) ] T .
Here R denotes the remaining term which contains the factor (I -P 0 )g. Straightforward calculation gives

V(ξ) = 3 j=1 V j ξ j = V 11 V 12 V 21 V 22 , with V 11 (ξ) =                    0 ξ 1 ξ 2 ξ 3 0 ξ 1 0 0 0 a 1 ξ 1 ξ 2 0 0 0 a 1 ξ 2 ξ 3 0 0 0 a 1 ξ 3 0 a 1 ξ 1 a 1 ξ 2 a 1 ξ 3 0                    , and
V 21 (ξ) = V 12 (ξ) T =                                    0 a 21 ξ 1 a 22 ξ 2 a 23 ξ 3 0 0 a 31 ξ 1 a 32 ξ 2 a 33 ξ 3 0 0 ξ 2 ξ 1 0 0 0 0 ξ 3 ξ 2 0 0 ξ 3 0 ξ 1 0 0 0 0 0 a 4 ξ 1 0 0 0 0 a 4 ξ 2 0 0 0 0 a 4 ξ 3                                    , where a 1 = 2 3 , a k j = √ 2c k j , k = 2, 3, j = 1, 2, 3
, and a 4 = 3 5 . By setting

R(ξ) = 3 j=1 R j ξ j = α R11 V 12 -V 21 0 , with R11 =                    0 ξ 1 ξ 2 ξ 3 0 -ξ 1 0 0 0 0 -ξ 2 0 0 0 0 -ξ 3 0 0 0 0 0 0 0 0 0                    .
It was shown in [START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF] that there exist positive constants c 1 and c 2 such that

Re R(ω)V(ω)W, W ≥ c 1 |W I | 2 -c 2 4 k=2 |W II | 2 ,
for any ω ∈ S 2 with the constant α suitably chosen. Here •, • represents the standard inner product in C 13 .

Hence, a compensating function S (ω) can be defined as follows. For any given ω ∈ S 2 , set R(ω) ≡ {r i j (ω)} 4 i, j=1 and let

S (ω)g ≡ 4 k,ℓ=1 λr kℓ (ω)(g, e ℓ ) L 2 (R 3 v ) e k , f ∈ L 2 (R 3 ).
When the parameter λ > 0 is chosen small enough, it was shown in [START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF] that the estimate (6.2) holds because of the dissipation of L on the space N ⊥ . To obtain the L p -L q estimate, by taking the Fourier transform in the variable x, the equation (6.1) yields

(6.3) ĝt + i|ξ|(v • ω) ĝ + Lĝ = ĥ,
where ω = ξ |ξ| . Take the inner product of (6.3) with ((1 + |ξ| 2 ) -iκS (ω)) ĝ and use the properties of the compensating function, to get

((1 + |ξ| 2 ) ĝ 2 L 2 (R 3 v ) -κ|ξ|(iS (ω) ĝ, ĝ) L 2 (R 3 v ) ) t + δ 0 ((1 + |ξ| 2 )|||(I -P) ĝ||| 2 + |ξ| 2 ||Pĝ 2 L 2 (R 3 v ) ) ≤ C(1 + |ξ| 2 )Re( ĝ, ĥ) L 2 (R 3 v ) , which implies that E( ĝ) t + δ 0 |ξ| 2 1 + |ξ| 2 E( ĝ) ≤ C ĥ 2 L 2 (R 3 v )
, where

E( ĝ) = ĝ 2 L 2 (R 3 v ) -κ |ξ| 1 + |ξ| 2 (iS (ω) ĝ, ĝ) L 2 (R 3 v ) ∼ ĝ 2 L 2 (R 3 v )
, when κ is chosen to be small. And this estimate yields

(6.4) ĝ 2 L 2 (R 3 v ) ≤ C exp{- δ 0 |ξ| 2 |t 1 + |ξ| 2 } ĝ0 2 L 2 (R 3 v ) + C t 0 exp{- δ 0 |ξ| 2 |(t -s) 1 + |ξ| 2 } ĥ 2 L 2 (R 3 v ) (s)ds.
Based on (6.4), we have the following L p -L q estimate on the solution operator of (6.1) obtained in [START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF].

Lemma 6.2. Let k ≥ k 1 ≥ 0 and N ≥ 4. Assume that (i) g 0 ∈ H N (R 6 ) ∩ Z q , (ii) h ∈ C 0 ([0, ∞[; H N ∩ Z q ), (iii) Ph(t, x, v) = 0 for all (t, x, v) ∈ [0, ∞) × R 3 × R 3 . (iv) g(t, x, v) ∈ C 0 ([0, ∞[; H N (R 6 )) ∩ C 1 ([0, ∞[; H N-1 (R 6 )
) is a solution of (6.1). Then we have 2 ds, (2.11) for any integer m = kk 1 ≥ 0, where q ∈ [1, 2] and

∇ k x g 2 L 2 (R 6 ) ≤ C(1 + t) -2σ q,m ( ∇ k 1 x g 0 Z q (R 6 ) + ∇ k x g 0 L 2 (R 6 ) ) 2 + t 0 (1 + t -s) -2σ q,m ( ∇ k 1 x h Z q (R 6 ) + ∇ k x h L 2 (R 6 ) )
σ q,m = 3 2 1 q - 1 2 + m 2 .
We now recall the energy estimates obtained for the global existence of solutions for the hard potential in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF]. Firstly, we have when N ≥ 6 and l > 3/2 + 2s + γ, ) . We claim that the following energy estimate also holds (6.5)

d dt E 1 + D ≤ C ∇ x Pg 2 L 2 x,v (R 6 )
,

where E 1 = ∇ x Pg 2 H N-1 (R 6 ) + (I -P)g 2 H N l (R 6 ) . In fact, for the energy estimate on the macroscopic component of the solution, by Lemma 4.4 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] and by taking the sum over 1 ≤ |α| ≤ N -1, we have

∇ x A 2 H N-1 (R 3 x ) - d dt 1≤|α|≤N-1 (∂ α r, ∇ x ∂ α (a, -b, c)) L 2 (R 3 x ) + (∂ α b, ∇ x ∂ α a) L 2 (R 3
x ) (6.6)

+ g 2 2 H N (R 3 x ;L 2 (R 3 v )) + E 1 2 D,
where we use the same notations used in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] except that we replace E N,1 D N,0 by E 1 2 D in an obvious way. On the other hand, for the energy estimate on the microscopic component without weight, one can follow the proof of Lemma 4.5 in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] except for α = 0, we take the L 2 (R 6

x,v ) inner product of (6. Moreover, for ℓ ≥ 0 and β 0, (4.10) in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] gives

d dt W ℓ ∂ α β g 2 2 L 2 (R 6 ) + ||W ℓ ∂ α β g 2 || B 0 0 g 2 2 H N (R 3 x ,L 2 (R 3 v )) + E 1/2
N,ℓ D N,ℓ (6.8)

+ δ 0 ∇ x A 2 H N-1 (R 3 ) + |α ′ |=|α|+1,|β ′ |=|β|-1 ∂ α ′ β ′ g 2 2 B 0 ℓ .
Here δ 0 > 0 is a small constant. By using induction on |β| and |α| + |β|, a suitable linear combination of (6.6), (6.7) and (6.8) gives (6.5) for sufficiently small E.

In the following, we also need some L p estimate on the nonlinear collision operator. Recall that Lemma 4.7 implies that Γ( f, g) L 2 (R 3 v )

f L 2 (R 3 v ) g H 2s (γ+2s) + (R 3 v ) . Hence, by using the fact that N ≥ 6 and ℓ > 3/2 + 2s + γ, Sobolev imbedding implies Γ(g, g) 2 L 2 (R 6

x,v ) + ∇ x Γ(g, g) 2 L 2 (R 6

x,v )

E 2 E 1 E + Pg 4 L 2 (R 6
x,v ) , Γ(g, g) 2

Z 1 E 1 E + Pg 4 L 2 (R 6 x,v ) . Define M(t) = sup 0≤s≤t {(1 + s) 5 2 E 1 (s)}, M 0 (t) = sup 0≤s≤t {(1 + s) 3 2 g(s) 2 L 2 (R 6
x,v ) }. Then by the L p -L q estimate, we have

∇ x g(t) 2 L 2 (R 6 x,v ) (1 + t) -5 2 ( g 0 2 Z 1 (R 6 ) + ∇ x g 0 2 L 2 (R 6
x,v ) ) By assumption, η + δ is small. The above estimate and the continuity argument give M 0 (t) ≤ C η,δ , and then M(t) ≤ Cη,δ , where C η,δ and Cη,δ are two constants depending on η and δ only. This completes the proof of the first part in Theorem 1.4. 6.2. Soft Potential. Finally, in this subsection, we will prove the second part of Theorem 1.4 t for the soft potential case, that is, when 2s + γ ≤ 0.

+ t 0 (1 + t -s) -
As for the case with angular cutoff, here we need to apply the following basic inequality from [START_REF] Deckelnick | Decay estimates for the compressible Navier-Stokes equations in unbounded domains[END_REF]. Now it remains to find the appropriate functionals f (t) and a(t) that satisfy the above differential inequality. First of all, the basic energy estimate derived in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF] for the global existence is d dt E N,ℓ + c 0 D N,ℓ ≤ 0, where c 0 > 0 is a constant. Here,

Lemma
E N,ℓ ∼ A 2 H N (R 3 ) + g 2 2 H N ℓ (R 6 )
,

D N,ℓ = ∇ x A 2 H N-1 (R 3 ) + g 2 2 B N ℓ (R 6 )
, and

B N ℓ (R 6 ) = g ∈ S ′ (R 6 ); ||g|| 2 B N ℓ (R 6 ) = |α|+|β|≤N R 3 x ||| Wℓ-|β| ∂ α β g(x, • )||| 2 Φ γ dx < +∞ .
Later we introduce another functional ĒN-1,ℓ-1 that has the following property ĒN-1,ℓ-1

∼ ∇ x A 2 H N-2 (R 3 ) + ∇ x g 2 2 H N-2 ℓ-1 (R 6 )
. (6.9) Clearly, by the property of the ||| • |||, ĒN-1,ℓ-1 D N,ℓ so that ∞ 0 ĒN-1,ℓ-1 (t)dt < ∞. Note that this functional contains spatial differentiation of at least one order, and the maximum order of differentiation is N -1. The reason for this is to have the time integrability of the functional coming from the dissipation effect. And the functional excludes the N-th order differentiation because we need to estimate the term like

R 3 A L 2 (R 3 x ) ∂ α x A L 2 (R 3 x ) |||∂ α x g 2 |||dx A L 6 (R 3 x ) ∂ α x A L 3 (R 3 x ) ||∂ α x g 2 || X 0 (R 6 ) ∇ x A L 2 (R 3 x ) ∂ α x A H 1 (R 3 x ) ||∂ α x g 2 || X 0 (R 6 )
, where

X N (R 6 ) = g ∈ S ′ (R 6 ); ||g|| 2 X N (R 6 ) = |α|≤N R 3 x |||∂ α x g||| 2 Φ γ dx < +∞ .
Hence, since the maximum order of differentiation is N, the above estimate requires that |α| ≤ N -1.

We now construct ĒN-1,ℓ-1 following the argument used in Lemma 6.3, Lemma 6.4 and (6.15) in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF]. Firstly, by taking 1 ≤ |α| ≤ N -2 in Lemma 6.3 of [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], we have

∇ x ∂ α A 2 L 2 (R 3 x ) ≤ - d dt (∂ α r, ∇ x ∂ α (a, -b, c)) L 2 (R 3 x ) + (∂ α b, ∇ x ∂ α a) L 2 (R 3
x ) (6.10)

+ ∇ x g 2 2 X N-2 (R 6 ) + A H N-1 (R 3 x ) + g 2 H N (R 3 x ;L 2 (R 3 v )) DN-1 + ∇ x A 2 H N-1 (R 3 x ) ∇ x A 2 H N-2 (R 3 x ) + g 2 2 L 2 (R 6 ) , where DN-1 = 1≤α≤N-2 ∇ x ∂ α A 2 L 2 (R 3 x ) + ∇ x g 2 2 X N-2 .
Note that DN-1 is different from D N-1 defined in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF]. In particular, in DN-1 , the usual dissipation terms

∇ x A 2 L 2 (R 3
x ) and g 2 2 L 2 (R 6 ) are not included. And this is also why there is the last term on the right hand side of (6.10).

Next, following the argument used for Lemma 6.4 in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], we can derive

d dt ĒN-1 + c 0 ∇ x g 2 2 X N-2 E 1 2 N DN-1 + 0≤|α|≤N-2 ∇ x ∂ α A 4 L 2 (R 3 x )
,

where E N = 0≤|α|≤N-1 ∂ α x g 2 L 2 (R 6 ) and ĒN-1 = 1≤|α|≤N-2 ∂ α x g 2 L 2 (R 6 ) . Finally, corresponding to the weighted estimate (6.15) in [START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential[END_REF], one can show that for |α| ≥ 1 and |β| ≥ 1 with |α + β| ≤ N -1, it holds

d dt ∂ α β g 2 L 2 ℓ-1-|β| (R 6 ) + (∂ α r, ∇ x ∂ α (a, -b, c)) L 2 (R 3 x ) + (∂ α b, ∇ x ∂ α a) L 2 (R 3 x ) + c 0 ||| Wℓ-1-|β| ∂ α β g 2 ||| Φ γ 2 L 2 (R 3 x ) ||∂ α g 2 || 2 L 2
s+γ/2 (R 6 ) + E 1/2 N,ℓ-1 DN-1,ℓ-1

+ |α ′ |=|α|+1,|β ′ |=|β|-1 ||| Wℓ-1-(|β|-1) ∂ α ′ x ∂ β ′ v g 2 ||| Φ γ 2 L 2 (R 3 x ) + δ 0 ∇ x g 2 2 B N-1 ℓ (R 6 ) + ∇ x A 2 H N-1 (R 3 x ) ∇ x A 2 H N-2 (R 3 x ) + g 2 2 L 2 (R 6 )
, where δ 0 > 0 is a small constant, and

DN-1,ℓ-1 = 1≤|α|≤N-2 ∇ x ∂A 2 L 2 (R 3 x ) + ∇ x g 2 2 B N-2 ℓ-1 (R 6 ) .
Here, we have used the assumption that ℓ -1 ≥ N. where c 1 > 0 and c α,β > 0 are small constants which can be chosen so that ĒN,ℓ-1 satisfies (6.9). It is straightforward to check that by induction on |β|, we have d dt ĒN-1,ℓ-1 + η 0 DN,ℓ-1

∇ x A 2 H N-1 (R 3 x ) ∇ x A 2 H N-2 (R 3 x ) + g 2 2 L 2 (R 6 ) ∇ x A 2 H N-1 (R 3
x ) ĒN-1,ℓ-1 , where η 0 > 0 is a constant. Here, we have used the fact that E N,ℓ (t) is sufficiently small for all time from the global existence. Since ∞ 0 ( ĒN-1,ℓ-1 + ∇ x A 2 H N-1 (R 3

x ) )dt < ∞, Lemma 6.3 implies that ĒN-1,ℓ-1 (1 + t) -1 .

By using the fact that ℓ -1 ≥ N, the Sobolev imbedding theorem implies the decay estimate given in the second part of Theorem 1.4.

Proposition 3 . 8 .

 38 Let s ≤ λ < 1. Suppose that, for any cutoff functions ϕ ∈ C ∞ 0 (]0, T [), ψ ∈ C ∞ 0 (R 3 ), any α ∈ N, |α| ≤ 5 and all ℓ ∈ N,(3.15)

Definition 6 . 1 .

 61 where h is a given function and L is the linearized Boltzmann collision operator.Let us now recall the definition of compensating function introduced by Kawashima[START_REF] Kawashima | The Boltzmann equation and thirteen moments[END_REF]. S (ω) is called a compensating function if it has the following properties: (i) S (•) is C ∞ on S 2 (the unit sphere in R 3 ) with values in the space of bounded linear operators on L 2 (R 3 ), and S (-ω) = -S (ω) for all ω ∈ S 2 .

  , and W = [W 1 , ..., W 13 ] T . For later use, set W I = [W 1 , ..., W 5 ] T , and W II = [W 6 , ..., W 13 ] T . Then we have

  and D = ∇ x Pg 2 H N-1 (R 6 ) + |||(I -P)g||| 2 B N l (R 6

1 1≤|α|≤N- 2 (

 12 Now we can define the functional ĒN-1,ℓ-1 as follows:ĒN-1,ℓ-1 = ĒN-1 + c ∂ α r, ∇ x ∂ α (a, -b, c)) L 2 (R 3 x ) + (∂ α b, ∇ x ∂ α a) L 2 (R 3 x ) + |α|,|β|≥1,|α+β|≤N-1 c α,β ∂ α β g 2 L 2 ℓ-1-|β| (R 6 ) + (∂ α r, ∇ x ∂ α (a, -b, c)) L 2 (R 3 x ) + (∂ α b, ∇ x ∂ α a) L 2 (R 3 x ) ,

  2s-1) + +γ + +ǫ ||g|| H λ (2s-1) + +γ + ||h|| H s ′ and moreover if λ ≥ 3/2 then we have the same estimate with f H s ′ replaced by f H λ-s ′ .

  This together with the estimate given in Lemma 4.5 of[START_REF] Alexandre | Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential[END_REF] for 1 ≤ |α| ≤ N gives

	to have								1) with (I -P)g = g 2
				d dt	g 2	2 L 2 (R 6 x,v ) + |||g 2 ||| 2 B 0 0 (R 6 )	∇ x Pg 2 L 2 (R 6 x,v ) + E	1 2 D.
	(6.7)	d dt	g 2	2 L 2 (R 6 x,v ) + ∇ x g 2 H N-1 (R 3 x ;L 2 (R 3 v )) + |||g 2 ||| 2 B 0 0 (R 6 ) +	1≤|α|≤N	|||∂ α g||| 2 B 0 0 (R 6 )	∇ x Pg 2 + E	1 2 D.

  5 2 ( Γ(g, g) Z 1 (R 6 ) + ∇ x Γ(g, g) L 2 (R 6x,v ) ) 2 ds (R6 ) . Here, we use δ > 0 to denote the upper bound of E for all time.Thus, we haveE 1 (t) ≤ E 1 (0)e -t +

					t
					0	e -(t-s) ∇ x g 2 L 2 (R 6 x,v ) (s)ds
			δe -t + η(1 + t) -5 2 + δ(1 + t) -5 2 M(t) + (1 + t) -5 2 M 2 0 (t),
	that is,
			M(t) (δ + η) + δM(t) + M 2 0 (t).
	By applying the L p -L q estimate again, we have g(t) 2 L 2 (R 6 x,v ) (1 + t) -3 2 ( g 0	2 Z 1 (R 6 ) + g 0	2 L 2 (R 6 x,v ) )
			+	0	t	(1 + t -s) -3 2 ( Γ(g, g) Z 1 (R 6 ) + Γ(g, g) L 2 (R 6 x,v ) ) 2 (s)ds
			η(1 + t) -3 2 +	0	t	(1 + t -s) -3 2 (EE 1 + Pg 4 L 2 (R 6 x,v ) )(s)ds
			η(1 + t) -3 2 + δ(1 + t) -3 2 M(t) + (1 + t) -3 2 M 2 0 (t).
	Hence,
				M 0 (t)	η + δM(t) + M 2 0 (t)
					(η + δ) + M 2 0 (t).
			η(1 + t) -5 2 +	0	t	(1 + t -s) -5 2 (EE 1 + Pg 4 L 2 (R 6 x,v ) )(s)ds
			η(1 + t) -5 2 + δM(t)	0	t	(1 + t -s) -5 2 (1 + s) -5 2 ds
			+M 2 0 (t)	0	t	(1 + t -s) -5 2 (1 + s) -3 ds
					-5 2 M 2 0 (t),
	where η = g 0	2 Z 1 (R 6 ) + g 0	2 H N

η(1 + t) -5 2 + δ(1 + t) -5 2 M(t) + (1 + t) ℓ